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University of California - Statistics 100 Homework 6




Question;University of California, Los Angeles;Department of Statistics;Statistics 100C Instructor: Nicolas Christou;Homework 6;Exercise 1;Please refer to homework 5, exercise 4.;a <- read.table(";restaurant.txt", header=TRUE);Consider the multiple regression model:costi=0+1foodi+2decoti+3seri+i, or in matrix form;y=X+, withE() =0andcov() =2I. Suppose we want to estimate the vector= (0, 1, 2, 3)0;subject to the following three constraints;0+ 51+ 202+3= 0;0+1+2+3=????25;0????31????2+3=????40;Answer the following questions;a. Find the constrained least squares estimate of the vector. The set of the three constraint above can;be expressed in matrix form asC????d=0. Please show all the steps.;b. UseRto nd the constrained least squares estimate of the vector. To check your answer, the vector;^must satisfy the three constraints.;Exercise 2;An exercise on centering and scaling. Access the data inRas follows;a names(a);[1] "x" "y" "Landuse" "Rock" "Cd;[6] "Co" "Cr" "Cu" "Ni" "Pb;[11] "Zn;The variablesx, yare the coordinates. Landuse and Rock represent type of land use (forest, pasture, meadow;tillage) and rock type (Argovian, Kimmeridgian, Sequanina, Portlandian, and Quaternary). The other vari-;ables are concerntrations in ppm of the following chemical elements;Cd: Cadmium;Co: Cobalt;Cr: Chromium;Cu: Copper;Ni: Nickel;Pb: Lead;Zn: Zinc;UseCdas your response variable. UseCo,Cr,Cu,Ni,Pb,Znas your predictors to answer the following;questions.;a. Run the multiple regression ofCdonCo,Cr,Cu,Ni,Pb,Zn. Print the R output.;b. Center and scale the predictors and run the multiple regression ofCdon the centered and scaled;versions ofCo,Cr,Cu,Ni,Pb,Zn.;c. Compute the variance covariance matrix of ^of the model of question (a).;d. Using the variance covariance matrix of ^nd the variance covariance matrix of ^of the model of;question (b).;e. Compute the correlation matrix of the predictors.;f. ComputeV IF1(corresponds to Co).


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