Principal Components Analysis
Introduction
The objective of this analysis is to apply Principal Components Analysis to a (large) dataset from the United States Department of Agriculture Food Composition Database. We will walk-through fitting PCA to the dataset and determine how many components would be adequately useful to represent the data in a lower dimension.
Data
The data set we will be using comes from the United States Department of Agriculture Food Composition Database. The USDA National Nutrient Database for Standard Reference (SR) is the major source of food composition data in the United States. It provides the foundation for most food composition databases in the public and private sectors. As information is updated, new versions of the database are released. This version, Release 28 (SR28), contains data on 8,789 food items and up to 150 food components. It replaces SR27, initially issued in August 2014 and revised in May 2015.
There are 8789 different types of food in the data. The columns for the food include various vitamin/mineral content, nutrient profiles, and other food related characteristics.
Exploratory Data Analysis
Let us import the data set from a text file:
setwd('C:/users/crzys/Documents')
SR = read.table("ABBREV.txt", header=F, row.names=1, sep="^", quote="~")Let us take a look at our data set:
summary(SR)## V2
## BEEF,CHUCK,UNDER BLADE CNTR STEAK,BNLESS,DENVER CUT,LN,0" FA: 2
## OIL,INDUSTRIAL,PALM KERNEL (HYDROGENATED),CONFECTION FAT : 2
## POPCORN,OIL-POPPED,LOFAT : 2
## ABALONE,MIXED SPECIES,RAW : 1
## ABALONE,MXD SP,CKD,FRIED : 1
## ABIYUCH,RAW : 1
## (Other) :8781
## V3 V4 V5 V6
## Min. : 0.00 Min. : 0.0 Min. : 0.00 Min. : 0.00
## 1st Qu.: 29.70 1st Qu.: 91.0 1st Qu.: 2.38 1st Qu.: 0.95
## Median : 63.08 Median :191.0 Median : 8.00 Median : 5.14
## Mean : 54.11 Mean :226.3 Mean :11.34 Mean : 10.55
## 3rd Qu.: 77.96 3rd Qu.:337.0 3rd Qu.:19.88 3rd Qu.: 13.72
## Max. :100.00 Max. :902.0 Max. :88.32 Max. :100.00
## NA's :1
## V7 V8 V9 V10
## Min. : 0.000 Min. : 0.00 Min. : 0.000 Min. : 0.000
## 1st Qu.: 0.840 1st Qu.: 0.05 1st Qu.: 0.000 1st Qu.: 0.000
## Median : 1.200 Median : 9.34 Median : 0.700 Median : 1.840
## Mean : 1.784 Mean : 22.13 Mean : 2.187 Mean : 8.543
## 3rd Qu.: 2.090 3rd Qu.: 34.91 3rd Qu.: 2.600 3rd Qu.: 9.287
## Max. :99.800 Max. :100.00 Max. :79.000 Max. :99.800
## NA's :325 NA's :594 NA's :1832
## V11 V12 V13 V14
## Min. : 0.00 Min. : 0.00 Min. : 0.0 Min. : 0.0
## 1st Qu.: 10.00 1st Qu.: 0.54 1st Qu.: 12.0 1st Qu.: 51.0
## Median : 21.00 Median : 1.38 Median : 21.0 Median : 145.0
## Mean : 76.74 Mean : 2.70 Mean : 35.3 Mean : 165.1
## 3rd Qu.: 69.00 3rd Qu.: 2.60 3rd Qu.: 30.0 3rd Qu.: 220.0
## Max. :7364.00 Max. :123.60 Max. :781.0 Max. :9918.0
## NA's :348 NA's :144 NA's :739 NA's :579
## V15 V16 V17 V18
## Min. : 0.0 Min. : 0.0 Min. : 0.000 Min. : 0.000
## 1st Qu.: 127.0 1st Qu.: 41.0 1st Qu.: 0.340 1st Qu.: 0.052
## Median : 229.5 Median : 88.0 Median : 0.990 Median : 0.090
## Mean : 279.5 Mean : 312.5 Mean : 2.117 Mean : 0.196
## 3rd Qu.: 336.0 3rd Qu.: 404.5 3rd Qu.: 2.920 3rd Qu.: 0.160
## Max. :16500.0 Max. :38758.0 Max. :90.950 Max. :15.050
## NA's :426 NA's :83 NA's :706 NA's :1257
## V19 V20 V21 V22
## Min. : 0.0000 Min. : 0.00 Min. : 0.000 Min. : 0.0000
## 1st Qu.: 0.0150 1st Qu.: 1.30 1st Qu.: 0.000 1st Qu.: 0.0400
## Median : 0.0820 Median : 10.10 Median : 0.000 Median : 0.0840
## Mean : 0.6582 Mean : 15.59 Mean : 9.231 Mean : 0.2231
## 3rd Qu.: 0.3260 3rd Qu.: 24.30 3rd Qu.: 3.500 3rd Qu.: 0.2500
## Max. :328.0000 Max. :1917.00 Max. :2732.000 Max. :23.3750
## NA's :2160 NA's :1700 NA's :818 NA's :634
## V23 V24 V25 V26
## Min. : 0.0000 Min. : 0.000 Min. : 0.0000 Min. : 0.0000
## 1st Qu.: 0.0630 1st Qu.: 0.552 1st Qu.: 0.1950 1st Qu.: 0.0510
## Median : 0.1700 Median : 2.580 Median : 0.4350 Median : 0.1500
## Mean : 0.2522 Mean : 3.658 Mean : 0.6510 Mean : 0.2915
## 3rd Qu.: 0.2700 3rd Qu.: 5.276 3rd Qu.: 0.7322 3rd Qu.: 0.3900
## Max. :17.5000 Max. :127.500 Max. :34.5000 Max. :12.0000
## NA's :616 NA's :637 NA's :2242 NA's :905
## V27 V28 V29 V30
## Min. : 0.0 Min. : 0.00 Min. : 0.00 Min. : 0.00
## 1st Qu.: 5.0 1st Qu.: 0.00 1st Qu.: 4.00 1st Qu.: 4.00
## Median : 12.0 Median : 0.00 Median : 9.00 Median : 11.00
## Mean : 51.6 Mean : 23.17 Mean : 24.56 Mean : 64.28
## 3rd Qu.: 46.0 3rd Qu.: 0.00 3rd Qu.: 21.00 3rd Qu.: 41.00
## Max. :3786.0 Max. :2993.00 Max. :2340.00 Max. :5881.00
## NA's :1261 NA's :2039 NA's :1768 NA's :2057
## V31 V32 V33 V34
## Min. : 0.00 Min. : 0.000 Min. : 0.0 Min. : 0.0
## 1st Qu.: 9.00 1st Qu.: 0.000 1st Qu.: 0.0 1st Qu.: 0.0
## Median : 24.75 Median : 0.240 Median : 32.0 Median : 4.0
## Mean : 43.60 Mean : 1.393 Mean : 735.8 Mean : 111.7
## 3rd Qu.: 69.67 3rd Qu.: 1.610 3rd Qu.: 252.5 3rd Qu.: 33.0
## Max. :2403.30 Max. :98.890 Max. :100000.0 Max. :30000.0
## NA's :4016 NA's :1193 NA's :711 NA's :1535
## V35 V36 V37 V38
## Min. : 0.00 Min. : 0.0 Min. : 0.0 Min. : 0.00
## 1st Qu.: 0.00 1st Qu.: 0.0 1st Qu.: 0.0 1st Qu.: 0.00
## Median : 0.00 Median : 0.0 Median : 0.0 Median : 0.00
## Mean : 89.77 Mean : 31.5 Mean : 221.3 Mean : 10.62
## 3rd Qu.: 12.00 3rd Qu.: 0.0 3rd Qu.: 21.0 3rd Qu.: 0.00
## Max. :30000.00 Max. :14251.0 Max. :42891.0 Max. :6252.00
## NA's :1806 NA's :3258 NA's :3162 NA's :3270
## V39 V40 V41 V42
## Min. : 0 Min. : 0 Min. : 0.000 Min. : 0.000
## 1st Qu.: 0 1st Qu.: 0 1st Qu.: 0.120 1st Qu.: 0.000
## Median : 0 Median : 0 Median : 0.300 Median : 0.000
## Mean : 157 Mean : 194 Mean : 1.331 Mean : 0.579
## 3rd Qu.: 0 3rd Qu.: 30 3rd Qu.: 0.800 3rd Qu.: 0.200
## Max. :46260 Max. :19697 Max. :149.400 Max. :250.000
## NA's :3292 NA's :3315 NA's :2889 NA's :3262
## V43 V44 V45 V46
## Min. : 0.00 Min. : 0.00 Min. : 0.000 Min. : 0.0000
## 1st Qu.: 0.00 1st Qu.: 0.20 1st Qu.: 0.220 1st Qu.: 0.2358
## Median : 0.00 Median : 1.60 Median : 1.592 Median : 1.9385
## Mean : 22.99 Mean : 15.63 Mean : 3.576 Mean : 4.0810
## 3rd Qu.: 6.00 3rd Qu.: 5.20 3rd Qu.: 4.345 3rd Qu.: 5.1150
## Max. :10000.00 Max. :1714.50 Max. :95.600 Max. :83.6890
## NA's :3211 NA's :3563 NA's :349 NA's :666
## V47 V48 V49 V50
## Min. : 0.000 Min. : 0.00 Min. : 0.1 3 oz :1409
## 1st Qu.: 0.220 1st Qu.: 0.00 1st Qu.: 29.0 1 cup :1182
## Median : 0.696 Median : 4.00 Median : 85.0 1 oz : 889
## Mean : 2.289 Mean : 40.61 Mean : 96.4 1 tbsp : 343
## 3rd Qu.: 2.081 3rd Qu.: 67.00 3rd Qu.: 130.0 : 299
## Max. :74.623 Max. :3100.00 Max. :1184.0 1 fl oz: 268
## NA's :665 NA's :410 NA's :300 (Other):4400
## V51
## Min. : 0.4
## 1st Qu.: 42.0
## Median : 164.0
## Mean : 250.7
## 3rd Qu.: 305.0
## Max. :5717.0
## NA's :3965
## V52
## :3965
## 1 cup : 298
## 1 steak : 267
## 1 lb : 260
## 1 piece, cooked, excluding refuse (yield from 1 lb raw meat with refuse): 201
## 1 oz : 198
## (Other) :3601
## V53
## Min. : 0.000
## 1st Qu.: 0.000
## Median : 0.000
## Mean : 4.964
## 3rd Qu.: 0.000
## Max. :81.000
## NA's :50We see that there are a lot of NA’s for many of the variables. In addition, the names for the variables are not very useful.
Let’s pre-process the data set and simplify it a little:
SR = na.omit(SR) # remove rows with missing values
SR = SR[row.names(SR) != "13352",] # remove "duplicate" entryRe-name the rows:
R = SR[row.names(SR) != "13352",] # remove "duplicate" entry
row.names(SR) = SR[,1] # set more meaningful row names
SR = SR[,-1]
names(SR) = c("Water_(g)", "Energ_Kcal", "Protein_(g)", "Lipid_Tot_(g)", "Ash_(g)", "Carbohydrt_(g)", "Fiber_TD_(g)", "Sugar_Tot_(g)", "Calcium_(mg)", "Iron_(mg)", "Magnesium_(mg)", "Phosphorus_(mg)", "Potassium_(mg)", "Sodium_(mg)", "Zinc_(mg)", "Copper_(mg)", "Manganese_(mg)", "Selenium_(µg)", "Vit_C_(mg)", "Thiamin_(mg)", "Riboflavin_(mg)", "Niacin_(mg)", "Panto_Acid_(mg)", "Vit_B6_(mg)", "Folate_Tot_(µg)", "Folic_Acid_(µg)", "Food_Folate_(µg)", "Folate_DFE_(µg)", "Choline_Tot_(mg)", "Vit_B12_(µg)", "Vit_A_IU", "Vit_A_RAE", "Retinol_(µg)", "Alpha_Carot_(µg)", "Beta_Carot_(µg)", "Beta_Crypt_(µg)", "Lycopene_(µg)", "Lut+Zea_(µg)", "Vit_E_(mg)", "Vit_D_µg", "Vit_D_IU", "Vit_K_(µg)", "FA_Sat_(g)", "FA_Mono_(g)", "FA_Poly_(g)", "Cholestrl_(mg)", "GmWt_1", "GmWt_Desc1", "GmWt_2", "GmWt_Desc2", "Refuse_Pct")We just want to use the nutrient variables:
SRp = SR[,c(1:46)] # restrict to just the nutrient variablesWe will utilize the SRp version of the data for our analysis.
summary(SRp)## Water_(g) Energ_Kcal Protein_(g) Lipid_Tot_(g)
## Min. : 0.00 Min. : 0.0 Min. : 0.00 Min. : 0.00
## 1st Qu.:41.00 1st Qu.: 83.0 1st Qu.: 2.21 1st Qu.: 0.74
## Median :64.96 Median :182.0 Median :10.08 Median : 5.34
## Mean :57.57 Mean :210.9 Mean :12.85 Mean : 10.15
## 3rd Qu.:79.00 3rd Qu.:292.0 3rd Qu.:22.46 3rd Qu.: 13.37
## Max. :99.90 Max. :902.0 Max. :85.60 Max. :100.00
## Ash_(g) Carbohydrt_(g) Fiber_TD_(g) Sugar_Tot_(g)
## Min. : 0.000 Min. : 0.00 Min. : 0.000 Min. : 0.000
## 1st Qu.: 0.840 1st Qu.: 0.00 1st Qu.: 0.000 1st Qu.: 0.000
## Median : 1.130 Median : 5.50 Median : 0.000 Median : 1.070
## Mean : 1.868 Mean :17.55 Mean : 1.763 Mean : 7.182
## 3rd Qu.: 1.900 3rd Qu.:22.70 3rd Qu.: 2.100 3rd Qu.: 6.370
## Max. :99.800 Max. :99.98 Max. :53.200 Max. :99.800
## Calcium_(mg) Iron_(mg) Magnesium_(mg) Phosphorus_(mg)
## Min. : 0.00 Min. : 0.000 Min. : 0.00 Min. : 0.0
## 1st Qu.: 11.00 1st Qu.: 0.445 1st Qu.: 12.00 1st Qu.: 50.0
## Median : 19.00 Median : 1.080 Median : 21.00 Median : 152.0
## Mean : 76.16 Mean : 1.890 Mean : 33.58 Mean : 177.0
## 3rd Qu.: 56.00 3rd Qu.: 2.340 3rd Qu.: 28.00 3rd Qu.: 221.5
## Max. :7364.00 Max. :89.800 Max. :711.00 Max. :9918.0
## Potassium_(mg) Sodium_(mg) Zinc_(mg) Copper_(mg)
## Min. : 0.0 Min. : 0 Min. : 0.000 Min. :0.0000
## 1st Qu.: 135.0 1st Qu.: 41 1st Qu.: 0.350 1st Qu.:0.0490
## Median : 241.0 Median : 76 Median : 1.000 Median :0.0810
## Mean : 298.6 Mean : 339 Mean : 2.185 Mean :0.1716
## 3rd Qu.: 339.0 3rd Qu.: 349 3rd Qu.: 3.100 3rd Qu.:0.1370
## Max. :16500.0 Max. :38758 Max. :90.950 Max. :6.1000
## Manganese_(mg) Selenium_(µg) Vit_C_(mg) Thiamin_(mg)
## Min. : 0.0000 Min. : 0.00 Min. : 0.000 Min. : 0.0000
## 1st Qu.: 0.0130 1st Qu.: 1.30 1st Qu.: 0.000 1st Qu.: 0.0360
## Median : 0.0550 Median : 13.00 Median : 0.000 Median : 0.0710
## Mean : 0.4587 Mean : 17.08 Mean : 6.356 Mean : 0.1753
## 3rd Qu.: 0.2800 3rd Qu.: 26.85 3rd Qu.: 1.800 3rd Qu.: 0.1725
## Max. :125.0000 Max. :1917.00 Max. :1900.000 Max. :10.9900
## Riboflavin_(mg) Niacin_(mg) Panto_Acid_(mg) Vit_B6_(mg)
## Min. :0.000 Min. : 0.000 Min. : 0.0000 Min. :0.0000
## 1st Qu.:0.057 1st Qu.: 0.438 1st Qu.: 0.1970 1st Qu.:0.0500
## Median :0.175 Median : 2.486 Median : 0.4330 Median :0.1520
## Mean :0.212 Mean : 3.317 Mean : 0.5864 Mean :0.2565
## 3rd Qu.:0.265 3rd Qu.: 5.279 3rd Qu.: 0.7320 3rd Qu.:0.4035
## Max. :4.000 Max. :45.249 Max. :22.6240 Max. :5.8990
## Folate_Tot_(µg) Folic_Acid_(µg) Food_Folate_(µg) Folate_DFE_(µg)
## Min. : 0.00 Min. : 0.000 Min. : 0.00 Min. : 0.0
## 1st Qu.: 3.00 1st Qu.: 0.000 1st Qu.: 3.00 1st Qu.: 3.0
## Median : 9.00 Median : 0.000 Median : 8.00 Median : 9.0
## Mean : 31.08 Mean : 9.052 Mean : 22.01 Mean : 37.4
## 3rd Qu.: 28.00 3rd Qu.: 0.000 3rd Qu.: 20.00 3rd Qu.: 31.0
## Max. :2340.00 Max. :1286.000 Max. :2340.00 Max. :2340.0
## Choline_Tot_(mg) Vit_B12_(µg) Vit_A_IU Vit_A_RAE
## Min. : 0.00 Min. : 0.000 Min. : 0.0 Min. : 0.00
## 1st Qu.: 10.60 1st Qu.: 0.000 1st Qu.: 1.0 1st Qu.: 0.00
## Median : 29.10 Median : 0.200 Median : 16.0 Median : 3.00
## Mean : 46.63 Mean : 0.961 Mean : 499.5 Mean : 48.21
## 3rd Qu.: 70.45 3rd Qu.: 1.390 3rd Qu.: 142.5 3rd Qu.: 22.00
## Max. :2403.30 Max. :20.090 Max. :49254.0 Max. :4220.00
## Retinol_(µg) Alpha_Carot_(µg) Beta_Carot_(µg) Beta_Crypt_(µg)
## Min. : 0.00 Min. : 0.0 Min. : 0.0 Min. : 0.00
## 1st Qu.: 0.00 1st Qu.: 0.0 1st Qu.: 0.0 1st Qu.: 0.00
## Median : 0.00 Median : 0.0 Median : 0.0 Median : 0.00
## Mean : 27.89 Mean : 26.8 Mean : 225.2 Mean : 10.92
## 3rd Qu.: 8.00 3rd Qu.: 0.0 3rd Qu.: 12.0 3rd Qu.: 0.00
## Max. :4220.00 Max. :4342.0 Max. :26162.0 Max. :6186.00
## Lycopene_(µg) Lut+Zea_(µg) Vit_E_(mg) Vit_D_µg
## Min. : 0.0 Min. : 0.0 Min. : 0.000 Min. : 0.0000
## 1st Qu.: 0.0 1st Qu.: 0.0 1st Qu.: 0.110 1st Qu.: 0.0000
## Median : 0.0 Median : 0.0 Median : 0.250 Median : 0.0000
## Mean : 162.4 Mean : 173.9 Mean : 0.999 Mean : 0.5171
## 3rd Qu.: 0.0 3rd Qu.: 26.5 3rd Qu.: 0.630 3rd Qu.: 0.2000
## Max. :45902.0 Max. :19697.0 Max. :41.080 Max. :31.9000
## Vit_D_IU Vit_K_(µg) FA_Sat_(g) FA_Mono_(g)
## Min. : 0.00 Min. : 0.00 Min. : 0.000 Min. : 0.0000
## 1st Qu.: 0.00 1st Qu.: 0.10 1st Qu.: 0.136 1st Qu.: 0.1215
## Median : 0.00 Median : 1.50 Median : 1.617 Median : 1.8820
## Mean : 20.68 Mean : 13.27 Mean : 3.396 Mean : 3.9440
## 3rd Qu.: 7.00 3rd Qu.: 4.25 3rd Qu.: 4.249 3rd Qu.: 5.2865
## Max. :1276.00 Max. :1714.50 Max. :86.503 Max. :83.6890
## FA_Poly_(g) Cholestrl_(mg)
## Min. : 0.000 Min. : 0.0
## 1st Qu.: 0.164 1st Qu.: 0.0
## Median : 0.559 Median : 8.0
## Mean : 1.910 Mean : 42.2
## 3rd Qu.: 1.619 3rd Qu.: 73.0
## Max. :57.740 Max. :3100.0We now have useful variable names and there are no NA’s.
Let us see the means of the columns:
apply(SRp, 2, mean)## Water_(g) Energ_Kcal Protein_(g) Lipid_Tot_(g)
## 57.5733648 210.8547009 12.8536302 10.1525641
## Ash_(g) Carbohydrt_(g) Fiber_TD_(g) Sugar_Tot_(g)
## 1.8684480 17.5526361 1.7632029 7.1816374
## Calcium_(mg) Iron_(mg) Magnesium_(mg) Phosphorus_(mg)
## 76.1605938 1.8895322 33.5802969 176.9505173
## Potassium_(mg) Sodium_(mg) Zinc_(mg) Copper_(mg)
## 298.5600540 338.9874044 2.1851192 0.1715866
## Manganese_(mg) Selenium_(µg) Vit_C_(mg) Thiamin_(mg)
## 0.4586914 17.0785425 6.3556905 0.1753297
## Riboflavin_(mg) Niacin_(mg) Panto_Acid_(mg) Vit_B6_(mg)
## 0.2120067 3.3172353 0.5863590 0.2565119
## Folate_Tot_(µg) Folic_Acid_(µg) Food_Folate_(µg) Folate_DFE_(µg)
## 31.0800720 9.0517319 22.0107962 37.3958614
## Choline_Tot_(mg) Vit_B12_(µg) Vit_A_IU Vit_A_RAE
## 46.6255511 0.9609717 499.4898785 48.2069276
## Retinol_(µg) Alpha_Carot_(µg) Beta_Carot_(µg) Beta_Crypt_(µg)
## 27.8884390 26.7953216 225.1785875 10.9221772
## Lycopene_(µg) Lut+Zea_(µg) Vit_E_(mg) Vit_D_µg
## 162.4291498 173.9185785 0.9990463 0.5170490
## Vit_D_IU Vit_K_(µg) FA_Sat_(g) FA_Mono_(g)
## 20.6842105 13.2708052 3.3955789 3.9440193
## FA_Poly_(g) Cholestrl_(mg)
## 1.9098399 42.1983806The mean varies a lot between the variables and they are in different units, like mg and g.
How about the variance of the columns?
apply(SRp, 2, var)## Water_(g) Energ_Kcal Protein_(g) Lipid_Tot_(g)
## 8.349523e+02 2.516972e+04 1.284037e+02 2.060145e+02
## Ash_(g) Carbohydrt_(g) Fiber_TD_(g) Sugar_Tot_(g)
## 1.596343e+01 6.076003e+02 1.622151e+01 2.164031e+02
## Calcium_(mg) Iron_(mg) Magnesium_(mg) Phosphorus_(mg)
## 7.473869e+04 1.505961e+01 3.197176e+03 9.443834e+04
## Potassium_(mg) Sodium_(mg) Zinc_(mg) Copper_(mg)
## 2.602524e+05 1.918025e+06 1.438623e+01 1.584133e-01
## Manganese_(mg) Selenium_(µg) Vit_C_(mg) Thiamin_(mg)
## 1.076104e+01 1.998693e+03 2.032894e+03 1.268544e-01
## Riboflavin_(mg) Niacin_(mg) Panto_Acid_(mg) Vit_B6_(mg)
## 8.725530e-02 1.292544e+01 9.630418e-01 9.814600e-02
## Folate_Tot_(µg) Folic_Acid_(µg) Food_Folate_(µg) Folate_DFE_(µg)
## 6.855954e+03 2.064629e+03 4.682490e+03 1.083398e+04
## Choline_Tot_(mg) Vit_B12_(µg) Vit_A_IU Vit_A_RAE
## 5.790941e+03 3.266657e+00 5.098720e+06 3.547239e+04
## Retinol_(µg) Alpha_Carot_(µg) Beta_Carot_(µg) Beta_Crypt_(µg)
## 2.411153e+04 5.225111e+04 1.490032e+06 2.447177e+04
## Lycopene_(µg) Lut+Zea_(µg) Vit_E_(mg) Vit_D_µg
## 2.469441e+06 1.226786e+06 1.050709e+01 4.662972e+00
## Vit_D_IU Vit_K_(µg) FA_Sat_(g) FA_Mono_(g)
## 7.462127e+03 5.920390e+03 3.401565e+01 4.299196e+01
## FA_Poly_(g) Cholestrl_(mg)
## 1.995847e+01 1.160956e+04Not as spread compared to the mean of the variables but there is still some difference among the variance.
We will need to scale the variables to mean zero and standard deviation one.
If we do not scale the variables before performing PCA, then most of the principal components that we observed would be driven by the variable with the largest mean and variance. Thus, it is important to standardize the variables to have mean zero and standard deviation one before performing PCA.
What is the new structure of our data?
str(SRp)## 'data.frame': 2223 obs. of 46 variables:
## $ Water_(g) : num 15.87 16.72 0.24 42.41 41.11 ...
## $ Energ_Kcal : int 717 718 876 353 371 334 300 404 394 98 ...
## $ Protein_(g) : num 0.85 0.49 0.28 21.4 23.24 ...
## $ Lipid_Tot_(g) : num 81.1 78.3 99.5 28.7 29.7 ...
## $ Ash_(g) : num 2.11 1.62 0 5.11 3.18 2.7 3.68 3.71 3.36 1.41 ...
## $ Carbohydrt_(g) : num 0.06 2.87 0 2.34 2.79 0.45 0.46 3.09 2.57 3.38 ...
## $ Fiber_TD_(g) : num 0 0 0 0 0 0 0 0 0 0 ...
## $ Sugar_Tot_(g) : num 0.06 0.06 0 0.5 0.51 0.45 0.46 0.48 0.52 2.67 ...
## $ Calcium_(mg) : int 24 23 4 528 674 184 388 710 685 83 ...
## $ Iron_(mg) : num 0.02 0.05 0 0.31 0.43 0.5 0.33 0.14 0.76 0.07 ...
## $ Magnesium_(mg) : int 2 1 0 23 24 20 20 27 26 8 ...
## $ Phosphorus_(mg) : int 24 24 3 387 451 188 347 455 457 159 ...
## $ Potassium_(mg) : int 24 41 5 256 136 152 187 76 127 104 ...
## $ Sodium_(mg) : int 643 583 2 1146 560 629 842 653 604 364 ...
## $ Zinc_(mg) : num 0.09 0.05 0.01 2.66 2.6 2.38 2.38 3.64 3.07 0.4 ...
## $ Copper_(mg) : num 0 0.01 0.001 0.04 0.024 0.019 0.021 0.03 0.042 0.029 ...
## $ Manganese_(mg) : num 0 0.001 0 0.009 0.012 0.034 0.038 0.027 0.012 0.002 ...
## $ Selenium_(µg) : num 1 0 0 14.5 14.5 14.5 14.5 28.5 14.5 9.7 ...
## $ Vit_C_(mg) : num 0 0 0 0 0 0 0 0 0 0 ...
## $ Thiamin_(mg) : num 0.005 0.007 0.001 0.029 0.014 0.07 0.028 0.029 0.015 0.027 ...
## $ Riboflavin_(mg) : num 0.034 0.064 0.005 0.382 0.351 0.52 0.488 0.428 0.375 0.163 ...
## $ Niacin_(mg) : num 0.042 0.022 0.003 1.016 0.118 ...
## $ Panto_Acid_(mg) : num 0.11 0.097 0.01 1.729 0.288 ...
## $ Vit_B6_(mg) : num 0.003 0.008 0.001 0.166 0.065 0.235 0.227 0.066 0.079 0.046 ...
## $ Folate_Tot_(µg) : int 3 4 0 36 20 65 62 27 18 12 ...
## $ Folic_Acid_(µg) : int 0 0 0 0 0 0 0 0 0 0 ...
## $ Food_Folate_(µg): int 3 4 0 36 20 65 62 27 18 12 ...
## $ Folate_DFE_(µg) : int 3 4 0 36 20 65 62 27 18 12 ...
## $ Choline_Tot_(mg): num 18.8 18.8 22.3 15.4 15.4 15.4 15.4 16.5 15.4 18.4 ...
## $ Vit_B12_(µg) : num 0.17 0.07 0.01 1.22 1.26 1.65 1.3 1.1 0.83 0.43 ...
## $ Vit_A_IU : int 2499 2468 3069 721 1080 592 820 1242 994 140 ...
## $ Vit_A_RAE : int 684 683 840 198 292 174 241 330 264 37 ...
## $ Retinol_(µg) : int 671 671 824 192 286 173 240 330 257 36 ...
## $ Alpha_Carot_(µg): int 0 1 0 0 0 0 0 0 0 0 ...
## $ Beta_Carot_(µg) : int 158 135 193 74 76 9 12 85 82 12 ...
## $ Beta_Crypt_(µg) : int 0 6 0 0 0 0 0 0 0 0 ...
## $ Lycopene_(µg) : int 0 0 0 0 0 0 0 0 0 0 ...
## $ Lut+Zea_(µg) : int 0 13 0 0 0 0 0 0 0 0 ...
## $ Vit_E_(mg) : num 2.32 1.37 2.8 0.25 0.26 0.24 0.21 0.71 0.28 0.08 ...
## $ Vit_D_µg : num 0 0 0 0.5 0.5 0.5 0.4 0.6 0.6 0.1 ...
## $ Vit_D_IU : int 0 0 0 21 22 20 18 24 24 3 ...
## $ Vit_K_(µg) : num 7 4.6 8.6 2.4 2.5 2.3 2 2.4 2.7 0 ...
## $ FA_Sat_(g) : num 51.4 45.4 61.9 18.7 18.8 ...
## $ FA_Mono_(g) : num 21.02 19.87 28.73 7.78 8.6 ...
## $ FA_Poly_(g) : num 3.043 3.331 3.694 0.8 0.784 ...
## $ Cholestrl_(mg) : int 215 225 256 75 94 100 72 99 95 17 ...We have 2223 observations with 46 variables.
We now perform principal components analysis using the prcomp() function, which is one of several functions in R that perform PCA.
pca = prcomp(SRp, scale = TRUE)By default, the prcomp function already sets the mean to zero for the variables, to set the standard deviation to one we let scale be true.
Loadings for each variable(Warning: long output)
pca$rotation## PC1 PC2 PC3 PC4
## Water_(g) 0.25460314 -0.030649880 0.275735871 -0.0244049168
## Energ_Kcal -0.23217104 0.123951819 -0.304485403 -0.0885494474
## Protein_(g) -0.13832714 0.263225869 0.188174696 -0.0688227059
## Lipid_Tot_(g) -0.16063707 0.185105935 -0.278239852 -0.2414205566
## Ash_(g) -0.10962435 -0.025576868 -0.047623246 0.0002234890
## Carbohydrt_(g) -0.12539693 -0.185198377 -0.236180806 0.1988645570
## Fiber_TD_(g) -0.18792946 -0.216355955 -0.107442527 0.0292432574
## Sugar_Tot_(g) -0.02067393 -0.115511278 -0.207879778 0.1159356556
## Calcium_(mg) -0.11297690 -0.033148530 -0.033195333 -0.0155171005
## Iron_(mg) -0.22064970 -0.084132847 0.049783200 0.0421019451
## Magnesium_(mg) -0.24623296 -0.049370783 -0.092055908 -0.0114746967
## Phosphorus_(mg) -0.15250627 0.068673518 -0.001799448 -0.0249868167
## Potassium_(mg) -0.13673839 -0.050692745 0.013868072 -0.0075262814
## Sodium_(mg) -0.03433893 -0.005676561 -0.048013167 0.0069608166
## Zinc_(mg) -0.10883508 0.154934364 0.117896372 -0.0534759842
## Copper_(mg) -0.19511597 -0.016186944 -0.053962392 -0.0056339073
## Manganese_(mg) -0.13726796 -0.061552421 0.016416080 0.0224209614
## Selenium_(µg) -0.07660596 0.120797800 0.041985399 -0.0519902736
## Vit_C_(mg) -0.03601389 -0.103087627 0.047192615 0.0280217798
## Thiamin_(mg) -0.21526854 -0.033963895 0.098325139 0.1969802063
## Riboflavin_(mg) -0.23710138 0.020026061 0.160042492 0.0437181249
## Niacin_(mg) -0.19045488 0.120644279 0.212208264 0.0617048684
## Panto_Acid_(mg) -0.21396980 0.062012980 0.167206020 0.0222886178
## Vit_B6_(mg) -0.18119480 0.058106066 0.234629534 0.0002396016
## Folate_Tot_(µg) -0.24681845 -0.137512630 0.089586301 0.2337041384
## Folic_Acid_(µg) -0.15198931 -0.080650638 0.060582038 0.1908837991
## Food_Folate_(µg) -0.19771823 -0.112855664 0.068209754 0.1560383817
## Folate_DFE_(µg) -0.24280039 -0.134038070 0.089819786 0.2442552465
## Choline_Tot_(mg) -0.10566587 0.206805861 0.173680905 -0.1362816065
## Vit_B12_(µg) -0.07176037 0.196802588 0.234479124 -0.1080010479
## Vit_A_IU -0.03631815 -0.322392453 0.110969469 -0.3368634324
## Vit_A_RAE -0.08498682 -0.191037744 0.132309805 -0.3034870112
## Retinol_(µg) -0.09205135 0.003266532 0.095721606 -0.1478413038
## Alpha_Carot_(µg) 0.01449421 -0.177530371 0.054075487 -0.1738005689
## Beta_Carot_(µg) -0.01530264 -0.328913585 0.090635342 -0.3058131949
## Beta_Crypt_(µg) -0.04679916 -0.202713609 0.050628744 -0.2235796300
## Lycopene_(µg) 0.01163187 -0.038522773 -0.004412801 0.0216547697
## Lut+Zea_(µg) -0.01611010 -0.248993851 0.073083451 -0.1947269929
## Vit_E_(mg) -0.18235994 -0.057388829 -0.111262091 -0.1617291642
## Vit_D_µg -0.05017610 0.097758135 0.162345669 -0.0632476186
## Vit_D_IU -0.05018942 0.097730767 0.162415829 -0.0632852690
## Vit_K_(µg) -0.04979110 -0.198271839 0.026651249 -0.1069872891
## FA_Sat_(g) -0.08590427 0.168427030 -0.201919177 -0.1963835342
## FA_Mono_(g) -0.14316517 0.177481192 -0.237823128 -0.2148891831
## FA_Poly_(g) -0.15657290 0.068684400 -0.233389253 -0.1488748831
## Cholestrl_(mg) -0.05175551 0.204613549 0.146429411 -0.1502284746
## PC5 PC6 PC7 PC8
## Water_(g) 0.0158023856 -0.120826592 0.019556963 -0.188041437
## Energ_Kcal 0.1153632829 0.084026633 -0.033839902 0.094826029
## Protein_(g) -0.0599307489 -0.166649221 -0.077501332 0.099163557
## Lipid_Tot_(g) 0.1490077278 0.003200098 -0.068918565 -0.133808821
## Ash_(g) -0.4797746490 0.150294202 -0.231443440 -0.126778140
## Carbohydrt_(g) 0.0037956474 0.186974324 0.093410743 0.277529828
## Fiber_TD_(g) -0.0645653415 -0.191893953 0.174540153 -0.037501846
## Sugar_Tot_(g) 0.0127277010 0.231098735 0.093329068 0.333074498
## Calcium_(mg) -0.4203592608 0.150206085 -0.064522928 -0.045263605
## Iron_(mg) -0.0862671324 -0.052696468 0.147985789 0.153930130
## Magnesium_(mg) -0.1279497614 -0.213266482 0.236869228 0.005979502
## Phosphorus_(mg) -0.3960676060 0.065427256 -0.084827002 -0.049705718
## Potassium_(mg) -0.2620469026 -0.105839281 0.089523557 0.049787146
## Sodium_(mg) -0.3074885157 0.176692400 -0.281711039 -0.141470036
## Zinc_(mg) -0.0433215869 -0.223237487 0.040332111 0.283692209
## Copper_(mg) -0.0811970985 -0.250650506 0.227721692 0.100023075
## Manganese_(mg) -0.0320443698 0.003108003 0.217548733 0.069224890
## Selenium_(µg) -0.0233959546 -0.075044785 0.016307486 0.010799916
## Vit_C_(mg) 0.0055538464 -0.041692899 0.033554447 -0.017684716
## Thiamin_(mg) 0.1382478058 -0.004238438 -0.164966338 -0.156911684
## Riboflavin_(mg) 0.0095021371 0.058798903 -0.082826103 0.113693276
## Niacin_(mg) 0.0769717076 -0.081927018 -0.064888754 0.069995554
## Panto_Acid_(mg) 0.0329036979 -0.013886704 -0.032339594 -0.054851392
## Vit_B6_(mg) 0.0435808550 -0.108120615 -0.049960399 0.124949363
## Folate_Tot_(µg) 0.1661587221 0.043521210 -0.130747035 -0.189180916
## Folic_Acid_(µg) 0.1592278322 0.224988654 -0.043810805 0.134768433
## Food_Folate_(µg) 0.0952917063 -0.096737870 -0.129079844 -0.318357611
## Folate_DFE_(µg) 0.1807939426 0.103367276 -0.117329342 -0.109277022
## Choline_Tot_(mg) -0.0243953027 0.023315617 -0.053253452 0.006045273
## Vit_B12_(µg) -0.0180520902 0.043491017 0.044323209 0.244383220
## Vit_A_IU 0.0470248632 0.038041692 -0.081781438 0.061358481
## Vit_A_RAE 0.0924278434 0.267118314 -0.065176768 0.106797528
## Retinol_(µg) 0.0935006194 0.355581905 -0.023556532 0.101818682
## Alpha_Carot_(µg) 0.0298560864 0.014030115 -0.086576462 0.065165407
## Beta_Carot_(µg) 0.0255779625 -0.046174330 -0.074560281 0.029274493
## Beta_Crypt_(µg) 0.0001772272 -0.050575164 -0.033558355 0.105065154
## Lycopene_(µg) -0.0374455782 -0.026343502 0.045732406 -0.005256978
## Lut+Zea_(µg) 0.0004372259 -0.092633419 -0.006729845 -0.113209759
## Vit_E_(mg) 0.0655053144 -0.085050497 0.098222784 -0.146429308
## Vit_D_µg -0.0169969413 0.321928761 0.464343942 -0.250409842
## Vit_D_IU -0.0171054741 0.321893637 0.464372200 -0.250261014
## Vit_K_(µg) -0.0565340864 -0.130384417 0.094080817 -0.148185282
## FA_Sat_(g) 0.1098178894 0.104458834 -0.129057274 0.005947157
## FA_Mono_(g) 0.1399168101 -0.035228529 -0.048256010 -0.137525373
## FA_Poly_(g) 0.0968297259 -0.078703050 0.035306458 -0.213172156
## Cholestrl_(mg) 0.0124605083 0.070142451 -0.115936118 0.017223496
## PC9 PC10 PC11 PC12
## Water_(g) 0.17236567 -0.039852757 0.067173005 -0.061890731
## Energ_Kcal -0.09933829 0.033389101 -0.085357687 -0.006384096
## Protein_(g) -0.08643103 0.107018461 -0.068776879 -0.050227434
## Lipid_Tot_(g) 0.06838099 0.025514755 -0.053005388 -0.056267693
## Ash_(g) 0.04223017 0.113591609 0.016049397 0.195523833
## Carbohydrt_(g) -0.21184993 -0.036362063 -0.021504053 0.083692379
## Fiber_TD_(g) -0.04350149 -0.068674241 0.113833827 0.078387284
## Sugar_Tot_(g) -0.30290957 -0.115466167 -0.025202001 -0.017465496
## Calcium_(mg) 0.09993492 -0.084479287 -0.082180083 -0.316075907
## Iron_(mg) 0.18460401 0.002675676 -0.227730320 0.135161876
## Magnesium_(mg) 0.04898012 -0.055344871 0.080183968 0.015328729
## Phosphorus_(mg) 0.01713432 -0.011109140 0.033302305 -0.318900682
## Potassium_(mg) -0.15287683 -0.024702344 0.049819277 -0.262766247
## Sodium_(mg) 0.06252956 0.180385392 0.022405981 0.493906939
## Zinc_(mg) 0.19909002 -0.062410680 0.039481713 0.202211756
## Copper_(mg) 0.12437795 -0.117001827 0.242680966 0.173763449
## Manganese_(mg) 0.26702081 -0.025807871 0.102160768 0.031166189
## Selenium_(µg) -0.04481206 0.051075879 0.056887015 0.129256608
## Vit_C_(mg) -0.09371019 -0.032572148 -0.121416548 -0.270493052
## Thiamin_(mg) -0.06747351 0.075171895 0.057991752 -0.057763367
## Riboflavin_(mg) -0.08967849 -0.018842614 -0.064039642 -0.128509660
## Niacin_(mg) -0.04034297 0.337016900 -0.116813184 -0.081731914
## Panto_Acid_(mg) -0.14292437 -0.094708289 0.189017873 -0.080516143
## Vit_B6_(mg) -0.06614108 0.284295010 -0.133327533 -0.150923515
## Folate_Tot_(µg) 0.03205372 -0.107424589 0.037768076 0.046536858
## Folic_Acid_(µg) 0.22577989 0.202681737 -0.199326425 0.120821058
## Food_Folate_(µg) -0.11114297 -0.264667629 0.178091587 -0.023912779
## Folate_DFE_(µg) 0.09447718 -0.023559953 -0.030870811 0.073936073
## Choline_Tot_(mg) -0.30646811 -0.310671990 0.007179353 0.210503728
## Vit_B12_(µg) 0.14983465 -0.072431775 -0.017243592 0.095379758
## Vit_A_IU -0.05995189 0.072308530 0.069628089 0.010251970
## Vit_A_RAE 0.20401111 -0.123807448 0.152391651 -0.090079520
## Retinol_(µg) 0.34926190 -0.243365734 0.161069002 -0.140067440
## Alpha_Carot_(µg) -0.07785001 0.162082861 0.322904551 0.024193442
## Beta_Carot_(µg) -0.13543230 0.111915322 0.003785514 0.041611068
## Beta_Crypt_(µg) -0.20319280 0.228353757 0.036579167 0.053501574
## Lycopene_(µg) -0.00814251 -0.032596544 0.131866313 -0.033742856
## Lut+Zea_(µg) -0.06114091 -0.135595446 -0.427435611 0.069540764
## Vit_E_(mg) 0.04164395 0.160700287 0.077706684 0.009363860
## Vit_D_µg -0.13017467 0.135604477 -0.020924112 0.055179310
## Vit_D_IU -0.13002261 0.135452978 -0.020767104 0.055357366
## Vit_K_(µg) 0.11472382 -0.240491408 -0.515465434 0.061942205
## FA_Sat_(g) 0.04401531 -0.081389742 -0.130289947 -0.117966908
## FA_Mono_(g) 0.07150091 0.070119438 -0.044387294 -0.038924257
## FA_Poly_(g) 0.05584598 0.085554496 0.082894463 0.037146487
## Cholestrl_(mg) -0.26914748 -0.351740500 -0.037430180 0.214225207
## PC13 PC14 PC15 PC16
## Water_(g) 0.027361032 0.028802314 0.006208485 -0.088613649
## Energ_Kcal -0.043787747 -0.038262059 0.025148445 0.055653210
## Protein_(g) -0.069534187 0.006895857 -0.016664066 0.190771439
## Lipid_Tot_(g) -0.009381487 -0.061332277 0.056295576 -0.027002179
## Ash_(g) 0.128891639 -0.092110010 -0.015412166 0.021100622
## Carbohydrt_(g) -0.021811517 0.017792681 -0.025991875 0.028746327
## Fiber_TD_(g) -0.025900200 0.081637453 -0.012390222 -0.035394691
## Sugar_Tot_(g) 0.062024094 -0.135866155 -0.079554620 0.032031805
## Calcium_(mg) -0.237735307 0.075704094 0.054667518 -0.163068425
## Iron_(mg) -0.071767369 0.131090855 0.151275014 -0.146312351
## Magnesium_(mg) 0.017854772 0.094384387 -0.065849854 0.120146640
## Phosphorus_(mg) -0.232939011 0.110591836 0.027640396 -0.110651179
## Potassium_(mg) 0.095172710 -0.028064142 0.117492287 0.137535489
## Sodium_(mg) 0.262318057 -0.167748973 -0.113699683 0.069443610
## Zinc_(mg) -0.174597550 -0.374115879 0.057442302 -0.149955619
## Copper_(mg) -0.027960688 -0.236784327 -0.023041212 -0.000343847
## Manganese_(mg) 0.177046619 0.351186433 -0.168630332 0.001097793
## Selenium_(µg) -0.101572117 0.217433915 -0.116539522 0.638078162
## Vit_C_(mg) 0.457109978 -0.310092442 -0.217249841 -0.181448214
## Thiamin_(mg) -0.080779370 -0.084521796 0.014152100 0.137109341
## Riboflavin_(mg) 0.074665249 -0.060610411 -0.038107144 0.067808606
## Niacin_(mg) 0.059112969 0.059050869 0.024129686 0.092706413
## Panto_Acid_(mg) 0.184216238 -0.002925710 -0.144827261 -0.070922418
## Vit_B6_(mg) 0.187062644 0.037754050 -0.062858801 -0.050332142
## Folate_Tot_(µg) -0.117393404 -0.058884226 0.058476916 -0.035851293
## Folic_Acid_(µg) -0.040510114 0.193553188 0.137566852 -0.147812946
## Food_Folate_(µg) -0.115160951 -0.199860437 -0.020530943 0.054785711
## Folate_DFE_(µg) -0.105747339 0.012251118 0.088530057 -0.073698731
## Choline_Tot_(mg) 0.070341930 0.248418270 0.035649008 -0.158943377
## Vit_B12_(µg) -0.037805115 -0.296760934 0.029167607 -0.081990999
## Vit_A_IU -0.088557201 -0.026446378 0.025195395 0.032831092
## Vit_A_RAE 0.075511900 0.001963557 -0.054276417 0.143498405
## Retinol_(µg) 0.187067497 0.026001522 -0.101025820 0.180158508
## Alpha_Carot_(µg) -0.259521903 0.024520173 0.096260516 -0.100116431
## Beta_Carot_(µg) -0.122354201 -0.039848228 0.044561969 0.008104839
## Beta_Crypt_(µg) 0.013931228 0.029851259 0.001632166 -0.130582918
## Lycopene_(µg) 0.386812787 -0.016597969 0.851282833 0.157887445
## Lut+Zea_(µg) 0.014992897 -0.084659562 -0.014117973 0.140460484
## Vit_E_(mg) 0.208988452 0.114937819 -0.058085625 -0.285432147
## Vit_D_µg -0.093061916 -0.131244182 0.034824206 0.011929956
## Vit_D_IU -0.093146239 -0.131369734 0.034878440 0.011790423
## Vit_K_(µg) 0.017034223 0.002464280 0.006798459 0.125601737
## FA_Sat_(g) -0.107597006 -0.177188614 0.110419355 0.061050382
## FA_Mono_(g) 0.015946368 -0.036380468 0.049840761 -0.068454638
## FA_Poly_(g) 0.089737922 0.094887572 -0.052106038 -0.063737664
## Cholestrl_(mg) 0.044180021 0.276273250 0.068454688 -0.219929796
## PC17 PC18 PC19 PC20
## Water_(g) -0.018421336 0.0707406056 0.020474959 3.082089e-02
## Energ_Kcal -0.009877687 -0.0648216959 -0.018597424 7.692856e-03
## Protein_(g) 0.068205697 -0.1710762132 -0.035763783 -1.835528e-02
## Lipid_Tot_(g) -0.048701738 -0.0433346589 0.023178192 6.382243e-02
## Ash_(g) -0.009516064 -0.0613404601 0.030049504 -7.028874e-03
## Carbohydrt_(g) 0.019851409 0.0327886745 -0.026713133 -6.462344e-02
## Fiber_TD_(g) 0.062587684 -0.1422753641 0.304389874 -4.121189e-02
## Sugar_Tot_(g) 0.037676307 0.1090898851 -0.249132483 9.858038e-03
## Calcium_(mg) -0.025776453 0.1786913985 -0.061772035 6.859202e-02
## Iron_(mg) -0.057632950 -0.1013515555 0.213254178 -2.237166e-01
## Magnesium_(mg) -0.049276155 -0.0911532442 0.010037110 -6.757503e-02
## Phosphorus_(mg) -0.003900714 0.2187619648 -0.078308547 9.203901e-02
## Potassium_(mg) 0.017771819 -0.2248870390 0.223993986 -1.577811e-01
## Sodium_(mg) -0.006627813 -0.1157855725 -0.025080519 1.208283e-03
## Zinc_(mg) -0.014829978 0.1297289510 -0.056513399 1.661495e-01
## Copper_(mg) -0.036059526 0.0940218658 -0.060616713 4.536897e-02
## Manganese_(mg) -0.040009288 -0.2941189289 -0.182325725 5.780262e-01
## Selenium_(µg) -0.324224406 0.4828713391 0.165870930 8.029713e-02
## Vit_C_(mg) -0.562553483 0.1096983193 0.274044150 1.028654e-01
## Thiamin_(mg) 0.022160676 -0.0293588302 -0.009604800 5.091345e-02
## Riboflavin_(mg) 0.048684698 0.0259139826 -0.158482390 -9.721392e-02
## Niacin_(mg) 0.077236198 -0.1484782705 -0.076586972 -8.759815e-02
## Panto_Acid_(mg) 0.091536408 0.0606047804 -0.374407786 1.659316e-01
## Vit_B6_(mg) 0.011857413 -0.0519799252 -0.020955636 -4.740665e-02
## Folate_Tot_(µg) -0.009102698 0.0351667684 0.069714282 3.598119e-02
## Folic_Acid_(µg) -0.182354237 0.1574296057 0.043328009 8.270567e-02
## Food_Folate_(µg) 0.110049328 -0.0619921938 0.055597030 -1.129484e-02
## Folate_DFE_(µg) -0.062943470 0.0760676682 0.068676025 5.395109e-02
## Choline_Tot_(mg) -0.079024007 -0.0236910499 0.035225330 -1.839716e-02
## Vit_B12_(µg) 0.041112106 0.1273264803 0.001968244 -1.244105e-01
## Vit_A_IU -0.035510470 -0.0141535239 -0.038740182 3.636816e-02
## Vit_A_RAE 0.070155387 -0.0198106051 0.061959003 -1.168460e-01
## Retinol_(µg) 0.133020071 -0.0164916082 0.123955355 -2.017934e-01
## Alpha_Carot_(µg) -0.512663598 -0.2534210731 -0.236857882 -1.736631e-01
## Beta_Carot_(µg) -0.049613267 0.0002274929 -0.076559457 9.199495e-02
## Beta_Crypt_(µg) 0.378730863 0.1875833705 0.379470779 2.495721e-01
## Lycopene_(µg) -0.013112521 0.1133018339 -0.116516019 1.037795e-01
## Lut+Zea_(µg) 0.096987379 0.1251181696 -0.157300880 1.679650e-01
## Vit_E_(mg) 0.129723141 0.2805225263 -0.103008046 -1.689613e-01
## Vit_D_µg -0.011446829 -0.0329153395 0.015281448 3.180696e-02
## Vit_D_IU -0.011534658 -0.0327811232 0.015357874 3.184933e-02
## Vit_K_(µg) -0.058574720 -0.0988729463 -0.247747317 -1.749407e-01
## FA_Sat_(g) -0.064303032 -0.2390481955 0.171621340 3.672485e-01
## FA_Mono_(g) -0.031000597 -0.0072890557 -0.001425355 -5.178278e-02
## FA_Poly_(g) -0.021680279 0.2095506870 -0.151448255 -2.103806e-01
## Cholestrl_(mg) -0.110737412 -0.0081167080 0.121278113 8.798673e-05
## PC21 PC22 PC23 PC24
## Water_(g) -0.041147753 0.124612117 0.0424817013 0.038401022
## Energ_Kcal 0.040624797 -0.073844957 -0.0234985060 -0.031380051
## Protein_(g) -0.039156404 -0.158539448 -0.0529366357 -0.181747261
## Lipid_Tot_(g) 0.006372160 0.041477091 0.0260705535 0.014764674
## Ash_(g) 0.063154623 0.036470194 -0.0153718343 0.023896854
## Carbohydrt_(g) 0.054426118 -0.107747848 -0.0400787268 0.027412333
## Fiber_TD_(g) -0.222245943 -0.125865043 -0.1035252658 0.095064837
## Sugar_Tot_(g) -0.051888381 -0.003922464 -0.1712683801 0.057544718
## Calcium_(mg) -0.210115828 -0.012930917 -0.0178374705 -0.019855119
## Iron_(mg) -0.297630606 0.109124074 0.0159856680 0.041474781
## Magnesium_(mg) -0.071514964 -0.107083564 0.1149221092 0.002285444
## Phosphorus_(mg) 0.085886032 -0.256223016 0.0009629337 0.047863762
## Potassium_(mg) 0.623669395 0.403579873 -0.0042470479 -0.020242428
## Sodium_(mg) -0.072577125 0.008713030 -0.0040800306 0.019223041
## Zinc_(mg) 0.142573837 -0.050260167 -0.0821387149 -0.059954574
## Copper_(mg) -0.001690185 -0.005000995 0.4926311636 0.086623486
## Manganese_(mg) 0.092556088 -0.020711484 -0.2750239820 -0.246730758
## Selenium_(µg) -0.070759574 0.168216745 -0.1725681599 0.105793024
## Vit_C_(mg) -0.049836924 -0.166101300 -0.0005721713 -0.102081168
## Thiamin_(mg) -0.112553630 -0.122011046 0.1031010141 -0.347473163
## Riboflavin_(mg) -0.302291603 0.412063286 0.2042648934 -0.477802656
## Niacin_(mg) 0.032143636 -0.248290355 0.0237929687 0.213080775
## Panto_Acid_(mg) -0.061019942 0.271154901 0.1553238340 0.393343829
## Vit_B6_(mg) -0.060706003 -0.133122993 -0.0763433182 0.311541263
## Folate_Tot_(µg) 0.104653731 0.009746240 -0.1138679681 0.054145312
## Folic_Acid_(µg) 0.235324307 0.114859323 0.2182060564 0.027650107
## Food_Folate_(µg) -0.029686093 -0.064378842 -0.2827653547 0.047131519
## Folate_DFE_(µg) 0.155079777 0.042905794 -0.0239721806 0.051534375
## Choline_Tot_(mg) 0.064648695 -0.071658943 0.0593938807 -0.035512125
## Vit_B12_(µg) 0.075536766 0.013316453 -0.4154283798 -0.042073208
## Vit_A_IU 0.037317698 -0.045688708 0.0349589736 -0.013941447
## Vit_A_RAE 0.015593070 -0.110787194 0.0582546391 0.022692353
## Retinol_(µg) -0.009900807 -0.121347971 0.0543813303 0.045197356
## Alpha_Carot_(µg) -0.069777037 0.077730169 -0.1155078059 0.014730576
## Beta_Carot_(µg) 0.057065757 -0.031815311 0.0348657401 -0.022991836
## Beta_Crypt_(µg) -0.101965124 0.072117379 0.0168380751 -0.082765950
## Lycopene_(µg) -0.118323343 -0.114629002 -0.0645405262 -0.018314347
## Lut+Zea_(µg) 0.194781045 -0.124319764 0.0854956387 -0.049646831
## Vit_E_(mg) -0.041874924 0.256051072 -0.3242966397 0.035340061
## Vit_D_µg -0.008368863 -0.015089219 0.0181429567 -0.008039319
## Vit_D_IU -0.008404713 -0.015342226 0.0181521343 -0.007845153
## Vit_K_(µg) -0.071317834 0.082907709 -0.1077920057 0.092071213
## FA_Sat_(g) -0.175052073 0.215241181 0.0498196822 0.339214504
## FA_Mono_(g) 0.038065977 0.072196880 -0.0895153045 -0.224179482
## FA_Poly_(g) 0.186170721 -0.229510440 0.1357096240 -0.072167752
## Cholestrl_(mg) 0.025290491 -0.056033588 0.0571044289 -0.022197227
## PC25 PC26 PC27 PC28
## Water_(g) -0.0722959427 -0.047356728 -0.1811259579 0.0988556394
## Energ_Kcal 0.0251053116 0.057434957 0.1180857089 -0.0737833387
## Protein_(g) 0.3586080420 0.144502409 -0.0400287272 -0.3533479070
## Lipid_Tot_(g) -0.0723794787 0.037146275 -0.0575521348 0.0704319895
## Ash_(g) -0.0421300993 -0.032158243 -0.0133009555 0.0097466155
## Carbohydrt_(g) -0.0313301718 -0.028449664 0.2652700620 0.0001423372
## Fiber_TD_(g) 0.1263340911 0.077197363 0.2883222431 0.2701125922
## Sugar_Tot_(g) -0.0798481490 -0.122597019 -0.4521353991 -0.0088998511
## Calcium_(mg) -0.0352777576 0.007265212 0.0088853758 -0.0093397867
## Iron_(mg) -0.2857985114 -0.047950069 0.0729240278 -0.1781200890
## Magnesium_(mg) 0.2867581107 0.070398273 -0.3157808861 0.2217416716
## Phosphorus_(mg) 0.0605882873 -0.053648595 -0.0030488569 0.0239328661
## Potassium_(mg) -0.1527556134 -0.034366078 -0.0112664561 -0.0014236876
## Sodium_(mg) 0.0432522619 0.009510301 -0.0546052987 0.0270699215
## Zinc_(mg) -0.1430796663 -0.144916574 -0.0371028184 -0.2246645899
## Copper_(mg) 0.0375286120 -0.177666103 -0.0821584199 0.0031199796
## Manganese_(mg) -0.1296783593 0.036985463 -0.0153598437 -0.0126518450
## Selenium_(µg) -0.0779772751 -0.049010393 0.0233104360 -0.0147103546
## Vit_C_(mg) 0.0518543622 0.039595882 0.0849790329 -0.1624839099
## Thiamin_(mg) -0.4857682663 -0.319794606 0.1214894820 0.1613781275
## Riboflavin_(mg) 0.1255954404 0.209670643 -0.0375602746 0.1193375056
## Niacin_(mg) 0.0026807817 -0.155304485 0.1375507250 -0.1874259883
## Panto_Acid_(mg) -0.0418079060 0.099459296 0.3503647937 -0.0575300573
## Vit_B6_(mg) -0.1553749061 -0.063866723 -0.3447391822 0.3983370433
## Folate_Tot_(µg) 0.1130006893 0.071347439 -0.1305665969 -0.0585808601
## Folic_Acid_(µg) 0.1842484500 0.059755539 -0.0346532354 0.0586228724
## Food_Folate_(µg) 0.0145345175 0.046381750 -0.1350598961 -0.1098107868
## Folate_DFE_(µg) 0.1461802724 0.074877754 -0.1144645735 -0.0286924372
## Choline_Tot_(mg) -0.0136039850 -0.053479116 -0.0692239153 -0.0351861590
## Vit_B12_(µg) -0.0207433011 0.297306900 0.1987159835 0.3984841453
## Vit_A_IU 0.0246282411 0.003188483 0.0392612054 0.0117942927
## Vit_A_RAE 0.0125271592 -0.066636264 -0.0003629327 -0.0574688282
## Retinol_(µg) -0.0032630284 -0.099725719 -0.0347402578 -0.0939479949
## Alpha_Carot_(µg) -0.0267733681 0.017658688 -0.0543366004 -0.0343235443
## Beta_Carot_(µg) 0.0386873822 0.016984592 0.0701303415 0.0531494989
## Beta_Crypt_(µg) -0.1268229618 0.156389529 -0.2000149163 -0.2008002235
## Lycopene_(µg) 0.0293489847 0.038198112 0.0055658820 -0.0153638912
## Lut+Zea_(µg) 0.1190071843 -0.101321292 0.1414111727 0.2086978955
## Vit_E_(mg) 0.1701401710 -0.292742861 0.0603690928 -0.1545459772
## Vit_D_µg 0.0006401993 -0.009981052 -0.0148394206 -0.0225152284
## Vit_D_IU 0.0003637848 -0.010829608 -0.0150512340 -0.0237247933
## Vit_K_(µg) -0.1495683318 0.042419569 -0.1007815107 -0.2087959491
## FA_Sat_(g) -0.0433860237 0.086354650 -0.0503114226 0.0336258421
## FA_Mono_(g) 0.1564966844 -0.395190763 -0.0242122008 0.1613004675
## FA_Poly_(g) -0.3650160691 0.524073113 -0.0829648753 -0.0473967337
## Cholestrl_(mg) -0.0462176540 -0.104080864 -0.0202126379 0.0823442241
## PC29 PC30 PC31 PC32
## Water_(g) 0.0793717447 0.0009116682 -0.022793211 -0.0436202011
## Energ_Kcal -0.0306043388 -0.0182306681 0.033782103 0.0524427227
## Protein_(g) -0.1024470253 0.0036687644 -0.094139049 0.1104117282
## Lipid_Tot_(g) 0.0519261480 -0.0295121131 0.052699155 0.0020991550
## Ash_(g) 0.0109884816 -0.0175078266 0.011270547 -0.0083240273
## Carbohydrt_(g) -0.0798070341 0.0160590145 0.038205627 -0.0006872018
## Fiber_TD_(g) -0.3407455827 -0.0626724746 0.165410856 -0.1873980066
## Sugar_Tot_(g) 0.1448561734 -0.0196498483 -0.058958608 -0.0267515560
## Calcium_(mg) -0.0257291160 -0.0921836312 0.084977383 -0.0412868036
## Iron_(mg) 0.3203812380 -0.0563930313 -0.068376790 0.4743789799
## Magnesium_(mg) 0.3262704486 0.1278683216 -0.112627847 -0.2278806966
## Phosphorus_(mg) 0.0144636761 0.0949487320 -0.055840930 0.0021947629
## Potassium_(mg) -0.0260847719 0.0061729283 -0.025765938 -0.0085587401
## Sodium_(mg) -0.0035877496 0.0031088782 -0.011020310 0.0017402928
## Zinc_(mg) -0.3018677803 -0.1173601474 0.141495843 -0.1009392214
## Copper_(mg) -0.0018600723 0.0535987913 -0.096221920 0.0660711734
## Manganese_(mg) 0.0264651716 -0.0239420604 -0.023578127 0.0714716242
## Selenium_(µg) -0.0071616520 -0.0432684751 0.046264942 0.0270531566
## Vit_C_(mg) 0.0723073771 0.0060305600 -0.002488762 -0.0216935807
## Thiamin_(mg) -0.0009771528 0.1488914841 -0.198503287 -0.2179549556
## Riboflavin_(mg) -0.1788731751 -0.0428787729 -0.008573513 0.0193746311
## Niacin_(mg) 0.2520044914 0.0866054005 -0.050462055 -0.2482749347
## Panto_Acid_(mg) 0.1919621292 -0.0334587982 0.252686682 -0.0125245411
## Vit_B6_(mg) -0.3381129811 -0.1445878099 0.114252470 0.2215483093
## Folate_Tot_(µg) 0.0296042983 -0.0332756659 0.018798915 0.0566595613
## Folic_Acid_(µg) -0.0521968318 0.0189164884 0.027205380 -0.1636762902
## Food_Folate_(µg) 0.0705650936 -0.0529106706 0.004722086 0.1772200088
## Folate_DFE_(µg) 0.0076826193 -0.0207761859 0.023260731 -0.0048260111
## Choline_Tot_(mg) -0.0566314822 -0.0318980365 0.097655766 0.0852442755
## Vit_B12_(µg) 0.2860408805 0.1332499650 -0.099594041 -0.1115199158
## Vit_A_IU 0.0730957946 -0.2927236847 -0.129451996 -0.0193780887
## Vit_A_RAE -0.0299169925 -0.0687122805 -0.006582915 -0.0057286546
## Retinol_(µg) -0.1073921271 0.1553991728 0.103330389 0.0086647311
## Alpha_Carot_(µg) -0.0824162065 0.4514435734 0.189545153 0.0413719859
## Beta_Carot_(µg) 0.1080498508 -0.4262444390 -0.205953755 -0.0173237076
## Beta_Crypt_(µg) 0.1231566823 0.3034454972 0.289217474 -0.1583705213
## Lycopene_(µg) -0.0200389921 -0.0021125850 0.028275151 0.0019542113
## Lut+Zea_(µg) -0.0437019818 0.4400698225 0.034646902 0.3843022305
## Vit_E_(mg) -0.2037808968 0.1474753269 -0.426087970 -0.0158366393
## Vit_D_µg -0.0481728541 -0.0238251092 0.016931981 0.0042290457
## Vit_D_IU -0.0489896460 -0.0241755762 0.017081446 0.0046015919
## Vit_K_(µg) -0.0984028937 -0.0177329480 0.152244287 -0.4362757397
## FA_Sat_(g) -0.1017561089 0.1417839212 -0.316907191 -0.0447197479
## FA_Mono_(g) 0.2522043712 -0.1751123423 0.468248036 0.0101396210
## FA_Poly_(g) -0.0650369042 -0.0188795428 -0.058074817 0.0566368899
## Cholestrl_(mg) 0.0067031491 0.0235662230 -0.184208095 -0.1911368908
## PC33 PC34 PC35 PC36
## Water_(g) -0.088509881 -0.083894784 0.121350782 0.0988253826
## Energ_Kcal 0.056242044 0.061545872 -0.100896393 -0.0858855885
## Protein_(g) 0.084550636 0.257054654 0.379467397 -0.2041151502
## Lipid_Tot_(g) -0.015088541 -0.044009033 0.052637743 0.0684308451
## Ash_(g) 0.020974669 -0.058559909 0.048742934 0.0278444124
## Carbohydrt_(g) 0.068952938 0.012011055 -0.340617759 -0.0586821027
## Fiber_TD_(g) -0.089442737 -0.092941444 0.387394861 0.1675135696
## Sugar_Tot_(g) -0.080884590 -0.072055700 0.451979704 0.1261633631
## Calcium_(mg) 0.000679246 -0.193406831 0.178357547 -0.1062444672
## Iron_(mg) -0.072191380 0.013802884 0.049966833 -0.0600915907
## Magnesium_(mg) -0.366610303 -0.036602036 -0.237129827 -0.3232168061
## Phosphorus_(mg) 0.054248525 0.212050681 -0.196177738 0.1841760850
## Potassium_(mg) -0.015455229 -0.027477225 0.085958346 -0.0253782806
## Sodium_(mg) -0.039560201 0.018979554 -0.010812262 -0.0187895185
## Zinc_(mg) -0.319824107 -0.251802996 -0.106402666 -0.1723599530
## Copper_(mg) 0.461045862 0.135109277 0.065316985 0.2470581499
## Manganese_(mg) 0.133969769 -0.041968749 -0.025011010 0.0971346725
## Selenium_(µg) 0.024680842 -0.057657831 -0.002277248 0.0326031358
## Vit_C_(mg) -0.003934352 0.022642231 -0.002904127 -0.0004927224
## Thiamin_(mg) -0.133973179 0.271678893 0.138361277 -0.1415182383
## Riboflavin_(mg) 0.018274892 -0.253084658 -0.193363598 0.2087953481
## Niacin_(mg) 0.006348039 -0.456547326 -0.021735042 0.3818870828
## Panto_Acid_(mg) -0.109954292 0.104541061 0.112544497 -0.2511118513
## Vit_B6_(mg) 0.142965131 0.098862056 -0.138320183 -0.1381865115
## Folate_Tot_(µg) 0.051442631 -0.027460840 -0.015982450 0.0202115970
## Folic_Acid_(µg) -0.060843923 0.116743414 0.177111016 -0.0190927288
## Food_Folate_(µg) 0.102554003 -0.110656818 -0.136829798 0.0369388959
## Folate_DFE_(µg) 0.022230634 0.013758611 0.041414843 0.0100204482
## Choline_Tot_(mg) -0.410083433 0.279471167 -0.081456565 0.4388212456
## Vit_B12_(µg) 0.124828778 0.164847397 0.044559321 0.1093567044
## Vit_A_IU -0.017247159 0.026304133 -0.012799739 -0.0005303386
## Vit_A_RAE -0.031187749 0.004727798 0.003700649 -0.0246913238
## Retinol_(µg) -0.030422612 -0.016277277 0.016643066 -0.0357264957
## Alpha_Carot_(µg) 0.039429248 -0.029347057 0.011048423 -0.0096963519
## Beta_Carot_(µg) -0.023043514 0.030954243 -0.014958424 0.0111694590
## Beta_Crypt_(µg) 0.112239639 0.088422538 -0.074537366 -0.0168449148
## Lycopene_(µg) 0.010522475 0.013118164 -0.023634398 -0.0261730080
## Lut+Zea_(µg) -0.099003125 -0.195906127 0.111969406 -0.0389574722
## Vit_E_(mg) -0.049742900 0.050983154 -0.026094306 -0.0045908754
## Vit_D_µg 0.004734483 -0.019632054 -0.014201172 -0.0294162771
## Vit_D_IU 0.003448209 -0.020423872 -0.014378643 -0.0308557653
## Vit_K_(µg) 0.175976328 0.236005633 -0.093780211 0.0517627073
## FA_Sat_(g) -0.102988614 0.005161271 -0.030802568 0.0988381010
## FA_Mono_(g) 0.094908418 0.010696796 0.058034563 0.0146025859
## FA_Poly_(g) -0.042011377 -0.146234427 0.132569163 0.0228600108
## Cholestrl_(mg) 0.387936664 -0.320298911 0.045825125 -0.3388158099
## PC37 PC38 PC39 PC40
## Water_(g) -0.087166129 -0.0358333483 1.397059e-01 6.468417e-02
## Energ_Kcal 0.059846855 0.0009920859 -6.340626e-01 2.829830e-01
## Protein_(g) 0.013743576 -0.0361712480 2.550920e-01 -5.992446e-02
## Lipid_Tot_(g) -0.029181465 -0.0080465795 -5.065693e-02 -8.253504e-01
## Ash_(g) 0.092782731 -0.7433551002 -1.112564e-02 1.037330e-02
## Carbohydrt_(g) 0.100645176 -0.0156570225 5.449289e-01 -1.240138e-01
## Fiber_TD_(g) -0.191624915 -0.0013198263 -4.211971e-02 1.483538e-02
## Sugar_Tot_(g) -0.159019661 -0.0171682097 -2.696323e-02 5.171371e-03
## Calcium_(mg) 0.554750662 0.2625381323 9.505016e-03 1.759757e-03
## Iron_(mg) -0.183464384 0.0006575262 1.605699e-03 -2.685370e-03
## Magnesium_(mg) 0.061927475 -0.0349417923 -3.996860e-03 1.706885e-05
## Phosphorus_(mg) -0.580067979 0.0598526125 -8.080612e-03 8.772446e-04
## Potassium_(mg) -0.007449825 0.1900995239 3.910642e-03 9.979479e-05
## Sodium_(mg) -0.121842737 0.5711873055 2.448749e-02 -5.679336e-04
## Zinc_(mg) -0.121478371 -0.0179300536 3.814612e-03 -4.255791e-03
## Copper_(mg) 0.177734120 0.0361691039 -1.352802e-02 6.859803e-03
## Manganese_(mg) 0.024109445 0.0098068394 1.525791e-03 2.334403e-03
## Selenium_(µg) 0.001105878 0.0070822376 -3.196597e-03 -2.786812e-03
## Vit_C_(mg) -0.006289094 0.0018945489 6.356357e-04 -1.326943e-05
## Thiamin_(mg) 0.038567664 0.0015399205 4.638714e-04 8.003433e-04
## Riboflavin_(mg) -0.151699503 -0.0362174020 -5.265855e-03 3.391033e-03
## Niacin_(mg) 0.074771173 0.0526598394 -2.772841e-03 1.107285e-02
## Panto_Acid_(mg) -0.100093953 -0.0251483884 3.254324e-04 -7.932855e-03
## Vit_B6_(mg) 0.083645068 -0.0046475014 -1.452616e-03 -1.739975e-03
## Folate_Tot_(µg) 0.019597412 0.0049694373 1.123253e-03 -3.891060e-04
## Folic_Acid_(µg) -0.014336268 -0.0121319651 -4.481112e-03 -1.811124e-03
## Food_Folate_(µg) 0.033027076 0.0140967449 4.206595e-03 7.013434e-05
## Folate_DFE_(µg) 0.011000525 0.0003050978 -5.977357e-04 -1.270149e-03
## Choline_Tot_(mg) 0.258505539 0.0184859678 3.275491e-03 8.714117e-03
## Vit_B12_(µg) 0.109269928 0.0098092198 1.429700e-03 3.904636e-03
## Vit_A_IU -0.003335435 0.0016424125 2.959317e-03 8.520154e-04
## Vit_A_RAE -0.012153418 -0.0039500462 1.502767e-03 -3.028698e-04
## Retinol_(µg) -0.014795983 -0.0069886482 -6.082496e-04 -7.826665e-04
## Alpha_Carot_(µg) 0.005950496 -0.0026471428 -3.475066e-03 -1.547120e-03
## Beta_Carot_(µg) -0.001912158 0.0030990872 4.393444e-03 2.471362e-03
## Beta_Crypt_(µg) 0.031008443 0.0074029975 -3.433818e-03 1.459487e-04
## Lycopene_(µg) -0.005110033 0.0127588689 4.505721e-04 5.286756e-04
## Lut+Zea_(µg) 0.013338729 0.0017314122 -3.547164e-03 -6.701708e-04
## Vit_E_(mg) 0.073637257 -0.0044567680 -1.155880e-03 -8.360449e-04
## Vit_D_µg -0.021466963 -0.0073697711 8.236550e-04 -1.400859e-03
## Vit_D_IU -0.021107165 -0.0058453039 2.124602e-03 7.579625e-04
## Vit_K_(µg) -0.038844771 -0.0075388048 -9.749889e-06 1.122532e-03
## FA_Sat_(g) -0.009448123 -0.0161175335 2.791647e-01 2.807126e-01
## FA_Mono_(g) -0.059908466 0.0043698229 2.926651e-01 2.950248e-01
## FA_Poly_(g) -0.024909845 0.0040279753 2.155936e-01 2.212853e-01
## Cholestrl_(mg) -0.148499348 0.0025819626 2.572089e-03 -4.449529e-04
## PC41 PC42 PC43 PC44
## Water_(g) -7.745814e-01 -9.699277e-04 -3.524119e-04 -6.458704e-04
## Energ_Kcal -4.714079e-01 -3.921920e-04 -2.614109e-04 -4.245799e-04
## Protein_(g) -1.480405e-01 -3.937682e-04 -7.037654e-05 -3.353801e-04
## Lipid_Tot_(g) -1.150911e-01 1.261177e-03 5.813583e-04 -7.782514e-04
## Ash_(g) -2.493854e-02 7.163873e-04 9.274405e-05 1.556101e-04
## Carbohydrt_(g) -3.568793e-01 -7.792409e-04 -2.730772e-04 -2.712571e-04
## Fiber_TD_(g) -2.534843e-02 2.132576e-04 4.144647e-05 -6.132843e-06
## Sugar_Tot_(g) -9.318598e-03 4.105216e-05 1.217832e-04 -3.683451e-05
## Calcium_(mg) -2.722189e-02 -7.237008e-04 -1.913924e-04 -7.262487e-06
## Iron_(mg) 1.336901e-03 -2.969678e-04 3.894966e-05 -7.090426e-05
## Magnesium_(mg) -1.250745e-03 -1.997935e-04 3.521309e-05 1.831093e-05
## Phosphorus_(mg) -1.200880e-02 3.532959e-04 1.153143e-04 -7.710118e-05
## Potassium_(mg) -2.197899e-02 -2.088687e-04 -3.102231e-05 -9.726836e-05
## Sodium_(mg) -7.723914e-02 -7.299998e-04 -1.199787e-04 -1.792839e-04
## Zinc_(mg) 6.994345e-03 -1.236684e-03 -1.350672e-04 -4.128135e-06
## Copper_(mg) -1.560111e-02 4.067509e-04 1.243772e-05 6.324060e-06
## Manganese_(mg) -2.152726e-03 1.804961e-04 3.654048e-05 -3.038462e-05
## Selenium_(µg) -5.516817e-04 -6.120392e-05 8.802852e-07 -7.901456e-06
## Vit_C_(mg) 7.813579e-05 -7.642109e-05 2.694772e-07 -1.669589e-05
## Thiamin_(mg) -1.618032e-04 -6.011257e-05 1.408565e-05 5.007039e-05
## Riboflavin_(mg) -7.353199e-04 4.295351e-04 6.114166e-05 -6.146424e-05
## Niacin_(mg) -6.627113e-03 8.805325e-05 -9.015555e-05 -1.285892e-04
## Panto_Acid_(mg) 5.673421e-03 -8.326293e-05 7.466745e-05 -1.700635e-05
## Vit_B6_(mg) 1.084770e-03 -6.260468e-05 -2.440647e-05 8.432161e-05
## Folate_Tot_(µg) -1.922773e-04 -1.607429e-01 7.927117e-01 2.988696e-03
## Folic_Acid_(µg) 9.429182e-04 2.958401e-02 -1.517611e-01 -3.917737e-04
## Food_Folate_(µg) -5.363091e-04 8.066287e-02 -4.042053e-01 -1.403525e-03
## Folate_DFE_(µg) 2.822753e-04 7.927977e-02 -3.816778e-01 -1.635723e-03
## Choline_Tot_(mg) -1.602472e-03 -7.075633e-05 -1.005698e-04 1.039399e-03
## Vit_B12_(µg) -2.180999e-03 9.467855e-04 1.421093e-04 -1.071556e-04
## Vit_A_IU 6.158621e-04 -1.193712e-02 -5.747796e-03 6.830073e-01
## Vit_A_RAE 1.439305e-03 -1.036761e-04 1.113731e-03 1.091212e-01
## Retinol_(µg) 1.290852e-03 2.354823e-03 3.206590e-04 -2.464471e-01
## Alpha_Carot_(µg) -3.724210e-04 9.930793e-04 3.945817e-04 -6.316159e-02
## Beta_Carot_(µg) 2.130547e-03 1.070840e-02 4.604957e-03 -6.743079e-01
## Beta_Crypt_(µg) -4.172804e-03 8.224213e-04 2.781002e-04 -4.320130e-02
## Lycopene_(µg) -1.109269e-03 -7.188599e-05 1.339671e-05 -2.144607e-05
## Lut+Zea_(µg) -1.187794e-03 -6.536927e-05 -6.087041e-06 -1.312597e-05
## Vit_E_(mg) -2.156968e-04 -2.842977e-04 -3.505751e-05 3.729263e-06
## Vit_D_µg 2.285977e-03 -6.929751e-01 -1.404538e-01 -1.210590e-02
## Vit_D_IU 6.209264e-04 6.928253e-01 1.404735e-01 1.213845e-02
## Vit_K_(µg) 4.585577e-04 2.393979e-04 1.549846e-05 1.264921e-05
## FA_Sat_(g) 4.960000e-02 -4.794724e-04 -2.508621e-04 3.656333e-04
## FA_Mono_(g) 4.909922e-02 -4.674991e-04 -1.474063e-04 3.826164e-04
## FA_Poly_(g) 3.598844e-02 -3.725550e-04 -3.617560e-04 2.136160e-04
## Cholestrl_(mg) 2.588582e-04 -1.143872e-04 2.883091e-05 -1.582452e-04
## PC45 PC46
## Water_(g) -2.655622e-05 -4.825874e-05
## Energ_Kcal -1.080079e-04 -1.273608e-04
## Protein_(g) 9.539227e-06 5.356825e-05
## Lipid_Tot_(g) 1.703748e-04 1.257370e-04
## Ash_(g) 3.745134e-05 4.854239e-05
## Carbohydrt_(g) 8.439780e-05 2.883045e-05
## Fiber_TD_(g) -3.391581e-05 3.750611e-06
## Sugar_Tot_(g) 2.365790e-05 2.267522e-05
## Calcium_(mg) -8.411550e-05 -5.456168e-05
## Iron_(mg) 3.001109e-05 -9.926471e-05
## Magnesium_(mg) -2.237052e-05 -2.305642e-05
## Phosphorus_(mg) 2.716133e-05 1.166685e-05
## Potassium_(mg) -5.487332e-06 1.503099e-06
## Sodium_(mg) -1.725164e-05 -4.297549e-05
## Zinc_(mg) -2.143926e-05 -5.423469e-05
## Copper_(mg) -7.210187e-07 6.286773e-05
## Manganese_(mg) 3.143399e-05 -1.410344e-05
## Selenium_(µg) -5.337063e-06 2.619346e-07
## Vit_C_(mg) 1.414778e-05 5.464298e-06
## Thiamin_(mg) -1.172043e-05 1.105011e-05
## Riboflavin_(mg) 1.030832e-05 -2.612334e-05
## Niacin_(mg) 5.128329e-05 1.042351e-05
## Panto_Acid_(mg) -1.039745e-05 -6.733073e-05
## Vit_B6_(mg) 3.379042e-05 -2.867595e-05
## Folate_Tot_(µg) 5.949628e-05 2.867534e-03
## Folic_Acid_(µg) 2.861150e-01 -4.430456e-01
## Food_Folate_(µg) 2.534715e-01 -3.935202e-01
## Folate_DFE_(µg) -3.856616e-01 5.949767e-01
## Choline_Tot_(mg) -7.747696e-05 1.568174e-05
## Vit_B12_(µg) 6.499915e-05 6.466321e-05
## Vit_A_IU 3.177742e-01 2.057591e-01
## Vit_A_RAE -6.323786e-01 -4.088654e-01
## Retinol_(µg) 4.485162e-01 2.899405e-01
## Alpha_Carot_(µg) 5.129183e-03 3.325947e-03
## Beta_Carot_(µg) 5.524289e-02 3.545293e-02
## Beta_Crypt_(µg) 3.574520e-03 2.316464e-03
## Lycopene_(µg) 2.129339e-05 4.863814e-05
## Lut+Zea_(µg) -2.217173e-05 -5.557383e-05
## Vit_E_(mg) -3.331782e-06 2.767053e-06
## Vit_D_µg -2.692475e-03 3.548610e-04
## Vit_D_IU 2.713812e-03 -3.862952e-04
## Vit_K_(µg) 2.072978e-05 6.647431e-05
## FA_Sat_(g) -7.474717e-05 -3.168058e-05
## FA_Mono_(g) -6.235300e-05 1.536868e-05
## FA_Poly_(g) -5.661687e-05 -9.407731e-05
## Cholestrl_(mg) -4.857557e-05 -5.822621e-05Some rows were omitted because there are so many variables and components.
Here is a plot of the components:
biplot(pca, scale = 0)
Seems like there are so many loadings and scores that we are not able to get a clear plot.
Let us see how much variation is described by each component:
pr.var=pca$sdev^2
pve=pr.var/sum(pr.var)
pve## [1] 1.585931e-01 9.422046e-02 9.050765e-02 7.542896e-02 5.717441e-02
## [6] 4.990966e-02 3.889550e-02 3.585142e-02 3.204046e-02 3.187225e-02
## [11] 2.846698e-02 2.492714e-02 2.412063e-02 2.259442e-02 2.146365e-02
## [16] 2.065716e-02 1.957926e-02 1.901520e-02 1.699657e-02 1.586561e-02
## [21] 1.441477e-02 1.240928e-02 1.144438e-02 1.010251e-02 9.391225e-03
## [26] 8.285616e-03 7.223179e-03 6.888471e-03 6.410823e-03 6.015062e-03
## [31] 5.657776e-03 4.906323e-03 4.444971e-03 4.035838e-03 3.589334e-03
## [36] 3.077027e-03 2.691105e-03 5.981964e-04 1.417999e-04 5.544217e-05
## [41] 3.420745e-05 1.106810e-06 8.623930e-07 1.218490e-07 3.829398e-08
## [46] 3.732215e-08We can also figure out the propotion of variance and cumulative variance using the summary function:
summary(pca)## Importance of components:
## PC1 PC2 PC3 PC4 PC5 PC6
## Standard deviation 2.7010 2.08186 2.04043 1.86272 1.62173 1.51520
## Proportion of Variance 0.1586 0.09422 0.09051 0.07543 0.05717 0.04991
## Cumulative Proportion 0.1586 0.25281 0.34332 0.41875 0.47592 0.52583
## PC7 PC8 PC9 PC10 PC11 PC12
## Standard deviation 1.3376 1.28420 1.21403 1.21084 1.14433 1.07082
## Proportion of Variance 0.0389 0.03585 0.03204 0.03187 0.02847 0.02493
## Cumulative Proportion 0.5647 0.60058 0.63262 0.66449 0.69296 0.71789
## PC13 PC14 PC15 PC16 PC17 PC18
## Standard deviation 1.05335 1.01948 0.99364 0.97480 0.94902 0.93525
## Proportion of Variance 0.02412 0.02259 0.02146 0.02066 0.01958 0.01902
## Cumulative Proportion 0.74201 0.76460 0.78607 0.80672 0.82630 0.84532
## PC19 PC20 PC21 PC22 PC23 PC24
## Standard deviation 0.8842 0.85429 0.81430 0.75553 0.72556 0.6817
## Proportion of Variance 0.0170 0.01587 0.01441 0.01241 0.01144 0.0101
## Cumulative Proportion 0.8623 0.87818 0.89260 0.90500 0.91645 0.9265
## PC25 PC26 PC27 PC28 PC29 PC30
## Standard deviation 0.65726 0.61736 0.57643 0.56291 0.54304 0.52602
## Proportion of Variance 0.00939 0.00829 0.00722 0.00689 0.00641 0.00602
## Cumulative Proportion 0.93594 0.94423 0.95145 0.95834 0.96475 0.97077
## PC31 PC32 PC33 PC34 PC35 PC36
## Standard deviation 0.51015 0.47507 0.45218 0.43087 0.40634 0.37622
## Proportion of Variance 0.00566 0.00491 0.00444 0.00404 0.00359 0.00308
## Cumulative Proportion 0.97642 0.98133 0.98577 0.98981 0.99340 0.99648
## PC37 PC38 PC39 PC40 PC41 PC42
## Standard deviation 0.35184 0.1659 0.08076 0.05050 0.03967 0.007135
## Proportion of Variance 0.00269 0.0006 0.00014 0.00006 0.00003 0.000000
## Cumulative Proportion 0.99917 0.9998 0.99991 0.99996 1.00000 1.000000
## PC43 PC44 PC45 PC46
## Standard deviation 0.006298 0.002367 0.001327 0.00131
## Proportion of Variance 0.000000 0.000000 0.000000 0.00000
## Cumulative Proportion 1.000000 1.000000 1.000000 1.00000We see that the proportion of variance expained by each component is very small and it looks like we will need a large number of components to explain a reasonable amount of variance.
Let us try to find the ‘elbow’ on the plot of the propotion of variance explained:
plot(pve, xlab="Principal Component", ylab="Proportion of Variance Explained", ylim=c(0,1),type='b')
plot(cumsum(pve), xlab="Principal Component", ylab="Cumulative Proportion of Variance Explained", ylim=c(0,1),type='b')
We see that there really is not much improvement when you add an additional component. There is not discernible ‘elbow’ but if you look very carefully at the plots, you can see that there is a slight bend near the sixth component. From the previous summary output, we see that the sixth component has a cumulative proportion of variance explained of 52.583%, which is not very great. This number of components does not adequately explain the variation in the data. This goes to show that the first few components are not able to capture any interpretable combinations of nutrients.
Conclusion
How many components would we need to explain a good amount of the variance? It depends on what we can consider a ‘good’ amount. For 75% of the variance explained we will need 14 components which explains 76.46% of it. For 90%+ we need at least 22 components which would explain 90.5% of the variance. Therefore, we see that PCA is able to represent the data in a lower dimension - although using a lot of components - compared to the original list of 46 variables. Lowering dimensions from 46 to 22 is a great change.