The document discusses steps for identifying and building ARIMA models for time series data. It describes ARIMA models as consisting of three components - identification, estimation, and diagnostic checking. For identification, it explains how to determine the p, d, and q values by examining the autocorrelation and partial autocorrelation functions of stationary differenced time series data. It then discusses using the method of moments to estimate ARIMA model parameters by equating sample statistics to population parameters.
This document provides an overview of ARMA, ARIMA, and SARIMA models. It describes the components of each model, including the autoregressive, integrated, and moving average parts. It also outlines the steps for identifying, estimating, and evaluating these models, including determining stationarity and selecting parameter values. The key assumptions of these times series models are that the data must be stationary or made stationary through differencing.
This document provides an introduction to ARIMA (AutoRegressive Integrated Moving Average) models. It discusses key assumptions of ARIMA including stationarity. ARIMA models combine autoregressive (AR) terms, differences or integrations (I), and moving averages (MA). The document outlines the Box-Jenkins approach for ARIMA modeling including identifying a model through correlograms and partial correlograms, estimating parameters, and diagnostic checking to validate the model prior to forecasting.
Time series forecasting involves analyzing sequential data measured over time. A time series can be univariate (containing a single variable) or multivariate (containing multiple variables). It can also be continuous or discrete. Key components of time series include trends, cyclical variations, seasonal variations, and irregular variations. Time series analysis involves fitting a model to the data. Stationarity, where the statistical properties do not depend on time, is required for forecasting. Common forecasting models include ARMA, ARIMA, and SARIMA stochastic models as well as artificial neural networks and support vector machines. Each approach has strengths for modeling nonlinear relationships and generalizing to make predictions.
The document provides an overview of time series analysis. It discusses key concepts like components of a time series, stationarity, autocorrelation functions, and various forecasting models including AR, MA, ARMA, and ARIMA. It also covers exponential smoothing and how to decompose, validate, and test the accuracy of forecasting models. Examples are given of different time series patterns and how to make non-stationary data stationary.
The document provides an overview of a time series analysis and forecasting course. It discusses key topics that will be covered including descriptive statistics, correlation, regression, hypothesis testing, clustering, time series analysis and forecasting techniques like TCSI and ARIMA models. It notes that the presentation serves as class notes and contains informal high-level summaries intended to aid the author, and encourages readers to check the website for updated versions of the document.
This document provides an overview of time series analysis techniques including moving average (MA) models, exponential smoothing, and ARMA models. It describes the key components of MA models including the MA(q) notation and theoretical properties. Exponential smoothing is presented as a weighted moving average for smoothing and short-term forecasting. The ARMA model is introduced as combining autoregressive and moving average terms to model a time series.
ARIMA models provide another approach to time series forecasting. Exponential smoothing and ARIMA models are the two most widely-used approaches to time series forecasting, and provide complementary approaches to the problem. While exponential smoothing models were based on a description of trend and seasonality in the data, ARIMA models aim to describe the autocorrelations in the data.
This document provides an overview of ARMA, ARIMA, and SARIMA models. It describes the components of each model, including the autoregressive, integrated, and moving average parts. It also outlines the steps for identifying, estimating, and evaluating these models, including determining stationarity and selecting parameter values. The key assumptions of these times series models are that the data must be stationary or made stationary through differencing.
This document provides an introduction to ARIMA (AutoRegressive Integrated Moving Average) models. It discusses key assumptions of ARIMA including stationarity. ARIMA models combine autoregressive (AR) terms, differences or integrations (I), and moving averages (MA). The document outlines the Box-Jenkins approach for ARIMA modeling including identifying a model through correlograms and partial correlograms, estimating parameters, and diagnostic checking to validate the model prior to forecasting.
Time series forecasting involves analyzing sequential data measured over time. A time series can be univariate (containing a single variable) or multivariate (containing multiple variables). It can also be continuous or discrete. Key components of time series include trends, cyclical variations, seasonal variations, and irregular variations. Time series analysis involves fitting a model to the data. Stationarity, where the statistical properties do not depend on time, is required for forecasting. Common forecasting models include ARMA, ARIMA, and SARIMA stochastic models as well as artificial neural networks and support vector machines. Each approach has strengths for modeling nonlinear relationships and generalizing to make predictions.
The document provides an overview of time series analysis. It discusses key concepts like components of a time series, stationarity, autocorrelation functions, and various forecasting models including AR, MA, ARMA, and ARIMA. It also covers exponential smoothing and how to decompose, validate, and test the accuracy of forecasting models. Examples are given of different time series patterns and how to make non-stationary data stationary.
The document provides an overview of a time series analysis and forecasting course. It discusses key topics that will be covered including descriptive statistics, correlation, regression, hypothesis testing, clustering, time series analysis and forecasting techniques like TCSI and ARIMA models. It notes that the presentation serves as class notes and contains informal high-level summaries intended to aid the author, and encourages readers to check the website for updated versions of the document.
This document provides an overview of time series analysis techniques including moving average (MA) models, exponential smoothing, and ARMA models. It describes the key components of MA models including the MA(q) notation and theoretical properties. Exponential smoothing is presented as a weighted moving average for smoothing and short-term forecasting. The ARMA model is introduced as combining autoregressive and moving average terms to model a time series.
ARIMA models provide another approach to time series forecasting. Exponential smoothing and ARIMA models are the two most widely-used approaches to time series forecasting, and provide complementary approaches to the problem. While exponential smoothing models were based on a description of trend and seasonality in the data, ARIMA models aim to describe the autocorrelations in the data.
1) To understand the underlying structure of Time Series represented by sequence of observations by breaking it down to its components.
2) To fit a mathematical model and proceed to forecast the future.
Time Series basic concepts and ARIMA family of models. There is an associated video session along with code in github: https://github.com/bhaskatripathi/timeseries-autoregressive-models
https://drive.google.com/file/d/1yXffXQlL6i4ufQLSpFFrJgymhHNXL1Mf/view?usp=sharing
This document discusses forecasting gasoline prices in the United States using an ARIMA model. It provides background on gasoline, including its consumption and retail prices. The objective is to understand price volatility due to supply and demand constraints. Data on US gasoline prices from 1993-2014 is obtained from the EIA. After checking for stationarity and transforming the data, an ARIMA(1,1,3) model is identified as best. This model reveals gasoline prices are significantly related to past prices and unobserved factors. The validated model is used to forecast future gasoline prices.
This document discusses autocorrelation models and their applications in Python. It describes the autocorrelation function (ACF) and partial autocorrelation function (PACF), and how they are used to identify autoregressive (AR) and moving average (MA) time series models. AR models regress the current value on prior values, while MA models regress the current value on prior noise terms. The document demonstrates how to interpret ACF and PACF plots to select AR or MA models, and how to fit these models in Python.
This document provides examples and explanations of various ARIMA models. It discusses:
- Examples of common ARIMA models including ARIMA(0,1,0), ARIMA(1,1,0), and ARIMA(2,1,2)
- That ARIMA models are used to make non-stationary time series data stationary through differencing
- The Box-Jenkins methodology is an iterative 4 step process used to identify, estimate, and select the best ARIMA model for forecasting a time series
ARCH/GARCH model.ARCH/GARCH is a method to measure the volatility of the series, to model the noise term of ARIMA model. ARCH/GARCH incorporates new information and analyze the series based on the conditional variance where users can forecast future values with updated information. Here we used ARIMA-ARCH model to forecast moments. And forecast error 0.9%
This document discusses stationarity in time series analysis. It defines stationarity as a time series having a constant mean, constant variance, and constant autocorrelation structure over time. Non-stationary time series can be identified through run sequence plots, summary statistics, histograms, and augmented Dickey-Fuller tests. Common transformations like removing trends, heteroscedasticity through logging, differencing to remove autocorrelation, and removing seasonality can be used to make non-stationary time series data stationary. Python is used to demonstrate identifying and transforming non-stationary time series data.
Businesses use forecasting extensively to make predictions such as demand, capacity, budgets and revenue. Among these different forecasting models identifying seasonal patterns in data can go a long way by providing seasonal insights to the business decision makers so that they can strategist for seasonal effects.
This document provides an overview of time series analysis and forecasting using neural networks. It discusses key concepts like time series components, smoothing methods, and applications. Examples are provided on using neural networks to forecast stock prices and economic time series. The agenda covers introduction to time series, importance, components, smoothing methods, applications, neural network issues, examples, and references.
Time series forecasting with machine learningDr Wei Liu
An introduction of developing and application time series forecast models with both traditional time series methods and machine learning techniques. Case study for a challenging very short-term electrical price forecasting project was presented.
Arima Forecasting - Presentation by Sera Cresta, Nora Alosaimi and Puneet MahanaAmrinder Arora
Arima Forecasting - Presentation by Sera Cresta, Nora Alosaimi and Puneet Mahana. Presentation for CS 6212 final project in GWU during Fall 2015 (Prof. Arora's class)
The explosion of sensors in all types of devices from “smart” consumer wearables and appliances to complex machines on manufacturing floors has given rise to a requirement to quickly analyze vast quantities of sensor metrics to provide meaningful insights. From exploratory to predictive analytics, analyzing time-series data is essential to address inefficiencies, identify risks and improve operations.
In this presentation, we will see how you can conduct exploratory analytics of time-series data rapidly to gain insights into the performance of the machines being monitored. We will talk about how to look at data from multiple metrics together in a holistic way to hone in on anomalies and identify potential problems. Finally, we will cover algorithms and techniques to predict future trends for time-series metrics. Along the way, we will discuss useful tools and technologies to perform time-series data analysis in minutes.
Data Science - Part X - Time Series ForecastingDerek Kane
This lecture provides an overview of Time Series forecasting techniques and the process of creating effective forecasts. We will go through some of the popular statistical methods including time series decomposition, exponential smoothing, Holt-Winters, ARIMA, and GLM Models. These topics will be discussed in detail and we will go through the calibration and diagnostics effective time series models on a number of diverse datasets.
1) The document discusses time series analysis and forecasting of financial time series data. It covers topics like identifying patterns in time series data, various time series models including AR, MA, ARMA and ARIMA models.
2) The analysis of MRF monthly return data identified it as a stationary time series. The ACF and PACF plots suggested an ARIMA(1,0,1) model which was fitted to the data.
3) The parameters of the ARIMA(1,0,1) model were estimated with the constant at 0.03219, AR coefficient at 0.858689 and MA coefficient at 0.998545. This provided the best fit for modeling
This document discusses methods for selecting the order of an autoregressive (AR) model. It explains that AR models depend only on previous outputs and have poles but no zeros. Several criteria for selecting the optimal AR model order are presented, including the Akaike Information Criterion (AIC) and Finite Prediction Error (FPE) criterion. Higher order models fit the data better but can introduce spurious peaks, so the goal is to minimize criteria like AIC or FPE to find the best balance. The document concludes that while these criteria provide guidance, the optimal order depends on the specific data, and inconsistencies can exist between the different methods.
Time series data are observations collected over time on one or more variables. Time series data can be used to analyze problems involving changes over time, such as stock prices, GDP, and exchange rates. Time series data must be stationary, meaning that its statistical properties like mean and variance do not change over time, to avoid spurious regressions. Non-stationary time series can be transformed to become stationary through differencing, removing trends, or taking logs. Common time series models like ARIMA rely on stationary data.
This document provides an overview of time series analysis and its key components. It discusses that a time series is a set of data measured at successive times joined together by time order. The main components of a time series are trends, seasonal variations, cyclical variations, and irregular variations. Time series analysis is important for business forecasting, understanding past behavior, and facilitating comparison. There are two main mathematical models used - the additive model which assumes data is the sum of its components, and the multiplicative model which assumes data is the product of its components. Decomposition of a time series involves discovering, measuring, and isolating these different components.
The document discusses using a vector autoregression (VAR) model to forecast two time series - leads and binds - that interact with each other. A 5-period VAR model is found to best capture the weekly periodicity between the series. The model is shown to accurately forecast leads 1-11 days in advance, within 2% error, and binds within 5% error over a two week period, indicating the interaction between the series can be used to predict each going forward. Some conclusions drawn are that the VAR model performs well but could be improved by trying other techniques or adding external variables.
This document defines time series and its components. A time series is a set of observations recorded over successive time intervals. It has four main components: trend, seasonality, cycles, and irregular variations. Trend refers to the overall increasing or decreasing tendency over time. Seasonality refers to predictable changes that occur around the same time each year. Cycles have periods longer than a year. Irregular variations are random fluctuations. The document also discusses methods for analyzing time series components including additive, multiplicative, and mixed models.
This Presentation describes, in short, Introduction to Time Series and the overall procedure required for Time Series Modelling including general terminologies and algorithms. However the detailed Mathematics is excluded in the slides, this ppt means to give a start to understanding the Time Series Modelling before going into detailed Statistics.
Business Analytics Foundation with R tool - Part 5Beamsync
The current presentation published by Beamsync.
If you are looking for analytics training in Bangalore, consult Beamsync Training Centre.
For upcoming schedules please visit: http://beamsync.com/business-analytics-training-bangalore/
1) To understand the underlying structure of Time Series represented by sequence of observations by breaking it down to its components.
2) To fit a mathematical model and proceed to forecast the future.
Time Series basic concepts and ARIMA family of models. There is an associated video session along with code in github: https://github.com/bhaskatripathi/timeseries-autoregressive-models
https://drive.google.com/file/d/1yXffXQlL6i4ufQLSpFFrJgymhHNXL1Mf/view?usp=sharing
This document discusses forecasting gasoline prices in the United States using an ARIMA model. It provides background on gasoline, including its consumption and retail prices. The objective is to understand price volatility due to supply and demand constraints. Data on US gasoline prices from 1993-2014 is obtained from the EIA. After checking for stationarity and transforming the data, an ARIMA(1,1,3) model is identified as best. This model reveals gasoline prices are significantly related to past prices and unobserved factors. The validated model is used to forecast future gasoline prices.
This document discusses autocorrelation models and their applications in Python. It describes the autocorrelation function (ACF) and partial autocorrelation function (PACF), and how they are used to identify autoregressive (AR) and moving average (MA) time series models. AR models regress the current value on prior values, while MA models regress the current value on prior noise terms. The document demonstrates how to interpret ACF and PACF plots to select AR or MA models, and how to fit these models in Python.
This document provides examples and explanations of various ARIMA models. It discusses:
- Examples of common ARIMA models including ARIMA(0,1,0), ARIMA(1,1,0), and ARIMA(2,1,2)
- That ARIMA models are used to make non-stationary time series data stationary through differencing
- The Box-Jenkins methodology is an iterative 4 step process used to identify, estimate, and select the best ARIMA model for forecasting a time series
ARCH/GARCH model.ARCH/GARCH is a method to measure the volatility of the series, to model the noise term of ARIMA model. ARCH/GARCH incorporates new information and analyze the series based on the conditional variance where users can forecast future values with updated information. Here we used ARIMA-ARCH model to forecast moments. And forecast error 0.9%
This document discusses stationarity in time series analysis. It defines stationarity as a time series having a constant mean, constant variance, and constant autocorrelation structure over time. Non-stationary time series can be identified through run sequence plots, summary statistics, histograms, and augmented Dickey-Fuller tests. Common transformations like removing trends, heteroscedasticity through logging, differencing to remove autocorrelation, and removing seasonality can be used to make non-stationary time series data stationary. Python is used to demonstrate identifying and transforming non-stationary time series data.
Businesses use forecasting extensively to make predictions such as demand, capacity, budgets and revenue. Among these different forecasting models identifying seasonal patterns in data can go a long way by providing seasonal insights to the business decision makers so that they can strategist for seasonal effects.
This document provides an overview of time series analysis and forecasting using neural networks. It discusses key concepts like time series components, smoothing methods, and applications. Examples are provided on using neural networks to forecast stock prices and economic time series. The agenda covers introduction to time series, importance, components, smoothing methods, applications, neural network issues, examples, and references.
Time series forecasting with machine learningDr Wei Liu
An introduction of developing and application time series forecast models with both traditional time series methods and machine learning techniques. Case study for a challenging very short-term electrical price forecasting project was presented.
Arima Forecasting - Presentation by Sera Cresta, Nora Alosaimi and Puneet MahanaAmrinder Arora
Arima Forecasting - Presentation by Sera Cresta, Nora Alosaimi and Puneet Mahana. Presentation for CS 6212 final project in GWU during Fall 2015 (Prof. Arora's class)
The explosion of sensors in all types of devices from “smart” consumer wearables and appliances to complex machines on manufacturing floors has given rise to a requirement to quickly analyze vast quantities of sensor metrics to provide meaningful insights. From exploratory to predictive analytics, analyzing time-series data is essential to address inefficiencies, identify risks and improve operations.
In this presentation, we will see how you can conduct exploratory analytics of time-series data rapidly to gain insights into the performance of the machines being monitored. We will talk about how to look at data from multiple metrics together in a holistic way to hone in on anomalies and identify potential problems. Finally, we will cover algorithms and techniques to predict future trends for time-series metrics. Along the way, we will discuss useful tools and technologies to perform time-series data analysis in minutes.
Data Science - Part X - Time Series ForecastingDerek Kane
This lecture provides an overview of Time Series forecasting techniques and the process of creating effective forecasts. We will go through some of the popular statistical methods including time series decomposition, exponential smoothing, Holt-Winters, ARIMA, and GLM Models. These topics will be discussed in detail and we will go through the calibration and diagnostics effective time series models on a number of diverse datasets.
1) The document discusses time series analysis and forecasting of financial time series data. It covers topics like identifying patterns in time series data, various time series models including AR, MA, ARMA and ARIMA models.
2) The analysis of MRF monthly return data identified it as a stationary time series. The ACF and PACF plots suggested an ARIMA(1,0,1) model which was fitted to the data.
3) The parameters of the ARIMA(1,0,1) model were estimated with the constant at 0.03219, AR coefficient at 0.858689 and MA coefficient at 0.998545. This provided the best fit for modeling
This document discusses methods for selecting the order of an autoregressive (AR) model. It explains that AR models depend only on previous outputs and have poles but no zeros. Several criteria for selecting the optimal AR model order are presented, including the Akaike Information Criterion (AIC) and Finite Prediction Error (FPE) criterion. Higher order models fit the data better but can introduce spurious peaks, so the goal is to minimize criteria like AIC or FPE to find the best balance. The document concludes that while these criteria provide guidance, the optimal order depends on the specific data, and inconsistencies can exist between the different methods.
Time series data are observations collected over time on one or more variables. Time series data can be used to analyze problems involving changes over time, such as stock prices, GDP, and exchange rates. Time series data must be stationary, meaning that its statistical properties like mean and variance do not change over time, to avoid spurious regressions. Non-stationary time series can be transformed to become stationary through differencing, removing trends, or taking logs. Common time series models like ARIMA rely on stationary data.
This document provides an overview of time series analysis and its key components. It discusses that a time series is a set of data measured at successive times joined together by time order. The main components of a time series are trends, seasonal variations, cyclical variations, and irregular variations. Time series analysis is important for business forecasting, understanding past behavior, and facilitating comparison. There are two main mathematical models used - the additive model which assumes data is the sum of its components, and the multiplicative model which assumes data is the product of its components. Decomposition of a time series involves discovering, measuring, and isolating these different components.
The document discusses using a vector autoregression (VAR) model to forecast two time series - leads and binds - that interact with each other. A 5-period VAR model is found to best capture the weekly periodicity between the series. The model is shown to accurately forecast leads 1-11 days in advance, within 2% error, and binds within 5% error over a two week period, indicating the interaction between the series can be used to predict each going forward. Some conclusions drawn are that the VAR model performs well but could be improved by trying other techniques or adding external variables.
This document defines time series and its components. A time series is a set of observations recorded over successive time intervals. It has four main components: trend, seasonality, cycles, and irregular variations. Trend refers to the overall increasing or decreasing tendency over time. Seasonality refers to predictable changes that occur around the same time each year. Cycles have periods longer than a year. Irregular variations are random fluctuations. The document also discusses methods for analyzing time series components including additive, multiplicative, and mixed models.
This Presentation describes, in short, Introduction to Time Series and the overall procedure required for Time Series Modelling including general terminologies and algorithms. However the detailed Mathematics is excluded in the slides, this ppt means to give a start to understanding the Time Series Modelling before going into detailed Statistics.
Business Analytics Foundation with R tool - Part 5Beamsync
The current presentation published by Beamsync.
If you are looking for analytics training in Bangalore, consult Beamsync Training Centre.
For upcoming schedules please visit: http://beamsync.com/business-analytics-training-bangalore/
This document provides an introduction to ARIMA (AutoRegressive Integrated Moving Average) models. It discusses key assumptions of ARIMA including stationarity. ARIMA models combine autoregressive (AR) terms, differences or integrations (I), and moving averages (MA). The document outlines the Box-Jenkins approach for ARIMA modeling including identifying a model through correlograms and partial correlograms, estimating parameters, and diagnostic checking to validate the model prior to forecasting.
This document summarizes a lecture on regularization techniques for reducing overfitting in machine learning models. It introduces regularization as modifying the loss function to penalize model complexity. Specifically, it covers ridge and lasso regression. Ridge regression adds an L2 penalty on coefficient magnitudes while lasso adds an L1 penalty, encouraging sparsity. Graphs show the geometric interpretations of regularization shrinking coefficients. Steps for choosing the regularization parameter using validation sets or cross-validation are provided.
New tools from the bandit literature to improve A/B Testingrecsysfr
- The document discusses improving A/B testing using tools from multi-armed bandit literature.
- It presents an optimal algorithm called Track-and-Stop for identifying the best arm that minimizes sample complexity. This algorithm adaptively tracks optimal sampling proportions and stops testing using a Chernoff rule.
- The algorithm is shown to be asymptotically optimal, achieving the fundamental lower bound on sample complexity for best arm identification problems.
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The "Great Lakes" data set is an example of a non-seasonal, non-stationary time series that
experiences a slight upward linear trend. The series is differenced and transformed using
"Box-Cox" in order to stabilize the mean and variance, correcting for stationarity. The best
model fitted for the data was an ARIMA(4,1,0) found by observing the partial and auto
correlation functions. The fit suggested the best estimates for the coefficients via the AIC.
Verified as independent random variables, the residuals of the fitted model were tested for
normality using the McLeod-Li, Ljung-Box, and Shapiro-Wilk test. The model proved to be
an adequate representation of the data providing reasonable predictions for precipitation.
Recent developments in the field of reduced order modeling - and in particular, active subspace construction - have made it possible to efficiently approximate complex models by constructing low-order response surfaces based upon a small subspace of the original high dimensional parameter space. These methods rely upon the fact that the response tends to vary more prominently in a few dominant directions defined by linear combinations of the original inputs, allowing for a rotation of the coordinate axis and a consequent transformation of the parameters. In this talk, we discuss a gradient free active subspace algorithm that is feasible for high dimensional parameter spaces where finite-difference techniques are impractical. We illustrate an initialized gradient-free active subspace algorithm for a neutronics example implemented with SCALE6.1.
Axial compressor theory - stage-wise isentropic efficiency - 18th March 2010CangTo Cheah
This document discusses the theory of stage-wise isentropic efficiency in axial compressors. It covers determining the pitch and chord between blades, calculating the number of blades based on pitch, and formulas for static pressure rise and forces acting on cascades, including lift and drag forces. Graphs are presented for mean deflection and stagnation pressure loss as functions of incidence angle. The document aims to calculate coefficients for drag, lift, and loss based on these graphs and the presented formulas.
Algorithm Selection for Preferred Extensions EnumerationFederico Cerutti
The document discusses algorithms for enumerating preferred extensions in abstract argumentation frameworks. It compares the performance of four algorithms: AspartixM, NAD-Alg, PrefSAT, and SCC-P. It finds that algorithm selection based on graph features can accurately predict runtime, with up to 80% accuracy in classification, and improves performance over a single best solver by 2-3 times. Key discriminating features include density, number of arguments, number of strongly connected components, and features related to computing graph properties.
This document discusses autoregressive models for financial time series analysis. It introduces autoregressive (AR) and moving average (MA) processes. The autoregressive integrated moving average (ARIMA) model is presented as a way to fit time series data that accounts for correlation between observations. The document outlines the Box-Jenkins methodology for identifying and fitting an ARIMA model to time series data, including checking for stationarity, identifying orders using autocorrelation and partial autocorrelation functions, and selecting the best model. It applies this process to Shanghai Stock Exchange index data, finding that an ARIMA(48,1,0) model provided the best fit.
The paper presents an efficient algorithm for path profiling programs with arbitrary control flow graphs. It first assigns values to edges such that the sum along each path is unique. It then selects a minimal set of edges for instrumentation. Each instrumented edge updates a register to encode the current execution path. This allows precisely tracking paths with low overhead. Previously, path profiling was assumed too costly but this shows overhead is only twice edge profiling. It also describes handling cycles by breaking at backedges.
The document discusses converting a schedule validation problem into a polygraph problem to leverage existing polygraph algorithms. It defines what a polygraph is - a triple of vertices, arcs, and choices. It then explains how to construct a polygraph P(s) from a schedule s by making transactions vertices and enforcing ordering/conflict relationships with arcs and choices. Testing if a schedule is conflict-serializable amounts to checking if the corresponding polygraph P(s) has a compatible acyclic directed graph - if so, it is serializable. An example converts a sample schedule into a polygraph to check for cycles.
Dynamics of actin filaments in the contractile ringPrafull Sharma
This document is a summer internship report submitted by Prafull Kumar Sharma to Karsten Kruse at the University of Saarland in Germany. It summarizes Prafull's work analyzing the dynamics of actin filaments in the contractile ring numerically. Prafull first learned techniques for numerically solving diffusion equations. He then analyzed models of actin filament dynamics both without and with bipolar filaments, studying stability and performing simulations. Key results included verifying that stability only depends on the wavenumber k=2pi/L. Prafull gained experience with programming in MATLAB and solving nonlinear equations during the productive internship.
This document introduces several concepts in estimation theory, including Bayesian parameter estimation, non-Bayesian parameter estimation, maximum likelihood estimation, and the Cramér-Rao lower bound. It provides examples of estimating parameters for linear and nonlinear models from observed data using different cost functions and derivation of the mean square error, maximum a posteriori, and maximum likelihood estimates.
The document discusses pushdown automata (PDA). It defines a PDA as a 7-tuple that includes a set of states, input alphabet, stack alphabet, initial/start state, starting stack symbol, set of final/accepting states, and a transition function. PDAs operate on an input tape with a stack, and can accept languages that finite automata cannot, such as anbn. The document provides examples of designing PDAs for specific languages and converting between context-free grammars and PDAs.
Natural Gas Time Series Analysis
The author analyzes natural gas price data from 1996 to 2016 using R. After differencing to achieve stationarity, ARIMA models are fitted and the SARIMA(1,0,0)×(2,1,1)12 model is identified as best based on having the lowest AIC value and significant coefficients. Forecasting with this model shows the predicted values follow a similar decreasing trend as the actual later data. Diagnostic checks confirm the residuals exhibit white noise. The analysis provides useful prediction of natural gas prices.
Time Series Analysis - 2 | Time Series in R | ARIMA Model Forecasting | Data ...Simplilearn
This Time Series Analysis (Part-2) in R presentation will help you understand what is ARIMA model, what is correlation & auto-correlation and you will alose see a use case implementation in which we forecast sales of air-tickets using ARIMA and at the end, we will also how to validate a model using Ljung-Box text. A time series is a sequence of data being recorded at specific time intervals. The past values are analyzed to forecast a future which is time-dependent. Compared to other forecast algorithms, with time series we deal with a single variable which is dependent on time. So, lets deep dive into this presentation and understand what is time series and how to implement time series using R.
Below topics are explained in this " Time Series in R presentation " -
1. Introduction to ARIMA model
2. Auto-correlation & partial auto-correlation
3. Use case - Forecast the sales of air-tickets using ARIMA
4. Model validating using Ljung-Box test
Become an expert in data analytics using the R programming language in this data science certification training course. You’ll master data exploration, data visualization, predictive analytics and descriptive analytics techniques with the R language. With this data science course, you’ll get hands-on practice on R CloudLab by implementing various real-life, industry-based projects in the domains of healthcare, retail, insurance, finance, airlines, music industry, and unemployment.
Why learn Data Science with R?
1. This course forms an ideal package for aspiring data analysts aspiring to build a successful career in analytics/data science. By the end of this training, participants will acquire a 360-degree overview of business analytics and R by mastering concepts like data exploration, data visualization, predictive analytics, etc
2. According to marketsandmarkets.com, the advanced analytics market will be worth $29.53 Billion by 2019
3. Wired.com points to a report by Glassdoor that the average salary of a data scientist is $118,709
4. Randstad reports that pay hikes in the analytics industry are 50% higher than IT
The Data Science with R is recommended for:
1. IT professionals looking for a career switch into data science and analytics
2. Software developers looking for a career switch into data science and analytics
3. Professionals working in data and business analytics
4. Graduates looking to build a career in analytics and data science
5. Anyone with a genuine interest in the data science field
6. Experienced professionals who would like to harness data science in their fields
Learn more at: https://www.simplilearn.com/
This document discusses using R for customer segmentation. It outlines using purchase behavior data and survey response data to create actionable customer segments. The goal is to improve customer lifetime value by sending targeted messages. The document demonstrates building RFM (recency, frequency, monetary) metrics and segments from purchase data, including data aggregation, metric calculation, and segment assignment. Visualizations of customers in the RFM space are shown to understand segment distributions.
The document discusses direct marketing analytics using R. It introduces direct marketing, the types of data and questions it involves, and common metrics used. It outlines developing R modules for direct marketing including classes to handle data structures, functions for metrics, testing, segmentation, and modeling. The goal is to finalize class structures and methods to support independent and triggered campaign analysis.
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What goes with what (Market Basket Analysis)Kumar P
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Amazon Aurora 클러스터를 초당 수백만 건의 쓰기 트랜잭션으로 확장하고 페타바이트 규모의 데이터를 관리할 수 있으며, 사용자 지정 애플리케이션 로직을 생성하거나 여러 데이터베이스를 관리할 필요 없이 Aurora에서 관계형 데이터베이스 워크로드를 단일 Aurora 라이터 인스턴스의 한도 이상으로 확장할 수 있는 Amazon Aurora Limitless Database를 소개합니다.
The Night Patrol Car Robot is an advanced robotic system engineered for night...yeshwanth27naidu
night patrol car robot
The Night Patrol Car Robot is an advanced robotic system engineered for night time surveillance and security tasks. This robot is equipped with cutting-edge night-vision cameras, infrared sensors, and efficient communication modules, enabling it to operate effectively in low-light environments.
The future of policing may well be shaped by these technological advancements. Sensor Improvement Enhance the capabilities of sensors for better night vision, thermal imaging, and environmental awareness. Improve algorithms for real-time decision-making, threat assessment, and predictive policing
Amazon DocumentDB(MongoDB와 호환됨)는 빠르고 안정적이며 완전 관리형 데이터베이스 서비스입니다. Amazon DocumentDB를 사용하면 클라우드에서 MongoDB 호환 데이터베이스를 쉽게 설치, 운영 및 규모를 조정할 수 있습니다. Amazon DocumentDB를 사용하면 MongoDB에서 사용하는 것과 동일한 애플리케이션 코드를 실행하고 동일한 드라이버와 도구를 사용하는 것을 실습합니다.
4. A non seasonal ARIMA model is classified as an "ARIMA(p,d,q)" model, where:
•p is the number of autoregressive terms,
•d is the number of non seasonal differences needed for stationarity, and
•q is the number of lagged forecast errors in the prediction equation.
•Stationary Series: A stationary series has no trend, its variations around its mean have a constant
amplitude. A non stationary series is made stationary by differencing
ARIMA MODEL
5. To identify an ARIMA(p,d,q) we use extensively
the autocorrelation function
{ρh : -∞ < h < ∞}
and
the partial autocorrelation function,
{Φkk: 0 ≤ k < ∞}.
6. The definition of the sample covariance function
{Cx(h) : -∞ < h < ∞}
and the sample autocorrelation function
{rh: -∞ < h < ∞}
are given below:
( ) ( )( )∑
−
=
+ −−=
hT
t
httx xxxx
T
hC
1
1
( )
( )0
and
x
x
h
C
hC
r = The divisor is T, some
statisticians use T – h
(If T is large, both give
approximately the
same results.)
7. It can be shown that:
( ) ∑
∞
−∞=
++ ≈
t
kttkhh
T
rrCov ρρ
1
,
Thus
( )
+≈≈ ∑∑ =
∞
−∞=
q
t
t
t
th r
TT
rVar
1
22
21
11
ρ
Assuming ρk = 0 for k > q
∑=
+=
q
t
tr r
T
s h
1
2
21
1
Let
8. The sample partial autocorrelation function is defined
by:
1
1
1
1
1
ˆ
21
21
11
21
21
11
−−
−
−
−−
=Φ
kk
k
k
kkk
kk
rr
rr
rr
rrr
rr
rr
9. It can be shown that:
( ) T
Var kk
1ˆ ≈Φ
T
s
kk
1
Let ˆ =Φ
10. Identification of an ARIMA process
Determining the values of p,d,q
Steps for ARIMA MODEL
• Visualization
• ACF and PCF plot
• Seasonal variation modelling
• Stationary check
• Identifying p,d,q for non seasonal series
• Model development
• Validating accuracy
• Selecting best model
11. • Recall that if a process is stationary one of the
roots of the autoregressive operator is equal to
one.
• This will cause the limiting value of the
autocorrelation function to be non-zero.
• Thus a nonstationary process is identified by
an autocorrelation function that does not tail
away to zero quickly or cut-off after a finite
number of steps.
12. To determine the value of d
Note: the autocorrelation function for a stationary ARMA
time series satisfies the following difference equation
1 1 2 2h h h p h pρ β ρ β ρ β ρ− − −= + + +
The solution to this equation has general form
1 2
1 2
1 1 1
h ph h h
p
c c c
r r r
ρ = + + +
where r1, r2, r1, … rp, are the roots of the polynomial
( ) 2
1 21 p
px x x xβ β β β= − − − −
13. For a stationary ARMA time series
Therefore
1 2
1 2
1 1 1
0 ash ph h h
p
c c c h
r r r
ρ = + + + → → ∞
The roots r1, r2, r1, … rp, have absolute value greater than 1.
If the ARMA time series is non-stationary
some of the roots r1, r2, r1, … rp, have absolute value
equal to 1, and
1 2
1 2
1 1 1
0 ash ph h h
p
c c c a h
r r r
ρ = + + + → ≠ → ∞
15. • If the process is non-stationary then first
differences of the series are computed to
determine if that operation results in a
stationary series.
• The process is continued until a stationary
time series is found.
• This then determines the value of d.
17. To determine the value of p and q we use the
graphical properties of the autocorrelation
function and the partial autocorrelation function.
Again recall the following:
Auto-correlation
function
Partial
Autocorrelation
function
Cuts off
Cuts off
Infinite. Tails off.
Damped Exponentials
and/or Cosine waves
Infinite. Tails off.
Infinite. Tails off.Infinite. Tails off.
Dominated by damped
Exponentials & Cosine
waves.
Dominated by damped
Exponentials & Cosine waves
Damped Exponentials
and/or Cosine waves
after q-p.
after p-q.
Process MA(q) AR(p) ARMA(p,q)
Properties of the ACF and PACF of MA, AR and ARMA Series
18. Summary: To determine p and q.
Use the following table.
MA(q) AR(p) ARMA(p,q)
ACF Cuts after q Tails off Tails off
PACF Tails off Cuts after p Tails off
Note: Usually p + q ≤ 4. There is no harm in over
identifying the time series. (allowing more parameters in
the model than necessary. We can always test to
determine if the extra parameters are zero.)
19. Examples Using R
IMPORTANT PACKAGES:forecast, tseries, TTR, fpp
Reference link:
https://www.otexts.org/fpp
20. DATA
Time Series:
Start = 1
End = 72
Frequency = 1
USAccDeaths:
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
1973 9007 8106 8928 9137 10017 10826 11317 10744 9713 9938 9161 8927
1974 7750 6981 8038 8422 8714 9512 10120 9823 8743 9129 8710 8680
1975 8162 7306 8124 7870 9387 9556 10093 9620 8285 8466 8160 8034
1976 7717 7461 7767 7925 8623 8945 10078 9179 8037 8488 7874 8647
1977 7792 6957 7726 8106 8890 9299 10625 9302 8314 8850 8265 8796
1978 7836 6892 7791 8192 9115 9434 10484 9827 9110 9070 8633 9240
24. Exponential Smoothing modelling using HoltWinters methods
R Code:
USAccforecasts <-HoltWinters(USAccDeaths$USAccDeaths, beta=FALSE,
gamma=FALSE)
print(USAccforecasts)
plot(USAccforecasts)
Holt-Winters exponential smoothing without trend and without seasonal component.
Call:
HoltWinters(x = USAccDeaths$USAccDeaths, beta = FALSE, gamma = FALSE)
Smoothing parameters:
alpha: 0.9999339
beta : FALSE
gamma: FALSE
Coefficients:
[,1]
a 9239.96
25. ACF and PACF plot
s) Augmented Dickey-Fuller Test data: USAccDeaths Dickey-Fuller = -3.8221, Lag order = 4, p-value = 0.02268 alternative h
hs) Augmented Dickey-Fuller Test data: USAccDeaths Dickey-Fuller = -3.8221, Lag order = 4, p-value = 0.02268 alternative h
Test of Stationarity: Augumented Dicky fuller test
(adf test)
adf.test(USAccDeaths)-R code
R output:
Augmented Dickey-Fuller Test
data: USAccDeaths
Dickey-Fuller = -3.8221, Lag order = 4, p-value =
0.02268
alternative hypothesis: stationary
* Since p-value=0.02268 < 0.05 , hence series
stationary
26. ACF and PACF plot
After taking first difference to remove seasonality
32. Estimation of parameters of an MA(q) series
The theoretical autocorrelation function in terms the
parameters of an MA(q) process is given by.
>
≤≤
++++
+++
=
−+
qh
qh
q
qhqhh
h
0
1
1 22
2
2
1
11
ααα
ααααα
ρ
To estimate α1, α2, … , αq we solve the system of
equations:
qhr
q
qhqhh
h ≤≤
++++
+++
=
−+
1
ˆˆˆ1
ˆˆˆˆˆ
22
2
2
1
11
ααα
ααααα
33. This set of equations is non-linear and generally very
difficult to solve
For q = 1 the equation becomes:
Thus
2
1
1
1
ˆ1
ˆ
α
α
+
=r
( ) 0ˆˆ1 11
2
1 =−+ αα r
or 0ˆˆ 11
2
11 =+− rr αα
This equation has the two solutions
1
4
1
2
1
ˆ 2
11
1 −±=
rr
α
One solution will result in the MA(1) time series being invertible
35. Estimation of parameters of an
ARMA(p,q) series
We use a similar technique.
Namely: Obtain an expression for ρh in terms β1,
β2 , ... , βp ; α1, α1, ... , αq of and set up q + p
equations for the estimates of β1, β2 , ... , βp ; α1,
α2, ... , αq by replacing ρh by rh.
36. Estimation of parameters of an ARMA(p,q) series
( )( )
112
11
2
1
1111
1
21
1
βρρ
βαα
βαβα
ρ
=
++
++
=
Example: The ARMA(1,1) process
The expression for ρ1 and ρ2 in terms of β1 and α1
are:
Further
( ) ( )0
21
1
11
2
1
2
12
xtuVar σ
βαα
β
σ
++
−
==
38. ( ) ( )( )111111
2
11
1
2
1
ˆˆˆˆ1ˆˆ2ˆ1
andˆ
βαβαβαα
β
++=++
=
r
r
r
Hence
or
+
+=
++
1
2
1
1
2
1
1
2
1
2
11
ˆˆ1ˆ2ˆ1
r
r
r
r
r
r
r αααα
This is a quadratic equation which can be solved
0ˆ12ˆ
1
2
112
1
2
2
2
2
1
1
2
1 =
−+
−−+
−
r
r
r
r
r
r
r
r
r αα
39. Example: For ARIMA
the time series was identified as either an
ARIMA(1,0,1) time series or an ARIMA(0,1,1)
series.
If we use the first identification then series xt is an
ARMA(1,1) series.
40. Identifying the series xt is an ARMA(1,1) series.
The autocorrelation at lag 1 is r1 = 0.570 and the
autocorrelation at lag 2 is r2 = 0.495 .
Thus the estimate of β1 is 0.495/0.570 = 0.87.
Also the quadratic equation
becomes
0ˆ12ˆ
1
2
112
1
2
2
2
2
1
1
2
1 =
−+
−−+
−
r
r
r
r
r
r
r
r
r αα
02984.0ˆ7642.0ˆ2984.0 1
2
1 =++ αα
which has the two solutions -0.48 and -2.08. Again we select
as our estimate of α1 to be the solution -0.48, resulting in an
invertible estimated series.
41. Since δ = µ(1 - β1) the estimate of δ can be computed as
follows:
Thus the identified model in this case is
xt = 0.87 xt-1 + ut - 0.48 ut-1 + 2.25
( ) 25.2)87.01(062.17ˆ1ˆ
1 =−=−= βδ x
42. If we use the second identification then series
∆xt = xt – xt-1 is an MA(1) series.
Thus the estimate of α1 is:
1
4
1
2
1
ˆ 2
11
1 −±=
rr
α
The value of r1 = -0.413.
Thus the estimate of α1 is:
( ) ( )
−
−
=−
−
±
−
=
53.0
89.1
1
413.04
1
413.02
1
ˆ 21α
The estimate of α1 = -0.53, corresponds to an invertible
time series. This is the solution that we will choose
43. The estimate of the parameter µ is the sample mean.
Thus the identified model in this case is:
∆xt = ut - 0.53 ut-1 + 0.002 or
xt = xt-1 + ut - 0.53 ut-1 + 0.002
This compares with the other identification:
xt = 0.87 xt-1 + ut - 0.48 ut-1 + 2.25
(An ARIMA(1,0,1) model)
(An ARIMA(0,1,1) model)
45. ( )
pp ρβρβ
σ
σ
−−−
=
11
2
1
0
111 1 −++= pp ρββρ
2112 −++= pp ρβρβρ
and
111 ppp βρβρ ++= −
The regression coefficients β1, β2, …., βp and the auto correlation function ρh satisfy the Yule-Walker equations:
46. ( ) ( )ppx rrC ββσ ˆˆ10ˆ 11
2
−−−×=
111
ˆ1ˆ
−++= pprr ββ
2112
ˆˆ
−++= pprrr ββ
and
1ˆˆ
11 ppp rr ββ ++= −
The Yule-Walker equations can be used to estimate the
regression coefficients β1, β2, …., βp using the sample auto
correlation function rh by replacing ρh with rh.
47. Example
Considering the data in example 1 (Sunspot Data) the time series
was identified as an AR(2) time series .
The autocorrelation at lag 1 is r1 = 0.807 and the autocorrelation
at lag 2 is r2 = 0.429 .
The equations for the estimators of the parameters of this series
are
4290ˆ0001ˆ8070
8070ˆ8070ˆ0001
21
21
...
...
=+
=+
ββ
ββ
which has solution
6370ˆ
321.1ˆ
2
1
.−=
=
β
β
Since δ = µ( 1 -β1 - β2) then it can be estimated as follows:
48. Thus the identified model in this case is
xt = 1.321 xt-1 -0.637 xt-2 + ut +14.9
( ) ( ) 9.14637.0321.11590.46ˆˆ1ˆ
21 =+−=−−= x ββδ
50. The method of Maximum Likelihood
Estimation selects as estimators of a set of
parameters θ1,θ2, ... , θk , the values that
maximize
L(θ1,θ2, ... , θk) = f(x1,x2, ... , xN;θ1,θ2, ... , θk)
where f(x1,x2, ... , xN;θ1,θ2, ... , θk) is the joint
density function of the observations x1,x2, ... , xN.
L(θ1,θ2, ... , θk) is called the Likelihood function.
51. It is important to note that:
finding the values -θ1,θ2, ... , θk- to maximize
L(θ1,θ2, ... , θk) is equivalent to finding the
values to maximize l(θ1,θ2, ... , θk) = ln L(θ1,θ2,
... , θk).
l(θ1,θ2, ... , θk) is called the log-Likelihood
function.
52. Again let {ut : t ∈T} be identically distributed
and uncorrelated with mean zero. In addition
assume that each is normally distributed .
Consider the time series {xt : t ∈T} defined by
the equation:
(*) xt = β1xt-1 + β2xt-2 +... +βpxt-p + δ + ut
+α1ut-1 + α2ut-2 +... +αqut-q
53. Assume that x1, x2, ...,xN are observations on the
time series up to time t = N.
To estimate the p + q + 2 parameters β1, β2, ...
,βp ; α1, α2, ... ,αq ; δ , σ2
by the method of
Maximum Likelihood estimation we need to find
the joint density function of x1, x2, ...,xN
f(x1, x2, ..., xN |β1, β2, ... ,βp ; α1, α2, ... ,αq , δ, σ2
)
= f(x| β, α, δ ,σ2
).
54. We know that u1, u2, ...,uN are independent
normal with mean zero and variance σ2
.
Thus the joint density function of u1, u2, ...,uN is
g(u1, u2, ...,uN ; σ2
) = g(u ; σ2
) is given by.
( ) ( )
−
== ∑=
N
t
t
n
N uguug
1
2
2
22
1
2
1
exp
2
1
;;,
σσπ
σσ u
55. It is difficult to determine the exact density
function of x1,x2, ... , xN from this information
however if we assume that p starting values on
the x-process x* = (x1-p,x2-p, ... , xo) and q starting
values on the u-process u* = (u1-q,u2-q, ... , uo) have
been observed then the conditional distribution
of x = (x1,x2, ... , xN) given x* = (x1-p,x2-p, ... , xo) and
u* = (u1-q,u2-q, ... , uo) can easily be determined.
57. can be solved for:
u1 = u1 (x, x*, u*; β, α, δ)
u2 = u2 (x, x*, u*; β, α, δ)
...
uN = uN (x, x*, u*; β, α, δ)
(The jacobian of the transformation is 1)
58. Then the joint density of x given x* and u* is
given by:
( )2
,,,*,*, σδαβuxxf
( )
−
= ∑=
N
t
t
n
u
1
2
2
,,*,*,
2
1
exp
2
1
δ
σσπ
αβux
( )
−
= δ
σσπ
,,*
2
1
exp
2
1
2
αβS
n
( ) ( )∑=
=
N
t
tuS
1
2
,,*,*,,,*where δδ αβuxαβ
59. Let:
( )2
**,
,,, σδαβuxx
L
( )
−
= ∑=
N
t
t
n
u
1
2
2
,,*,*,
2
1
exp
2
1
δ
σσπ
αβux
( )
−
= δ
σσπ
,,*
2
1
exp
2
1
2
αβS
n
( ) ( )∑=
=
N
t
tuS
1
2
,,*,*,,,*again δδ αβuxαβ
= “conditional likelihood function”
61. ( ) ( )2
**,
2
**,
,,,and,,, σδσδ αβαβ uxxuxx
Ll
( ) ( )∑=
=
N
t
tuS
1
2
,,*,*,,,* δδ αβuxαβ
The values that maximize
are the values
that minimize
δˆ,ˆ,ˆ αβ
( ) ( )δδσ ˆ,ˆ,ˆ*
1ˆ,ˆ,ˆ*,*,
1
ˆ
1
22
αβαβux S
n
u
n
N
t
t == ∑=
with
62. ( ) ( )∑=
=
N
t
tuS
1
2
,,*,*,,,* δδ αβuxαβ
Comment:
Requires a iterative numerical minimization
procedure to find:
The minimization of:
δˆ,ˆ,ˆ αβ
• Steepest descent
• Simulated annealing
• etc
63. ( ) ( )∑=
=
N
t
tuS
1
2
,,*,*,,,* δδ αβuxαβ
Comment:
for specific values of
The computation of:
can be achieved by using the forecast equations
δ,,αβ
( )1ˆ 1−−= ttt xxu
64. ( ) ( )∑=
=
N
t
tuS
1
2
,,*,*,,,* δδ αβuxαβ
Comment:
assumes we know the value of starting values of the
time series {xt| t T} and {ut| t T}
The minimization of :
Namely x* and u*.
66. Backcasting:
If the time series {xt|t T} satisfies the equation:
2211 qtqttt uuuu −−− +++++ ααα
2211 δβββ ++++= −−− ptpttt xxxx
It can also be shown to satisfy the equation:
2211 qtqttt uuuu +++ +++++ ααα
2211 δβββ ++++= +++ ptpttt xxxx
Both equations result in a time series with the same
mean, variance and autocorrelation function:
In the same way that the first equation can be used to
forecast into the future the second equation can be used
to backcast into the past:
67. *ofcomponentsfor the0
*ofcomponentsfor the
u
xx
Approaches to handling starting values of the series {xt|t T} and {ut|t T}
1. Initially start with the values:
2. Estimate the parameters of the model using
Maximum Likelihood estimation and the
conditional Likelihood function.
3. Use the estimated parameters to backcast the
components of x*. The backcasted components of
u* will still be zero.
68. 4. Repeat steps 2 and 3 until the estimates stablize.
This algorithm is an application of the E-M algorithm
This general algorithm is frequently used when there
are missing values.
The E stands for Expectation (using a model to estimate
the missing values)
The M stands for Maximum Likelihood Estimation, the
process used to estimate the parameters of the model.
69. ARIMA+X=ARIMAX
ARIMA with environmental variable is very important in the
case when external variable start impacting the series
Ex. Flight delay prediction depends not only historical time
series data but external variables like weather condition
(temperature , pressure, humidity, visibility, arrival of other
flights, weighting time etc.)
70. ARIMA+X=ARIMAX
An ARMAX model simply adds in the covariate on the right hand side:
yt=βxt+ϕ1yt−1+⋯+ϕpyt−p–θ1zt−1–…–θqzt−q+zt
Covariate xt
R function:
riod = NA), xreg = NULL, include.mean = TRUE, transform.pars = TRUE, fixed = NULL, init = NULL, method = c("CSS-ML", "ML", "CSS"), n.cond, SSinit = c("Gardner1980", "Rossign
arima(x, order = c(0L, 0L, 0L),seasonal = list(order = c(0L, 0L, 0L), period = NA),
xreg = xt)