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Dimitris N. Politis - Model-Free Prediction and Regression

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Dimitris N. Politis Model-Free Prediction and Regression
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The book is intended for those who want to learn how to use Azure Machine Learning. Perhaps you already know a bit about Machine Learning, but have never used ML Studio in Azure; or perhaps you are an absolute newbie. In either case, this book will get you up-and-running quickly.;Cover -- Copyright -- Credits -- About the Author -- Acknowledgments -- About the Reviewers -- www.PacktPub.com -- Table of Contents -- Preface -- Chapter 1: Introduction -- Introduction to predictive analytics -- Problem definition and scoping -- Data collection -- Data exploration and preparation -- Model development -- Model deployment -- Machine learning -- Kinds of machine learning problems -- Classification -- Regression -- Clustering -- Common machine learning techniques/algorithms -- Linear regression -- Logistic regression -- Decision tree-based ensemble models -- Neural networks and deep learning -- Introduction to Azure Machine Learning -- ML Studio -- Summary -- Chapter 2: ML Studio Inside Out -- Introduction to ML Studio -- Getting started with Microsoft Azure -- Microsoft account and subscription -- Creating and managing ML workspaces -- Inside ML Studio -- Experiments -- Creating and editing an experiment -- Running an experiment -- Creating and running an experiment -- do it yourself -- Workspace as a collaborative environment -- Summary -- Chapter 3: Data Exploration and Visualization -- The basic concepts -- The mean -- The median -- Standard deviation and variance -- Understanding a histogram -- The box and whiskers plot -- The outliers -- A scatter plot -- Data exploration in ML Studio -- Visualizing an automobile price dataset -- A histogram -- The box and whiskers plot -- Comparing features -- A snapshot -- Do it yourself -- Summary -- Chapter 4: Getting Data in and out of ML Studio -- Getting data in ML Studio -- Uploading data from a PC -- The Enter Data module -- The Data Reader module -- Getting data from the Web -- Getting data from Azure -- Data format conversion -- Getting data from ML Studio -- Saving dataset in a PC -- Saving results in ML Studio -- The Writer module -- Summary -- Chapter 5: Data Preparation.

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Part I
The Model-Free Prediction Principle
The Author 2015
Dimitris N. Politis Model-Free Prediction and Regression Frontiers in Probability and the Statistical Sciences 10.1007/978-3-319-21347-7_1
1. Prediction: Some Heuristic Notions
Dimitris N. Politis 1
(1)
Department of Mathematics, University of California, San Diego, La Jolla, CA, USA
1.1 To Explain or to Predict?
Statistics is the scientific discipline that enables us to draw inferences about the real world on the basis of observed data. Statistical inference comes in two general flavors:
A.
Explaining/modeling the world . Here the role of the statistician is much like the role of a natural scientist trying to find how the observed/observable quantity Y depends on another observed/observable quantity X . Natural scientists may have additional information at their disposal, e.g., physical laws of conservation, etc., but the statistician must typically answer this question solely on the basis of the data at hand. Nonetheless, insights provided by the science behind the data can help the statistician formulate a better model.
In a question asked this way, Y is called the response variable, and X is called a regressor or predictor variable. The data are often pairs: Model-Free Prediction and Regression - image 1 ( Y n , X n ), where Y i is the measured response associated with a regressor value given by X i . Figure a shows an example of a scatterplot associated with such a dataset.
The goal is to find a so-called regression function, say (), such that Y ( X ). The relation Y ( X ) is written as an approximation because either the association between X and Y is not exact and/or the observation of X and Y is corrupted by measurement error. The inexactness of the association and the possible measurement errors are combined in the discrepancy defined as Model-Free Prediction and Regression - image 2 ; the last equation can then be re-written as:
Model-Free Prediction and Regression - image 3
(1.1)
In the above, ( X ) is the part of Y that is explainable by X , and is an unexplainable, error term.
Is Eq.().
In addition to assumptions on the error term , typical model assumptions specify the allowed type for function (). This is done by specifying a family of functions, say Picture 4 and insisting that () must belong to If is finite-dimensional then it is called a parametric family A popular - photo 5 . If is finite-dimensional then it is called a parametric family A popular - photo 6 is finite-dimensional, then it is called a parametric family. A popular two-dimensional example corresponds to
12 which is the usual straight-line regression with slope 1 and intercept 0 - photo 7
(1.2)
which is the usual straight-line regression with slope 1 and intercept 0. If Picture 8 is not finite-dimensional, then it is called a nonparametric (sometimes also called infinite-parametric) family. For instance, Model-Free Prediction and Regression - image 9 could be the family of all functions that are (say) twice continuously differentiable over their support.
Under such model assumptions, the task of the statistician is to use the available data Model-Free Prediction and Regression - image 10 in order to (a) optimally estimate the function Picture 11 , and (b) to quantify the statistical/stochastic accuracy of the estimator.
Part (a) can be accomplished after formulating an appropriate optimality criterion; the oldest and most popular such criterion is Least Squares (LS) . The LS estimator of is the function say that minimizes the sum of squared errors among all - photo 12 is the function, say that minimizes the sum of squared errors among all If - photo 13 , that minimizes the sum of squared errors among all If happens to be the two-parameter family of straight-line - photo 14 among all Picture 15 If Picture 16 happens to be the two-parameter family of straight-line regression functions, then it is sufficient to obtain LS estimates, say Picture 17 and Picture 18 , of the intercept and slope 0 and 1, respectively. Under a correctly specified model, the LS estimators Picture 19 and Picture 20 have minimum variance among all unbiased estimators that are linear functions of the data.
To address part (b) before the 1980s statisticians often resorted to further restrictive model assumptions such as an (exact) Gaussian distribution for the errors i . Fortunately, the bootstrap and other computer-intensive methods have rendered such unrealistic/unverifiable assumptions obsolete; see, e.g., Efron ().
To fix ideas, consider a toy example involving n =20 patients taken from a group of people with borderline high blood pressure, i.e., (systolic) blood pressure of about 140. A drug for lowering blood pressure may be under consideration, and the question is to empirically see how (systolic) blood pressure Y corresponds to dosage X where the latter is measured as units of the drug taken daily. The Y responses were as follows: (145,148,133,137) for X =0; (140,132,137,128) for X =0.25; (123,131,118,125) for X =0.5; (115,118,120,126) for X =1; and (108,115,111,112) for X =2. Figure b shows the scatterplot of the 20 data pairs ( Y i , X i ), one for each patient, having superimposed both the LS straight-line regression function (with estimated intercept and slope equal to 136.5 and 13.7, respectively), as well as an estimated nonparametric regression function based on a smoothing spline, i.e., piecewise cubic function, under the assumption that the true function () is smooth, e.g., twice continuously differentiable.
Fig 11 a Systolic blood pressure vs daily dosage of a drug b Same - photo 21
Fig. 1.1
( a ) Systolic blood pressure vs. daily dosage of a drug. ( b ) Same data with superimposed straight-line regression as well as nonparametric regression based on a smoothing spline
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