Ridge model

library(rvfl)

Example 1: MPG Prediction (mtcars dataset)

Load and prepare data

data(mtcars)

set.seed(1243)
train_idx <- sample(nrow(mtcars), size = floor(0.8 * nrow(mtcars)))
train_data <- mtcars[train_idx, ]
test_data <- mtcars[-train_idx, -1]

Fit models

# Fit regular linear model
start <- proc.time()[3]
lm_model <- lm(mpg ~ ., data = train_data)
print(proc.time()[3] - start)

## elapsed 
##    0.01

print(summary(lm_model))

## 
## Call:
## lm(formula = mpg ~ ., data = train_data)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.5211 -0.9792 -0.0324  1.1808  4.9814 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)  
## (Intercept) -5.054416  25.456900  -0.199   0.8455  
## cyl          0.695392   1.396506   0.498   0.6262  
## disp         0.005254   0.017342   0.303   0.7664  
## hp          -0.007610   0.027723  -0.274   0.7877  
## drat         4.128157   2.724353   1.515   0.1520  
## wt          -1.621396   2.139071  -0.758   0.4610  
## qsec         0.064356   0.932144   0.069   0.9459  
## vs           0.138716   3.421183   0.041   0.9682  
## am          -0.498476   2.956568  -0.169   0.8685  
## gear         4.402648   2.287816   1.924   0.0749 .
## carb        -1.999389   1.299580  -1.538   0.1462  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.464 on 14 degrees of freedom
## Multiple R-squared:  0.8938, Adjusted R-squared:  0.818 
## F-statistic: 11.79 on 10 and 14 DF,  p-value: 3.4e-05

print(confint(lm_model))

##                    2.5 %      97.5 %
## (Intercept) -59.65403559 49.54520296
## cyl          -2.29981561  3.69060001
## disp         -0.03194096  0.04244882
## hp           -0.06707095  0.05185084
## drat         -1.71500030  9.97131342
## wt           -6.20924769  2.96645550
## qsec         -1.93489537  2.06360651
## vs           -7.19899241  7.47642359
## am           -6.83968216  5.84273112
## gear         -0.50422869  9.30952400
## carb         -4.78671119  0.78793282

# Fit calibrated model 
start <- proc.time()[3]
ridge_model <- rvfl::calibmodel(lambda=10**seq(-10, 10, length.out=100), x = as.matrix(train_data[,-1]), y = train_data$mpg)
print(proc.time()[3] - start)

## elapsed 
##   0.019

print(summary(ridge_model))

## $model_summary
## 
## Call:
## engine(formula = y ~ . - 1, data = df_train)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.5715 -1.6454 -0.6596  1.9582  5.6548 
## 
## Coefficients:
##      Estimate Std. Error t value Pr(>|t|)
## cyl   -0.4956     0.7586  -0.653    0.524
## disp  -0.5709     0.7473  -0.764    0.458
## hp    -0.6475     0.7508  -0.862    0.403
## drat   0.8272     0.6951   1.190    0.254
## wt    -0.7784     0.7239  -1.075    0.300
## qsec   0.2927     0.7384   0.396    0.698
## vs     0.4345     0.7133   0.609    0.552
## am     0.7154     0.6849   1.045    0.314
## gear   0.4152     0.7179   0.578    0.572
## carb  -0.7023     0.6358  -1.105    0.288
## 
## Residual standard error: 2.99 on 14 degrees of freedom
## Multiple R-squared:  0.7422, Adjusted R-squared:  0.5581 
## F-statistic: 4.031 on 10 and 14 DF,  p-value: 0.009055
## 
## 
## $derivatives_summary
##      Effect.mean of x    Std.Error       p.value     CI.lower     CI.upper
## cyl      -0.272562363 1.251420e-11 1.261186e-223 -0.272562363 -0.272562363
## disp     -0.004417317 1.602004e-11 5.557056e-180 -0.004417317 -0.004417317
## hp       -0.011728958 1.766407e-11 9.254360e-189 -0.011728958 -0.011728958
## drat      1.594838830 1.851980e-11 2.340861e-237  1.594838830  1.594838830
## wt       -0.748782818 1.224970e-11 6.207613e-234 -0.748782819 -0.748782818
## qsec      0.188050094 3.700743e-12 4.350927e-232  0.188050094  0.188050094
## vs        0.857626338 1.185444e-11 1.287167e-235  0.857626338  0.857626338
## am        1.460396700 6.124848e-12 1.572499e-247  1.460396700  1.460396700
## gear      0.638082724 1.510165e-11 3.031228e-230  0.638082724  0.638082724
## carb     -0.532426743 1.504239e-11 1.780484e-228 -0.532426743 -0.532426743
## 
## $all_derivatives
##              cyl         disp          hp     drat         wt      qsec
##  [1,] -0.2725624 -0.004417317 -0.01172896 1.594839 -0.7487828 0.1880501
##  [2,] -0.2725624 -0.004417317 -0.01172896 1.594839 -0.7487828 0.1880501
##  [3,] -0.2725624 -0.004417317 -0.01172896 1.594839 -0.7487828 0.1880501
##  [4,] -0.2725624 -0.004417317 -0.01172896 1.594839 -0.7487828 0.1880501
##  [5,] -0.2725624 -0.004417317 -0.01172896 1.594839 -0.7487828 0.1880501
##  [6,] -0.2725624 -0.004417317 -0.01172896 1.594839 -0.7487828 0.1880501
##  [7,] -0.2725624 -0.004417317 -0.01172896 1.594839 -0.7487828 0.1880501
##  [8,] -0.2725624 -0.004417317 -0.01172896 1.594839 -0.7487828 0.1880501
##  [9,] -0.2725624 -0.004417317 -0.01172896 1.594839 -0.7487828 0.1880501
## [10,] -0.2725624 -0.004417317 -0.01172896 1.594839 -0.7487828 0.1880501
## [11,] -0.2725624 -0.004417317 -0.01172896 1.594839 -0.7487828 0.1880501
## [12,] -0.2725624 -0.004417317 -0.01172896 1.594839 -0.7487828 0.1880501
## [13,] -0.2725624 -0.004417317 -0.01172896 1.594839 -0.7487828 0.1880501
## [14,] -0.2725624 -0.004417317 -0.01172896 1.594839 -0.7487828 0.1880501
## [15,] -0.2725624 -0.004417317 -0.01172896 1.594839 -0.7487828 0.1880501
## [16,] -0.2725624 -0.004417317 -0.01172896 1.594839 -0.7487828 0.1880501
## [17,] -0.2725624 -0.004417317 -0.01172896 1.594839 -0.7487828 0.1880501
## [18,] -0.2725624 -0.004417317 -0.01172896 1.594839 -0.7487828 0.1880501
## [19,] -0.2725624 -0.004417317 -0.01172896 1.594839 -0.7487828 0.1880501
## [20,] -0.2725624 -0.004417317 -0.01172896 1.594839 -0.7487828 0.1880501
## [21,] -0.2725624 -0.004417317 -0.01172896 1.594839 -0.7487828 0.1880501
## [22,] -0.2725624 -0.004417317 -0.01172896 1.594839 -0.7487828 0.1880501
## [23,] -0.2725624 -0.004417317 -0.01172896 1.594839 -0.7487828 0.1880501
## [24,] -0.2725624 -0.004417317 -0.01172896 1.594839 -0.7487828 0.1880501
##              vs       am      gear       carb
##  [1,] 0.8576263 1.460397 0.6380827 -0.5324267
##  [2,] 0.8576263 1.460397 0.6380827 -0.5324267
##  [3,] 0.8576263 1.460397 0.6380827 -0.5324267
##  [4,] 0.8576263 1.460397 0.6380827 -0.5324267
##  [5,] 0.8576263 1.460397 0.6380827 -0.5324267
##  [6,] 0.8576263 1.460397 0.6380827 -0.5324267
##  [7,] 0.8576263 1.460397 0.6380827 -0.5324267
##  [8,] 0.8576263 1.460397 0.6380827 -0.5324267
##  [9,] 0.8576263 1.460397 0.6380827 -0.5324267
## [10,] 0.8576263 1.460397 0.6380827 -0.5324267
## [11,] 0.8576263 1.460397 0.6380827 -0.5324267
## [12,] 0.8576263 1.460397 0.6380827 -0.5324267
## [13,] 0.8576263 1.460397 0.6380827 -0.5324267
## [14,] 0.8576263 1.460397 0.6380827 -0.5324267
## [15,] 0.8576263 1.460397 0.6380827 -0.5324267
## [16,] 0.8576263 1.460397 0.6380827 -0.5324267
## [17,] 0.8576263 1.460397 0.6380827 -0.5324267
## [18,] 0.8576263 1.460397 0.6380827 -0.5324267
## [19,] 0.8576263 1.460397 0.6380827 -0.5324267
## [20,] 0.8576263 1.460397 0.6380827 -0.5324267
## [21,] 0.8576263 1.460397 0.6380827 -0.5324267
## [22,] 0.8576263 1.460397 0.6380827 -0.5324267
## [23,] 0.8576263 1.460397 0.6380827 -0.5324267
## [24,] 0.8576263 1.460397 0.6380827 -0.5324267

print(plot(ridge_model))

## NULL

Example 2: Boston Housing Price Prediction

Load and prepare data

library(MASS)
data(Boston)

set.seed(1243)
train_idx <- sample(nrow(Boston), size = floor(0.8 * nrow(Boston)))
train_data <- Boston[train_idx, ]
test_data <- Boston[-train_idx, -14]  # -14 removes 'medv' (target variable)

Fit models

# Fit regular linear model
start <- proc.time()[3]
lm_model <- lm(medv ~ ., data = train_data)
print(proc.time()[3] - start)

## elapsed 
##    0.01

print(summary(lm_model))

## 
## Call:
## lm(formula = medv ~ ., data = train_data)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -15.220  -2.757  -0.494   1.863  26.961 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  35.832339   5.763210   6.217 1.30e-09 ***
## crim         -0.095389   0.034717  -2.748 0.006282 ** 
## zn            0.042689   0.016086   2.654 0.008283 ** 
## indus        -0.033013   0.073521  -0.449 0.653657    
## chas          2.506064   0.939731   2.667 0.007977 ** 
## nox         -17.521010   4.237379  -4.135 4.35e-05 ***
## rm            3.966727   0.477640   8.305 1.66e-15 ***
## age           0.006479   0.014922   0.434 0.664410    
## dis          -1.463187   0.232348  -6.297 8.17e-10 ***
## rad           0.253984   0.075379   3.369 0.000828 ***
## tax          -0.009853   0.004350  -2.265 0.024068 *  
## ptratio      -1.002914   0.147016  -6.822 3.44e-11 ***
## black         0.008723   0.002984   2.923 0.003664 ** 
## lstat        -0.501984   0.057704  -8.699  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.835 on 390 degrees of freedom
## Multiple R-squared:  0.7403, Adjusted R-squared:  0.7316 
## F-statistic: 85.51 on 13 and 390 DF,  p-value: < 2.2e-16

print(confint(lm_model))

##                    2.5 %      97.5 %
## (Intercept)  24.50149090 47.16318623
## crim         -0.16364441 -0.02713269
## zn            0.01106383  0.07431451
## indus        -0.17755967  0.11153316
## chas          0.65849254  4.35363615
## nox         -25.85197459 -9.19004505
## rm            3.02765548  4.90579898
## age          -0.02285933  0.03581667
## dis          -1.91999833 -1.00637601
## rad           0.10578379  0.40218383
## tax          -0.01840515 -0.00129986
## ptratio      -1.29195676 -0.71387117
## black         0.00285660  0.01458927
## lstat        -0.61543411 -0.38853422

# Fit calibrated model 
start <- proc.time()[3]
ridge_model <- rvfl::calibmodel(lambda=10**seq(-10, 10, length.out=100), x = as.matrix(train_data[,-14]), y = train_data$medv)
print(proc.time()[3] - start)

## elapsed 
##   0.011

print(summary(ridge_model))

## $model_summary
## 
## Call:
## engine(formula = y ~ . - 1, data = df_train)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -10.8939  -2.9240  -0.5968   1.9927  26.0705 
## 
## Coefficients:
##         Estimate Std. Error t value Pr(>|t|)    
## crim     -1.0059     0.5000  -2.012 0.045575 *  
## zn        0.8264     0.4991   1.656 0.099298 .  
## indus     0.1430     0.7574   0.189 0.850401    
## chas      0.6056     0.3571   1.696 0.091434 .  
## nox      -1.8962     0.6871  -2.760 0.006311 ** 
## rm        2.7488     0.4796   5.731 3.56e-08 ***
## age       0.7043     0.6016   1.171 0.243052    
## dis      -2.1701     0.6775  -3.203 0.001579 ** 
## rad       1.9496     0.9275   2.102 0.036785 *  
## tax      -1.3630     1.0753  -1.268 0.206424    
## ptratio  -1.7350     0.4696  -3.694 0.000283 ***
## black     0.7928     0.4409   1.798 0.073660 .  
## lstat    -3.9392     0.5793  -6.800 1.15e-10 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 5.065 on 203 degrees of freedom
## Multiple R-squared:  0.6897, Adjusted R-squared:  0.6698 
## F-statistic: 34.71 on 13 and 203 DF,  p-value: < 2.2e-16
## 
## 
## $derivatives_summary
##         Effect.mean of x    Std.Error p.value      CI.lower      CI.upper
## crim        -0.109407657 1.326001e-11       0  -0.109407657  -0.109407657
## zn           0.034950695 1.356861e-11       0   0.034950695   0.034950695
## indus        0.020962278 6.519300e-12       0   0.020962278   0.020962278
## chas         2.272330710 1.209542e-11       0   2.272330710   2.272330710
## nox        -16.268105574 1.475402e-11       0 -16.268105574 -16.268105574
## rm           3.876061598 1.534257e-11       0   3.876061598   3.876061598
## age          0.024769829 7.315992e-12       0   0.024769829   0.024769829
## dis         -1.020603896 1.194942e-11       0  -1.020603896  -1.020603896
## rad          0.222028558 1.236129e-11       0   0.222028558   0.222028558
## tax         -0.007978888 5.183059e-12       0  -0.007978888  -0.007978888
## ptratio     -0.793111734 1.383592e-11       0  -0.793111734  -0.793111734
## black        0.008408181 1.089693e-11       0   0.008408181   0.008408181
## lstat       -0.539244502 9.324810e-12       0  -0.539244502  -0.539244502
## 
## $all_derivatives
##              crim         zn      indus     chas       nox       rm        age
##   [1,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##   [2,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##   [3,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##   [4,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##   [5,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##   [6,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##   [7,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##   [8,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##   [9,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [10,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [11,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [12,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [13,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [14,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [15,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [16,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [17,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [18,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [19,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [20,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [21,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [22,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [23,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [24,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [25,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [26,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [27,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [28,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [29,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [30,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [31,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [32,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [33,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [34,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [35,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [36,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [37,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [38,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [39,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [40,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [41,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [42,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [43,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [44,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [45,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [46,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [47,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [48,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [49,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [50,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [51,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [52,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [53,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [54,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [55,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [56,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [57,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [58,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [59,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [60,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [61,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [62,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [63,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [64,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [65,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [66,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [67,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [68,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [69,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [70,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [71,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [72,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [73,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [74,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [75,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [76,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [77,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [78,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [79,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [80,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [81,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [82,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [83,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [84,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [85,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [86,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [87,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [88,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [89,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [90,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [91,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [92,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [93,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [94,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [95,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [96,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [97,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [98,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##  [99,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [100,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [101,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [102,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [103,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [104,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [105,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [106,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [107,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [108,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [109,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [110,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [111,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [112,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [113,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [114,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [115,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [116,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [117,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [118,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [119,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [120,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [121,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [122,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [123,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [124,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [125,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [126,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [127,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [128,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [129,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [130,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [131,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [132,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [133,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [134,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [135,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [136,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [137,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [138,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [139,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [140,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [141,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [142,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [143,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [144,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [145,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [146,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [147,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [148,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [149,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [150,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [151,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [152,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [153,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [154,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [155,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [156,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [157,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [158,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [159,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [160,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [161,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [162,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [163,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [164,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [165,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [166,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [167,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [168,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [169,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [170,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [171,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [172,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [173,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [174,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [175,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [176,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [177,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [178,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [179,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [180,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [181,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [182,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [183,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [184,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [185,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [186,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [187,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [188,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [189,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [190,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [191,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [192,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [193,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [194,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [195,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [196,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [197,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [198,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [199,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [200,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [201,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [202,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [203,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [204,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [205,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [206,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [207,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [208,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [209,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [210,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [211,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [212,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [213,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [214,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [215,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
## [216,] -0.1094077 0.03495069 0.02096228 2.272331 -16.26811 3.876062 0.02476983
##              dis       rad          tax    ptratio       black      lstat
##   [1,] -1.020604 0.2220286 -0.007978888 -0.7931117 0.008408181 -0.5392445
##   [2,] -1.020604 0.2220286 -0.007978888 -0.7931117 0.008408181 -0.5392445
##   [3,] -1.020604 0.2220286 -0.007978888 -0.7931117 0.008408181 -0.5392445
##   [4,] -1.020604 0.2220286 -0.007978888 -0.7931117 0.008408180 -0.5392445
##   [5,] -1.020604 0.2220286 -0.007978888 -0.7931117 0.008408181 -0.5392445
##   [6,] -1.020604 0.2220286 -0.007978888 -0.7931117 0.008408181 -0.5392445
##   [7,] -1.020604 0.2220286 -0.007978888 -0.7931117 0.008408181 -0.5392445
##   [8,] -1.020604 0.2220286 -0.007978888 -0.7931117 0.008408180 -0.5392445
##   [9,] -1.020604 0.2220286 -0.007978888 -0.7931117 0.008408181 -0.5392445
##  [10,] -1.020604 0.2220286 -0.007978888 -0.7931117 0.008408181 -0.5392445
##  [11,] -1.020604 0.2220286 -0.007978888 -0.7931117 0.008408181 -0.5392445
##  [12,] -1.020604 0.2220286 -0.007978888 -0.7931117 0.008408181 -0.5392445
##  [13,] -1.020604 0.2220286 -0.007978888 -0.7931117 0.008408181 -0.5392445
##  [14,] -1.020604 0.2220286 -0.007978888 -0.7931117 0.008408181 -0.5392445
##  [15,] -1.020604 0.2220286 -0.007978888 -0.7931117 0.008408181 -0.5392445
##  [16,] -1.020604 0.2220286 -0.007978888 -0.7931117 0.008408181 -0.5392445
##  [17,] -1.020604 0.2220286 -0.007978888 -0.7931117 0.008408181 -0.5392445
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## [211,] -1.020604 0.2220286 -0.007978888 -0.7931117 0.008408181 -0.5392445
## [212,] -1.020604 0.2220286 -0.007978888 -0.7931117 0.008408181 -0.5392445
## [213,] -1.020604 0.2220286 -0.007978888 -0.7931117 0.008408181 -0.5392445
## [214,] -1.020604 0.2220286 -0.007978888 -0.7931117 0.008408181 -0.5392445
## [215,] -1.020604 0.2220286 -0.007978888 -0.7931117 0.008408181 -0.5392445
## [216,] -1.020604 0.2220286 -0.007978888 -0.7931117 0.008408181 -0.5392445

print(plot(ridge_model))

## NULL

Example 3: Economic Indicators (Longley dataset)

Load and prepare data

data(longley)

set.seed(1243)
train_idx <- sample(nrow(longley), size = floor(0.8 * nrow(longley)))
train_data <- longley[train_idx, ]
test_data <- longley[-train_idx, -7]  # -7 removes 'Employed' (target variable)

Fit models

# Fit regular linear model
start <- proc.time()[3]
lm_model <- lm(Employed ~ ., data = train_data)
print(proc.time()[3] - start)

## elapsed 
##   0.003

print(summary(lm_model))

## 
## Call:
## lm(formula = Employed ~ ., data = train_data)
## 
## Residuals:
##     1948     1951     1962     1960     1953     1958     1956     1947 
## -0.15209  0.23935 -0.15902 -0.15702  0.04017 -0.11637  0.35418  0.07762 
##     1954     1955     1952     1961 
## -0.03138 -0.23164 -0.18207  0.31828 
## 
## Coefficients:
##                Estimate Std. Error t value Pr(>|t|)   
## (Intercept)  -3.314e+03  9.930e+02  -3.338  0.02061 * 
## GNP.deflator -7.877e-02  1.033e-01  -0.763  0.48016   
## GNP          -6.186e-03  4.032e-02  -0.153  0.88407   
## Unemployed   -1.578e-02  6.060e-03  -2.604  0.04802 * 
## Armed.Forces -1.074e-02  2.478e-03  -4.336  0.00745 **
## Population   -3.256e-01  2.798e-01  -1.164  0.29697   
## Year          1.758e+00  5.066e-01   3.470  0.01784 * 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.3053 on 5 degrees of freedom
## Multiple R-squared:  0.9962, Adjusted R-squared:  0.9916 
## F-statistic: 216.2 on 6 and 5 DF,  p-value: 7.153e-06

print(confint(lm_model))

##                      2.5 %        97.5 %
## (Intercept)  -5.866601e+03 -7.615995e+02
## GNP.deflator -3.443182e-01  1.867776e-01
## GNP          -1.098296e-01  9.745798e-02
## Unemployed   -3.135588e-02 -2.021522e-04
## Armed.Forces -1.711223e-02 -4.374897e-03
## Population   -1.044745e+00  3.935168e-01
## Year          4.557539e-01  3.060217e+00

# Fit calibrated model 
start <- proc.time()[3]
ridge_model <- rvfl::calibmodel(lambda=10**seq(-10, 10, length.out=100), x = as.matrix(train_data[,-7]), y = train_data$Employed)
print(proc.time()[3] - start)

## elapsed 
##   0.008

# print(summary(ridge_model))
print(plot(ridge_model))

## NULL

Example 4: US Crime Rate Analysis

Load and prepare data

data(UScrime)

set.seed(1243)
train_idx <- sample(nrow(UScrime), size = floor(0.8 * nrow(UScrime)))
train_data <- UScrime[train_idx, ]
test_data <- UScrime[-train_idx, -16]  # -16 removes 'y' (crime rate)

Fit models

# Fit regular linear model
start <- proc.time()[3]
lm_model <- lm(y ~ ., data = train_data)
print(proc.time()[3] - start)

## elapsed 
##   0.004

print(summary(lm_model))

## 
## Call:
## lm(formula = y ~ ., data = train_data)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -391.18 -114.34  -14.92  117.15  460.03 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)   
## (Intercept) -6991.5860  2219.2203  -3.150  0.00483 **
## M               7.1029     5.4076   1.313  0.20319   
## So            -89.2556   183.9512  -0.485  0.63255   
## Ed             22.1862     7.7816   2.851  0.00956 **
## Po1            19.4504    15.2637   1.274  0.21648   
## Po2           -10.7837    16.0263  -0.673  0.50837   
## LF             -1.3616     1.8522  -0.735  0.47038   
## M.F             2.9017     2.6491   1.095  0.28576   
## Pop            -0.9371     1.6448  -0.570  0.57492   
## NW              0.8155     0.9873   0.826  0.41810   
## U1             -8.4003     5.8482  -1.436  0.16562   
## U2             19.8835    11.0640   1.797  0.08671 . 
## GDP             0.7120     1.2620   0.564  0.57861   
## Ineq            8.0589     2.8019   2.876  0.00904 **
## Prob        -3533.2550  2862.6770  -1.234  0.23074   
## Time            1.3920     9.4699   0.147  0.88454   
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 234.8 on 21 degrees of freedom
## Multiple R-squared:  0.758,  Adjusted R-squared:  0.5851 
## F-statistic: 4.385 on 15 and 21 DF,  p-value: 0.001081

print(confint(lm_model))

##                     2.5 %       97.5 %
## (Intercept) -11606.707241 -2376.464825
## M               -4.142866    18.348646
## So            -471.803028   293.291831
## Ed               6.003559    38.368826
## Po1            -12.292291    51.193066
## Po2            -44.112293    22.544935
## LF              -5.213545     2.490260
## M.F             -2.607377     8.410782
## Pop             -4.357700     2.483540
## NW              -1.237650     2.868551
## U1             -20.562297     3.761712
## U2              -3.125406    42.892337
## GDP             -1.912548     3.336557
## Ineq             2.232069    13.885810
## Prob         -9486.517778  2420.007809
## Time           -18.301822    21.085767

# Fit calibrated model 
start <- proc.time()[3]
ridge_model <- rvfl::calibmodel(lambda=10**seq(-10, 10, length.out=100), x = as.matrix(train_data[,-16]), y = train_data$y)
print(proc.time()[3] - start)

## elapsed 
##   0.009

print(summary(ridge_model))

## $model_summary
## 
## Call:
## engine(formula = y ~ . - 1, data = df_train)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -362.44 -182.76  -80.78   49.85  441.14 
## 
## Coefficients:
##      Estimate Std. Error t value Pr(>|t|)
## M      80.823     73.867   1.094    0.287
## So    -44.123     88.038  -0.501    0.622
## Ed      8.284     82.799   0.100    0.921
## Po1    86.606     89.029   0.973    0.342
## Po2    90.128     88.143   1.023    0.319
## LF     29.463     79.163   0.372    0.714
## M.F    52.984     65.431   0.810    0.428
## Pop   -42.968     79.616  -0.540    0.595
## NW     97.504     86.985   1.121    0.276
## U1     -8.699     72.594  -0.120    0.906
## U2     26.426     75.343   0.351    0.729
## GDP    67.582     86.059   0.785    0.441
## Ineq   81.123     82.098   0.988    0.335
## Prob  -66.384     64.063  -1.036    0.312
## Time   41.257     63.728   0.647    0.525
## 
## Residual standard error: 245 on 20 degrees of freedom
## Multiple R-squared:  0.4321, Adjusted R-squared:  0.006091 
## F-statistic: 1.014 on 15 and 20 DF,  p-value: 0.4792
## 
## 
## $derivatives_summary
##      Effect.mean of x    Std.Error       p.value      CI.lower      CI.upper
## M           6.4106715 4.932428e-09 1.983781e-285     6.4106715     6.4106715
## So        -91.1675631 4.286112e-09  0.000000e+00   -91.1675631   -91.1675631
## Ed          0.7733606 3.859431e-09 8.034392e-258     0.7733605     0.7733606
## Po1         2.9747341 3.103584e-09 6.226907e-281     2.9747341     2.9747341
## Po2         3.2426017 4.912254e-09 2.001309e-275     3.2426017     3.2426017
## LF          0.7449199 3.897834e-09 4.021792e-257     0.7449199     0.7449199
## M.F         1.9354222 4.072264e-09 1.419697e-270     1.9354222     1.9354222
## Pop        -1.0406821 4.674003e-09 2.232342e-259    -1.0406821    -1.0406821
## NW          1.0284186 5.206657e-09 1.310319e-257     1.0284186     1.0284187
## U1         -0.5081944 4.912254e-09 4.641734e-248    -0.5081944    -0.5081944
## U2          3.0572692 4.544743e-09 1.052257e-275     3.0572692     3.0572692
## GDP         0.7398920 5.366321e-09 2.663376e-252     0.7398920     0.7398920
## Ineq        2.1276257 3.103584e-09 5.534414e-276     2.1276257     2.1276257
## Prob    -2939.4151255 1.850918e-09  0.000000e+00 -2939.4151255 -2939.4151255
## Time        5.6384259 4.229996e-09 8.401203e-286     5.6384259     5.6384259
## 
## $all_derivatives
##              M        So        Ed      Po1      Po2        LF      M.F
##  [1,] 6.410672 -91.16756 0.7733606 2.974734 3.242602 0.7449199 1.935422
##  [2,] 6.410672 -91.16756 0.7733605 2.974734 3.242602 0.7449199 1.935422
##  [3,] 6.410672 -91.16756 0.7733605 2.974734 3.242602 0.7449199 1.935422
##  [4,] 6.410672 -91.16756 0.7733605 2.974734 3.242602 0.7449198 1.935422
##  [5,] 6.410672 -91.16756 0.7733605 2.974734 3.242602 0.7449199 1.935422
##  [6,] 6.410672 -91.16756 0.7733605 2.974734 3.242602 0.7449199 1.935422
##  [7,] 6.410672 -91.16756 0.7733605 2.974734 3.242602 0.7449199 1.935422
##  [8,] 6.410672 -91.16756 0.7733605 2.974734 3.242602 0.7449199 1.935422
##  [9,] 6.410672 -91.16756 0.7733605 2.974734 3.242602 0.7449199 1.935422
## [10,] 6.410672 -91.16756 0.7733606 2.974734 3.242602 0.7449199 1.935422
## [11,] 6.410672 -91.16756 0.7733606 2.974734 3.242602 0.7449199 1.935422
## [12,] 6.410672 -91.16756 0.7733605 2.974734 3.242602 0.7449199 1.935422
## [13,] 6.410671 -91.16756 0.7733605 2.974734 3.242602 0.7449199 1.935422
## [14,] 6.410672 -91.16756 0.7733605 2.974734 3.242602 0.7449199 1.935422
## [15,] 6.410671 -91.16756 0.7733606 2.974734 3.242602 0.7449199 1.935422
## [16,] 6.410672 -91.16756 0.7733605 2.974734 3.242602 0.7449199 1.935422
## [17,] 6.410672 -91.16756 0.7733605 2.974734 3.242602 0.7449198 1.935422
## [18,] 6.410672 -91.16756 0.7733605 2.974734 3.242602 0.7449199 1.935422
## [19,] 6.410672 -91.16756 0.7733605 2.974734 3.242602 0.7449199 1.935422
## [20,] 6.410671 -91.16756 0.7733606 2.974734 3.242602 0.7449199 1.935422
## [21,] 6.410672 -91.16756 0.7733606 2.974734 3.242602 0.7449199 1.935422
## [22,] 6.410672 -91.16756 0.7733605 2.974734 3.242602 0.7449199 1.935422
## [23,] 6.410672 -91.16756 0.7733605 2.974734 3.242602 0.7449199 1.935422
## [24,] 6.410672 -91.16756 0.7733606 2.974734 3.242602 0.7449199 1.935422
## [25,] 6.410672 -91.16756 0.7733605 2.974734 3.242602 0.7449199 1.935422
## [26,] 6.410672 -91.16756 0.7733606 2.974734 3.242602 0.7449199 1.935422
## [27,] 6.410672 -91.16756 0.7733605 2.974734 3.242602 0.7449199 1.935422
## [28,] 6.410672 -91.16756 0.7733606 2.974734 3.242602 0.7449199 1.935422
## [29,] 6.410672 -91.16756 0.7733606 2.974734 3.242602 0.7449199 1.935422
## [30,] 6.410672 -91.16756 0.7733606 2.974734 3.242602 0.7449199 1.935422
## [31,] 6.410672 -91.16756 0.7733606 2.974734 3.242602 0.7449199 1.935422
## [32,] 6.410672 -91.16756 0.7733606 2.974734 3.242602 0.7449199 1.935422
## [33,] 6.410672 -91.16756 0.7733606 2.974734 3.242602 0.7449199 1.935422
## [34,] 6.410672 -91.16756 0.7733605 2.974734 3.242602 0.7449198 1.935422
## [35,] 6.410672 -91.16756 0.7733606 2.974734 3.242602 0.7449199 1.935422
##             Pop       NW         U1       U2       GDP     Ineq      Prob
##  [1,] -1.040682 1.028419 -0.5081944 3.057269 0.7398920 2.127626 -2939.415
##  [2,] -1.040682 1.028419 -0.5081944 3.057269 0.7398920 2.127626 -2939.415
##  [3,] -1.040682 1.028419 -0.5081944 3.057269 0.7398920 2.127626 -2939.415
##  [4,] -1.040682 1.028419 -0.5081944 3.057269 0.7398921 2.127626 -2939.415
##  [5,] -1.040682 1.028419 -0.5081944 3.057269 0.7398920 2.127626 -2939.415
##  [6,] -1.040682 1.028419 -0.5081944 3.057269 0.7398920 2.127626 -2939.415
##  [7,] -1.040682 1.028419 -0.5081944 3.057269 0.7398920 2.127626 -2939.415
##  [8,] -1.040682 1.028419 -0.5081944 3.057269 0.7398920 2.127626 -2939.415
##  [9,] -1.040682 1.028419 -0.5081944 3.057269 0.7398920 2.127626 -2939.415
## [10,] -1.040682 1.028419 -0.5081944 3.057269 0.7398920 2.127626 -2939.415
## [11,] -1.040682 1.028419 -0.5081944 3.057269 0.7398920 2.127626 -2939.415
## [12,] -1.040682 1.028419 -0.5081944 3.057269 0.7398920 2.127626 -2939.415
## [13,] -1.040682 1.028419 -0.5081944 3.057269 0.7398919 2.127626 -2939.415
## [14,] -1.040682 1.028419 -0.5081944 3.057269 0.7398920 2.127626 -2939.415
## [15,] -1.040682 1.028419 -0.5081944 3.057269 0.7398920 2.127626 -2939.415
## [16,] -1.040682 1.028419 -0.5081944 3.057269 0.7398920 2.127626 -2939.415
## [17,] -1.040682 1.028419 -0.5081944 3.057269 0.7398920 2.127626 -2939.415
## [18,] -1.040682 1.028419 -0.5081944 3.057269 0.7398920 2.127626 -2939.415
## [19,] -1.040682 1.028419 -0.5081944 3.057269 0.7398920 2.127626 -2939.415
## [20,] -1.040682 1.028419 -0.5081944 3.057269 0.7398920 2.127626 -2939.415
## [21,] -1.040682 1.028419 -0.5081944 3.057269 0.7398920 2.127626 -2939.415
## [22,] -1.040682 1.028419 -0.5081944 3.057269 0.7398920 2.127626 -2939.415
## [23,] -1.040682 1.028419 -0.5081944 3.057269 0.7398920 2.127626 -2939.415
## [24,] -1.040682 1.028419 -0.5081944 3.057269 0.7398920 2.127626 -2939.415
## [25,] -1.040682 1.028419 -0.5081944 3.057269 0.7398920 2.127626 -2939.415
## [26,] -1.040682 1.028419 -0.5081944 3.057269 0.7398920 2.127626 -2939.415
## [27,] -1.040682 1.028419 -0.5081944 3.057269 0.7398920 2.127626 -2939.415
## [28,] -1.040682 1.028419 -0.5081944 3.057269 0.7398920 2.127626 -2939.415
## [29,] -1.040682 1.028419 -0.5081944 3.057269 0.7398920 2.127626 -2939.415
## [30,] -1.040682 1.028419 -0.5081944 3.057269 0.7398920 2.127626 -2939.415
## [31,] -1.040682 1.028419 -0.5081944 3.057269 0.7398920 2.127626 -2939.415
## [32,] -1.040682 1.028419 -0.5081944 3.057269 0.7398920 2.127626 -2939.415
## [33,] -1.040682 1.028419 -0.5081944 3.057269 0.7398920 2.127626 -2939.415
## [34,] -1.040682 1.028419 -0.5081944 3.057269 0.7398920 2.127626 -2939.415
## [35,] -1.040682 1.028419 -0.5081944 3.057269 0.7398920 2.127626 -2939.415
##           Time
##  [1,] 5.638426
##  [2,] 5.638426
##  [3,] 5.638426
##  [4,] 5.638426
##  [5,] 5.638426
##  [6,] 5.638426
##  [7,] 5.638426
##  [8,] 5.638426
##  [9,] 5.638426
## [10,] 5.638426
## [11,] 5.638426
## [12,] 5.638426
## [13,] 5.638426
## [14,] 5.638426
## [15,] 5.638426
## [16,] 5.638426
## [17,] 5.638426
## [18,] 5.638426
## [19,] 5.638426
## [20,] 5.638426
## [21,] 5.638426
## [22,] 5.638426
## [23,] 5.638426
## [24,] 5.638426
## [25,] 5.638426
## [26,] 5.638426
## [27,] 5.638426
## [28,] 5.638426
## [29,] 5.638426
## [30,] 5.638426
## [31,] 5.638426
## [32,] 5.638426
## [33,] 5.638426
## [34,] 5.638426
## [35,] 5.638426

print(plot(ridge_model))

## NULL

Example 5: Car Price Analysis (Cars93 dataset)

Load and prepare data

data(Cars93, package = "MASS")

# Remove rows with missing values
Cars93 <- na.omit(Cars93)

# Select numeric predictors and price as response
predictors <- c("MPG.city", "MPG.highway", "EngineSize", "Horsepower", 
                "RPM", "Rev.per.mile", "Fuel.tank.capacity", "Length", 
                "Wheelbase", "Width", "Turn.circle", "Weight")
car_data <- Cars93[, c(predictors, "Price")]

set.seed(1243)
train_idx <- sample(nrow(car_data), size = floor(0.8 * nrow(car_data)))
train_data <- car_data[train_idx, ]
test_data <- car_data[-train_idx, -which(names(car_data) == "Price")]

Fit models

# Fit regular linear model
start <- proc.time()[3]
lm_model <- lm(Price ~ ., data = train_data)
print(proc.time()[3] - start)

## elapsed 
##   0.003

print(summary(lm_model))

## 
## Call:
## lm(formula = Price ~ ., data = train_data)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -8.9444 -3.4879 -0.0823  2.5740 10.7036 
## 
## Coefficients:
##                      Estimate Std. Error t value Pr(>|t|)  
## (Intercept)         6.0285088 31.8235292   0.189   0.8505  
## MPG.city           -0.3069383  0.4798163  -0.640   0.5252  
## MPG.highway        -0.0041568  0.4591254  -0.009   0.9928  
## EngineSize          2.4810783  2.4779636   1.001   0.3213  
## Horsepower          0.1016741  0.0446071   2.279   0.0268 *
## RPM                 0.0001602  0.0023006   0.070   0.9448  
## Rev.per.mile        0.0049762  0.0026868   1.852   0.0697 .
## Fuel.tank.capacity -0.1866149  0.5198553  -0.359   0.7211  
## Length              0.0203242  0.1333518   0.152   0.8795  
## Wheelbase           0.4888949  0.2655782   1.841   0.0713 .
## Width              -0.8957689  0.4446575  -2.015   0.0491 *
## Turn.circle        -0.3835124  0.3579624  -1.071   0.2889  
## Weight              0.0041302  0.0059593   0.693   0.4914  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.731 on 52 degrees of freedom
## Multiple R-squared:  0.7886, Adjusted R-squared:  0.7398 
## F-statistic: 16.16 on 12 and 52 DF,  p-value: 1.548e-13

print(confint(lm_model))

##                            2.5 %       97.5 %
## (Intercept)        -5.783007e+01 69.887091986
## MPG.city           -1.269760e+00  0.655883438
## MPG.highway        -9.254594e-01  0.917145804
## EngineSize         -2.491319e+00  7.453475962
## Horsepower          1.216348e-02  0.191184768
## RPM                -4.456410e-03  0.004776746
## Rev.per.mile       -4.152533e-04  0.010367616
## Fuel.tank.capacity -1.229781e+00  0.856550981
## Length             -2.472657e-01  0.287914102
## Wheelbase          -4.402679e-02  1.021816593
## Width              -1.788039e+00 -0.003498354
## Turn.circle        -1.101817e+00  0.334791684
## Weight             -7.828121e-03  0.016088453

# Fit calibrated model 
start <- proc.time()[3]
ridge_model <- rvfl::calibmodel(lambda=10**seq(-10, 10, length.out=100), x = as.matrix(train_data[,-which(names(train_data) == "Price")]), 
                               y = train_data$Price)
print(proc.time()[3] - start)

## elapsed 
##   0.009

print(summary(ridge_model))

## $model_summary
## 
## Call:
## engine(formula = y ~ . - 1, data = df_train)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -7.7848 -2.9438 -0.5461  1.0213 13.0730 
## 
## Coefficients:
##                    Estimate Std. Error t value Pr(>|t|)  
## MPG.city           -0.90230    1.49983  -0.602   0.5516  
## MPG.highway        -0.33641    1.39992  -0.240   0.8116  
## EngineSize          2.56961    1.62400   1.582   0.1231  
## Horsepower          2.43635    1.35158   1.803   0.0806 .
## RPM                 0.90504    1.05839   0.855   0.3987  
## Rev.per.mile        1.30430    1.26257   1.033   0.3091  
## Fuel.tank.capacity  0.24579    1.53119   0.161   0.8734  
## Length              0.48894    1.63807   0.298   0.7672  
## Wheelbase           1.73442    1.50930   1.149   0.2588  
## Width              -1.03940    1.50301  -0.692   0.4941  
## Turn.circle        -0.04015    1.23036  -0.033   0.9742  
## Weight              1.26377    1.72926   0.731   0.4701  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.635 on 33 degrees of freedom
## Multiple R-squared:  0.7162, Adjusted R-squared:  0.613 
## F-statistic: 6.941 on 12 and 33 DF,  p-value: 4.626e-06
## 
## 
## $derivatives_summary
##                    Effect.mean of x    Std.Error       p.value     CI.lower
## MPG.city               -0.167966274 4.572614e-11  0.000000e+00 -0.167966274
## MPG.highway            -0.070195128 4.167413e-11  0.000000e+00 -0.070195129
## EngineSize              2.469188497 4.459692e-11  0.000000e+00  2.469188497
## Horsepower              0.047182250 4.584989e-11  0.000000e+00  0.047182250
## RPM                     0.001491490 4.694898e-11 1.399465e-295  0.001491490
## Rev.per.mile            0.002555233 2.245672e-11 5.833927e-320  0.002555233
## Fuel.tank.capacity      0.079054422 4.167413e-11  0.000000e+00  0.079054422
## Length                  0.032229864 3.461836e-11  0.000000e+00  0.032229864
## Wheelbase               0.264582497 3.441316e-11  0.000000e+00  0.264582497
## Width                  -0.272013951 2.905658e-11  0.000000e+00 -0.272013951
## Turn.circle            -0.012610575 2.740055e-11  0.000000e+00 -0.012610575
## Weight                  0.002223263 2.479512e-11 2.080750e-315  0.002223263
##                        CI.upper
## MPG.city           -0.167966274
## MPG.highway        -0.070195128
## EngineSize          2.469188497
## Horsepower          0.047182251
## RPM                 0.001491490
## Rev.per.mile        0.002555233
## Fuel.tank.capacity  0.079054422
## Length              0.032229864
## Wheelbase           0.264582497
## Width              -0.272013951
## Turn.circle        -0.012610575
## Weight              0.002223263
## 
## $all_derivatives
##         MPG.city MPG.highway EngineSize Horsepower        RPM Rev.per.mile
##  [1,] -0.1679663 -0.07019513   2.469188 0.04718225 0.00149149  0.002555233
##  [2,] -0.1679663 -0.07019513   2.469188 0.04718225 0.00149149  0.002555233
##  [3,] -0.1679663 -0.07019513   2.469188 0.04718225 0.00149149  0.002555233
##  [4,] -0.1679663 -0.07019513   2.469188 0.04718225 0.00149149  0.002555233
##  [5,] -0.1679663 -0.07019513   2.469188 0.04718225 0.00149149  0.002555233
##  [6,] -0.1679663 -0.07019513   2.469188 0.04718225 0.00149149  0.002555233
##  [7,] -0.1679663 -0.07019513   2.469188 0.04718225 0.00149149  0.002555233
##  [8,] -0.1679663 -0.07019513   2.469188 0.04718225 0.00149149  0.002555233
##  [9,] -0.1679663 -0.07019513   2.469188 0.04718225 0.00149149  0.002555233
## [10,] -0.1679663 -0.07019513   2.469188 0.04718225 0.00149149  0.002555233
## [11,] -0.1679663 -0.07019513   2.469188 0.04718225 0.00149149  0.002555233
## [12,] -0.1679663 -0.07019513   2.469188 0.04718225 0.00149149  0.002555233
## [13,] -0.1679663 -0.07019513   2.469188 0.04718225 0.00149149  0.002555233
## [14,] -0.1679663 -0.07019513   2.469188 0.04718225 0.00149149  0.002555233
## [15,] -0.1679663 -0.07019513   2.469188 0.04718225 0.00149149  0.002555233
## [16,] -0.1679663 -0.07019513   2.469188 0.04718225 0.00149149  0.002555233
## [17,] -0.1679663 -0.07019513   2.469188 0.04718225 0.00149149  0.002555233
## [18,] -0.1679663 -0.07019513   2.469188 0.04718225 0.00149149  0.002555233
## [19,] -0.1679663 -0.07019513   2.469188 0.04718225 0.00149149  0.002555232
## [20,] -0.1679663 -0.07019513   2.469188 0.04718225 0.00149149  0.002555233
## [21,] -0.1679663 -0.07019513   2.469188 0.04718225 0.00149149  0.002555233
## [22,] -0.1679663 -0.07019513   2.469188 0.04718225 0.00149149  0.002555233
## [23,] -0.1679663 -0.07019513   2.469188 0.04718225 0.00149149  0.002555233
## [24,] -0.1679663 -0.07019513   2.469188 0.04718225 0.00149149  0.002555233
## [25,] -0.1679663 -0.07019513   2.469188 0.04718225 0.00149149  0.002555233
## [26,] -0.1679663 -0.07019513   2.469188 0.04718225 0.00149149  0.002555233
## [27,] -0.1679663 -0.07019513   2.469188 0.04718225 0.00149149  0.002555233
## [28,] -0.1679663 -0.07019513   2.469188 0.04718225 0.00149149  0.002555233
## [29,] -0.1679663 -0.07019513   2.469188 0.04718225 0.00149149  0.002555233
## [30,] -0.1679663 -0.07019513   2.469188 0.04718225 0.00149149  0.002555233
## [31,] -0.1679663 -0.07019513   2.469188 0.04718225 0.00149149  0.002555233
## [32,] -0.1679663 -0.07019513   2.469188 0.04718225 0.00149149  0.002555233
## [33,] -0.1679663 -0.07019513   2.469188 0.04718225 0.00149149  0.002555233
## [34,] -0.1679663 -0.07019513   2.469188 0.04718225 0.00149149  0.002555233
## [35,] -0.1679663 -0.07019513   2.469188 0.04718225 0.00149149  0.002555233
## [36,] -0.1679663 -0.07019513   2.469188 0.04718225 0.00149149  0.002555233
## [37,] -0.1679663 -0.07019513   2.469188 0.04718225 0.00149149  0.002555233
## [38,] -0.1679663 -0.07019513   2.469188 0.04718225 0.00149149  0.002555233
## [39,] -0.1679663 -0.07019513   2.469188 0.04718225 0.00149149  0.002555233
## [40,] -0.1679663 -0.07019513   2.469188 0.04718225 0.00149149  0.002555233
## [41,] -0.1679663 -0.07019513   2.469188 0.04718225 0.00149149  0.002555233
## [42,] -0.1679663 -0.07019513   2.469188 0.04718225 0.00149149  0.002555233
## [43,] -0.1679663 -0.07019513   2.469188 0.04718225 0.00149149  0.002555233
## [44,] -0.1679663 -0.07019513   2.469188 0.04718225 0.00149149  0.002555233
## [45,] -0.1679663 -0.07019513   2.469188 0.04718225 0.00149149  0.002555233
##       Fuel.tank.capacity     Length Wheelbase     Width Turn.circle      Weight
##  [1,]         0.07905442 0.03222986 0.2645825 -0.272014 -0.01261057 0.002223263
##  [2,]         0.07905442 0.03222986 0.2645825 -0.272014 -0.01261058 0.002223263
##  [3,]         0.07905442 0.03222986 0.2645825 -0.272014 -0.01261057 0.002223263
##  [4,]         0.07905442 0.03222986 0.2645825 -0.272014 -0.01261057 0.002223263
##  [5,]         0.07905442 0.03222986 0.2645825 -0.272014 -0.01261057 0.002223263
##  [6,]         0.07905442 0.03222986 0.2645825 -0.272014 -0.01261057 0.002223263
##  [7,]         0.07905442 0.03222986 0.2645825 -0.272014 -0.01261057 0.002223263
##  [8,]         0.07905442 0.03222986 0.2645825 -0.272014 -0.01261058 0.002223263
##  [9,]         0.07905442 0.03222986 0.2645825 -0.272014 -0.01261057 0.002223263
## [10,]         0.07905442 0.03222986 0.2645825 -0.272014 -0.01261057 0.002223263
## [11,]         0.07905442 0.03222986 0.2645825 -0.272014 -0.01261058 0.002223263
## [12,]         0.07905442 0.03222986 0.2645825 -0.272014 -0.01261057 0.002223263
## [13,]         0.07905442 0.03222986 0.2645825 -0.272014 -0.01261058 0.002223263
## [14,]         0.07905442 0.03222986 0.2645825 -0.272014 -0.01261057 0.002223263
## [15,]         0.07905442 0.03222986 0.2645825 -0.272014 -0.01261058 0.002223263
## [16,]         0.07905442 0.03222986 0.2645825 -0.272014 -0.01261057 0.002223263
## [17,]         0.07905442 0.03222986 0.2645825 -0.272014 -0.01261058 0.002223263
## [18,]         0.07905442 0.03222986 0.2645825 -0.272014 -0.01261058 0.002223263
## [19,]         0.07905442 0.03222986 0.2645825 -0.272014 -0.01261057 0.002223263
## [20,]         0.07905442 0.03222986 0.2645825 -0.272014 -0.01261057 0.002223263
## [21,]         0.07905442 0.03222986 0.2645825 -0.272014 -0.01261058 0.002223263
## [22,]         0.07905442 0.03222986 0.2645825 -0.272014 -0.01261057 0.002223263
## [23,]         0.07905442 0.03222986 0.2645825 -0.272014 -0.01261057 0.002223263
## [24,]         0.07905442 0.03222986 0.2645825 -0.272014 -0.01261057 0.002223263
## [25,]         0.07905442 0.03222986 0.2645825 -0.272014 -0.01261057 0.002223263
## [26,]         0.07905442 0.03222986 0.2645825 -0.272014 -0.01261058 0.002223263
## [27,]         0.07905442 0.03222986 0.2645825 -0.272014 -0.01261057 0.002223263
## [28,]         0.07905442 0.03222986 0.2645825 -0.272014 -0.01261058 0.002223263
## [29,]         0.07905442 0.03222986 0.2645825 -0.272014 -0.01261057 0.002223263
## [30,]         0.07905442 0.03222986 0.2645825 -0.272014 -0.01261057 0.002223263
## [31,]         0.07905442 0.03222986 0.2645825 -0.272014 -0.01261057 0.002223263
## [32,]         0.07905442 0.03222986 0.2645825 -0.272014 -0.01261058 0.002223263
## [33,]         0.07905442 0.03222986 0.2645825 -0.272014 -0.01261058 0.002223263
## [34,]         0.07905442 0.03222986 0.2645825 -0.272014 -0.01261057 0.002223263
## [35,]         0.07905442 0.03222986 0.2645825 -0.272014 -0.01261057 0.002223263
## [36,]         0.07905442 0.03222986 0.2645825 -0.272014 -0.01261057 0.002223263
## [37,]         0.07905442 0.03222986 0.2645825 -0.272014 -0.01261057 0.002223263
## [38,]         0.07905442 0.03222986 0.2645825 -0.272014 -0.01261057 0.002223263
## [39,]         0.07905442 0.03222986 0.2645825 -0.272014 -0.01261057 0.002223263
## [40,]         0.07905442 0.03222986 0.2645825 -0.272014 -0.01261057 0.002223263
## [41,]         0.07905442 0.03222986 0.2645825 -0.272014 -0.01261057 0.002223263
## [42,]         0.07905442 0.03222986 0.2645825 -0.272014 -0.01261057 0.002223263
## [43,]         0.07905442 0.03222986 0.2645825 -0.272014 -0.01261057 0.002223263
## [44,]         0.07905442 0.03222986 0.2645825 -0.272014 -0.01261057 0.002223263
## [45,]         0.07905442 0.03222986 0.2645825 -0.272014 -0.01261057 0.002223263

print(plot(ridge_model))

## NULL