Learning PCA
Jorge
1. Before PCA
Load the data
# load the data
help(mtcars)
head(mtcars)Visualize the correlation between variables
Some questions to guide your analyses:
- How many variables do you have?
- Is there a dimensionality problem?
- Are the variables related?
# visualize the correlation
#install.packages("corrplot")
library(corrplot)## corrplot 0.84 loadedM<-cor(mtcars)
head(round(M,2))## mpg cyl disp hp drat wt qsec vs am gear carb
## mpg 1.00 -0.85 -0.85 -0.78 0.68 -0.87 0.42 0.66 0.60 0.48 -0.55
## cyl -0.85 1.00 0.90 0.83 -0.70 0.78 -0.59 -0.81 -0.52 -0.49 0.53
## disp -0.85 0.90 1.00 0.79 -0.71 0.89 -0.43 -0.71 -0.59 -0.56 0.39
## hp -0.78 0.83 0.79 1.00 -0.45 0.66 -0.71 -0.72 -0.24 -0.13 0.75
## drat 0.68 -0.70 -0.71 -0.45 1.00 -0.71 0.09 0.44 0.71 0.70 -0.09
## wt -0.87 0.78 0.89 0.66 -0.71 1.00 -0.17 -0.55 -0.69 -0.58 0.43# Morrelation Matrix
corrplot(M, method="color")
Set up an objective: Can you predict how fast is a car?
# simple regressions: Choose a dependent variables to demonstrate the objective
summary(lm(formula = "qsec ~ mpg", data = mtcars))##
## Call:
## lm(formula = "qsec ~ mpg", data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.8161 -1.0287 0.0954 0.8623 4.7149
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 15.35477 1.02978 14.911 2.05e-15 ***
## mpg 0.12414 0.04916 2.525 0.0171 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.65 on 30 degrees of freedom
## Multiple R-squared: 0.1753, Adjusted R-squared: 0.1478
## F-statistic: 6.377 on 1 and 30 DF, p-value: 0.01708summary(lm(formula = "qsec ~ mpg+cyl", data = mtcars))##
## Call:
## lm(formula = "qsec ~ mpg+cyl", data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.5454 -0.9929 0.4851 0.7705 3.4270
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 25.00212 3.28710 7.606 2.19e-08 ***
## mpg -0.09220 0.08311 -1.109 0.2764
## cyl -0.85673 0.28047 -3.055 0.0048 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.459 on 29 degrees of freedom
## Multiple R-squared: 0.376, Adjusted R-squared: 0.333
## F-statistic: 8.739 on 2 and 29 DF, p-value: 0.001071summary(lm(formula = "qsec ~ mpg+cyl+disp", data = mtcars))##
## Call:
## lm(formula = "qsec ~ mpg+cyl+disp", data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.6385 -1.0082 0.1268 0.8874 3.2562
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 24.553482 3.270432 7.508 3.54e-08 ***
## mpg -0.053003 0.087735 -0.604 0.55062
## cyl -1.158244 0.364015 -3.182 0.00356 **
## disp 0.006617 0.005172 1.279 0.21123
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.444 on 28 degrees of freedom
## Multiple R-squared: 0.4105, Adjusted R-squared: 0.3474
## F-statistic: 6.5 on 3 and 28 DF, p-value: 0.001777summary(lm(formula = "qsec ~ mpg+cyl+disp+hp", data = mtcars))##
## Call:
## lm(formula = "qsec ~ mpg+cyl+disp+hp", data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.3832 -0.6694 -0.1863 0.6997 3.4004
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 25.652179 2.684761 9.555 3.74e-10 ***
## mpg -0.105556 0.072893 -1.448 0.159106
## cyl -0.695276 0.320273 -2.171 0.038900 *
## disp 0.008186 0.004241 1.930 0.064168 .
## hp -0.022288 0.005751 -3.875 0.000615 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.178 on 27 degrees of freedom
## Multiple R-squared: 0.6212, Adjusted R-squared: 0.5651
## F-statistic: 11.07 on 4 and 27 DF, p-value: 1.91e-05summary(lm(formula = "qsec ~ mpg+cyl+disp+hp+drat", data = mtcars))##
## Call:
## lm(formula = "qsec ~ mpg+cyl+disp+hp+drat", data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.9439 -0.4343 -0.1150 0.5630 3.3673
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 30.083312 3.441864 8.740 3.23e-09 ***
## mpg -0.070857 0.071790 -0.987 0.33274
## cyl -0.880359 0.320099 -2.750 0.01069 *
## disp 0.006118 0.004184 1.462 0.15562
## hp -0.017102 0.006109 -2.800 0.00952 **
## drat -1.186305 0.616077 -1.926 0.06516 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.123 on 26 degrees of freedom
## Multiple R-squared: 0.6685, Adjusted R-squared: 0.6047
## F-statistic: 10.49 on 5 and 26 DF, p-value: 1.362e-05summary(lm(formula = "qsec ~ mpg+cyl+disp+hp+drat+wt", data = mtcars))##
## Call:
## lm(formula = "qsec ~ mpg+cyl+disp+hp+drat+wt", data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.60648 -0.52041 -0.02188 0.48331 2.91008
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 22.048508 3.991422 5.524 9.67e-06 ***
## mpg 0.058756 0.075605 0.777 0.44436
## cyl -0.654615 0.288404 -2.270 0.03210 *
## disp -0.002837 0.004680 -0.606 0.54982
## hp -0.013752 0.005432 -2.532 0.01801 *
## drat -0.973094 0.541001 -1.799 0.08415 .
## wt 1.504983 0.493764 3.048 0.00538 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.9783 on 25 degrees of freedom
## Multiple R-squared: 0.7583, Adjusted R-squared: 0.7003
## F-statistic: 13.07 on 6 and 25 DF, p-value: 1.153e-06summary(lm(formula = "qsec ~ mpg+cyl+disp+hp+drat+wt+vs", data = mtcars))##
## Call:
## lm(formula = "qsec ~ mpg+cyl+disp+hp+drat+wt+vs", data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.2465 -0.3602 -0.1187 0.2469 2.9281
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 18.045363 3.647146 4.948 4.74e-05 ***
## mpg 0.046355 0.064921 0.714 0.48210
## cyl -0.221732 0.282468 -0.785 0.44015
## disp -0.001670 0.004028 -0.415 0.68214
## hp -0.013322 0.004658 -2.860 0.00863 **
## drat -0.644776 0.475146 -1.357 0.18741
## wt 1.289547 0.428645 3.008 0.00608 **
## vs 1.722758 0.543967 3.167 0.00416 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.8385 on 24 degrees of freedom
## Multiple R-squared: 0.8295, Adjusted R-squared: 0.7798
## F-statistic: 16.68 on 7 and 24 DF, p-value: 8.256e-08summary(lm(formula = "qsec ~ mpg+cyl+disp+hp+drat+wt+vs+am", data = mtcars))##
## Call:
## lm(formula = "qsec ~ mpg+cyl+disp+hp+drat+wt+vs+am", data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.23411 -0.32872 0.00477 0.26556 2.35665
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 18.379009 3.396731 5.411 1.69e-05 ***
## mpg 0.076138 0.061936 1.229 0.2314
## cyl -0.397493 0.274961 -1.446 0.1618
## disp -0.002548 0.003770 -0.676 0.5058
## hp -0.008650 0.004837 -1.788 0.0869 .
## drat -0.372726 0.459443 -0.811 0.4255
## wt 1.109605 0.407305 2.724 0.0121 *
## vs 1.164650 0.567495 2.052 0.0517 .
## am -1.188121 0.546546 -2.174 0.0403 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.7801 on 23 degrees of freedom
## Multiple R-squared: 0.8586, Adjusted R-squared: 0.8094
## F-statistic: 17.46 on 8 and 23 DF, p-value: 4.566e-08summary(lm(formula = "qsec ~ mpg+cyl+disp+hp+drat+wt+vs+am+gear", data = mtcars))##
## Call:
## lm(formula = "qsec ~ mpg+cyl+disp+hp+drat+wt+vs+am+gear", data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.9749 -0.3050 -0.0830 0.2615 2.4843
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 19.661451 3.557495 5.527 1.49e-05 ***
## mpg 0.077595 0.061550 1.261 0.22063
## cyl -0.469658 0.280432 -1.675 0.10814
## disp -0.003821 0.003908 -0.978 0.33889
## hp -0.005600 0.005500 -1.018 0.31968
## drat -0.261221 0.466851 -0.560 0.58145
## wt 1.147468 0.406045 2.826 0.00984 **
## vs 1.100808 0.566617 1.943 0.06495 .
## am -0.925655 0.589838 -1.569 0.13084
## gear -0.439419 0.385541 -1.140 0.26665
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.7751 on 22 degrees of freedom
## Multiple R-squared: 0.8665, Adjusted R-squared: 0.8119
## F-statistic: 15.86 on 9 and 22 DF, p-value: 1.099e-07summary(lm(formula = "qsec ~ mpg+cyl+disp+hp+drat+wt+vs+am+gear+carb", data = mtcars))##
## Call:
## lm(formula = "qsec ~ mpg+cyl+disp+hp+drat+wt+vs+am+gear+carb",
## data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.93377 -0.33421 -0.03696 0.31389 2.38743
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 17.776177 3.875998 4.586 0.00016 ***
## mpg 0.069048 0.061462 1.123 0.27394
## cyl -0.362678 0.292621 -1.239 0.22887
## disp -0.007501 0.004985 -1.505 0.14730
## hp -0.001563 0.006449 -0.242 0.81089
## drat -0.131064 0.476002 -0.275 0.78574
## wt 1.496332 0.500469 2.990 0.00698 **
## vs 0.970035 0.572767 1.694 0.10512
## am -0.901186 0.585218 -1.540 0.13851
## gear -0.201285 0.432798 -0.465 0.64666
## carb -0.273598 0.233143 -1.174 0.25373
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.7685 on 21 degrees of freedom
## Multiple R-squared: 0.8747, Adjusted R-squared: 0.815
## F-statistic: 14.66 on 10 and 21 DF, p-value: 2.438e-072. Perform PCA
# install.packages("factoextra")
library(factoextra)## Warning: package 'factoextra' was built under R version 3.5.2## Welcome! Want to learn more? See two factoextra-related books at https://goo.gl/ve3WBaPreliminaries
Remove the dependent variable!
drops <- c("qsec")
data <- mtcars[ , !(names(mtcars) %in% drops)]Compute PCA
The main and built-in function to do PCA is prcomp. If you want to learn more, type: help(prcomp)
res.pca <- prcomp(data, scale = TRUE)
summary(res.pca)## Importance of components:
## PC1 PC2 PC3 PC4 PC5 PC6 PC7
## Standard deviation 2.5265 1.4507 0.71431 0.51819 0.47059 0.42359 0.3674
## Proportion of Variance 0.6383 0.2105 0.05102 0.02685 0.02215 0.01794 0.0135
## Cumulative Proportion 0.6383 0.8488 0.89982 0.92667 0.94882 0.96676 0.9803
## PC8 PC9 PC10
## Standard deviation 0.33929 0.23856 0.15935
## Proportion of Variance 0.01151 0.00569 0.00254
## Cumulative Proportion 0.99177 0.99746 1.00000Scree plot
How many components should I use?
Use a scree plot to visualize eigenvalues. Show the percentage of variances explained by each principal component.
fviz_eig(res.pca)
Data plot by PCA
Graph of individuals. Individuals with a similar profile are grouped together.
fviz_pca_ind(res.pca,
col.ind = "cos2", # Color by the quality of representation
gradient.cols = c("#00AFBB", "#E7B800", "#FC4E07"),
repel = TRUE # Avoid text overlapping
)
Variable plot by PCA
Graph of variables. Positive correlated variables point to the same side of the plot. Negative correlated variables point to opposite sides of the graph.
fviz_pca_var(res.pca,
col.var = "contrib", # Color by contributions to the PC
gradient.cols = c("#00AFBB", "#E7B800", "#FC4E07"),
repel = TRUE # Avoid text overlapping
)
Biplot of individuals and variables
Visualize both the data points and the variables by PCA
fviz_pca_biplot(res.pca, repel = TRUE,
col.var = "#2E9FDF", # Variables color
col.ind = "#696969" # Individuals color
)
Extracting the values
How do you get the actual results? The components? the variance explained?
# Results for individuals
res.ind <- get_pca_ind(res.pca)
res.ind$coord # Coordinates## Dim.1 Dim.2 Dim.3 Dim.4 Dim.5
## Mazda RX4 -0.88932503 -1.365132502 -0.7165487 0.04671982 0.86552907
## Mazda RX4 Wag -0.79500151 -1.346100834 -0.6333614 0.10600580 0.84591971
## Datsun 710 -2.69205912 0.366502004 -0.1131377 -0.24013425 -0.48934831
## Hornet 4 Drive -0.02805671 2.188755138 0.1719630 -0.45602689 -0.49977078
## Hornet Sportabout 1.92240577 0.859828167 -0.9403429 0.07094159 -0.15641251
## Valiant 0.32838706 2.366562980 0.3273906 -0.96404089 -0.29162066
## Duster 360 2.77769801 -0.193415173 -0.1381697 0.01890594 -0.06310678
## Merc 240D -1.75392677 1.213754998 1.0251084 -0.11484231 0.32543053
## Merc 230 -1.66824215 0.920045509 1.1589415 0.22566967 0.16646668
## Merc 280 -0.46335344 0.022369158 1.6480994 0.18536325 0.41016049
## Merc 280C -0.37753002 0.010889763 1.7010238 0.19393874 0.44445170
## Merc 450SE 2.23368912 0.573400285 -0.3282147 -0.14804885 0.38974352
## Merc 450SL 2.05275224 0.555404338 -0.4731539 -0.23260964 0.39384498
## Merc 450SLC 2.19998216 0.541916946 -0.3774561 -0.20812171 0.44143682
## Cadillac Fleetwood 3.95373834 0.240517052 0.3603170 0.22752849 -0.26445662
## Lincoln Continental 3.98770271 0.168893252 0.4522916 0.35122738 -0.29871529
## Chrysler Imperial 3.57019362 0.007438769 0.3474257 0.63184614 -0.41382047
## Fiat 128 -3.67464746 0.448720063 -0.4592981 0.01109641 -0.39121030
## Honda Civic -4.21749294 -0.306129629 -0.1273632 1.20409564 0.11619807
## Toyota Corolla -4.01112383 0.366880951 -0.5969032 0.13165380 -0.35430570
## Toyota Corona -1.57266468 1.973518077 0.4386018 0.12542926 0.24063905
## Dodge Challenger 2.13158194 1.158538246 -0.8678132 -0.59360347 0.23751811
## AMC Javelin 1.84825333 0.950109422 -0.7817894 -0.00151445 0.33009556
## Camaro Z28 2.60476490 -0.447347975 0.1166588 0.91837052 0.01153168
## Pontiac Firebird 2.20367482 0.931206103 -0.8783982 0.12701030 -0.36106342
## Fiat X1-9 -3.45911838 0.387134033 -0.3532148 -0.01867936 -0.24709148
## Porsche 914-2 -2.87320660 -1.469674939 -0.9241287 0.49309618 0.43501263
## Lotus Europa -3.54917651 -0.701190903 -0.1388198 -1.09901703 -0.64840939
## Ford Pantera L 0.85012758 -3.053206346 -0.3100419 0.61276640 -1.08936937
## Ferrari Dino -0.40324800 -2.878891350 0.1603252 -0.97747899 0.77063499
## Maserati Bora 2.11458829 -4.342275621 0.8378585 -0.86436393 -0.46347964
## Volvo 142E -2.35136674 -0.149019983 0.4121501 0.23681645 -0.39243286
## Dim.6 Dim.7 Dim.8 Dim.9
## Mazda RX4 -0.29123992 0.412191499 0.042123003 -0.136187219
## Mazda RX4 Wag -0.45569630 0.406747419 0.110602298 -0.135424646
## Datsun 710 -0.33776704 0.538024050 -0.596181503 0.068693086
## Hornet 4 Drive 0.12422860 0.042461824 0.092296226 -0.164530041
## Hornet Sportabout 0.35704630 -0.346922689 0.098264233 -0.063709227
## Valiant -0.18378292 0.263529711 -0.268906985 -0.177026513
## Duster 360 0.76522300 0.184051522 0.002236971 0.357025259
## Merc 240D -0.36557793 -0.688790065 0.141053958 0.120426485
## Merc 230 -0.06821101 -0.577696934 -0.134842169 0.307244897
## Merc 280 0.30661465 -0.185097532 0.082526904 -0.431028769
## Merc 280C 0.28299028 -0.116598225 -0.108248117 -0.450036087
## Merc 450SE 0.08089811 0.040278973 0.157664645 0.006833937
## Merc 450SL 0.31536038 0.003502477 0.189000003 0.018036162
## Merc 450SLC 0.24767747 0.105183971 -0.083735215 -0.010325290
## Cadillac Fleetwood -0.89143185 0.111859860 0.049719091 -0.017622440
## Lincoln Continental -0.89894330 0.136173595 0.075259821 0.053291039
## Chrysler Imperial -0.57353686 -0.030732340 0.590069286 0.201217085
## Fiat 128 -0.15470300 0.093226484 0.639165383 -0.005578360
## Honda Civic 0.43583118 0.300820709 0.349780811 -0.255963807
## Toyota Corolla 0.15718664 0.036125687 0.721990030 -0.007384398
## Toyota Corona 0.54046334 0.284525827 -0.386464231 0.612541706
## Dodge Challenger 0.01693834 -0.126119554 -0.282843679 -0.202489220
## AMC Javelin 0.21732474 -0.096442124 -0.408388102 -0.250659873
## Camaro Z28 0.76306788 0.229217358 -0.138147440 0.279683988
## Pontiac Firebird 0.02538886 -0.449777090 0.311459127 -0.074286572
## Fiat X1-9 -0.07027262 0.347878043 -0.126763847 -0.075823423
## Porsche 914-2 -0.53405226 -0.898520209 -0.464651559 0.361579963
## Lotus Europa 0.20961194 -0.512044495 0.122836491 0.066947379
## Ford Pantera L 0.21996998 -0.447299912 -0.716660716 -0.400361865
## Ferrari Dino -0.23806774 -0.044166385 0.065472635 0.088870333
## Maserati Bora 0.43559541 0.275262285 0.318548396 0.201777860
## Volvo 142E -0.43813432 0.709146262 -0.444235749 0.114268571
## Dim.10
## Mazda RX4 -0.187999469
## Mazda RX4 Wag -0.061209605
## Datsun 710 0.080227441
## Hornet 4 Drive -0.175182955
## Hornet Sportabout -0.136139445
## Valiant 0.020534587
## Duster 360 -0.227206670
## Merc 240D -0.083838570
## Merc 230 0.079226798
## Merc 280 0.052498872
## Merc 280C 0.032395223
## Merc 450SE 0.409924947
## Merc 450SL 0.253795570
## Merc 450SLC 0.248500853
## Cadillac Fleetwood -0.269053922
## Lincoln Continental -0.064445970
## Chrysler Imperial 0.160188571
## Fiat 128 0.223186546
## Honda Civic -0.253056492
## Toyota Corolla 0.116008466
## Toyota Corona -0.097466279
## Dodge Challenger -0.069451487
## AMC Javelin 0.004358759
## Camaro Z28 0.005957542
## Pontiac Firebird -0.153496927
## Fiat X1-9 0.016554829
## Porsche 914-2 0.055597021
## Lotus Europa -0.146731481
## Ford Pantera L 0.130600079
## Ferrari Dino -0.082957767
## Maserati Bora 0.006035215
## Volvo 142E 0.112645720res.ind$contrib # Contributions to the PCs## Dim.1 Dim.2 Dim.3 Dim.4
## Mazda RX4 0.3871838061 2.767188e+00 3.14461150 2.540276e-02
## Mazda RX4 Wag 0.3094083787 2.690570e+00 2.45685073 1.307789e-01
## Datsun 710 3.5478507632 1.994535e-01 0.07839541 6.710992e-01
## Hornet 4 Drive 0.0003853627 7.113501e+00 0.18111129 2.420246e+00
## Hornet Sportabout 1.8091987512 1.097773e+00 5.41561818 5.857070e-02
## Valiant 0.0527920132 8.316205e+00 0.65645965 1.081607e+01
## Duster 360 3.7771672300 5.554822e-02 0.11692331 4.159826e-03
## Merc 240D 1.5059795311 2.187515e+00 6.43598705 1.534908e-01
## Merc 230 1.3624304461 1.256920e+00 8.22618772 5.926863e-01
## Merc 280 0.1051044087 7.429992e-04 16.63574950 3.998764e-01
## Merc 280C 0.0697748884 1.760864e-04 17.72133162 4.377314e-01
## Merc 450SE 2.4425396020 4.882080e-01 0.65976849 2.550872e-01
## Merc 450SL 2.0628574986 4.580444e-01 1.37113562 6.297004e-01
## Merc 450SLC 2.3693786892 4.360683e-01 0.87258625 5.040961e-01
## Cadillac Fleetwood 7.6526527247 8.589755e-02 0.79514251 6.024904e-01
## Lincoln Continental 7.7846968513 4.235587e-02 1.25288920 1.435672e+00
## Chrysler Imperial 6.2399297185 8.216583e-05 0.73926377 4.646235e+00
## Fiat 128 6.6103966124 2.989785e-01 1.29200686 1.432990e-03
## Honda Civic 8.7077286287 1.391553e-01 0.09934894 1.687332e+01
## Toyota Corolla 7.8764096748 1.998662e-01 2.18214390 2.017183e-01
## Toyota Corona 1.2107888898 5.783241e+00 1.17819286 1.830949e-01
## Dodge Challenger 2.2243351349 1.993011e+00 4.61241155 4.100827e+00
## AMC Javelin 1.6723191044 1.340406e+00 3.74330391 2.669245e-05
## Camaro Z28 3.3214920816 2.971529e-01 0.08335102 9.815545e+00
## Pontiac Firebird 2.3773393441 1.287599e+00 4.72561603 1.877398e-01
## Fiat X1-9 5.8576984268 2.225419e-01 0.76410554 4.060716e-03
## Porsche 914-2 4.0413816884 3.207241e+00 5.23046758 2.829709e+00
## Lotus Europa 6.1666791673 7.300651e-01 0.11802619 1.405683e+01
## Ford Pantera L 0.3538053276 1.384208e+01 0.58873035 4.369869e+00
## Ferrari Dino 0.0796050477 1.230664e+01 0.15742692 1.111971e+01
## Maserati Bora 2.1890102981 2.799779e+01 4.29948977 8.695044e+00
## Volvo 142E 2.7066799096 3.297449e-02 1.04036678 6.526830e-01
## Dim.5 Dim.6 Dim.7 Dim.8
## Mazda RX4 10.571240500 1.47728489 3.934096e+00 4.816748e-02
## Mazda RX4 Wag 10.097664340 3.61670802 3.830862e+00 3.320807e-01
## Datsun 710 3.379082678 1.98699594 6.702708e+00 9.648777e+00
## Hornet 4 Drive 3.524555521 0.26878537 4.174882e-02 2.312508e-01
## Hornet Sportabout 0.345227686 2.22029926 2.786842e+00 2.621237e-01
## Valiant 1.200049643 0.58826466 1.608074e+00 1.962996e+00
## Duster 360 0.056197205 10.19855084 7.843786e-01 1.358426e-04
## Merc 240D 1.494442569 2.32767523 1.098552e+01 5.401145e-01
## Merc 230 0.391036479 0.08103477 7.727642e+00 4.935904e-01
## Merc 280 2.373942998 1.63737534 7.933195e-01 1.848870e-01
## Merc 280C 2.787480098 1.39477922 3.147971e-01 3.180944e-01
## Merc 450SE 2.143485035 0.11398276 3.756675e-02 6.748137e-01
## Merc 450SL 2.188836262 1.73211488 2.840518e-04 9.697033e-01
## Merc 450SLC 2.749791402 1.06840383 2.561805e-01 1.903407e-01
## Cadillac Fleetwood 0.986896286 13.84008437 2.897313e-01 6.710604e-02
## Lincoln Continental 1.259150237 14.07430750 4.293710e-01 1.537594e-01
## Chrysler Imperial 2.416498808 5.72908293 2.186947e-02 9.451947e+00
## Fiat 128 2.159649184 0.41683091 2.012452e-01 1.109026e+01
## Honda Civic 0.190528720 3.30825538 2.095378e+00 3.321293e+00
## Toyota Corolla 1.771409255 0.43032217 3.021893e-02 1.415069e+01
## Toyota Corona 0.817136916 5.08738868 1.874521e+00 4.054468e+00
## Dodge Challenger 0.796078839 0.00499694 3.683085e-01 2.171742e+00
## AMC Javelin 1.537595154 0.82258499 2.153676e-01 4.527531e+00
## Camaro Z28 0.001876496 10.14118684 1.216583e+00 5.180849e-01
## Pontiac Firebird 1.839626397 0.01122661 4.684266e+00 2.633403e+00
## Fiat X1-9 0.861545409 0.08600718 2.802212e+00 4.362205e-01
## Porsche 914-2 2.670338886 4.96740938 1.869402e+01 5.860978e+00
## Lotus Europa 5.932820872 0.76523440 6.071029e+00 4.096096e-01
## Ford Pantera L 16.746072290 0.84273161 4.632810e+00 1.394256e+01
## Ferrari Dino 8.380310104 0.98710546 4.516797e-02 1.163684e-01
## Maserati Bora 3.031265409 3.30467706 1.754447e+00 2.754648e+00
## Volvo 142E 2.173168321 3.34331257 1.164444e+01 5.357254e+00
## Dim.9 Dim.10
## Mazda RX4 1.018447128 4.349955043
## Mazda RX4 Wag 1.007073574 0.461116517
## Datsun 710 0.259114574 0.792168199
## Hornet 4 Drive 1.486470156 3.777071606
## Hornet Sportabout 0.222879668 2.281074525
## Valiant 1.720847614 0.051897144
## Duster 360 6.999446020 6.353509421
## Merc 240D 0.796360669 0.865085927
## Merc 230 5.183643669 0.772530675
## Merc 280 10.201835071 0.339212200
## Merc 280C 11.121425615 0.129161613
## Merc 450SE 0.002564530 20.681413681
## Merc 450SL 0.017862986 7.927561070
## Merc 450SLC 0.005854237 7.600239754
## Cadillac Fleetwood 0.017052885 8.909435287
## Lincoln Continental 0.155945973 0.511167289
## Chrysler Imperial 2.223286787 3.158163269
## Fiat 128 0.001708751 6.130664434
## Honda Civic 3.597683079 7.881456617
## Toyota Corolla 0.002994305 1.656345344
## Toyota Corona 20.603315352 1.169177081
## Dodge Challenger 2.251487788 0.593655675
## AMC Javelin 3.450129637 0.002338284
## Camaro Z28 4.295374067 0.004368233
## Pontiac Firebird 0.303030556 2.899819013
## Fiat X1-9 0.315698529 0.033730356
## Porsche 914-2 7.179174244 0.380429827
## Lotus Europa 0.246112089 2.649830671
## Ford Pantera L 8.801793812 2.099222150
## Ferrari Dino 0.433689986 0.847004325
## Maserati Bora 2.235696284 0.004482880
## Volvo 142E 0.717000367 1.561711889res.ind$cos2 # Quality of representation ## Dim.1 Dim.2 Dim.3 Dim.4
## Mazda RX4 0.1869897289 4.406019e-01 0.1213914670 5.160586e-04
## Mazda RX4 Wag 0.1588247663 4.553418e-01 0.1008058000 2.823850e-03
## Datsun 710 0.8564813396 1.587455e-02 0.0015127382 6.814860e-03
## Hornet 4 Drive 0.0001468001 8.934024e-01 0.0055147103 3.878231e-02
## Hornet Sportabout 0.6565625298 1.313437e-01 0.1570935835 8.941036e-04
## Valiant 0.0153236622 7.958416e-01 0.0152308093 1.320631e-01
## Duster 360 0.8997809855 4.362625e-03 0.0022263441 4.168342e-05
## Merc 240D 0.4830120178 2.313114e-01 0.1649964588 2.070803e-03
## Merc 230 0.5052209356 1.536674e-01 0.2438294780 9.245056e-03
## Merc 280 0.0620932316 1.447167e-04 0.7855725439 9.937263e-03
## Merc 280C 0.0398096860 3.312246e-05 0.8081771172 1.050546e-02
## Merc 450SE 0.8601158022 5.667967e-02 0.0185706659 3.778519e-03
## Merc 450SL 0.8173732882 5.983646e-02 0.0434262392 1.049549e-02
## Merc 450SLC 0.8557859175 5.192689e-02 0.0251918241 7.658815e-03
## Cadillac Fleetwood 0.9291645398 3.438496e-03 0.0077169589 3.077148e-03
## Lincoln Continental 0.9252340781 1.659704e-03 0.0119026367 7.177659e-03
## Chrysler Imperial 0.8987813584 3.901869e-06 0.0085112701 2.815096e-02
## Fiat 128 0.9274375848 1.382943e-02 0.0144891394 8.457034e-06
## Honda Civic 0.8941516203 4.711001e-03 0.0008154365 7.288269e-02
## Toyota Corolla 0.9308853294 7.787793e-03 0.0206144556 1.002838e-03
## Toyota Corona 0.3279692915 5.164677e-01 0.0255094877 2.086213e-03
## Dodge Challenger 0.6319609007 1.866839e-01 0.1047462861 4.900936e-02
## AMC Javelin 0.6415015412 1.695204e-01 0.1147768326 4.307096e-07
## Camaro Z28 0.7913019890 2.333975e-02 0.0015872326 9.836511e-02
## Pontiac Firebird 0.6966823852 1.244032e-01 0.1106936422 2.314290e-03
## Fiat X1-9 0.9611162763 1.203837e-02 0.0100212724 2.802652e-05
## Porsche 914-2 0.6280702830 1.643301e-01 0.0649739822 1.849856e-02
## Lotus Europa 0.8351505090 3.259734e-02 0.0012776514 8.007896e-02
## Ford Pantera L 0.0571655153 7.373577e-01 0.0076033813 2.969996e-02
## Ferrari Dino 0.0160945332 8.203222e-01 0.0025441188 9.456900e-02
## Maserati Bora 0.1760529010 7.423801e-01 0.0276396727 2.941608e-02
## Volvo 142E 0.8072570001 3.242352e-03 0.0248017308 8.188320e-03
## Dim.5 Dim.6 Dim.7 Dim.8
## Mazda RX4 1.771169e-01 2.005389e-02 4.016935e-02 4.195033e-04
## Mazda RX4 Wag 1.798211e-01 5.218351e-02 4.157497e-02 3.074048e-03
## Datsun 710 2.829990e-02 1.348290e-02 3.420992e-02 4.200546e-02
## Hornet 4 Drive 4.657946e-02 2.878034e-03 3.362405e-04 1.588622e-03
## Hornet Sportabout 4.346393e-03 2.264827e-02 2.138215e-02 1.715445e-03
## Valiant 1.208446e-02 4.799553e-03 9.868467e-03 1.027530e-02
## Duster 360 4.644285e-04 6.828775e-02 3.950441e-03 5.835627e-07
## Merc 240D 1.662844e-02 2.098432e-02 7.449185e-02 3.123961e-03
## Merc 230 5.030577e-03 8.446410e-04 6.058474e-02 3.300765e-03
## Merc 280 4.865496e-02 2.718975e-02 9.908793e-03 1.969748e-03
## Merc 280C 5.517405e-02 2.236808e-02 3.797254e-03 3.272853e-03
## Merc 450SE 2.618601e-02 1.128205e-03 2.796844e-04 4.285289e-03
## Merc 450SL 3.008832e-02 1.929131e-02 2.379567e-06 6.929007e-03
## Merc 450SLC 3.445593e-02 1.084675e-02 1.956257e-03 1.239776e-03
## Cadillac Fleetwood 4.157054e-03 4.723382e-02 7.437472e-04 1.469343e-04
## Lincoln Continental 5.191836e-03 4.701871e-02 1.078927e-03 3.295584e-04
## Chrysler Imperial 1.207520e-02 2.319494e-02 6.659809e-05 2.455142e-02
## Fiat 128 1.051172e-02 1.643807e-03 5.969417e-04 2.805949e-02
## Honda Civic 6.787350e-04 9.548585e-03 4.549021e-03 6.150273e-03
## Toyota Corolla 7.263071e-03 1.429537e-03 7.550858e-05 3.015966e-02
## Toyota Corona 7.678796e-03 3.873409e-02 1.073506e-02 1.980521e-02
## Dodge Challenger 7.846565e-03 3.990505e-05 2.212336e-03 1.112703e-02
## AMC Javelin 2.046228e-02 8.869376e-03 1.746659e-03 3.131994e-02
## Camaro Z28 1.550923e-05 6.790966e-02 6.127738e-03 2.225824e-03
## Pontiac Firebird 1.870281e-02 9.247539e-05 2.902248e-02 1.391688e-02
## Fiat X1-9 4.904115e-03 3.966590e-04 9.720735e-03 1.290731e-03
## Porsche 914-2 1.439721e-02 2.169912e-02 6.142289e-02 1.642591e-02
## Lotus Europa 2.787457e-02 2.913007e-03 1.738301e-02 1.000379e-03
## Ford Pantera L 9.386771e-02 3.827302e-03 1.582572e-02 4.062498e-02
## Ferrari Dino 5.878024e-02 5.609641e-03 1.930713e-04 4.242810e-04
## Maserati Bora 8.457711e-03 7.470645e-03 2.983218e-03 3.995235e-03
## Volvo 142E 2.248546e-02 2.802758e-02 7.342479e-02 2.881364e-02
## Dim.9 Dim.10
## Mazda RX4 4.384998e-03 8.356222e-03
## Mazda RX4 Wag 4.608692e-03 9.415025e-04
## Datsun 710 5.576668e-04 7.606670e-04
## Hornet 4 Drive 5.048277e-03 5.723167e-03
## Hornet Sportabout 7.210919e-04 3.292713e-03
## Valiant 4.453148e-03 5.991871e-05
## Duster 360 1.486499e-02 6.020172e-03
## Merc 240D 2.277084e-03 1.103628e-03
## Merc 230 1.713691e-02 1.139482e-03
## Merc 280 5.373188e-02 7.971121e-04
## Merc 280C 5.656925e-02 2.931215e-04
## Merc 450SE 8.051066e-06 2.896811e-02
## Merc 450SL 6.310091e-05 1.249440e-02
## Merc 450SLC 1.885087e-05 1.091899e-02
## Cadillac Fleetwood 1.845903e-05 4.302842e-03
## Lincoln Continental 1.652397e-04 2.416560e-04
## Chrysler Imperial 2.854961e-03 1.809396e-03
## Fiat 128 2.137304e-06 3.421286e-03
## Honda Civic 3.293514e-03 3.219122e-03
## Toyota Corolla 3.154963e-06 7.786520e-04
## Toyota Corona 4.975449e-02 1.259707e-03
## Dodge Challenger 5.702824e-03 6.708868e-04
## AMC Javelin 1.179898e-02 3.567799e-06
## Camaro Z28 9.123057e-03 4.139420e-06
## Pontiac Firebird 7.916996e-04 3.380175e-03
## Fiat X1-9 4.617978e-04 2.201377e-05
## Porsche 914-2 9.946797e-03 2.351676e-04
## Lotus Europa 2.971507e-04 1.427434e-03
## Ford Pantera L 1.267860e-02 1.349127e-03
## Ferrari Dino 7.817134e-04 6.811583e-04
## Maserati Bora 1.603019e-03 1.434092e-06
## Volvo 142E 1.906447e-03 1.852680e-03dataafter <- data.frame(qsec=mtcars$qsec, res.ind$coord )
dataafter3. After PCA
After doing PCA, we can use the dimensions in our regression to ‘explain’ how fast cars are.
# simple regressions:
summary(lm(formula = "qsec ~ Dim.1", data = dataafter))##
## Call:
## lm(formula = "qsec ~ Dim.1", data = dataafter)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.0874 -1.0936 0.0239 0.8545 4.5384
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 17.8488 0.2892 61.721 <2e-16 ***
## Dim.1 -0.3074 0.1163 -2.644 0.0129 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.636 on 30 degrees of freedom
## Multiple R-squared: 0.189, Adjusted R-squared: 0.1619
## F-statistic: 6.99 on 1 and 30 DF, p-value: 0.01291summary(lm(formula = "qsec ~ Dim.1+Dim.2", data = dataafter))##
## Call:
## lm(formula = "qsec ~ Dim.1+Dim.2", data = dataafter)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.4271 -0.5386 -0.0752 0.4271 3.7342
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 17.84875 0.18111 98.549 < 2e-16 ***
## Dim.1 -0.30745 0.07283 -4.221 0.000219 ***
## Dim.2 0.87405 0.12684 6.891 1.43e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.025 on 29 degrees of freedom
## Multiple R-squared: 0.6925, Adjusted R-squared: 0.6713
## F-statistic: 32.65 on 2 and 29 DF, p-value: 3.751e-08summary(lm(formula = "qsec ~ Dim.1+Dim.2+Dim.3", data = dataafter))##
## Call:
## lm(formula = "qsec ~ Dim.1+Dim.2+Dim.3", data = dataafter)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.3493 -0.3696 -0.1109 0.3834 2.7168
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 17.84875 0.14272 125.059 < 2e-16 ***
## Dim.1 -0.30745 0.05739 -5.357 1.05e-05 ***
## Dim.2 0.87405 0.09996 8.744 1.71e-09 ***
## Dim.3 0.87785 0.20300 4.324 0.000175 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.8074 on 28 degrees of freedom
## Multiple R-squared: 0.8156, Adjusted R-squared: 0.7959
## F-statistic: 41.29 on 3 and 28 DF, p-value: 2.069e-10