Simple regression methods
02 November, 2024
About
This is mainly a replication of:
- Dana, J., & Dawes, R. M. (2004). The superiority of simple alternatives to regression for social science predictions. Journal of Educational and Behavioral Statistics, 29(3), 317-331.
Init
#global options
options(
digits = 2,
contrasts = c("contr.treatment", "contr.treatment")
)
#packages
library(kirkegaard)## Loading required package: tidyverse## ── Attaching core tidyverse packages ──────────────────────── tidyverse 2.0.0 ──
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## ✔ lubridate 1.9.3 ✔ tidyr 1.3.1
## ✔ purrr 1.0.2
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## +load_packages(
broom,
furrr
)## Loading required package: future#ggplot2
theme_set(theme_bw())
#multi-threading
plan(multisession)Functions
#generate random correlation matrices
rand_cor_mat_pca_method = function(p, latent_dims = 2, loadings = "runif", rnorm_sd = 0.2, lower_bound = -1, upper_bound = 1, noise = 0) {
# browser()
#generate random data for the PCA
latent_x = MASS::mvrnorm(1000, mu = rep(0, latent_dims), Sigma = diag(latent_dims))
#loadings method
if (is.numeric(loadings)) {
loadings = matrix(loadings, nrow = latent_dims, ncol = latent_dims)
} else if (loadings == "runif") {
loadings = matrix(runif(p*latent_dims, min = lower_bound, max = upper_bound), nrow = p, ncol = latent_dims)
} else if (loadings == "rtnorm") {
loadings = matrix(TruncatedNormal::rtnorm(p*latent_dims, lb = lower_bound, ub = upper_bound, sd = rnorm_sd), nrow = p, ncol = latent_dims)
}
#begin with uncorrelated mvrnorm data
x = MASS::mvrnorm(1000, mu = rep(0, p), Sigma = diag(p)) * noise
#multiply by loadings
x2 = (latent_x %*% t(loadings)) + x
#get correlations
cor(x2)
}
#simulate multivariate normal data from correlation matrix
sim_data = function(n, preds = 5, y_noise = 1, x_noise = 1) {
#generate random correlation matrix
corr_mat <- rand_cor_mat_pca_method(preds, noise = x_noise)
#generate data
x = MASS::mvrnorm(n, mu = rep(0, ncol(corr_mat)), Sigma = corr_mat)
colnames(x) = paste0("x", 1:ncol(x))
#add y based on random weights
y_betas = rnorm(ncol(x), sd = 0.30)
y = standardize(x %*% y_betas) + rnorm(n, sd = y_noise)
#data frame
x = bind_cols(
tibble(y = y), #unit variance
as_tibble(x)
)
#return but move y to position 1
x %>%
select(y, everything())
}
#fit models
#function that fits the various models we wish to compare
fit_models = function(x, x_test) {
# browser()
#subsets
x_mat = x[, -1] %>% as.matrix()
x_test_mat = x_test[, -1] %>% as.matrix()
#OLS
ols = lm(y ~ ., data = x)
ols_preds = predict(ols)
#ridge regression
ridge = glmnet::cv.glmnet(x_mat, x$y, alpha = 0)
ridge_preds = predict(ridge, s = "lambda.min", x_mat)
#correlation weights
cors = cor(x)
cor_weights_preds = x_mat %*% cors[1, -1]
#unit weights, based on correlation signs
cor_unit_weights_preds = x_mat %*% sign(cors[1, -1])
#unit weights, based on regression signs
reg_unit_weights_preds = x_mat %*% sign(coef(ols)[-1])
#do the same for the test set
ols_preds_test = predict(ols, newdata = x_test)
ridge_preds_test = predict(ridge, s = "lambda.min", newx = x_test_mat)
cor_weights_preds_test = x_test_mat %*% cors[1, -1]
cor_unit_weights_preds_test = x_test_mat %*% sign(cors[1, -1])
reg_unit_weights_preds_test = x_test_mat %*% sign(coef(ols)[-1])
#return correlations
res = tibble(
#empirical metadata
mean_abs_cor = mean(abs(MAT_half(cors[-1, -1]))),
#training set
ols_train = cor(x$y, ols_preds) %>% as.numeric(),
ridge_train = cor(x$y, ridge_preds) %>% as.numeric(),
cor_weights_train = cor(x$y, cor_weights_preds) %>% as.numeric(),
cor_unit_weights_train = cor(x$y, cor_unit_weights_preds) %>% as.numeric(),
reg_unit_weights_train = cor(x$y, reg_unit_weights_preds) %>% as.numeric(),
#test set
ols_test = cor(x_test$y, ols_preds_test) %>% as.numeric(),
ridge_test = cor(x_test$y, ridge_preds_test) %>% as.numeric(),
cor_weights_test = cor(x_test$y, cor_weights_preds_test) %>% as.numeric(),
cor_unit_weights_test = cor(x_test$y, cor_unit_weights_preds_test) %>% as.numeric(),
reg_unit_weights_test = cor(x_test$y, reg_unit_weights_preds_test) %>% as.numeric()
)
res
}Simulations
#simulation parameters
sims = expand_grid(
n = seq(50, 500, by = 25),
preds = c(5, 10, 15, 20, 25),
y_noise = c(0.5, 1, 1.5, 2, 3, 4),
x_noise = c(0.5, 1, 2, 3),
reps = 1:50
)
#run simulations
set.seed(1)
sim_results = future_map_dfr(1:nrow(sims), ~{
#simulate data
x = sim_data(
n = sims$n[.x] + 500, #last 500 cases are test set
preds = sims$preds[.x],
y_noise = sims$y_noise[.x],
x_noise = sims$x_noise[.x]
)
#split into train and test
x_train = x[1:sims$n[.x], ]
x_test = x[(sims$n[.x] + 1):(sims$n[.x] + 500), ]
#fit models on train
res = fit_models(
x_train,
x_test
)
res
}, .options = furrr_options(seed = T), .progress = T)## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in 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standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, 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standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero## Warning in cor(x$y, ridge_preds): the standard deviation is zero## Warning in cor(x_test$y, ridge_preds_test): the standard deviation is zero#combine
sim_final = bind_cols(sims, sim_results)Analysis
#which method is better as a function of sample size?
sim_final %>%
group_by(n) %>%
summarise(
ols = mean(ols_test, na.rm = T),
ridge = mean(ridge_test, na.rm = T),
cor_weights = mean(cor_weights_test, na.rm = T),
cor_unit_weights = mean(cor_unit_weights_test, na.rm = T),
reg_unit_weights = mean(reg_unit_weights_test, na.rm = T)
) %>%
gather(model, cor, -n) %>%
ggplot(aes(n, cor, color = model)) +
geom_line(alpha = .5) +
labs(
title = "Correlation between predicted and actual y",
x = "Sample size",
y = "Correlation"
)
GG_save("figs/sample_size.png")
#in hard situations
sim_final %>%
filter(y_noise == 4, preds == 25, x_noise == 0.5) %>%
group_by(n) %>%
summarise(
ols = mean(ols_test, na.rm = T),
ridge = mean(ridge_test, na.rm = T),
cor_weights = mean(cor_weights_test, na.rm = T),
cor_unit_weights = mean(cor_unit_weights_test, na.rm = T),
reg_unit_weights = mean(reg_unit_weights_test, na.rm = T)
) %>%
gather(model, cor, -n) %>%
ggplot(aes(n, cor, color = model)) +
geom_line() +
labs(
title = "Correlation between predicted and actual y",
x = "Sample size",
y = "Correlation"
)
GG_save("figs/hard_situation.png")
#compare across levels of y_noise
sim_final %>%
filter(n == 50, preds == 25, x_noise == 0.5) %>%
group_by(y_noise) %>%
summarise(
ols = mean(ols_test, na.rm = T),
ridge = mean(ridge_test, na.rm = T),
cor_weights = mean(cor_weights_test, na.rm = T),
cor_unit_weights = mean(cor_unit_weights_test, na.rm = T),
reg_unit_weights = mean(reg_unit_weights_test, na.rm = T)
) %>%
gather(model, cor, -y_noise) %>%
ggplot(aes(y_noise, cor, color = model)) +
geom_line() +
labs(
title = "Correlation between predicted and actual y",
x = "Noise in y",
y = "Correlation"
)
GG_save("figs/y_noise.png")
#x_noise effects
sim_final %>%
filter(n == 50, preds == 25, y_noise == 2) %>%
group_by(x_noise) %>%
summarise(
ols = mean(ols_test, na.rm = T),
ridge = mean(ridge_test, na.rm = T),
cor_weights = mean(cor_weights_test, na.rm = T),
cor_unit_weights = mean(cor_unit_weights_test, na.rm = T),
reg_unit_weights = mean(reg_unit_weights_test, na.rm = T)
) %>%
gather(model, cor, -x_noise) %>%
ggplot(aes(x_noise, cor, color = model)) +
geom_line() +
labs(
title = "Correlation between predicted and actual y",
x = "Noise in x",
y = "Correlation"
)
#preds effects
sim_final %>%
filter(n == 50, y_noise == 2) %>%
group_by(preds) %>%
summarise(
ols = mean(ols_test, na.rm = T),
ridge = mean(ridge_test, na.rm = T),
cor_weights = mean(cor_weights_test, na.rm = T),
cor_unit_weights = mean(cor_unit_weights_test, na.rm = T),
reg_unit_weights = mean(reg_unit_weights_test, na.rm = T)
) %>%
gather(model, cor, -preds) %>%
ggplot(aes(preds, cor, color = model)) +
geom_line() +
labs(
title = "Correlation between predicted and actual y",
x = "Number of predictors",
y = "Correlation"
)
GG_save("figs/preds.png")Meta
write_sessioninfo()## R version 4.4.1 (2024-06-14)
## Platform: x86_64-pc-linux-gnu
## Running under: Linux Mint 21.1
##
## Matrix products: default
## BLAS: /usr/lib/x86_64-linux-gnu/blas/libblas.so.3.10.0
## LAPACK: /usr/lib/x86_64-linux-gnu/lapack/liblapack.so.3.10.0
##
## locale:
## [1] LC_CTYPE=en_DK.UTF-8 LC_NUMERIC=C
## [3] LC_TIME=en_DK.UTF-8 LC_COLLATE=en_DK.UTF-8
## [5] LC_MONETARY=en_DK.UTF-8 LC_MESSAGES=en_DK.UTF-8
## [7] LC_PAPER=en_DK.UTF-8 LC_NAME=C
## [9] LC_ADDRESS=C LC_TELEPHONE=C
## [11] LC_MEASUREMENT=en_DK.UTF-8 LC_IDENTIFICATION=C
##
## time zone: Europe/Copenhagen
## tzcode source: system (glibc)
##
## attached base packages:
## [1] stats graphics grDevices utils datasets methods base
##
## other attached packages:
## [1] furrr_0.3.1 future_1.34.0 broom_1.0.7
## [4] kirkegaard_2024-09-27 psych_2.4.6.26 assertthat_0.2.1
## [7] weights_1.0.4 Hmisc_5.1-3 magrittr_2.0.3
## [10] lubridate_1.9.3 forcats_1.0.0 stringr_1.5.1
## [13] dplyr_1.1.4 purrr_1.0.2 readr_2.1.5
## [16] tidyr_1.3.1 tibble_3.2.1 ggplot2_3.5.1
## [19] tidyverse_2.0.0
##
## loaded via a namespace (and not attached):
## [1] tidyselect_1.2.1 farver_2.1.2 fastmap_1.2.0 digest_0.6.37
## [5] rpart_4.1.23 timechange_0.3.0 lifecycle_1.0.4 cluster_2.1.6
## [9] survival_3.7-0 gdata_3.0.1 compiler_4.4.1 rlang_1.1.4
## [13] sass_0.4.9 tools_4.4.1 utf8_1.2.4 yaml_2.3.10
## [17] data.table_1.16.2 knitr_1.48 labeling_0.4.3 htmlwidgets_1.6.4
## [21] mnormt_2.1.1 withr_3.0.1 foreign_0.8-86 nnet_7.3-19
## [25] grid_4.4.1 fansi_1.0.6 jomo_2.7-6 colorspace_2.1-1
## [29] mice_3.16.0 globals_0.16.3 scales_1.3.0 gtools_3.9.5
## [33] iterators_1.0.14 MASS_7.3-61 cli_3.6.3 rmarkdown_2.28
## [37] ragg_1.3.3 generics_0.1.3 rstudioapi_0.17.1 tzdb_0.4.0
## [41] minqa_1.2.8 cachem_1.1.0 splines_4.4.1 parallel_4.4.1
## [45] base64enc_0.1-3 vctrs_0.6.5 boot_1.3-30 glmnet_4.1-8
## [49] Matrix_1.6-5 jsonlite_1.8.9 hms_1.1.3 mitml_0.4-5
## [53] Formula_1.2-5 htmlTable_2.4.3 listenv_0.9.1 systemfonts_1.1.0
## [57] foreach_1.5.2 jquerylib_0.1.4 parallelly_1.38.0 glue_1.8.0
## [61] nloptr_2.1.1 pan_1.9 codetools_0.2-19 stringi_1.8.4
## [65] gtable_0.3.6 shape_1.4.6.1 lme4_1.1-35.5 munsell_0.5.1
## [69] pillar_1.9.0 htmltools_0.5.8.1 R6_2.5.1 textshaping_0.4.0
## [73] evaluate_1.0.1 lattice_0.22-5 highr_0.11 backports_1.5.0
## [77] bslib_0.8.0 Rcpp_1.0.13 gridExtra_2.3 nlme_3.1-165
## [81] checkmate_2.3.2 xfun_0.48 pkgconfig_2.0.3#upload to OSF
#avoid uploading the data in case they freak out again
if (F) {
library(osfr)
#auth
osf_auth(readr::read_lines("~/.config/osf_token"))
#the project we will use
osf_proj = osf_retrieve_node("https://osf.io/XXX/")
#upload files
#overwrite existing (versioning)
osf_upload(osf_proj, conflicts = "overwrite",
path = c(
"figs",
"data",
"notebook.html",
"notebook.Rmd",
))
}