Math 185 Final Project (
Problem 1
)
Due December 8
The baseball dataset consists of the statistics of 263 players in Major League Baseball in the season 1986. The dataset (hitters.csv) consist of 20 variables:
Variable
Description
AtBat
Number of times at bat in 1986
Hits
Number of hits in 1986
HmRun
Number of home runs in 1986
Runs
Number of runs in 1986
RBI
Number of runs batted in in 1986
Walks
Number of walks in 1986
Years
Number of years in major leagues
CAtBat
Number of times at bat during his career
CHits
Number of hits during his career
CHmRun
Number of home runs during his career
CRuns
Number of runs during his career
CRBI
Number of runs batted in during his career
CWalks
Number of walks during his career
League
A
A factor with levels (coded as 1) and (coded as 2) indicating
player’s league at the end of 1986
N
Division
A factor with levels (coded as 1) and (coded as 2) indicating
player’s division at the end of 1986
E
W
PutOuts
Number of put outs in 1986
Assists
Number of assists in 1986
Errors
Number of errors in 1986
Salary
1987 annual salary on opening day in thousands of dollars
NewLeague
A
A factor with levels (coded as 1) and (coded as 2) indicating
player’s league at the beginning of 1987
N
In this problem, we use Salary as the response variable, and the rest 19 variables as predictors/covariates, which measure the performance of each player in season
1986 and his whole career. Write R functions to perform variable selection using best subset selection partnered with BIC (Bayesian Information Criterion):
1) Starting from the null model, apply the forward stepwise selection algorithm to produce a sequence of sub-models iteratively, and select a single best model using the BIC. Plot the “BIC vs Number of Variables” curve. Present the selected model with the corresponding BIC.
2) Starting from the full model (that is, the one obtained from minimizing the MSE/RSS using all the predictors), apply the backward stepwise selection algorithm to produce a sequence of sub-models iteratively, and select a single best model using the BIC. Plot the “BIC vs Number of Variables” curve. Present the selected model with the corresponding BIC.
3) Are the selected models from 1) and 2) the same?
Problem 2
In this problem, we fit ridge regression on the same dataset as in Problem 1. First, standardize the variables so that they are on the same scale. Next, choose a grid of 𝜆 values ranging from 𝜆 = 1010 to 𝜆 = 10−2, essentially covering the full range of scenarios from the null model containing only the intercept, to the least squares fit. For example:
> grid = 10^seq(10, -2, length=100)
1) Write an R function to do the following: associated with each value of 𝜆, compute a vector of ridge regression coefficients (including the intercept), stored in a 20 × 100 matrix, with 20 rows (one for each predictor, plus an intercept) and 100 columns (one for each value of 𝜆).
2) To find the “best” 𝜆 , use ten-fold cross-validation to choose the tuning parameter from the previous grid of values. Set a random seed – set.seed(1), first so your results will be reproducible, since the choice of the cross-validation folds is random. Plot the “Cross-Validation Error versus 𝜆” curve, and report the selected 𝜆.
3) Finally, refit the ridge regression model on the full dataset, using the value of 𝜆 chosen by cross-validation, and report the coefficient estimates.
Remark: You should expect that none of the coefficients are zero – ridge regression does not perform variable selection.
Problem 3
In this problem, we revisit the best subset selection problem. Given a response
vector 𝑌=(𝑦 ,…,𝑦 )𝑇and an 𝑛×𝑝 design matrix 𝑋=(𝑥 ,…,𝑥 )𝑇 with 𝑥 =
1𝑛 1𝑛𝑖 ̂̂
𝑇
(𝑥𝑖1, … , 𝑥𝑖𝑝) . For 1 ≤ 𝑘 ≤ 𝑝, let 𝛽0, 𝛽 be the solution to the following sparsity-
constrained least squares problem:
min ∥𝑌−𝛽 −𝑋𝛽∥2 = min ∑𝑛 (𝑦 −𝛽 −𝑥𝑇𝛽)2.
𝛽0,𝛽:∥𝛽∥0=𝑘 0 2 𝛽0,𝛽:∥𝛽∥0=𝑘 𝑖=1 𝑖 0 𝑖
̂𝑇̂
Basedontheproperty𝛽0 =𝑦̅−𝑥̅ 𝛽,wecancenter𝑌and𝑋firsttogetridofthe
intercept, and solve
̃̃2 min ∥𝑌−𝑋𝛽∥2,
𝛽:∥𝛽∥0=𝑘
̃̃
where 𝑌 and 𝑋 represent the centered 𝑌 and 𝑋, respectively. To solve this, we
introduce the Gradient Hard Thresholding Pursuit (GraHTP) algorithm. Let 𝑓(𝛽) = ̃̃2
∥ 𝑌 − 𝑋𝛽 ∥2⁄(2𝑛) be the objective function. GraHTP Algorithm.
̃̃
Input: 𝑌, 𝑋, sparsity 𝑘, stepsize 𝜂 > 0
̃
(Hint: normalize the columns of 𝑋 to have variance 1).
Initialization: 𝛽0 = 0, 𝑡 = 1. repeat
3) Compute 𝛽𝑡 = argmin{𝑓(𝛽);supp(𝛽) ⊆ 𝒮𝑡}; 𝑡 = 𝑡 + 1;
until convergence, i.e. ∥ 𝛽𝑡 − 𝛽𝑡−1 ∥2< 10−4. Output: 𝛽𝑡.
1) Write an R function to implement the above GraHTP algorithm.
2) Consider again the baseball dataset in Problem 1 with 𝑛 = 263, 𝑝 = 19. For 𝑘 =
1, ... , 𝑝, use the above function to find the best 𝑘-sparse model, denoted by M . 𝑘
Then use BIC to select a single best model among M ,...,M . 1𝑝
3) Compare your result with those obtained in Problem 1.
̃𝑡 𝑡−1 1) Compute 𝛽 = 𝛽
𝑡−1 𝑡̃𝑡 ̃𝑡
− 𝜂∇𝑓(𝛽
2) Let 𝒮 = supp(𝛽 , 𝑘) be the indices of 𝛽 with the largest 𝑘 absolute values;
);