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Machine Learning Strategies for Time
Series Prediction
Machine Learning Summer School
(Hammamet, 2013)
Gianluca Bontempi
Machine Learning Group, Computer Science Department
Boulevard de Triomphe - CP 212
http://www.ulb.ac.be/di
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 1/128
Introducing myself
• 1992: Computer science engineer (Politecnico di Milano, Italy),
• 1994: Researcher in robotics in IRST, Trento, Italy,
• 1995: Researcher in IRIDIA, ULB Artificial Intelligence Lab, Brussels,
• 1996-97: Researcher in IDSIA, Artificial Intelligence Lab, Lugano,
Switzerland,
• 1998-2000: Marie Curie fellowship in IRIDIA, ULB Artificial Intelligence
Lab, Brussels,
• 2000-2001: Scientist in Philips Research, Eindhoven, The Netherlands,
• 2001-2002: Scientist in IMEC, Microelectronics Institute, Leuven,
Belgium,
• since 2002: professor in Machine Learning, Modeling and Simulation,
Bioinformatics in ULB Computer Science Dept., Brussels,
• since 2004: head of the ULB Machine Learning Group (MLG).
• since 2013: director of the Interuniversity Institute of Bioinformatics in
Brussels (IB)2
, ibsquare.be.
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 2/128
... and in terms of distances
According to MathSciNet
•• Distance from Erdos= 5
• Distance from Zaiane= 6
• Distance from Deisenroth= 8
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 3/128
ULB Machine Learning Group (MLG)
• 3 professors, 10 PhD students, 5 postdocs.
• Research topics: Knowledge discovery from data, Classification, Computational
statistics, Data mining, Regression, Time series prediction, Sensor networks,
Bioinformatics, Network inference.
• Computing facilities: high-performing cluster for analysis of massive datasets,
Wireless Sensor Lab.
• Website: mlg.ulb.ac.be.
• Scientific collaborations in ULB: Hopital Jules Bordet, Laboratoire de Médecine
experimentale, Laboratoire d’Anatomie, Biomécanique et Organogénèse (LABO),
Service d’Anesthesie (ERASME).
• Scientific collaborations outside ULB: Harvard Dana Farber (US), UCL Machine
Learning Group (B), Politecnico di Milano (I), Universitá del Sannio (I), Inst Rech
Cliniques Montreal (CAN).
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 4/128
ULB-MLG: recent projects
1. Machine Learning for Question Answering (2013-2014).
2. Adaptive real-time machine learning for credit card fraud detection (2012-2013).
3. Epigenomic and Transcriptomic Analysis of Breast Cancer (2012-2015).
4. Discovery of the molecular pathways regulating pancreatic beta cell dysfunction
and apoptosis in diabetes using functional genomics and bioinformatics: ARC
(2010-2015)
5. ICT4REHAB - Advanced ICT Platform for Rehabilitation (2011-2013)
6. Integrating experimental and theoretical approaches to decipher the molecular
networks of nitrogen utilisation in yeast: ARC (2006-2010).
7. TANIA - Système d’aide à la conduite de l’anesthésie. WALEO II project funded
by the Région Wallonne (2006-2010)
8. "COMP2
SYS" (COMPutational intelligence methods for COMPlex SYStems)
MARIE CURIE Early Stage Research Training funded by the EU (2004-2008).
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 5/128
What you are supposed to know
• Basic notions of probability and statistics
• Random variable
• Expectation, variance, covariance
• Least-squares
What you are expected to get acquainted with
• Foundations of statistical machine learning
• How to build a predictive model from data
• Strategies for forecasting
What will remain
• Interest, curiosity for machine learning
• taste for prediction
• Contacts
• Companion webpage
http://www.ulb.ac.be/di/map/gbonte/mlss.html
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 6/128
Outline
• Notions of time series (30 mins)
• conditional probability
• Machine learning for prediction (45 mins)
• bias/variance
• parametric and structural identification
• validation
• model selection
• feature selection
• COFFEE BREAK
• Local learning (15 mins)
• Forecasting: one-step and multi-step-ahed (30 mins)
• Some applications (15 mins)
• time series competitions
• wireless sensor
• biomedical
• Future directions and perspectives (15 mins)
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 7/128
What is machine learning?
Machine learning is that domain of computational intelligence which is
concerned with the question of how to construct computer programs that
automatically improve with experience. [16]
Reductionist attitude: ML is just a buzzword which equates to statistics plus
marketing
Positive attitude: ML paved the way to the treatment of real problems related
to data analysis, sometimes overlooked by statisticians (nonlinearity,
classification, pattern recognition, missing variables, adaptivity,
optimization, massive datasets, data management, causality,
representation of knowledge, parallelisation)
Interdisciplinary attitude: ML should have its roots on statistics and
complements it by focusing on: algorithmic issues, computational
efficiency, data engineering.
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 8/128
Why study machine learning?
• Machine learning is cool.
• Practical way to understand: All models are wrong but some are useful...
• The fastest way to become a data scientist ... the sexiest job in the 21st
century
• Someone who knows statistics better than a computer scientists and
programs better than a statistician...
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 9/128
Notion of time series
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 10/128
Time series
Definition A time series is a sequence of observations st ∈ R, usually
ordered in time.
Examples of time series in every scientific and applied domain:
• Meteorology: weather variables, like temperature, pressure, wind.
• Economy and finance: economic factors (GNP), financial indexes,
exchange rate, spread.
• Marketing: activity of business, sales.
• Industry: electric load, power consumption, voltage, sensors.
• Biomedicine: physiological signals (EEG), heart-rate, patient
temperature.
• Web: clicks, logs.
• Genomics: time series of gene expression during cell cycle.
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 11/128
Why studying time series?
There are various reasons:
Prediction of the future based on the past.
Control of the process producing the series.
Understanding of the mechanism generating the series.
Description of the salient features of the series.
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 12/128
Univariate discrete time series
• Quantities, like temperature and voltage, change in a continuous way.
• In practice, however, the digital recording is made discretely in time.
• We shall confine ourselves to discrete time series (which however take
continuous values).
• Moreover we will consider univariate time series, where one type of
measurement is made repeatedly on the same object or individual.
• Multivariate time series are out of the scope of this presentation but
represent an important topic in the domain.
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A general model
Let an observed discrete univariate time series be s1, . . . , sT . This means
that we have T numbers which are observations on some variable made at T
equally distant time points, which for convenience we label 1, 2, . . . , T.
A fairly general model for the time series can be written
st = g(t) + ϕt t = 1, . . . , T
The observed series is made of two components
Systematic part: g(t), also called signal or trend, which is a determistic
function of time
Stochastic sequence: a residual term ϕt, also called noise, which follows a
probability law.
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 14/128
Types of variation
Traditional methods of time-series analysis are mainly concerned with
decomposing the variation of a series st into:
Trend : this is a long-term change in the mean level, eg. an increasing trend.
Seasonal effect : many time series (sale figures, temperature readings) exhibit
variation which is seasonal (e.g. annual) in period. The measure and the
removal of such variation brings to deseasonalized data.
Irregular fluctuations : after trend and cyclic variations have been removed
from a set of data, we are left with a series of residuals, which may or
may not be completely random.
We will assume here that once we have detrended and deseasonalized the
series, we can still extract information about the dependency between the
past and the future. Henceforth ϕt will denote the detrended and
deseasonalized series.
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 15/128
320340360
observed
320340360
trend
−3−1123
seasonal
−0.50.00.5
1960 1970 1980 1990
random
Time
Decomposition of additive time series
Decomposition returned by the R package forecast.
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 16/128
Probability and dependency
• Forecasting a time series is possible since future depends on the past or
analogously because there is a relationship between the future and the
past. However this relation is not deterministic and can hardly be written
in an analytical form.
• An effective way to describe a nondeterministic relation between two
variables is provided by the probability formalism.
• Consider two continuous random variables ϕ1 and ϕ2 representing for
instance the temperature today (time t1) and tomorrow (t2). We tend to
believe that ϕ1 could be used as a predictor of ϕ2 with some degree of
uncertainty.
• The stochastic dependency between ϕ1 and ϕ2 is resumed by the joint
density p(ϕ1, ϕ2) or equivalently by the conditional probability
p(ϕ2|ϕ1) =
p(ϕ1, ϕ2)
p(ϕ1)
• If p(ϕ2|ϕ1) = p(ϕ2) then ϕ1 and ϕ2 are not independent or equivalently
the knowledge of the value of ϕ1 reduces the uncertainty about ϕ2.
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 17/128
Stochastic processes
• The stochastic approach to time series makes the assumption that a
time series is a realization of a stochastic process (like tossing an
unbiased coin is the realization of a discrete random variable with equal
head/tail probability).
• A discrete-time stochastic process is a collection of random variables ϕt,
t = 1, . . . , T defined by a joint density
p(ϕ1, . . . , ϕT )
• Statistical time-series analysis is concerned with evaluating the
properties of the probability model which generated the observed time
series.
• Statistical time-series modeling is concerned with inferring the properties
of the probability model which generated the observed time series from a
limited set of observations.
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 18/128
Strictly stationary processes
• Predicting a time series is possible if and only if the dependence
between values existing in the past is preserved also in the future.
• In other terms, though measures change, the stochastic rule underlying
their realization does not. This aspect is formalized by the notion of
stationarity.
• Definition A stochastic process is said to be strictly stationary if the joint
distribution of ϕt1
, ϕt2
, . . . , ϕtn
is the same as the joint distribution of
ϕt1+k, ϕt2+k, . . . , ϕtn+k for all n, t1, . . . , tn and k.
• In other words shifting the time origin by an amount k has no effect on
the joint distribution which depends only on the intervals between
t1, . . . , tn.
• This implies that the (marginal) distribution of ϕt is the same for all t.
• The definition holds for any value of n.
• Let us see what does it mean in practice for n = 1 and n = 2.
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 19/128
Properties
n=1 : If ϕt is strictly stationary and its first two moments are finite, we have
E[ϕt] = µt = µ, Var [ϕt] = σ2
t = σ2
n=2 : Furthermore the autocovariance function γ(t1, t2) depends only on the
lag k = t2 − t1 and may be written by
γ(k) = Cov[ϕt, ϕt+k] = E (ϕt − µ)(ϕt+k − µ)
• In order to avoid scaling effects, it is useful to introduce the
autocorrelation function
ρ(k) =
γ(k)
σ2
=
γ(k)
γ(0)
• Another relevant function is the the partial autocorrelation function π(k)
where π(k), k > 1 measures the degree of association between ϕt and
ϕt−k when the effects of the intermediary lags 1, . . . , k − 1 are removed
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 20/128
Weak stationarity
• A less restricted definition of stationarity concerns only the first two
moments of ϕt
Definition A process is called second-order stationary or weakly stationary
if its mean is constant and its autocovariance function depends only on
the lag.
• No assumptions are made about higher moments than those of second
order.
• Strict stationarity implies weak stationarity but not viceversa in general.
Definition A process is called normal is the joint distribution of
ϕt1
, ϕt2
, . . . , ϕtn
is multivariate normal for all t1, . . . , tn.
• In the special case of normal processes, weak stationarity implies strict
stationarity. This is due to the fact that a normal process is completely
specified by the mean and the autocovariance function.
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 21/128
Estimators of first moments
Here you will find some common estimators of the two first moments of a
time series:
• The empirical mean is given by
ˆµ =
T
t=1 ϕt
T
• The empirical autocovariance function is given by
ˆγ(k) =
T −k
t=1 (ϕt − ˆµ)(ϕt+k − ˆµ)
T − k − 1
, k < T − 2
• The empirical autocorrelation function is given by
ˆρ(k) =
ˆγ(k)
ˆγ(0)
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 22/128
Some examples of stochastic
processes
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 23/128
Purely random processes
• It consists of a sequence of random variables ϕt which are mutually
independent and identically distributed. For each t and k
p(ϕt+k|ϕt) = p(ϕt+k)
• It follows that this process has constant mean and variance. Also
γ(k) = Cov[ϕt, ϕt+k] = 0
for k = ±1, ±2, . . . .
• A purely random process is strictly stationary.
• A purely random process is sometimes called white noise by engineers.
• An example of purely random process is the series of numbers drawn by
a roulette wheel in a casino.
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 24/128
Example: Gaussian purely random
0 200 400 600 800 1000
−3−2−10123
White noise
t
y
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 25/128
Example: autocorrelation function
0 5 10 15 20 25 30
0.00.20.40.60.81.0
Lag
ACF
Series y
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 26/128
Random walk
• Suppose that wt is a discrete, purely random process with mean µ and
variance σ2
w.
• A process ϕt is said to be a random walk if
ϕt = ϕt−1 + wt
• The next value of a random walk is obtained by summing a random
shock to the latest value.
• If ϕ0 = 0 then
ϕt =
t
i=1
wi
• E[ϕt] = tµ and Var [ϕt] = tσ2
w.
• As the mean and variance change with t the process is non-stationary.
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 27/128
Random walk (II)
• The first differences of a random walk given by
∇ϕt = ϕt − ϕt−1
form a purely random process, which is stationary.
• Examples of time series which behave like random walks are
• stock prices on successive days.
• the path traced by a molecule as it travels in a liquid or a gas,
• the search path of a foraging animal
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 28/128
Ten random walks
Let w ∼ N(0, 1).
0 50 100 150 200 250 300 350 400 450 500
−40
−30
−20
−10
0
10
20
30
40
50
60
Random walks
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 29/128
Autoregressive processes
• Suppose that wt is a purely random process with mean zero and
variance σ2
w.
• A process ϕt is said to be an autoregressive process of order n (also an
AR(n) process) if
ϕt = α1ϕt−1 + · · · + αnϕt−n + wt
• This means that the next value is a linear weighted sum of the past n
values plus a random shock.
• Finite memory filter.
• If w is a normal variable, ϕt will be normal too.
• Note that this is like a linear regression model where ϕ is regressed not
on independent variables but on its past values (hence the prefix “auto”).
• The properties of stationarity depends on the values αi, i = 1, . . . , n.
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 30/128
First order AR(1) process
If n = 1, we have the so-called Markov process AR(1)
ϕt = αϕt−1 + wt
By substitution it can be shown that
ϕt = α(αϕt−2 + wt−1) + wt = α2
(αϕt−3 + wt−2) + αwt−1 + wt =
= wt + αwt−1 + α2
wt−2 + . . .
Then
E[ϕt] = 0 Var [ϕt] = σ2
w(1 + α2
+ α4
+ . . . )
Then if |α| < 1 the variance if finite and equals
Var [ϕt] = σ2
ϕ = σ2
w/(1 − α2
)
and the autocorrelation is
ρ(k) = αk
k = 0, . . . , 1, 2
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 31/128
General order AR(n) process
• It has been shown that condition necessary and sufficient for the
stationarity is that the complex roots of the equation
φ(z) = 1 − α1z − · · · − αnzn
= 0
lie outside the unit circle.
• The autocorrelation function of an AR(n) attenuates slowly with the lag k
(exponential decay or damped sine wave pattern).
• On the contrary the partial autocorrelation function cuts off at k > n, i.e.
it is not significantly different from zero beyond the lag n.
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 32/128
Example: AR(2)
0 200 400 600 800 1000
051015
AR(2)
t
y
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 33/128
Example: AR(2)
0 5 10 15 20 25 30
−0.50.00.51.0
Lag
ACF
Series y
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 34/128
Partial autocorrelation in AR(2)
0 5 10 15 20 25 30
−0.50.00.5
Lag
PartialACF
Series y
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 35/128
Fitting an autoregressive process
The estimation of an autoregressive process to a set of data
DT = {ϕ1, . . . , ϕT } demands the resolution of two problems:
1. The estimation of the order n of the process. This is typically supported
by the analysis of the partial autocorrelation function.
2. The estimation of the set of parameters {α1, . . . , αn}.
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 36/128
Estimation of AR(n) parameters
Suppose we have an AR(n) process of order n
ϕt = α1ϕt−1 + · · · + αpϕt−n + wt
Given T observations, the parameters may be estimated by least-squares by
minimizing
ˆα = arg min
α
T
t=n+1
[ϕt − α1ϕt−1 − · · · − αnϕt−n]
2
In matrix form this amounts to solve the multiple least-squares problem
Y = Xα where
X =







ϕT −1 ϕT −2 . . . ϕT −n−1
ϕT −2 ϕT −3 . . . ϕT −n−2
...
...
...
...
ϕn ϕn−1 . . . ϕ1







Y =







ϕT
ϕT −1
...
ϕn+1







(1)
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 37/128
Least-squares estimation of AR(n) parms
• Let N be the number of rows of X. In order to estimate the AR(n)
parameters we compute the least-squares estimator
ˆα = arg min
a
N
i=1
(yi − xT
i a)2
= arg min
a
(Y − Xa)T
(Y − Xa)
• It can be shown that
ˆα = (XT
X)−1
XT
Y
where the XT
X matrix is a symmetric [n × n] matrix which plays an
important role in multiple linear regression.
• Conventional linear regression theory provides also confidence interval
and significativity tests for the AR(n) coefficients.
• A recursive version of least-squares, i.e. where time samples arrive
sequentially, is provided by the RLS algorithm.
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 38/128
From linear to nonlinear setting
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 39/128
The NAR representation
• AR models assume that the relation between past and future is linear
• Once we assume that the linear assumption does not hold, we may
extend the AR formulation to a Nonlinear Auto Regressive (NAR)
formulation
ϕt = f (ϕt−1, ϕt−2, . . . , ϕt−n) + w(t)
where the missing information is lumped into a noise term w.
• In what follows we will consider this relationship as a particular instance
of a dependence
y = f(x) + w
between a multidimensional input x ∈ X ⊂ Rn
and a scalar output y ∈ R.
• NOTA BENE. In what follows y will denote the next value ϕt to be
predicted and
x = [ϕt−1, ϕt−2, . . . , ϕt−n]
will denote the n-dimensional input vector also known as embedding
vector.
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 40/128
Nonlinear vs. linear time series
The advantage of linear models are numerous:
• the least-squares ˆα estimate can be expressed in an analytical form
• the least-squares ˆα estimate can be easily calculated through matrix
computation.
• statistical properties of the estimator can be easily defined.
• recursive formulation for sequential updating are avaialble.
• the relation between empirical and generalization error is known,
BUT...
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 41/128
Nonlinear vs. linear time series
• linear methods interpret all the structure in a time series through linear
correlation
• deterministic linear dynamics can only lead to simple exponential or
periodically oscillating behavior, so all irregular behavior is attributed to
external noise while deterministic nonlinear equations could produce
very irregular data,
• in real problems it is extremely unlikely that the variables are linked by a
linear relation.
In practice, the form of the relation is often unknown and only a limited
amount of samples is available.
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 42/128
Machine learning for prediction
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 43/128
Supervised learning
TRAINING
DATASET
UNKNOWN
DEPENDENCY
INPUT OUTPUT
ERROR
PREDICTION
MODEL
PREDICTION
From now on we consider the prediction problem as a problem of supervised
learning problem, where we have to infer from historical data the possibly
nonlinear dependance between the input (past embedding vector) and the
output (future value).
Statistical machine learning is the discipline concerned with this problem.
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 44/128
The regression plus noise form
• A typical way of representing the unknown input/output relation is the
regression plus noise form
y = f(x) + w
where f(·) is a deterministic function and the term w represents the
noise or random error. It is typically assumed that w is independent of x
and E[w] = 0.
• Suppose that we have available a training set { xi, yi : i = 1, . . . , N},
where xi = (xi1, . . . , xin) and yi, generated according to the previous
model.
• The goal of a learning procedure is to estimate a model ˆf(x) which is
able to give a good approximation of the unknown function f(x).
• But how to choose ˆf, if we do not know the probability distribution
underlying the data and we have only a limited training set?
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 45/128
A simple example with n = 1
−2 −1 0 1 2
−5051015
x
Y
NOTA BENE: this is NOT a time series ! y = ϕt, x = ϕt−1. The horizontal axis
does not represent time but the past value of the series.
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 46/128
Model degree 1
−2 −1 0 1 2
−5051015
x
Y
Training error= 2 degree= 1
ˆf(x) = α0 + α1x
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 47/128
Model degree 3
−2 −1 0 1 2
−5051015
x
Y
Training error= 0.92 degree= 3
ˆf(x) = α0 + α1x + · · · + α3x3
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 48/128
Model degree 18
−2 −1 0 1 2
−5051015
x
Y
Training error= 0.4 degree= 18
ˆf(x) = α0 + α1x + · · · + α18x1
8
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 49/128
Generalization and overfitting
• How to estimate the quality of a model? Is the training error a good
measure of the quality?
• The goal of learning is to find a model which is able to generalize, i.e.
able to return good predictions for input values independent of the
training set.
• In a nonlinear setting, it is possible to find models with such a complicate
structure that they have null training errors. Are these models good?
• Typically NOT. Since doing very well on the training set could mean
doing badly on new data.
• This is the phenomenon of overfitting.
• Using the same data for training a model and assessing it is typically a
wrong procedure, since this returns an over optimistic assessment of the
model generalization capability.
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 50/128
Bias and variance of a model
• A fundamental result of estimation theory shows that the
mean-squared-error, i.e. a measure of the generalization quality of an
estimator can be decomposed into three terms:
MISE = σ2
w + squared bias + variance
where the intrinsic noise term reflects the target alone, the bias reflects
the target’s relation with the learning algorithm and the variance term
reflects the learning algorithm alone.
• This result is purely theoretical since these quantities cannot be
measured on the basis of a finite amount of data.
• However, this result provides insight about what makes accurate a
learning process.
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 51/128
The bias/variance trade-off
• The first term is the variance of y around its true mean f(x) and cannot
be avoided no matter how well we estimate f(x), unless σ2
w = 0.
• The bias measures the difference in x between the average of the
outputs of the hypothesis functions ˆf over the set of possible DN and
the regression function value f(x)
• The variance reflects the variability of the guessed ˆf(x, αN ) as one
varies over training sets of fixed dimension N. This quantity measures
how sensitive the algorithm is to changes in the data set, regardless to
the target.
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 52/128
The bias/variance dilemma
• The designer of a learning machine has not access to the term MISE but
can only estimate it on the basis of the training set. Hence, the
bias/variance decomposition is relevant in practical learning since it
provides a useful hint about the features to control in order to make the
error MISE small.
• The bias term measures the lack of representational power of the class
of hypotheses. To reduce the bias term we should consider complex
hypotheses which can approximate a large number of input/output
mappings.
• The variance term warns us against an excessive complexity of the
approximator. This means that a class of too powerful hypotheses runs
the risk of being excessively sensitive to the noise affecting the training
set; therefore, our class could contain the target but it could be
practically impossible to find it out on the basis of the available dataset.
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 53/128
• In other terms, it is commonly said that an hypothesis with large bias but
low variance underfits the data while an hypothesis with low bias but
large variance overfits the data.
• In both cases, the hypothesis gives a poor representation of the target
and a reasonable trade-off needs to be found.
• The task of the model designer is to search for the optimal trade-off
between the variance and the bias term, on the basis of the available
training set.
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 54/128
Bias/variance trade-off
complexity
generalization
error
Bias
Variance
Underfitting Overfitting
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 55/128
The learning procedure
A learning procedure aims at two main goals:
1. to choose a parametric family of hypothesis ˆf(x, α) which contains or
gives good approximation of the unknown function f (structural
identification).
2. within the family ˆf(x, α), to estimate on the basis of the training set DN
the parameter αN which best approximates f (parametric identification).
In order to accomplish that, a learning procedure is made of two nested
loops:
1. an external structural identification loop which goes through different
model structures
2. an inner parametric identification loop which searches for the best
parameter vector within the family structure.
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 56/128
Parametric identification
The parametric identification of the hypothesis is done according to ERM
(Empirical Risk Minimization) principle where
αN = α(DN ) = arg min
α∈Λ
MISEemp(α)
minimizes the training error
MISEemp(α) =
N
i=1 yi − ˆf(xi, α)
2
N
constructed on the basis of the training data set DN .
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 57/128
Parametric identification (II)
• The computation of αN requires a procedure of multivariate optimization
in the space of parameters.
• The complexity of the optimization depends on the form of ˆf(·).
• In some cases the parametric identification problem may be an NP-hard
problem.
• Thus, we must resort to some form of heuristic search.
• Examples of parametric identification procedure are linear least-squares
for linear models and backpropagated gradient-descent for feedforward
neural networks.
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 58/128
Model assessment
• We have seen before that the training error is not a good estimator (i.e. it
is too optimistic) of the generalization capability of the learned model.
• Two alternative exists:
1. Complexity-based penalty criteria
2. Data-driven validation techniques
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Complexity-based penalization
• In conventional statistics, various criteria have been developed, often in
the context of linear models, for assessing the generalization
performance of the learned hypothesis without the use of further
validation data.
• Such criteria take the form of a sum of two terms
ˆGPE = MISEemp + complexity term
where the complexity term represents a penalty which grows as the
number of free parameters in the model grows.
• This expression quantifies the qualitative consideration that simple
models return high training error with a reduced complexity term while
complex models have a low training error thanks to the high number of
parameters.
• The minimum for the criterion represents a trade-off between
performance on the training set and complexity.
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 60/128
Complexity-based penalty criteria
If the input/output relation is linear, well-known examples of complexity based
criteria are:
• the Final Prediction Error (FPE)
FPE = MISEemp(αN )
1 + p/N
1 − p/N
with p = n + 1,
• the Generalized Cross-Validation (GCV)
GCV = MISEemp(αN )
1
(1 − p
N )2
• the Akaike Information Criterion (AIC)
AIC =
p
N
−
1
N
L(αN )
where L(·) is the log-likelihood function,
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 61/128
• the Cp criterion proposed by Mallows
Cp =
MISEemp(αN )
ˆσ2
w
+ 2p − N
where ˆσ2
w is an estimate of the variance of noise,
• the Predicted Squared Error (PSE)
PSE = MISEemp(αN ) + 2ˆσ2
w
p
N
where ˆσ2
w is an estimate of the variance of noise.
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 62/128
Data-driven validation techniques
If no (e.g. linear) assumptions are made, how to measure MISE in a reliable
way on a finite dataset? The most common techniques to return an estimate
MISE are
Testing: a testing sequence independent of DN and distributed according to
the same probability distribution is used to assess the quality. In
practice, unfortunately, an additional set of input/output observations is
rarely available.
Holdout: The holdout method, sometimes called test sample estimation,
partitions the data DN into two mutually exclusive subsets, the training
set Dtr and the holdout or test set DNts .
k-fold Cross-validation: the set DN is randomly divided into k mutually
exclusive test partitions of approximately equal size. The cases not
found in each test partition are independently used for selecting the
hypothesis which will be tested on the partition itself. The average error
over all the k partitions is the cross-validated error rate.
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The K-fold cross-validation
This is the algorithm in detail:
1. split the dataset DN into k roughly equal-sized parts.
2. For the kth part k = 1, . . . , K, fit the model to the other K − 1 parts of the
data, and calculate the prediction error of the fitted model when
predicting the k-th part of the data.
3. Do the above for k = 1, . . . , K and average the K estimates of prediction
error.
Let k(i) be the part of DN containing the ith sample. Then the
cross-validation estimate of the MISE prediction error is
MISECV =
1
N
N
i=1
(yi − ˆy
−k(i)
i )2
=
1
N
N
i=1
yi − ˆf(xi, α−k(i)
)
2
where ˆy
−k(i)
i denotes the fitted value for the ith observation returned by the
model estimated with the k(i)th part of the data removed.
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10-fold cross-validation
K = 10: at each iteration 90% of data are used for training and the remaining
10% for the test.
90%
10%
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 65/128
Leave-one-out cross validation
• The cross-validation algorithm where K = N is also called the
leave-one-out algorithm.
• This means that for each ith sample, i = 1, . . . , N,
1. we carry out the parametric identification, leaving that observation
out of the training set,
2. we compute the predicted value for the ith observation, denoted by
ˆy−i
i
The corresponding estimate of the MISE prediction error is
MISELOO =
1
N
N
i=1
(yi − ˆy−i
i )2
=
1
N
N
i=1
(yi − ˆf xi, α−i
)
2
where α−i
is the set of parameters returned by the parametric identification
perfomed on the training set with the ith sample set aside.
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 66/128
Model selection
• Model selection concerns the final choice of the model structure
• By structure we mean:
• family of the approximator (e.g. linear, non linear) and if nonlinear
which kind of learner (e.g. neural networks, support vector
machines, nearest-neighbours, regression trees)
• the value of hyper parameters (e.g. number of hidden layers, number
of hidden nodes in NN, number of neighbors in KNN, number of
levels in trees)
• number and set of input variables
• this choice is typically the result of a compromise between different
factors, like the quantitative measures, the personal experience of the
designer and the effort required to implement a particular model in
practice.
• Here we will consider only quantitative criteria. Two are the possible
approaches:
1. the winner-takes-all approach
2. the combination of estimators approach.
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 67/128
Model selection
N
REALIZATION
STOCHASTIC
PROCESS
VALIDATION
CLASSES of HYPOTHESIS
LEARNED MODEL
TRAINING
SET
MODEL SELECTION
PARAMETRIC IDENTIFICATION
IDENTIFICATION
,,
, ,
,
,
STRUCTURAL
α
?
ΛSΛ2Λ1
GN
1
α
1
N
α
1
N
α
2
N
α
2
N
αN
s
αN
s
GN
2
GN
S
GN
2
GN
1
GN
S
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 68/128
Winner-takes-all
The best hypothesis is selected in the set {αs
N }, with s = 1, . . . , S, according
to
˜s = arg min
s=1,...,S
MISE
s
A model with complexity ˜s is trained on the whole dataset DN and used for
future predictions.
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 69/128
Winner-takes-all pseudo-code
1. for s = 1, . . . , S: (Structural loop)
• for j = 1, . . . , N
(a) Inner parametric identification (for l-o-o):
αs
N−1 = arg min
α∈Λs
i=1:N,i=j
(yi − ˆf(xi, α))2
(b) ej = yj − ˆf(xj, αs
N−1)
• MISELOO(s) = 1
N
N
j=1 e2
j
2. Model selection: ˜s = arg mins=1,...,S MISELOO(s)
3. Final parametric identification:
α˜s
N = arg minα∈Λ˜s
N
i=1(yi − ˆf(xi, α))2
4. The output prediction model is ˆf(·, α˜s
N )
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 70/128
Model combination
• The winner-takes-all approach is intuitively the approach which should
work the best.
• However, recent results in machine learning show that the performance
of the final model can be improved not by choosing the model structure
which is expected to predict the best but by creating a model whose
output is the combination of the output of models having different
structures.
• The reason is that in reality any chosen hypothesis ˆf(·, αN ) is only an
estimate of the real target and, like any estimate, is affected by a bias
and a variance term.
• Theoretical results on the combination of estimators show that the
combination of unbiased estimators leads an unbiased estimator with
reduced variance.
• This principle is at the basis of approaches like bagging or boosting.
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 71/128
Feature selection problem
• Machine learning algorithms are known to degrade in performance
(prediction accuracy) when faced with many inputs (aka features) that
are not necessary for predicting the desired output.
• In the feature selection problem, a learning algorithm is faced with the
problem of selecting some subset of features upon which to focus its
attention, while ignoring the rest.
• Using all available features may negatively affect generalization
performance, especially in the presence of irrelevant or redundant
features.
• Feature selection can be seen as an instance of model selection
problem.
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 72/128
Benefits and drawbacks of feature selection
There are many potential benefits of feature selection:
• facilitating data visualization and data understanding,
• reducing the measurement and storage requirements,
• reducing training and utilization times of the final model,
• defying the curse of dimensionality to improve prediction performance.
Drawbacks are
• the search for a subset of relevant features introduces an additional layer
of complexity in the modelling task. The search in the model hypothesis
space is augmented by another dimension: the one of finding the
optimal subset of relevant features.
• additional time for learning.
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 73/128
Curse of dimensionality
• The error of the best model decreases with n but the mean integrated
squared error of models increases faster than linearly in n.
• In high dimensions, all data sets are sparse.
• In high dimensions, the number of possible models to consider
increases superexponenetially in n.
• In high dimensions, all datasets show multicollinearity.
• As n increases the amount of local data goes to zero.
• For a uniform distribution around a query point xq the amount of data
that are contained in a ball of radius r < 1 centered in xq grows like rn
.
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 74/128
Local density for large n
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
r
fractionoflocal
points
n=1
n=2
n=3
n=100
The size of the neighborhood on which we can estimate local features of the
output (e.g. E[y|x]) increases with dimension n, making the estimation
coarser and coarser.
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 75/128
Methods of feature selection
Two are the main approaches to feature selection:
Filter methods: they are preprocessing methods. They attempt to assess the
merits of features from the data, ignoring the effects of the selected
feature subset on the performance of the learning algorithm. Examples
are methods that select variables by ranking them through compression
techniques (like PCA or clustering) or by computing correlation with the
output.
Wrapper methods: these methods assess subsets of variables according to
their usefulness to a given predictor. The method conducts a search for
a good subset using the learning algorithm itself as part of the evaluation
function. The problem boils down to a problem of stochastic state space
search. Example are the stepwise methods proposed in linear
regression analysis.
Embedded methods: they perform variable selection as part of the learning
procedure and are usually specific to given learning machines.
Examples are classification trees, random forests, and methods based
on regularization techniques (e.g. lasso)
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 76/128
Local learning
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Local modeling procedure
The learning of a local model in xq ∈ Rn
can be summarized in these steps:
1. Compute the distance between the query xq and the training samples
according to a predefined metric.
2. Rank the neighbors on the basis of their distance to the query.
3. Select a subset of the k nearest neighbors according to the bandwidth
which measures the size of the neighborhood.
4. Fit a local model (e.g. constant, linear,...).
Each of the local approaches has one or more structural (or smoothing)
parameters that control the amount of smoothing performed.
Let us focus on the bandwidth selection.
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 78/128
The bandwidth trade-off: overfit
e
q
0011
0011
0011
01
01
0011
01
0011
0011
00
00
11
11 0
0
1
1
0011
0011
00001111
01
00001111
0011
01
0011
00001111
0011
0011
000000000000000000000000000000000000000000011111111111111111111111111111111111111111110
00
00
0
00
00
000000000000000000000
1
11
11
1
11
11
111111111111111111111
x
y
0011
0011
0011
01
01
0011
01
0011
0011
00
00
11
11 0
0
1
1
0011
000
111
00001111
01
00001111
0011
01
0011
00001111
0011
000000
111111
0011
000000000000000000000000000000000000000000011111111111111111111111111111111111111111110
00
00
0
00
00
000000000000000000000
1
11
11
1
11
11
111111111111111111111
x
y
Too narrow bandwidth ⇒ overfitting ⇒ large prediction error e.
In terms of bias/variance trade-off, this is typically a situation of high variance.
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 79/128
The bandwidth trade-off: underfit
e
q
0011
0011
0011
01
01
0011
01
0011
0011
00
00
11
11 0
0
1
1
0011
00001111
01
00001111
0011
01
0011
00001111
0011
0011
000000000000000000000000000000000000000000011111111111111111111111111111111111111111110
00
00
0
00
00
000000000000000000000
1
11
11
1
11
11
111111111111111111111
x
y
0011
0011
0011
01
01
0011
000000
111111
0
00
1
11 00
0000
11
1111
0011
000
111
00001111
000
111
01
001100001111
00110011
000000
111111 0011
00001111
000000
111111
0011
000
111
0
00
1
11
000000
111111
00001111
000000000000000000000000000000000000000000011111111111111111111111111111111111111111110
00
00
0
00
00
000000000000000000000
1
11
11
1
11
11
111111111111111111111
x
y
Too large bandwidth ⇒ underfitting ⇒ large prediction error e
In terms of bias/variance trade-off, this is typically a situation of high bias.
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 80/128
Bandwidth and bias/variance trade-off
Mean Squared Error
1/Bandwith
FEW NEIGHBORSMANY NEIGHBORS
Bias
Variance
Underfitting Overfitting
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 81/128
The PRESS statistic
• Cross-validation can provide a reliable estimate of the algorithm
generalization error but it requires the training process to be repeated K
times, which sometimes means a large computational effort.
• In the case of linear models there exists a powerful statistical procedure
to compute the leave-one-out cross-validation measure at a reduced
computational cost
• It is the PRESS (Prediction Sum of Squares) statistic, a simple formula
which returns the leave-one-out (l-o-o) as a by-product of the
least-squares.
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 82/128
Leave-one-out for linear models
PARAMETRIC IDENTIFICATION ON N-1 SAMPLES
PUT THE j-th SAMPLE ASIDE
TEST ON THE j-th SAMPLE
PARAMETRIC IDENTIFICATION
ON N SAMPLES
N TIMES
TRAINING SET
PRESS STATISTIC
LEAVE-ONE-OUT
The leave-one-out error can be computed in two equivalent ways: the slowest
way (on the right) which repeats N times the training and the test procedure;
the fastest way (on the left) which performs only once the parametric
identification and the computation of the PRESS statistic.
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 83/128
The PRESS statistic
• This allows a fast cross-validation without repeating N times the
leave-one-out procedure. The PRESS procedure can be described as
follows:
1. we use the whole training set to estimate the linear regression
coefficients
ˆα = (XT
X)−1
XT
Y
2. This procedure is performed only once on the N samples and
returns as by product the Hat matrix
H = X(XT
X)−1
XT
3. we compute the residual vector e, whose jth
term is ej = yj − xT
j ˆα,
4. we use the PRESS statistic to compute eloo
j as
eloo
j =
ej
1 − Hjj
where Hjj is the jth
diagonal term of the matrix H.
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 84/128
The PRESS statistic
Thus, the leave-one-out estimate of the local mean integrated squared error
is:
MISELOO =
1
N
N
i=1
yi − ˆyi
1 − Hii
2
Note that PRESS is not an approximation of the loo error but simply a faster
way of computing it.
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 85/128
Selection of the number of neighbours
• For a given query point xq, we can compute a set of predictions
ˆyq(k) = xT
q ˆα(k)
, together with a set of associated leave-one-out error vectors
MISELOO(k) for a number of neighbors ranging in [kmin, kmax].
• If the selection paradigm, frequently called winner-takes-all, is adopted,
the most natural way to extract a final prediction ˆyq, consists in
comparing the prediction obtained for each value of k on the basis of the
classical mean square error criterion:
ˆyq = xT
q ˆα(ˆk), with ˆk = arg min
k
MISELOO(k)
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 86/128
Local Model combination
• As an alternative to the winner-takes-all paradigm, we can use a
combination of estimates.
• The final prediction of the value yq is obtained as a weighted average of
the best b models, where b is a parameter of the algorithm.
• Suppose the predictions ˆyq(k) and the loo errors MISELOO(k) have been
ordered creating a sequence of integers {ki} so that
MISELOO(ki) ≤ MISELOO(kj), ∀i < j. The prediction of ˆyq is given by
ˆyq =
b
i=1 ζi ˆyq(ki)
b
i=1 ζi
,
where the weights are the inverse of the mean square errors:
ζi = 1/MISELOO(ki).
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Forecasting
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 88/128
One step-ahead and iterated prediction
• Once a model of the embedding mapping is available, it can be used for
two objectives: one-step-ahead prediction and iterated prediction.
• In one-step-ahead prediction, the n previous values of the series are
available and the forecasting problem can be cast in the form of a
generic regression problem
• In literature a number of supervised learning approaches have been
used with success to perform one-step-ahead forecasting on the basis
of historical data.
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One step-ahead prediction
f
ϕt-2
z-1
z-1
z-1
ϕt-3
ϕt-n
ϕt-1
z-1
ϕt
The approximator ˆf returns the prediction of the value of the time series at
time t + 1 as a function of the n previous values (the rectangular box
containing z−1
represents a unit delay operator, i.e., ϕt−1
= z−1
ϕt
).
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 90/128
Nearest-neighbor one-step-ahead forecasts
t
−−
y
t−11t−16 t−1t−6
t
We want to predict at time ¯t − 1 the next value of the series y of order n = 6.
The pattern y¯t−16, y¯t−15, . . . , y¯t−11 is the most similar to the pattern
{y¯t−6, y¯t−5, . . . , yˆt−1}. Then, the prediction ˆy¯t = y¯t−10 is returned.
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 91/128
Multi-step ahead prediction
• The prediction of the value of a time series H > 1 steps ahead is called
H-step-ahead prediction.
• We classify the methods for H-step-ahead prediction according to two
features: the horizon of the training criterion and the single-output or
multi-output nature of the predictor.
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 92/128
Multi-step ahead prediction strategies
The most common strategies are
1. Iterated: the model predicts H steps ahead by iterating a
one-step-ahead predictor whose parameters are optimized to minimize
the training error on one-step-ahead forecast (one-step-ahead training
criterion).
2. Iterated strategy where parameters are optimized to minimize the
training error on the iterated htr-step-ahead forecast (htr-step-ahead
training criterion) where 1 < htr ≤ H.
3. Direct: the model makes a direct forecast at time t + h − 1, h = 1, . . . , H
by modeling the time series in a multi-input single-output form
4. Direc: direct forecast but the input vector is extended at each step with
predicted values.
5. MIMO: the model returns a vectorial forecast by modeling the time
series in a multi-input multi-output form
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 93/128
Iterated (or recursive) prediction
• In the case of iterated prediction, the predicted output is fed back as
input for the next prediction.
• Here, the inputs consist of predicted values as opposed to actual
observations of the original time series.
• As the feedback values are typically distorted by the errors made by the
predictor in previous steps, the iterative procedure may produce
undesired effects of accumulation of the error.
• Low performance is expected in long horizon tasks. This is due to the
fact that they are essentially models tuned with a one-step-ahead
criterion which is not capable of taking temporal behavior into account.
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 94/128
Iterated prediction
f
ϕt-2
z-1
z-1
z-1
z-1
ϕt-3
ϕt-n
ϕt-1
z-1
ϕt
The approximator ˆf returns the prediction of the value of the time series at
time t + 1 by iterating the predictions obtained in the previous steps (the
rectangular box containing z−1
represents a unit delay operator, i.e.,
ˆϕt−1
= z−1
ˆϕt
).
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 95/128
Iterated with h-step training criterion
• This strategy adopts one-step-ahead predictors but adapts the model
selection criterion in order to take into account the multi-step-ahead
objective.
• Methods like Recurrent Neural Networks belong to such class. Their
recurrent architecture and the associated training algorithm (temporal
backpropagation) are suitable to handle the time-dependent nature of
the data.
• In [4] we proposed an adaptation of the Lazy Learning algorithm where
the number of neighbors is optimized in order to minimize the
leave-one-out error over an horizon larger than one. This technique
ranked second in the 1998 KULeuven Time Series competition.
• A similar technique has been proposed by [14] who won the competition.
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 96/128
Conventional and iterated leave-one-out
a)
3
1
2
4
5
3
e (3)
cv
1
2
3
4
5
e (3)
b)
it
1
2
4
5
3
1
2 3
4
5
3
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 97/128
Santa Fe time series A
0 200 400 600 800 1000
050100150200250
Santa Fe time series A
t
y
The A chaotic time series has a training set of 1000 values: the task is to
predict the continuation for 100 steps, starting from different points.
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 98/128
One-step assessment criterion
0 10 20 30 40 50 60 70 80 90 100
0
50
100
150
200
250
300
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Multi-step assessment criterion
0 10 20 30 40 50 60 70 80 90 100
0
50
100
150
200
250
300
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 100/128
Direct strategy
• The Direct strategy [22, 17, 7] learns independently H models fh
ϕt+h−1 = fh(ϕt−1, . . . , ϕt−n) + wt+h−1
with h ∈ {1, . . . , H} and returns a multi-step forecast by concatenating
the H predictions.
• Several machine learning models have been used to implement the
Direct strategy for multi-step forecasting tasks, for instance neural
networks [10], nearest neighbors [17] and decision trees [21].
• Since the Direct strategy does not use any approximated values to
compute the forecasts, it is not prone to any accumulation of errors,
since each model is tailored for the horizon it is supposed to predict.
• Notwithstanding, it has some weaknesses.
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 101/128
Direct strategy limitations
• Since the H models are learned independently no statistical
dependencies between the predictions ˆϕt+h−1, h = 1, . . . , H [3, 5, 10] is
guaranteed.
• Direct methods often require higher functional complexity [20] than
iterated ones in order to model the stochastic dependency between two
series values at two distant instants [9].
• This strategy demands a large computational time since the number of
models to learn is equal to the size of the horizon.
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 102/128
DirRec strategy
• The DirRec strategy [18] combines the architectures and the principles
underlying the Direct and the Recursive strategies.
• DirRec computes the forecasts with different models for every
horizon (like the Direct strategy) and, at each time step, it enlarges the
set of inputs by adding variables corresponding to the forecasts of the
previous step (like the Recursive strategy).
• Unlike the previous strategies, the embedding size n is not the same for
all the horizons. In other terms, the DirRec strategy learns H models fh
from the time series where
ϕt+h−1 = fh(ϕt+h−1, . . . , ϕt−n) + wt+h−1
with h ∈ {1, . . . , H}.
• The technique is prone to the curse of dimensionality. The use of feature
selection is recommended for large h.
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 103/128
MIMO strategy
• This strategy [3, 5] (also known as Joint strategy [10]) avoids the
simplistic assumption of conditional independence between future
values made by the Direct strategy [3, 5] by learning a single
multiple-output model
[ϕt+H−1, . . . , ϕt] = F(ϕt−1, . . . , ϕt−n) + w
where F : Rd
→ RH
is a vector-valued function [15], and w ∈ RH
is a
noise vector with a covariance that is not necessarily diagonal [13].
• The forecasts are returned in one step by a multiple-input
multiple-output regression model.
• In [5] we proposed a multi-output extension of the local learning
algorithm.
• Other multi-output regression model could be taken into consideration
like multi-output neural networks or partial least squares.
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 104/128
Time series dependencies
t+3
ϕ ϕ
ϕ ϕ
ϕ
t−1 t
t+1 t+2
n = 2 NAR dependency ϕt = f(ϕt−1, ϕt−2) + w(t).
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 105/128
Iterated modeling of dependencies
t+3
ϕ ϕ
ϕ
t+1ϕ ϕt+2
tt−1
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 106/128
Direct modeling of dependencies
t+3
ϕ ϕ
ϕ ϕ
ϕ
t−1 t
t+1 t+2
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 107/128
MIMO strategy
• The rationale of the MIMO strategy is to model, between the predicted
values, the stochastic dependency characterizing the time series. This
strategy avoids the conditional independence assumption made by the
Direct strategy as well as the accumulation of errors which plagues the
Recursive strategy.
• So far, this strategy has been successfully applied to several real-world
multi-step time series forecasting tasks [3, 5, 19, 2].
• However, the wish to preserve the stochastic dependencies constrains
all the horizons to be forecasted with the same model structure. Since
this constraint could reduce the flexibility of the forecasting
approach [19], a variant of the MIMO strategy (called DIRMO) has been
proposed in [19, 2] .
• Extensive validation on the 111 times series of the NN5 competition
showed that MIMO are invariably better than single-output approaches.
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 108/128
Validation of time series methods
• The huge variety of strategies and algorithms that can be used to infer a
predictor from observed data asks for a rigorous procedure of
comparison and assessment.
• Assessment demands benchmarks and benchmarking procedure.
• Benchmarks can be defined by using
• Simulated data obtained by simulating AR, NAR and other stochastic
processes. This is particular useful for validating theoretical
properties in terms of bias/variance.
• Public domain benchmarks, like the one provided by Time Series
Competitions.
• Real measured data
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 109/128
Competitions
• Santa Fe Time Series Prediction and Analysis Competition (1994) [22]:
• International Workshop on Advanced Black-box techniques for nonlinear
modeling Competition (Leuven, Belgium; 1998)
• NN3 competition [8]: 111 monthly time series drawn from homogeneous
population of empirical business time series.
• NN5 competition [1]: 111 time series of the daily retirement amounts
from independent cash machines at different, randomly selected
locations across England.
• Kaggle competition.
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 110/128
Accuracy measures
Let
et+h = ϕt+h − ˆϕt+h
represent the error of the forecast ˆϕt+h at the horizon h = 1, . . . , H. A
conventional measure of accuracy is the Normalized Mean Squared Error
NMSE =
H
h=1(ϕt+h − ˆϕt+h)2
H
h=1(ϕt+h − ˆµ)2
This quantity is smaller than one if the predictor performs better than the
naivest predictor, i.e. the average ˆµ.
Other measures rely on relative or percentage error
pet+h = 100
ϕt+h − ˆϕt+h
ϕt+h
like
MAPE =
H
h=1 |pet+h|
H
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 111/128
Applications in my lab
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 112/128
MLG projects on forecasting
• Low-energy streaming of wireless sensor data
• Decision support in anesthesia
• Side-channel attack
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 113/128
Low-energy streaming of wireless sensor data
• In monitoring applications, only an approximation to sensor readings is
sufficient (e.g. ±0.5C, ±2% humidity, ..). In this context a Dual Prediction
Scheme is effective.
• A sensor node is provided with a time series prediction model
ϕt = f(ϕt−1, α) (e.g. autoregressive models) and a learning method for
identifying the best set of parameters α (e.g. recursive least squares).
• The sensor node then sends the parameters of the model instead of the
actual data to the recipient. The recipient node then runs the models to
reconstruct an approximation of the data stream collected on the distant
node.
• The sensor node also runs the prediction model. When its predictions
diverges by more than ±ǫ from the actual reading, a new model is sent
to the recipient.
• This allows to reduce the communication effort if an appropriate model is
run by the sensor node.
• Since the cost of transmission is much larger than the cost of
computation, in realistic situations this scheme allows economy of power
consumption. MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 114/128
Low-energy streaming of wireless sensor data
0 5 10 15 20
202530354045
Accuracy: 2°C
Constant model
Time (Hour)
Temperature(°C)
G GG GGG GG G G GG G G G G G G G G G
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 115/128
Adaptive model selection
Tradeoff: more complex models predict better measurements but have a
higher number of parameters
Model complexity
Metric
Communication costs
Model error
AR(p) : ˆsi[t] =
p
j=1
θjsi[t − j]
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 116/128
Adaptive Model Selection
We proposed an Adaptive Model Selection strategy [11] that
• takes into account the cost of sending model parameters,
• allows sensor nodes to determine autonomously the model which best
fits their measurements,
• provides a statistically sound selection mechanism to discard poorly
performing models,
• gave in experimental results about 45% of communication savings on
average,
• was implemented in TinyOS, the reference operating system.
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 117/128
Predictive modeling in anesthesiology
• During surgery, anesthesiologists controls the depth of anesthesia by
means of types of drugs
• Anesthesiologists observe the patient state by observing
unconsciousness signal in real-time which are collected by monitors
connected via electrodes to the patient’s forehead
• The bispectral index BIS monitor (by Aspect Medical Systems) is a
single dimensionless number between 0 to 100 where 0 equals EEG
silence, 100 is the expected value for a fully awake adult, and between
40 and 60 indicates a recommended level.
• It remains difficult for the anesthesiologist, especially if inexperienced, to
predict how the BIS signal could vary after a change in the administered
anesthetic agents. This is generally due to inter-individual variability
problem
• We designed a ML system [6] to predict multi-step-ahead the evolution
of the BIS on the basis of historical data.
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 118/128
Predictive modeling in anesthesiology
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 119/128
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 120/128
Side channel attack
• In cryptography, a side channel attack is any attack based on the
analysis of measurements related to the physical implementation of a
cryptosystem.
• Side channel attacks take advantage of the fact that instantaneous
power consumption, encryption time or/and electromagnetic leaks of a
cryptographic device depend on the processed data and the performed
operations.
• Power analysis attacks are an instance of side-channel attacks which
assume that different encryption keys imply a different power
consumptions.
• Nowadays, the possibility of collecting a large amount of power traces
(i.e. time series) paves the way to the adoption of machine learning
techniques.
• Effective side channel attacks enables effective countermeasures.
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 121/128
SCA and time series classification
• The power consumption of a crypto device using a secret key
Q ∈ {0, 1}k
(size k = 8) can be modeled as a time series T(Q)
of order n
T
(Q)
(t+1) = f(T
(Q)
(t) , T
(Q)
(t−1), ..., T
(Q)
(t−n)) + ǫ
• For each key Qj we infer a predictive model ˆf [12] such that
T
(Qj )
(t+1) = ˆfQj (T
(Qj )
(t) , T
(Qj )
(t−1), ..., T
(Qj)
(t−n)) + ǫ
• These models can be used to classify an unlabeled time series T and
predict the associated key by computing a distance for each Qj
D (Qj, T) =
1
N − n + 1
N
t=n
ˆfQj T(t−1), T(t−2), ..., T(t−n−1) − T(t)
2
and choosing the key minimizing it
ˆQ = arg min
j∈[0,2(k−1)]
D (Qj, T)
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 122/128
Conclusions
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 123/128
Open-source software
Many commercial solutions exist but only open-source software can cope with
• fast integration of new algorithms
• portability over several platforms
• new paradigms of data storage (e.g. Hadoop)
• integration with different data formats and architectures
A de-facto standard in computational statistics, machine learning,
bioinformatics, geostatistics or more general analytics is nowadays
Highly recommended !
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 124/128
All that we didn’t have time to discuss
• ARIMA models
• GARCH models
• Frequence space representation
• Nonstationarity
• Vectorial time series
• Spatio temporal time series
• Time series classification
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 125/128
Suggestions for PhD topics
• Time series and big data
• streaming data (environmental data)
• large vectorial time series (weather data)
• Spatio-temporal time series and graphical models
• Beyond cross-validation for model/input selection
• Long term forecasting (effective integration of iterated and directed
models)
• Causality and time-series
• Scalable machine learning
Suggestion: use methods and models to solve problems... not problems to
sanctify methods or models...
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 126/128
Conclusion
• Popper claimed that, if a theory is falsifiable (i.e. it can be contradicted
by an observation or the outcome of a physical experiment.), then it is
scientific. Since prediction is the most falsifiable aspect of science it is
also the most scientific one.
• Effective machine learning is an extension of statistics, in no way an
alternative.
• Simplest (i.e. linear) model first.
• Local learning techniques represent an effective trade-off between
linearity and nonlinearity.
• Modelling is more an art than an automatic process... then experience
data analysts are more valuable than expensive tools.
• Expert knowledge matters..., data too
• Understanding what is predictable is as important as trying to predict it.
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 127/128
Forecasting is difficult, especially the future
• "Computers in the future may weigh no more than 1.5 tons." –Popular
Mechanics, forecasting the relentless march of science, 1949
• "I think there is a world market for maybe five computers." –Chairman of
IBM, 1943
• "Stocks have reached what looks like a permanently high plateau."
–Professor of Economics, Yale University, 1929.
• "Airplanes are interesting toys but of no military value." –Professor of
Strategy, Ecole Superieure de Guerre.
• "Everything that can be invented has been invented." –Commissioner,
U.S. Office of Patents, 1899.
• "Louis Pasteur’s theory of germs is ridiculous fiction". –Professor of
Physiology at Toulouse, 1872
• "640K ought to be enough for anybody." – Bill Gates, 1981
MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 128/128
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128-4

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Machine Learning Strategies for Time Series Prediction

  • 1. Machine Learning Strategies for Time Series Prediction Machine Learning Summer School (Hammamet, 2013) Gianluca Bontempi Machine Learning Group, Computer Science Department Boulevard de Triomphe - CP 212 http://www.ulb.ac.be/di MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 1/128
  • 2. Introducing myself • 1992: Computer science engineer (Politecnico di Milano, Italy), • 1994: Researcher in robotics in IRST, Trento, Italy, • 1995: Researcher in IRIDIA, ULB Artificial Intelligence Lab, Brussels, • 1996-97: Researcher in IDSIA, Artificial Intelligence Lab, Lugano, Switzerland, • 1998-2000: Marie Curie fellowship in IRIDIA, ULB Artificial Intelligence Lab, Brussels, • 2000-2001: Scientist in Philips Research, Eindhoven, The Netherlands, • 2001-2002: Scientist in IMEC, Microelectronics Institute, Leuven, Belgium, • since 2002: professor in Machine Learning, Modeling and Simulation, Bioinformatics in ULB Computer Science Dept., Brussels, • since 2004: head of the ULB Machine Learning Group (MLG). • since 2013: director of the Interuniversity Institute of Bioinformatics in Brussels (IB)2 , ibsquare.be. MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 2/128
  • 3. ... and in terms of distances According to MathSciNet •• Distance from Erdos= 5 • Distance from Zaiane= 6 • Distance from Deisenroth= 8 MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 3/128
  • 4. ULB Machine Learning Group (MLG) • 3 professors, 10 PhD students, 5 postdocs. • Research topics: Knowledge discovery from data, Classification, Computational statistics, Data mining, Regression, Time series prediction, Sensor networks, Bioinformatics, Network inference. • Computing facilities: high-performing cluster for analysis of massive datasets, Wireless Sensor Lab. • Website: mlg.ulb.ac.be. • Scientific collaborations in ULB: Hopital Jules Bordet, Laboratoire de Médecine experimentale, Laboratoire d’Anatomie, Biomécanique et Organogénèse (LABO), Service d’Anesthesie (ERASME). • Scientific collaborations outside ULB: Harvard Dana Farber (US), UCL Machine Learning Group (B), Politecnico di Milano (I), Universitá del Sannio (I), Inst Rech Cliniques Montreal (CAN). MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 4/128
  • 5. ULB-MLG: recent projects 1. Machine Learning for Question Answering (2013-2014). 2. Adaptive real-time machine learning for credit card fraud detection (2012-2013). 3. Epigenomic and Transcriptomic Analysis of Breast Cancer (2012-2015). 4. Discovery of the molecular pathways regulating pancreatic beta cell dysfunction and apoptosis in diabetes using functional genomics and bioinformatics: ARC (2010-2015) 5. ICT4REHAB - Advanced ICT Platform for Rehabilitation (2011-2013) 6. Integrating experimental and theoretical approaches to decipher the molecular networks of nitrogen utilisation in yeast: ARC (2006-2010). 7. TANIA - Système d’aide à la conduite de l’anesthésie. WALEO II project funded by the Région Wallonne (2006-2010) 8. "COMP2 SYS" (COMPutational intelligence methods for COMPlex SYStems) MARIE CURIE Early Stage Research Training funded by the EU (2004-2008). MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 5/128
  • 6. What you are supposed to know • Basic notions of probability and statistics • Random variable • Expectation, variance, covariance • Least-squares What you are expected to get acquainted with • Foundations of statistical machine learning • How to build a predictive model from data • Strategies for forecasting What will remain • Interest, curiosity for machine learning • taste for prediction • Contacts • Companion webpage http://www.ulb.ac.be/di/map/gbonte/mlss.html MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 6/128
  • 7. Outline • Notions of time series (30 mins) • conditional probability • Machine learning for prediction (45 mins) • bias/variance • parametric and structural identification • validation • model selection • feature selection • COFFEE BREAK • Local learning (15 mins) • Forecasting: one-step and multi-step-ahed (30 mins) • Some applications (15 mins) • time series competitions • wireless sensor • biomedical • Future directions and perspectives (15 mins) MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 7/128
  • 8. What is machine learning? Machine learning is that domain of computational intelligence which is concerned with the question of how to construct computer programs that automatically improve with experience. [16] Reductionist attitude: ML is just a buzzword which equates to statistics plus marketing Positive attitude: ML paved the way to the treatment of real problems related to data analysis, sometimes overlooked by statisticians (nonlinearity, classification, pattern recognition, missing variables, adaptivity, optimization, massive datasets, data management, causality, representation of knowledge, parallelisation) Interdisciplinary attitude: ML should have its roots on statistics and complements it by focusing on: algorithmic issues, computational efficiency, data engineering. MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 8/128
  • 9. Why study machine learning? • Machine learning is cool. • Practical way to understand: All models are wrong but some are useful... • The fastest way to become a data scientist ... the sexiest job in the 21st century • Someone who knows statistics better than a computer scientists and programs better than a statistician... MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 9/128
  • 10. Notion of time series MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 10/128
  • 11. Time series Definition A time series is a sequence of observations st ∈ R, usually ordered in time. Examples of time series in every scientific and applied domain: • Meteorology: weather variables, like temperature, pressure, wind. • Economy and finance: economic factors (GNP), financial indexes, exchange rate, spread. • Marketing: activity of business, sales. • Industry: electric load, power consumption, voltage, sensors. • Biomedicine: physiological signals (EEG), heart-rate, patient temperature. • Web: clicks, logs. • Genomics: time series of gene expression during cell cycle. MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 11/128
  • 12. Why studying time series? There are various reasons: Prediction of the future based on the past. Control of the process producing the series. Understanding of the mechanism generating the series. Description of the salient features of the series. MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 12/128
  • 13. Univariate discrete time series • Quantities, like temperature and voltage, change in a continuous way. • In practice, however, the digital recording is made discretely in time. • We shall confine ourselves to discrete time series (which however take continuous values). • Moreover we will consider univariate time series, where one type of measurement is made repeatedly on the same object or individual. • Multivariate time series are out of the scope of this presentation but represent an important topic in the domain. MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 13/128
  • 14. A general model Let an observed discrete univariate time series be s1, . . . , sT . This means that we have T numbers which are observations on some variable made at T equally distant time points, which for convenience we label 1, 2, . . . , T. A fairly general model for the time series can be written st = g(t) + ϕt t = 1, . . . , T The observed series is made of two components Systematic part: g(t), also called signal or trend, which is a determistic function of time Stochastic sequence: a residual term ϕt, also called noise, which follows a probability law. MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 14/128
  • 15. Types of variation Traditional methods of time-series analysis are mainly concerned with decomposing the variation of a series st into: Trend : this is a long-term change in the mean level, eg. an increasing trend. Seasonal effect : many time series (sale figures, temperature readings) exhibit variation which is seasonal (e.g. annual) in period. The measure and the removal of such variation brings to deseasonalized data. Irregular fluctuations : after trend and cyclic variations have been removed from a set of data, we are left with a series of residuals, which may or may not be completely random. We will assume here that once we have detrended and deseasonalized the series, we can still extract information about the dependency between the past and the future. Henceforth ϕt will denote the detrended and deseasonalized series. MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 15/128
  • 16. 320340360 observed 320340360 trend −3−1123 seasonal −0.50.00.5 1960 1970 1980 1990 random Time Decomposition of additive time series Decomposition returned by the R package forecast. MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 16/128
  • 17. Probability and dependency • Forecasting a time series is possible since future depends on the past or analogously because there is a relationship between the future and the past. However this relation is not deterministic and can hardly be written in an analytical form. • An effective way to describe a nondeterministic relation between two variables is provided by the probability formalism. • Consider two continuous random variables ϕ1 and ϕ2 representing for instance the temperature today (time t1) and tomorrow (t2). We tend to believe that ϕ1 could be used as a predictor of ϕ2 with some degree of uncertainty. • The stochastic dependency between ϕ1 and ϕ2 is resumed by the joint density p(ϕ1, ϕ2) or equivalently by the conditional probability p(ϕ2|ϕ1) = p(ϕ1, ϕ2) p(ϕ1) • If p(ϕ2|ϕ1) = p(ϕ2) then ϕ1 and ϕ2 are not independent or equivalently the knowledge of the value of ϕ1 reduces the uncertainty about ϕ2. MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 17/128
  • 18. Stochastic processes • The stochastic approach to time series makes the assumption that a time series is a realization of a stochastic process (like tossing an unbiased coin is the realization of a discrete random variable with equal head/tail probability). • A discrete-time stochastic process is a collection of random variables ϕt, t = 1, . . . , T defined by a joint density p(ϕ1, . . . , ϕT ) • Statistical time-series analysis is concerned with evaluating the properties of the probability model which generated the observed time series. • Statistical time-series modeling is concerned with inferring the properties of the probability model which generated the observed time series from a limited set of observations. MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 18/128
  • 19. Strictly stationary processes • Predicting a time series is possible if and only if the dependence between values existing in the past is preserved also in the future. • In other terms, though measures change, the stochastic rule underlying their realization does not. This aspect is formalized by the notion of stationarity. • Definition A stochastic process is said to be strictly stationary if the joint distribution of ϕt1 , ϕt2 , . . . , ϕtn is the same as the joint distribution of ϕt1+k, ϕt2+k, . . . , ϕtn+k for all n, t1, . . . , tn and k. • In other words shifting the time origin by an amount k has no effect on the joint distribution which depends only on the intervals between t1, . . . , tn. • This implies that the (marginal) distribution of ϕt is the same for all t. • The definition holds for any value of n. • Let us see what does it mean in practice for n = 1 and n = 2. MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 19/128
  • 20. Properties n=1 : If ϕt is strictly stationary and its first two moments are finite, we have E[ϕt] = µt = µ, Var [ϕt] = σ2 t = σ2 n=2 : Furthermore the autocovariance function γ(t1, t2) depends only on the lag k = t2 − t1 and may be written by γ(k) = Cov[ϕt, ϕt+k] = E (ϕt − µ)(ϕt+k − µ) • In order to avoid scaling effects, it is useful to introduce the autocorrelation function ρ(k) = γ(k) σ2 = γ(k) γ(0) • Another relevant function is the the partial autocorrelation function π(k) where π(k), k > 1 measures the degree of association between ϕt and ϕt−k when the effects of the intermediary lags 1, . . . , k − 1 are removed MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 20/128
  • 21. Weak stationarity • A less restricted definition of stationarity concerns only the first two moments of ϕt Definition A process is called second-order stationary or weakly stationary if its mean is constant and its autocovariance function depends only on the lag. • No assumptions are made about higher moments than those of second order. • Strict stationarity implies weak stationarity but not viceversa in general. Definition A process is called normal is the joint distribution of ϕt1 , ϕt2 , . . . , ϕtn is multivariate normal for all t1, . . . , tn. • In the special case of normal processes, weak stationarity implies strict stationarity. This is due to the fact that a normal process is completely specified by the mean and the autocovariance function. MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 21/128
  • 22. Estimators of first moments Here you will find some common estimators of the two first moments of a time series: • The empirical mean is given by ˆµ = T t=1 ϕt T • The empirical autocovariance function is given by ˆγ(k) = T −k t=1 (ϕt − ˆµ)(ϕt+k − ˆµ) T − k − 1 , k < T − 2 • The empirical autocorrelation function is given by ˆρ(k) = ˆγ(k) ˆγ(0) MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 22/128
  • 23. Some examples of stochastic processes MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 23/128
  • 24. Purely random processes • It consists of a sequence of random variables ϕt which are mutually independent and identically distributed. For each t and k p(ϕt+k|ϕt) = p(ϕt+k) • It follows that this process has constant mean and variance. Also γ(k) = Cov[ϕt, ϕt+k] = 0 for k = ±1, ±2, . . . . • A purely random process is strictly stationary. • A purely random process is sometimes called white noise by engineers. • An example of purely random process is the series of numbers drawn by a roulette wheel in a casino. MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 24/128
  • 25. Example: Gaussian purely random 0 200 400 600 800 1000 −3−2−10123 White noise t y MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 25/128
  • 26. Example: autocorrelation function 0 5 10 15 20 25 30 0.00.20.40.60.81.0 Lag ACF Series y MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 26/128
  • 27. Random walk • Suppose that wt is a discrete, purely random process with mean µ and variance σ2 w. • A process ϕt is said to be a random walk if ϕt = ϕt−1 + wt • The next value of a random walk is obtained by summing a random shock to the latest value. • If ϕ0 = 0 then ϕt = t i=1 wi • E[ϕt] = tµ and Var [ϕt] = tσ2 w. • As the mean and variance change with t the process is non-stationary. MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 27/128
  • 28. Random walk (II) • The first differences of a random walk given by ∇ϕt = ϕt − ϕt−1 form a purely random process, which is stationary. • Examples of time series which behave like random walks are • stock prices on successive days. • the path traced by a molecule as it travels in a liquid or a gas, • the search path of a foraging animal MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 28/128
  • 29. Ten random walks Let w ∼ N(0, 1). 0 50 100 150 200 250 300 350 400 450 500 −40 −30 −20 −10 0 10 20 30 40 50 60 Random walks MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 29/128
  • 30. Autoregressive processes • Suppose that wt is a purely random process with mean zero and variance σ2 w. • A process ϕt is said to be an autoregressive process of order n (also an AR(n) process) if ϕt = α1ϕt−1 + · · · + αnϕt−n + wt • This means that the next value is a linear weighted sum of the past n values plus a random shock. • Finite memory filter. • If w is a normal variable, ϕt will be normal too. • Note that this is like a linear regression model where ϕ is regressed not on independent variables but on its past values (hence the prefix “auto”). • The properties of stationarity depends on the values αi, i = 1, . . . , n. MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 30/128
  • 31. First order AR(1) process If n = 1, we have the so-called Markov process AR(1) ϕt = αϕt−1 + wt By substitution it can be shown that ϕt = α(αϕt−2 + wt−1) + wt = α2 (αϕt−3 + wt−2) + αwt−1 + wt = = wt + αwt−1 + α2 wt−2 + . . . Then E[ϕt] = 0 Var [ϕt] = σ2 w(1 + α2 + α4 + . . . ) Then if |α| < 1 the variance if finite and equals Var [ϕt] = σ2 ϕ = σ2 w/(1 − α2 ) and the autocorrelation is ρ(k) = αk k = 0, . . . , 1, 2 MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 31/128
  • 32. General order AR(n) process • It has been shown that condition necessary and sufficient for the stationarity is that the complex roots of the equation φ(z) = 1 − α1z − · · · − αnzn = 0 lie outside the unit circle. • The autocorrelation function of an AR(n) attenuates slowly with the lag k (exponential decay or damped sine wave pattern). • On the contrary the partial autocorrelation function cuts off at k > n, i.e. it is not significantly different from zero beyond the lag n. MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 32/128
  • 33. Example: AR(2) 0 200 400 600 800 1000 051015 AR(2) t y MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 33/128
  • 34. Example: AR(2) 0 5 10 15 20 25 30 −0.50.00.51.0 Lag ACF Series y MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 34/128
  • 35. Partial autocorrelation in AR(2) 0 5 10 15 20 25 30 −0.50.00.5 Lag PartialACF Series y MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 35/128
  • 36. Fitting an autoregressive process The estimation of an autoregressive process to a set of data DT = {ϕ1, . . . , ϕT } demands the resolution of two problems: 1. The estimation of the order n of the process. This is typically supported by the analysis of the partial autocorrelation function. 2. The estimation of the set of parameters {α1, . . . , αn}. MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 36/128
  • 37. Estimation of AR(n) parameters Suppose we have an AR(n) process of order n ϕt = α1ϕt−1 + · · · + αpϕt−n + wt Given T observations, the parameters may be estimated by least-squares by minimizing ˆα = arg min α T t=n+1 [ϕt − α1ϕt−1 − · · · − αnϕt−n] 2 In matrix form this amounts to solve the multiple least-squares problem Y = Xα where X =        ϕT −1 ϕT −2 . . . ϕT −n−1 ϕT −2 ϕT −3 . . . ϕT −n−2 ... ... ... ... ϕn ϕn−1 . . . ϕ1        Y =        ϕT ϕT −1 ... ϕn+1        (1) MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 37/128
  • 38. Least-squares estimation of AR(n) parms • Let N be the number of rows of X. In order to estimate the AR(n) parameters we compute the least-squares estimator ˆα = arg min a N i=1 (yi − xT i a)2 = arg min a (Y − Xa)T (Y − Xa) • It can be shown that ˆα = (XT X)−1 XT Y where the XT X matrix is a symmetric [n × n] matrix which plays an important role in multiple linear regression. • Conventional linear regression theory provides also confidence interval and significativity tests for the AR(n) coefficients. • A recursive version of least-squares, i.e. where time samples arrive sequentially, is provided by the RLS algorithm. MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 38/128
  • 39. From linear to nonlinear setting MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 39/128
  • 40. The NAR representation • AR models assume that the relation between past and future is linear • Once we assume that the linear assumption does not hold, we may extend the AR formulation to a Nonlinear Auto Regressive (NAR) formulation ϕt = f (ϕt−1, ϕt−2, . . . , ϕt−n) + w(t) where the missing information is lumped into a noise term w. • In what follows we will consider this relationship as a particular instance of a dependence y = f(x) + w between a multidimensional input x ∈ X ⊂ Rn and a scalar output y ∈ R. • NOTA BENE. In what follows y will denote the next value ϕt to be predicted and x = [ϕt−1, ϕt−2, . . . , ϕt−n] will denote the n-dimensional input vector also known as embedding vector. MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 40/128
  • 41. Nonlinear vs. linear time series The advantage of linear models are numerous: • the least-squares ˆα estimate can be expressed in an analytical form • the least-squares ˆα estimate can be easily calculated through matrix computation. • statistical properties of the estimator can be easily defined. • recursive formulation for sequential updating are avaialble. • the relation between empirical and generalization error is known, BUT... MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 41/128
  • 42. Nonlinear vs. linear time series • linear methods interpret all the structure in a time series through linear correlation • deterministic linear dynamics can only lead to simple exponential or periodically oscillating behavior, so all irregular behavior is attributed to external noise while deterministic nonlinear equations could produce very irregular data, • in real problems it is extremely unlikely that the variables are linked by a linear relation. In practice, the form of the relation is often unknown and only a limited amount of samples is available. MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 42/128
  • 43. Machine learning for prediction MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 43/128
  • 44. Supervised learning TRAINING DATASET UNKNOWN DEPENDENCY INPUT OUTPUT ERROR PREDICTION MODEL PREDICTION From now on we consider the prediction problem as a problem of supervised learning problem, where we have to infer from historical data the possibly nonlinear dependance between the input (past embedding vector) and the output (future value). Statistical machine learning is the discipline concerned with this problem. MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 44/128
  • 45. The regression plus noise form • A typical way of representing the unknown input/output relation is the regression plus noise form y = f(x) + w where f(·) is a deterministic function and the term w represents the noise or random error. It is typically assumed that w is independent of x and E[w] = 0. • Suppose that we have available a training set { xi, yi : i = 1, . . . , N}, where xi = (xi1, . . . , xin) and yi, generated according to the previous model. • The goal of a learning procedure is to estimate a model ˆf(x) which is able to give a good approximation of the unknown function f(x). • But how to choose ˆf, if we do not know the probability distribution underlying the data and we have only a limited training set? MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 45/128
  • 46. A simple example with n = 1 −2 −1 0 1 2 −5051015 x Y NOTA BENE: this is NOT a time series ! y = ϕt, x = ϕt−1. The horizontal axis does not represent time but the past value of the series. MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 46/128
  • 47. Model degree 1 −2 −1 0 1 2 −5051015 x Y Training error= 2 degree= 1 ˆf(x) = α0 + α1x MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 47/128
  • 48. Model degree 3 −2 −1 0 1 2 −5051015 x Y Training error= 0.92 degree= 3 ˆf(x) = α0 + α1x + · · · + α3x3 MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 48/128
  • 49. Model degree 18 −2 −1 0 1 2 −5051015 x Y Training error= 0.4 degree= 18 ˆf(x) = α0 + α1x + · · · + α18x1 8 MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 49/128
  • 50. Generalization and overfitting • How to estimate the quality of a model? Is the training error a good measure of the quality? • The goal of learning is to find a model which is able to generalize, i.e. able to return good predictions for input values independent of the training set. • In a nonlinear setting, it is possible to find models with such a complicate structure that they have null training errors. Are these models good? • Typically NOT. Since doing very well on the training set could mean doing badly on new data. • This is the phenomenon of overfitting. • Using the same data for training a model and assessing it is typically a wrong procedure, since this returns an over optimistic assessment of the model generalization capability. MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 50/128
  • 51. Bias and variance of a model • A fundamental result of estimation theory shows that the mean-squared-error, i.e. a measure of the generalization quality of an estimator can be decomposed into three terms: MISE = σ2 w + squared bias + variance where the intrinsic noise term reflects the target alone, the bias reflects the target’s relation with the learning algorithm and the variance term reflects the learning algorithm alone. • This result is purely theoretical since these quantities cannot be measured on the basis of a finite amount of data. • However, this result provides insight about what makes accurate a learning process. MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 51/128
  • 52. The bias/variance trade-off • The first term is the variance of y around its true mean f(x) and cannot be avoided no matter how well we estimate f(x), unless σ2 w = 0. • The bias measures the difference in x between the average of the outputs of the hypothesis functions ˆf over the set of possible DN and the regression function value f(x) • The variance reflects the variability of the guessed ˆf(x, αN ) as one varies over training sets of fixed dimension N. This quantity measures how sensitive the algorithm is to changes in the data set, regardless to the target. MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 52/128
  • 53. The bias/variance dilemma • The designer of a learning machine has not access to the term MISE but can only estimate it on the basis of the training set. Hence, the bias/variance decomposition is relevant in practical learning since it provides a useful hint about the features to control in order to make the error MISE small. • The bias term measures the lack of representational power of the class of hypotheses. To reduce the bias term we should consider complex hypotheses which can approximate a large number of input/output mappings. • The variance term warns us against an excessive complexity of the approximator. This means that a class of too powerful hypotheses runs the risk of being excessively sensitive to the noise affecting the training set; therefore, our class could contain the target but it could be practically impossible to find it out on the basis of the available dataset. MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 53/128
  • 54. • In other terms, it is commonly said that an hypothesis with large bias but low variance underfits the data while an hypothesis with low bias but large variance overfits the data. • In both cases, the hypothesis gives a poor representation of the target and a reasonable trade-off needs to be found. • The task of the model designer is to search for the optimal trade-off between the variance and the bias term, on the basis of the available training set. MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 54/128
  • 55. Bias/variance trade-off complexity generalization error Bias Variance Underfitting Overfitting MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 55/128
  • 56. The learning procedure A learning procedure aims at two main goals: 1. to choose a parametric family of hypothesis ˆf(x, α) which contains or gives good approximation of the unknown function f (structural identification). 2. within the family ˆf(x, α), to estimate on the basis of the training set DN the parameter αN which best approximates f (parametric identification). In order to accomplish that, a learning procedure is made of two nested loops: 1. an external structural identification loop which goes through different model structures 2. an inner parametric identification loop which searches for the best parameter vector within the family structure. MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 56/128
  • 57. Parametric identification The parametric identification of the hypothesis is done according to ERM (Empirical Risk Minimization) principle where αN = α(DN ) = arg min α∈Λ MISEemp(α) minimizes the training error MISEemp(α) = N i=1 yi − ˆf(xi, α) 2 N constructed on the basis of the training data set DN . MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 57/128
  • 58. Parametric identification (II) • The computation of αN requires a procedure of multivariate optimization in the space of parameters. • The complexity of the optimization depends on the form of ˆf(·). • In some cases the parametric identification problem may be an NP-hard problem. • Thus, we must resort to some form of heuristic search. • Examples of parametric identification procedure are linear least-squares for linear models and backpropagated gradient-descent for feedforward neural networks. MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 58/128
  • 59. Model assessment • We have seen before that the training error is not a good estimator (i.e. it is too optimistic) of the generalization capability of the learned model. • Two alternative exists: 1. Complexity-based penalty criteria 2. Data-driven validation techniques MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 59/128
  • 60. Complexity-based penalization • In conventional statistics, various criteria have been developed, often in the context of linear models, for assessing the generalization performance of the learned hypothesis without the use of further validation data. • Such criteria take the form of a sum of two terms ˆGPE = MISEemp + complexity term where the complexity term represents a penalty which grows as the number of free parameters in the model grows. • This expression quantifies the qualitative consideration that simple models return high training error with a reduced complexity term while complex models have a low training error thanks to the high number of parameters. • The minimum for the criterion represents a trade-off between performance on the training set and complexity. MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 60/128
  • 61. Complexity-based penalty criteria If the input/output relation is linear, well-known examples of complexity based criteria are: • the Final Prediction Error (FPE) FPE = MISEemp(αN ) 1 + p/N 1 − p/N with p = n + 1, • the Generalized Cross-Validation (GCV) GCV = MISEemp(αN ) 1 (1 − p N )2 • the Akaike Information Criterion (AIC) AIC = p N − 1 N L(αN ) where L(·) is the log-likelihood function, MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 61/128
  • 62. • the Cp criterion proposed by Mallows Cp = MISEemp(αN ) ˆσ2 w + 2p − N where ˆσ2 w is an estimate of the variance of noise, • the Predicted Squared Error (PSE) PSE = MISEemp(αN ) + 2ˆσ2 w p N where ˆσ2 w is an estimate of the variance of noise. MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 62/128
  • 63. Data-driven validation techniques If no (e.g. linear) assumptions are made, how to measure MISE in a reliable way on a finite dataset? The most common techniques to return an estimate MISE are Testing: a testing sequence independent of DN and distributed according to the same probability distribution is used to assess the quality. In practice, unfortunately, an additional set of input/output observations is rarely available. Holdout: The holdout method, sometimes called test sample estimation, partitions the data DN into two mutually exclusive subsets, the training set Dtr and the holdout or test set DNts . k-fold Cross-validation: the set DN is randomly divided into k mutually exclusive test partitions of approximately equal size. The cases not found in each test partition are independently used for selecting the hypothesis which will be tested on the partition itself. The average error over all the k partitions is the cross-validated error rate. MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 63/128
  • 64. The K-fold cross-validation This is the algorithm in detail: 1. split the dataset DN into k roughly equal-sized parts. 2. For the kth part k = 1, . . . , K, fit the model to the other K − 1 parts of the data, and calculate the prediction error of the fitted model when predicting the k-th part of the data. 3. Do the above for k = 1, . . . , K and average the K estimates of prediction error. Let k(i) be the part of DN containing the ith sample. Then the cross-validation estimate of the MISE prediction error is MISECV = 1 N N i=1 (yi − ˆy −k(i) i )2 = 1 N N i=1 yi − ˆf(xi, α−k(i) ) 2 where ˆy −k(i) i denotes the fitted value for the ith observation returned by the model estimated with the k(i)th part of the data removed. MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 64/128
  • 65. 10-fold cross-validation K = 10: at each iteration 90% of data are used for training and the remaining 10% for the test. 90% 10% MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 65/128
  • 66. Leave-one-out cross validation • The cross-validation algorithm where K = N is also called the leave-one-out algorithm. • This means that for each ith sample, i = 1, . . . , N, 1. we carry out the parametric identification, leaving that observation out of the training set, 2. we compute the predicted value for the ith observation, denoted by ˆy−i i The corresponding estimate of the MISE prediction error is MISELOO = 1 N N i=1 (yi − ˆy−i i )2 = 1 N N i=1 (yi − ˆf xi, α−i ) 2 where α−i is the set of parameters returned by the parametric identification perfomed on the training set with the ith sample set aside. MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 66/128
  • 67. Model selection • Model selection concerns the final choice of the model structure • By structure we mean: • family of the approximator (e.g. linear, non linear) and if nonlinear which kind of learner (e.g. neural networks, support vector machines, nearest-neighbours, regression trees) • the value of hyper parameters (e.g. number of hidden layers, number of hidden nodes in NN, number of neighbors in KNN, number of levels in trees) • number and set of input variables • this choice is typically the result of a compromise between different factors, like the quantitative measures, the personal experience of the designer and the effort required to implement a particular model in practice. • Here we will consider only quantitative criteria. Two are the possible approaches: 1. the winner-takes-all approach 2. the combination of estimators approach. MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 67/128
  • 68. Model selection N REALIZATION STOCHASTIC PROCESS VALIDATION CLASSES of HYPOTHESIS LEARNED MODEL TRAINING SET MODEL SELECTION PARAMETRIC IDENTIFICATION IDENTIFICATION ,, , , , , STRUCTURAL α ? ΛSΛ2Λ1 GN 1 α 1 N α 1 N α 2 N α 2 N αN s αN s GN 2 GN S GN 2 GN 1 GN S MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 68/128
  • 69. Winner-takes-all The best hypothesis is selected in the set {αs N }, with s = 1, . . . , S, according to ˜s = arg min s=1,...,S MISE s A model with complexity ˜s is trained on the whole dataset DN and used for future predictions. MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 69/128
  • 70. Winner-takes-all pseudo-code 1. for s = 1, . . . , S: (Structural loop) • for j = 1, . . . , N (a) Inner parametric identification (for l-o-o): αs N−1 = arg min α∈Λs i=1:N,i=j (yi − ˆf(xi, α))2 (b) ej = yj − ˆf(xj, αs N−1) • MISELOO(s) = 1 N N j=1 e2 j 2. Model selection: ˜s = arg mins=1,...,S MISELOO(s) 3. Final parametric identification: α˜s N = arg minα∈Λ˜s N i=1(yi − ˆf(xi, α))2 4. The output prediction model is ˆf(·, α˜s N ) MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 70/128
  • 71. Model combination • The winner-takes-all approach is intuitively the approach which should work the best. • However, recent results in machine learning show that the performance of the final model can be improved not by choosing the model structure which is expected to predict the best but by creating a model whose output is the combination of the output of models having different structures. • The reason is that in reality any chosen hypothesis ˆf(·, αN ) is only an estimate of the real target and, like any estimate, is affected by a bias and a variance term. • Theoretical results on the combination of estimators show that the combination of unbiased estimators leads an unbiased estimator with reduced variance. • This principle is at the basis of approaches like bagging or boosting. MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 71/128
  • 72. Feature selection problem • Machine learning algorithms are known to degrade in performance (prediction accuracy) when faced with many inputs (aka features) that are not necessary for predicting the desired output. • In the feature selection problem, a learning algorithm is faced with the problem of selecting some subset of features upon which to focus its attention, while ignoring the rest. • Using all available features may negatively affect generalization performance, especially in the presence of irrelevant or redundant features. • Feature selection can be seen as an instance of model selection problem. MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 72/128
  • 73. Benefits and drawbacks of feature selection There are many potential benefits of feature selection: • facilitating data visualization and data understanding, • reducing the measurement and storage requirements, • reducing training and utilization times of the final model, • defying the curse of dimensionality to improve prediction performance. Drawbacks are • the search for a subset of relevant features introduces an additional layer of complexity in the modelling task. The search in the model hypothesis space is augmented by another dimension: the one of finding the optimal subset of relevant features. • additional time for learning. MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 73/128
  • 74. Curse of dimensionality • The error of the best model decreases with n but the mean integrated squared error of models increases faster than linearly in n. • In high dimensions, all data sets are sparse. • In high dimensions, the number of possible models to consider increases superexponenetially in n. • In high dimensions, all datasets show multicollinearity. • As n increases the amount of local data goes to zero. • For a uniform distribution around a query point xq the amount of data that are contained in a ball of radius r < 1 centered in xq grows like rn . MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 74/128
  • 75. Local density for large n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 r fractionoflocal points n=1 n=2 n=3 n=100 The size of the neighborhood on which we can estimate local features of the output (e.g. E[y|x]) increases with dimension n, making the estimation coarser and coarser. MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 75/128
  • 76. Methods of feature selection Two are the main approaches to feature selection: Filter methods: they are preprocessing methods. They attempt to assess the merits of features from the data, ignoring the effects of the selected feature subset on the performance of the learning algorithm. Examples are methods that select variables by ranking them through compression techniques (like PCA or clustering) or by computing correlation with the output. Wrapper methods: these methods assess subsets of variables according to their usefulness to a given predictor. The method conducts a search for a good subset using the learning algorithm itself as part of the evaluation function. The problem boils down to a problem of stochastic state space search. Example are the stepwise methods proposed in linear regression analysis. Embedded methods: they perform variable selection as part of the learning procedure and are usually specific to given learning machines. Examples are classification trees, random forests, and methods based on regularization techniques (e.g. lasso) MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 76/128
  • 77. Local learning MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 77/128
  • 78. Local modeling procedure The learning of a local model in xq ∈ Rn can be summarized in these steps: 1. Compute the distance between the query xq and the training samples according to a predefined metric. 2. Rank the neighbors on the basis of their distance to the query. 3. Select a subset of the k nearest neighbors according to the bandwidth which measures the size of the neighborhood. 4. Fit a local model (e.g. constant, linear,...). Each of the local approaches has one or more structural (or smoothing) parameters that control the amount of smoothing performed. Let us focus on the bandwidth selection. MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 78/128
  • 79. The bandwidth trade-off: overfit e q 0011 0011 0011 01 01 0011 01 0011 0011 00 00 11 11 0 0 1 1 0011 0011 00001111 01 00001111 0011 01 0011 00001111 0011 0011 000000000000000000000000000000000000000000011111111111111111111111111111111111111111110 00 00 0 00 00 000000000000000000000 1 11 11 1 11 11 111111111111111111111 x y 0011 0011 0011 01 01 0011 01 0011 0011 00 00 11 11 0 0 1 1 0011 000 111 00001111 01 00001111 0011 01 0011 00001111 0011 000000 111111 0011 000000000000000000000000000000000000000000011111111111111111111111111111111111111111110 00 00 0 00 00 000000000000000000000 1 11 11 1 11 11 111111111111111111111 x y Too narrow bandwidth ⇒ overfitting ⇒ large prediction error e. In terms of bias/variance trade-off, this is typically a situation of high variance. MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 79/128
  • 80. The bandwidth trade-off: underfit e q 0011 0011 0011 01 01 0011 01 0011 0011 00 00 11 11 0 0 1 1 0011 00001111 01 00001111 0011 01 0011 00001111 0011 0011 000000000000000000000000000000000000000000011111111111111111111111111111111111111111110 00 00 0 00 00 000000000000000000000 1 11 11 1 11 11 111111111111111111111 x y 0011 0011 0011 01 01 0011 000000 111111 0 00 1 11 00 0000 11 1111 0011 000 111 00001111 000 111 01 001100001111 00110011 000000 111111 0011 00001111 000000 111111 0011 000 111 0 00 1 11 000000 111111 00001111 000000000000000000000000000000000000000000011111111111111111111111111111111111111111110 00 00 0 00 00 000000000000000000000 1 11 11 1 11 11 111111111111111111111 x y Too large bandwidth ⇒ underfitting ⇒ large prediction error e In terms of bias/variance trade-off, this is typically a situation of high bias. MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 80/128
  • 81. Bandwidth and bias/variance trade-off Mean Squared Error 1/Bandwith FEW NEIGHBORSMANY NEIGHBORS Bias Variance Underfitting Overfitting MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 81/128
  • 82. The PRESS statistic • Cross-validation can provide a reliable estimate of the algorithm generalization error but it requires the training process to be repeated K times, which sometimes means a large computational effort. • In the case of linear models there exists a powerful statistical procedure to compute the leave-one-out cross-validation measure at a reduced computational cost • It is the PRESS (Prediction Sum of Squares) statistic, a simple formula which returns the leave-one-out (l-o-o) as a by-product of the least-squares. MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 82/128
  • 83. Leave-one-out for linear models PARAMETRIC IDENTIFICATION ON N-1 SAMPLES PUT THE j-th SAMPLE ASIDE TEST ON THE j-th SAMPLE PARAMETRIC IDENTIFICATION ON N SAMPLES N TIMES TRAINING SET PRESS STATISTIC LEAVE-ONE-OUT The leave-one-out error can be computed in two equivalent ways: the slowest way (on the right) which repeats N times the training and the test procedure; the fastest way (on the left) which performs only once the parametric identification and the computation of the PRESS statistic. MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 83/128
  • 84. The PRESS statistic • This allows a fast cross-validation without repeating N times the leave-one-out procedure. The PRESS procedure can be described as follows: 1. we use the whole training set to estimate the linear regression coefficients ˆα = (XT X)−1 XT Y 2. This procedure is performed only once on the N samples and returns as by product the Hat matrix H = X(XT X)−1 XT 3. we compute the residual vector e, whose jth term is ej = yj − xT j ˆα, 4. we use the PRESS statistic to compute eloo j as eloo j = ej 1 − Hjj where Hjj is the jth diagonal term of the matrix H. MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 84/128
  • 85. The PRESS statistic Thus, the leave-one-out estimate of the local mean integrated squared error is: MISELOO = 1 N N i=1 yi − ˆyi 1 − Hii 2 Note that PRESS is not an approximation of the loo error but simply a faster way of computing it. MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 85/128
  • 86. Selection of the number of neighbours • For a given query point xq, we can compute a set of predictions ˆyq(k) = xT q ˆα(k) , together with a set of associated leave-one-out error vectors MISELOO(k) for a number of neighbors ranging in [kmin, kmax]. • If the selection paradigm, frequently called winner-takes-all, is adopted, the most natural way to extract a final prediction ˆyq, consists in comparing the prediction obtained for each value of k on the basis of the classical mean square error criterion: ˆyq = xT q ˆα(ˆk), with ˆk = arg min k MISELOO(k) MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 86/128
  • 87. Local Model combination • As an alternative to the winner-takes-all paradigm, we can use a combination of estimates. • The final prediction of the value yq is obtained as a weighted average of the best b models, where b is a parameter of the algorithm. • Suppose the predictions ˆyq(k) and the loo errors MISELOO(k) have been ordered creating a sequence of integers {ki} so that MISELOO(ki) ≤ MISELOO(kj), ∀i < j. The prediction of ˆyq is given by ˆyq = b i=1 ζi ˆyq(ki) b i=1 ζi , where the weights are the inverse of the mean square errors: ζi = 1/MISELOO(ki). MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 87/128
  • 88. Forecasting MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 88/128
  • 89. One step-ahead and iterated prediction • Once a model of the embedding mapping is available, it can be used for two objectives: one-step-ahead prediction and iterated prediction. • In one-step-ahead prediction, the n previous values of the series are available and the forecasting problem can be cast in the form of a generic regression problem • In literature a number of supervised learning approaches have been used with success to perform one-step-ahead forecasting on the basis of historical data. MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 89/128
  • 90. One step-ahead prediction f ϕt-2 z-1 z-1 z-1 ϕt-3 ϕt-n ϕt-1 z-1 ϕt The approximator ˆf returns the prediction of the value of the time series at time t + 1 as a function of the n previous values (the rectangular box containing z−1 represents a unit delay operator, i.e., ϕt−1 = z−1 ϕt ). MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 90/128
  • 91. Nearest-neighbor one-step-ahead forecasts t −− y t−11t−16 t−1t−6 t We want to predict at time ¯t − 1 the next value of the series y of order n = 6. The pattern y¯t−16, y¯t−15, . . . , y¯t−11 is the most similar to the pattern {y¯t−6, y¯t−5, . . . , yˆt−1}. Then, the prediction ˆy¯t = y¯t−10 is returned. MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 91/128
  • 92. Multi-step ahead prediction • The prediction of the value of a time series H > 1 steps ahead is called H-step-ahead prediction. • We classify the methods for H-step-ahead prediction according to two features: the horizon of the training criterion and the single-output or multi-output nature of the predictor. MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 92/128
  • 93. Multi-step ahead prediction strategies The most common strategies are 1. Iterated: the model predicts H steps ahead by iterating a one-step-ahead predictor whose parameters are optimized to minimize the training error on one-step-ahead forecast (one-step-ahead training criterion). 2. Iterated strategy where parameters are optimized to minimize the training error on the iterated htr-step-ahead forecast (htr-step-ahead training criterion) where 1 < htr ≤ H. 3. Direct: the model makes a direct forecast at time t + h − 1, h = 1, . . . , H by modeling the time series in a multi-input single-output form 4. Direc: direct forecast but the input vector is extended at each step with predicted values. 5. MIMO: the model returns a vectorial forecast by modeling the time series in a multi-input multi-output form MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 93/128
  • 94. Iterated (or recursive) prediction • In the case of iterated prediction, the predicted output is fed back as input for the next prediction. • Here, the inputs consist of predicted values as opposed to actual observations of the original time series. • As the feedback values are typically distorted by the errors made by the predictor in previous steps, the iterative procedure may produce undesired effects of accumulation of the error. • Low performance is expected in long horizon tasks. This is due to the fact that they are essentially models tuned with a one-step-ahead criterion which is not capable of taking temporal behavior into account. MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 94/128
  • 95. Iterated prediction f ϕt-2 z-1 z-1 z-1 z-1 ϕt-3 ϕt-n ϕt-1 z-1 ϕt The approximator ˆf returns the prediction of the value of the time series at time t + 1 by iterating the predictions obtained in the previous steps (the rectangular box containing z−1 represents a unit delay operator, i.e., ˆϕt−1 = z−1 ˆϕt ). MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 95/128
  • 96. Iterated with h-step training criterion • This strategy adopts one-step-ahead predictors but adapts the model selection criterion in order to take into account the multi-step-ahead objective. • Methods like Recurrent Neural Networks belong to such class. Their recurrent architecture and the associated training algorithm (temporal backpropagation) are suitable to handle the time-dependent nature of the data. • In [4] we proposed an adaptation of the Lazy Learning algorithm where the number of neighbors is optimized in order to minimize the leave-one-out error over an horizon larger than one. This technique ranked second in the 1998 KULeuven Time Series competition. • A similar technique has been proposed by [14] who won the competition. MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 96/128
  • 97. Conventional and iterated leave-one-out a) 3 1 2 4 5 3 e (3) cv 1 2 3 4 5 e (3) b) it 1 2 4 5 3 1 2 3 4 5 3 MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 97/128
  • 98. Santa Fe time series A 0 200 400 600 800 1000 050100150200250 Santa Fe time series A t y The A chaotic time series has a training set of 1000 values: the task is to predict the continuation for 100 steps, starting from different points. MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 98/128
  • 99. One-step assessment criterion 0 10 20 30 40 50 60 70 80 90 100 0 50 100 150 200 250 300 MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 99/128
  • 100. Multi-step assessment criterion 0 10 20 30 40 50 60 70 80 90 100 0 50 100 150 200 250 300 MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 100/128
  • 101. Direct strategy • The Direct strategy [22, 17, 7] learns independently H models fh ϕt+h−1 = fh(ϕt−1, . . . , ϕt−n) + wt+h−1 with h ∈ {1, . . . , H} and returns a multi-step forecast by concatenating the H predictions. • Several machine learning models have been used to implement the Direct strategy for multi-step forecasting tasks, for instance neural networks [10], nearest neighbors [17] and decision trees [21]. • Since the Direct strategy does not use any approximated values to compute the forecasts, it is not prone to any accumulation of errors, since each model is tailored for the horizon it is supposed to predict. • Notwithstanding, it has some weaknesses. MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 101/128
  • 102. Direct strategy limitations • Since the H models are learned independently no statistical dependencies between the predictions ˆϕt+h−1, h = 1, . . . , H [3, 5, 10] is guaranteed. • Direct methods often require higher functional complexity [20] than iterated ones in order to model the stochastic dependency between two series values at two distant instants [9]. • This strategy demands a large computational time since the number of models to learn is equal to the size of the horizon. MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 102/128
  • 103. DirRec strategy • The DirRec strategy [18] combines the architectures and the principles underlying the Direct and the Recursive strategies. • DirRec computes the forecasts with different models for every horizon (like the Direct strategy) and, at each time step, it enlarges the set of inputs by adding variables corresponding to the forecasts of the previous step (like the Recursive strategy). • Unlike the previous strategies, the embedding size n is not the same for all the horizons. In other terms, the DirRec strategy learns H models fh from the time series where ϕt+h−1 = fh(ϕt+h−1, . . . , ϕt−n) + wt+h−1 with h ∈ {1, . . . , H}. • The technique is prone to the curse of dimensionality. The use of feature selection is recommended for large h. MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 103/128
  • 104. MIMO strategy • This strategy [3, 5] (also known as Joint strategy [10]) avoids the simplistic assumption of conditional independence between future values made by the Direct strategy [3, 5] by learning a single multiple-output model [ϕt+H−1, . . . , ϕt] = F(ϕt−1, . . . , ϕt−n) + w where F : Rd → RH is a vector-valued function [15], and w ∈ RH is a noise vector with a covariance that is not necessarily diagonal [13]. • The forecasts are returned in one step by a multiple-input multiple-output regression model. • In [5] we proposed a multi-output extension of the local learning algorithm. • Other multi-output regression model could be taken into consideration like multi-output neural networks or partial least squares. MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 104/128
  • 105. Time series dependencies t+3 ϕ ϕ ϕ ϕ ϕ t−1 t t+1 t+2 n = 2 NAR dependency ϕt = f(ϕt−1, ϕt−2) + w(t). MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 105/128
  • 106. Iterated modeling of dependencies t+3 ϕ ϕ ϕ t+1ϕ ϕt+2 tt−1 MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 106/128
  • 107. Direct modeling of dependencies t+3 ϕ ϕ ϕ ϕ ϕ t−1 t t+1 t+2 MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 107/128
  • 108. MIMO strategy • The rationale of the MIMO strategy is to model, between the predicted values, the stochastic dependency characterizing the time series. This strategy avoids the conditional independence assumption made by the Direct strategy as well as the accumulation of errors which plagues the Recursive strategy. • So far, this strategy has been successfully applied to several real-world multi-step time series forecasting tasks [3, 5, 19, 2]. • However, the wish to preserve the stochastic dependencies constrains all the horizons to be forecasted with the same model structure. Since this constraint could reduce the flexibility of the forecasting approach [19], a variant of the MIMO strategy (called DIRMO) has been proposed in [19, 2] . • Extensive validation on the 111 times series of the NN5 competition showed that MIMO are invariably better than single-output approaches. MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 108/128
  • 109. Validation of time series methods • The huge variety of strategies and algorithms that can be used to infer a predictor from observed data asks for a rigorous procedure of comparison and assessment. • Assessment demands benchmarks and benchmarking procedure. • Benchmarks can be defined by using • Simulated data obtained by simulating AR, NAR and other stochastic processes. This is particular useful for validating theoretical properties in terms of bias/variance. • Public domain benchmarks, like the one provided by Time Series Competitions. • Real measured data MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 109/128
  • 110. Competitions • Santa Fe Time Series Prediction and Analysis Competition (1994) [22]: • International Workshop on Advanced Black-box techniques for nonlinear modeling Competition (Leuven, Belgium; 1998) • NN3 competition [8]: 111 monthly time series drawn from homogeneous population of empirical business time series. • NN5 competition [1]: 111 time series of the daily retirement amounts from independent cash machines at different, randomly selected locations across England. • Kaggle competition. MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 110/128
  • 111. Accuracy measures Let et+h = ϕt+h − ˆϕt+h represent the error of the forecast ˆϕt+h at the horizon h = 1, . . . , H. A conventional measure of accuracy is the Normalized Mean Squared Error NMSE = H h=1(ϕt+h − ˆϕt+h)2 H h=1(ϕt+h − ˆµ)2 This quantity is smaller than one if the predictor performs better than the naivest predictor, i.e. the average ˆµ. Other measures rely on relative or percentage error pet+h = 100 ϕt+h − ˆϕt+h ϕt+h like MAPE = H h=1 |pet+h| H MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 111/128
  • 112. Applications in my lab MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 112/128
  • 113. MLG projects on forecasting • Low-energy streaming of wireless sensor data • Decision support in anesthesia • Side-channel attack MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 113/128
  • 114. Low-energy streaming of wireless sensor data • In monitoring applications, only an approximation to sensor readings is sufficient (e.g. ±0.5C, ±2% humidity, ..). In this context a Dual Prediction Scheme is effective. • A sensor node is provided with a time series prediction model ϕt = f(ϕt−1, α) (e.g. autoregressive models) and a learning method for identifying the best set of parameters α (e.g. recursive least squares). • The sensor node then sends the parameters of the model instead of the actual data to the recipient. The recipient node then runs the models to reconstruct an approximation of the data stream collected on the distant node. • The sensor node also runs the prediction model. When its predictions diverges by more than ±ǫ from the actual reading, a new model is sent to the recipient. • This allows to reduce the communication effort if an appropriate model is run by the sensor node. • Since the cost of transmission is much larger than the cost of computation, in realistic situations this scheme allows economy of power consumption. MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 114/128
  • 115. Low-energy streaming of wireless sensor data 0 5 10 15 20 202530354045 Accuracy: 2°C Constant model Time (Hour) Temperature(°C) G GG GGG GG G G GG G G G G G G G G G MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 115/128
  • 116. Adaptive model selection Tradeoff: more complex models predict better measurements but have a higher number of parameters Model complexity Metric Communication costs Model error AR(p) : ˆsi[t] = p j=1 θjsi[t − j] MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 116/128
  • 117. Adaptive Model Selection We proposed an Adaptive Model Selection strategy [11] that • takes into account the cost of sending model parameters, • allows sensor nodes to determine autonomously the model which best fits their measurements, • provides a statistically sound selection mechanism to discard poorly performing models, • gave in experimental results about 45% of communication savings on average, • was implemented in TinyOS, the reference operating system. MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 117/128
  • 118. Predictive modeling in anesthesiology • During surgery, anesthesiologists controls the depth of anesthesia by means of types of drugs • Anesthesiologists observe the patient state by observing unconsciousness signal in real-time which are collected by monitors connected via electrodes to the patient’s forehead • The bispectral index BIS monitor (by Aspect Medical Systems) is a single dimensionless number between 0 to 100 where 0 equals EEG silence, 100 is the expected value for a fully awake adult, and between 40 and 60 indicates a recommended level. • It remains difficult for the anesthesiologist, especially if inexperienced, to predict how the BIS signal could vary after a change in the administered anesthetic agents. This is generally due to inter-individual variability problem • We designed a ML system [6] to predict multi-step-ahead the evolution of the BIS on the basis of historical data. MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 118/128
  • 119. Predictive modeling in anesthesiology MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 119/128
  • 120. MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 120/128
  • 121. Side channel attack • In cryptography, a side channel attack is any attack based on the analysis of measurements related to the physical implementation of a cryptosystem. • Side channel attacks take advantage of the fact that instantaneous power consumption, encryption time or/and electromagnetic leaks of a cryptographic device depend on the processed data and the performed operations. • Power analysis attacks are an instance of side-channel attacks which assume that different encryption keys imply a different power consumptions. • Nowadays, the possibility of collecting a large amount of power traces (i.e. time series) paves the way to the adoption of machine learning techniques. • Effective side channel attacks enables effective countermeasures. MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 121/128
  • 122. SCA and time series classification • The power consumption of a crypto device using a secret key Q ∈ {0, 1}k (size k = 8) can be modeled as a time series T(Q) of order n T (Q) (t+1) = f(T (Q) (t) , T (Q) (t−1), ..., T (Q) (t−n)) + ǫ • For each key Qj we infer a predictive model ˆf [12] such that T (Qj ) (t+1) = ˆfQj (T (Qj ) (t) , T (Qj ) (t−1), ..., T (Qj) (t−n)) + ǫ • These models can be used to classify an unlabeled time series T and predict the associated key by computing a distance for each Qj D (Qj, T) = 1 N − n + 1 N t=n ˆfQj T(t−1), T(t−2), ..., T(t−n−1) − T(t) 2 and choosing the key minimizing it ˆQ = arg min j∈[0,2(k−1)] D (Qj, T) MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 122/128
  • 123. Conclusions MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 123/128
  • 124. Open-source software Many commercial solutions exist but only open-source software can cope with • fast integration of new algorithms • portability over several platforms • new paradigms of data storage (e.g. Hadoop) • integration with different data formats and architectures A de-facto standard in computational statistics, machine learning, bioinformatics, geostatistics or more general analytics is nowadays Highly recommended ! MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 124/128
  • 125. All that we didn’t have time to discuss • ARIMA models • GARCH models • Frequence space representation • Nonstationarity • Vectorial time series • Spatio temporal time series • Time series classification MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 125/128
  • 126. Suggestions for PhD topics • Time series and big data • streaming data (environmental data) • large vectorial time series (weather data) • Spatio-temporal time series and graphical models • Beyond cross-validation for model/input selection • Long term forecasting (effective integration of iterated and directed models) • Causality and time-series • Scalable machine learning Suggestion: use methods and models to solve problems... not problems to sanctify methods or models... MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 126/128
  • 127. Conclusion • Popper claimed that, if a theory is falsifiable (i.e. it can be contradicted by an observation or the outcome of a physical experiment.), then it is scientific. Since prediction is the most falsifiable aspect of science it is also the most scientific one. • Effective machine learning is an extension of statistics, in no way an alternative. • Simplest (i.e. linear) model first. • Local learning techniques represent an effective trade-off between linearity and nonlinearity. • Modelling is more an art than an automatic process... then experience data analysts are more valuable than expensive tools. • Expert knowledge matters..., data too • Understanding what is predictable is as important as trying to predict it. MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 127/128
  • 128. Forecasting is difficult, especially the future • "Computers in the future may weigh no more than 1.5 tons." –Popular Mechanics, forecasting the relentless march of science, 1949 • "I think there is a world market for maybe five computers." –Chairman of IBM, 1943 • "Stocks have reached what looks like a permanently high plateau." –Professor of Economics, Yale University, 1929. • "Airplanes are interesting toys but of no military value." –Professor of Strategy, Ecole Superieure de Guerre. • "Everything that can be invented has been invented." –Commissioner, U.S. Office of Patents, 1899. • "Louis Pasteur’s theory of germs is ridiculous fiction". –Professor of Physiology at Toulouse, 1872 • "640K ought to be enough for anybody." – Bill Gates, 1981 MLSS 2013, Hammamet - Machine Learning Strategies for Prediction – p. 128/128
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