DOCSLIB.ORG

Matrix Approach to Linear Regression

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Matrix Approach to Linear Regression Matrix Approach to Linear Regression Dr. Frank Wood Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 1 Random Vectors and Matrices • Let’s say we have a vector consisting of three random variables The expectation of a random vector is defined Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 2 Expectation of a Random Matrix • The expectation of a random matrix is defined similarly Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 3 Covariance Matrix of a Random Vector • The collection of variances and covariances of and between the elements of a random vector can be collection into a matrix called the covariance matrix remember so the covariance matrix is symmetric Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 4 Derivation of Covariance Matrix • In vector terms the covariance matrix is defined by because verify first entry Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 5 Regression Example • Take a regression example with n=3 with constant error terms σ {ǫi} = σ and are ≠ uncorrelated so that σ {ǫi, ǫj} = 0 for all i j • The covariance matrix for the random vector ǫǫǫ is which can be written as Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 6 Basic Results • If A is a constant matrix and Y is a random matrix then is a random matrix Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 7 Multivariate Normal Density • Let Y be a vector of p observations • Let µ be a vector of p means for each of the p observations Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 8 Multivariate Normal Density • Let § be the covariance matrix of Y • Then the multivariate normal density is given by Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 9 Example 2d Multivariate Normal Distribution 0.04 0.02 0 10 mvnpdf([0 0], [10 2;2 2]) 8 6 4 2 10 0 8 6 -2 4 -4 2 0 -6 -2 -4 y -8 -6 Run multivariate_normal_plots.m -8 -10 x -10 Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 10 Matrix Simple Linear Regression • Nothing new – only matrix formalism for previous results • Remember the normal error regression model • This implies Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 11 Regression Matrices • If we identify the following matrices • We can write the linear regression equations in a compact form Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 12 Regression Matrices • Of course, in the normal regression model the expected value of each of the ǫi’s is zero, we can write • This is because Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 13 Error Covariance • Because the error terms are independent and have constant variance σ Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 14 Matrix Normal Regression Model • In matrix terms the normal regression model can be written as where and Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 15 Least Squares Estimation • Starting from the normal equations you have derived we can see that these equations are equivalent to the following matrix operations with demonstrate this on board Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 16 Estimation • We can solve this equation (if the inverse of X’X exists) by the following and since we have Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 17 Least Squares Solution • The matrix normal equations can be derived directly from the minimization of w.r.t. to βββ Do this on board. Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 18 Fitted Values and Residuals • Let the vector of the fitted values be in matrix notation we then have Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 19 Hat Matrix – Puts hat on Y • We can also directly express the fitted values in terms of only the X and Y matrices and we can further define H, the “hat matrix” • The hat matrix plans an important role in diagnostics for regression analysis. write H on board Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 20 Hat Matrix Properties • The hat matrix is symmetric • The hat matrix is idempotent, i.e. demonstrate on board Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 21 Residuals • The residuals, like the fitted values of \hat{Y_i} can be expressed as linear combinations of the response variable observations Y i Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 22 Covariance of Residuals • Starting with we see that but which means that and since I-H is idempotent (check) we have we can plug in MSE for σ as an estimate Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 23 ANOVA • We can express the ANOVA results in matrix form as well, starting with where leaving J is matrix of all ones, do 3x3 example Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 24 SSE • Remember • We have derive this on board and this b • Simplified Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 25 SSR • It can be shown that – for instance, remember SSR = SSTO-SSE write these on board Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 26 Tests and Inference • The ANOVA tests and inferences we can perform are the same as before • Only the algebraic method of getting the quantities changes • Matrix notation is a writing short-cut, not a computational shortcut Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 27 Quadratic Forms • The ANOVA sums of squares can be shown to be quadratic forms. An example of a quadratic form is given by • Note that this can be expressed in matrix notation as (where A is a symmetric matrix) do on board Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 28 Quadratic Forms • In general, a quadratic form is defined by A is the matrix of the quadratic form. • The ANOVA sums SSTO, SSE, and SSR are all quadratic forms. Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 29 ANOVA quadratic forms • Consider the following rexpression of b’X’ • With this it is easy to see that Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 30 Inference • We can derive the sampling variance of the β vector estimator by remembering that where A is a constant matrix which yields Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 31 Variance of b • Since is symmetric we can write and thus Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 32 Variance of b • Of course this assumes that we know σ. If we don’t, we, as usual, replace it with the MSE. Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 33 Mean Response • To estimate the mean response we can create the following matrix • The fit (or prediction) is then since Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 34 Variance of Mean Response • Is given by and is arrived at in the same way as for the variance of \beta • Similarly the estimated variance in matrix notation is given by Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 35 Wrap-Up • Expectation and variance of random vector and matrices • Simple linear regression in matrix form • Next: multiple regression Frank Wood, [email protected] Linear Regression Models Lecture 11, Slide 36.
Load more

Recommended publications
  • LECTURES 2 - 3 : Stochastic Processes, Autocorrelation Function
    LECTURES 2 - 3 : Stochastic Processes, Autocorrelation function. Stationarity. Important points of Lecture 1: A time series fXtg is a series of observations taken sequentially over time: xt is an observation recorded at a specific time t. Characteristics of times series data: observations are dependent, become available at equally spaced time points and are time-ordered. This is a discrete time series. The purposes of time series analysis are to model and to predict or forecast future values of a series based on the history of that series. 2.2 Some descriptive techniques. (Based on [BD] x1.3 and x1.4) ......................................................................................... Take a step backwards: how do we describe a r.v. or a random vector? ² for a r.v. X: 2 d.f. FX (x) := P (X · x), mean ¹ = EX and variance σ = V ar(X). ² for a r.vector (X1;X2): joint d.f. FX1;X2 (x1; x2) := P (X1 · x1;X2 · x2), marginal d.f.FX1 (x1) := P (X1 · x1) ´ FX1;X2 (x1; 1) 2 2 mean vector (¹1; ¹2) = (EX1; EX2), variances σ1 = V ar(X1); σ2 = V ar(X2), and covariance Cov(X1;X2) = E(X1 ¡ ¹1)(X2 ¡ ¹2) ´ E(X1X2) ¡ ¹1¹2. Often we use correlation = normalized covariance: Cor(X1;X2) = Cov(X1;X2)=fσ1σ2g ......................................................................................... To describe a process X1;X2;::: we define (i) Def. Distribution function: (fi-di) d.f. Ft1:::tn (x1; : : : ; xn) = P (Xt1 · x1;:::;Xtn · xn); i.e. this is the joint d.f. for the vector (Xt1 ;:::;Xtn ). (ii) First- and Second-order moments. ² Mean: ¹X (t) = EXt 2 2 2 2 ² Variance: σX (t) = E(Xt ¡ ¹X (t)) ´ EXt ¡ ¹X (t) 1 ² Autocovariance function: γX (t; s) = Cov(Xt;Xs) = E[(Xt ¡ ¹X (t))(Xs ¡ ¹X (s))] ´ E(XtXs) ¡ ¹X (t)¹X (s) (Note: this is an infinite matrix).
    [Show full text]
  • Spatial Autocorrelation: Covariance and Semivariance Semivariance
    Spatial Autocorrelation: Covariance and Semivariancence Lily Housese ­P eters GEOG 593 November 10, 2009 Quantitative Terrain Descriptorsrs Covariance and Semivariogram areare numericc methods used to describe the character of the terrain (ex. Terrain roughness, irregularity) Terrain character has important implications for: 1. the type of sampling strategy chosen 2. estimating DTM accuracy (after sampling and reconstruction) Spatial Autocorrelationon The First Law of Geography ““ Everything is related to everything else, but near things are moo re related than distant things.” (Waldo Tobler) The value of a variable at one point in space is related to the value of that same variable in a nearby location Ex. Moranan ’s I, Gearyary ’s C, LISA Positive Spatial Autocorrelation (Neighbors are similar) Negative Spatial Autocorrelation (Neighbors are dissimilar) R(d) = correlation coefficient of all the points with horizontal interval (d) Covariance The degree of similarity between pairs of surface points The value of similarity is an indicator of the complexity of the terrain surface Smaller the similarity = more complex the terrain surface V = Variance calculated from all N points Cov (d) = Covariance of all points with horizontal interval d Z i = Height of point i M = average height of all points Z i+d = elevation of the point with an interval of d from i Semivariancee Expresses the degree of relationship between points on a surface Equal to half the variance of the differences between all possible points spaced a constant distance apart
    [Show full text]
  • Fast Estimation of the Median Covariation Matrix with Application to Online Robust Principal Components Analysis
    Fast Estimation of the Median Covariation Matrix with Application to Online Robust Principal Components Analysis Hervé Cardot, Antoine Godichon-Baggioni Institut de Mathématiques de Bourgogne, Université de Bourgogne Franche-Comté, 9, rue Alain Savary, 21078 Dijon, France July 12, 2016 Abstract The geometric median covariation matrix is a robust multivariate indicator of dis- persion which can be extended without any difficulty to functional data. We define estimators, based on recursive algorithms, that can be simply updated at each new observation and are able to deal rapidly with large samples of high dimensional data without being obliged to store all the data in memory. Asymptotic convergence prop- erties of the recursive algorithms are studied under weak conditions. The computation of the principal components can also be performed online and this approach can be useful for online outlier detection. A simulation study clearly shows that this robust indicator is a competitive alternative to minimum covariance determinant when the dimension of the data is small and robust principal components analysis based on projection pursuit and spherical projections for high dimension data. An illustration on a large sample and high dimensional dataset consisting of individual TV audiences measured at a minute scale over a period of 24 hours confirms the interest of consider- ing the robust principal components analysis based on the median covariation matrix. All studied algorithms are available in the R package Gmedian on CRAN. Keywords. Averaging, Functional data, Geometric median, Online algorithms, Online principal components, Recursive robust estimation, Stochastic gradient, Weiszfeld’s algo- arXiv:1504.02852v5 [math.ST] 9 Jul 2016 rithm.
    [Show full text]
  • Characteristics and Statistics of Digital Remote Sensing Imagery (1)
    Characteristics and statistics of digital remote sensing imagery (1) Digital Images: 1 Digital Image • With raster data structure, each image is treated as an array of values of the pixels. • Image data is organized as rows and columns (or lines and pixels) start from the upper left corner of the image. • Each pixel (picture element) is treated as a separate unite. Statistics of Digital Images Help: • Look at the frequency of occurrence of individual brightness values in the image displayed • View individual pixel brightness values at specific locations or within a geographic area; • Compute univariate descriptive statistics to determine if there are unusual anomalies in the image data; and • Compute multivariate statistics to determine the amount of between-band correlation (e.g., to identify redundancy). 2 Statistics of Digital Images It is necessary to calculate fundamental univariate and multivariate statistics of the multispectral remote sensor data. This involves identification and calculation of – maximum and minimum value –the range, mean, standard deviation – between-band variance-covariance matrix – correlation matrix, and – frequencies of brightness values The results of the above can be used to produce histograms. Such statistics provide information necessary for processing and analyzing remote sensing data. A “population” is an infinite or finite set of elements. A “sample” is a subset of the elements taken from a population used to make inferences about certain characteristics of the population. (e.g., training signatures) 3 Large samples drawn randomly from natural populations usually produce a symmetrical frequency distribution. Most values are clustered around the central value, and the frequency of occurrence declines away from this central point.
    [Show full text]
  • Covariance of Cross-Correlations: Towards Efficient Measures for Large-Scale Structure
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by RERO DOC Digital Library Mon. Not. R. Astron. Soc. 400, 851–865 (2009) doi:10.1111/j.1365-2966.2009.15490.x Covariance of cross-correlations: towards efficient measures for large-scale structure Robert E. Smith Institute for Theoretical Physics, University of Zurich, Zurich CH 8037, Switzerland Accepted 2009 August 4. Received 2009 July 17; in original form 2009 June 13 ABSTRACT We study the covariance of the cross-power spectrum of different tracers for the large-scale structure. We develop the counts-in-cells framework for the multitracer approach, and use this to derive expressions for the full non-Gaussian covariance matrix. We show that for the usual autopower statistic, besides the off-diagonal covariance generated through gravitational mode- coupling, the discreteness of the tracers and their associated sampling distribution can generate strong off-diagonal covariance, and that this becomes the dominant source of covariance as spatial frequencies become larger than the fundamental mode of the survey volume. On comparison with the derived expressions for the cross-power covariance, we show that the off-diagonal terms can be suppressed, if one cross-correlates a high tracer-density sample with a low one. Taking the effective estimator efficiency to be proportional to the signal-to-noise ratio (S/N), we show that, to probe clustering as a function of physical properties of the sample, i.e. cluster mass or galaxy luminosity, the cross-power approach can outperform the autopower one by factors of a few.
    [Show full text]
  • On the Use of the Autocorrelation and Covariance Methods for Feedforward Control of Transverse Angle and Position Jitter in Linear Particle Beam Accelerators*
    '^C 7 ON THE USE OF THE AUTOCORRELATION AND COVARIANCE METHODS FOR FEEDFORWARD CONTROL OF TRANSVERSE ANGLE AND POSITION JITTER IN LINEAR PARTICLE BEAM ACCELERATORS* Dean S. Ban- Advanced Photon Source, Argonne National Laboratory, 9700 S. Cass Ave., Argonne, IL 60439 ABSTRACT It is desired to design a predictive feedforward transverse jitter control system to control both angle and position jitter m pulsed linear accelerators. Such a system will increase the accuracy and bandwidth of correction over that of currently available feedback correction systems. Intrapulse correction is performed. An offline process actually "learns" the properties of the jitter, and uses these properties to apply correction to the beam. The correction weights calculated offline are downloaded to a real-time analog correction system between macropulses. Jitter data were taken at the Los Alamos National Laboratory (LANL) Ground Test Accelerator (GTA) telescope experiment at Argonne National Laboratory (ANL). The experiment consisted of the LANL telescope connected to the ANL ZGS proton source and linac. A simulation of the correction system using this data was shown to decrease the average rms jitter by a factor of two over that of a comparable standard feedback correction system. The system also improved the correction bandwidth. INTRODUCTION Figure 1 shows the standard setup for a feedforward transverse jitter control system. Note that one pickup #1 and two kickers are needed to correct beam position jitter, while two pickup #l's and one kicker are needed to correct beam trajectory-angle jitter. pickup #1 kicker pickup #2 Beam Fast loop Slow loop Processor Figure 1. Feedforward Transverse Jitter Control System It is assumed that the beam is fast enough to beat the correction signal (through the fast loop) to the kicker.
    [Show full text]
  • Lecture 4 Multivariate Normal Distribution and Multivariate CLT
    Lecture 4 Multivariate normal distribution and multivariate CLT. T We start with several simple observations. If X = (x1; : : : ; xk) is a k 1 random vector then its expectation is × T EX = (Ex1; : : : ; Exk) and its covariance matrix is Cov(X) = E(X EX)(X EX)T : − − Notice that a covariance matrix is always symmetric Cov(X)T = Cov(X) and nonnegative definite, i.e. for any k 1 vector a, × a T Cov(X)a = Ea T (X EX)(X EX)T a T = E a T (X EX) 2 0: − − j − j � We will often use that for any vector X its squared length can be written as X 2 = XT X: If we multiply a random k 1 vector X by a n k matrix A then the covariancej j of Y = AX is a n n matrix × × × Cov(Y ) = EA(X EX)(X EX)T AT = ACov(X)AT : − − T Multivariate normal distribution. Let us consider a k 1 vector g = (g1; : : : ; gk) of i.i.d. standard normal random variables. The covariance of g is,× obviously, a k k identity × matrix, Cov(g) = I: Given a n k matrix A, the covariance of Ag is a n n matrix × × � := Cov(Ag) = AIAT = AAT : Definition. The distribution of a vector Ag is called a (multivariate) normal distribution with covariance � and is denoted N(0; �): One can also shift this disrtibution, the distribution of Ag + a is called a normal distri­ bution with mean a and covariance � and is denoted N(a; �): There is one potential problem 23 with the above definition - we assume that the distribution depends only on covariance ma­ trix � and does not depend on the construction, i.e.
    [Show full text]
  • Lecture 2 Covariance Is the Statistical Measure That Indicates The
    Lecture 2 Covariance is the statistical measure that indicates the interactive risk of a security relative to others in a portfolio of securities. In other words, the way security returns vary with each other affects the overall risk of the portfolio. The covariance between two securities X and Y may be calculated using the following formula: Where: Covxy = Covariance between x and y. Rx = Return of security x. Ry = Return of security y Rx = Expected or mean return of security x. Ry = Expected or mean return of security y. N = Number of observations. Calculation of Covariance Year Rx Deviation R…3 Deviation Product of deviations Rx – Rx y Ry – Ry (Rx – Rx) (Ry – Ry) 1 10 -4 17 5 -20 2 12 -2 13 1 -2 3 16 2 10 -2 -4 4 18 4 8 -4 -16 = -42 / 4 = -10.5 The covariance is a measure of how returns of two securities move together. If the returns of the two securities move in the same direction consistently the covariance would be positive. If the returns of the two securities move in opposite direction consistently the covariance would be negative. If the movements of returns are independent of each other, covariance would be close to zero. Covariance is an absolute measure of interactive risk between two securities. To facilitate comparison, covariance can be standardized. Dividing the covariance between two securities by product of the standard deviation of each security gives such a standardised measure. This measure is called the coefficient of correlation. This may be expressed as: Where Rxy = Coefficient of correlation between x and y Covxy = Covariance between x and y.
    [Show full text]
  • Week 7: Multiple Regression
    Week 7: Multiple Regression Brandon Stewart1 Princeton October 24, 26, 2016 1These slides are heavily influenced by Matt Blackwell, Adam Glynn, Jens Hainmueller and Danny Hidalgo. Stewart (Princeton) Week 7: Multiple Regression October 24, 26, 2016 1 / 145 Where We've Been and Where We're Going... Last Week I regression with two variables I omitted variables, multicollinearity, interactions This Week I Monday: F matrix form of linear regression I Wednesday: F hypothesis tests Next Week I break! I then ::: regression in social science Long Run I probability ! inference ! regression Questions? Stewart (Princeton) Week 7: Multiple Regression October 24, 26, 2016 2 / 145 1 Matrix Algebra Refresher 2 OLS in matrix form 3 OLS inference in matrix form 4 Inference via the Bootstrap 5 Some Technical Details 6 Fun With Weights 7 Appendix 8 Testing Hypotheses about Individual Coefficients 9 Testing Linear Hypotheses: A Simple Case 10 Testing Joint Significance 11 Testing Linear Hypotheses: The General Case 12 Fun With(out) Weights Stewart (Princeton) Week 7: Multiple Regression October 24, 26, 2016 3 / 145 Why Matrices and Vectors? Here's one way to write the full multiple regression model: yi = β0 + xi1β1 + xi2β2 + ··· + xiK βK + ui Notation is going to get needlessly messy as we add variables Matrices are clean, but they are like a foreign language You need to build intuitions over a long period of time (and they will return in Soc504) Reminder of Parameter Interpretation: β1 is the effect of a one-unit change in xi1 conditional on all other xik . We are going to review the key points quite quickly just to refresh the basics.
    [Show full text]
  • Stat 111 Spring 2011 Week 3 Central Limit Theorem, Covariance, T Distribution Presentations 1. Prove the Central Limit Theorem I
    Stat 111 Spring 2011 Week 3 Central Limit Theorem, Covariance, t distribution Presentations 1. Prove The Central Limit Theorem in the case where X1;:::;Xn are i.i.d. random variables, all with mean 0, variance σ2, and MGF M (t). Define S = P X and p x n i Zn = Sn=(σ n). a) Find the MGF for Zn and show that it converges to the MGF for a standard Normal r.v. as n ! 1. That is, show that Zn converges in distribution to N(0; 1). b) The CLT implies that certain distributions will be approximately Normal for large values of certain parameters. Give a list of examples. c) Describe the Normal Quantile Plot for assessing the normality of a set of data values. Use this to demonstrate some of the approximations in part b. 2. The covariance between random variables X and Y is defined as Cov(X; Y ) = σxy = E((X − µx)(Y − µY )) = E(XY ) − E(X)E(Y ): a) Show that, for constants a1; a2; b1 and b2, Cov(a1+b1X; a2+b2Y ) = b1b2Cov(X; Y ). 2 2 b) Show that Var(X + Y ) = σx + σy + 2σxy; and that, more generally, n ! n ! 0 1 X X X Var Xi = Var(Xi) + 2 @ Cov(Xi;Xj)A : i=1 i=1 i<j What is V ar(X − Y )? c) The correlation between X and Y is defined as ρ = σxy : Use the fact that xy σxσy V ar(X +Y ) and V ar(X −Y ) are both non-negative to prove that −1 ≤ ρxy ≤ 1.
    [Show full text]
  • Linear Regression in Matrix Form
    Statistics 512: Applied Linear Models Topic 3 Topic Overview This topic will cover • thinking in terms of matrices • regression on multiple predictor variables • case study: CS majors • Text Example (KNNL 236) Chapter 5: Linear Regression in Matrix Form The SLR Model in Scalar Form iid 2 Yi = β0 + β1Xi + i where i ∼ N(0,σ ) Consider now writing an equation for each observation: Y1 = β0 + β1X1 + 1 Y2 = β0 + β1X2 + 2 . Yn = β0 + β1Xn + n TheSLRModelinMatrixForm Y1 β0 + β1X1 1 Y2 β0 + β1X2 2 . = . + . . . . Yn β0 + β1Xn n Y X 1 1 1 1 Y X 2 1 2 β0 2 . = . + . . . β1 . Yn 1 Xn n (I will try to use bold symbols for matrices. At first, I will also indicate the dimensions as a subscript to the symbol.) 1 • X is called the design matrix. • β is the vector of parameters • is the error vector • Y is the response vector The Design Matrix 1 X1 1 X2 Xn×2 = . . 1 Xn Vector of Parameters β0 β2×1 = β1 Vector of Error Terms 1 2 n×1 = . . n Vector of Responses Y1 Y2 Yn×1 = . . Yn Thus, Y = Xβ + Yn×1 = Xn×2β2×1 + n×1 2 Variance-Covariance Matrix In general, for any set of variables U1,U2,... ,Un,theirvariance-covariance matrix is defined to be 2 σ {U1} σ{U1,U2} ··· σ{U1,Un} . σ{U ,U } σ2{U } ... σ2{ } 2 1 2 . U = . . .. .. σ{Un−1,Un} 2 σ{Un,U1} ··· σ{Un,Un−1} σ {Un} 2 where σ {Ui} is the variance of Ui,andσ{Ui,Uj} is the covariance of Ui and Uj.
    [Show full text]
  • The Truth About Linear Regression
    14:51 Friday 18th January, 2013 Chapter 2 The Truth about Linear Regression We need to say some more about how linear regression, and especially about how it really works and how it can fail. Linear regression is important because 1. it’s a fairly straightforward technique which often works reasonably well for prediction; 2. it’s a simple foundation for some more sophisticated techniques; 3. it’s a standard method so people use it to communicate; and 4. it’s a standard method so people have come to confuse it with prediction and even with causal inference as such. We need to go over (1)–(3), and provide prophylaxis against (4). A very good resource on regression is Berk (2004). It omits technical details, but is superb on the high-level picture, and especially on what must be assumed in order to do certain things with regression, and what cannot be done under any assumption. 2.1 Optimal Linear Prediction: Multiple Variables We have a response variable Y and a p-dimensional vector of predictor variables or features X~ . To simplify the book-keeping, we’ll take these to be centered — we can al- ways un-center them later. We would like to predict Y using X~ . We saw last time that the best predictor we could use, at least in a mean-squared sense, is the conditional expectation, r (~x)=E Y X~ = ~x (2.1) | î ó Instead of using the optimal predictor r (~x), let’s try to predict as well as possible while using only a linear function of ~x, say ~x β.
    [Show full text]