14 Linear Model - Ridge & LASSO

Use the built-in R dataset, “diamonds”, using “price” as our outcome variable and all other features in the dataset as “predictors” or features.

install_if_not <- function( list.of.packages ) {

  new.packages <- list.of.packages[!(list.of.packages %in% installed.packages()[,"Package"])]
  
  if(length(new.packages)) { install.packages(new.packages) } else { print(paste0("the package '", list.of.packages , "' is already installed")) }
}


library(tidyverse)

── Attaching core tidyverse packages ──────────────────────── tidyverse 2.0.0 ──
✔ dplyr     1.1.4     ✔ readr     2.1.5
✔ forcats   1.0.0     ✔ stringr   1.5.1
✔ ggplot2   3.5.1     ✔ tibble    3.2.1
✔ lubridate 1.9.3     ✔ tidyr     1.3.1
✔ purrr     1.0.2     
── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
✖ dplyr::filter() masks stats::filter()
✖ dplyr::lag()    masks stats::lag()
ℹ Use the conflicted package (<http://conflicted.r-lib.org/>) to force all conflicts to become errors

library(caret)

Loading required package: lattice

Attaching package: 'caret'

The following object is masked from 'package:purrr':

    lift

# Take a look at the data
install_if_not('glmnet')

[1] "the package 'glmnet' is already installed"

library(glmnet)

Loading required package: Matrix

Attaching package: 'Matrix'

The following objects are masked from 'package:tidyr':

    expand, pack, unpack

Loaded glmnet 4.1-8

glimpse(diamonds)

Rows: 53,940
Columns: 10
$ carat   <dbl> 0.23, 0.21, 0.23, 0.29, 0.31, 0.24, 0.24, 0.26, 0.22, 0.23, 0.…
$ cut     <ord> Ideal, Premium, Good, Premium, Good, Very Good, Very Good, Ver…
$ color   <ord> E, E, E, I, J, J, I, H, E, H, J, J, F, J, E, E, I, J, J, J, I,…
$ clarity <ord> SI2, SI1, VS1, VS2, SI2, VVS2, VVS1, SI1, VS2, VS1, SI1, VS1, …
$ depth   <dbl> 61.5, 59.8, 56.9, 62.4, 63.3, 62.8, 62.3, 61.9, 65.1, 59.4, 64…
$ table   <dbl> 55, 61, 65, 58, 58, 57, 57, 55, 61, 61, 55, 56, 61, 54, 62, 58…
$ price   <int> 326, 326, 327, 334, 335, 336, 336, 337, 337, 338, 339, 340, 34…
$ x       <dbl> 3.95, 3.89, 4.05, 4.20, 4.34, 3.94, 3.95, 4.07, 3.87, 4.00, 4.…
$ y       <dbl> 3.98, 3.84, 4.07, 4.23, 4.35, 3.96, 3.98, 4.11, 3.78, 4.05, 4.…
$ z       <dbl> 2.43, 2.31, 2.31, 2.63, 2.75, 2.48, 2.47, 2.53, 2.49, 2.39, 2.…

14.0.1 Split into test and training sets

# Split the data into training and test set
set.seed(8675309)

training.samples <- diamonds$price %>%
  createDataPartition(p = 0.7, list = FALSE)

train.data  <- diamonds[training.samples, ]
test.data <- diamonds[-training.samples, ]

14.0.2 Create `lambda` tuning grid

Set up a range of lambda values for cross-validation (that is, so you the optimal lambda is chosen).

lambda <- 10^seq(-3, 3, length = 100)

14.1 LASSO Regression

# Build the model using 10-fold cross-validation

lasso <- train(
  price ~., 
  data = train.data, 
  method = "glmnet",
  trControl = trainControl("cv", number = 10),
  tuneGrid = expand.grid(alpha = 1, lambda = lambda), # alpha=1 means LASSO
  preProcess = c('center','scale')
  )

# Investigate model coefficients

# Model coefficients
# You will see that the variable "y" was zeroed out of the model
coef_lasso <- coef(lasso$finalModel, lasso$bestTune$lambda)

coef_lasso <- as_tibble(as.matrix(coef_lasso), rownames='feature') %>%
  rename(estimate = s1) 


predictions <- lasso %>% predict(test.data)

test.scored.lasso <- cbind(test.data, predictions)

library('yardstick')


Attaching package: 'yardstick'

The following objects are masked from 'package:caret':

    precision, recall, sensitivity, specificity

The following object is masked from 'package:readr':

    spec

rmse(test.scored.lasso, price, predictions)

# A tibble: 1 × 3
  .metric .estimator .estimate
  <chr>   <chr>          <dbl>
1 rmse    standard       1139.

14.2 Ridge Regression

# Build the model using 10-fold cross-validation


ridge <- train(
  price ~., 
  data = train.data, 
  method = "glmnet",
  trControl = trainControl("cv", number = 10),
  tuneGrid = expand.grid(alpha = 0, lambda = lambda), #alpha=0 means Ridge
  preProcess = c('center','scale')
  )

plot(ridge)

# Model coefficients
coef_ridge <- coef(ridge$finalModel, ridge$bestTune$lambda)
coef_ridge <- as_tibble(as.matrix(coef_ridge), rownames='feature') %>%
  rename(estimate = s1) 
  
coef_ridge

# A tibble: 24 × 2
   feature     estimate
   <chr>          <dbl>
 1 (Intercept)  3932.  
 2 carat        2147.  
 3 cut.L         190.  
 4 cut.Q        -112.  
 5 cut.C          26.4 
 6 cut^4          -2.62
 7 color.L      -449.  
 8 color.Q      -146.  
 9 color.C       -49.7 
10 color^4         3.15
# ℹ 14 more rows

14.2.1 Score test data

# Predict price using the estimated ridge model and test data
predictions <- ridge %>% predict(test.data)

test.scored.ridge <- cbind(test.data, predictions)

rmse(test.scored.ridge, price, predictions)

# A tibble: 1 × 3
  .metric .estimator .estimate
  <chr>   <chr>          <dbl>
1 rmse    standard       1373.

14.3 GLM

glm_model <- train(price ~., 
                   data = train.data, 
                   method = "glm",
                   trControl = trainControl("cv", number = 10),
                   preProcess = c('center','scale')
                   )

predictions <- glm_model %>% predict(test.data)

test.scored.glm <- cbind(test.data, predictions)


test.scored_Stacked <- bind_rows( test.scored.ridge %>% mutate(model = 'ridge'),
                                  test.scored.lasso %>% mutate(model = 'lasso'),
                                  test.scored.glm %>% mutate(model = 'glm')
                                  )


test.scored_Stacked %>%
  group_by(model) %>%
  rmse(truth=price, predictions) %>%
  arrange(.estimate)

# A tibble: 3 × 4
  model .metric .estimator .estimate
  <chr> <chr>   <chr>          <dbl>
1 lasso rmse    standard       1139.
2 glm   rmse    standard       1141.
3 ridge rmse    standard       1373.

coef_glm <- as_tibble(glm_model$finalModel$coefficients,
                      rownames='feature') %>%
  rename(estimate = value)

coef_compare <- bind_rows(coef_glm %>% mutate(model = 'glm'),
                          coef_lasso %>% mutate(model = 'lasso'),
                          coef_ridge %>% mutate(model = 'ridge')
                          )

any_column_NA <- function(x){
    any(is.na(x))
}

replace_NA_0 <- function(x){
    if_else(is.na(x),0,x)
}

coef_compare %>%
  pivot_wider(names_from = model, 
              values_from = estimate) %>%
  mutate_if(any_column_NA, replace_NA_0) %>%
  mutate(lasso_pct_diff_glm = (glm-lasso)/glm) %>%
  mutate(ridge_pct_diff_glm = (glm-ridge)/glm) %>%
  arrange(-lasso_pct_diff_glm) %>%
  knitr::kable()

feature	glm	lasso	ridge	lasso_pct_diff_glm	ridge_pct_diff_glm
cut^4	0.0000000	-5.749140	-2.615307	Inf	Inf
color^5	0.0000000	-32.099308	-21.247125	Inf	Inf
color^6	0.0000000	-19.508480	-21.264867	Inf	Inf
clarity^4	0.0000000	-129.224723	-35.276472	Inf	Inf
`cut^4`	-12.5826932	0.000000	0.000000	1.0000000	1.0000000
`color^4`	8.3376384	0.000000	0.000000	1.0000000	1.0000000
`color^5`	-34.9618857	0.000000	0.000000	1.0000000	1.0000000
`color^6`	-21.3028796	0.000000	0.000000	1.0000000	1.0000000
`clarity^4`	-137.0286078	0.000000	0.000000	1.0000000	1.0000000
`clarity^5`	92.1936626	0.000000	0.000000	1.0000000	1.0000000
`clarity^6`	-0.1474419	0.000000	0.000000	1.0000000	1.0000000
`clarity^7`	39.2175527	0.000000	0.000000	1.0000000	1.0000000
y	165.2820422	0.000000	569.528183	1.0000000	-2.4457959
x	-1068.0369153	-816.584124	647.774976	0.2354346	1.6065099
z	-225.1271735	-196.577485	596.050824	0.1268158	3.6476183
cut.C	60.1095711	56.200869	26.409631	0.0650263	0.5606418
depth	-59.7719605	-56.554788	-10.247484	0.0538241	0.8285570
color.C	-65.2773861	-61.903237	-49.688592	0.0516894	0.2388085
cut.Q	-142.1065994	-135.024557	-112.192345	0.0498361	0.2105057
table	-61.2639120	-58.348170	-33.411300	0.0475931	0.4546333
clarity.C	302.2403654	290.104646	88.484396	0.0401525	0.7072383
clarity.Q	-469.7390115	-458.474010	-272.481912	0.0239814	0.4199291
color.Q	-233.1298779	-227.768585	-146.265139	0.0229970	0.3726023
carat	5309.0675020	5191.334788	2146.939906	0.0221758	0.5956089
cut.L	206.9329173	203.424317	190.182371	0.0169553	0.0809467
color.L	-634.9561138	-628.962264	-448.988299	0.0094398	0.2928829
clarity.L	1044.5822449	1036.337828	798.588125	0.0078925	0.2354952
(Intercept)	3931.5103949	3931.510395	3931.510395	0.0000000	0.0000000
color^4	0.0000000	7.305958	3.152159	-Inf	-Inf
clarity^5	0.0000000	85.434511	20.131680	-Inf	-Inf
clarity^7	0.0000000	36.719177	50.483468	-Inf	-Inf
clarity^6	0.0000000	0.000000	9.785508	NaN	-Inf

coef_compare %>%
  group_by(model) %>%
  ggplot(aes(x=feature,
             y=abs(estimate),
             fill=model)) +
  geom_bar(stat = 'identity',position = 'dodge') +
  coord_flip()

14.4 Ridge Versus LASSO

As a final note, we can test both lasso and ridge within the same cross-validated model by expanding our tuning grid further:

alpha = c(0,1)

lasso_v_ridge <- train( price ~ . ,
  data = train.data,
  method = "glmnet",
  trControl = trainControl("cv", number = 10),
  tuneGrid = expand.grid(alpha = alpha, lambda = lambda),
  preProcess = c('center','scale')
)

lasso_v_ridge

glmnet 

37759 samples
    9 predictor

Pre-processing: centered (23), scaled (23) 
Resampling: Cross-Validated (10 fold) 
Summary of sample sizes: 33983, 33982, 33984, 33982, 33983, 33984, ... 
Resampling results across tuning parameters:

  alpha  lambda        RMSE      Rsquared   MAE      
  0      1.000000e-03  1328.233  0.8905538   940.5504
  0      1.149757e-03  1328.233  0.8905538   940.5504
  0      1.321941e-03  1328.233  0.8905538   940.5504
  0      1.519911e-03  1328.233  0.8905538   940.5504
  0      1.747528e-03  1328.233  0.8905538   940.5504
  0      2.009233e-03  1328.233  0.8905538   940.5504
  0      2.310130e-03  1328.233  0.8905538   940.5504
  0      2.656088e-03  1328.233  0.8905538   940.5504
  0      3.053856e-03  1328.233  0.8905538   940.5504
  0      3.511192e-03  1328.233  0.8905538   940.5504
  0      4.037017e-03  1328.233  0.8905538   940.5504
  0      4.641589e-03  1328.233  0.8905538   940.5504
  0      5.336699e-03  1328.233  0.8905538   940.5504
  0      6.135907e-03  1328.233  0.8905538   940.5504
  0      7.054802e-03  1328.233  0.8905538   940.5504
  0      8.111308e-03  1328.233  0.8905538   940.5504
  0      9.326033e-03  1328.233  0.8905538   940.5504
  0      1.072267e-02  1328.233  0.8905538   940.5504
  0      1.232847e-02  1328.233  0.8905538   940.5504
  0      1.417474e-02  1328.233  0.8905538   940.5504
  0      1.629751e-02  1328.233  0.8905538   940.5504
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  0      2.154435e-02  1328.233  0.8905538   940.5504
  0      2.477076e-02  1328.233  0.8905538   940.5504
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  0      4.328761e-02  1328.233  0.8905538   940.5504
  0      4.977024e-02  1328.233  0.8905538   940.5504
  0      5.722368e-02  1328.233  0.8905538   940.5504
  0      6.579332e-02  1328.233  0.8905538   940.5504
  0      7.564633e-02  1328.233  0.8905538   940.5504
  0      8.697490e-02  1328.233  0.8905538   940.5504
  0      1.000000e-01  1328.233  0.8905538   940.5504
  0      1.149757e-01  1328.233  0.8905538   940.5504
  0      1.321941e-01  1328.233  0.8905538   940.5504
  0      1.519911e-01  1328.233  0.8905538   940.5504
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  0      2.656088e-01  1328.233  0.8905538   940.5504
  0      3.053856e-01  1328.233  0.8905538   940.5504
  0      3.511192e-01  1328.233  0.8905538   940.5504
  0      4.037017e-01  1328.233  0.8905538   940.5504
  0      4.641589e-01  1328.233  0.8905538   940.5504
  0      5.336699e-01  1328.233  0.8905538   940.5504
  0      6.135907e-01  1328.233  0.8905538   940.5504
  0      7.054802e-01  1328.233  0.8905538   940.5504
  0      8.111308e-01  1328.233  0.8905538   940.5504
  0      9.326033e-01  1328.233  0.8905538   940.5504
  0      1.072267e+00  1328.233  0.8905538   940.5504
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  0      1.629751e+00  1328.233  0.8905538   940.5504
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  0      3.764936e+00  1328.233  0.8905538   940.5504
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  0      4.977024e+00  1328.233  0.8905538   940.5504
  0      5.722368e+00  1328.233  0.8905538   940.5504
  0      6.579332e+00  1328.233  0.8905538   940.5504
  0      7.564633e+00  1328.233  0.8905538   940.5504
  0      8.697490e+00  1328.233  0.8905538   940.5504
  0      1.000000e+01  1328.233  0.8905538   940.5504
  0      1.149757e+01  1328.233  0.8905538   940.5504
  0      1.321941e+01  1328.233  0.8905538   940.5504
  0      1.519911e+01  1328.233  0.8905538   940.5504
  0      1.747528e+01  1328.233  0.8905538   940.5504
  0      2.009233e+01  1328.233  0.8905538   940.5504
  0      2.310130e+01  1328.233  0.8905538   940.5504
  0      2.656088e+01  1328.233  0.8905538   940.5504
  0      3.053856e+01  1328.233  0.8905538   940.5504
  0      3.511192e+01  1328.233  0.8905538   940.5504
  0      4.037017e+01  1328.233  0.8905538   940.5504
  0      4.641589e+01  1328.233  0.8905538   940.5504
  0      5.336699e+01  1328.233  0.8905538   940.5504
  0      6.135907e+01  1328.233  0.8905538   940.5504
  0      7.054802e+01  1328.233  0.8905538   940.5504
  0      8.111308e+01  1328.233  0.8905538   940.5504
  0      9.326033e+01  1328.233  0.8905538   940.5504
  0      1.072267e+02  1328.233  0.8905538   940.5504
  0      1.232847e+02  1328.233  0.8905538   940.5504
  0      1.417474e+02  1328.233  0.8905538   940.5504
  0      1.629751e+02  1328.233  0.8905538   940.5504
  0      1.873817e+02  1328.233  0.8905538   940.5504
  0      2.154435e+02  1328.233  0.8905538   940.5504
  0      2.477076e+02  1328.233  0.8905538   940.5504
  0      2.848036e+02  1328.233  0.8905538   940.5504
  0      3.274549e+02  1328.233  0.8905538   940.5504
  0      3.764936e+02  1331.202  0.8901090   942.4954
  0      4.328761e+02  1348.075  0.8875674   953.3319
  0      4.977024e+02  1365.185  0.8850186   963.5800
  0      5.722368e+02  1382.631  0.8824611   973.2589
  0      6.579332e+02  1400.457  0.8799049   982.3759
  0      7.564633e+02  1418.812  0.8773460   991.0442
  0      8.697490e+02  1437.914  0.8747747   999.2806
  0      1.000000e+03  1457.959  0.8721833  1007.1196
  1      1.000000e-03  1128.530  0.9197241   740.8260
  1      1.149757e-03  1128.530  0.9197241   740.8260
  1      1.321941e-03  1128.530  0.9197241   740.8260
  1      1.519911e-03  1128.530  0.9197241   740.8260
  1      1.747528e-03  1128.530  0.9197241   740.8260
  1      2.009233e-03  1128.530  0.9197241   740.8260
  1      2.310130e-03  1128.530  0.9197241   740.8260
  1      2.656088e-03  1128.530  0.9197241   740.8260
  1      3.053856e-03  1128.530  0.9197241   740.8260
  1      3.511192e-03  1128.530  0.9197241   740.8260
  1      4.037017e-03  1128.530  0.9197241   740.8260
  1      4.641589e-03  1128.530  0.9197241   740.8260
  1      5.336699e-03  1128.530  0.9197241   740.8260
  1      6.135907e-03  1128.530  0.9197241   740.8260
  1      7.054802e-03  1128.530  0.9197241   740.8260
  1      8.111308e-03  1128.530  0.9197241   740.8260
  1      9.326033e-03  1128.530  0.9197241   740.8260
  1      1.072267e-02  1128.530  0.9197241   740.8260
  1      1.232847e-02  1128.530  0.9197241   740.8260
  1      1.417474e-02  1128.530  0.9197241   740.8260
  1      1.629751e-02  1128.530  0.9197241   740.8260
  1      1.873817e-02  1128.530  0.9197241   740.8260
  1      2.154435e-02  1128.530  0.9197241   740.8260
  1      2.477076e-02  1128.530  0.9197241   740.8260
  1      2.848036e-02  1128.530  0.9197241   740.8260
  1      3.274549e-02  1128.530  0.9197241   740.8260
  1      3.764936e-02  1128.530  0.9197241   740.8260
  1      4.328761e-02  1128.530  0.9197241   740.8260
  1      4.977024e-02  1128.530  0.9197241   740.8260
  1      5.722368e-02  1128.530  0.9197241   740.8260
  1      6.579332e-02  1128.530  0.9197241   740.8260
  1      7.564633e-02  1128.530  0.9197241   740.8260
  1      8.697490e-02  1128.530  0.9197241   740.8260
  1      1.000000e-01  1128.530  0.9197241   740.8260
  1      1.149757e-01  1128.530  0.9197241   740.8260
  1      1.321941e-01  1128.530  0.9197241   740.8260
  1      1.519911e-01  1128.530  0.9197241   740.8260
  1      1.747528e-01  1128.530  0.9197241   740.8260
  1      2.009233e-01  1128.530  0.9197241   740.8260
  1      2.310130e-01  1128.530  0.9197241   740.8260
  1      2.656088e-01  1128.530  0.9197241   740.8260
  1      3.053856e-01  1128.530  0.9197241   740.8260
  1      3.511192e-01  1128.530  0.9197241   740.8260
  1      4.037017e-01  1128.530  0.9197241   740.8260
  1      4.641589e-01  1128.530  0.9197241   740.8260
  1      5.336699e-01  1128.530  0.9197241   740.8260
  1      6.135907e-01  1128.530  0.9197241   740.8260
  1      7.054802e-01  1128.530  0.9197241   740.8260
  1      8.111308e-01  1128.530  0.9197241   740.8260
  1      9.326033e-01  1128.530  0.9197241   740.8260
  1      1.072267e+00  1128.530  0.9197241   740.8260
  1      1.232847e+00  1128.530  0.9197241   740.8260
  1      1.417474e+00  1128.530  0.9197241   740.8260
  1      1.629751e+00  1128.530  0.9197241   740.8260
  1      1.873817e+00  1128.554  0.9197212   740.8719
  1      2.154435e+00  1128.651  0.9197077   741.1256
  1      2.477076e+00  1128.754  0.9196929   741.4981
  1      2.848036e+00  1128.893  0.9196732   741.9429
  1      3.274549e+00  1129.064  0.9196490   742.4901
  1      3.764936e+00  1129.278  0.9196187   743.1335
  1      4.328761e+00  1129.594  0.9195747   743.8847
  1      4.977024e+00  1129.986  0.9195199   744.7839
  1      5.722368e+00  1130.502  0.9194480   745.8555
  1      6.579332e+00  1131.182  0.9193532   747.1605
  1      7.564633e+00  1132.061  0.9192305   748.7196
  1      8.697490e+00  1133.229  0.9190674   750.6287
  1      1.000000e+01  1134.764  0.9188529   752.9746
  1      1.149757e+01  1136.798  0.9185679   755.9350
  1      1.321941e+01  1139.499  0.9181888   759.6902
  1      1.519911e+01  1143.124  0.9176791   764.5271
  1      1.747528e+01  1147.632  0.9170412   770.3485
  1      2.009233e+01  1153.172  0.9162504   777.0174
  1      2.310130e+01  1159.992  0.9152696   784.4810
  1      2.656088e+01  1166.065  0.9143955   790.1091
  1      3.053856e+01  1169.270  0.9139619   789.8811
  1      3.511192e+01  1173.084  0.9134470   789.6917
  1      4.037017e+01  1176.863  0.9129486   789.4904
  1      4.641589e+01  1180.521  0.9124875   788.6588
  1      5.336699e+01  1184.966  0.9119282   787.8838
  1      6.135907e+01  1190.816  0.9111811   787.6081
  1      7.054802e+01  1198.061  0.9102534   787.8937
  1      8.111308e+01  1205.069  0.9094053   788.0329
  1      9.326033e+01  1213.517  0.9083886   788.7301
  1      1.072267e+02  1224.594  0.9070231   790.5540
  1      1.232847e+02  1239.086  0.9051857   794.0593
  1      1.417474e+02  1257.024  0.9028673   799.1376
  1      1.629751e+02  1276.842  0.9003378   804.5367
  1      1.873817e+02  1301.890  0.8970313   812.8215
  1      2.154435e+02  1331.716  0.8930051   823.6523
  1      2.477076e+02  1370.108  0.8875213   839.9880
  1      2.848036e+02  1415.244  0.8808129   859.6805
  1      3.274549e+02  1465.793  0.8730583   882.9116
  1      3.764936e+02  1503.311  0.8681694   901.0994
  1      4.328761e+02  1549.897  0.8617489   929.5854
  1      4.977024e+02  1608.610  0.8529723   976.3468
  1      5.722368e+02  1647.380  0.8495885  1008.8091
  1      6.579332e+02  1679.058  0.8495885  1036.6043
  1      7.564633e+02  1720.039  0.8495885  1076.7016
  1      8.697490e+02  1772.761  0.8495885  1131.7484
  1      1.000000e+03  1840.141  0.8495885  1200.8500

RMSE was used to select the optimal model using the smallest value.
The final values used for the model were alpha = 1 and lambda = 1.629751.

plot(lasso_v_ridge)

14.5 General Elastic Net Penalty

alpha = seq(0, 1, length.out = length(lambda))

Elastic_Net <- train( price ~ . ,
  data = train.data,
  method = "glmnet",
  trControl = trainControl("cv", number = 10),
  tuneGrid = expand.grid(alpha = alpha, lambda = lambda),
  preProcess = c('center','scale')
)

Elastic_Net

glmnet 

37759 samples
    9 predictor

Pre-processing: centered (23), scaled (23) 
Resampling: Cross-Validated (10 fold) 
Summary of sample sizes: 33983, 33983, 33983, 33982, 33984, 33983, ... 
Resampling results across tuning parameters:

  alpha       lambda        RMSE      Rsquared   MAE      
  0.00000000  1.000000e-03  1327.815  0.8906263   940.5751
  0.00000000  1.149757e-03  1327.815  0.8906263   940.5751
  0.00000000  1.321941e-03  1327.815  0.8906263   940.5751
  0.00000000  1.519911e-03  1327.815  0.8906263   940.5751
  0.00000000  1.747528e-03  1327.815  0.8906263   940.5751
  0.00000000  2.009233e-03  1327.815  0.8906263   940.5751
  0.00000000  2.310130e-03  1327.815  0.8906263   940.5751
  0.00000000  2.656088e-03  1327.815  0.8906263   940.5751
  0.00000000  3.053856e-03  1327.815  0.8906263   940.5751
  0.00000000  3.511192e-03  1327.815  0.8906263   940.5751
  0.00000000  4.037017e-03  1327.815  0.8906263   940.5751
  0.00000000  4.641589e-03  1327.815  0.8906263   940.5751
  0.00000000  5.336699e-03  1327.815  0.8906263   940.5751
  0.00000000  6.135907e-03  1327.815  0.8906263   940.5751
  0.00000000  7.054802e-03  1327.815  0.8906263   940.5751
  0.00000000  8.111308e-03  1327.815  0.8906263   940.5751
  0.00000000  9.326033e-03  1327.815  0.8906263   940.5751
  0.00000000  1.072267e-02  1327.815  0.8906263   940.5751
  0.00000000  1.232847e-02  1327.815  0.8906263   940.5751
  0.00000000  1.417474e-02  1327.815  0.8906263   940.5751
  0.00000000  1.629751e-02  1327.815  0.8906263   940.5751
  0.00000000  1.873817e-02  1327.815  0.8906263   940.5751
  0.00000000  2.154435e-02  1327.815  0.8906263   940.5751
  0.00000000  2.477076e-02  1327.815  0.8906263   940.5751
  0.00000000  2.848036e-02  1327.815  0.8906263   940.5751
  0.00000000  3.274549e-02  1327.815  0.8906263   940.5751
  0.00000000  3.764936e-02  1327.815  0.8906263   940.5751
  0.00000000  4.328761e-02  1327.815  0.8906263   940.5751
  0.00000000  4.977024e-02  1327.815  0.8906263   940.5751
  0.00000000  5.722368e-02  1327.815  0.8906263   940.5751
  0.00000000  6.579332e-02  1327.815  0.8906263   940.5751
  0.00000000  7.564633e-02  1327.815  0.8906263   940.5751
  0.00000000  8.697490e-02  1327.815  0.8906263   940.5751
  0.00000000  1.000000e-01  1327.815  0.8906263   940.5751
  0.00000000  1.149757e-01  1327.815  0.8906263   940.5751
  0.00000000  1.321941e-01  1327.815  0.8906263   940.5751
  0.00000000  1.519911e-01  1327.815  0.8906263   940.5751
  0.00000000  1.747528e-01  1327.815  0.8906263   940.5751
  0.00000000  2.009233e-01  1327.815  0.8906263   940.5751
  0.00000000  2.310130e-01  1327.815  0.8906263   940.5751
  0.00000000  2.656088e-01  1327.815  0.8906263   940.5751
  0.00000000  3.053856e-01  1327.815  0.8906263   940.5751
  0.00000000  3.511192e-01  1327.815  0.8906263   940.5751
  0.00000000  4.037017e-01  1327.815  0.8906263   940.5751
  0.00000000  4.641589e-01  1327.815  0.8906263   940.5751
  0.00000000  5.336699e-01  1327.815  0.8906263   940.5751
  0.00000000  6.135907e-01  1327.815  0.8906263   940.5751
  0.00000000  7.054802e-01  1327.815  0.8906263   940.5751
  0.00000000  8.111308e-01  1327.815  0.8906263   940.5751
  0.00000000  9.326033e-01  1327.815  0.8906263   940.5751
  0.00000000  1.072267e+00  1327.815  0.8906263   940.5751
  0.00000000  1.232847e+00  1327.815  0.8906263   940.5751
  0.00000000  1.417474e+00  1327.815  0.8906263   940.5751
  0.00000000  1.629751e+00  1327.815  0.8906263   940.5751
  0.00000000  1.873817e+00  1327.815  0.8906263   940.5751
  0.00000000  2.154435e+00  1327.815  0.8906263   940.5751
  0.00000000  2.477076e+00  1327.815  0.8906263   940.5751
  0.00000000  2.848036e+00  1327.815  0.8906263   940.5751
  0.00000000  3.274549e+00  1327.815  0.8906263   940.5751
  0.00000000  3.764936e+00  1327.815  0.8906263   940.5751
  0.00000000  4.328761e+00  1327.815  0.8906263   940.5751
  0.00000000  4.977024e+00  1327.815  0.8906263   940.5751
  0.00000000  5.722368e+00  1327.815  0.8906263   940.5751
  0.00000000  6.579332e+00  1327.815  0.8906263   940.5751
  0.00000000  7.564633e+00  1327.815  0.8906263   940.5751
  0.00000000  8.697490e+00  1327.815  0.8906263   940.5751
  0.00000000  1.000000e+01  1327.815  0.8906263   940.5751
  0.00000000  1.149757e+01  1327.815  0.8906263   940.5751
  0.00000000  1.321941e+01  1327.815  0.8906263   940.5751
  0.00000000  1.519911e+01  1327.815  0.8906263   940.5751
  0.00000000  1.747528e+01  1327.815  0.8906263   940.5751
  0.00000000  2.009233e+01  1327.815  0.8906263   940.5751
  0.00000000  2.310130e+01  1327.815  0.8906263   940.5751
  0.00000000  2.656088e+01  1327.815  0.8906263   940.5751
  0.00000000  3.053856e+01  1327.815  0.8906263   940.5751
  0.00000000  3.511192e+01  1327.815  0.8906263   940.5751
  0.00000000  4.037017e+01  1327.815  0.8906263   940.5751
  0.00000000  4.641589e+01  1327.815  0.8906263   940.5751
  0.00000000  5.336699e+01  1327.815  0.8906263   940.5751
  0.00000000  6.135907e+01  1327.815  0.8906263   940.5751
  0.00000000  7.054802e+01  1327.815  0.8906263   940.5751
  0.00000000  8.111308e+01  1327.815  0.8906263   940.5751
  0.00000000  9.326033e+01  1327.815  0.8906263   940.5751
  0.00000000  1.072267e+02  1327.815  0.8906263   940.5751
  0.00000000  1.232847e+02  1327.815  0.8906263   940.5751
  0.00000000  1.417474e+02  1327.815  0.8906263   940.5751
  0.00000000  1.629751e+02  1327.815  0.8906263   940.5751
  0.00000000  1.873817e+02  1327.815  0.8906263   940.5751
  0.00000000  2.154435e+02  1327.815  0.8906263   940.5751
  0.00000000  2.477076e+02  1327.815  0.8906263   940.5751
  0.00000000  2.848036e+02  1327.815  0.8906263   940.5751
  0.00000000  3.274549e+02  1327.815  0.8906263   940.5751
  0.00000000  3.764936e+02  1330.784  0.8901804   942.5138
  0.00000000  4.328761e+02  1347.669  0.8876382   953.3585
  0.00000000  4.977024e+02  1364.791  0.8850885   963.6058
  0.00000000  5.722368e+02  1382.250  0.8825295   973.2895
  0.00000000  6.579332e+02  1400.089  0.8799716   982.4024
  0.00000000  7.564633e+02  1418.457  0.8774104   991.0736
  0.00000000  8.697490e+02  1437.572  0.8748369   999.3063
  0.00000000  1.000000e+03  1457.630  0.8722428  1007.1440
  0.01010101  1.000000e-03  1145.196  0.9174339   776.2811
  0.01010101  1.149757e-03  1145.196  0.9174339   776.2811
  0.01010101  1.321941e-03  1145.196  0.9174339   776.2811
  0.01010101  1.519911e-03  1145.196  0.9174339   776.2811
  0.01010101  1.747528e-03  1145.196  0.9174339   776.2811
  0.01010101  2.009233e-03  1145.196  0.9174339   776.2811
  0.01010101  2.310130e-03  1145.196  0.9174339   776.2811
  0.01010101  2.656088e-03  1145.196  0.9174339   776.2811
  0.01010101  3.053856e-03  1145.196  0.9174339   776.2811
  0.01010101  3.511192e-03  1145.196  0.9174339   776.2811
  0.01010101  4.037017e-03  1145.196  0.9174339   776.2811
  0.01010101  4.641589e-03  1145.196  0.9174339   776.2811
  0.01010101  5.336699e-03  1145.196  0.9174339   776.2811
  0.01010101  6.135907e-03  1145.196  0.9174339   776.2811
  0.01010101  7.054802e-03  1145.196  0.9174339   776.2811
  0.01010101  8.111308e-03  1145.196  0.9174339   776.2811
  0.01010101  9.326033e-03  1145.196  0.9174339   776.2811
  0.01010101  1.072267e-02  1145.196  0.9174339   776.2811
  0.01010101  1.232847e-02  1145.196  0.9174339   776.2811
  0.01010101  1.417474e-02  1145.196  0.9174339   776.2811
  0.01010101  1.629751e-02  1145.196  0.9174339   776.2811
  0.01010101  1.873817e-02  1145.196  0.9174339   776.2811
  0.01010101  2.154435e-02  1145.196  0.9174339   776.2811
  0.01010101  2.477076e-02  1145.196  0.9174339   776.2811
  0.01010101  2.848036e-02  1145.196  0.9174339   776.2811
  0.01010101  3.274549e-02  1145.196  0.9174339   776.2811
  0.01010101  3.764936e-02  1145.196  0.9174339   776.2811
  0.01010101  4.328761e-02  1145.196  0.9174339   776.2811
  0.01010101  4.977024e-02  1145.196  0.9174339   776.2811
  0.01010101  5.722368e-02  1145.196  0.9174339   776.2811
  0.01010101  6.579332e-02  1145.196  0.9174339   776.2811
  0.01010101  7.564633e-02  1145.196  0.9174339   776.2811
  0.01010101  8.697490e-02  1145.196  0.9174339   776.2811
  0.01010101  1.000000e-01  1145.196  0.9174339   776.2811
  0.01010101  1.149757e-01  1145.196  0.9174339   776.2811
  0.01010101  1.321941e-01  1145.196  0.9174339   776.2811
  0.01010101  1.519911e-01  1145.196  0.9174339   776.2811
  0.01010101  1.747528e-01  1145.196  0.9174339   776.2811
  0.01010101  2.009233e-01  1145.196  0.9174339   776.2811
  0.01010101  2.310130e-01  1145.196  0.9174339   776.2811
  0.01010101  2.656088e-01  1145.196  0.9174339   776.2811
  0.01010101  3.053856e-01  1145.196  0.9174339   776.2811
  0.01010101  3.511192e-01  1145.196  0.9174339   776.2811
  0.01010101  4.037017e-01  1145.196  0.9174339   776.2811
  0.01010101  4.641589e-01  1145.196  0.9174339   776.2811
  0.01010101  5.336699e-01  1145.196  0.9174339   776.2811
  0.01010101  6.135907e-01  1145.196  0.9174339   776.2811
  0.01010101  7.054802e-01  1145.196  0.9174339   776.2811
  0.01010101  8.111308e-01  1145.196  0.9174339   776.2811
  0.01010101  9.326033e-01  1145.196  0.9174339   776.2811
  0.01010101  1.072267e+00  1145.196  0.9174339   776.2811
  0.01010101  1.232847e+00  1145.196  0.9174339   776.2811
  0.01010101  1.417474e+00  1145.196  0.9174339   776.2811
  0.01010101  1.629751e+00  1145.196  0.9174339   776.2811
  0.01010101  1.873817e+00  1145.196  0.9174339   776.2811
  0.01010101  2.154435e+00  1145.196  0.9174339   776.2811
  0.01010101  2.477076e+00  1145.196  0.9174339   776.2811
  0.01010101  2.848036e+00  1145.196  0.9174339   776.2811
  0.01010101  3.274549e+00  1145.196  0.9174339   776.2811
  0.01010101  3.764936e+00  1145.196  0.9174339   776.2811
  0.01010101  4.328761e+00  1145.196  0.9174339   776.2811
  0.01010101  4.977024e+00  1145.196  0.9174339   776.2811
  0.01010101  5.722368e+00  1145.196  0.9174339   776.2811
  0.01010101  6.579332e+00  1145.196  0.9174339   776.2811
  0.01010101  7.564633e+00  1145.196  0.9174339   776.2811
  0.01010101  8.697490e+00  1145.196  0.9174339   776.2811
  0.01010101  1.000000e+01  1145.196  0.9174339   776.2811
  0.01010101  1.149757e+01  1145.196  0.9174339   776.2811
  0.01010101  1.321941e+01  1145.196  0.9174339   776.2811
  0.01010101  1.519911e+01  1145.196  0.9174339   776.2811
  0.01010101  1.747528e+01  1145.196  0.9174339   776.2811
  0.01010101  2.009233e+01  1145.196  0.9174339   776.2811
  0.01010101  2.310130e+01  1145.196  0.9174339   776.2811
  0.01010101  2.656088e+01  1145.196  0.9174339   776.2811
  0.01010101  3.053856e+01  1145.196  0.9174339   776.2811
  0.01010101  3.511192e+01  1145.196  0.9174339   776.2811
  0.01010101  4.037017e+01  1148.172  0.9170292   780.4327
  0.01010101  4.641589e+01  1152.816  0.9163933   786.5969
  0.01010101  5.336699e+01  1157.978  0.9156848   793.0128
  0.01010101  6.135907e+01  1163.886  0.9148744   799.8341
  0.01010101  7.054802e+01  1171.203  0.9138541   808.0782
  0.01010101  8.111308e+01  1179.499  0.9126881   817.0005
  0.01010101  9.326033e+01  1188.950  0.9113475   826.6651
  0.01010101  1.072267e+02  1199.432  0.9098468   836.9103
  0.01010101  1.232847e+02  1211.016  0.9081720   847.7645
  0.01010101  1.417474e+02  1223.560  0.9063410   858.9966
  0.01010101  1.629751e+02  1237.129  0.9043419   870.6435
  0.01010101  1.873817e+02  1251.446  0.9022148   882.4581
  0.01010101  2.154435e+02  1266.596  0.8999460   894.5552
  0.01010101  2.477076e+02  1282.268  0.8975862   906.6084
  0.01010101  2.848036e+02  1298.593  0.8951166   918.6532
  0.01010101  3.274549e+02  1315.206  0.8926014   930.3157
  0.01010101  3.764936e+02  1332.324  0.8900113   941.8185
  0.01010101  4.328761e+02  1349.648  0.8874070   952.7689
  0.01010101  4.977024e+02  1367.417  0.8847590   963.2694
  0.01010101  5.722368e+02  1385.250  0.8821565   973.0584
  0.01010101  6.579332e+02  1403.593  0.8795396   982.5810
  0.01010101  7.564633e+02  1422.453  0.8769289   991.5796
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  0.09090909  4.977024e-02  1128.104  0.9197228   742.8603
  0.09090909  5.722368e-02  1128.104  0.9197228   742.8603
  0.09090909  6.579332e-02  1128.104  0.9197228   742.8603
  0.09090909  7.564633e-02  1128.104  0.9197228   742.8603
  0.09090909  8.697490e-02  1128.104  0.9197228   742.8603
  0.09090909  1.000000e-01  1128.104  0.9197228   742.8603
  0.09090909  1.149757e-01  1128.104  0.9197228   742.8603
  0.09090909  1.321941e-01  1128.104  0.9197228   742.8603
  0.09090909  1.519911e-01  1128.104  0.9197228   742.8603
  0.09090909  1.747528e-01  1128.104  0.9197228   742.8603
  0.09090909  2.009233e-01  1128.104  0.9197228   742.8603
  0.09090909  2.310130e-01  1128.104  0.9197228   742.8603
  0.09090909  2.656088e-01  1128.104  0.9197228   742.8603
  0.09090909  3.053856e-01  1128.104  0.9197228   742.8603
  0.09090909  3.511192e-01  1128.104  0.9197228   742.8603
  0.09090909  4.037017e-01  1128.104  0.9197228   742.8603
  0.09090909  4.641589e-01  1128.104  0.9197228   742.8603
  0.09090909  5.336699e-01  1128.104  0.9197228   742.8603
  0.09090909  6.135907e-01  1128.104  0.9197228   742.8603
  0.09090909  7.054802e-01  1128.104  0.9197228   742.8603
  0.09090909  8.111308e-01  1128.104  0.9197228   742.8603
  0.09090909  9.326033e-01  1128.104  0.9197228   742.8603
  0.09090909  1.072267e+00  1128.104  0.9197228   742.8603
  0.09090909  1.232847e+00  1128.104  0.9197228   742.8603
  0.09090909  1.417474e+00  1128.104  0.9197228   742.8603
  0.09090909  1.629751e+00  1128.104  0.9197228   742.8603
  0.09090909  1.873817e+00  1128.104  0.9197228   742.8603
  0.09090909  2.154435e+00  1128.104  0.9197228   742.8603
  0.09090909  2.477076e+00  1128.104  0.9197228   742.8603
  0.09090909  2.848036e+00  1128.104  0.9197228   742.8603
  0.09090909  3.274549e+00  1128.104  0.9197228   742.8603
  0.09090909  3.764936e+00  1128.104  0.9197228   742.8603
  0.09090909  4.328761e+00  1128.163  0.9197150   743.1620
  0.09090909  4.977024e+00  1128.314  0.9196950   743.8440
  0.09090909  5.722368e+00  1128.516  0.9196682   744.6331
  0.09090909  6.579332e+00  1128.776  0.9196338   745.5391
  0.09090909  7.564633e+00  1129.094  0.9195917   746.5555
  0.09090909  8.697490e+00  1129.487  0.9195397   747.7146
  0.09090909  1.000000e+01  1129.979  0.9194745   749.0745
  0.09090909  1.149757e+01  1130.609  0.9193909   750.6686
  0.09090909  1.321941e+01  1131.407  0.9192847   752.5151
  0.09090909  1.519911e+01  1132.413  0.9191506   754.6539
  0.09090909  1.747528e+01  1133.679  0.9189814   757.1360
  0.09090909  2.009233e+01  1135.263  0.9187690   760.0310
  0.09090909  2.310130e+01  1137.235  0.9185037   763.3966
  0.09090909  2.656088e+01  1139.672  0.9181744   767.2812
  0.09090909  3.053856e+01  1142.657  0.9177693   771.7678
  0.09090909  3.511192e+01  1146.299  0.9172725   776.9220
  0.09090909  4.037017e+01  1150.716  0.9166662   782.8303
  0.09090909  4.641589e+01  1154.944  0.9160919   787.8469
  0.09090909  5.336699e+01  1157.603  0.9157703   789.4575
  0.09090909  6.135907e+01  1162.280  0.9151567   793.7944
  0.09090909  7.054802e+01  1169.039  0.9142403   800.5132
  0.09090909  8.111308e+01  1177.363  0.9130911   808.8849
  0.09090909  9.326033e+01  1186.948  0.9117554   818.0948
  0.09090909  1.072267e+02  1197.778  0.9102313   828.0232
  0.09090909  1.232847e+02  1209.957  0.9084999   838.6767
  0.09090909  1.417474e+02  1223.473  0.9065574   850.0062
  0.09090909  1.629751e+02  1238.301  0.9044026   861.8916
  0.09090909  1.873817e+02  1254.288  0.9020562   874.2106
  0.09090909  2.154435e+02  1271.100  0.8995727   886.8666
  0.09090909  2.477076e+02  1288.708  0.8969610   899.8359
  0.09090909  2.848036e+02  1306.937  0.8942507   912.9559
  0.09090909  3.274549e+02  1326.236  0.8913681   926.2558
  0.09090909  3.764936e+02  1346.316  0.8883710   939.3972
  0.09090909  4.328761e+02  1366.202  0.8854574   951.8999
  0.09090909  4.977024e+02  1386.553  0.8825280   963.8516
  0.09090909  5.722368e+02  1407.973  0.8794915   975.5694
  0.09090909  6.579332e+02  1430.919  0.8762812   987.2467
  0.09090909  7.564633e+02  1455.778  0.8728405   999.0495
  0.09090909  8.697490e+02  1483.041  0.8690984  1011.2741
  0.09090909  1.000000e+03  1512.973  0.8650031  1024.0647
  0.10101010  1.000000e-03  1128.028  0.9197329   742.4345
  0.10101010  1.149757e-03  1128.028  0.9197329   742.4345
  0.10101010  1.321941e-03  1128.028  0.9197329   742.4345
  0.10101010  1.519911e-03  1128.028  0.9197329   742.4345
  0.10101010  1.747528e-03  1128.028  0.9197329   742.4345
  0.10101010  2.009233e-03  1128.028  0.9197329   742.4345
  0.10101010  2.310130e-03  1128.028  0.9197329   742.4345
  0.10101010  2.656088e-03  1128.028  0.9197329   742.4345
  0.10101010  3.053856e-03  1128.028  0.9197329   742.4345
  0.10101010  3.511192e-03  1128.028  0.9197329   742.4345
  0.10101010  4.037017e-03  1128.028  0.9197329   742.4345
  0.10101010  4.641589e-03  1128.028  0.9197329   742.4345
  0.10101010  5.336699e-03  1128.028  0.9197329   742.4345
  0.10101010  6.135907e-03  1128.028  0.9197329   742.4345
  0.10101010  7.054802e-03  1128.028  0.9197329   742.4345
  0.10101010  8.111308e-03  1128.028  0.9197329   742.4345
  0.10101010  9.326033e-03  1128.028  0.9197329   742.4345
  0.10101010  1.072267e-02  1128.028  0.9197329   742.4345
  0.10101010  1.232847e-02  1128.028  0.9197329   742.4345
  0.10101010  1.417474e-02  1128.028  0.9197329   742.4345
  0.10101010  1.629751e-02  1128.028  0.9197329   742.4345
  0.10101010  1.873817e-02  1128.028  0.9197329   742.4345
  0.10101010  2.154435e-02  1128.028  0.9197329   742.4345
  0.10101010  2.477076e-02  1128.028  0.9197329   742.4345
  0.10101010  2.848036e-02  1128.028  0.9197329   742.4345
  0.10101010  3.274549e-02  1128.028  0.9197329   742.4345
  0.10101010  3.764936e-02  1128.028  0.9197329   742.4345
  0.10101010  4.328761e-02  1128.028  0.9197329   742.4345
  0.10101010  4.977024e-02  1128.028  0.9197329   742.4345
  0.10101010  5.722368e-02  1128.028  0.9197329   742.4345
  0.10101010  6.579332e-02  1128.028  0.9197329   742.4345
  0.10101010  7.564633e-02  1128.028  0.9197329   742.4345
  0.10101010  8.697490e-02  1128.028  0.9197329   742.4345
  0.10101010  1.000000e-01  1128.028  0.9197329   742.4345
  0.10101010  1.149757e-01  1128.028  0.9197329   742.4345
  0.10101010  1.321941e-01  1128.028  0.9197329   742.4345
  0.10101010  1.519911e-01  1128.028  0.9197329   742.4345
  0.10101010  1.747528e-01  1128.028  0.9197329   742.4345
  0.10101010  2.009233e-01  1128.028  0.9197329   742.4345
  0.10101010  2.310130e-01  1128.028  0.9197329   742.4345
  0.10101010  2.656088e-01  1128.028  0.9197329   742.4345
  0.10101010  3.053856e-01  1128.028  0.9197329   742.4345
  0.10101010  3.511192e-01  1128.028  0.9197329   742.4345
  0.10101010  4.037017e-01  1128.028  0.9197329   742.4345
  0.10101010  4.641589e-01  1128.028  0.9197329   742.4345
  0.10101010  5.336699e-01  1128.028  0.9197329   742.4345
  0.10101010  6.135907e-01  1128.028  0.9197329   742.4345
  0.10101010  7.054802e-01  1128.028  0.9197329   742.4345
  0.10101010  8.111308e-01  1128.028  0.9197329   742.4345
  0.10101010  9.326033e-01  1128.028  0.9197329   742.4345
  0.10101010  1.072267e+00  1128.028  0.9197329   742.4345
  0.10101010  1.232847e+00  1128.028  0.9197329   742.4345
  0.10101010  1.417474e+00  1128.028  0.9197329   742.4345
  0.10101010  1.629751e+00  1128.028  0.9197329   742.4345
  0.10101010  1.873817e+00  1128.028  0.9197329   742.4345
  0.10101010  2.154435e+00  1128.028  0.9197329   742.4345
  0.10101010  2.477076e+00  1128.028  0.9197329   742.4345
  0.10101010  2.848036e+00  1128.028  0.9197329   742.4345
  0.10101010  3.274549e+00  1128.028  0.9197329   742.4345
  0.10101010  3.764936e+00  1128.051  0.9197298   742.5724
  0.10101010  4.328761e+00  1128.166  0.9197146   743.1595
  0.10101010  4.977024e+00  1128.322  0.9196939   743.8477
  0.10101010  5.722368e+00  1128.532  0.9196660   744.6488
  0.10101010  6.579332e+00  1128.790  0.9196319   745.5385
  0.10101010  7.564633e+00  1129.108  0.9195898   746.5561
  0.10101010  8.697490e+00  1129.500  0.9195379   747.7216
  0.10101010  1.000000e+01  1130.001  0.9194716   749.1003
  0.10101010  1.149757e+01  1130.642  0.9193864   750.7090
  0.10101010  1.321941e+01  1131.451  0.9192787   752.5660
  0.10101010  1.519911e+01  1132.471  0.9191426   754.7215
  0.10101010  1.747528e+01  1133.754  0.9189709   757.2226
  0.10101010  2.009233e+01  1135.359  0.9187556   760.1356
  0.10101010  2.310130e+01  1137.364  0.9184856   763.5360
  0.10101010  2.656088e+01  1139.832  0.9181519   767.4417
  0.10101010  3.053856e+01  1142.876  0.9177384   771.9875
  0.10101010  3.511192e+01  1146.573  0.9172335   777.1877
  0.10101010  4.037017e+01  1151.084  0.9166136   783.1879
  0.10101010  4.641589e+01  1155.039  0.9160795   787.6763
  0.10101010  5.336699e+01  1157.542  0.9157836   788.8960
  0.10101010  6.135907e+01  1162.282  0.9151624   793.2629
  0.10101010  7.054802e+01  1168.859  0.9142749   799.6294
  0.10101010  8.111308e+01  1177.185  0.9131279   807.9049
  0.10101010  9.326033e+01  1186.822  0.9117873   817.0995
  0.10101010  1.072267e+02  1197.709  0.9102580   826.9771
  0.10101010  1.232847e+02  1210.051  0.9085057   837.6961
  0.10101010  1.417474e+02  1223.672  0.9065507   849.0076
  0.10101010  1.629751e+02  1238.749  0.9043607   860.9624
  0.10101010  1.873817e+02  1254.757  0.9020180   873.3029
  0.10101010  2.154435e+02  1271.841  0.8994987   886.1113
  0.10101010  2.477076e+02  1289.348  0.8969130   899.1087
  0.10101010  2.848036e+02  1308.083  0.8941269   912.5423
  0.10101010  3.274549e+02  1327.618  0.8912138   925.8908
  0.10101010  3.764936e+02  1347.900  0.8881991   939.2016
  0.10101010  4.328761e+02  1367.801  0.8853064   951.6117
  0.10101010  4.977024e+02  1388.573  0.8823263   963.6953
  0.10101010  5.722368e+02  1410.561  0.8792126   975.6015
  0.10101010  6.579332e+02  1434.455  0.8758539   987.7355
  0.10101010  7.564633e+02  1460.307  0.8722563   999.9861
  0.10101010  8.697490e+02  1488.926  0.8682807  1012.9159
  0.10101010  1.000000e+03  1519.962  0.8640079  1026.1459
  0.11111111  1.000000e-03  1127.974  0.9197400   742.0945
  0.11111111  1.149757e-03  1127.974  0.9197400   742.0945
  0.11111111  1.321941e-03  1127.974  0.9197400   742.0945
  0.11111111  1.519911e-03  1127.974  0.9197400   742.0945
  0.11111111  1.747528e-03  1127.974  0.9197400   742.0945
  0.11111111  2.009233e-03  1127.974  0.9197400   742.0945
  0.11111111  2.310130e-03  1127.974  0.9197400   742.0945
  0.11111111  2.656088e-03  1127.974  0.9197400   742.0945
  0.11111111  3.053856e-03  1127.974  0.9197400   742.0945
  0.11111111  3.511192e-03  1127.974  0.9197400   742.0945
  0.11111111  4.037017e-03  1127.974  0.9197400   742.0945
  0.11111111  4.641589e-03  1127.974  0.9197400   742.0945
  0.11111111  5.336699e-03  1127.974  0.9197400   742.0945
  0.11111111  6.135907e-03  1127.974  0.9197400   742.0945
  0.11111111  7.054802e-03  1127.974  0.9197400   742.0945
  0.11111111  8.111308e-03  1127.974  0.9197400   742.0945
  0.11111111  9.326033e-03  1127.974  0.9197400   742.0945
  0.11111111  1.072267e-02  1127.974  0.9197400   742.0945
  0.11111111  1.232847e-02  1127.974  0.9197400   742.0945
  0.11111111  1.417474e-02  1127.974  0.9197400   742.0945
  0.11111111  1.629751e-02  1127.974  0.9197400   742.0945
  0.11111111  1.873817e-02  1127.974  0.9197400   742.0945
  0.11111111  2.154435e-02  1127.974  0.9197400   742.0945
  0.11111111  2.477076e-02  1127.974  0.9197400   742.0945
  0.11111111  2.848036e-02  1127.974  0.9197400   742.0945
  0.11111111  3.274549e-02  1127.974  0.9197400   742.0945
  0.11111111  3.764936e-02  1127.974  0.9197400   742.0945
  0.11111111  4.328761e-02  1127.974  0.9197400   742.0945
  0.11111111  4.977024e-02  1127.974  0.9197400   742.0945
  0.11111111  5.722368e-02  1127.974  0.9197400   742.0945
  0.11111111  6.579332e-02  1127.974  0.9197400   742.0945
  0.11111111  7.564633e-02  1127.974  0.9197400   742.0945
  0.11111111  8.697490e-02  1127.974  0.9197400   742.0945
  0.11111111  1.000000e-01  1127.974  0.9197400   742.0945
  0.11111111  1.149757e-01  1127.974  0.9197400   742.0945
  0.11111111  1.321941e-01  1127.974  0.9197400   742.0945
  0.11111111  1.519911e-01  1127.974  0.9197400   742.0945
  0.11111111  1.747528e-01  1127.974  0.9197400   742.0945
  0.11111111  2.009233e-01  1127.974  0.9197400   742.0945
  0.11111111  2.310130e-01  1127.974  0.9197400   742.0945
  0.11111111  2.656088e-01  1127.974  0.9197400   742.0945
  0.11111111  3.053856e-01  1127.974  0.9197400   742.0945
  0.11111111  3.511192e-01  1127.974  0.9197400   742.0945
  0.11111111  4.037017e-01  1127.974  0.9197400   742.0945
  0.11111111  4.641589e-01  1127.974  0.9197400   742.0945
  0.11111111  5.336699e-01  1127.974  0.9197400   742.0945
  0.11111111  6.135907e-01  1127.974  0.9197400   742.0945
  0.11111111  7.054802e-01  1127.974  0.9197400   742.0945
  0.11111111  8.111308e-01  1127.974  0.9197400   742.0945
  0.11111111  9.326033e-01  1127.974  0.9197400   742.0945
  0.11111111  1.072267e+00  1127.974  0.9197400   742.0945
  0.11111111  1.232847e+00  1127.974  0.9197400   742.0945
  0.11111111  1.417474e+00  1127.974  0.9197400   742.0945
  0.11111111  1.629751e+00  1127.974  0.9197400   742.0945
  0.11111111  1.873817e+00  1127.974  0.9197400   742.0945
  0.11111111  2.154435e+00  1127.974  0.9197400   742.0945
  0.11111111  2.477076e+00  1127.974  0.9197400   742.0945
  0.11111111  2.848036e+00  1127.974  0.9197400   742.0945
  0.11111111  3.274549e+00  1127.974  0.9197400   742.0945
  0.11111111  3.764936e+00  1128.051  0.9197298   742.5733
  0.11111111  4.328761e+00  1128.172  0.9197138   743.1714
  0.11111111  4.977024e+00  1128.334  0.9196923   743.8545
  0.11111111  5.722368e+00  1128.546  0.9196641   744.6585
  0.11111111  6.579332e+00  1128.805  0.9196299   745.5449
  0.11111111  7.564633e+00  1129.123  0.9195878   746.5576
  0.11111111  8.697490e+00  1129.517  0.9195356   747.7429
  0.11111111  1.000000e+01  1130.028  0.9194679   749.1384
  0.11111111  1.149757e+01  1130.675  0.9193819   750.7508
  0.11111111  1.321941e+01  1131.492  0.9192729   752.6186
  0.11111111  1.519911e+01  1132.526  0.9191350   754.7754
  0.11111111  1.747528e+01  1133.832  0.9189602   757.3107
  0.11111111  2.009233e+01  1135.461  0.9187415   760.2482
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  0.98989899  5.722368e+02  1646.649  0.8495414  1008.3795
  0.98989899  6.579332e+02  1678.160  0.8495408  1035.9602
  0.98989899  7.564633e+02  1718.899  0.8495408  1075.7161
  0.98989899  8.697490e+02  1771.273  0.8495408  1130.3429
  0.98989899  1.000000e+03  1838.162  0.8495408  1198.9810
  1.00000000  1.000000e-03  1128.134  0.9197161   740.9368
  1.00000000  1.149757e-03  1128.134  0.9197161   740.9368
  1.00000000  1.321941e-03  1128.134  0.9197161   740.9368
  1.00000000  1.519911e-03  1128.134  0.9197161   740.9368
  1.00000000  1.747528e-03  1128.134  0.9197161   740.9368
  1.00000000  2.009233e-03  1128.134  0.9197161   740.9368
  1.00000000  2.310130e-03  1128.134  0.9197161   740.9368
  1.00000000  2.656088e-03  1128.134  0.9197161   740.9368
  1.00000000  3.053856e-03  1128.134  0.9197161   740.9368
  1.00000000  3.511192e-03  1128.134  0.9197161   740.9368
  1.00000000  4.037017e-03  1128.134  0.9197161   740.9368
  1.00000000  4.641589e-03  1128.134  0.9197161   740.9368
  1.00000000  5.336699e-03  1128.134  0.9197161   740.9368
  1.00000000  6.135907e-03  1128.134  0.9197161   740.9368
  1.00000000  7.054802e-03  1128.134  0.9197161   740.9368
  1.00000000  8.111308e-03  1128.134  0.9197161   740.9368
  1.00000000  9.326033e-03  1128.134  0.9197161   740.9368
  1.00000000  1.072267e-02  1128.134  0.9197161   740.9368
  1.00000000  1.232847e-02  1128.134  0.9197161   740.9368
  1.00000000  1.417474e-02  1128.134  0.9197161   740.9368
  1.00000000  1.629751e-02  1128.134  0.9197161   740.9368
  1.00000000  1.873817e-02  1128.134  0.9197161   740.9368
  1.00000000  2.154435e-02  1128.134  0.9197161   740.9368
  1.00000000  2.477076e-02  1128.134  0.9197161   740.9368
  1.00000000  2.848036e-02  1128.134  0.9197161   740.9368
  1.00000000  3.274549e-02  1128.134  0.9197161   740.9368
  1.00000000  3.764936e-02  1128.134  0.9197161   740.9368
  1.00000000  4.328761e-02  1128.134  0.9197161   740.9368
  1.00000000  4.977024e-02  1128.134  0.9197161   740.9368
  1.00000000  5.722368e-02  1128.134  0.9197161   740.9368
  1.00000000  6.579332e-02  1128.134  0.9197161   740.9368
  1.00000000  7.564633e-02  1128.134  0.9197161   740.9368
  1.00000000  8.697490e-02  1128.134  0.9197161   740.9368
  1.00000000  1.000000e-01  1128.134  0.9197161   740.9368
  1.00000000  1.149757e-01  1128.134  0.9197161   740.9368
  1.00000000  1.321941e-01  1128.134  0.9197161   740.9368
  1.00000000  1.519911e-01  1128.134  0.9197161   740.9368
  1.00000000  1.747528e-01  1128.134  0.9197161   740.9368
  1.00000000  2.009233e-01  1128.134  0.9197161   740.9368
  1.00000000  2.310130e-01  1128.134  0.9197161   740.9368
  1.00000000  2.656088e-01  1128.134  0.9197161   740.9368
  1.00000000  3.053856e-01  1128.134  0.9197161   740.9368
  1.00000000  3.511192e-01  1128.134  0.9197161   740.9368
  1.00000000  4.037017e-01  1128.134  0.9197161   740.9368
  1.00000000  4.641589e-01  1128.134  0.9197161   740.9368
  1.00000000  5.336699e-01  1128.134  0.9197161   740.9368
  1.00000000  6.135907e-01  1128.134  0.9197161   740.9368
  1.00000000  7.054802e-01  1128.134  0.9197161   740.9368
  1.00000000  8.111308e-01  1128.134  0.9197161   740.9368
  1.00000000  9.326033e-01  1128.134  0.9197161   740.9368
  1.00000000  1.072267e+00  1128.134  0.9197161   740.9368
  1.00000000  1.232847e+00  1128.134  0.9197161   740.9368
  1.00000000  1.417474e+00  1128.134  0.9197161   740.9368
  1.00000000  1.629751e+00  1128.134  0.9197161   740.9368
  1.00000000  1.873817e+00  1128.131  0.9197163   740.9606
  1.00000000  2.154435e+00  1128.179  0.9197097   741.2045
  1.00000000  2.477076e+00  1128.292  0.9196944   741.5695
  1.00000000  2.848036e+00  1128.420  0.9196769   742.0015
  1.00000000  3.274549e+00  1128.593  0.9196534   742.5332
  1.00000000  3.764936e+00  1128.819  0.9196226   743.1610
  1.00000000  4.328761e+00  1129.112  0.9195825   743.9119
  1.00000000  4.977024e+00  1129.504  0.9195290   744.8111
  1.00000000  5.722368e+00  1130.024  0.9194576   745.8989
  1.00000000  6.579332e+00  1130.704  0.9193642   747.1905
  1.00000000  7.564633e+00  1131.599  0.9192408   748.7591
  1.00000000  8.697490e+00  1132.776  0.9190783   750.6825
  1.00000000  1.000000e+01  1134.343  0.9188610   753.1154
  1.00000000  1.149757e+01  1136.338  0.9185838   756.0578
  1.00000000  1.321941e+01  1138.977  0.9182152   759.7844
  1.00000000  1.519911e+01  1142.496  0.9177218   764.5263
  1.00000000  1.747528e+01  1146.989  0.9170885   770.3611
  1.00000000  2.009233e+01  1152.532  0.9163015   776.9343
  1.00000000  2.310130e+01  1159.300  0.9153319   784.3320
  1.00000000  2.656088e+01  1165.736  0.9144080   790.1445
  1.00000000  3.053856e+01  1168.882  0.9139819   789.7961
  1.00000000  3.511192e+01  1172.804  0.9134484   789.6553
  1.00000000  4.037017e+01  1176.545  0.9129552   789.4366
  1.00000000  4.641589e+01  1180.237  0.9124873   788.6182
  1.00000000  5.336699e+01  1184.682  0.9119256   787.8517
  1.00000000  6.135907e+01  1190.536  0.9111759   787.5838
  1.00000000  7.054802e+01  1197.826  0.9102376   787.8639
  1.00000000  8.111308e+01  1204.880  0.9093785   788.0277
  1.00000000  9.326033e+01  1213.321  0.9083592   788.7385
  1.00000000  1.072267e+02  1224.390  0.9069909   790.5527
  1.00000000  1.232847e+02  1238.872  0.9051502   794.0588
  1.00000000  1.417474e+02  1256.761  0.9028341   799.1109
  1.00000000  1.629751e+02  1276.574  0.9002995   804.5334
  1.00000000  1.873817e+02  1301.610  0.8969887   812.8292
  1.00000000  2.154435e+02  1331.412  0.8929595   823.6454
  1.00000000  2.477076e+02  1369.783  0.8874720   839.9699
  1.00000000  2.848036e+02  1414.920  0.8807605   859.6584
  1.00000000  3.274549e+02  1465.396  0.8730197   882.8598
  1.00000000  3.764936e+02  1502.953  0.8681271   901.0542
  1.00000000  4.328761e+02  1549.550  0.8617084   929.5449
  1.00000000  4.977024e+02  1608.358  0.8529148   976.3727
  1.00000000  5.722368e+02  1647.096  0.8495408  1008.7435
  1.00000000  6.579332e+02  1678.793  0.8495408  1036.5440
  1.00000000  7.564633e+02  1719.799  0.8495408  1076.6325
  1.00000000  8.697490e+02  1772.550  0.8495408  1131.6820
  1.00000000  1.000000e+03  1839.965  0.8495408  1200.7836

RMSE was used to select the optimal model using the smallest value.
The final values used for the model were alpha = 0.2323232 and lambda
 = 2.477076.

plot(Elastic_Net)

Elastic_Net$results %>%
  glimpse()

Rows: 10,000
Columns: 8
$ alpha      <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,…
$ lambda     <dbl> 0.001000000, 0.001149757, 0.001321941, 0.001519911, 0.00174…
$ RMSE       <dbl> 1327.815, 1327.815, 1327.815, 1327.815, 1327.815, 1327.815,…
$ Rsquared   <dbl> 0.8906263, 0.8906263, 0.8906263, 0.8906263, 0.8906263, 0.89…
$ MAE        <dbl> 940.5751, 940.5751, 940.5751, 940.5751, 940.5751, 940.5751,…
$ RMSESD     <dbl> 30.42761, 30.42761, 30.42761, 30.42761, 30.42761, 30.42761,…
$ RsquaredSD <dbl> 0.004456371, 0.004456371, 0.004456371, 0.004456371, 0.00445…
$ MAESD      <dbl> 13.31655, 13.31655, 13.31655, 13.31655, 13.31655, 13.31655,…

Elastic_Net$results %>%
  ggplot(aes(x=alpha, y=RMSE, color = lambda )) +
  geom_point() + 
  geom_line()

Elastic_Net$results %>%
  ggplot(aes(x=alpha, y=lambda, z=RMSE, color = RMSE)) + 
  geom_density_2d()

Warning: The following aesthetics were dropped during statistical transformation: z and
colour.
ℹ This can happen when ggplot fails to infer the correct grouping structure in
  the data.
ℹ Did you forget to specify a `group` aesthetic or to convert a numerical
  variable into a factor?