Machine Learning Strategies for Time Series Prediction

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.

... and in terms of distances
According to MathSciNet
•• Distance from Erdos= 5
• Distance from Zaiane= 6
• Distance from Deisenroth= 8

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).

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).

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

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)

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.

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...

Notion of time series

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.

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.

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.

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.

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.

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.

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.

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.

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.

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

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.

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)

Some examples of stochastic
processes

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.

Example: Gaussian purely random
0 200 400 600 800 1000
−3−2−10123
White noise
t
y

Example: autocorrelation function
0 5 10 15 20 25 30
0.00.20.40.60.81.0
Lag
ACF
Series y

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.

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

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

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.

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

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.

Example: AR(2)
0 200 400 600 800 1000
051015
AR(2)
t
y

Example: AR(2)
0 5 10 15 20 25 30
−0.50.00.51.0
Lag
ACF
Series y

Partial autocorrelation in AR(2)
0 5 10 15 20 25 30
−0.50.00.5
Lag
PartialACF
Series y

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}.

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)

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.

From linear to nonlinear setting

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.

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...

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.

Machine learning for prediction

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.

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?

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.

Model degree 1
−2 −1 0 1 2
−5051015
x
Y
Training error= 2 degree= 1
ˆf(x) = α0 + α1x

Model degree 3
−2 −1 0 1 2
−5051015
x
Y
Training error= 0.92 degree= 3
ˆf(x) = α0 + α1x + · · · + α3x3

Model degree 18
−2 −1 0 1 2
−5051015
x
Y
Training error= 0.4 degree= 18
ˆf(x) = α0 + α1x + · · · + α18x1
8

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.

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.

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.

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.

• 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.

Bias/variance trade-off
complexity
generalization
error
Bias
Variance
Underfitting Overfitting

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.

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 .

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.

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

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.

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,

• 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.

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.

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.

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%

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.

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.

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

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.

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 )

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.

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.

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.

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
.

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.

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)

Local learning

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.

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.

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.

Bandwidth and bias/variance trade-off
Mean Squared Error
1/Bandwith
FEW NEIGHBORSMANY NEIGHBORS
Bias
Variance
Underfitting Overfitting

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.

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.

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.

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.

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)

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).

Forecasting

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.

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
).

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.

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.

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

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.

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
).

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.

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

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.

One-step assessment criterion
0 10 20 30 40 50 60 70 80 90 100
0
50
100
150
200
250
300

Multi-step assessment criterion
0 10 20 30 40 50 60 70 80 90 100
0
50
100
150
200
250
300

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.

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.

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.

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.

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).

Iterated modeling of dependencies
t+3
ϕ ϕ
ϕ
t+1ϕ ϕt+2
tt−1

Direct modeling of dependencies
t+3
ϕ ϕ
ϕ ϕ
ϕ
t−1 t
t+1 t+2

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.

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

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.

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

Applications in my lab

MLG projects on forecasting
• Low-energy streaming of wireless sensor data
• Decision support in anesthesia
• Side-channel attack

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

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]

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.

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.

Predictive modeling in anesthesiology

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.

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)

Conclusions

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 !

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

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...

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.

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

References
[1] Robert R. Andrawis, Amir F. Atiya, and Hisham El-
Shishiny. Forecast combinations of computational intel-
ligence and linear models for the NN5 time series fore-
casting competition. International Journal of Forecasting,
January 2011.
[2] S. Ben Taieb, A. Sorjamaa, and G. Bontempi. Multiple-
output modelling for multi-step-ahead forecasting. Neuro-
computing, 73:1950–1957, 2010.
[3] G. Bontempi. Long term time series prediction with multi-
input multi-output local learning. In Proceedings of the 2nd
European Symposium on Time Series Prediction (TSP),
ESTSP08, pages 145–154, Helsinki, Finland, February
2008.
[4] G. Bontempi, M. Birattari, and H. Bersini. Local learning
for iterated time-series prediction. In I. Bratko and S. Dze-
roski, editors, Machine Learning: Proceedings of the Six-
teenth International Conference, pages 32–38, San Fran-
cisco, CA, 1999. Morgan Kaufmann Publishers.
[5] G. Bontempi and S. Ben Taieb. Conditionally dependent
strategies for multiple-step-ahead prediction in local learn-
128-1

ing. International Journal of Forecasting, 27(3):689–699,
2011.
[6] Olivier Caelen, Gianluca Bontempi, and Luc Barvais. Ma-
chine learning techniques for decision support in anesthe-
sia. In Artiﬁcial Intelligence in Medicine, pages 165–169.
Springer Berlin Heidelberg, 2007.
[7] Haibin Cheng, Pang-Ning Tan, Jing Gao, and Jerry
Scripps. Multistep-ahead time series prediction. In
PAKDD, pages 765–774, 2006.
[8] Sven F. Crone, Michèle Hibon, and Konstantinos
Nikolopoulos. Advances in forecasting with neural net-
works? empirical evidence from the nn3 competition on
time series prediction. International Journal of Forecast-
ing, 27, 2011.
[9] M. Guo, Z. Bai, and H.Z. An. Multi-step prediction for non-
linear autoregressive models based on empirical distribu-
tions. Statistica Sinica, pages 559–570, 1999.
[10] D. M. Kline. Methods for multi-step time series forecasting
with neural networks. In G. Peter Zhang, editor, Neural
Networks in Business Forecasting, pages 226–250. Infor-
mation Science Publishing, 2004.
128-2

[11] Yann-Aël Le Borgne, Silvia Santini, and Gianluca Bon-
tempi. Adaptive model selection for time series pre-
diction in wireless sensor networks. Signal Processing,
87(12):3010–3020, 2007.
[12] Liran Lerman, Gianluca Bontempi, Souhaib Ben Taieb,
and Olivier Markowitch. A time series approach for pro-
ﬁling attack. In SPACE, pages 75–94, 2013.
[13] José M. Matías. Multi-output nonparametric regression.
In Carlos Bento, Amílcar Cardoso, and Gaël Dias, edi-
tors, EPIA, volume 3808 of Lecture Notes in Computer
Science, pages 288–292. Springer, 2005.
[14] J. McNames. A nearest trajectory strategy for time series
prediction. In Proceedings of the InternationalWorkshop
on Advanced Black-Box Techniques for Nonlinear Model-
ing, pages 112–128, Belgium, 1998. K.U. Leuven.
[15] Charles A. Micchelli and Massimiliano A. Pontil. On learn-
ing vector-valued functions. Neural Comput., 17(1):177–
204, 2005.
[16] T. M. Mitchell. Machine Learning. McGraw Hill, 1997.
[17] A. Sorjamaa, J. Hao, N. Reyhani, Y. Ji, and A. Lendasse.
Methodology for long-term prediction of time series. Neu-
rocomputing, 70(16-18):2861–2869, October 2007.
128-3

[18] A. Sorjamaa and A. Lendasse. Time series prediction us-
ing dirrec strategy. In M. Verleysen, editor, ESANN06, Eu-
ropean Symposium on Artiﬁcial Neural Networks, pages
143–148, Bruges, Belgium, April 26-28 2006. European
Symposium on Artiﬁcial Neural Networks.
[19] Souhaib Ben Taieb, Gianluca Bontempi, Antti Sorjamaa,
and Amaury Lendasse. Long-term prediction of time se-
ries by combining direct and mimo strategies. Interna-
tional Joint Conference on Neural Networks, 2009.
[20] H. Tong. Threshold models in Nonlinear Time Series Anal-
ysis. Springer Verlag, Berlin, 1983.
[21] Van Tung Tran, Bo-Suk Yang, and Andy Chit Chiow Tan.
Multi-step ahead direct prediction for the machine con-
dition prognosis using regression trees and neuro-fuzzy
systems. Expert Syst. Appl., 36(5):9378–9387, 2009.
[22] A.S. Weigend and N.A. Gershenfeld. Time Series Predic-
tion: forecasting the future and understanding the past.
Addison Wesley, Harlow, UK, 1994.
128-4

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