Statistics: Running Simulations for Five Methods Given a linear model Y=β0∗+X∗⊤β∗+ε, ε∼N (0,σ∗2). The summation ϵ is an independent and identically distributed or an iid with normal random variables having a mean of 0 and a variance of σ2. Arguably, this model comprises of three fixed parameters that can be estimated: β0, β1, and σ2 unknown constants. In the above linear model, there is a slight modification of the notation here. We have applied the notation to be Y since the model will be fitted using a set of n data points ranging from i=1, 2… onwards. 1. Running simulations for five methods: • LASSO (Tibshirani; 1996) Summary • true model is y=Xβ+ϵy=Xβ+ϵ • where ϵ∼Nn (0, I) ϵ∼Nn (0, I) • • • o Example 1 Small signal. Lots of noise. β=(115…,04085)Tβ=(1,...,1⏟15,0,...,0⏟4085)T p=5000>n=1000p=5000>n=1000 Uncorrelated predictors: Xi∼iidN(0,I) Generating Data library(MASS) # Package needed to generate correlated predictors library(glmnet) # Package to fit ridge/lasso/elastic net models # Generate data set.seed(19875) # Set seed for reproducibility n <- 1000 # Number of observations p <- 5000 # Number of predictors included in model real_p <- 15 # Number of true predictors x <- matrix(rnorm(n*p), nrow=n, ncol=p) y <- apply(x[,1:real_p], 1, sum) + rnorm(n) # Split data into train (2/3) and test (1/3) sets train_rows <- sample(1:n, .66*n) x.train <- x[train_rows, ] x.test <- x[-train_rows, ] y.train <- y[train_rows] y.test ...
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