data <- read.csv("/Users/robertwu0428/Downloads/AQI.csv", # 資料檔名
header=T, # 資料中的第一列，作為欄位名稱
sep=",") # 將逗號視為分隔符號來讀取資料
head(data)
## aqi so2 co o3 o3_8hr pm10 pm2.5 no2 nox no windspeed winddirec co_8hr
## 1 26 0.7 0.13 26.7 25.0 14 6 1.0 1.6 0.6 2.8 175 0.1
## 2 27 0.6 0.13 24.4 24.8 15 9 1.3 1.9 0.5 2.7 175 0.1
## 3 27 0.7 0.13 30.6 25.2 13 10 1.0 1.6 0.5 3.3 195 0.1
## 4 29 0.9 0.13 31.9 26.4 16 12 0.7 1.2 0.5 3.5 193 0.1
## 5 30 0.9 0.13 31.9 27.4 16 9 0.6 1.0 0.4 3.8 193 0.1
## 6 27 0.6 0.13 28.3 24.7 20 9 0.7 1.2 0.4 3.6 185 0.1
## pm2.5_avg pm10_avg so2_avg
## 1 8.0 15 0
## 2 8.2 15 0
## 3 8.3 15 0
## 4 8.8 15 0
## 5 9.2 15 0
## 6 8.4 16 0
summary(data)
## aqi so2 co o3
## Min. : 26.00 Min. :0.000 Min. :0.0400 Min. : 3.60
## 1st Qu.: 60.00 1st Qu.:0.800 1st Qu.:0.2100 1st Qu.:22.30
## Median : 70.00 Median :1.400 Median :0.2800 Median :29.60
## Mean : 68.33 Mean :1.517 Mean :0.3123 Mean :31.31
## 3rd Qu.: 78.00 3rd Qu.:1.900 3rd Qu.:0.3600 3rd Qu.:38.40
## Max. :106.00 Max. :7.700 Max. :2.0500 Max. :81.40
## o3_8hr pm10 pm2.5 no2
## Min. : 7.50 Min. : 11.00 Min. : 4.00 Min. : 0.40
## 1st Qu.:18.20 1st Qu.: 31.00 1st Qu.:18.00 1st Qu.: 5.20
## Median :25.20 Median : 37.00 Median :24.00 Median : 9.00
## Mean :26.85 Mean : 37.99 Mean :23.45 Mean :10.26
## 3rd Qu.:33.20 3rd Qu.: 45.00 3rd Qu.:29.00 3rd Qu.:13.60
## Max. :61.30 Max. :162.00 Max. :49.00 Max. :40.50
## nox no windspeed winddirec
## Min. : 0.40 Min. : 0.000 Min. :0.000 Min. : 0.0
## 1st Qu.: 6.20 1st Qu.: 1.000 1st Qu.:1.000 1st Qu.:171.0
## Median :10.70 Median : 1.700 Median :1.600 Median :213.0
## Mean :13.17 Mean : 2.868 Mean :1.789 Mean :207.4
## 3rd Qu.:16.70 3rd Qu.: 2.900 3rd Qu.:2.400 3rd Qu.:263.0
## Max. :79.40 Max. :42.100 Max. :6.000 Max. :359.0
## co_8hr pm2.5_avg pm10_avg so2_avg
## Min. :0.0000 Min. : 8.00 Min. :14.00 Min. :0.000
## 1st Qu.:0.2000 1st Qu.:19.30 1st Qu.:30.00 1st Qu.:0.000
## Median :0.2000 Median :23.10 Median :36.00 Median :1.000
## Mean :0.2482 Mean :22.66 Mean :35.51 Mean :0.867
## 3rd Qu.:0.3000 3rd Qu.:26.40 3rd Qu.:41.00 3rd Qu.:1.000
## Max. :1.0000 Max. :37.30 Max. :57.00 Max. :3.000
pca <- prcomp(formula = ~ aqi+so2+co+o3+o3_8hr+pm10+pm2.5+no2+nox+no+windspeed+co_8hr+pm2.5_avg+pm10_avg+so2_avg, #選擇15個變數
data = data, # 資料
scale = TRUE) # 正規化資料
# 這就是我們的主成份
print(pca)
## Standard deviations (1, .., p=15):
## [1] 2.459951744 1.883156495 1.146092835 0.981515320 0.903275378 0.758430493
## [7] 0.724369659 0.589409883 0.529974710 0.486611701 0.385974498 0.327727167
## [13] 0.293724168 0.043419833 0.003817483
##
## Rotation (n x k) = (15 x 15):
## PC1 PC2 PC3 PC4 PC5
## aqi 0.28576417 -0.31333522 -0.2305332753 0.04878961 -0.090980118
## so2 0.16912470 -0.26023222 0.4825883383 0.12605827 0.062243783
## co 0.33055470 0.18855497 0.1312730296 -0.27859213 0.003190805
## o3 -0.12527274 -0.39731692 0.0573814880 -0.41766630 -0.037436066
## o3_8hr -0.17992304 -0.26921040 -0.0318141804 -0.53752336 -0.439292908
## pm10 0.25199967 -0.25495733 -0.0005477708 0.12855389 0.300149711
## pm2.5 0.29932309 -0.22361539 -0.2567626760 0.13328960 0.102182335
## no2 0.35506083 0.18349445 -0.0238764019 -0.02341463 -0.024615547
## nox 0.35009790 0.23857164 0.0455364075 -0.12936391 0.021221550
## no 0.27024539 0.28652322 0.1568206628 -0.28861191 0.096791361
## windspeed -0.14942935 -0.18927145 0.3604905640 -0.25472357 0.658628165
## co_8hr 0.30791258 0.09574847 0.1017593888 -0.34686910 -0.159722348
## pm2.5_avg 0.28453119 -0.31516475 -0.2350165634 0.04758370 -0.080175029
## pm10_avg 0.22808667 -0.34919129 0.0167514492 -0.03241606 0.044743088
## so2_avg 0.08669161 -0.11944948 0.6351692404 0.34256271 -0.463433099
## PC6 PC7 PC8 PC9 PC10
## aqi 0.23879382 -0.20286502 0.12861609 -0.093889100 0.14785110
## so2 -0.44537029 -0.33546072 -0.09045241 -0.534653978 -0.08176617
## co -0.23885664 -0.01878401 0.05221415 0.044392830 0.14598452
## o3 -0.27805414 -0.14726467 -0.29381106 0.323903644 0.20022287
## o3_8hr -0.08126282 0.10234096 0.38963199 -0.044181943 -0.37295182
## pm10 -0.35721822 0.57563288 -0.01167854 0.301960436 -0.10594078
## pm2.5 -0.11027783 -0.34643555 0.02283464 0.430800719 -0.10723739
## no2 0.03004875 -0.04287894 -0.02056473 -0.001065343 -0.61648740
## nox -0.00114229 -0.03472361 0.15675158 0.023719744 -0.22970543
## no -0.05487163 -0.01353701 0.43453676 0.063123245 0.49272966
## windspeed 0.45321277 -0.15227157 0.13002752 0.105410555 -0.19667051
## co_8hr 0.30976107 0.07329528 -0.68668962 -0.008292699 0.02891557
## pm2.5_avg 0.23492455 -0.21246828 0.11975191 -0.100417564 0.13345274
## pm10_avg 0.21434074 0.53195094 0.05304779 -0.363186446 0.11997283
## so2_avg 0.23976272 0.05026633 0.11972290 0.402501823 0.03024850
## PC11 PC12 PC13 PC14 PC15
## aqi -0.03400608 0.016341964 0.338514465 0.7035201394 2.542432e-03
## so2 0.08926116 0.172041809 -0.041732759 0.0100473841 4.745585e-04
## co 0.30827311 -0.734505561 0.205447498 -0.0130762260 -3.624936e-05
## o3 -0.55595364 -0.088432455 0.007183784 -0.0046434390 -3.966181e-04
## o3_8hr 0.28870145 0.135806498 -0.022244111 0.0030722309 1.370671e-04
## pm10 0.14268819 0.254303122 0.344016315 0.0012014658 -8.727906e-06
## pm2.5 0.29945751 -0.006811553 -0.588776004 0.0099805361 3.228659e-04
## no2 -0.39745647 -0.076695039 0.040990731 -0.0003664727 5.383130e-01
## nox -0.30773063 0.106624140 -0.030281260 0.0038976032 -7.843161e-01
## no -0.09007476 0.404965026 -0.147187712 -0.0021381396 3.083041e-01
## windspeed 0.09880162 -0.058189396 0.069045305 0.0041580452 2.546169e-05
## co_8hr 0.29292498 0.282205206 -0.023711636 0.0076805765 9.876053e-05
## pm2.5_avg -0.02440068 0.038527319 0.326677919 -0.7102217454 -2.976023e-03
## pm10_avg -0.18749222 -0.264634333 -0.489361571 -0.0048382393 2.011418e-04
## so2_avg -0.03573846 -0.062185046 0.011077906 -0.0109957765 -4.829834e-05
# 使用plot()函式
plot(pca, # 放pca
type="line", # 用直線連結每個點
main="AQI") # 主標題
# 用藍線標示出特徵值=1的地方
abline(h=1, col="blue") # Kaiser eigenvalue-greater-than-one rule
vars <- (pca$sdev)^2 # 從pca中取出標準差(pca$sdev)後再平方，計算variance(特徵值)
# 計算每個主成分的解釋比例 = 各個主成分的特徵值/總特徵值
props <- vars / sum(vars)
cumulative.props <- cumsum(props) # 累加前n個元素的值
summary(pca)
## Importance of components:
## PC1 PC2 PC3 PC4 PC5 PC6 PC7
## Standard deviation 2.4600 1.8832 1.14609 0.98152 0.90328 0.75843 0.72437
## Proportion of Variance 0.4034 0.2364 0.08757 0.06422 0.05439 0.03835 0.03498
## Cumulative Proportion 0.4034 0.6398 0.72741 0.79164 0.84603 0.88438 0.91936
## PC8 PC9 PC10 PC11 PC12 PC13 PC14
## Standard deviation 0.58941 0.52997 0.48661 0.38597 0.32773 0.29372 0.04342
## Proportion of Variance 0.02316 0.01872 0.01579 0.00993 0.00716 0.00575 0.00013
## Cumulative Proportion 0.94252 0.96124 0.97703 0.98696 0.99412 0.99987 1.00000
## PC15
## Standard deviation 0.003817
## Proportion of Variance 0.000000
## Cumulative Proportion 1.000000
cumulative.props[4]
## [1] 0.7916361
plot(cumulative.props)
# pca$rotation
top4_pca.data <- pca$x[, 1:4]
top4_pca.data
## PC1 PC2 PC3 PC4
## 1 -5.503202410 3.0714178220 0.5410075199 -0.2867388750
## 2 -5.293513394 3.0524578067 0.3219418705 -0.1238655136
## 3 -5.454315181 2.7187597040 0.5714511577 -0.4706064133
## 4 -5.305547210 2.3526064991 0.6164327810 -0.5105471395
## 5 -5.460706286 2.2986736375 0.7835495802 -0.6697589487
## 6 -5.378423157 2.5684652197 0.6516385263 -0.3968586447
## 7 -5.358075000 2.3911533951 0.6456184415 -0.5569413999
## 8 -5.384197525 2.2220742485 0.5567719614 -0.7494682649
## 9 -5.654796529 2.1534940577 0.9447932276 -1.0365872464
## 10 -5.569387976 2.0810376059 0.7233822297 -1.0580546612
## 11 -5.240317941 2.2993218588 0.1594496903 -0.5436556122
## 12 -5.208776263 2.6954820591 -0.0770558924 -0.2730262069
## 13 -3.852480618 1.9588436863 -0.4703357264 -0.2704339965
## 14 -3.972592512 1.9622638733 -1.2609832812 0.1929918755
## 15 -3.557712781 1.3450899525 -0.4331248467 -0.4868077370
## 16 -3.901861676 1.8716326058 -1.1474165878 0.3600861040
## 17 -2.823032054 2.2351381196 -1.0388393137 -0.8240707016
## 18 -3.832634733 1.8152765355 -1.4319188465 0.1984975174
## 19 -4.208523397 2.1227667046 -0.9239125905 0.2911932173
## 20 -4.023625081 1.9402334288 -1.2206605720 0.7877248578
## 21 -2.642738209 2.8081798559 -0.8905747519 -0.7284754622
## 22 -2.778350820 2.7001535689 -0.5837539390 -0.9535862450
## 23 -3.128298262 2.2063619071 -1.0052737875 -0.6455576162
## 24 -3.296445825 2.1453454360 -0.6692939205 -0.9133369586
## 25 -3.253083548 1.3609169262 -0.2591915783 -0.8981761404
## 26 -2.839514419 2.2852913758 -0.8239151047 -1.0268122128
## 27 -2.949127662 1.3154509191 -0.8063903295 -0.7009005406
## 28 -2.755598207 1.5348917470 -1.0274777332 -0.5002221151
## 29 -2.286686010 1.7292308536 -1.6818729058 0.2554602994
## 30 -2.155516848 1.9657841856 -0.9114052317 -0.7679102756
## 31 -2.614439564 1.8285099693 -1.2485441932 0.1742072037
## 32 -2.456595000 1.3475249942 -1.4426389852 0.2473312491
## 33 -3.298055991 1.6918690944 -1.0913196316 0.6154745544
## 34 -3.060776484 1.4966518178 -0.9088996698 -0.2498527975
## 35 -2.332612314 1.1337614017 -0.9004437666 -0.8494902226
## 36 -4.903175067 2.2069601377 -0.1312323165 -0.0086037593
## 37 -4.230313219 2.9277791881 -0.0279141016 -0.4858478309
## 38 -4.170166798 2.4202552047 0.1651651292 -0.6452283165
## 39 -4.373242944 1.8599955791 -0.1368416411 -0.4539759110
## 40 -5.041423683 1.8200031012 -0.1169643953 -0.5966578667
## 41 -2.898064475 1.1385534904 -0.3245238595 -0.9385575204
## 42 -3.609693290 1.5181354518 -0.5955082315 -0.6465345133
## 43 -3.235993980 0.8084138638 -0.4764445984 -0.6392772118
## 44 -2.276577270 0.1690103127 -0.9969507662 -0.3789331620
## 45 -5.360184828 2.0329486267 0.1139991746 0.0622273345
## 46 -4.849768536 2.1297510733 -0.1719798849 -0.0013758847
## 47 -3.465485146 2.3991967788 -0.4993651111 -0.2269783186
## 48 -4.062887399 1.6443157065 -0.1134541763 -0.3739879513
## 49 -3.931667948 2.6457254156 -0.0745382139 -0.6636888854
## 50 -3.608868361 2.3320313115 -0.1146823483 -0.5845241095
## 51 -3.696241036 1.3662633189 -0.7771092569 -0.1583295805
## 52 -2.927870058 0.8953434544 -1.0767009813 0.0012736132
## 53 -3.114757340 0.9431262803 -0.7834784824 -0.4461017142
## 54 1.822766286 3.0334188889 -2.0792082551 0.7690228675
## 55 -4.869173079 1.7478588611 0.3345675994 -0.2403782017
## 56 -3.827043424 2.5219075512 -0.3946279110 -0.0662699880
## 57 -3.097248742 2.7485167676 -0.4089269891 -0.3397955644
## 58 -4.897902969 1.5600278819 0.6935778499 -0.9377781606
## 59 -4.662097349 1.5849556278 -0.3975953209 -0.4059543255
## 60 -3.134139765 1.2796467788 -1.0979711098 0.7085583783
## 61 -2.842154810 1.9303630121 -0.6230266786 -0.3597345560
## 62 -2.775092706 1.2271234591 -0.9856953884 -0.0104396935
## 63 1.293674403 3.8129234561 -1.4362125794 -0.1103104490
## 64 -4.389803513 1.6133777733 -0.7084547065 -0.2043467877
## 65 -2.791508099 1.6439047950 -0.4852797339 -0.4703162885
## 66 -2.495916488 2.1286010279 -0.3758438418 -0.8314940504
## 67 -2.227835120 1.2880316451 -1.4004135613 0.4315034723
## 68 -2.957445551 -0.4261141160 -1.1848693058 -1.8468747230
## 69 -2.927291300 -0.4015720773 -1.3564317150 -1.5952047297
## 70 -1.799958585 0.0864462657 -1.9084900099 -0.1891133036
## 71 -4.565120066 1.4570851096 0.2545219364 -0.7267821711
## 72 -4.215565187 1.2592484615 -0.0056794789 -0.5547227169
## 73 -3.935770314 0.9110157099 -0.2824523544 -0.9512976289
## 74 -3.085122257 1.9542094734 -1.2361977960 0.4622494063
## 75 -3.468552614 0.8887236954 -0.7175157389 -0.1766621310
## 76 -3.913192816 0.6770303090 -0.3097277954 -0.8162938942
## 77 -2.133915764 1.5522233853 -1.0347171139 0.1559727220
## 78 -2.205651484 1.8240422163 -1.5525852308 0.7076859517
## 79 -2.930035668 -0.6799305100 -1.0070677952 -2.0870208665
## 80 -1.923508449 -0.1762779211 -1.5161150752 -0.5312624584
## 81 3.881042642 4.1159768032 -1.3725563944 -0.4612844256
## 82 -3.655752363 1.4457615834 -0.8778719110 -0.0273371669
## 83 -4.014429422 0.6683643777 -0.4508463785 -0.7958557946
## 84 -3.866096718 0.2162949858 -0.3664196311 -0.9374338822
## 85 -2.885630435 0.4673577343 -1.7893862452 -0.2843806411
## 86 -0.970307188 1.7508239644 -1.8584953075 1.6471692629
## 87 -0.150612398 1.4709079223 -1.5384699879 0.5444300762
## 88 -2.857861849 0.2203289317 -1.0657081578 -0.5745867996
## 89 -3.001175702 0.1224438296 -1.0779697490 -0.6968034447
## 90 -2.765974895 0.1846119855 -1.1449029097 -0.2906431307
## 91 -0.563260143 1.6538375282 -1.8798321417 1.1053238632
## 92 -2.703626563 -0.6326929815 -1.3595360886 -1.8372628508
## 93 1.310671770 2.8005167877 -1.0160842199 -0.1991724845
## 94 0.225147652 0.7655440643 -1.6721445749 0.2336436814
## 95 0.054569074 1.8503096194 -1.3992124985 -0.0912954495
## 96 -1.217430057 -0.8555533278 -1.8214330072 -0.5251242668
## 97 2.680304514 2.5428468353 -1.2882638852 -0.2866798994
## 98 0.181875603 0.4098090928 -1.6350790207 1.4730035701
## 99 -0.434700508 -0.4094894369 -2.2301765557 0.8115843162
## 100 -1.434692683 -1.6756985114 -1.6508384762 -0.2451018681
## 101 -1.178506377 -1.1800494216 -1.9047302958 0.2068359720
## 102 -2.816857788 0.1830290364 -0.7628467652 -0.1976632631
## 103 -1.753538418 -1.3214710616 -1.2722164320 -1.7103811517
## 104 -1.770774918 0.2049298749 -1.0088645833 -0.1066060695
## 105 -1.379720251 0.8908603585 -1.4161111234 0.4617160286
## 106 1.501217680 1.9772826217 -1.3063317205 -0.0930072982
## 107 0.693109898 1.9573680722 -1.0473606384 -0.2545614155
## 108 0.018180519 1.5363532098 -1.5653634828 0.2562185996
## 109 -0.696500409 1.0698857360 -1.2846482635 -0.5116936459
## 110 1.086203214 0.6713920607 -1.2059833164 0.0025675474
## 111 1.293234756 3.2979288952 -0.2209402228 -0.5747445139
## 112 -2.245785982 -1.0378993996 -1.4476253327 -1.7855359923
## 113 -1.411812206 -0.7878122022 -0.9926901016 -0.4272773997
## 114 1.550900211 1.8885158334 -0.9085182084 -0.0532846651
## 115 0.785555402 1.7647581614 -1.4559873890 0.9984366512
## 116 0.568651262 1.0883732867 -1.0802618803 -0.1879572006
## 117 1.625716760 2.6256082096 -1.8556088897 -0.1925225256
## 118 0.860633179 1.3640065000 -2.0612329109 0.4233765789
## 119 0.254604434 0.2008997454 -0.7891535662 0.1936368009
## 120 0.846047367 1.3324910505 -1.6333135824 -0.0020583660
## 121 -0.549716152 0.7617971086 -2.0498742941 0.5839831350
## 122 -0.159840534 0.1840166659 -1.1590542071 0.3464956805
## 123 0.238454228 0.1605406095 -2.8625714177 0.3378003595
## 124 -2.480437815 0.1280646354 -0.9828860231 -0.0311719278
## 125 3.453433843 3.8971499214 0.1640152885 -1.2349374044
## 126 0.453276777 0.1468133309 -1.8675932154 0.4151291382
## 127 3.246680048 3.3406853700 -0.6944394180 -1.3037193202
## 128 1.713374535 2.0919549984 -1.8059327516 0.0331012174
## 129 0.979172492 0.2528071526 -0.9379508266 -0.1629694651
## 130 0.757590960 0.3085349614 -2.2698478149 0.3645318316
## 131 0.267966943 0.5083987226 -1.9272549051 -0.2819176764
## 132 -0.388805148 0.6644166710 -1.8136580336 -0.3238398524
## 133 3.985052388 3.8234775758 -0.1056645782 -0.8949796306
## 134 3.245454756 2.5670402893 -1.0143021443 -0.3710888289
## 135 3.053709813 2.2470704694 -1.0687354080 -0.4811995124
## 136 0.486172833 0.4542352310 -2.3486595532 0.1442760081
## 137 0.150477319 0.1547263417 -1.8241236142 -0.3239290756
## 138 0.272613021 0.2605329150 -1.9321081843 0.0653752002
## 139 0.062390911 -0.8218925995 -2.1510274147 -0.3364233523
## 140 0.437858378 -0.1676094973 -2.8646806925 0.2011881922
## 141 -0.081285563 -0.6536028970 -2.5183703683 0.1047980509
## 142 -2.476215749 0.6542706749 -1.0721272835 -1.2372586774
## 143 -3.151288923 0.4701370887 -0.8955403731 -1.3581738672
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## 670 -0.230431837 -1.7831701724 -1.1485161977 -0.7630871429
## 671 2.767579154 0.9155327463 -1.0285767208 0.4108660449
## 672 2.509717550 0.2687082441 -1.1883288634 0.6568814812
## 673 2.476679825 -2.2424646472 0.5700717957 0.5837984849
## 674 3.587425632 -0.0104498036 -0.9064691318 0.7335270168
## 675 5.061631375 1.5969143398 -0.2838074779 -0.4050411159
## 676 1.945279362 -1.3235112382 -0.3663632819 1.0525338223
## 677 5.362715878 1.6525548315 -0.5308547341 0.1490977297
## 678 0.240944250 -3.2523907644 0.5167949636 -0.8709991391
## 679 -0.248856168 -3.6689249898 -1.4374343977 -1.0033348952
## 680 5.455943867 1.2587882340 -1.1023545034 0.5241132410
## 681 -2.007380529 -1.0342328332 0.9433140911 -1.0003673749
## 682 1.361051976 -3.2308696890 0.8258663242 -1.0775257985
## 683 0.638908316 -2.4922280870 1.1417185551 -1.4492808354
## 684 1.449925577 -0.7756498317 -0.0703137326 1.9128113278
## 685 0.149247091 -3.0747317379 0.7644177754 0.7392689560
## 686 0.303150244 -2.9897708570 -0.1195193415 -0.7735398047
## 687 0.115640835 -2.8764756309 0.4353699957 -1.3664396386
## 688 1.964611747 -2.1985443239 -0.4379835162 1.5152928923
## 689 5.540832972 1.5727108170 -0.0511189405 -0.6856897060
## 690 -0.057736978 -2.1051441966 -0.8486981402 -0.5304715375
## 691 1.599700295 -2.6637351406 -0.3273224497 0.7828918257
## 692 4.781809212 0.2597080370 -1.2704763991 1.0590246244
## 693 2.690432592 -2.8000579918 -0.6862252258 0.5669508074
## 694 -0.265444030 -3.7206293335 -1.2308802234 -1.2081615599
## 695 1.772639149 -4.0117240814 -0.0708577764 -0.7815411506
## 696 1.024839918 -2.4427485147 -1.1316056922 0.4664421972
## 697 0.115396254 -4.1986387347 -0.5647497775 -1.1864135369
## 698 -2.030565771 -0.7010257608 0.3753127485 0.6207623772
## 699 1.134305547 -1.7965782368 1.2508493628 -0.0257549393
## 700 0.015930830 -0.9561291366 -0.1416006715 -0.9089552667
## 701 -0.450077721 -1.5918405068 0.2382008369 -0.8407280225
## 702 -0.064565500 -1.8041685777 0.3120161578 -1.2620519595
## 703 -0.696254351 -1.4034541630 -0.7900215678 0.2763825045
## 704 -1.259921915 -1.1592563841 -0.7039054186 0.0948741902
## 705 0.522711826 -1.7400127053 0.0876360045 -0.5420744483
## 706 -0.415026793 -2.9715761859 0.6087427107 -0.6256387047
## 707 2.987472976 -0.6194342514 0.4327889177 0.1191015887
## 708 4.457559821 1.3366858019 -0.4633114100 -0.2872657843
## 709 5.599656980 1.9192702283 0.0478198177 -0.6843007672
## 710 1.067195295 -2.7880826370 -0.5448003312 -0.0104747473
## 711 0.339534057 -3.5003292720 -0.5967651143 -1.0688114846
## 712 0.691341310 -3.2986001095 0.7732832654 -0.5066427080
## 713 2.633013225 -2.9208768928 1.5516377260 0.0002152087
## 714 2.830664294 -0.3540042059 -1.3951218397 1.1345108875
## 715 1.264821943 -3.5738675205 -0.2395258192 -0.9621081860
## 716 3.938106560 -1.1461241805 -1.2354178929 0.7511027193
## 717 3.731361256 -0.5898880466 -1.1018031432 0.8335750978
## 718 3.895964375 -0.0993707256 -1.5560065746 0.9399168070
## 719 3.384540046 -1.2768293760 -1.3290974650 0.5665846564
## 720 1.305321452 -2.4720457297 0.3827931507 -0.3297559788
## 721 0.124225194 -2.8460546329 -0.7741034964 -0.2039938276
## 722 -0.505144890 -2.2354842532 -0.5087375869 0.0096687317
## 723 1.503791687 -1.6710761999 -0.6112370999 1.6689562664
## 724 -0.493734340 -2.5983219747 -0.4696443337 -0.0416599604
## 725 2.457856824 -2.4051240992 1.0134548138 -0.4199057381
## 726 0.753299394 -3.9298312955 0.0838700964 -0.8325138356
## 727 -0.239730309 -2.4597623691 -0.4798889105 -0.7628710415
## 728 0.108340041 -3.0176307796 -0.4779345113 -0.7915011974
## 729 1.997208269 -2.3908804468 -0.9754536425 1.1784099665
## 730 4.383356684 -0.2924518062 -1.0037908806 1.0853549590
## 731 2.394511136 -4.6255398729 0.0582242754 -1.3596386819
## 732 3.345216663 -0.8091817790 -1.5294824433 1.1260756325
## 733 -0.327552126 -2.6133149507 -0.4895987595 0.0290626565
## 734 3.457978936 -0.9160801009 -1.8403046115 1.3597860196
## 735 1.283865457 -2.5922840549 0.6865497696 -0.5183363837
## 736 0.605639437 -1.4883035224 -0.5197133143 0.0444185031
## 737 1.674992659 -2.7881993377 -0.0757823932 1.2509990276
## 738 0.515255174 -3.5694970951 -0.4300348769 0.1160608495
## 739 3.360046419 -3.2330934168 -1.0172297055 1.4628557099
## 740 1.872709920 -4.2680564590 0.5438918955 -1.5776410920
## 741 5.764269540 0.7384856233 -1.0023620126 0.1584676698
## 742 6.320066631 0.6095948466 -1.5620014597 0.4754234888
## 743 -1.161070628 -1.9344935285 0.4314182078 -1.1784448599
## 744 1.310614262 -2.6879875466 1.1185278915 -0.8526845243
## 745 0.253745221 -3.3986029731 -0.0154736419 -0.9100174037
## 746 2.873209372 -1.4850739320 -0.9921362591 -0.5204094172
## 747 -0.027356297 -2.8335561898 0.4254723140 -1.2543400458
## 748 1.123733317 -1.5529309518 -0.6480874903 -0.2048896070
## 749 1.533920529 -3.7782227310 -0.0757174286 0.5598759649
## 750 2.201647448 -3.4792324873 -0.3387362455 0.3562129670
## 751 3.221797752 -2.6647072198 -0.7367007994 -0.4597791435
## 752 1.519367560 -3.4582127692 -0.9461490690 1.5422674287
## 753 1.793878536 -4.3913869567 0.5143528015 -0.8033529068
## 754 -1.819081754 -2.8458545406 0.6751204378 -0.3408774627
## 755 -1.158969827 -3.3812199392 0.7718392465 -0.2344108183
## 756 0.923303645 -3.3601783746 0.3879668915 -0.9693453920
## 757 1.767754845 -2.6949689722 0.1865050760 -0.1889276301
## 758 2.962797025 -3.5805691123 -0.6890545466 0.8622201297
## 759 -0.666588669 -3.1912425411 1.2580255019 -0.1928398549
## 760 1.678559202 -2.3321382387 -0.3006971801 -0.2214299768
## 761 5.393905885 -2.4391122578 0.9427910621 1.6723469124
## 762 2.660779232 -3.4241570623 0.2343355430 0.0360434321
## 763 2.069051850 -4.9452035407 0.7642593764 1.2184353949
## 764 -2.215007737 2.8359550532 1.5948383202 1.5011527374
## 765 -1.973530575 1.3625859886 1.9197739900 0.8988884143
## 766 -1.632952837 1.4229499474 1.6705605140 1.1170170335
## 767 -1.829921705 1.0644624637 2.4297550108 0.4415737710
## 768 -1.650539547 1.2924596225 1.8066446772 0.9575207463
## 769 -1.409700042 1.0830392400 1.1523390810 1.4132545604
## 770 -1.772800264 1.1438906041 0.8276394941 1.7532552291
## 771 -1.230217678 1.1607462731 0.5977854401 1.6361252071
## 772 -2.221540029 1.3714238935 1.6367056090 0.9572755074
## 773 -1.681201944 0.9481855934 1.7441873555 0.5948363457
## 774 1.163135343 1.1099795035 0.4920686134 1.9540731029
## 775 2.854444058 3.1470848757 0.6359985251 0.5555745734
## 776 -2.281540411 1.4738385982 1.6032377504 0.8787114737
## 777 -0.755174969 0.7534087037 0.6797603451 1.6291809666
## 778 -0.976906099 0.3058410268 1.1076922026 1.0308280191
## 779 -1.936726243 -0.4927486934 1.6006093377 0.4094646129
## 780 -1.829149839 -0.2672314356 1.2872892360 0.6832044636
## 781 0.063921984 1.1165499565 0.8891177628 1.5539040996
## 782 2.802738519 2.0598504544 1.6826850018 0.6380944597
## 783 3.020502526 2.2729851422 0.5614826855 1.2186161461
## 784 -1.933143838 1.1627490475 1.4943551624 0.9559117609
## 785 -1.925149284 0.7519428481 1.9321299943 0.3897705951
## 786 -1.197326960 0.7210519626 1.8039322692 0.9407589082
## 787 -1.536981100 -0.5530348342 1.1994555411 0.4251695119
## 788 -1.463416373 -0.5207508569 1.0597959096 0.2855027468
## 789 -1.701819688 -0.4952371773 1.3173862197 0.3975229037
## 790 0.065100868 1.0261440058 1.6702971576 1.2596504635
## 791 0.163813063 0.7725845004 1.5417398378 1.3660296905
## 792 2.282342133 1.3393620522 3.1131950307 0.5806104707
## 793 0.164428821 2.4107595233 2.1873372324 -0.4783959095
## 794 0.056051007 2.5082057883 2.3930384956 -0.5596140573
## 795 -1.702380684 0.7744747888 1.3981183998 0.9020637178
## 796 -0.946295748 0.3563499252 1.9046754379 1.0250883638
## 797 -1.202585759 -0.3124756188 1.3595879724 0.4834486010
## 798 0.936683468 0.6816552508 3.0270582471 0.6596014952
## 799 0.243170443 0.1721058179 0.7994952573 1.8550305796
## 800 -0.351385520 2.2142388938 2.3444041680 -0.7422485827
## 801 -0.515854741 -0.0242891522 1.1280453896 1.0571947728
## 802 -1.251700485 -0.0883928573 1.1674807424 0.5475284353
## 803 -1.623816408 -0.6802252948 1.5846746924 0.3891365288
## 804 -0.137703064 0.6347370324 2.0245236708 0.4294999898
## 805 0.131663381 0.5865933686 2.1656176388 0.6740186660
## 806 -1.735839527 -0.3249484574 1.7287742340 0.1324896890
## 807 0.027969208 0.2943495606 0.8945460047 2.2653294892
## 808 -0.393179523 1.3752630840 1.9763198474 -0.4131550215
## 809 -0.691535293 0.0945364576 0.7962643270 1.0684972998
## 810 0.167871692 0.2759969717 0.9994702843 2.1165840703
## 811 0.870363426 -0.2623268312 2.5269251215 1.5398237723
## 812 1.167955033 0.1737363835 1.4263225061 1.2415534114
## 813 0.855594051 0.9929287958 1.6970665985 1.0127132375
## 814 0.399928519 -0.2819338471 1.7681148278 0.9330884419
## 815 -1.353868438 -0.5690373288 2.3489136617 -0.2448302756
## 816 0.183154290 0.6681980395 1.5756347829 0.6351160743
## 817 -0.479002514 0.7602678172 1.9939135077 -0.3788964100
## 818 1.502453804 0.1551046408 2.4973046800 0.4042540076
## 819 3.045584737 -0.0066719576 2.4541510699 0.5187763814
## 820 0.739002578 -1.1282758510 1.5760733287 0.7447605887
## 821 -0.060575985 -0.7623585478 2.2372372597 0.1084780260
## 822 1.119867030 -0.5137542484 1.9750707499 0.4228162798
## 823 2.548272495 -0.2187540711 2.4314852200 0.4998887749
## 824 1.087207651 -1.2024816598 1.2285049537 1.7763704482
## 825 3.390159406 -0.0115015197 4.2725866651 -0.0154308389
## 826 2.018279923 0.1637073614 3.9061972269 -0.2409957814
## 827 0.915090840 -1.7603431422 1.3117365879 0.6420359109
## 828 3.081199404 0.0008056045 -0.0098839880 1.3458222461
## 829 1.659336818 -1.0462339756 0.8657504415 2.9947378612
## 830 -1.419491706 0.0683525781 1.4898288991 0.4705529974
## 831 1.387402694 -2.5112609841 0.8011594515 0.8986158516
## 832 1.924005560 -1.6067072783 1.2518894930 2.4716240887
## 833 1.476856254 -1.1229004514 0.8805592925 0.0854616837
## 834 -1.493989879 -0.8717937865 2.0125104540 -0.1609090837
## 835 -1.432453752 -0.5640977183 1.9045590004 0.3499541190
## 836 -1.691987615 -0.7371884872 2.2604842646 -0.2812551566
## 837 2.507019635 -2.3667923477 1.0548468731 2.5424057180
## 838 -1.061270898 -1.6679286389 1.8482872004 -0.1790395404
## 839 0.153147085 -4.1389720763 2.1941357038 0.6284067748
## 840 2.603356264 -4.8924437904 0.9691732993 -0.1305440063
## 841 -0.301053328 -2.1758133291 2.6523972978 -0.5464539222
## 842 2.486287620 -2.0140999979 3.2676721069 0.1031007790
## 843 1.356173850 -2.2249275786 3.2356936105 -0.2692799900
## 844 2.075602588 -4.9394402461 0.8944976034 -0.4971826396
## 845 3.128260333 -3.9101967773 0.5017829389 2.5495439231
## 846 1.223457074 -4.9326609703 2.4030018064 0.8313321050
## 847 1.241551982 -4.8567066805 1.6314326035 0.4777296149
## 848 2.916909793 -5.3775348754 1.9973501138 2.1311463673
## 849 3.500369375 -6.3033336346 2.6687482847 1.7922710779
## 850 3.201540602 -6.0679592728 2.7594478374 0.5751660996
## 851 0.185736946 0.2629243674 2.2123305289 2.4680555121
## 852 2.223474885 -0.2202965934 4.3647269350 2.8789190654
## 853 0.655313735 -0.4908655231 2.6390615627 2.1597215987
## 854 1.794868719 -0.9607494537 4.4682554056 2.3337444044
## 855 0.686012668 -3.7231443953 4.2058444280 1.9447956382
## 856 0.320263051 -3.2569504105 2.9649674156 1.8259510083
## 857 2.423102502 -1.7563611936 3.7750698517 1.4397490327
top4.pca.eigenvector <- pca$rotation[, 1:4]
top4.pca.eigenvector
## PC1 PC2 PC3 PC4
## aqi 0.28576417 -0.31333522 -0.2305332753 0.04878961
## so2 0.16912470 -0.26023222 0.4825883383 0.12605827
## co 0.33055470 0.18855497 0.1312730296 -0.27859213
## o3 -0.12527274 -0.39731692 0.0573814880 -0.41766630
## o3_8hr -0.17992304 -0.26921040 -0.0318141804 -0.53752336
## pm10 0.25199967 -0.25495733 -0.0005477708 0.12855389
## pm2.5 0.29932309 -0.22361539 -0.2567626760 0.13328960
## no2 0.35506083 0.18349445 -0.0238764019 -0.02341463
## nox 0.35009790 0.23857164 0.0455364075 -0.12936391
## no 0.27024539 0.28652322 0.1568206628 -0.28861191
## windspeed -0.14942935 -0.18927145 0.3604905640 -0.25472357
## co_8hr 0.30791258 0.09574847 0.1017593888 -0.34686910
## pm2.5_avg 0.28453119 -0.31516475 -0.2350165634 0.04758370
## pm10_avg 0.22808667 -0.34919129 0.0167514492 -0.03241606
## so2_avg 0.08669161 -0.11944948 0.6351692404 0.34256271
first.pca <- top4.pca.eigenvector[, 1] # 第一主成份
second.pca <- top4.pca.eigenvector[, 2] # 第二主成份
third.pca <- top4.pca.eigenvector[, 3] # 第三主成份
fourth.pca<-top4.pca.eigenvector[, 4] # 第四主成份
# 第一主成份：由小到大排序原變數的係數
first.pca[order(first.pca, decreasing=FALSE)]
## o3_8hr windspeed o3 so2_avg so2 pm10_avg
## -0.17992304 -0.14942935 -0.12527274 0.08669161 0.16912470 0.22808667
## pm10 no pm2.5_avg aqi pm2.5 co_8hr
## 0.25199967 0.27024539 0.28453119 0.28576417 0.29932309 0.30791258
## co nox no2
## 0.33055470 0.35009790 0.35506083
# 使用dotchart，繪製主成份負荷圖
dotchart(first.pca[order(first.pca, decreasing=FALSE)] , # 排序後的係數
main="Loading Plot for PC1", # 主標題
xlab="Variable Loadings", # x軸的標題
col="red") # 顏色
# 第二主成份：由小到大排序原變數的係數
second.pca[order(second.pca, decreasing=FALSE)]
## o3 pm10_avg pm2.5_avg aqi o3_8hr so2
## -0.39731692 -0.34919129 -0.31516475 -0.31333522 -0.26921040 -0.26023222
## pm10 pm2.5 windspeed so2_avg co_8hr no2
## -0.25495733 -0.22361539 -0.18927145 -0.11944948 0.09574847 0.18349445
## co nox no
## 0.18855497 0.23857164 0.28652322
# 使用dotchart，繪製主成份負荷圖
dotchart(second.pca[order(second.pca, decreasing=FALSE)] , # 排序後的係數
main="Loading Plot for PC2", # 主標題
xlab="Variable Loadings", # x軸的標題
col="red")
# 第三主成份：由小到大排序原變數的係數
third.pca[order(third.pca, decreasing=FALSE)]
## pm2.5 pm2.5_avg aqi o3_8hr no2
## -0.2567626760 -0.2350165634 -0.2305332753 -0.0318141804 -0.0238764019
## pm10 pm10_avg nox o3 co_8hr
## -0.0005477708 0.0167514492 0.0455364075 0.0573814880 0.1017593888
## co no windspeed so2 so2_avg
## 0.1312730296 0.1568206628 0.3604905640 0.4825883383 0.6351692404
# 使用dotchart，繪製主成份負荷圖
dotchart(third.pca[order(third.pca, decreasing=FALSE)] , # 排序後的係數
main="Loading Plot for PC3", # 主標題
xlab="Variable Loadings", # x軸的標題
col="red")
# 第四主成份：由小到大排序原變數的係數
fourth.pca[order(fourth.pca, decreasing=FALSE)]
## o3_8hr o3 co_8hr no co windspeed
## -0.53752336 -0.41766630 -0.34686910 -0.28861191 -0.27859213 -0.25472357
## nox pm10_avg no2 pm2.5_avg aqi so2
## -0.12936391 -0.03241606 -0.02341463 0.04758370 0.04878961 0.12605827
## pm10 pm2.5 so2_avg
## 0.12855389 0.13328960 0.34256271
# 使用dotchart，繪製主成份負荷圖
dotchart(fourth.pca[order(fourth.pca, decreasing=FALSE)] , # 排序後的係數
main="Loading Plot for PC4", # 主標題
xlab="Variable Loadings", # x軸的標題
col="red")
biplot(pca, choices = c(1, 2))
biplot(pca, choices = c(1, 3))
biplot(pca, choices = c(1, 4))
biplot(pca, choices = c(2, 3))
biplot(pca, choices = c(2, 4))
biplot(pca, choices = c(3, 4))
#變數的相關係數矩陣
df=data
df.cor=cor(df)
df.cor
## aqi so2 co o3 o3_8hr
## aqi 1.00000000 0.41716779 0.2876116 0.15631819 -0.01342465
## so2 0.41716779 1.00000000 0.2569859 0.26702558 -0.03260585
## co 0.28761163 0.25698593 1.0000000 -0.36750727 -0.39634661
## o3 0.15631819 0.26702558 -0.3675073 1.00000000 0.66141565
## o3_8hr -0.01342465 -0.03260585 -0.3963466 0.66141565 1.00000000
## pm10 0.59024114 0.47632325 0.3356653 0.12953943 -0.14426981
## pm2.5 0.83143779 0.40319119 0.3939310 0.06314233 -0.19965222
## no2 0.40734939 0.18055579 0.7981513 -0.51437678 -0.51014520
## nox 0.32161459 0.15271968 0.8750721 -0.54412034 -0.52368670
## no 0.10648168 0.07201407 0.8315966 -0.48548875 -0.44089708
## windspeed -0.14208833 0.14734563 -0.3467670 0.40819158 0.23173280
## winddirec 0.14451938 0.11380896 -0.1025109 0.23519148 0.16145739
## co_8hr 0.39443102 0.18023708 0.7260969 -0.22766722 -0.28322669
## pm2.5_avg 0.99793188 0.41992750 0.2819030 0.15963583 -0.01282740
## pm10_avg 0.74766933 0.46102310 0.2039364 0.24162728 0.09373832
## so2_avg 0.16456499 0.48292608 0.0881739 0.01393583 -0.02061374
## pm10 pm2.5 no2 nox no
## aqi 0.59024114 0.83143779 0.40734939 0.32161459 0.10648168
## so2 0.47632325 0.40319119 0.18055579 0.15271968 0.07201407
## co 0.33566528 0.39393100 0.79815128 0.87507211 0.83159656
## o3 0.12953943 0.06314233 -0.51437678 -0.54412034 -0.48548875
## o3_8hr -0.14426981 -0.19965222 -0.51014520 -0.52368670 -0.44089708
## pm10 1.00000000 0.64620945 0.35363530 0.29967440 0.14409017
## pm2.5 0.64620945 1.00000000 0.50210172 0.41881162 0.18808130
## no2 0.35363530 0.50210172 1.00000000 0.95892491 0.69252912
## nox 0.29967440 0.41881162 0.95892491 1.00000000 0.86868232
## no 0.14409017 0.18808130 0.69252912 0.86868232 1.00000000
## windspeed -0.05046396 -0.20129507 -0.42913321 -0.39641595 -0.25899797
## winddirec 0.10678433 0.08230062 -0.06202293 -0.08963199 -0.11994446
## co_8hr 0.27390878 0.36352867 0.71636725 0.72919445 0.60353759
## pm2.5_avg 0.59070864 0.83539674 0.40498438 0.31779257 0.10092003
## pm10_avg 0.72733412 0.54378553 0.24765171 0.18455764 0.03643177
## so2_avg 0.16535537 0.06530585 0.09048669 0.07725804 0.03784066
## windspeed winddirec co_8hr pm2.5_avg pm10_avg so2_avg
## aqi -0.14208833 0.14451938 0.39443102 0.9979319 0.74766933 0.16456499
## so2 0.14734563 0.11380896 0.18023708 0.4199275 0.46102310 0.48292608
## co -0.34676701 -0.10251087 0.72609692 0.2819030 0.20393638 0.08817390
## o3 0.40819158 0.23519148 -0.22766722 0.1596358 0.24162728 0.01393583
## o3_8hr 0.23173280 0.16145739 -0.28322669 -0.0128274 0.09373832 -0.02061374
## pm10 -0.05046396 0.10678433 0.27390878 0.5907086 0.72733412 0.16535537
## pm2.5 -0.20129507 0.08230062 0.36352867 0.8353967 0.54378553 0.06530585
## no2 -0.42913321 -0.06202293 0.71636725 0.4049844 0.24765171 0.09048669
## nox -0.39641595 -0.08963199 0.72919445 0.3177926 0.18455764 0.07725804
## no -0.25899797 -0.11994446 0.60353759 0.1009200 0.03643177 0.03784066
## windspeed 1.00000000 0.19153305 -0.25053245 -0.1360094 0.06347450 0.04377843
## winddirec 0.19153305 1.00000000 0.06504472 0.1524620 0.19471646 0.01804516
## co_8hr -0.25053245 0.06504472 1.00000000 0.3919759 0.34612380 0.16368890
## pm2.5_avg -0.13600941 0.15246205 0.39197586 1.0000000 0.74511636 0.15429102
## pm10_avg 0.06347450 0.19471646 0.34612380 0.7451164 1.00000000 0.26182878
## so2_avg 0.04377843 0.01804516 0.16368890 0.1542910 0.26182878 1.00000000