PCA (Principal Component Analysis) on Credit Card Transaction Data
Stephen Situ
2022-11-22
This is a project on conducting PCA (Principal Component Analysis) On Credit Card Transaction Data. The goal of PCA is to reduce the dimensionality/features of a dataset to capture the maximum variance in the least number of features. We find that using PCA, we can capture the statistically significant portion of the data in 4 Principal Components, down from 31 features. Original dataset can be found here: https://www.kaggle.com/datasets/mlg-ulb/creditcardfraud
Install packages
install.packages('tidyverse', repos = "http://cran.us.r-project.org")## Installing package into 'C:/Users/Steve/AppData/Local/R/win-library/4.2'
## (as 'lib' is unspecified)## package 'tidyverse' successfully unpacked and MD5 sums checked
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## The downloaded binary packages are in
## C:\Users\Steve\AppData\Local\Temp\RtmpyWFmpx\downloaded_packageslibrary(tidyverse)## ── Attaching packages
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## tidyverse 1.3.2 ──## ✔ ggplot2 3.4.0 ✔ purrr 0.3.5
## ✔ tibble 3.1.8 ✔ dplyr 1.0.10
## ✔ tidyr 1.2.1 ✔ stringr 1.4.1
## ✔ readr 2.1.3 ✔ forcats 0.5.2
## ── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
## ✖ dplyr::filter() masks stats::filter()
## ✖ dplyr::lag() masks stats::lag()Read CSV
credit_df <- read_csv('creditcard.csv')## Rows: 284807 Columns: 31
## ── Column specification ────────────────────────────────────────────────────────
## Delimiter: ","
## dbl (31): Time, V1, V2, V3, V4, V5, V6, V7, V8, V9, V10, V11, V12, V13, V14,...
##
## ℹ Use `spec()` to retrieve the full column specification for this data.
## ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.Structure
str(credit_df)## spc_tbl_ [284,807 × 31] (S3: spec_tbl_df/tbl_df/tbl/data.frame)
## $ Time : num [1:284807] 0 0 1 1 2 2 4 7 7 9 ...
## $ V1 : num [1:284807] -1.36 1.192 -1.358 -0.966 -1.158 ...
## $ V2 : num [1:284807] -0.0728 0.2662 -1.3402 -0.1852 0.8777 ...
## $ V3 : num [1:284807] 2.536 0.166 1.773 1.793 1.549 ...
## $ V4 : num [1:284807] 1.378 0.448 0.38 -0.863 0.403 ...
## $ V5 : num [1:284807] -0.3383 0.06 -0.5032 -0.0103 -0.4072 ...
## $ V6 : num [1:284807] 0.4624 -0.0824 1.8005 1.2472 0.0959 ...
## $ V7 : num [1:284807] 0.2396 -0.0788 0.7915 0.2376 0.5929 ...
## $ V8 : num [1:284807] 0.0987 0.0851 0.2477 0.3774 -0.2705 ...
## $ V9 : num [1:284807] 0.364 -0.255 -1.515 -1.387 0.818 ...
## $ V10 : num [1:284807] 0.0908 -0.167 0.2076 -0.055 0.7531 ...
## $ V11 : num [1:284807] -0.552 1.613 0.625 -0.226 -0.823 ...
## $ V12 : num [1:284807] -0.6178 1.0652 0.0661 0.1782 0.5382 ...
## $ V13 : num [1:284807] -0.991 0.489 0.717 0.508 1.346 ...
## $ V14 : num [1:284807] -0.311 -0.144 -0.166 -0.288 -1.12 ...
## $ V15 : num [1:284807] 1.468 0.636 2.346 -0.631 0.175 ...
## $ V16 : num [1:284807] -0.47 0.464 -2.89 -1.06 -0.451 ...
## $ V17 : num [1:284807] 0.208 -0.115 1.11 -0.684 -0.237 ...
## $ V18 : num [1:284807] 0.0258 -0.1834 -0.1214 1.9658 -0.0382 ...
## $ V19 : num [1:284807] 0.404 -0.146 -2.262 -1.233 0.803 ...
## $ V20 : num [1:284807] 0.2514 -0.0691 0.525 -0.208 0.4085 ...
## $ V21 : num [1:284807] -0.01831 -0.22578 0.248 -0.1083 -0.00943 ...
## $ V22 : num [1:284807] 0.27784 -0.63867 0.77168 0.00527 0.79828 ...
## $ V23 : num [1:284807] -0.11 0.101 0.909 -0.19 -0.137 ...
## $ V24 : num [1:284807] 0.0669 -0.3398 -0.6893 -1.1756 0.1413 ...
## $ V25 : num [1:284807] 0.129 0.167 -0.328 0.647 -0.206 ...
## $ V26 : num [1:284807] -0.189 0.126 -0.139 -0.222 0.502 ...
## $ V27 : num [1:284807] 0.13356 -0.00898 -0.05535 0.06272 0.21942 ...
## $ V28 : num [1:284807] -0.0211 0.0147 -0.0598 0.0615 0.2152 ...
## $ Amount: num [1:284807] 149.62 2.69 378.66 123.5 69.99 ...
## $ Class : num [1:284807] 0 0 0 0 0 0 0 0 0 0 ...
## - attr(*, "spec")=
## .. cols(
## .. Time = col_double(),
## .. V1 = col_double(),
## .. V2 = col_double(),
## .. V3 = col_double(),
## .. V4 = col_double(),
## .. V5 = col_double(),
## .. V6 = col_double(),
## .. V7 = col_double(),
## .. V8 = col_double(),
## .. V9 = col_double(),
## .. V10 = col_double(),
## .. V11 = col_double(),
## .. V12 = col_double(),
## .. V13 = col_double(),
## .. V14 = col_double(),
## .. V15 = col_double(),
## .. V16 = col_double(),
## .. V17 = col_double(),
## .. V18 = col_double(),
## .. V19 = col_double(),
## .. V20 = col_double(),
## .. V21 = col_double(),
## .. V22 = col_double(),
## .. V23 = col_double(),
## .. V24 = col_double(),
## .. V25 = col_double(),
## .. V26 = col_double(),
## .. V27 = col_double(),
## .. V28 = col_double(),
## .. Amount = col_double(),
## .. Class = col_double()
## .. )
## - attr(*, "problems")=<externalptr>summary(credit_df)## Time V1 V2 V3
## Min. : 0 Min. :-56.40751 Min. :-72.71573 Min. :-48.3256
## 1st Qu.: 54202 1st Qu.: -0.92037 1st Qu.: -0.59855 1st Qu.: -0.8904
## Median : 84692 Median : 0.01811 Median : 0.06549 Median : 0.1799
## Mean : 94814 Mean : 0.00000 Mean : 0.00000 Mean : 0.0000
## 3rd Qu.:139321 3rd Qu.: 1.31564 3rd Qu.: 0.80372 3rd Qu.: 1.0272
## Max. :172792 Max. : 2.45493 Max. : 22.05773 Max. : 9.3826
## V4 V5 V6 V7
## Min. :-5.68317 Min. :-113.74331 Min. :-26.1605 Min. :-43.5572
## 1st Qu.:-0.84864 1st Qu.: -0.69160 1st Qu.: -0.7683 1st Qu.: -0.5541
## Median :-0.01985 Median : -0.05434 Median : -0.2742 Median : 0.0401
## Mean : 0.00000 Mean : 0.00000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.74334 3rd Qu.: 0.61193 3rd Qu.: 0.3986 3rd Qu.: 0.5704
## Max. :16.87534 Max. : 34.80167 Max. : 73.3016 Max. :120.5895
## V8 V9 V10 V11
## Min. :-73.21672 Min. :-13.43407 Min. :-24.58826 Min. :-4.79747
## 1st Qu.: -0.20863 1st Qu.: -0.64310 1st Qu.: -0.53543 1st Qu.:-0.76249
## Median : 0.02236 Median : -0.05143 Median : -0.09292 Median :-0.03276
## Mean : 0.00000 Mean : 0.00000 Mean : 0.00000 Mean : 0.00000
## 3rd Qu.: 0.32735 3rd Qu.: 0.59714 3rd Qu.: 0.45392 3rd Qu.: 0.73959
## Max. : 20.00721 Max. : 15.59500 Max. : 23.74514 Max. :12.01891
## V12 V13 V14 V15
## Min. :-18.6837 Min. :-5.79188 Min. :-19.2143 Min. :-4.49894
## 1st Qu.: -0.4056 1st Qu.:-0.64854 1st Qu.: -0.4256 1st Qu.:-0.58288
## Median : 0.1400 Median :-0.01357 Median : 0.0506 Median : 0.04807
## Mean : 0.0000 Mean : 0.00000 Mean : 0.0000 Mean : 0.00000
## 3rd Qu.: 0.6182 3rd Qu.: 0.66251 3rd Qu.: 0.4931 3rd Qu.: 0.64882
## Max. : 7.8484 Max. : 7.12688 Max. : 10.5268 Max. : 8.87774
## V16 V17 V18
## Min. :-14.12985 Min. :-25.16280 Min. :-9.498746
## 1st Qu.: -0.46804 1st Qu.: -0.48375 1st Qu.:-0.498850
## Median : 0.06641 Median : -0.06568 Median :-0.003636
## Mean : 0.00000 Mean : 0.00000 Mean : 0.000000
## 3rd Qu.: 0.52330 3rd Qu.: 0.39968 3rd Qu.: 0.500807
## Max. : 17.31511 Max. : 9.25353 Max. : 5.041069
## V19 V20 V21
## Min. :-7.213527 Min. :-54.49772 Min. :-34.83038
## 1st Qu.:-0.456299 1st Qu.: -0.21172 1st Qu.: -0.22839
## Median : 0.003735 Median : -0.06248 Median : -0.02945
## Mean : 0.000000 Mean : 0.00000 Mean : 0.00000
## 3rd Qu.: 0.458949 3rd Qu.: 0.13304 3rd Qu.: 0.18638
## Max. : 5.591971 Max. : 39.42090 Max. : 27.20284
## V22 V23 V24
## Min. :-10.933144 Min. :-44.80774 Min. :-2.83663
## 1st Qu.: -0.542350 1st Qu.: -0.16185 1st Qu.:-0.35459
## Median : 0.006782 Median : -0.01119 Median : 0.04098
## Mean : 0.000000 Mean : 0.00000 Mean : 0.00000
## 3rd Qu.: 0.528554 3rd Qu.: 0.14764 3rd Qu.: 0.43953
## Max. : 10.503090 Max. : 22.52841 Max. : 4.58455
## V25 V26 V27
## Min. :-10.29540 Min. :-2.60455 Min. :-22.565679
## 1st Qu.: -0.31715 1st Qu.:-0.32698 1st Qu.: -0.070840
## Median : 0.01659 Median :-0.05214 Median : 0.001342
## Mean : 0.00000 Mean : 0.00000 Mean : 0.000000
## 3rd Qu.: 0.35072 3rd Qu.: 0.24095 3rd Qu.: 0.091045
## Max. : 7.51959 Max. : 3.51735 Max. : 31.612198
## V28 Amount Class
## Min. :-15.43008 Min. : 0.00 Min. :0.000000
## 1st Qu.: -0.05296 1st Qu.: 5.60 1st Qu.:0.000000
## Median : 0.01124 Median : 22.00 Median :0.000000
## Mean : 0.00000 Mean : 88.35 Mean :0.001728
## 3rd Qu.: 0.07828 3rd Qu.: 77.17 3rd Qu.:0.000000
## Max. : 33.84781 Max. :25691.16 Max. :1.000000Filter Class not equal to 0
Class can only be 0 or 1
view(filter(credit_df, Class != 0))Create correlation matrix to check PCA eligibility
Average correlation is 0.0259
round(cor(credit_df),3)## Time V1 V2 V3 V4 V5 V6 V7 V8 V9
## Time 1.000 0.117 -0.011 -0.420 -0.105 0.173 -0.063 0.085 -0.037 -0.009
## V1 0.117 1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## V2 -0.011 0.000 1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## V3 -0.420 0.000 0.000 1.000 0.000 0.000 0.000 0.000 0.000 0.000
## V4 -0.105 0.000 0.000 0.000 1.000 0.000 0.000 0.000 0.000 0.000
## V5 0.173 0.000 0.000 0.000 0.000 1.000 0.000 0.000 0.000 0.000
## V6 -0.063 0.000 0.000 0.000 0.000 0.000 1.000 0.000 0.000 0.000
## V7 0.085 0.000 0.000 0.000 0.000 0.000 0.000 1.000 0.000 0.000
## V8 -0.037 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.000 0.000
## V9 -0.009 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.000
## V10 0.031 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## V11 -0.248 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## V12 0.124 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## V13 -0.066 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## V14 -0.099 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## V15 -0.183 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## V16 0.012 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## V17 -0.073 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## V18 0.090 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## V19 0.029 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## V20 -0.051 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## V21 0.045 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## V22 0.144 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## V23 0.051 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## V24 -0.016 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## V25 -0.233 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## V26 -0.041 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## V27 -0.005 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## V28 -0.009 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## Amount -0.011 -0.228 -0.531 -0.211 0.099 -0.386 0.216 0.397 -0.103 -0.044
## Class -0.012 -0.101 0.091 -0.193 0.133 -0.095 -0.044 -0.187 0.020 -0.098
## V10 V11 V12 V13 V14 V15 V16 V17 V18 V19
## Time 0.031 -0.248 0.124 -0.066 -0.099 -0.183 0.012 -0.073 0.090 0.029
## V1 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## V2 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## V3 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## V4 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## V5 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## V6 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## V7 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## V8 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## V9 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## V10 1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## V11 0.000 1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## V12 0.000 0.000 1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## V13 0.000 0.000 0.000 1.000 0.000 0.000 0.000 0.000 0.000 0.000
## V14 0.000 0.000 0.000 0.000 1.000 0.000 0.000 0.000 0.000 0.000
## V15 0.000 0.000 0.000 0.000 0.000 1.000 0.000 0.000 0.000 0.000
## V16 0.000 0.000 0.000 0.000 0.000 0.000 1.000 0.000 0.000 0.000
## V17 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.000 0.000 0.000
## V18 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.000 0.000
## V19 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.000
## V20 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## V21 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## V22 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## V23 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## V24 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## V25 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## V26 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## V27 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## V28 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## Amount -0.102 0.000 -0.010 0.005 0.034 -0.003 -0.004 0.007 0.036 -0.056
## Class -0.217 0.155 -0.261 -0.005 -0.303 -0.004 -0.197 -0.326 -0.111 0.035
## V20 V21 V22 V23 V24 V25 V26 V27 V28 Amount
## Time -0.051 0.045 0.144 0.051 -0.016 -0.233 -0.041 -0.005 -0.009 -0.011
## V1 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 -0.228
## V2 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 -0.531
## V3 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 -0.211
## V4 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.099
## V5 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 -0.386
## V6 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.216
## V7 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.397
## V8 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 -0.103
## V9 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 -0.044
## V10 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 -0.102
## V11 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## V12 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 -0.010
## V13 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.005
## V14 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.034
## V15 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 -0.003
## V16 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 -0.004
## V17 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.007
## V18 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.036
## V19 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 -0.056
## V20 1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.339
## V21 0.000 1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.106
## V22 0.000 0.000 1.000 0.000 0.000 0.000 0.000 0.000 0.000 -0.065
## V23 0.000 0.000 0.000 1.000 0.000 0.000 0.000 0.000 0.000 -0.113
## V24 0.000 0.000 0.000 0.000 1.000 0.000 0.000 0.000 0.000 0.005
## V25 0.000 0.000 0.000 0.000 0.000 1.000 0.000 0.000 0.000 -0.048
## V26 0.000 0.000 0.000 0.000 0.000 0.000 1.000 0.000 0.000 -0.003
## V27 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.000 0.000 0.029
## V28 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.000 0.010
## Amount 0.339 0.106 -0.065 -0.113 0.005 -0.048 -0.003 0.029 0.010 1.000
## Class 0.020 0.040 0.001 -0.003 -0.007 0.003 0.004 0.018 0.010 0.006
## Class
## Time -0.012
## V1 -0.101
## V2 0.091
## V3 -0.193
## V4 0.133
## V5 -0.095
## V6 -0.044
## V7 -0.187
## V8 0.020
## V9 -0.098
## V10 -0.217
## V11 0.155
## V12 -0.261
## V13 -0.005
## V14 -0.303
## V15 -0.004
## V16 -0.197
## V17 -0.326
## V18 -0.111
## V19 0.035
## V20 0.020
## V21 0.040
## V22 0.001
## V23 -0.003
## V24 -0.007
## V25 0.003
## V26 0.004
## V27 0.018
## V28 0.010
## Amount 0.006
## Class 1.000mean(cor(credit_df)) ## [1] 0.0259556Conduct PCA on scaled correlation matrix since we have data on different scales
We use PCA on covariance matrix when data is on same scale
PCA = princomp(credit_df, cor=TRUE, scale=TRUE)## Warning: In princomp.default(credit_df, cor = TRUE, scale = TRUE) :
## extra argument 'scale' will be disregardedprint(PCA)## Call:
## princomp(x = credit_df, cor = TRUE, scale = TRUE)
##
## Standard deviations:
## Comp.1 Comp.2 Comp.3 Comp.4 Comp.5 Comp.6 Comp.7 Comp.8
## 1.3994101 1.3137433 1.2960734 1.0000000 1.0000000 1.0000000 1.0000000 1.0000000
## Comp.9 Comp.10 Comp.11 Comp.12 Comp.13 Comp.14 Comp.15 Comp.16
## 1.0000000 1.0000000 1.0000000 1.0000000 1.0000000 1.0000000 1.0000000 1.0000000
## Comp.17 Comp.18 Comp.19 Comp.20 Comp.21 Comp.22 Comp.23 Comp.24
## 1.0000000 1.0000000 1.0000000 1.0000000 1.0000000 1.0000000 1.0000000 1.0000000
## Comp.25 Comp.26 Comp.27 Comp.28 Comp.29 Comp.30 Comp.31
## 1.0000000 1.0000000 1.0000000 1.0000000 0.5620611 0.5271344 0.2052813
##
## 31 variables and 284807 observations.Summary of PCA
summary(PCA)## Importance of components:
## Comp.1 Comp.2 Comp.3 Comp.4 Comp.5
## Standard deviation 1.39941006 1.31374332 1.2960734 1.00000000 1.00000000
## Proportion of Variance 0.06317253 0.05567489 0.0541873 0.03225806 0.03225806
## Cumulative Proportion 0.06317253 0.11884742 0.1730347 0.20529278 0.23755085
## Comp.6 Comp.7 Comp.8 Comp.9 Comp.10
## Standard deviation 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
## Proportion of Variance 0.03225806 0.03225806 0.03225806 0.03225806 0.03225806
## Cumulative Proportion 0.26980891 0.30206698 0.33432504 0.36658310 0.39884117
## Comp.11 Comp.12 Comp.13 Comp.14 Comp.15
## Standard deviation 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
## Proportion of Variance 0.03225806 0.03225806 0.03225806 0.03225806 0.03225806
## Cumulative Proportion 0.43109923 0.46335730 0.49561536 0.52787343 0.56013149
## Comp.16 Comp.17 Comp.18 Comp.19 Comp.20
## Standard deviation 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
## Proportion of Variance 0.03225806 0.03225806 0.03225806 0.03225806 0.03225806
## Cumulative Proportion 0.59238956 0.62464762 0.65690569 0.68916375 0.72142181
## Comp.21 Comp.22 Comp.23 Comp.24 Comp.25
## Standard deviation 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
## Proportion of Variance 0.03225806 0.03225806 0.03225806 0.03225806 0.03225806
## Cumulative Proportion 0.75367988 0.78593794 0.81819601 0.85045407 0.88271214
## Comp.26 Comp.27 Comp.28 Comp.29 Comp.30
## Standard deviation 1.00000000 1.00000000 1.00000000 0.56206107 0.52713441
## Proportion of Variance 0.03225806 0.03225806 0.03225806 0.01019073 0.00896357
## Cumulative Proportion 0.91497020 0.94722827 0.97948633 0.98967706 0.99864063
## Comp.31
## Standard deviation 0.205281338
## Proportion of Variance 0.001359369
## Cumulative Proportion 1.000000000names
names(PCA)## [1] "sdev" "loadings" "center" "scale" "n.obs" "scores" "call"Use loading matrix to evaluate if PCA captures the essence of original variables
Loadings show the eigenvectors of the correlation matrix
Eigen_vectors <- PCA$loadings
Eigen_vectors##
## Loadings:
## Comp.1 Comp.2 Comp.3 Comp.4 Comp.5 Comp.6 Comp.7 Comp.8 Comp.9 Comp.10
## Time 0.207 0.675
## V1 0.173 0.122 0.127 0.105 0.101
## V2 0.390 -0.102 -0.108 -0.111 0.314
## V3 0.145 -0.481 -0.183 -0.311 0.273 0.134
## V4 -0.152 -0.119 0.173
## V5 0.292 0.128 0.132 0.310 -0.138 0.151 -0.102
## V6 -0.161 0.254 -0.371 -0.164 0.130
## V7 -0.287 0.210 -0.129 0.112 0.187 -0.124 0.580
## V8 0.131 0.197 -0.157
## V9 0.121 -0.123 -0.162 0.295 0.228
## V10 0.208 -0.212 -0.172 -0.129 -0.350
## V11 -0.215 -0.200 0.122 -0.199 0.105 0.389 -0.312
## V12 0.278 -0.622
## V13 0.684
## V14 0.255 -0.187 0.114 -0.283
## V15 -0.184 -0.141 0.164 -0.116 -0.390 -0.442 -0.170
## V16 0.187 0.265 0.206 -0.217 -0.135 -0.200
## V17 0.284 -0.170 0.150 0.441 0.217
## V18 0.131 -0.665
## V19 0.357 -0.194 0.261 0.276
## V20 -0.252 0.138 -0.136
## V21 -0.491 -0.454 -0.104
## V22 0.141 -0.426 -0.419 0.123 -0.125
## V23 0.129 0.301
## V24 -0.159 0.138
## V25 -0.232 -0.110 -0.394 0.205 0.128
## V26 -0.105 -0.333 0.115 0.271
## V27 0.157 0.336 0.314 -0.283
## V28 -0.107 -0.119 0.102 0.468 -0.139
## Amount -0.707
## Class -0.677 0.202
## Comp.11 Comp.12 Comp.13 Comp.14 Comp.15 Comp.16 Comp.17 Comp.18 Comp.19
## Time
## V1 -0.105 -0.114 0.112 -0.415 -0.479 -0.338 -0.136
## V2 -0.167 -0.198 -0.317 -0.265 0.133 0.167
## V3 -0.178 0.152 0.115
## V4 0.360 -0.209 -0.105 -0.346 0.301 0.333
## V5 0.367 0.321 -0.244 0.260 0.201 -0.149
## V6 -0.430 0.131 -0.247 -0.294 0.184
## V7 0.242
## V8 0.363 0.333 -0.141 -0.325 0.207 -0.400 -0.100
## V9 -0.108 0.336 0.264 0.237 0.370 -0.344 -0.308
## V10 -0.149 -0.163 0.130 0.168 0.105 -0.295 0.224
## V11 0.355 -0.149 -0.147 0.256 -0.188 -0.167
## V12
## V13 0.100
## V14 0.137 -0.374 -0.300 -0.191 0.159
## V15 0.227 0.143 -0.198 0.123 -0.222 0.102
## V16 0.403 0.107 -0.139 -0.177 0.225 0.189 0.183
## V17 -0.189 -0.165 0.119
## V18 -0.129 0.156 0.145 -0.186 0.122
## V19 -0.221 0.262 -0.280 0.193
## V20 -0.292 0.239 -0.339 -0.198 -0.122 -0.438
## V21 0.297 -0.129
## V22 -0.173 -0.164 -0.208 0.245
## V23 0.465 -0.113 0.288 -0.131 0.256 -0.130 -0.243 0.325 -0.135
## V24 0.115 -0.132 0.429 -0.268 0.257 0.258 0.415
## V25 0.218 0.216 -0.386 0.123
## V26 0.454 -0.206 -0.108 0.266 -0.117 -0.397 0.184
## V27 0.105 -0.153 -0.298 0.422
## V28 0.181 0.252 -0.313 0.206 0.128 -0.197
## Amount
## Class
## Comp.20 Comp.21 Comp.22 Comp.23 Comp.24 Comp.25 Comp.26 Comp.27 Comp.28
## Time
## V1 -0.159 -0.397 0.113 -0.230 -0.114
## V2 0.274 0.226 -0.181 0.148 -0.253
## V3 -0.218 -0.112 0.133 -0.190 0.129
## V4 -0.483 0.145 -0.132 0.140 -0.170 -0.141
## V5 -0.237 0.203 0.151 -0.195
## V6 0.218 0.341 -0.176 -0.245 0.129 0.105
## V7 0.141 -0.243 -0.140 -0.213 -0.138 0.194 0.127
## V8 0.505 -0.102 0.156
## V9 0.274 0.142 0.136 0.217 0.101
## V10 0.156 -0.251 0.186 -0.326 0.370 0.170
## V11 -0.270 0.180 0.180 0.206
## V12 0.660
## V13 0.679
## V14 0.339 0.249 0.289 0.177 -0.174 -0.164
## V15 0.326 0.179 -0.160 -0.200 0.261
## V16 -0.155 -0.239 0.249 0.151 0.312 -0.257
## V17 0.172 -0.186 -0.125 -0.139 -0.225 -0.508
## V18 -0.117 0.131 0.112 -0.520 -0.221
## V19 -0.139 0.250 0.454 -0.272 0.226
## V20 -0.177 0.410 0.235 -0.201
## V21 0.131 -0.117 0.114 -0.337 -0.109 -0.456
## V22 -0.227 0.191 -0.152 -0.146 0.212 0.453
## V23 -0.106 0.170 -0.270 -0.364 0.107
## V24 0.164 0.345 0.299 0.138 -0.228 -0.158
## V25 -0.217 -0.549
## V26 0.108 -0.110 0.393 -0.156 0.142
## V27 0.134 0.209 -0.335 0.270 -0.240 0.183 -0.143
## V28 0.167 -0.250 0.309 -0.233 -0.201 0.351
## Amount
## Class
## Comp.29 Comp.30 Comp.31
## Time 0.707
## V1 -0.118 -0.102 0.167
## V2 0.393
## V3 0.439 -0.185 0.155
## V4 0.106 0.132
## V5 -0.175 0.284
## V6 -0.160
## V7 -0.182 -0.295
## V8
## V9
## V10 -0.213
## V11 0.253 0.155
## V12 -0.123 -0.257
## V13
## V14 0.109 -0.295
## V15 0.190
## V16 -0.193
## V17 -0.319
## V18 -0.110
## V19
## V20 -0.250
## V21
## V22 -0.149
## V23
## V24
## V25 0.241
## V26
## V27
## V28
## Amount 0.707
## Class -0.707
##
## Comp.1 Comp.2 Comp.3 Comp.4 Comp.5 Comp.6 Comp.7 Comp.8 Comp.9
## SS loadings 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
## Proportion Var 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032
## Cumulative Var 0.032 0.065 0.097 0.129 0.161 0.194 0.226 0.258 0.290
## Comp.10 Comp.11 Comp.12 Comp.13 Comp.14 Comp.15 Comp.16 Comp.17
## SS loadings 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
## Proportion Var 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032
## Cumulative Var 0.323 0.355 0.387 0.419 0.452 0.484 0.516 0.548
## Comp.18 Comp.19 Comp.20 Comp.21 Comp.22 Comp.23 Comp.24 Comp.25
## SS loadings 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
## Proportion Var 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032
## Cumulative Var 0.581 0.613 0.645 0.677 0.710 0.742 0.774 0.806
## Comp.26 Comp.27 Comp.28 Comp.29 Comp.30 Comp.31
## SS loadings 1.000 1.000 1.000 1.000 1.000 1.000
## Proportion Var 0.032 0.032 0.032 0.032 0.032 0.032
## Cumulative Var 0.839 0.871 0.903 0.935 0.968 1.000Eigen values are standard dev squared
Eigen_values <- PCA$sdev * PCA$sdev
Eigen_values## Comp.1 Comp.2 Comp.3 Comp.4 Comp.5 Comp.6 Comp.7
## 1.95834852 1.72592150 1.67980622 1.00000000 1.00000000 1.00000000 1.00000000
## Comp.8 Comp.9 Comp.10 Comp.11 Comp.12 Comp.13 Comp.14
## 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
## Comp.15 Comp.16 Comp.17 Comp.18 Comp.19 Comp.20 Comp.21
## 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
## Comp.22 Comp.23 Comp.24 Comp.25 Comp.26 Comp.27 Comp.28
## 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
## Comp.29 Comp.30 Comp.31
## 0.31591265 0.27787068 0.04214043Create correlation matrix between original data and Scores
Scores are the values projected onto the eigenvectors
round(cor(credit_df, PCA$scores), 3)## Comp.1 Comp.2 Comp.3 Comp.4 Comp.5 Comp.6 Comp.7 Comp.8 Comp.9 Comp.10
## Time 0.042 0.272 0.875 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## V1 0.242 0.160 0.103 0.127 -0.069 -0.034 0.105 -0.071 -0.076 0.101
## V2 0.546 -0.135 0.000 -0.069 -0.108 0.028 -0.111 -0.023 -0.020 0.314
## V3 0.203 0.072 -0.623 -0.097 -0.183 0.050 -0.311 0.273 0.134 0.089
## V4 -0.109 -0.200 -0.080 -0.119 -0.033 -0.070 0.173 -0.020 0.062 0.087
## V5 0.408 0.168 0.171 0.087 0.038 0.310 -0.138 0.151 -0.102 0.094
## V6 -0.225 0.037 -0.089 0.019 -0.020 0.254 -0.371 -0.164 0.130 -0.058
## V7 -0.402 0.275 0.053 0.044 -0.129 0.112 -0.062 0.187 -0.124 0.580
## V8 0.104 -0.042 -0.044 -0.020 0.009 0.131 -0.014 0.197 -0.157 0.003
## V9 0.047 0.115 -0.051 0.121 -0.123 -0.162 0.295 -0.095 -0.011 0.228
## V10 0.111 0.274 -0.048 0.064 -0.051 -0.212 0.080 -0.172 -0.129 -0.350
## V11 -0.014 -0.283 -0.259 -0.003 0.122 -0.199 0.105 0.389 -0.312 -0.003
## V12 0.021 0.366 0.059 -0.622 0.013 -0.043 0.013 0.010 0.028 -0.014
## V13 -0.008 -0.019 -0.086 0.684 -0.042 0.050 -0.055 -0.076 0.076 0.015
## V14 -0.033 0.335 -0.242 0.114 -0.058 -0.283 0.060 0.074 0.096 -0.016
## V15 -0.005 -0.064 -0.238 -0.086 -0.141 0.164 -0.116 -0.390 -0.442 -0.170
## V16 0.009 0.245 -0.061 0.050 0.265 0.206 -0.217 -0.135 -0.042 -0.200
## V17 -0.004 0.373 -0.220 0.150 0.441 -0.007 0.217 0.068 -0.086 0.037
## V18 -0.030 0.172 0.075 0.050 -0.665 0.008 -0.038 0.005 0.085 -0.067
## V19 0.058 -0.034 0.048 -0.012 0.357 -0.009 -0.194 -0.097 0.261 0.276
## V20 -0.353 -0.032 -0.044 -0.040 0.003 0.138 -0.136 0.033 -0.038 0.080
## V21 -0.108 -0.029 0.077 0.006 0.084 -0.491 -0.454 -0.010 -0.104 -0.026
## V22 0.073 0.051 0.183 0.089 -0.022 -0.426 -0.419 0.123 -0.125 -0.007
## V23 0.119 0.019 0.060 -0.030 0.129 -0.080 -0.059 -0.055 0.301 -0.008
## V24 -0.006 0.003 -0.023 -0.069 -0.008 -0.159 0.138 0.044 -0.066 0.075
## V25 0.039 -0.093 -0.301 -0.084 0.012 -0.110 0.035 -0.394 0.205 0.128
## V26 0.001 -0.021 -0.052 -0.028 0.026 -0.105 -0.017 -0.333 0.115 0.271
## V27 -0.030 -0.022 0.001 -0.001 0.035 0.157 -0.019 0.336 0.314 -0.283
## V28 -0.011 -0.015 -0.008 0.024 -0.107 -0.119 0.087 0.102 0.468 -0.139
## Amount -0.989 0.026 0.027 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## Class -0.021 -0.890 0.262 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## Comp.11 Comp.12 Comp.13 Comp.14 Comp.15 Comp.16 Comp.17 Comp.18 Comp.19
## Time 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## V1 -0.017 -0.105 -0.114 0.112 -0.415 -0.479 -0.338 -0.136 0.023
## V2 -0.167 -0.198 0.015 -0.317 0.000 -0.265 0.133 0.167 0.077
## V3 -0.178 -0.062 0.152 0.095 0.115 0.092 0.020 -0.068 -0.046
## V4 -0.096 0.360 0.031 -0.209 -0.105 0.040 -0.346 0.301 0.333
## V5 0.367 0.321 -0.244 0.260 -0.001 0.201 0.026 -0.007 -0.149
## V6 -0.042 0.049 -0.430 0.131 0.097 -0.247 -0.294 -0.038 0.184
## V7 0.088 0.086 0.071 -0.057 0.068 -0.066 0.072 0.242 0.086
## V8 0.016 0.363 0.333 -0.141 -0.325 0.207 -0.400 -0.039 -0.100
## V9 -0.108 0.336 0.264 0.237 0.370 -0.344 -0.020 -0.308 -0.038
## V10 -0.149 -0.163 0.130 0.168 0.098 0.105 -0.295 0.224 -0.084
## V11 0.355 -0.149 -0.147 0.256 0.097 -0.188 -0.167 0.068 -0.018
## V12 -0.020 0.032 -0.045 -0.020 -0.006 -0.025 0.018 -0.043 -0.038
## V13 -0.039 0.000 0.055 -0.053 -0.036 0.089 0.072 0.100 0.003
## V14 0.137 -0.001 -0.374 -0.300 -0.191 0.060 0.006 0.159 -0.075
## V15 0.227 0.143 -0.023 -0.198 0.123 -0.222 0.052 0.086 0.102
## V16 0.085 0.067 0.403 0.107 -0.139 -0.177 0.225 0.189 0.183
## V17 -0.037 -0.031 -0.041 -0.189 0.067 0.006 -0.049 -0.165 0.119
## V18 0.094 -0.129 0.085 0.156 0.027 0.145 -0.186 0.037 0.122
## V19 -0.221 -0.089 -0.039 0.262 0.060 0.055 -0.280 0.193 0.045
## V20 0.049 -0.292 0.239 -0.001 -0.339 -0.198 -0.078 -0.122 -0.438
## V21 -0.039 0.297 0.010 0.095 0.097 -0.031 0.014 -0.015 -0.129
## V22 0.030 0.002 0.082 -0.173 -0.164 -0.035 0.035 -0.208 0.245
## V23 0.465 -0.113 0.288 -0.131 0.256 -0.130 -0.243 0.325 -0.135
## V24 0.115 -0.132 0.081 0.429 -0.268 0.036 0.257 0.258 0.415
## V25 0.077 0.218 -0.033 0.216 -0.386 0.088 0.123 -0.089 -0.078
## V26 0.454 -0.206 0.088 -0.108 0.066 0.266 -0.117 -0.397 0.184
## V27 0.095 0.026 0.105 -0.040 -0.012 -0.153 -0.038 -0.298 0.422
## V28 0.181 0.252 -0.048 -0.052 -0.073 -0.313 0.206 0.128 -0.197
## Amount 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## Class 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## Comp.20 Comp.21 Comp.22 Comp.23 Comp.24 Comp.25 Comp.26 Comp.27 Comp.28
## Time 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## V1 -0.159 -0.397 0.113 0.084 -0.230 -0.114 0.033 -0.013 -0.015
## V2 0.274 0.226 -0.181 -0.057 0.148 -0.094 0.005 -0.253 0.021
## V3 -0.218 -0.112 0.078 0.133 -0.190 -0.100 0.067 0.129 -0.063
## V4 -0.483 0.145 -0.132 0.140 0.094 -0.170 0.057 -0.141 0.028
## V5 -0.237 0.203 -0.009 -0.018 0.151 -0.195 0.068 -0.021 -0.049
## V6 0.218 0.341 -0.176 0.080 -0.245 0.129 0.105 0.002 0.009
## V7 0.141 -0.243 -0.140 -0.213 0.007 -0.138 0.194 0.127 -0.043
## V8 0.505 0.051 0.066 0.097 -0.102 0.156 -0.061 0.098 0.032
## V9 -0.008 0.274 0.142 0.045 0.136 0.217 0.101 -0.017 -0.014
## V10 0.156 0.078 -0.251 -0.100 0.186 -0.326 0.370 0.170 -0.045
## V11 0.050 -0.093 -0.270 -0.009 0.180 0.180 -0.038 -0.074 0.206
## V12 -0.020 0.009 0.010 0.045 0.014 0.041 -0.012 -0.020 0.660
## V13 -0.027 0.020 0.067 0.032 -0.028 -0.053 0.022 -0.026 0.679
## V14 0.069 0.079 0.339 0.077 0.249 0.289 0.177 -0.174 -0.164
## V15 0.017 -0.086 0.326 -0.081 0.179 -0.160 -0.200 0.261 0.031
## V16 -0.086 -0.155 -0.239 0.249 0.151 0.312 0.048 -0.257 -0.098
## V17 0.000 0.172 -0.186 -0.125 -0.139 -0.225 -0.508 -0.038 -0.080
## V18 -0.013 -0.015 -0.117 -0.077 0.131 0.112 -0.520 -0.221 -0.041
## V19 0.056 -0.139 0.250 0.018 0.454 0.069 -0.272 0.226 0.010
## V20 -0.177 0.410 0.078 0.014 0.235 -0.201 -0.025 -0.082 -0.003
## V21 0.131 -0.117 0.114 0.043 -0.093 -0.337 -0.109 -0.456 -0.015
## V22 -0.227 0.191 -0.152 -0.146 0.051 0.212 -0.011 0.453 0.012
## V23 -0.106 0.092 0.170 -0.270 -0.364 0.107 0.010 -0.011 -0.004
## V24 0.164 0.345 0.299 0.138 -0.228 -0.158 -0.041 0.077 -0.010
## V25 0.034 -0.046 -0.217 -0.549 0.003 0.073 0.016 -0.045 0.024
## V26 0.108 -0.079 -0.110 0.393 0.090 -0.156 0.142 -0.026 0.006
## V27 0.134 -0.075 0.209 -0.335 0.270 -0.240 0.183 -0.143 0.041
## V28 0.167 -0.040 -0.250 0.309 0.095 -0.233 -0.201 0.351 -0.024
## Amount 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## Class 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
## Comp.29 Comp.30 Comp.31
## Time 0.397 0.005 0.000
## V1 -0.066 -0.054 0.034
## V2 0.006 0.046 0.081
## V3 0.247 -0.097 0.032
## V4 0.059 0.070 -0.015
## V5 -0.099 -0.051 0.058
## V6 0.037 -0.022 -0.033
## V7 -0.048 -0.096 -0.060
## V8 0.021 0.010 0.016
## V9 0.006 -0.050 0.007
## V10 -0.015 -0.112 0.015
## V11 0.142 0.082 0.000
## V12 -0.069 -0.135 0.001
## V13 0.038 -0.002 -0.001
## V14 0.061 -0.155 -0.005
## V15 0.107 -0.001 0.001
## V16 -0.004 -0.102 0.000
## V17 0.047 -0.168 -0.001
## V18 -0.051 -0.058 -0.006
## V19 -0.017 0.018 0.009
## V20 0.029 0.012 -0.051
## V21 -0.027 0.021 -0.016
## V22 -0.084 -0.001 0.010
## V23 -0.029 -0.002 0.017
## V24 0.009 -0.004 -0.001
## V25 0.135 0.003 0.007
## V26 0.024 0.003 0.001
## V27 0.003 0.009 -0.004
## V28 0.005 0.005 -0.002
## Amount 0.001 -0.002 0.145
## Class 0.009 -0.373 -0.001Use Scree plot to pick number of PCs to use
Use scree test to find that Elbow occurs at 4 components and these 4 components are statistically significant
screeplot(PCA, type='l',main='Screeplot for credit data')
abline(1,0, col ='red',lty=2)
Another method for PCA analysis using psych package
install.packages('psych', repos = "http://cran.us.r-project.org")## Installing package into 'C:/Users/Steve/AppData/Local/R/win-library/4.2'
## (as 'lib' is unspecified)## package 'psych' successfully unpacked and MD5 sums checked
##
## The downloaded binary packages are in
## C:\Users\Steve\AppData\Local\Temp\RtmpyWFmpx\downloaded_packageslibrary(psych)##
## Attaching package: 'psych'## The following objects are masked from 'package:ggplot2':
##
## %+%, alphaPCA for 4 PCs
PCA1 <- principal(credit_df, nfactors = 4, rotate='none')
PCA1## Principal Components Analysis
## Call: principal(r = credit_df, nfactors = 4, rotate = "none")
## Standardized loadings (pattern matrix) based upon correlation matrix
## PC1 PC2 PC3 PC4 h2 u2 com
## Time -0.04 0.27 -0.88 0.00 0.8420 0.158 1.2
## V1 -0.24 0.16 -0.10 0.07 0.1000 0.900 2.4
## V2 -0.55 -0.13 0.00 -0.04 0.3173 0.683 1.1
## V3 -0.20 0.07 0.62 -0.07 0.4395 0.560 1.3
## V4 0.11 -0.20 0.08 -0.08 0.0652 0.935 2.3
## V5 -0.41 0.17 -0.17 0.09 0.2313 0.769 1.8
## V6 0.22 0.04 0.09 -0.04 0.0618 0.938 1.5
## V7 0.40 0.28 -0.05 0.08 0.2467 0.753 1.9
## V8 -0.10 -0.04 0.04 -0.03 0.0155 0.985 1.9
## V9 -0.05 0.12 0.05 0.11 0.0296 0.970 2.7
## V10 -0.11 0.27 0.05 0.06 0.0934 0.907 1.5
## V11 0.01 -0.28 0.26 -0.09 0.1562 0.844 2.2
## V12 -0.02 0.37 -0.06 -0.64 0.5441 0.456 1.6
## V13 0.01 -0.02 0.09 0.69 0.4834 0.517 1.0
## V14 0.03 0.34 0.24 0.13 0.1881 0.812 2.2
## V15 0.00 -0.06 0.24 -0.06 0.0640 0.936 1.3
## V16 -0.01 0.25 0.06 0.07 0.0684 0.932 1.3
## V17 0.00 0.37 0.22 0.11 0.1996 0.800 1.8
## V18 0.03 0.17 -0.07 0.06 0.0398 0.960 1.7
## V19 -0.06 -0.03 -0.05 -0.02 0.0071 0.993 2.8
## V20 0.35 -0.03 0.04 -0.03 0.1286 0.871 1.1
## V21 0.11 -0.03 -0.08 0.00 0.0186 0.981 2.0
## V22 -0.07 0.05 -0.18 0.05 0.0438 0.956 1.6
## V23 -0.12 0.02 -0.06 -0.01 0.0181 0.982 1.6
## V24 0.01 0.00 0.02 -0.07 0.0049 0.995 1.3
## V25 -0.04 -0.09 0.30 -0.09 0.1093 0.891 1.4
## V26 0.00 -0.02 0.05 -0.04 0.0048 0.995 2.3
## V27 0.03 -0.02 0.00 0.01 0.0015 0.999 2.0
## V28 0.01 -0.01 0.01 0.03 0.0015 0.998 1.8
## Amount 0.99 0.03 -0.03 0.00 0.9789 0.021 1.0
## Class 0.02 -0.89 -0.26 0.00 0.8611 0.139 1.2
##
## PC1 PC2 PC3 PC4
## SS loadings 1.96 1.73 1.68 1.00
## Proportion Var 0.06 0.06 0.05 0.03
## Cumulative Var 0.06 0.12 0.17 0.21
## Proportion Explained 0.31 0.27 0.26 0.16
## Cumulative Proportion 0.31 0.58 0.84 1.00
##
## Mean item complexity = 1.7
## Test of the hypothesis that 4 components are sufficient.
##
## The root mean square of the residuals (RMSR) is 0.05
## with the empirical chi square 681022.4 with prob < 0
##
## Fit based upon off diagonal values = 0.37