# The number of engineers in the Swedish regions, feature importance

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In my last post, I analysed the feature importance for the per cent of engineers in Sweden who are women. I found that the size of the region is a feature that is significant for the per cent of engineers in Sweden who are women. The size of the region is correlated to the number of engineers that works in the region. In this post, I will analyse what predictors are best to forecast the number of engineers in the region. I will use models from the caret package in my analysis.

Statistics Sweden use NUTS (Nomenclature des Unités Territoriales Statistiques), which is the EU’s hierarchical regional division, to specify the regions.

Please send suggestions for improvement of the analysis to [email protected].

First, define libraries and functions.

`library (tidyverse)`

`## -- Attaching packages -------------------------------------------------- tidyverse 1.3.0 --`

```
## v ggplot2 3.3.0 v purrr 0.3.4
## v tibble 3.0.0 v dplyr 0.8.5
## v tidyr 1.0.2 v stringr 1.4.0
## v readr 1.3.1 v forcats 0.5.0
```

```
## -- Conflicts ----------------------------------------------------- tidyverse_conflicts() --
## x dplyr::filter() masks stats::filter()
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```

```
library (broom)
library (car)
```

`## Loading required package: carData`

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`library (caret) `

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`library (recipes) `

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`library (PerformanceAnalytics)`

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`library (ggpubr)`

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```
library (ipred)
library (iml)
library (DALEX)
```

```
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```

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`library (Metrics)`

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`library (auditor)`

```
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```
readfile <- function (file1){read_csv (file1, col_types = cols(), locale = readr::locale (encoding = "latin1"), na = c("..", "NA")) %>%
gather (starts_with("19"), starts_with("20"), key = "year", value = groupsize) %>%
drop_na() %>%
mutate (year_n = parse_number (year))
}
perc_women <- function(x){
ifelse (length(x) == 2, x[2] / (x[1] + x[2]), NA)
}
nuts <- read.csv("nuts.csv") %>%
mutate(NUTS2_sh = substr(NUTS2, 3, 4))
nuts %>%
distinct (NUTS2_en) %>%
knitr::kable(
booktabs = TRUE,
caption = 'Nomenclature des Unités Territoriales Statistiques (NUTS)')
```

NUTS2_en |
---|

SE11 Stockholm |

SE12 East-Central Sweden |

SE21 Småland and islands |

SE22 South Sweden |

SE23 West Sweden |

SE31 North-Central Sweden |

SE32 Central Norrland |

SE33 Upper Norrland |

```
bs <- function(formula, data, indices) {
d <- data[indices,] # allows boot to select sample
fit <- lm(formula, weights = tbnum_weights, data=d)
return(coef(fit))
}
```

The data tables are downloaded from Statistics Sweden. They are saved as a comma-delimited file without heading, UF0506A1.csv, http://www.statistikdatabasen.scb.se/pxweb/en/ssd/.

The tables:

UF0506A1_1.csv: Population 16-74 years of age by region, highest level of education, age and sex. Year 1985 – 2018 NUTS 2 level 2008- 10 year intervals (16-74)

000000CG_1: Average basic salary, monthly salary and women´s salary as a percentage of men´s salary by region, sector, occupational group (SSYK 2012) and sex. Year 2014 – 2018 Monthly salary All sectors.

000000CD_1.csv: Average basic salary, monthly salary and women´s salary as a percentage of men´s salary by region, sector, occupational group (SSYK 2012) and sex. Year 2014 – 2018 Number of employees All sectors-

The data is aggregated, the size of each group is in the column groupsize.

I have also included some calculated predictors from the original data.

perc_women: The percentage of women within each group defined by edulevel, region and year

perc_women_region: The percentage of women within each group defined by year and region

regioneduyears: The average number of education years per capita within each group defined by year and region

eduquotient: The quotient between regioneduyears for men and women

salaryquotient: The quotient between salary for men and women within each group defined by year and region

perc_women_eng_region: The percentage of women who are engineers within each group defined by year and region

```
numedulevel <- read.csv("edulevel_1.csv")
numedulevel[, 2] <- data.frame(c(8, 9, 10, 12, 13, 15, 22, NA))
tb <- readfile("000000CG_1.csv")
tb <- readfile("000000CD_1.csv") %>%
left_join(tb, by = c("region", "year", "sex", "sector","occuptional (SSYK 2012)")) %>%
filter(`occuptional (SSYK 2012)` == "214 Engineering professionals")
tb <- readfile("UF0506A1_1.csv") %>%
right_join(tb, by = c("region", "year", "sex")) %>%
right_join(numedulevel, by = c("level of education" = "level.of.education")) %>%
filter(!is.na(eduyears)) %>%
mutate(edulevel = `level of education`) %>%
group_by(edulevel, region, year, sex) %>%
mutate(groupsize_all_ages = sum(groupsize)) %>%
group_by(edulevel, region, year) %>%
mutate (perc_women = perc_women (groupsize_all_ages[1:2])) %>%
mutate (suming = sum(groupsize.x)) %>%
mutate (salary = (groupsize.y[2] * groupsize.x[2] + groupsize.y[1] * groupsize.x[1])/(groupsize.x[2] + groupsize.x[1])) %>%
group_by (sex, year, region) %>%
mutate(regioneduyears_sex = sum(groupsize * eduyears) / sum(groupsize)) %>%
mutate(regiongroupsize = sum(groupsize)) %>%
mutate(suming_sex = groupsize.x) %>%
group_by(region, year) %>%
mutate (sum_pop = sum(groupsize)) %>%
mutate (regioneduyears = sum(groupsize * eduyears) / sum(groupsize)) %>%
mutate (perc_women_region = perc_women (regiongroupsize[1:2])) %>%
mutate (eduquotient = regioneduyears_sex[2] / regioneduyears_sex[1]) %>%
mutate (salary_sex = groupsize.y) %>%
mutate (salaryquotient = salary_sex[2] / salary_sex[1]) %>%
mutate (perc_women_eng_region = perc_women(suming_sex[1:2])) %>%
left_join(nuts %>% distinct (NUTS2_en, NUTS2_sh), by = c("region" = "NUTS2_en")) %>%
drop_na()
summary(tb)
```

```
## region age level of education sex
## Length:532 Length:532 Length:532 Length:532
## Class :character Class :character Class :character Class :character
## Mode :character Mode :character Mode :character Mode :character
##
##
##
## year groupsize year_n sector
## Length:532 Min. : 405 Min. :2014 Length:532
## Class :character 1st Qu.: 20996 1st Qu.:2015 Class :character
## Mode :character Median : 43656 Median :2016 Mode :character
## Mean : 64760 Mean :2016
## 3rd Qu.:102394 3rd Qu.:2017
## Max. :271889 Max. :2018
## occuptional (SSYK 2012) groupsize.x year_n.x groupsize.y
## Length:532 Min. : 340 Min. :2014 Min. :34700
## Class :character 1st Qu.: 1700 1st Qu.:2015 1st Qu.:40300
## Mode :character Median : 3000 Median :2016 Median :42000
## Mean : 5850 Mean :2016 Mean :42078
## 3rd Qu.: 7475 3rd Qu.:2017 3rd Qu.:43925
## Max. :21400 Max. :2018 Max. :49400
## year_n.y eduyears edulevel groupsize_all_ages
## Min. :2014 Min. : 8.00 Length:532 Min. : 405
## 1st Qu.:2015 1st Qu.: 9.00 Class :character 1st Qu.: 20996
## Median :2016 Median :12.00 Mode :character Median : 43656
## Mean :2016 Mean :12.71 Mean : 64760
## 3rd Qu.:2017 3rd Qu.:15.00 3rd Qu.:102394
## Max. :2018 Max. :22.00 Max. :271889
## perc_women suming salary regioneduyears_sex
## Min. :0.3575 Min. : 2040 Min. :38376 Min. :11.18
## 1st Qu.:0.4338 1st Qu.: 3080 1st Qu.:40916 1st Qu.:11.61
## Median :0.4631 Median : 8950 Median :42866 Median :11.74
## Mean :0.4771 Mean :11701 Mean :42776 Mean :11.79
## 3rd Qu.:0.5132 3rd Qu.:20100 3rd Qu.:44444 3rd Qu.:12.04
## Max. :0.6423 Max. :29500 Max. :48429 Max. :12.55
## regiongroupsize suming_sex sum_pop regioneduyears
## Min. :128262 Min. : 340 Min. : 262870 Min. :11.39
## 1st Qu.:288058 1st Qu.: 1700 1st Qu.: 587142 1st Qu.:11.54
## Median :514608 Median : 3000 Median :1029820 Median :11.81
## Mean :453318 Mean : 5850 Mean : 906635 Mean :11.79
## 3rd Qu.:691870 3rd Qu.: 7475 3rd Qu.:1395157 3rd Qu.:11.90
## Max. :827940 Max. :21400 Max. :1655215 Max. :12.41
## perc_women_region eduquotient salary_sex salaryquotient
## Min. :0.4831 Min. :1.019 Min. :34700 Min. :0.8653
## 1st Qu.:0.4882 1st Qu.:1.029 1st Qu.:40300 1st Qu.:0.9329
## Median :0.4934 Median :1.034 Median :42000 Median :0.9395
## Mean :0.4923 Mean :1.034 Mean :42078 Mean :0.9447
## 3rd Qu.:0.4970 3rd Qu.:1.041 3rd Qu.:43925 3rd Qu.:0.9537
## Max. :0.5014 Max. :1.047 Max. :49400 Max. :1.0446
## perc_women_eng_region NUTS2_sh
## Min. :0.1566 Length:532
## 1st Qu.:0.1787 Class :character
## Median :0.2042 Mode :character
## Mean :0.2039
## 3rd Qu.:0.2216
## Max. :0.2746
```

Prepare the data using Tidyverse recipes package, i.e. centre, scale and make sure all predictors are numerical.

```
tbtemp <- ungroup(tb) %>% dplyr::select(region, salary, year_n, sum_pop, regioneduyears, suming, perc_women_region, salaryquotient, eduquotient, perc_women_eng_region)
tb_outliers_info <- unique(tbtemp)
tb_unique <- unique(dplyr::select(tbtemp, -region))
tbnum_weights <- tb_unique$suming
blueprint <- recipe(suming ~ ., data = tb_unique) %>%
step_integer(matches("Qual|Cond|QC|Qu")) %>%
step_center(all_numeric(), -all_outcomes()) %>%
step_scale(all_numeric(), -all_outcomes()) %>%
step_dummy(all_nominal(), -all_outcomes(), one_hot = TRUE)
prepare <- prep(blueprint, training = tb_unique)
tbnum <- bake(prepare, new_data = tb_unique)
```

The correlation chart shows that many predictors are correlated with the response variable but also that many predictors are correlated between each other. Some notable correlations are in a dedicated plot below.

`chart.Correlation(tbnum, histogram = TRUE, pch = 19)`

```
p1 <- tb %>%
ggscatter(x = "sum_pop", y = "suming",
add = "reg.line", conf.int = TRUE,
cor.coef = TRUE, cor.method = "pearson")
p2 <- tb %>%
ggscatter(x = "sum_pop", y = "perc_women_region",
add = "reg.line", conf.int = TRUE,
cor.coef = TRUE, cor.method = "pearson")
p3 <- tb %>%
ggscatter(x = "suming", y = "perc_women_eng_region",
add = "reg.line", conf.int = TRUE,
cor.coef = TRUE, cor.method = "pearson")
p4 <- tb %>%
ggscatter(x = "sum_pop", y = "perc_women_eng_region",
add = "reg.line", conf.int = TRUE,
cor.coef = TRUE, cor.method = "pearson")
gridExtra::grid.arrange(p1, p2, p3, p4, ncol = 2)
```

```
## `geom_smooth()` using formula 'y ~ x'
## `geom_smooth()` using formula 'y ~ x'
## `geom_smooth()` using formula 'y ~ x'
## `geom_smooth()` using formula 'y ~ x'
```

plot_model will show diagnostics for the model, a residual plot, scale location plot, residual density and the regression error characteristic curve. It will also show the residuals versus the actual response to help find outliers. It will plot the feature importance and feature effects. In addition, it will plot how strongly features interact with each other and the 2-way interactions between the feature with the strongest interaction and all other features. The interaction measure regards how much of the variance of f(x) is explained by the interaction. The measure is between 0 (no interaction) and 1 (= 100% of variance of f(x) due to interactions).

```
plot_model <- function(model){
invisible(capture.output(exp_model <- DALEX::explain(model, data = tbnum, y = tbnum$suming))) # Knit and DALEX::explain generates invalid rss feed
lm_mr <- model_residual(exp_model)
predictor <- Predictor$new(model,
data = dplyr::select(tbnum, -suming),
y = tbnum$suming)
writeLines("")
print(model)
print(postResample(pred = predict(model), obs = tbnum$suming))
p1 <- plot(lm_mr, type = "residual")
p2 <- plot(lm_mr, type = "scalelocation")
p3 <- plot(lm_mr, type = "residual_density")
p4 <- plot(lm_mr, type = "rec")
print(gridExtra::grid.arrange(p1, p2, p3, p4, ncol = 2))
print(plot_residual(lm_mr, variable = "_y_hat_", nlabel = 10))
print(plot (FeatureImp$new(predictor, loss = "mae")))
print(plot (FeatureEffects$new(predictor)))
interact <- Interaction$new(predictor)
p1 <- plot (interact)
p2 <- plot (Interaction$new(predictor, feature = as.character(arrange(interact$results, desc(.interaction))[1,1])))
print(gridExtra::grid.arrange(p1, p2, ncol = 2))
}
```

Fit the following models and plot inference and diagnostics. Principal component analysis (PCA) is used to transform the data into a smaller subspace where the new variables are uncorrelated with one another due to the high multicollinearity.

Linear Regression, Projection Pursuit Regression, Bagged MARS, Random Forest, Bagged CART, Boosted Tree, Conditional Inference Tree

```
modelcollection <- c("lm", "ppr", "bagEarth", "ranger", "treebag", "blackboost", "ctree")
for (model in modelcollection){
invisible(capture.output(model <- caret::train(
suming ~ .,
data = tbnum,
method = model,
preProc=c("pca"),
weights = tbnum_weights,
trControl = trainControl(method = "cv", number = 10)
)))
plot_model(model)
}
```

```
##
## Linear Regression
##
## 38 samples
## 8 predictor
##
## Pre-processing: principal component signal extraction (8), centered (8),
## scaled (8)
## Resampling: Cross-Validated (10 fold)
## Summary of sample sizes: 35, 34, 34, 35, 34, 34, ...
## Resampling results:
##
## RMSE Rsquared MAE
## 4237.874 0.8822043 3423.46
##
## Tuning parameter 'intercept' was held constant at a value of TRUE
## RMSE Rsquared MAE
## 3883.8558386 0.8463367 2767.1742230
```

```
## TableGrob (2 x 2) "arrange": 4 grobs
## z cells name grob
## 1 1 (1-1,1-1) arrange gtable[layout]
## 2 2 (1-1,2-2) arrange gtable[layout]
## 3 3 (2-2,1-1) arrange gtable[layout]
## 4 4 (2-2,2-2) arrange gtable[layout]
```

```
##
## Attaching package: 'patchwork'
```

```
## The following object is masked from 'package:MASS':
##
## area
```

```
## TableGrob (1 x 2) "arrange": 2 grobs
## z cells name grob
## 1 1 (1-1,1-1) arrange gtable[layout]
## 2 2 (1-1,2-2) arrange gtable[layout]
##
## Projection Pursuit Regression
##
## 38 samples
## 8 predictor
##
## Pre-processing: principal component signal extraction (8), centered (8),
## scaled (8)
## Resampling: Cross-Validated (10 fold)
## Summary of sample sizes: 34, 34, 34, 34, 34, 34, ...
## Resampling results across tuning parameters:
##
## nterms RMSE Rsquared MAE
## 1 4130.062 0.8121472 3256.277
## 2 4246.374 0.8418553 3408.002
## 3 3906.203 0.8768507 3171.871
##
## RMSE was used to select the optimal model using the smallest value.
## The final value used for the model was nterms = 3.
## RMSE Rsquared MAE
## 1402.1162486 0.9796416 1090.5426055
```

```
## TableGrob (2 x 2) "arrange": 4 grobs
## z cells name grob
## 1 1 (1-1,1-1) arrange gtable[layout]
## 2 2 (1-1,2-2) arrange gtable[layout]
## 3 3 (2-2,1-1) arrange gtable[layout]
## 4 4 (2-2,2-2) arrange gtable[layout]
```

```
## TableGrob (1 x 2) "arrange": 2 grobs
## z cells name grob
## 1 1 (1-1,1-1) arrange gtable[layout]
## 2 2 (1-1,2-2) arrange gtable[layout]
```

`## Loading required package: earth`

`## Loading required package: Formula`

`## Loading required package: plotmo`

`## Loading required package: plotrix`

```
##
## Attaching package: 'plotrix'
```

```
## The following object is masked from 'package:scales':
##
## rescale
```

`## Loading required package: TeachingDemos`

```
##
## Bagged MARS
##
## 38 samples
## 8 predictor
##
## Pre-processing: principal component signal extraction (8), centered (8),
## scaled (8)
## Resampling: Cross-Validated (10 fold)
## Summary of sample sizes: 34, 34, 34, 34, 35, 34, ...
## Resampling results across tuning parameters:
##
## nprune RMSE Rsquared MAE
## 2 4671.353 0.8678650 3762.492
## 7 4301.170 0.8466750 3379.813
## 13 4535.948 0.8434574 3713.592
##
## Tuning parameter 'degree' was held constant at a value of 1
## RMSE was used to select the optimal model using the smallest value.
## The final values used for the model were nprune = 7 and degree = 1.
## RMSE Rsquared MAE
## 2950.1269148 0.9075753 2178.7339222
```

```
## TableGrob (2 x 2) "arrange": 4 grobs
## z cells name grob
## 1 1 (1-1,1-1) arrange gtable[layout]
## 2 2 (1-1,2-2) arrange gtable[layout]
## 3 3 (2-2,1-1) arrange gtable[layout]
## 4 4 (2-2,2-2) arrange gtable[layout]
```

```
## TableGrob (1 x 2) "arrange": 2 grobs
## z cells name grob
## 1 1 (1-1,1-1) arrange gtable[layout]
## 2 2 (1-1,2-2) arrange gtable[layout]
##
## Random Forest
##
## 38 samples
## 8 predictor
##
## Pre-processing: principal component signal extraction (8), centered (8),
## scaled (8)
## Resampling: Cross-Validated (10 fold)
## Summary of sample sizes: 34, 34, 34, 34, 34, 34, ...
## Resampling results across tuning parameters:
##
## mtry splitrule RMSE Rsquared MAE
## 2 variance 6106.457 0.7899054 5600.253
## 2 extratrees 5899.990 0.8178129 5413.206
## 3 variance 5242.999 0.8372295 4583.812
## 3 extratrees 4978.308 0.8145967 4339.194
## 4 variance 4367.341 0.8690214 3526.925
## 4 extratrees 4461.317 0.8391514 3797.024
##
## Tuning parameter 'min.node.size' was held constant at a value of 5
## RMSE was used to select the optimal model using the smallest value.
## The final values used for the model were mtry = 4, splitrule = variance
## and min.node.size = 5.
## RMSE Rsquared MAE
## 1782.3243833 0.9749664 1275.2575351
```

```
## TableGrob (2 x 2) "arrange": 4 grobs
## z cells name grob
## 1 1 (1-1,1-1) arrange gtable[layout]
## 2 2 (1-1,2-2) arrange gtable[layout]
## 3 3 (2-2,1-1) arrange gtable[layout]
## 4 4 (2-2,2-2) arrange gtable[layout]
```

```
## TableGrob (1 x 2) "arrange": 2 grobs
## z cells name grob
## 1 1 (1-1,1-1) arrange gtable[layout]
## 2 2 (1-1,2-2) arrange gtable[layout]
##
## Bagged CART
##
## 38 samples
## 8 predictor
##
## Pre-processing: principal component signal extraction (8), centered (8),
## scaled (8)
## Resampling: Cross-Validated (10 fold)
## Summary of sample sizes: 35, 34, 34, 34, 34, 34, ...
## Resampling results:
##
## RMSE Rsquared MAE
## 4828.894 0.8094027 3527.702
##
## RMSE Rsquared MAE
## 3481.9118680 0.8750994 2582.9979981
```

```
## TableGrob (2 x 2) "arrange": 4 grobs
## z cells name grob
## 1 1 (1-1,1-1) arrange gtable[layout]
## 2 2 (1-1,2-2) arrange gtable[layout]
## 3 3 (2-2,1-1) arrange gtable[layout]
## 4 4 (2-2,2-2) arrange gtable[layout]
```

```
## TableGrob (1 x 2) "arrange": 2 grobs
## z cells name grob
## 1 1 (1-1,1-1) arrange gtable[layout]
## 2 2 (1-1,2-2) arrange gtable[layout]
```

```
## Warning in if (class(response) == "data.frame") response <- as.matrix(response):
## the condition has length > 1 and only the first element will be used
## Warning in if (class(response) == "data.frame") response <- as.matrix(response):
## the condition has length > 1 and only the first element will be used
```

```
##
## Boosted Tree
##
## 38 samples
## 8 predictor
##
## Pre-processing: principal component signal extraction (8), centered (8),
## scaled (8)
## Resampling: Cross-Validated (10 fold)
## Summary of sample sizes: 34, 34, 34, 34, 34, 35, ...
## Resampling results across tuning parameters:
##
## maxdepth mstop RMSE Rsquared MAE
## 1 50 4440.664 0.7708467 3558.175
## 1 100 4429.005 0.7943325 3512.541
## 1 150 4456.381 0.7981314 3487.719
## 2 50 4598.712 0.8028983 3497.232
## 2 100 4582.388 0.7786909 3482.167
## 2 150 4546.031 0.7990669 3437.192
## 3 50 4166.522 0.8926944 2940.884
## 3 100 4134.064 0.8935203 2910.923
## 3 150 4139.154 0.8939033 2918.382
##
## RMSE was used to select the optimal model using the smallest value.
## The final values used for the model were mstop = 100 and maxdepth = 3.
## RMSE Rsquared MAE
## 110.7432126 0.9998769 88.9875261
```

```
## TableGrob (2 x 2) "arrange": 4 grobs
## z cells name grob
## 1 1 (1-1,1-1) arrange gtable[layout]
## 2 2 (1-1,2-2) arrange gtable[layout]
## 3 3 (2-2,1-1) arrange gtable[layout]
## 4 4 (2-2,2-2) arrange gtable[layout]
```

```
## TableGrob (1 x 2) "arrange": 2 grobs
## z cells name grob
## 1 1 (1-1,1-1) arrange gtable[layout]
## 2 2 (1-1,2-2) arrange gtable[layout]
##
## Conditional Inference Tree
##
## 38 samples
## 8 predictor
##
## Pre-processing: principal component signal extraction (8), centered (8),
## scaled (8)
## Resampling: Cross-Validated (10 fold)
## Summary of sample sizes: 34, 34, 35, 34, 34, 34, ...
## Resampling results across tuning parameters:
##
## mincriterion RMSE Rsquared MAE
## 0.01 4244.251 0.7539372 2824.5
## 0.50 4244.251 0.7539372 2824.5
## 0.99 4244.251 0.7539372 2824.5
##
## RMSE was used to select the optimal model using the smallest value.
## The final value used for the model was mincriterion = 0.99.
## RMSE Rsquared MAE
## 0 1 0
```

`## Warning: Removed 38 rows containing missing values (geom_point).`

```
## TableGrob (2 x 2) "arrange": 4 grobs
## z cells name grob
## 1 1 (1-1,1-1) arrange gtable[layout]
## 2 2 (1-1,2-2) arrange gtable[layout]
## 3 3 (2-2,1-1) arrange gtable[layout]
## 4 4 (2-2,2-2) arrange gtable[layout]
```

```
## Warning in .subset2(public_bind_env, "initialize")(...): Model error is 0,
## switching from compare='ratio' to compare='difference'
```

```
## TableGrob (1 x 2) "arrange": 2 grobs
## z cells name grob
## 1 1 (1-1,1-1) arrange gtable[layout]
## 2 2 (1-1,2-2) arrange gtable[layout]
```

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