Math代写 Final Project代写 ridge regression model代写 R functions代写

Math 185 Final Project (Due December 8)

Math代写 The baseball dataset consists of the statistics of 263 players in Major LeagueBaseball in the season 1986.The dataset (hitters.csv)

Problem 1 Math代写

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:

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 algorithmto produce a sequence of sub-models iteratively, and select a single best model using the  Plot the “BIC vs Number of Variables” curve. Present the selected model with the corresponding BIC.Math代写

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

3)Are the selected models from 1) and 2) the same?

Problem 2 Math代写

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)Math代写

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 ofλ).

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), firstso your results will be reproducible, since the choice of the cross-validation folds is  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 ofλ chosen by cross-validation, and report the coefficient estimates.Math代写

Remark: You should expect that none of the coefficients are zero – ridge regression does not perform variable selection.

Problem 3 Math代写

In this problem, we revisit the best subset selection problem. Given a response

vector Y = (𝑦1, … ,𝑦n)^t and an n× p design matrix X= (x₁, … , x_n)^t with xi =

(x₁, … , xi_p) . For 1 ≤ k ≤ p, let β₀, β be the solution to the following sparsity-

constrained least squares problem:

Based on the property β0  = y − x^tβ, we can center Y and X first to get rid of the intercept, and solve

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) =

∥ y − xβ ∥²⁄(2n) be the objective function.

GraHTP Algorithm.

Input:y, x, sparsity k, stepsize n > 0Math代写

(Hint: normalize the columns of x to have variance 1). Initialization: β⁰ = 0, t = 1.

repeat

1)   Compute βt  = β^t−1 − n∇f(β^t−1);

2)   Let S^t = supp(βt, k) be the indices of βt  with the largest k absolute values; 3) Compute β^t = argmin{f(β); supp(β) ⊆ S^t};

t = t + 1;Math代写

until convergence, i.e. ∥ β^t − β^t−1 ∥2< 10⁻⁴.

Output: β^t.Math代写

1)Write an R function to implement the above GraHTP

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 ℳk.Then use BIC to select a single best model among ℳ1, … , ℳp.

3)Compare your result with those obtained in Problem1.

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