Principal Components Analysis

Chapter 14 Within-Subjects Designs - CMU Statistics

Chapter 14 Within-Subjects Designs ... although often the term repeated measures analysis is used in a narrower sense to indicate the speci c set of analyses discussed

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Chapter 9 Simple Linear Regression

Chapter 9 Simple Linear Regression An analysis appropriate for a quantitative outcome and a single quantitative ex-planatory variable. 9.1 …

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Lecture Notes 9 Asymptotic Theory (Chapter 9)

Lecture Notes 9 Asymptotic Theory (Chapter 9) In these notes we look at the large sample properties of estimators, especially the maxi-mum likelihood estimator.

2 Probability Theory and Classical Statistics

2 Probability Theory and Classical Statistics Statistical inference rests on probability theory, and so an in-depth under-standing of the basics of probability theory is necessary for acquiring a con-

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Ryan Tibshirani Data Mining: 36-462/36-662 January 22 2013

Ryan Tibshirani Data Mining: 36-462/36-662 January 22 2013 Optional reading: ESL 14.10 1. Information retrieval with the web Last time:information retrieval, learned how to compute similarity scores (distances) of documents to a given query string But what if …

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Ryan Tibshirani Data Mining: 36-462/36-662 April 25 2013

Boosting Boosting1 is similar to bagging in that we combine the results of several classi cation trees. However, boosting does something fundamentally di erent, and can work a lot better As usual, we start with training data (x

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Chapter 8 Threats to Your Experiment - CMU Statistics

This chapter discusses possible complaints about internal validity, external validity, construct validity, Type 1 error, and power. We are using \threats" to mean things that will reduce the impact of

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Advanced Data Analysis from an Elementary Point of View

Advanced Data Analysis from an Elementary Point of View Cosma Rohilla Shalizi

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Finding Informative Features - CMU Statistics

Similarly, our uncertainty about the class C, in the absence of any other information, is just the entropy of C: H[C] = X c Pr(C= c)log 2 Pr(C= c) Now suppose we observe the value of the feature X.

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Degrees of Freedom and Model Search - CMU Statistics

Degrees of Freedom and Model Search Ryan J. Tibshirani Abstract Degrees of freedom is a fundamental concept in statistical modeling, as it provides a quan-titative description of the amount of tting performed by a given procedure. But, despite this

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Kalman Filtering Tutorial

and is a symmetric n by n matrix and is positive definite unless there is a linear dependence among the components of x. The (i,j) th element of P xx is sx x i j 2 Interpreting a covariance matrix: diagonal elements are the variances, off-diagonal encode correlations.

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MATLAB Basic Functions Reference - MathWorks

zeros(m,n) Create m x n matrix of zeros ones(m,n) Create m x n matrix of ones eye(n) Create a n x n identity matrix A=diag(x) Create diagonal matrix from vector x=diag(A) Get diagonal elements of matrix meshgrid(x,y) Create 2D and 3D grids rand(m,n), randi Create uniformly distributed random numbers or integers

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Mathematics for Machine Learning - GitHub Pages

E[X] expected value of random variable X Var(X) variance of random variable X Cov(X;Y) covariance of random variables Xand Y Other notes: Vectors and matrices are in bold (e.g. x;A). This is true for vectors in Rn as well as for vectors in general vector spaces. We generally use Greek letters for scalars and capital Roman

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Gaussian Random Vectors - Math

Gaussian Random Vectors 1. The multivariate normal distribution Let X:= (X1 ￿￿￿￿￿X￿)￿ be a random vector. We say that X is a Gaussian random vector if we can write X = µ +AZ￿ where µ ∈ R￿, A is an ￿ × ￿ matrix and Z:= (Z1 ￿￿￿￿￿Z￿)￿ is a ￿-vector of i.i.d. standard normal random variables. Proposition 1.

  Matrix, Vector, Random, Random vectors

Negative Binomial Regression Models and Estimation …

where, exp 0 i is defined as a random intercept; 0 1 exp K iijj j x is the log-link between the Poisson mean and the covariates or independent variables xs; and s are the regression coefficients. As discussed in Appendix C, the relationship can also be formulated using vectors, such that exp(x'β) i i.

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Lecture 7 — Spectral methods 7.1 Linear algebra review

Suppose random vector X ∈Rd has mean µ and covariance matrix M. Then zTMz represents the variance of X in direction z: var(zTX) = E[(zT(X −µ))2] = E[zT(X −µ)(X −µ)Tz] = zTMz. Theorem 7 tells us that the direction of maximum variance is u1, and that of minimum variance is ud. Continuing with this example, suppose that we are ...

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3. The Multivariate Normal Distribution

(a) nd the mean and variance of the linear combination a0X 1 of the three components of X 1 where a= [a 1 a 2 a 3]0. (b)Consider two linear combinations of random vectors 1 2 X 1 + 1 2 X 2 + 1 2 X 3 + 1 2 X 4 and X 1 + X 2 + X 3 3X 4: Find the mean vector and covariance matrix for each linear combination of vectors and also the covariance ...

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