This document provides an introduction and outline for a machine learning summer school session on machine learning strategies for time series prediction. It introduces the speaker, Gianluca Bontempi, and his background in machine learning. It then discusses what attendees should know and expect to learn, including foundations of statistical machine learning and strategies for forecasting. The outline presents topics that will be covered, including notions of time series, machine learning for prediction, local learning approaches, forecasting techniques, applications, and future directions.
Local modeling in regression and time series predictionGianluca Bontempi
The document discusses global modeling versus local modeling approaches for regression and time series prediction problems. Global modeling fits a single analytical function to all input data, while local modeling performs separate fits to subsets of nearby data points. The document outlines the local modeling approach using lazy learning, which stores all training data and performs local fits when making predictions for new query points. It then applies lazy learning techniques to problems in regression, time series prediction, and feature selection.
Computational Intelligence for Time Series PredictionGianluca Bontempi
This document provides an overview of computational intelligence methods for time series prediction. It begins with introductions to time series analysis and machine learning approaches for prediction. Specific models discussed include autoregressive (AR), moving average (MA), and autoregressive moving average (ARMA) processes. Parameter estimation techniques for AR models are also covered. The document outlines applications in areas like forecasting, wireless sensors, and biomedicine and concludes with perspectives on future directions.
THM1: Formalizing a problem as a prediction problem is often the most important contribution of a data scientist.
THM2: A predictor is an estimator, i.e. an algorithm which takes data and returns a prediction. Reality is stochastic, so data and predictions are stochastic.
THM3: Learning is challenging since data must be used both to create prediction models and to assess them. Bias and variance must be balanced to achieve good generalization.
Application of discrete mathematics in ITShahidAbbas52
This document discusses discrete mathematics and its applications. It begins with defining discrete mathematics and providing examples of its different fields like graphs, networks, and logic. It then discusses various real-world applications of discrete mathematics in areas like computers, encryption, Google Maps, and scheduling. Discrete mathematical concepts like graphs, algorithms, and logic are widely used in fields like computer science, engineering, operations research, and social sciences.
In the past few years, India has witnessed exponential growth in the sector of Data Science. With the advent of digital transformation in businesses, the demand for data scientists is boosting every day with a ton of job opportunities machine learning course in mumbai’machine learning course in mumbais lying in their path. Boston Institute of Analytics provides data science courses in Mumbai. They train students under experienced industry professionals and make them industry ready. To know more about their courses check out their website https://www.biaclassroom.com/courses.
A Dynamic Factor Model: Inference and Empirical Application. Ioannis Vrontos SYRTO Project
The document describes a dynamic factor model to analyze how financial risks are interconnected within the Eurozone. It uses the model to examine risk dynamics using sovereign CDS and equity returns from 2007-2009 covering the US financial crisis and pre-sovereign crisis in Europe. The model relates asset returns to latent sector factors, macro factors, and covariates. Bayesian inference is applied using MCMC to estimate the time-varying parameters and latent factors.
The document discusses the history and importance of numerical algorithms. It began with early papers in the 1940s and projects like ENIAC for calculating trajectories. Over time, important people and projects advanced areas like linear algebra, differential equations, and approximation methods. Today, numerical algorithms are widely used in applications such as digital imaging, engineering, finance, and more. Understanding numerical methods is crucial for teaching mathematics with technology, as it ensures students have proper conceptual knowledge to analyze problems.
Local modeling in regression and time series predictionGianluca Bontempi
The document discusses global modeling versus local modeling approaches for regression and time series prediction problems. Global modeling fits a single analytical function to all input data, while local modeling performs separate fits to subsets of nearby data points. The document outlines the local modeling approach using lazy learning, which stores all training data and performs local fits when making predictions for new query points. It then applies lazy learning techniques to problems in regression, time series prediction, and feature selection.
Computational Intelligence for Time Series PredictionGianluca Bontempi
This document provides an overview of computational intelligence methods for time series prediction. It begins with introductions to time series analysis and machine learning approaches for prediction. Specific models discussed include autoregressive (AR), moving average (MA), and autoregressive moving average (ARMA) processes. Parameter estimation techniques for AR models are also covered. The document outlines applications in areas like forecasting, wireless sensors, and biomedicine and concludes with perspectives on future directions.
THM1: Formalizing a problem as a prediction problem is often the most important contribution of a data scientist.
THM2: A predictor is an estimator, i.e. an algorithm which takes data and returns a prediction. Reality is stochastic, so data and predictions are stochastic.
THM3: Learning is challenging since data must be used both to create prediction models and to assess them. Bias and variance must be balanced to achieve good generalization.
Application of discrete mathematics in ITShahidAbbas52
This document discusses discrete mathematics and its applications. It begins with defining discrete mathematics and providing examples of its different fields like graphs, networks, and logic. It then discusses various real-world applications of discrete mathematics in areas like computers, encryption, Google Maps, and scheduling. Discrete mathematical concepts like graphs, algorithms, and logic are widely used in fields like computer science, engineering, operations research, and social sciences.
In the past few years, India has witnessed exponential growth in the sector of Data Science. With the advent of digital transformation in businesses, the demand for data scientists is boosting every day with a ton of job opportunities machine learning course in mumbai’machine learning course in mumbais lying in their path. Boston Institute of Analytics provides data science courses in Mumbai. They train students under experienced industry professionals and make them industry ready. To know more about their courses check out their website https://www.biaclassroom.com/courses.
A Dynamic Factor Model: Inference and Empirical Application. Ioannis Vrontos SYRTO Project
The document describes a dynamic factor model to analyze how financial risks are interconnected within the Eurozone. It uses the model to examine risk dynamics using sovereign CDS and equity returns from 2007-2009 covering the US financial crisis and pre-sovereign crisis in Europe. The model relates asset returns to latent sector factors, macro factors, and covariates. Bayesian inference is applied using MCMC to estimate the time-varying parameters and latent factors.
The document discusses the history and importance of numerical algorithms. It began with early papers in the 1940s and projects like ENIAC for calculating trajectories. Over time, important people and projects advanced areas like linear algebra, differential equations, and approximation methods. Today, numerical algorithms are widely used in applications such as digital imaging, engineering, finance, and more. Understanding numerical methods is crucial for teaching mathematics with technology, as it ensures students have proper conceptual knowledge to analyze problems.
Calculus is used extensively in software engineering and related fields for several purposes:
1) In computer graphics, calculus is used to simulate light bouncing and rendering objects smoothly as well as powering physics engines.
2) In programming language design and semantics, calculus provides a functional predicate calculus to support formal reasoning about programs.
3) Calculus allows the creation of mathematical models to arrive at optimal solutions and is used in many areas of physics that relate to software such as motion, heat, and dynamics.
4) Fields like signal analysis, scientific computing, financial modeling, and robotics also employ calculus for tasks like examining functions, simulating systems, and modeling optimization.
The document discusses modelling and evaluation in machine learning. It defines what models are and how they are selected and trained for predictive and descriptive tasks. Specifically, it covers:
1) Models represent raw data in meaningful patterns and are selected based on the problem and data type, like regression for continuous numeric prediction.
2) Models are trained by assigning parameters to optimize an objective function and evaluate quality. Cross-validation is used to evaluate models.
3) Predictive models predict target values like classification to categorize data or regression for continuous targets. Descriptive models find patterns without targets for tasks like clustering.
4) Model performance can be affected by underfitting if too simple or overfitting if too complex,
Entropy and systemic risk measures
M. Billio, R. Casarin, M. Costola, A. Pasqualini
Ca’ Foscari Venice University
Final SYRTO Conference - Université Paris1 Panthéon-Sorbonne
February 19, 2016
The document discusses a simulation model for overcoming gender segregation in German high schools. It begins with an introduction to structuralist metatheory and non-statement views. It then presents a case study of using simulation to model the process of increasing female teachers in German gymnasia from 1950-1990. The simulation model was validated by comparing its output to empirical data and estimating the values of theoretical parameters that produced the best fit. A structuralist reconstruction of the model, its elements, and intended applications was presented as a way to help with verification, replication, validation, and measuring unobservable terms.
30.03.2017 Data Science Meetup - USER JOURNEY ANALYSIS, BETWEEN BUDGET ALLOCA...Zalando adtech lab
This document discusses user journey analysis for online marketing. It presents several approaches for analyzing user journey data, including sliding time windows, hand-crafted features, higher-order Markov models, and time series analysis. The objective is to inform micro-level decisions like digital ad bidding. Features are generated from online user behavior data within time windows. Models account for offline channels like TV and model awareness decay. Bayesian statistics are used to estimate models, which are optimized based on costs and benefits.
Machine Learning Strategies for Time Series PredictionGianluca Bontempi
This document introduces machine learning strategies for time series prediction. It begins with an introduction to the speaker and his background and research interests. It then provides an outline of the topics to be covered, including notions of time series, machine learning approaches for prediction, local learning methods, forecasting techniques, and applications and future directions. The document discusses what the audience should know coming into the course and what they will learn.
Machine Learning for Forecasting: From Data to DeploymentAnant Agarwal
Forecasting is everywhere. This talk covers:
• Fundamental concepts of time series
• Data preprocessing (imputation and outlier analysis)
• Feature engineering and EDA for time series
• Statistical and machine learning algorithms
• Model evaluation through backtesting
• Model explanation using SHAP
• Model monitoring and deployment considerations
This document provides an overview of techniques for temporal data mining. It discusses how temporal sequences arise in domains like engineering, science, finance, and healthcare. It covers approaches for representing temporal sequences, including keeping data in its original form, piecewise approximations, transformations to other domains, discretization, and generative models. It also discusses measuring similarity between sequences and techniques for classification and relation finding in temporal data, including frequent pattern detection and prediction. The goal is to classify and organize temporal data mining techniques to help practitioners address problems involving temporal data.
Time Series Weather Forecasting Techniques: Literature SurveyIRJET Journal
This document summarizes various time series forecasting techniques discussed in literature, including ARIMA, Prophet, and LSTM models. It reviews their applications in weather forecasting, analyzing COVID-19 data, real estate prices, bitcoin values, and more. The key techniques are compared based on their forecasting accuracy on different datasets. ARIMA is generally good at capturing trends but requires stationary data, while Prophet and LSTM can handle non-stationary data and seasonal effects better. Prophet achieved 91% accuracy on a COVID dataset, outperforming ARIMA. LSTM achieved 76% accuracy for rainfall forecasting. The document concludes different approaches are still being developed to address the unique challenges of weather data forecasting.
This document provides an overview of machine learning concepts including supervised learning, unsupervised learning, and reinforcement learning. It discusses common machine learning applications and challenges. Key topics covered include linear regression, classification, clustering, neural networks, bias-variance tradeoff, and model selection. Evaluation techniques like training error, validation error, and test error are also summarized.
The document describes a research team that studies emerging methodologies in economics and finance, including complex networks and machine learning. The team is based at Democritus University of Thrace in Greece and led by Drs. Periklis Gogas and Theophilos Papadimitriou. Their research focuses on using techniques like minimum dominating sets and support vector machines to analyze economic and financial networks and time series data. They have found these emerging methodologies can more accurately represent network structures and forecast macroeconomic and financial variables compared to traditional econometric methods.
Introduction AI ML& Mathematicals of ML.pdfGandhiMathy6
Machine learning uses probability theory to deal with uncertainty that arises from noisy data, limited data sets, and ambiguity. Probability theory provides a framework to quantify and manipulate uncertainty. It allows optimal predictions given available information, even if that information is incomplete. Key concepts in probability theory for machine learning include defining sample spaces and events, calculating probabilities, working with joint, conditional, and independent probabilities, and using Bayes' rule. These concepts help machine learning algorithms make inferences from data.
This document provides an overview of machine learning concepts and techniques including linear regression, logistic regression, unsupervised learning, and k-means clustering. It discusses how machine learning involves using data to train models that can then be used to make predictions on new data. Key machine learning types covered are supervised learning (regression, classification), unsupervised learning (clustering), and reinforcement learning. Example machine learning applications are also mentioned such as spam filtering, recommender systems, and autonomous vehicles.
This document discusses mathematical modeling and computer simulation. It begins by defining key terms such as mathematical model, computer modeling, and computer simulation. It then explains the process of creating a mathematical model to represent a real-world phenomenon and using that model to perform computer simulations. The success of a mathematical model depends on how accurately it predicts and explains the phenomenon. Examples of modeling a falling rock are provided to illustrate the modeling process.
FellowBuddy.com is an innovative platform that brings students together to share notes, exam papers, study guides, project reports and presentation for upcoming exams.
We connect Students who have an understanding of course material with Students who need help.
Benefits:-
# Students can catch up on notes they missed because of an absence.
# Underachievers can find peer developed notes that break down lecture and study material in a way that they can understand
# Students can earn better grades, save time and study effectively
Our Vision & Mission – Simplifying Students Life
Our Belief – “The great breakthrough in your life comes when you realize it, that you can learn anything you need to learn; to accomplish any goal that you have set for yourself. This means there are no limits on what you can be, have or do.”
Like Us - https://www.facebook.com/FellowBuddycom
The document discusses the matrix profile, a data structure that annotates time series data. It has many desirable properties, including being exact, simple, space efficient, and able to handle problems like motif discovery, anomaly detection, and segmentation in a trivial manner. The matrix profile records the distance of subsequences to their nearest neighbors, allowing patterns in time series data to be easily interpreted from the profile values. Both real and synthetic examples are used to demonstrate how the matrix profile visually represents motifs, anomalies, and other patterns in time series data.
This document provides an overview of machine learning using Python. It introduces machine learning applications and key Python concepts for machine learning like data types, variables, strings, dates, conditional statements, loops, and common machine learning libraries like NumPy, Matplotlib, and Pandas. It also covers important machine learning topics like statistics, probability, algorithms like linear regression, logistic regression, KNN, Naive Bayes, and clustering. It distinguishes between supervised and unsupervised learning, and highlights algorithm types like regression, classification, decision trees, and dimensionality reduction techniques. Finally, it provides examples of potential machine learning projects.
Stock Market Prediction using Long Short-Term MemoryIRJET Journal
This document discusses using a Long Short-Term Memory (LSTM) model to predict stock market prices. It begins by introducing the problem of predicting stock markets and how machine learning techniques like LSTM can help. It then discusses collecting stock price data and designing an LSTM model in Python using Keras and other libraries. The model is trained on historical stock price data to identify patterns and predict future prices. The document suggests LSTM models are well-suited for this due to their ability to use past data in predictions. It evaluates the model's predictions against actual prices to determine accuracy.
This document introduces the plm package for panel data econometrics in R. It discusses panel data models and estimation approaches, describes the software approach and functions in plm, and compares plm to other R packages for longitudinal data analysis. Key features of plm include functions for estimating linear panel models with fixed effects, random effects, and GMM, as well as data management and diagnostic testing capabilities for panel data. The package aims to make panel data analysis intuitive and straightforward for econometricians in R.
Significant Role of Statistics in Computational SciencesEditor IJCATR
This paper is focused on the issues related to optimizing statistical approaches in the emerging fields of Computer Science
and Information Technology. More emphasis has been given on the role of statistical techniques in modern data mining. Statistics is
the science of learning from data and of measuring, controlling, and communicating uncertainty. Statistical approaches can play a vital
role for providing significance contribution in the field of software engineering, neural network, data mining, bioinformatics and other
allied fields. Statistical techniques not only helps make scientific models but it quantifies the reliability, reproducibility and general
uncertainty associated with these models. In the current scenario, large amount of data is automatically recorded with computers and
managed with the data base management systems (DBMS) for storage and fast retrieval purpose. The practice of examining large preexisting
databases in order to generate new information is known as data mining. Presently, data mining has attracted substantial
attention in the research and commercial arena which involves applications of a variety of statistical techniques. Twenty years ago
mostly data was collected manually and the data set was in simple form but in present time, there have been considerable changes in
the nature of data. Statistical techniques and computer applications can be utilized to obtain maximum information with the fewest
possible measurements to reduce the cost of data collection.
The document discusses the importance of mathematics in petroleum engineering. It outlines how mathematical modeling allows for virtual experimentation and analysis of complex phenomena. Areas like differential equations, numerical analysis, statistics, and harmonic analysis play key roles in modeling processes for applications such as image processing, telecommunications, data transmission, and oil prospecting. Developing strong links between mathematics and petroleum engineering is important for addressing challenges through interdisciplinary collaboration.
Calculus is used extensively in software engineering and related fields for several purposes:
1) In computer graphics, calculus is used to simulate light bouncing and rendering objects smoothly as well as powering physics engines.
2) In programming language design and semantics, calculus provides a functional predicate calculus to support formal reasoning about programs.
3) Calculus allows the creation of mathematical models to arrive at optimal solutions and is used in many areas of physics that relate to software such as motion, heat, and dynamics.
4) Fields like signal analysis, scientific computing, financial modeling, and robotics also employ calculus for tasks like examining functions, simulating systems, and modeling optimization.
The document discusses modelling and evaluation in machine learning. It defines what models are and how they are selected and trained for predictive and descriptive tasks. Specifically, it covers:
1) Models represent raw data in meaningful patterns and are selected based on the problem and data type, like regression for continuous numeric prediction.
2) Models are trained by assigning parameters to optimize an objective function and evaluate quality. Cross-validation is used to evaluate models.
3) Predictive models predict target values like classification to categorize data or regression for continuous targets. Descriptive models find patterns without targets for tasks like clustering.
4) Model performance can be affected by underfitting if too simple or overfitting if too complex,
Entropy and systemic risk measures
M. Billio, R. Casarin, M. Costola, A. Pasqualini
Ca’ Foscari Venice University
Final SYRTO Conference - Université Paris1 Panthéon-Sorbonne
February 19, 2016
The document discusses a simulation model for overcoming gender segregation in German high schools. It begins with an introduction to structuralist metatheory and non-statement views. It then presents a case study of using simulation to model the process of increasing female teachers in German gymnasia from 1950-1990. The simulation model was validated by comparing its output to empirical data and estimating the values of theoretical parameters that produced the best fit. A structuralist reconstruction of the model, its elements, and intended applications was presented as a way to help with verification, replication, validation, and measuring unobservable terms.
30.03.2017 Data Science Meetup - USER JOURNEY ANALYSIS, BETWEEN BUDGET ALLOCA...Zalando adtech lab
This document discusses user journey analysis for online marketing. It presents several approaches for analyzing user journey data, including sliding time windows, hand-crafted features, higher-order Markov models, and time series analysis. The objective is to inform micro-level decisions like digital ad bidding. Features are generated from online user behavior data within time windows. Models account for offline channels like TV and model awareness decay. Bayesian statistics are used to estimate models, which are optimized based on costs and benefits.
Machine Learning Strategies for Time Series PredictionGianluca Bontempi
This document introduces machine learning strategies for time series prediction. It begins with an introduction to the speaker and his background and research interests. It then provides an outline of the topics to be covered, including notions of time series, machine learning approaches for prediction, local learning methods, forecasting techniques, and applications and future directions. The document discusses what the audience should know coming into the course and what they will learn.
Machine Learning for Forecasting: From Data to DeploymentAnant Agarwal
Forecasting is everywhere. This talk covers:
• Fundamental concepts of time series
• Data preprocessing (imputation and outlier analysis)
• Feature engineering and EDA for time series
• Statistical and machine learning algorithms
• Model evaluation through backtesting
• Model explanation using SHAP
• Model monitoring and deployment considerations
This document provides an overview of techniques for temporal data mining. It discusses how temporal sequences arise in domains like engineering, science, finance, and healthcare. It covers approaches for representing temporal sequences, including keeping data in its original form, piecewise approximations, transformations to other domains, discretization, and generative models. It also discusses measuring similarity between sequences and techniques for classification and relation finding in temporal data, including frequent pattern detection and prediction. The goal is to classify and organize temporal data mining techniques to help practitioners address problems involving temporal data.
Time Series Weather Forecasting Techniques: Literature SurveyIRJET Journal
This document summarizes various time series forecasting techniques discussed in literature, including ARIMA, Prophet, and LSTM models. It reviews their applications in weather forecasting, analyzing COVID-19 data, real estate prices, bitcoin values, and more. The key techniques are compared based on their forecasting accuracy on different datasets. ARIMA is generally good at capturing trends but requires stationary data, while Prophet and LSTM can handle non-stationary data and seasonal effects better. Prophet achieved 91% accuracy on a COVID dataset, outperforming ARIMA. LSTM achieved 76% accuracy for rainfall forecasting. The document concludes different approaches are still being developed to address the unique challenges of weather data forecasting.
This document provides an overview of machine learning concepts including supervised learning, unsupervised learning, and reinforcement learning. It discusses common machine learning applications and challenges. Key topics covered include linear regression, classification, clustering, neural networks, bias-variance tradeoff, and model selection. Evaluation techniques like training error, validation error, and test error are also summarized.
The document describes a research team that studies emerging methodologies in economics and finance, including complex networks and machine learning. The team is based at Democritus University of Thrace in Greece and led by Drs. Periklis Gogas and Theophilos Papadimitriou. Their research focuses on using techniques like minimum dominating sets and support vector machines to analyze economic and financial networks and time series data. They have found these emerging methodologies can more accurately represent network structures and forecast macroeconomic and financial variables compared to traditional econometric methods.
Introduction AI ML& Mathematicals of ML.pdfGandhiMathy6
Machine learning uses probability theory to deal with uncertainty that arises from noisy data, limited data sets, and ambiguity. Probability theory provides a framework to quantify and manipulate uncertainty. It allows optimal predictions given available information, even if that information is incomplete. Key concepts in probability theory for machine learning include defining sample spaces and events, calculating probabilities, working with joint, conditional, and independent probabilities, and using Bayes' rule. These concepts help machine learning algorithms make inferences from data.
This document provides an overview of machine learning concepts and techniques including linear regression, logistic regression, unsupervised learning, and k-means clustering. It discusses how machine learning involves using data to train models that can then be used to make predictions on new data. Key machine learning types covered are supervised learning (regression, classification), unsupervised learning (clustering), and reinforcement learning. Example machine learning applications are also mentioned such as spam filtering, recommender systems, and autonomous vehicles.
This document discusses mathematical modeling and computer simulation. It begins by defining key terms such as mathematical model, computer modeling, and computer simulation. It then explains the process of creating a mathematical model to represent a real-world phenomenon and using that model to perform computer simulations. The success of a mathematical model depends on how accurately it predicts and explains the phenomenon. Examples of modeling a falling rock are provided to illustrate the modeling process.
FellowBuddy.com is an innovative platform that brings students together to share notes, exam papers, study guides, project reports and presentation for upcoming exams.
We connect Students who have an understanding of course material with Students who need help.
Benefits:-
# Students can catch up on notes they missed because of an absence.
# Underachievers can find peer developed notes that break down lecture and study material in a way that they can understand
# Students can earn better grades, save time and study effectively
Our Vision & Mission – Simplifying Students Life
Our Belief – “The great breakthrough in your life comes when you realize it, that you can learn anything you need to learn; to accomplish any goal that you have set for yourself. This means there are no limits on what you can be, have or do.”
Like Us - https://www.facebook.com/FellowBuddycom
The document discusses the matrix profile, a data structure that annotates time series data. It has many desirable properties, including being exact, simple, space efficient, and able to handle problems like motif discovery, anomaly detection, and segmentation in a trivial manner. The matrix profile records the distance of subsequences to their nearest neighbors, allowing patterns in time series data to be easily interpreted from the profile values. Both real and synthetic examples are used to demonstrate how the matrix profile visually represents motifs, anomalies, and other patterns in time series data.
This document provides an overview of machine learning using Python. It introduces machine learning applications and key Python concepts for machine learning like data types, variables, strings, dates, conditional statements, loops, and common machine learning libraries like NumPy, Matplotlib, and Pandas. It also covers important machine learning topics like statistics, probability, algorithms like linear regression, logistic regression, KNN, Naive Bayes, and clustering. It distinguishes between supervised and unsupervised learning, and highlights algorithm types like regression, classification, decision trees, and dimensionality reduction techniques. Finally, it provides examples of potential machine learning projects.
Stock Market Prediction using Long Short-Term MemoryIRJET Journal
This document discusses using a Long Short-Term Memory (LSTM) model to predict stock market prices. It begins by introducing the problem of predicting stock markets and how machine learning techniques like LSTM can help. It then discusses collecting stock price data and designing an LSTM model in Python using Keras and other libraries. The model is trained on historical stock price data to identify patterns and predict future prices. The document suggests LSTM models are well-suited for this due to their ability to use past data in predictions. It evaluates the model's predictions against actual prices to determine accuracy.
This document introduces the plm package for panel data econometrics in R. It discusses panel data models and estimation approaches, describes the software approach and functions in plm, and compares plm to other R packages for longitudinal data analysis. Key features of plm include functions for estimating linear panel models with fixed effects, random effects, and GMM, as well as data management and diagnostic testing capabilities for panel data. The package aims to make panel data analysis intuitive and straightforward for econometricians in R.
Significant Role of Statistics in Computational SciencesEditor IJCATR
This paper is focused on the issues related to optimizing statistical approaches in the emerging fields of Computer Science
and Information Technology. More emphasis has been given on the role of statistical techniques in modern data mining. Statistics is
the science of learning from data and of measuring, controlling, and communicating uncertainty. Statistical approaches can play a vital
role for providing significance contribution in the field of software engineering, neural network, data mining, bioinformatics and other
allied fields. Statistical techniques not only helps make scientific models but it quantifies the reliability, reproducibility and general
uncertainty associated with these models. In the current scenario, large amount of data is automatically recorded with computers and
managed with the data base management systems (DBMS) for storage and fast retrieval purpose. The practice of examining large preexisting
databases in order to generate new information is known as data mining. Presently, data mining has attracted substantial
attention in the research and commercial arena which involves applications of a variety of statistical techniques. Twenty years ago
mostly data was collected manually and the data set was in simple form but in present time, there have been considerable changes in
the nature of data. Statistical techniques and computer applications can be utilized to obtain maximum information with the fewest
possible measurements to reduce the cost of data collection.
The document discusses the importance of mathematics in petroleum engineering. It outlines how mathematical modeling allows for virtual experimentation and analysis of complex phenomena. Areas like differential equations, numerical analysis, statistics, and harmonic analysis play key roles in modeling processes for applications such as image processing, telecommunications, data transmission, and oil prospecting. Developing strong links between mathematics and petroleum engineering is important for addressing challenges through interdisciplinary collaboration.
This document discusses using machine learning to analyze patterns in COVID waves. It presents on detecting COVID waves using machine learning. The methodology section discusses using an agile development model and tools like Jupyter Notebook, Anaconda Prompt, Python libraries for analysis. Various regression and time series algorithms like ARIMA are used for forecasting and predicting COVID test cases. The experiments show analysis of missing data, COVID deaths/cases in top states, testing rates, and ICU beds. Challenges in time series algorithms and conclusions are also provided.
This document proposes a new quantile-based fuzzy time series forecasting model. It begins by discussing how time series models have been used to predict things like enrollments, weather, accidents, and stock prices. It then provides background on quantile regression models and fuzzy time series forecasting. The proposed method develops a time variant quantile-based fuzzy time series forecasting approach based on predicting future data trends. The method converts statistical quantiles to fuzzy quantiles using membership functions and provides a fuzzy metric to calculate future values based on trend forecasts. The model is applied to TAIFEX forecasting and shows better performance than other models in terms of complexity and accuracy.
[Keynote] predictive technologies and the prediction of technology - Bob Will...PAPIs.io
I will examine predictive technologies in the light of the history of technology and its prediction. No matter how shiny and new, a new technology is still a technology, and there are general patterns that seem to recur. We can learn from those patterns if we pay attention to them.
In particular I will look at the challenge of predicting the impact of new technologies, talk about how they evolve, and the role that modularity, standards and interoperability play in their evolution.
I will talk more specifically about some of the particular challenges of making APIs and interfaces for predictive technologies such as machine learning, and speculate on the prospects for making machine learning a service, and more of a mature engineering discipline. In passing I will briefly demonstrate some recent machine learning work from NICTA.
Intro/Overview on Machine Learning PresentationAnkit Gupta
This document provides an overview of a presentation on machine learning given at Gurukul Kangri University in 2017. It defines machine learning as a field that allows computers to learn without being explicitly programmed. It discusses different machine learning algorithms including supervised learning, unsupervised learning, and semi-supervised learning. Examples of applications of machine learning discussed include data mining, natural language processing, image recognition, and expert systems. The document also contrasts artificial intelligence, machine learning, and deep learning.
Gen Z and the marketplaces - let's translate their needsLaura Szabó
The product workshop focused on exploring the requirements of Generation Z in relation to marketplace dynamics. We delved into their specific needs, examined the specifics in their shopping preferences, and analyzed their preferred methods for accessing information and making purchases within a marketplace. Through the study of real-life cases , we tried to gain valuable insights into enhancing the marketplace experience for Generation Z.
The workshop was held on the DMA Conference in Vienna June 2024.
HijackLoader Evolution: Interactive Process HollowingDonato Onofri
CrowdStrike researchers have identified a HijackLoader (aka IDAT Loader) sample that employs sophisticated evasion techniques to enhance the complexity of the threat. HijackLoader, an increasingly popular tool among adversaries for deploying additional payloads and tooling, continues to evolve as its developers experiment and enhance its capabilities.
In their analysis of a recent HijackLoader sample, CrowdStrike researchers discovered new techniques designed to increase the defense evasion capabilities of the loader. The malware developer used a standard process hollowing technique coupled with an additional trigger that was activated by the parent process writing to a pipe. This new approach, called "Interactive Process Hollowing", has the potential to make defense evasion stealthier.
Ready to Unlock the Power of Blockchain!Toptal Tech
Imagine a world where data flows freely, yet remains secure. A world where trust is built into the fabric of every transaction. This is the promise of blockchain, a revolutionary technology poised to reshape our digital landscape.
Toptal Tech is at the forefront of this innovation, connecting you with the brightest minds in blockchain development. Together, we can unlock the potential of this transformative technology, building a future of transparency, security, and endless possibilities.
Discover the benefits of outsourcing SEO to Indiadavidjhones387
"Discover the benefits of outsourcing SEO to India! From cost-effective services and expert professionals to round-the-clock work advantages, learn how your business can achieve digital success with Indian SEO solutions.
1. Machine Learning Strategies for Time
Series Prediction
Machine Learning Summer School
(Hammamet, 2013)
Gianluca Bontempi
Machine Learning Group, Computer Science Department
Boulevard de Triomphe - CP 212
http://www.ulb.ac.be/di
Machine Learning Strategies for Prediction – p. 1/128
2. Introducing myself
• 1992: Computer science engineer (Politecnico di Milano, Italy),
• 1994: Researcher in robotics in IRST, Trento, Italy,
• 1995: Researcher in IRIDIA, ULB Artificial Intelligence Lab, Brussels,
• 1996-97: Researcher in IDSIA, Artificial Intelligence Lab, Lugano,
Switzerland,
• 1998-2000: Marie Curie fellowship in IRIDIA, ULB Artificial Intelligence
Lab, Brussels,
• 2000-2001: Scientist in Philips Research, Eindhoven, The Netherlands,
• 2001-2002: Scientist in IMEC, Microelectronics Institute, Leuven,
Belgium,
• since 2002: professor in Machine Learning, Modeling and Simulation,
Bioinformatics in ULB Computer Science Dept., Brussels,
• since 2004: head of the ULB Machine Learning Group (MLG).
• since 2013: director of the Interuniversity Institute of Bioinformatics in
Brussels (IB)2
, ibsquare.be.
Machine Learning Strategies for Prediction – p. 2/128
3. ... and in terms of distances
According to MathSciNet
•• Distance from Erdos= 5
• Distance from Zaiane= 6
• Distance from Deisenroth= 8
Machine Learning Strategies for Prediction – p. 3/128
4. ULB Machine Learning Group (MLG)
• 3 professors, 10 PhD students, 5 postdocs.
• Research topics: Knowledge discovery from data, Classification, Computational
statistics, Data mining, Regression, Time series prediction, Sensor networks,
Bioinformatics, Network inference.
• Computing facilities: high-performing cluster for analysis of massive datasets,
Wireless Sensor Lab.
• Website: mlg.ulb.ac.be.
• Scientific collaborations in ULB: Hopital Jules Bordet, Laboratoire de Médecine
experimentale, Laboratoire d’Anatomie, Biomécanique et Organogénèse (LABO),
Service d’Anesthesie (ERASME).
• Scientific collaborations outside ULB: Harvard Dana Farber (US), UCL Machine
Learning Group (B), Politecnico di Milano (I), Universitá del Sannio (I), Inst Rech
Cliniques Montreal (CAN).
Machine Learning Strategies for Prediction – p. 4/128
5. ULB-MLG: recent projects
1. Machine Learning for Question Answering (2013-2014).
2. Adaptive real-time machine learning for credit card fraud detection (2012-2013).
3. Epigenomic and Transcriptomic Analysis of Breast Cancer (2012-2015).
4. Discovery of the molecular pathways regulating pancreatic beta cell dysfunction
and apoptosis in diabetes using functional genomics and bioinformatics: ARC
(2010-2015)
5. ICT4REHAB - Advanced ICT Platform for Rehabilitation (2011-2013)
6. Integrating experimental and theoretical approaches to decipher the molecular
networks of nitrogen utilisation in yeast: ARC (2006-2010).
7. TANIA - Système d’aide à la conduite de l’anesthésie. WALEO II project funded
by the Région Wallonne (2006-2010)
8. "COMP2
SYS" (COMPutational intelligence methods for COMPlex SYStems)
MARIE CURIE Early Stage Research Training funded by the EU (2004-2008).
Machine Learning Strategies for Prediction – p. 5/128
6. What you are supposed to know
• Basic notions of probability and statistics
• Random variable
• Expectation, variance, covariance
• Least-squares
What you are expected to get acquainted with
• Foundations of statistical machine learning
• How to build a predictive model from data
• Strategies for forecasting
What will remain
• Interest, curiosity for machine learning
• taste for prediction
• Contacts
• Companion webpage
http://www.ulb.ac.be/di/map/gbonte/mlss.html
Machine Learning Strategies for Prediction – p. 6/128
7. Outline
• Notions of time series (30 mins)
• conditional probability
• Machine learning for prediction (45 mins)
• bias/variance
• parametric and structural identification
• validation
• model selection
• feature selection
• COFFEE BREAK
• Local learning (15 mins)
• Forecasting: one-step and multi-step-ahed (30 mins)
• Some applications (15 mins)
• time series competitions
• wireless sensor
• biomedical
• Future directions and perspectives (15 mins)
Machine Learning Strategies for Prediction – p. 7/128
8. What is machine learning?
Machine learning is that domain of computational intelligence which is
concerned with the question of how to construct computer programs that
automatically improve with experience. [16]
Reductionist attitude: ML is just a buzzword which equates to statistics plus
marketing
Positive attitude: ML paved the way to the treatment of real problems related
to data analysis, sometimes overlooked by statisticians (nonlinearity,
classification, pattern recognition, missing variables, adaptivity,
optimization, massive datasets, data management, causality,
representation of knowledge, parallelisation)
Interdisciplinary attitude: ML should have its roots on statistics and
complements it by focusing on: algorithmic issues, computational
efficiency, data engineering.
Machine Learning Strategies for Prediction – p. 8/128
9. Why study machine learning?
• Machine learning is cool.
• Practical way to understand: All models are wrong but some are useful...
• The fastest way to become a data scientist ... the sexiest job in the 21st
century
• Someone who knows statistics better than a computer scientists and
programs better than a statistician...
Machine Learning Strategies for Prediction – p. 9/128
10. Notion of time series
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11. Time series
Definition A time series is a sequence of observations st ∈ R, usually
ordered in time.
Examples of time series in every scientific and applied domain:
• Meteorology: weather variables, like temperature, pressure, wind.
• Economy and finance: economic factors (GNP), financial indexes,
exchange rate, spread.
• Marketing: activity of business, sales.
• Industry: electric load, power consumption, voltage, sensors.
• Biomedicine: physiological signals (EEG), heart-rate, patient
temperature.
• Web: clicks, logs.
• Genomics: time series of gene expression during cell cycle.
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12. Why studying time series?
There are various reasons:
Prediction of the future based on the past.
Control of the process producing the series.
Understanding of the mechanism generating the series.
Description of the salient features of the series.
Machine Learning Strategies for Prediction – p. 12/128
13. Univariate discrete time series
• Quantities, like temperature and voltage, change in a continuous way.
• In practice, however, the digital recording is made discretely in time.
• We shall confine ourselves to discrete time series (which however take
continuous values).
• Moreover we will consider univariate time series, where one type of
measurement is made repeatedly on the same object or individual.
• Multivariate time series are out of the scope of this presentation but
represent an important topic in the domain.
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14. A general model
Let an observed discrete univariate time series be s1, . . . , sT . This means
that we have T numbers which are observations on some variable made at T
equally distant time points, which for convenience we label 1, 2, . . . , T.
A fairly general model for the time series can be written
st = g(t) + ϕt t = 1, . . . , T
The observed series is made of two components
Systematic part: g(t), also called signal or trend, which is a determistic
function of time
Stochastic sequence: a residual term ϕt, also called noise, which follows a
probability law.
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15. Types of variation
Traditional methods of time-series analysis are mainly concerned with
decomposing the variation of a series st into:
Trend : this is a long-term change in the mean level, eg. an increasing trend.
Seasonal effect : many time series (sale figures, temperature readings) exhibit
variation which is seasonal (e.g. annual) in period. The measure and the
removal of such variation brings to deseasonalized data.
Irregular fluctuations : after trend and cyclic variations have been removed
from a set of data, we are left with a series of residuals, which may or
may not be completely random.
We will assume here that once we have detrended and deseasonalized the
series, we can still extract information about the dependency between the
past and the future. Henceforth ϕt will denote the detrended and
deseasonalized series.
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17. Probability and dependency
• Forecasting a time series is possible since future depends on the past or
analogously because there is a relationship between the future and the
past. However this relation is not deterministic and can hardly be written
in an analytical form.
• An effective way to describe a nondeterministic relation between two
variables is provided by the probability formalism.
• Consider two continuous random variables ϕ1 and ϕ2 representing for
instance the temperature today (time t1) and tomorrow (t2). We tend to
believe that ϕ1 could be used as a predictor of ϕ2 with some degree of
uncertainty.
• The stochastic dependency between ϕ1 and ϕ2 is resumed by the joint
density p(ϕ1, ϕ2) or equivalently by the conditional probability
p(ϕ2|ϕ1) =
p(ϕ1, ϕ2)
p(ϕ1)
• If p(ϕ2|ϕ1) = p(ϕ2) then ϕ1 and ϕ2 are not independent or equivalently
the knowledge of the value of ϕ1 reduces the uncertainty about ϕ2.
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18. Stochastic processes
• The stochastic approach to time series makes the assumption that a
time series is a realization of a stochastic process (like tossing an
unbiased coin is the realization of a discrete random variable with equal
head/tail probability).
• A discrete-time stochastic process is a collection of random variables ϕt,
t = 1, . . . , T defined by a joint density
p(ϕ1, . . . , ϕT )
• Statistical time-series analysis is concerned with evaluating the
properties of the probability model which generated the observed time
series.
• Statistical time-series modeling is concerned with inferring the properties
of the probability model which generated the observed time series from a
limited set of observations.
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19. Strictly stationary processes
• Predicting a time series is possible if and only if the dependence
between values existing in the past is preserved also in the future.
• In other terms, though measures change, the stochastic rule underlying
their realization does not. This aspect is formalized by the notion of
stationarity.
• Definition A stochastic process is said to be strictly stationary if the joint
distribution of ϕt1
, ϕt2
, . . . , ϕtn
is the same as the joint distribution of
ϕt1+k, ϕt2+k, . . . , ϕtn+k for all n, t1, . . . , tn and k.
• In other words shifting the time origin by an amount k has no effect on
the joint distribution which depends only on the intervals between
t1, . . . , tn.
• This implies that the (marginal) distribution of ϕt is the same for all t.
• The definition holds for any value of n.
• Let us see what does it mean in practice for n = 1 and n = 2.
Machine Learning Strategies for Prediction – p. 19/128
20. Properties
n=1 : If ϕt is strictly stationary and its first two moments are finite, we have
E[ϕt] = µt = µ, Var [ϕt] = σ2
t = σ2
n=2 : Furthermore the autocovariance function γ(t1, t2) depends only on the
lag k = t2 − t1 and may be written by
γ(k) = Cov[ϕt, ϕt+k] = E (ϕt − µ)(ϕt+k − µ)
• In order to avoid scaling effects, it is useful to introduce the
autocorrelation function
ρ(k) =
γ(k)
σ2
=
γ(k)
γ(0)
• Another relevant function is the the partial autocorrelation function π(k)
where π(k), k > 1 measures the degree of association between ϕt and
ϕt−k when the effects of the intermediary lags 1, . . . , k − 1 are removed
Machine Learning Strategies for Prediction – p. 20/128
21. Weak stationarity
• A less restricted definition of stationarity concerns only the first two
moments of ϕt
Definition A process is called second-order stationary or weakly stationary
if its mean is constant and its autocovariance function depends only on
the lag.
• No assumptions are made about higher moments than those of second
order.
• Strict stationarity implies weak stationarity but not viceversa in general.
Definition A process is called normal is the joint distribution of
ϕt1
, ϕt2
, . . . , ϕtn
is multivariate normal for all t1, . . . , tn.
• In the special case of normal processes, weak stationarity implies strict
stationarity. This is due to the fact that a normal process is completely
specified by the mean and the autocovariance function.
Machine Learning Strategies for Prediction – p. 21/128
22. Estimators of first moments
Here you will find some common estimators of the two first moments of a
time series:
• The empirical mean is given by
ˆµ =
T
t=1 ϕt
T
• The empirical autocovariance function is given by
ˆγ(k) =
T −k
t=1 (ϕt − ˆµ)(ϕt+k − ˆµ)
T − k − 1
, k < T − 2
• The empirical autocorrelation function is given by
ˆρ(k) =
ˆγ(k)
ˆγ(0)
Machine Learning Strategies for Prediction – p. 22/128
23. Some examples of stochastic
processes
Machine Learning Strategies for Prediction – p. 23/128
24. Purely random processes
• It consists of a sequence of random variables ϕt which are mutually
independent and identically distributed. For each t and k
p(ϕt+k|ϕt) = p(ϕt+k)
• It follows that this process has constant mean and variance. Also
γ(k) = Cov[ϕt, ϕt+k] = 0
for k = ±1, ±2, . . . .
• A purely random process is strictly stationary.
• A purely random process is sometimes called white noise by engineers.
• An example of purely random process is the series of numbers drawn by
a roulette wheel in a casino.
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25. Example: Gaussian purely random
0 200 400 600 800 1000
−3−2−10123
White noise
t
y
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26. Example: autocorrelation function
0 5 10 15 20 25 30
0.00.20.40.60.81.0
Lag
ACF
Series y
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27. Random walk
• Suppose that wt is a discrete, purely random process with mean µ and
variance σ2
w.
• A process ϕt is said to be a random walk if
ϕt = ϕt−1 + wt
• The next value of a random walk is obtained by summing a random
shock to the latest value.
• If ϕ0 = 0 then
ϕt =
t
i=1
wi
• E[ϕt] = tµ and Var [ϕt] = tσ2
w.
• As the mean and variance change with t the process is non-stationary.
Machine Learning Strategies for Prediction – p. 27/128
28. Random walk (II)
• The first differences of a random walk given by
∇ϕt = ϕt − ϕt−1
form a purely random process, which is stationary.
• Examples of time series which behave like random walks are
• stock prices on successive days.
• the path traced by a molecule as it travels in a liquid or a gas,
• the search path of a foraging animal
Machine Learning Strategies for Prediction – p. 28/128
29. Ten random walks
Let w ∼ N(0, 1).
0 50 100 150 200 250 300 350 400 450 500
−40
−30
−20
−10
0
10
20
30
40
50
60
Random walks
Machine Learning Strategies for Prediction – p. 29/128
30. Autoregressive processes
• Suppose that wt is a purely random process with mean zero and
variance σ2
w.
• A process ϕt is said to be an autoregressive process of order n (also an
AR(n) process) if
ϕt = α1ϕt−1 + · · · + αnϕt−n + wt
• This means that the next value is a linear weighted sum of the past n
values plus a random shock.
• Finite memory filter.
• If w is a normal variable, ϕt will be normal too.
• Note that this is like a linear regression model where ϕ is regressed not
on independent variables but on its past values (hence the prefix “auto”).
• The properties of stationarity depends on the values αi, i = 1, . . . , n.
Machine Learning Strategies for Prediction – p. 30/128
31. First order AR(1) process
If n = 1, we have the so-called Markov process AR(1)
ϕt = αϕt−1 + wt
By substitution it can be shown that
ϕt = α(αϕt−2 + wt−1) + wt = α2
(αϕt−3 + wt−2) + αwt−1 + wt =
= wt + αwt−1 + α2
wt−2 + . . .
Then
E[ϕt] = 0 Var [ϕt] = σ2
w(1 + α2
+ α4
+ . . . )
Then if |α| < 1 the variance if finite and equals
Var [ϕt] = σ2
ϕ = σ2
w/(1 − α2
)
and the autocorrelation is
ρ(k) = αk
k = 0, . . . , 1, 2
Machine Learning Strategies for Prediction – p. 31/128
32. General order AR(n) process
• It has been shown that condition necessary and sufficient for the
stationarity is that the complex roots of the equation
φ(z) = 1 − α1z − · · · − αnzn
= 0
lie outside the unit circle.
• The autocorrelation function of an AR(n) attenuates slowly with the lag k
(exponential decay or damped sine wave pattern).
• On the contrary the partial autocorrelation function cuts off at k > n, i.e.
it is not significantly different from zero beyond the lag n.
Machine Learning Strategies for Prediction – p. 32/128
33. Example: AR(2)
0 200 400 600 800 1000
051015
AR(2)
t
y
Machine Learning Strategies for Prediction – p. 33/128
34. Example: AR(2)
0 5 10 15 20 25 30
−0.50.00.51.0
Lag
ACF
Series y
Machine Learning Strategies for Prediction – p. 34/128
35. Partial autocorrelation in AR(2)
0 5 10 15 20 25 30
−0.50.00.5
Lag
PartialACF
Series y
Machine Learning Strategies for Prediction – p. 35/128
36. Fitting an autoregressive process
The estimation of an autoregressive process to a set of data
DT = {ϕ1, . . . , ϕT } demands the resolution of two problems:
1. The estimation of the order n of the process. This is typically supported
by the analysis of the partial autocorrelation function.
2. The estimation of the set of parameters {α1, . . . , αn}.
Machine Learning Strategies for Prediction – p. 36/128
37. Estimation of AR(n) parameters
Suppose we have an AR(n) process of order n
ϕt = α1ϕt−1 + · · · + αpϕt−n + wt
Given T observations, the parameters may be estimated by least-squares by
minimizing
ˆα = arg min
α
T
t=n+1
[ϕt − α1ϕt−1 − · · · − αnϕt−n]
2
In matrix form this amounts to solve the multiple least-squares problem
Y = Xα where
X =
ϕT −1 ϕT −2 . . . ϕT −n−1
ϕT −2 ϕT −3 . . . ϕT −n−2
...
...
...
...
ϕn ϕn−1 . . . ϕ1
Y =
ϕT
ϕT −1
...
ϕn+1
(1)
Machine Learning Strategies for Prediction – p. 37/128
38. Least-squares estimation of AR(n) parms
• Let N be the number of rows of X. In order to estimate the AR(n)
parameters we compute the least-squares estimator
ˆα = arg min
a
N
i=1
(yi − xT
i a)2
= arg min
a
(Y − Xa)T
(Y − Xa)
• It can be shown that
ˆα = (XT
X)−1
XT
Y
where the XT
X matrix is a symmetric [n × n] matrix which plays an
important role in multiple linear regression.
• Conventional linear regression theory provides also confidence interval
and significativity tests for the AR(n) coefficients.
• A recursive version of least-squares, i.e. where time samples arrive
sequentially, is provided by the RLS algorithm.
Machine Learning Strategies for Prediction – p. 38/128
39. From linear to nonlinear setting
Machine Learning Strategies for Prediction – p. 39/128
40. The NAR representation
• AR models assume that the relation between past and future is linear
• Once we assume that the linear assumption does not hold, we may
extend the AR formulation to a Nonlinear Auto Regressive (NAR)
formulation
ϕt = f (ϕt−1, ϕt−2, . . . , ϕt−n) + w(t)
where the missing information is lumped into a noise term w.
• In what follows we will consider this relationship as a particular instance
of a dependence
y = f(x) + w
between a multidimensional input x ∈ X ⊂ Rn
and a scalar output y ∈ R.
• NOTA BENE. In what follows y will denote the next value ϕt to be
predicted and
x = [ϕt−1, ϕt−2, . . . , ϕt−n]
will denote the n-dimensional input vector also known as embedding
vector.
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41. Nonlinear vs. linear time series
The advantage of linear models are numerous:
• the least-squares ˆα estimate can be expressed in an analytical form
• the least-squares ˆα estimate can be easily calculated through matrix
computation.
• statistical properties of the estimator can be easily defined.
• recursive formulation for sequential updating are avaialble.
• the relation between empirical and generalization error is known,
BUT...
Machine Learning Strategies for Prediction – p. 41/128
42. Nonlinear vs. linear time series
• linear methods interpret all the structure in a time series through linear
correlation
• deterministic linear dynamics can only lead to simple exponential or
periodically oscillating behavior, so all irregular behavior is attributed to
external noise while deterministic nonlinear equations could produce
very irregular data,
• in real problems it is extremely unlikely that the variables are linked by a
linear relation.
In practice, the form of the relation is often unknown and only a limited
amount of samples is available.
Machine Learning Strategies for Prediction – p. 42/128
43. Machine learning for prediction
Machine Learning Strategies for Prediction – p. 43/128
44. Supervised learning
TRAINING
DATASET
UNKNOWN
DEPENDENCY
INPUT OUTPUT
ERROR
PREDICTION
MODEL
PREDICTION
From now on we consider the prediction problem as a problem of supervised
learning problem, where we have to infer from historical data the possibly
nonlinear dependance between the input (past embedding vector) and the
output (future value).
Statistical machine learning is the discipline concerned with this problem.
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45. The regression plus noise form
• A typical way of representing the unknown input/output relation is the
regression plus noise form
y = f(x) + w
where f(·) is a deterministic function and the term w represents the
noise or random error. It is typically assumed that w is independent of x
and E[w] = 0.
• Suppose that we have available a training set { xi, yi : i = 1, . . . , N},
where xi = (xi1, . . . , xin) and yi, generated according to the previous
model.
• The goal of a learning procedure is to estimate a model ˆf(x) which is
able to give a good approximation of the unknown function f(x).
• But how to choose ˆf, if we do not know the probability distribution
underlying the data and we have only a limited training set?
Machine Learning Strategies for Prediction – p. 45/128
46. A simple example with n = 1
−2 −1 0 1 2
−5051015
x
Y
NOTA BENE: this is NOT a time series ! y = ϕt, x = ϕt−1. The horizontal axis
does not represent time but the past value of the series.
Machine Learning Strategies for Prediction – p. 46/128
47. Model degree 1
−2 −1 0 1 2
−5051015
x
Y
Training error= 2 degree= 1
ˆf(x) = α0 + α1x
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48. Model degree 3
−2 −1 0 1 2
−5051015
x
Y
Training error= 0.92 degree= 3
ˆf(x) = α0 + α1x + · · · + α3x3
Machine Learning Strategies for Prediction – p. 48/128
49. Model degree 18
−2 −1 0 1 2
−5051015
x
Y
Training error= 0.4 degree= 18
ˆf(x) = α0 + α1x + · · · + α18x1
8
Machine Learning Strategies for Prediction – p. 49/128
50. Generalization and overfitting
• How to estimate the quality of a model? Is the training error a good
measure of the quality?
• The goal of learning is to find a model which is able to generalize, i.e.
able to return good predictions for input values independent of the
training set.
• In a nonlinear setting, it is possible to find models with such a complicate
structure that they have null training errors. Are these models good?
• Typically NOT. Since doing very well on the training set could mean
doing badly on new data.
• This is the phenomenon of overfitting.
• Using the same data for training a model and assessing it is typically a
wrong procedure, since this returns an over optimistic assessment of the
model generalization capability.
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51. Bias and variance of a model
• A fundamental result of estimation theory shows that the
mean-squared-error, i.e. a measure of the generalization quality of an
estimator can be decomposed into three terms:
MISE = σ2
w + squared bias + variance
where the intrinsic noise term reflects the target alone, the bias reflects
the target’s relation with the learning algorithm and the variance term
reflects the learning algorithm alone.
• This result is purely theoretical since these quantities cannot be
measured on the basis of a finite amount of data.
• However, this result provides insight about what makes accurate a
learning process.
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52. The bias/variance trade-off
• The first term is the variance of y around its true mean f(x) and cannot
be avoided no matter how well we estimate f(x), unless σ2
w = 0.
• The bias measures the difference in x between the average of the
outputs of the hypothesis functions ˆf over the set of possible DN and
the regression function value f(x)
• The variance reflects the variability of the guessed ˆf(x, αN ) as one
varies over training sets of fixed dimension N. This quantity measures
how sensitive the algorithm is to changes in the data set, regardless to
the target.
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53. The bias/variance dilemma
• The designer of a learning machine has not access to the term MISE but
can only estimate it on the basis of the training set. Hence, the
bias/variance decomposition is relevant in practical learning since it
provides a useful hint about the features to control in order to make the
error MISE small.
• The bias term measures the lack of representational power of the class
of hypotheses. To reduce the bias term we should consider complex
hypotheses which can approximate a large number of input/output
mappings.
• The variance term warns us against an excessive complexity of the
approximator. This means that a class of too powerful hypotheses runs
the risk of being excessively sensitive to the noise affecting the training
set; therefore, our class could contain the target but it could be
practically impossible to find it out on the basis of the available dataset.
Machine Learning Strategies for Prediction – p. 53/128
54. • In other terms, it is commonly said that an hypothesis with large bias but
low variance underfits the data while an hypothesis with low bias but
large variance overfits the data.
• In both cases, the hypothesis gives a poor representation of the target
and a reasonable trade-off needs to be found.
• The task of the model designer is to search for the optimal trade-off
between the variance and the bias term, on the basis of the available
training set.
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56. The learning procedure
A learning procedure aims at two main goals:
1. to choose a parametric family of hypothesis ˆf(x, α) which contains or
gives good approximation of the unknown function f (structural
identification).
2. within the family ˆf(x, α), to estimate on the basis of the training set DN
the parameter αN which best approximates f (parametric identification).
In order to accomplish that, a learning procedure is made of two nested
loops:
1. an external structural identification loop which goes through different
model structures
2. an inner parametric identification loop which searches for the best
parameter vector within the family structure.
Machine Learning Strategies for Prediction – p. 56/128
57. Parametric identification
The parametric identification of the hypothesis is done according to ERM
(Empirical Risk Minimization) principle where
αN = α(DN ) = arg min
α∈Λ
MISEemp(α)
minimizes the training error
MISEemp(α) =
N
i=1 yi − ˆf(xi, α)
2
N
constructed on the basis of the training data set DN .
Machine Learning Strategies for Prediction – p. 57/128
58. Parametric identification (II)
• The computation of αN requires a procedure of multivariate optimization
in the space of parameters.
• The complexity of the optimization depends on the form of ˆf(·).
• In some cases the parametric identification problem may be an NP-hard
problem.
• Thus, we must resort to some form of heuristic search.
• Examples of parametric identification procedure are linear least-squares
for linear models and backpropagated gradient-descent for feedforward
neural networks.
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59. Model assessment
• We have seen before that the training error is not a good estimator (i.e. it
is too optimistic) of the generalization capability of the learned model.
• Two alternative exists:
1. Complexity-based penalty criteria
2. Data-driven validation techniques
Machine Learning Strategies for Prediction – p. 59/128
60. Complexity-based penalization
• In conventional statistics, various criteria have been developed, often in
the context of linear models, for assessing the generalization
performance of the learned hypothesis without the use of further
validation data.
• Such criteria take the form of a sum of two terms
ˆGPE = MISEemp + complexity term
where the complexity term represents a penalty which grows as the
number of free parameters in the model grows.
• This expression quantifies the qualitative consideration that simple
models return high training error with a reduced complexity term while
complex models have a low training error thanks to the high number of
parameters.
• The minimum for the criterion represents a trade-off between
performance on the training set and complexity.
Machine Learning Strategies for Prediction – p. 60/128
61. Complexity-based penalty criteria
If the input/output relation is linear, well-known examples of complexity based
criteria are:
• the Final Prediction Error (FPE)
FPE = MISEemp(αN )
1 + p/N
1 − p/N
with p = n + 1,
• the Generalized Cross-Validation (GCV)
GCV = MISEemp(αN )
1
(1 − p
N )2
• the Akaike Information Criterion (AIC)
AIC =
p
N
−
1
N
L(αN )
where L(·) is the log-likelihood function,
Machine Learning Strategies for Prediction – p. 61/128
62. • the Cp criterion proposed by Mallows
Cp =
MISEemp(αN )
ˆσ2
w
+ 2p − N
where ˆσ2
w is an estimate of the variance of noise,
• the Predicted Squared Error (PSE)
PSE = MISEemp(αN ) + 2ˆσ2
w
p
N
where ˆσ2
w is an estimate of the variance of noise.
Machine Learning Strategies for Prediction – p. 62/128
63. Data-driven validation techniques
If no (e.g. linear) assumptions are made, how to measure MISE in a reliable
way on a finite dataset? The most common techniques to return an estimate
MISE are
Testing: a testing sequence independent of DN and distributed according to
the same probability distribution is used to assess the quality. In
practice, unfortunately, an additional set of input/output observations is
rarely available.
Holdout: The holdout method, sometimes called test sample estimation,
partitions the data DN into two mutually exclusive subsets, the training
set Dtr and the holdout or test set DNts .
k-fold Cross-validation: the set DN is randomly divided into k mutually
exclusive test partitions of approximately equal size. The cases not
found in each test partition are independently used for selecting the
hypothesis which will be tested on the partition itself. The average error
over all the k partitions is the cross-validated error rate.
Machine Learning Strategies for Prediction – p. 63/128
64. The K-fold cross-validation
This is the algorithm in detail:
1. split the dataset DN into k roughly equal-sized parts.
2. For the kth part k = 1, . . . , K, fit the model to the other K − 1 parts of the
data, and calculate the prediction error of the fitted model when
predicting the k-th part of the data.
3. Do the above for k = 1, . . . , K and average the K estimates of prediction
error.
Let k(i) be the part of DN containing the ith sample. Then the
cross-validation estimate of the MISE prediction error is
MISECV =
1
N
N
i=1
(yi − ˆy
−k(i)
i )2
=
1
N
N
i=1
yi − ˆf(xi, α−k(i)
)
2
where ˆy
−k(i)
i denotes the fitted value for the ith observation returned by the
model estimated with the k(i)th part of the data removed.
Machine Learning Strategies for Prediction – p. 64/128
65. 10-fold cross-validation
K = 10: at each iteration 90% of data are used for training and the remaining
10% for the test.
90%
10%
Machine Learning Strategies for Prediction – p. 65/128
66. Leave-one-out cross validation
• The cross-validation algorithm where K = N is also called the
leave-one-out algorithm.
• This means that for each ith sample, i = 1, . . . , N,
1. we carry out the parametric identification, leaving that observation
out of the training set,
2. we compute the predicted value for the ith observation, denoted by
ˆy−i
i
The corresponding estimate of the MISE prediction error is
MISELOO =
1
N
N
i=1
(yi − ˆy−i
i )2
=
1
N
N
i=1
(yi − ˆf xi, α−i
)
2
where α−i
is the set of parameters returned by the parametric identification
perfomed on the training set with the ith sample set aside.
Machine Learning Strategies for Prediction – p. 66/128
67. Model selection
• Model selection concerns the final choice of the model structure
• By structure we mean:
• family of the approximator (e.g. linear, non linear) and if nonlinear
which kind of learner (e.g. neural networks, support vector
machines, nearest-neighbours, regression trees)
• the value of hyper parameters (e.g. number of hidden layers, number
of hidden nodes in NN, number of neighbors in KNN, number of
levels in trees)
• number and set of input variables
• this choice is typically the result of a compromise between different
factors, like the quantitative measures, the personal experience of the
designer and the effort required to implement a particular model in
practice.
• Here we will consider only quantitative criteria. Two are the possible
approaches:
1. the winner-takes-all approach
2. the combination of estimators approach.
Machine Learning Strategies for Prediction – p. 67/128
68. Model selection
N
REALIZATION
STOCHASTIC
PROCESS
VALIDATION
CLASSES of HYPOTHESIS
LEARNED MODEL
TRAINING
SET
MODEL SELECTION
PARAMETRIC IDENTIFICATION
IDENTIFICATION
,,
, ,
,
,
STRUCTURAL
α
?
ΛSΛ2Λ1
GN
1
α
1
N
α
1
N
α
2
N
α
2
N
αN
s
αN
s
GN
2
GN
S
GN
2
GN
1
GN
S
Machine Learning Strategies for Prediction – p. 68/128
69. Winner-takes-all
The best hypothesis is selected in the set {αs
N }, with s = 1, . . . , S, according
to
˜s = arg min
s=1,...,S
MISE
s
A model with complexity ˜s is trained on the whole dataset DN and used for
future predictions.
Machine Learning Strategies for Prediction – p. 69/128
70. Winner-takes-all pseudo-code
1. for s = 1, . . . , S: (Structural loop)
• for j = 1, . . . , N
(a) Inner parametric identification (for l-o-o):
αs
N−1 = arg min
α∈Λs
i=1:N,i=j
(yi − ˆf(xi, α))2
(b) ej = yj − ˆf(xj, αs
N−1)
• MISELOO(s) = 1
N
N
j=1 e2
j
2. Model selection: ˜s = arg mins=1,...,S MISELOO(s)
3. Final parametric identification:
α˜s
N = arg minα∈Λ˜s
N
i=1(yi − ˆf(xi, α))2
4. The output prediction model is ˆf(·, α˜s
N )
Machine Learning Strategies for Prediction – p. 70/128
71. Model combination
• The winner-takes-all approach is intuitively the approach which should
work the best.
• However, recent results in machine learning show that the performance
of the final model can be improved not by choosing the model structure
which is expected to predict the best but by creating a model whose
output is the combination of the output of models having different
structures.
• The reason is that in reality any chosen hypothesis ˆf(·, αN ) is only an
estimate of the real target and, like any estimate, is affected by a bias
and a variance term.
• Theoretical results on the combination of estimators show that the
combination of unbiased estimators leads an unbiased estimator with
reduced variance.
• This principle is at the basis of approaches like bagging or boosting.
Machine Learning Strategies for Prediction – p. 71/128
72. Feature selection problem
• Machine learning algorithms are known to degrade in performance
(prediction accuracy) when faced with many inputs (aka features) that
are not necessary for predicting the desired output.
• In the feature selection problem, a learning algorithm is faced with the
problem of selecting some subset of features upon which to focus its
attention, while ignoring the rest.
• Using all available features may negatively affect generalization
performance, especially in the presence of irrelevant or redundant
features.
• Feature selection can be seen as an instance of model selection
problem.
Machine Learning Strategies for Prediction – p. 72/128
73. Benefits and drawbacks of feature selection
There are many potential benefits of feature selection:
• facilitating data visualization and data understanding,
• reducing the measurement and storage requirements,
• reducing training and utilization times of the final model,
• defying the curse of dimensionality to improve prediction performance.
Drawbacks are
• the search for a subset of relevant features introduces an additional layer
of complexity in the modelling task. The search in the model hypothesis
space is augmented by another dimension: the one of finding the
optimal subset of relevant features.
• additional time for learning.
Machine Learning Strategies for Prediction – p. 73/128
74. Curse of dimensionality
• The error of the best model decreases with n but the mean integrated
squared error of models increases faster than linearly in n.
• In high dimensions, all data sets are sparse.
• In high dimensions, the number of possible models to consider
increases superexponenetially in n.
• In high dimensions, all datasets show multicollinearity.
• As n increases the amount of local data goes to zero.
• For a uniform distribution around a query point xq the amount of data
that are contained in a ball of radius r < 1 centered in xq grows like rn
.
Machine Learning Strategies for Prediction – p. 74/128
75. Local density for large n
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
r
fractionoflocal
points
n=1
n=2
n=3
n=100
The size of the neighborhood on which we can estimate local features of the
output (e.g. E[y|x]) increases with dimension n, making the estimation
coarser and coarser.
Machine Learning Strategies for Prediction – p. 75/128
76. Methods of feature selection
Two are the main approaches to feature selection:
Filter methods: they are preprocessing methods. They attempt to assess the
merits of features from the data, ignoring the effects of the selected
feature subset on the performance of the learning algorithm. Examples
are methods that select variables by ranking them through compression
techniques (like PCA or clustering) or by computing correlation with the
output.
Wrapper methods: these methods assess subsets of variables according to
their usefulness to a given predictor. The method conducts a search for
a good subset using the learning algorithm itself as part of the evaluation
function. The problem boils down to a problem of stochastic state space
search. Example are the stepwise methods proposed in linear
regression analysis.
Embedded methods: they perform variable selection as part of the learning
procedure and are usually specific to given learning machines.
Examples are classification trees, random forests, and methods based
on regularization techniques (e.g. lasso)
Machine Learning Strategies for Prediction – p. 76/128
78. Local modeling procedure
The learning of a local model in xq ∈ Rn
can be summarized in these steps:
1. Compute the distance between the query xq and the training samples
according to a predefined metric.
2. Rank the neighbors on the basis of their distance to the query.
3. Select a subset of the k nearest neighbors according to the bandwidth
which measures the size of the neighborhood.
4. Fit a local model (e.g. constant, linear,...).
Each of the local approaches has one or more structural (or smoothing)
parameters that control the amount of smoothing performed.
Let us focus on the bandwidth selection.
Machine Learning Strategies for Prediction – p. 78/128
79. The bandwidth trade-off: overfit
e
q
0011
0011
0011
01
01
0011
01
0011
0011
00
00
11
11 0
0
1
1
0011
0011
00001111
01
00001111
0011
01
0011
00001111
0011
0011
000000000000000000000000000000000000000000011111111111111111111111111111111111111111110
00
00
0
00
00
000000000000000000000
1
11
11
1
11
11
111111111111111111111
x
y
0011
0011
0011
01
01
0011
01
0011
0011
00
00
11
11 0
0
1
1
0011
000
111
00001111
01
00001111
0011
01
0011
00001111
0011
000000
111111
0011
000000000000000000000000000000000000000000011111111111111111111111111111111111111111110
00
00
0
00
00
000000000000000000000
1
11
11
1
11
11
111111111111111111111
x
y
Too narrow bandwidth ⇒ overfitting ⇒ large prediction error e.
In terms of bias/variance trade-off, this is typically a situation of high variance.
Machine Learning Strategies for Prediction – p. 79/128
80. The bandwidth trade-off: underfit
e
q
0011
0011
0011
01
01
0011
01
0011
0011
00
00
11
11 0
0
1
1
0011
00001111
01
00001111
0011
01
0011
00001111
0011
0011
000000000000000000000000000000000000000000011111111111111111111111111111111111111111110
00
00
0
00
00
000000000000000000000
1
11
11
1
11
11
111111111111111111111
x
y
0011
0011
0011
01
01
0011
000000
111111
0
00
1
11 00
0000
11
1111
0011
000
111
00001111
000
111
01
001100001111
00110011
000000
111111 0011
00001111
000000
111111
0011
000
111
0
00
1
11
000000
111111
00001111
000000000000000000000000000000000000000000011111111111111111111111111111111111111111110
00
00
0
00
00
000000000000000000000
1
11
11
1
11
11
111111111111111111111
x
y
Too large bandwidth ⇒ underfitting ⇒ large prediction error e
In terms of bias/variance trade-off, this is typically a situation of high bias.
Machine Learning Strategies for Prediction – p. 80/128
81. Bandwidth and bias/variance trade-off
Mean Squared Error
1/Bandwith
FEW NEIGHBORSMANY NEIGHBORS
Bias
Variance
Underfitting Overfitting
Machine Learning Strategies for Prediction – p. 81/128
82. The PRESS statistic
• Cross-validation can provide a reliable estimate of the algorithm
generalization error but it requires the training process to be repeated K
times, which sometimes means a large computational effort.
• In the case of linear models there exists a powerful statistical procedure
to compute the leave-one-out cross-validation measure at a reduced
computational cost
• It is the PRESS (Prediction Sum of Squares) statistic, a simple formula
which returns the leave-one-out (l-o-o) as a by-product of the
least-squares.
Machine Learning Strategies for Prediction – p. 82/128
83. Leave-one-out for linear models
PARAMETRIC IDENTIFICATION ON N-1 SAMPLES
PUT THE j-th SAMPLE ASIDE
TEST ON THE j-th SAMPLE
PARAMETRIC IDENTIFICATION
ON N SAMPLES
N TIMES
TRAINING SET
PRESS STATISTIC
LEAVE-ONE-OUT
The leave-one-out error can be computed in two equivalent ways: the slowest
way (on the right) which repeats N times the training and the test procedure;
the fastest way (on the left) which performs only once the parametric
identification and the computation of the PRESS statistic.
Machine Learning Strategies for Prediction – p. 83/128
84. The PRESS statistic
• This allows a fast cross-validation without repeating N times the
leave-one-out procedure. The PRESS procedure can be described as
follows:
1. we use the whole training set to estimate the linear regression
coefficients
ˆα = (XT
X)−1
XT
Y
2. This procedure is performed only once on the N samples and
returns as by product the Hat matrix
H = X(XT
X)−1
XT
3. we compute the residual vector e, whose jth
term is ej = yj − xT
j ˆα,
4. we use the PRESS statistic to compute eloo
j as
eloo
j =
ej
1 − Hjj
where Hjj is the jth
diagonal term of the matrix H.
Machine Learning Strategies for Prediction – p. 84/128
85. The PRESS statistic
Thus, the leave-one-out estimate of the local mean integrated squared error
is:
MISELOO =
1
N
N
i=1
yi − ˆyi
1 − Hii
2
Note that PRESS is not an approximation of the loo error but simply a faster
way of computing it.
Machine Learning Strategies for Prediction – p. 85/128
86. Selection of the number of neighbours
• For a given query point xq, we can compute a set of predictions
ˆyq(k) = xT
q ˆα(k)
, together with a set of associated leave-one-out error vectors
MISELOO(k) for a number of neighbors ranging in [kmin, kmax].
• If the selection paradigm, frequently called winner-takes-all, is adopted,
the most natural way to extract a final prediction ˆyq, consists in
comparing the prediction obtained for each value of k on the basis of the
classical mean square error criterion:
ˆyq = xT
q ˆα(ˆk), with ˆk = arg min
k
MISELOO(k)
Machine Learning Strategies for Prediction – p. 86/128
87. Local Model combination
• As an alternative to the winner-takes-all paradigm, we can use a
combination of estimates.
• The final prediction of the value yq is obtained as a weighted average of
the best b models, where b is a parameter of the algorithm.
• Suppose the predictions ˆyq(k) and the loo errors MISELOO(k) have been
ordered creating a sequence of integers {ki} so that
MISELOO(ki) ≤ MISELOO(kj), ∀i < j. The prediction of ˆyq is given by
ˆyq =
b
i=1 ζi ˆyq(ki)
b
i=1 ζi
,
where the weights are the inverse of the mean square errors:
ζi = 1/MISELOO(ki).
Machine Learning Strategies for Prediction – p. 87/128
89. One step-ahead and iterated prediction
• Once a model of the embedding mapping is available, it can be used for
two objectives: one-step-ahead prediction and iterated prediction.
• In one-step-ahead prediction, the n previous values of the series are
available and the forecasting problem can be cast in the form of a
generic regression problem
• In literature a number of supervised learning approaches have been
used with success to perform one-step-ahead forecasting on the basis
of historical data.
Machine Learning Strategies for Prediction – p. 89/128
90. One step-ahead prediction
f
ϕt-2
z-1
z-1
z-1
ϕt-3
ϕt-n
ϕt-1
z-1
ϕt
The approximator ˆf returns the prediction of the value of the time series at
time t + 1 as a function of the n previous values (the rectangular box
containing z−1
represents a unit delay operator, i.e., ϕt−1
= z−1
ϕt
).
Machine Learning Strategies for Prediction – p. 90/128
91. Nearest-neighbor one-step-ahead forecasts
t
−−
y
t−11t−16 t−1t−6
t
We want to predict at time ¯t − 1 the next value of the series y of order n = 6.
The pattern y¯t−16, y¯t−15, . . . , y¯t−11 is the most similar to the pattern
{y¯t−6, y¯t−5, . . . , yˆt−1}. Then, the prediction ˆy¯t = y¯t−10 is returned.
Machine Learning Strategies for Prediction – p. 91/128
92. Multi-step ahead prediction
• The prediction of the value of a time series H > 1 steps ahead is called
H-step-ahead prediction.
• We classify the methods for H-step-ahead prediction according to two
features: the horizon of the training criterion and the single-output or
multi-output nature of the predictor.
Machine Learning Strategies for Prediction – p. 92/128
93. Multi-step ahead prediction strategies
The most common strategies are
1. Iterated: the model predicts H steps ahead by iterating a
one-step-ahead predictor whose parameters are optimized to minimize
the training error on one-step-ahead forecast (one-step-ahead training
criterion).
2. Iterated strategy where parameters are optimized to minimize the
training error on the iterated htr-step-ahead forecast (htr-step-ahead
training criterion) where 1 < htr ≤ H.
3. Direct: the model makes a direct forecast at time t + h − 1, h = 1, . . . , H
by modeling the time series in a multi-input single-output form
4. Direc: direct forecast but the input vector is extended at each step with
predicted values.
5. MIMO: the model returns a vectorial forecast by modeling the time
series in a multi-input multi-output form
Machine Learning Strategies for Prediction – p. 93/128
94. Iterated (or recursive) prediction
• In the case of iterated prediction, the predicted output is fed back as
input for the next prediction.
• Here, the inputs consist of predicted values as opposed to actual
observations of the original time series.
• As the feedback values are typically distorted by the errors made by the
predictor in previous steps, the iterative procedure may produce
undesired effects of accumulation of the error.
• Low performance is expected in long horizon tasks. This is due to the
fact that they are essentially models tuned with a one-step-ahead
criterion which is not capable of taking temporal behavior into account.
Machine Learning Strategies for Prediction – p. 94/128
95. Iterated prediction
f
ϕt-2
z-1
z-1
z-1
z-1
ϕt-3
ϕt-n
ϕt-1
z-1
ϕt
The approximator ˆf returns the prediction of the value of the time series at
time t + 1 by iterating the predictions obtained in the previous steps (the
rectangular box containing z−1
represents a unit delay operator, i.e.,
ˆϕt−1
= z−1
ˆϕt
).
Machine Learning Strategies for Prediction – p. 95/128
96. Iterated with h-step training criterion
• This strategy adopts one-step-ahead predictors but adapts the model
selection criterion in order to take into account the multi-step-ahead
objective.
• Methods like Recurrent Neural Networks belong to such class. Their
recurrent architecture and the associated training algorithm (temporal
backpropagation) are suitable to handle the time-dependent nature of
the data.
• In [4] we proposed an adaptation of the Lazy Learning algorithm where
the number of neighbors is optimized in order to minimize the
leave-one-out error over an horizon larger than one. This technique
ranked second in the 1998 KULeuven Time Series competition.
• A similar technique has been proposed by [14] who won the competition.
Machine Learning Strategies for Prediction – p. 96/128
97. Conventional and iterated leave-one-out
a)
3
1
2
4
5
3
e (3)
cv
1
2
3
4
5
e (3)
b)
it
1
2
4
5
3
1
2 3
4
5
3
Machine Learning Strategies for Prediction – p. 97/128
98. Santa Fe time series A
0 200 400 600 800 1000
050100150200250
Santa Fe time series A
t
y
The A chaotic time series has a training set of 1000 values: the task is to
predict the continuation for 100 steps, starting from different points.
Machine Learning Strategies for Prediction – p. 98/128
101. Direct strategy
• The Direct strategy [22, 17, 7] learns independently H models fh
ϕt+h−1 = fh(ϕt−1, . . . , ϕt−n) + wt+h−1
with h ∈ {1, . . . , H} and returns a multi-step forecast by concatenating
the H predictions.
• Several machine learning models have been used to implement the
Direct strategy for multi-step forecasting tasks, for instance neural
networks [10], nearest neighbors [17] and decision trees [21].
• Since the Direct strategy does not use any approximated values to
compute the forecasts, it is not prone to any accumulation of errors,
since each model is tailored for the horizon it is supposed to predict.
• Notwithstanding, it has some weaknesses.
Machine Learning Strategies for Prediction – p. 101/128
102. Direct strategy limitations
• Since the H models are learned independently no statistical
dependencies between the predictions ˆϕt+h−1, h = 1, . . . , H [3, 5, 10] is
guaranteed.
• Direct methods often require higher functional complexity [20] than
iterated ones in order to model the stochastic dependency between two
series values at two distant instants [9].
• This strategy demands a large computational time since the number of
models to learn is equal to the size of the horizon.
Machine Learning Strategies for Prediction – p. 102/128
103. DirRec strategy
• The DirRec strategy [18] combines the architectures and the principles
underlying the Direct and the Recursive strategies.
• DirRec computes the forecasts with different models for every
horizon (like the Direct strategy) and, at each time step, it enlarges the
set of inputs by adding variables corresponding to the forecasts of the
previous step (like the Recursive strategy).
• Unlike the previous strategies, the embedding size n is not the same for
all the horizons. In other terms, the DirRec strategy learns H models fh
from the time series where
ϕt+h−1 = fh(ϕt+h−1, . . . , ϕt−n) + wt+h−1
with h ∈ {1, . . . , H}.
• The technique is prone to the curse of dimensionality. The use of feature
selection is recommended for large h.
Machine Learning Strategies for Prediction – p. 103/128
104. MIMO strategy
• This strategy [3, 5] (also known as Joint strategy [10]) avoids the
simplistic assumption of conditional independence between future
values made by the Direct strategy [3, 5] by learning a single
multiple-output model
[ϕt+H−1, . . . , ϕt] = F(ϕt−1, . . . , ϕt−n) + w
where F : Rd
→ RH
is a vector-valued function [15], and w ∈ RH
is a
noise vector with a covariance that is not necessarily diagonal [13].
• The forecasts are returned in one step by a multiple-input
multiple-output regression model.
• In [5] we proposed a multi-output extension of the local learning
algorithm.
• Other multi-output regression model could be taken into consideration
like multi-output neural networks or partial least squares.
Machine Learning Strategies for Prediction – p. 104/128
105. Time series dependencies
t+3
ϕ ϕ
ϕ ϕ
ϕ
t−1 t
t+1 t+2
n = 2 NAR dependency ϕt = f(ϕt−1, ϕt−2) + w(t).
Machine Learning Strategies for Prediction – p. 105/128
106. Iterated modeling of dependencies
t+3
ϕ ϕ
ϕ
t+1ϕ ϕt+2
tt−1
Machine Learning Strategies for Prediction – p. 106/128
107. Direct modeling of dependencies
t+3
ϕ ϕ
ϕ ϕ
ϕ
t−1 t
t+1 t+2
Machine Learning Strategies for Prediction – p. 107/128
108. MIMO strategy
• The rationale of the MIMO strategy is to model, between the predicted
values, the stochastic dependency characterizing the time series. This
strategy avoids the conditional independence assumption made by the
Direct strategy as well as the accumulation of errors which plagues the
Recursive strategy.
• So far, this strategy has been successfully applied to several real-world
multi-step time series forecasting tasks [3, 5, 19, 2].
• However, the wish to preserve the stochastic dependencies constrains
all the horizons to be forecasted with the same model structure. Since
this constraint could reduce the flexibility of the forecasting
approach [19], a variant of the MIMO strategy (called DIRMO) has been
proposed in [19, 2] .
• Extensive validation on the 111 times series of the NN5 competition
showed that MIMO are invariably better than single-output approaches.
Machine Learning Strategies for Prediction – p. 108/128
109. Validation of time series methods
• The huge variety of strategies and algorithms that can be used to infer a
predictor from observed data asks for a rigorous procedure of
comparison and assessment.
• Assessment demands benchmarks and benchmarking procedure.
• Benchmarks can be defined by using
• Simulated data obtained by simulating AR, NAR and other stochastic
processes. This is particular useful for validating theoretical
properties in terms of bias/variance.
• Public domain benchmarks, like the one provided by Time Series
Competitions.
• Real measured data
Machine Learning Strategies for Prediction – p. 109/128
110. Competitions
• Santa Fe Time Series Prediction and Analysis Competition (1994) [22]:
• International Workshop on Advanced Black-box techniques for nonlinear
modeling Competition (Leuven, Belgium; 1998)
• NN3 competition [8]: 111 monthly time series drawn from homogeneous
population of empirical business time series.
• NN5 competition [1]: 111 time series of the daily retirement amounts
from independent cash machines at different, randomly selected
locations across England.
• Kaggle competition.
Machine Learning Strategies for Prediction – p. 110/128
111. Accuracy measures
Let
et+h = ϕt+h − ˆϕt+h
represent the error of the forecast ˆϕt+h at the horizon h = 1, . . . , H. A
conventional measure of accuracy is the Normalized Mean Squared Error
NMSE =
H
h=1(ϕt+h − ˆϕt+h)2
H
h=1(ϕt+h − ˆµ)2
This quantity is smaller than one if the predictor performs better than the
naivest predictor, i.e. the average ˆµ.
Other measures rely on relative or percentage error
pet+h = 100
ϕt+h − ˆϕt+h
ϕt+h
like
MAPE =
H
h=1 |pet+h|
H
Machine Learning Strategies for Prediction – p. 111/128
112. Applications in my lab
Machine Learning Strategies for Prediction – p. 112/128
113. MLG projects on forecasting
• Low-energy streaming of wireless sensor data
• Decision support in anesthesia
• Side-channel attack
Machine Learning Strategies for Prediction – p. 113/128
114. Low-energy streaming of wireless sensor data
• In monitoring applications, only an approximation to sensor readings is
sufficient (e.g. ±0.5C, ±2% humidity, ..). In this context a Dual Prediction
Scheme is effective.
• A sensor node is provided with a time series prediction model
ϕt = f(ϕt−1, α) (e.g. autoregressive models) and a learning method for
identifying the best set of parameters α (e.g. recursive least squares).
• The sensor node then sends the parameters of the model instead of the
actual data to the recipient. The recipient node then runs the models to
reconstruct an approximation of the data stream collected on the distant
node.
• The sensor node also runs the prediction model. When its predictions
diverges by more than ±ǫ from the actual reading, a new model is sent
to the recipient.
• This allows to reduce the communication effort if an appropriate model is
run by the sensor node.
• Since the cost of transmission is much larger than the cost of
computation, in realistic situations this scheme allows economy of power
consumption. Machine Learning Strategies for Prediction – p. 114/128
115. Low-energy streaming of wireless sensor data
0 5 10 15 20
202530354045
Accuracy: 2°C
Constant model
Time (Hour)
Temperature(°C)
G GG GGG GG G G GG G G G G G G G G G
Machine Learning Strategies for Prediction – p. 115/128
116. Adaptive model selection
Tradeoff: more complex models predict better measurements but have a
higher number of parameters
Model complexity
Metric
Communication costs
Model error
AR(p) : ˆsi[t] =
p
j=1
θjsi[t − j]
Machine Learning Strategies for Prediction – p. 116/128
117. Adaptive Model Selection
We proposed an Adaptive Model Selection strategy [11] that
• takes into account the cost of sending model parameters,
• allows sensor nodes to determine autonomously the model which best
fits their measurements,
• provides a statistically sound selection mechanism to discard poorly
performing models,
• gave in experimental results about 45% of communication savings on
average,
• was implemented in TinyOS, the reference operating system.
Machine Learning Strategies for Prediction – p. 117/128
118. Predictive modeling in anesthesiology
• During surgery, anesthesiologists controls the depth of anesthesia by
means of types of drugs
• Anesthesiologists observe the patient state by observing
unconsciousness signal in real-time which are collected by monitors
connected via electrodes to the patient’s forehead
• The bispectral index BIS monitor (by Aspect Medical Systems) is a
single dimensionless number between 0 to 100 where 0 equals EEG
silence, 100 is the expected value for a fully awake adult, and between
40 and 60 indicates a recommended level.
• It remains difficult for the anesthesiologist, especially if inexperienced, to
predict how the BIS signal could vary after a change in the administered
anesthetic agents. This is generally due to inter-individual variability
problem
• We designed a ML system [6] to predict multi-step-ahead the evolution
of the BIS on the basis of historical data.
Machine Learning Strategies for Prediction – p. 118/128
119. Predictive modeling in anesthesiology
Machine Learning Strategies for Prediction – p. 119/128
121. Side channel attack
• In cryptography, a side channel attack is any attack based on the
analysis of measurements related to the physical implementation of a
cryptosystem.
• Side channel attacks take advantage of the fact that instantaneous
power consumption, encryption time or/and electromagnetic leaks of a
cryptographic device depend on the processed data and the performed
operations.
• Power analysis attacks are an instance of side-channel attacks which
assume that different encryption keys imply a different power
consumptions.
• Nowadays, the possibility of collecting a large amount of power traces
(i.e. time series) paves the way to the adoption of machine learning
techniques.
• Effective side channel attacks enables effective countermeasures.
Machine Learning Strategies for Prediction – p. 121/128
122. SCA and time series classification
• The power consumption of a crypto device using a secret key
Q ∈ {0, 1}k
(size k = 8) can be modeled as a time series T(Q)
of order n
T
(Q)
(t+1) = f(T
(Q)
(t) , T
(Q)
(t−1), ..., T
(Q)
(t−n)) + ǫ
• For each key Qj we infer a predictive model ˆf [12] such that
T
(Qj )
(t+1) = ˆfQj (T
(Qj )
(t) , T
(Qj )
(t−1), ..., T
(Qj)
(t−n)) + ǫ
• These models can be used to classify an unlabeled time series T and
predict the associated key by computing a distance for each Qj
D (Qj, T) =
1
N − n + 1
N
t=n
ˆfQj T(t−1), T(t−2), ..., T(t−n−1) − T(t)
2
and choosing the key minimizing it
ˆQ = arg min
j∈[0,2(k−1)]
D (Qj, T)
Machine Learning Strategies for Prediction – p. 122/128
124. Open-source software
Many commercial solutions exist but only open-source software can cope with
• fast integration of new algorithms
• portability over several platforms
• new paradigms of data storage (e.g. Hadoop)
• integration with different data formats and architectures
A de-facto standard in computational statistics, machine learning,
bioinformatics, geostatistics or more general analytics is nowadays
Highly recommended !
Machine Learning Strategies for Prediction – p. 124/128
125. All that we didn’t have time to discuss
• ARIMA models
• GARCH models
• Frequence space representation
• Nonstationarity
• Vectorial time series
• Spatio temporal time series
• Time series classification
Machine Learning Strategies for Prediction – p. 125/128
126. Suggestions for PhD topics
• Time series and big data
• streaming data (environmental data)
• large vectorial time series (weather data)
• Spatio-temporal time series and graphical models
• Beyond cross-validation for model/input selection
• Long term forecasting (effective integration of iterated and directed
models)
• Causality and time-series
• Scalable machine learning
Suggestion: use methods and models to solve problems... not problems to
sanctify methods or models...
Machine Learning Strategies for Prediction – p. 126/128
127. Conclusion
• Popper claimed that, if a theory is falsifiable (i.e. it can be contradicted
by an observation or the outcome of a physical experiment.), then it is
scientific. Since prediction is the most falsifiable aspect of science it is
also the most scientific one.
• Effective machine learning is an extension of statistics, in no way an
alternative.
• Simplest (i.e. linear) model first.
• Local learning techniques represent an effective trade-off between
linearity and nonlinearity.
• Modelling is more an art than an automatic process... then experience
data analysts are more valuable than expensive tools.
• Expert knowledge matters..., data too
• Understanding what is predictable is as important as trying to predict it.
Machine Learning Strategies for Prediction – p. 127/128
128. Forecasting is difficult, especially the future
• "Computers in the future may weigh no more than 1.5 tons." –Popular
Mechanics, forecasting the relentless march of science, 1949
• "I think there is a world market for maybe five computers." –Chairman of
IBM, 1943
• "Stocks have reached what looks like a permanently high plateau."
–Professor of Economics, Yale University, 1929.
• "Airplanes are interesting toys but of no military value." –Professor of
Strategy, Ecole Superieure de Guerre.
• "Everything that can be invented has been invented." –Commissioner,
U.S. Office of Patents, 1899.
• "Louis Pasteur’s theory of germs is ridiculous fiction". –Professor of
Physiology at Toulouse, 1872
• "640K ought to be enough for anybody." – Bill Gates, 1981
Machine Learning Strategies for Prediction – p. 128/128
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