Stock Market data
We load and look at the data.
stock = read.table('stocks.txt')
head(stock)## V1 V2 V3 V4 V5
## 1 0.0130338 -0.0078431 -0.0031889 -0.0447693 0.0052151
## 2 0.0084862 0.0166886 -0.0062100 0.0119560 0.0134890
## 3 -0.0179153 -0.0086393 0.0100360 0.0000000 -0.0061428
## 4 0.0215589 -0.0034858 0.0174353 -0.0285917 -0.0069534
## 5 0.0108225 0.0037167 -0.0101345 0.0291900 0.0409751
## 6 0.0101713 -0.0121978 -0.0083768 0.0137083 0.0029895We begin factor analysis now.
First we use PC method
First we do using covariance matrix then correlation matrix.
First we use Covariance matrix
Calculating the covariance matrix and its EV-EV pairs.
covmat <- cov(stock)
eig = eigen(cov(stock))To check number of factors required we plot the eigenvalues and check screeplot.
plot(eig$values, type = "l", main = "screeplot")
It looks like 2 factors is good. We check percentages of variance explained by 1, 2 or 3 factors.
100*eig$values[1]/sum(eig$values)## [1] 52.92607100*sum(eig$values[c(1,2)])/sum(eig$values)## [1] 80.05936100*sum(eig$values[1:3])/sum(eig$values)## [1] 89.88095So 2 factors explain 80 percent.
Next we estimate the loading matrix using the two first two principal components.That is the matrix of eigenvectors scaled by square roots of eigenvalues. This is the 5x2 L matrix.
ev <- as.matrix(eig$vectors[,1:2])
lambda <- matrix(0,2,2)
diag(lambda) <- eig$values[1:2]
L = ev%*%sqrt(lambda)
L## [,1] [,2]
## [1,] 0.008240463 0.016555621
## [2,] 0.011364238 0.015103596
## [3,] 0.005725215 0.009122290
## [4,] 0.023630398 -0.006565506
## [5,] 0.024071832 -0.008522342Calculating communalities
commu <- (L[,1])^2 + (L[,2])^2
commu## [1] 0.0003419938 0.0003572645 0.0001159943 0.0006015016 0.0006520834Calculating uniqeness
uniq <- diag(covmat - L%*%t(L))
uniq## V1 V2 V3 V4 V5
## 9.127563e-05 8.145269e-05 1.079779e-04 1.209948e-04 1.135907e-04Next we use Correlation matrix
Calculating the correlation matrix and its EV-EV pairs.
cormat <- cor(stock)
eig = eigen(cor(stock))To check number of factors required we plot the eigenvalues and check screeplot.
plot(eig$values, type = "l", main = "screeplot")
It looks like 2 factors is good as only two eigenvalues larger than 1. We check percentages of variance explained by 1, 2 or 3 factors.
100*eig$values[1]/sum(eig$values)## [1] 48.74546100*sum(eig$values[c(1,2)])/sum(eig$values)## [1] 76.88572100*sum(eig$values[1:3])/sum(eig$values)## [1] 86.89597So 2 factors are good explaining 76%.
Next we estimate the loading matrix using the two first two principal components.That is the matrix of eigenvectors scaled by square roots of eigenvalues. This is the 5x2 L matrix.
ev <- as.matrix(eig$vectors[,1:2])
lambda <- matrix(0,2,2)
diag(lambda) <- eig$values[1:2]
L = ev%*%sqrt(lambda)
L## [,1] [,2]
## [1,] 0.7323218 0.4365209
## [2,] 0.8311791 0.2804859
## [3,] 0.7262022 0.3738582
## [4,] 0.6047155 -0.6939569
## [5,] 0.5630885 -0.7186401Calculating communalities
commu <- (L[,1])^2 + (L[,2])^2
commu## [1] 0.7268458 0.7695311 0.6671396 0.8472571 0.8335122Calculating uniqeness
uniq <- diag(cormat - L%*%t(L))
uniq## V1 V2 V3 V4 V5
## 0.2731542 0.2304689 0.3328604 0.1527429 0.1664878Next we use MLE method
We know the number of factors should be 2. We use factanal function which uses MLE method.
We try without rotation and check
stock.fa = factanal(stock, factors = 2, rotation = "none")
stock.fa##
## Call:
## factanal(x = stock, factors = 2, rotation = "none")
##
## Uniquenesses:
## V1 V2 V3 V4 V5
## 0.417 0.275 0.542 0.005 0.530
##
## Loadings:
## Factor1 Factor2
## V1 0.121 0.754
## V2 0.328 0.786
## V3 0.188 0.650
## V4 0.997
## V5 0.685
##
## Factor1 Factor2
## SS loadings 1.622 1.610
## Proportion Var 0.324 0.322
## Cumulative Var 0.324 0.646
##
## Test of the hypothesis that 2 factors are sufficient.
## The chi square statistic is 1.97 on 1 degree of freedom.
## The p-value is 0.16We also calulate communalities
commu <- (stock.fa$loadings[,1])^2 + (stock.fa$loadings[,2])^2
commu## V1 V2 V3 V4 V5
## 0.5834625 0.7253097 0.4579766 0.9950016 0.4701569We use varimax rotation and check
stock.fa = factanal(stock, factors = 2, rotation = "varimax")
stock.fa##
## Call:
## factanal(x = stock, factors = 2, rotation = "varimax")
##
## Uniquenesses:
## V1 V2 V3 V4 V5
## 0.417 0.275 0.542 0.005 0.530
##
## Loadings:
## Factor1 Factor2
## V1 0.763
## V2 0.819 0.232
## V3 0.668 0.108
## V4 0.113 0.991
## V5 0.108 0.677
##
## Factor1 Factor2
## SS loadings 1.725 1.507
## Proportion Var 0.345 0.301
## Cumulative Var 0.345 0.646
##
## Test of the hypothesis that 2 factors are sufficient.
## The chi square statistic is 1.97 on 1 degree of freedom.
## The p-value is 0.16We also calulate communalities
commu <- (stock.fa$loadings[,1])^2 + (stock.fa$loadings[,2])^2
commu## V1 V2 V3 V4 V5
## 0.5834625 0.7253097 0.4579766 0.9950016 0.4701569We calculate factor scores with varimax using regression
stock.fa = factanal(stock, factors = 2, rotation = "varimax", scores = "regression")
stock.fa$scores## Factor1 Factor2
## [1,] 0.16535864 -1.83427398
## [2,] 0.36753184 0.25550919
## [3,] -0.39519052 -0.10792854
## [4,] 0.62520403 -1.28789064
## [5,] -0.06003873 0.94945235
## [6,] -0.37607023 0.39962913
## [7,] 0.80260447 0.89563925
## [8,] 0.84651321 -0.01307322
## [9,] -0.89995399 -1.16280345
## [10,] 0.42882250 -0.95869583
## [11,] -0.32403193 -0.45354773
## [12,] 0.80088906 1.11551771
## [13,] -0.38178474 0.87939688
## [14,] -1.86615206 1.46024763
## [15,] -0.67075407 -1.04526763
## [16,] -0.70907326 0.19865391
## [17,] -1.62926480 1.19103502
## [18,] -0.42963361 -0.33251470
## [19,] 0.35399035 -0.91592798
## [20,] 0.91065169 1.07104730
## [21,] 0.15260540 -0.08971020
## [22,] 0.55035453 0.34825285
## [23,] -0.35566282 1.04861138
## [24,] -0.34547704 -0.22366635
## [25,] -0.35625414 -1.02993312
## [26,] -1.08456840 0.50813920
## [27,] -1.02216946 0.05605792
## [28,] 0.58452519 -2.07235509
## [29,] 0.55155962 0.35776843
## [30,] -1.14028640 -0.72101512
## [31,] 0.96876650 0.59001848
## [32,] 1.89079695 0.23362634
## [33,] 0.98094278 -0.24594299
## [34,] 0.12142680 0.69167953
## [35,] 0.15131130 -0.31703165
## [36,] -0.24937354 0.30538742
## [37,] -1.54674969 -0.92045439
## [38,] 0.90323075 0.49718139
## [39,] -0.34393133 0.37378690
## [40,] -0.87428582 0.03282412
## [41,] -1.91802392 0.27061568
## [42,] 2.08905617 0.61332353
## [43,] 1.72427065 0.46594471
## [44,] 0.53006979 0.73408768
## [45,] -1.99919361 -0.05697661
## [46,] 0.29316625 1.01168393
## [47,] 0.23694777 -0.37115572
## [48,] 0.16476325 -0.81271418
## [49,] 0.48927213 -0.20147260
## [50,] 1.33794035 0.76711974
## [51,] -0.48625758 0.04541153
## [52,] 0.22989070 -1.15337178
## [53,] -1.27083380 0.41734040
## [54,] -0.40334370 0.17506283
## [55,] 0.24001096 0.72409279
## [56,] 1.20380390 1.62989890
## [57,] -0.44873512 -0.26335806
## [58,] -1.20365580 1.60631036
## [59,] -0.01849556 0.41790397
## [60,] 0.21862178 1.17314319
## [61,] -0.99268886 -1.06031002
## [62,] -0.61782578 -0.44939689
## [63,] -1.76575905 -1.85425018
## [64,] -0.10255476 0.44321362
## [65,] 0.60528001 0.50408016
## [66,] -0.04450378 -1.49284903
## [67,] 0.51822782 0.53460927
## [68,] 1.20489103 -0.77024224
## [69,] -0.14068264 0.13080833
## [70,] -0.81308704 -1.86579512
## [71,] 2.17608899 0.17047205
## [72,] -0.69224355 1.06506654
## [73,] 0.16610640 -0.40750394
## [74,] -0.08126035 0.10424234
## [75,] -0.08605287 2.17101513
## [76,] -0.75823328 -0.07046082
## [77,] -0.60955093 1.36886226
## [78,] 0.06991938 -0.33766851
## [79,] 0.93413784 -1.40506862
## [80,] -1.21297215 -1.94028602
## [81,] -0.70803942 0.54642946
## [82,] -0.26110609 1.52227979
## [83,] -0.82333192 2.24389360
## [84,] 0.70419227 -1.79976830
## [85,] -1.39547480 -0.77399584
## [86,] 0.47921505 1.91765860
## [87,] 0.97954714 -1.10773682
## [88,] 0.48614059 0.25414120
## [89,] -0.76251065 -0.42179171
## [90,] -0.12336256 0.36290942
## [91,] 0.25514110 -1.79263378
## [92,] 0.03541405 -0.85993239
## [93,] 0.36140969 -2.21335606
## [94,] 1.53278476 1.84296537
## [95,] 0.26525280 0.27950060
## [96,] 2.45286948 -1.60666185
## [97,] -0.05597037 1.51916992
## [98,] 1.28414412 0.02525961
## [99,] -0.71641574 -0.13816356
## [100,] 0.04719622 -0.20425006
## [101,] 0.82432694 -0.84767865
## [102,] 0.13434943 -0.26347004
## [103,] -0.85866218 -0.24362687We calculate factor scores with varimax using WLS
stock.fa = factanal(stock, factors = 2, rotation = "varimax", scores = "Bartlett")
stock.fa$scores## Factor1 Factor2
## [1,] 0.25089936 -1.853670704
## [2,] 0.44303382 0.247877831
## [3,] -0.48078772 -0.098356805
## [4,] 0.79921292 -1.314978954
## [5,] -0.09859658 0.958816349
## [6,] -0.47081640 0.412851650
## [7,] 0.95854964 0.881739462
## [8,] 1.03632812 -0.035576617
## [9,] -1.07061981 -1.148516016
## [10,] 0.55016716 -0.977892504
## [11,] -0.38455811 -0.448689098
## [12,] 0.95063293 1.103464002
## [13,] -0.49050325 0.896699647
## [14,] -2.32247360 1.521579416
## [15,] -0.79322937 -1.036081606
## [16,] -0.87303590 0.219040848
## [17,] -2.02544295 1.243894726
## [18,] -0.51699805 -0.323870948
## [19,] 0.45745434 -0.932794556
## [20,] 1.08613951 1.055725397
## [21,] 0.18913545 -0.094482385
## [22,] 0.66432269 0.336544054
## [23,] -0.46301158 1.066608820
## [24,] -0.41688513 -0.216357723
## [25,] -0.40874286 -1.028942294
## [26,] -1.34076354 0.540995211
## [27,] -1.25243750 0.083560710
## [28,] 0.77018393 -2.104791538
## [29,] 0.66554575 0.346105681
## [30,] -1.37643257 -0.696751143
## [31,] 1.16998840 0.569219630
## [32,] 2.30781949 0.185514511
## [33,] 1.20700718 -0.273910102
## [34,] 0.13030502 0.694131438
## [35,] 0.19356598 -0.323631250
## [36,] -0.31326873 0.314486112
## [37,] -1.86859506 -0.887069856
## [38,] 1.09224046 0.477356145
## [39,] -0.43080038 0.385947479
## [40,] -1.07083942 0.056224054
## [41,] -2.35448128 0.323577356
## [42,] 2.54040794 0.563075854
## [43,] 2.09787430 0.424141269
## [44,] 0.62928967 0.726075360
## [45,] -2.44515145 -0.004550238
## [46,] 0.33201728 1.012212775
## [47,] 0.29980182 -0.380464269
## [48,] 0.22314315 -0.823729074
## [49,] 0.60411282 -0.216067308
## [50,] 1.61710621 0.738003971
## [51,] -0.59629461 0.058648422
## [52,] 0.31186044 -1.168899577
## [53,] -1.56631713 0.454380859
## [54,] -0.49825292 0.187167836
## [55,] 0.27457347 0.723672748
## [56,] 1.43012026 1.611397733
## [57,] -0.54220460 -0.253642581
## [58,] -1.51555992 1.651310560
## [59,] -0.03369184 0.421815741
## [60,] 0.23651630 1.176966545
## [61,] -1.18682255 -1.042729714
## [62,] -0.74421957 -0.436731312
## [63,] -2.11191814 -1.822718609
## [64,] -0.13723503 0.449556626
## [65,] 0.72741900 0.492194343
## [66,] -0.01496820 -1.503897281
## [67,] 0.62007483 0.525276622
## [68,] 1.49495164 -0.808428134
## [69,] -0.17563145 0.135601649
## [70,] -0.94571088 -1.859563091
## [71,] 2.65863729 0.114295004
## [72,] -0.87536213 1.092103711
## [73,] 0.21406621 -0.415235898
## [74,] -0.10220623 0.107245952
## [75,] -0.16275236 2.191071384
## [76,] -0.92607905 -0.050977159
## [77,] -0.78219858 1.396199553
## [78,] 0.09450269 -0.342283715
## [79,] 1.18039339 -1.441290076
## [80,] -1.43312862 -1.924084212
## [81,] -0.88097181 0.569637161
## [82,] -0.35982281 1.541654635
## [83,] -1.06697949 2.284053007
## [84,] 0.90942337 -1.833138440
## [85,] -1.68733626 -0.743414181
## [86,] 0.53573854 1.920684612
## [87,] 1.22809978 -1.142724675
## [88,] 0.58822613 0.243360596
## [89,] -0.92201857 -0.405072109
## [90,] -0.16057547 0.369145289
## [91,] 0.35967551 -1.814064875
## [92,] 0.06609189 -0.867911766
## [93,] 0.50086056 -2.241044227
## [94,] 1.82709743 1.817505223
## [95,] 0.31722765 0.274771739
## [96,] 3.04438550 -1.684715573
## [97,] -0.10869074 1.533091989
## [98,] 1.57089666 -0.008508333
## [99,] -0.87311056 -0.120340723
## [100,] 0.06316369 -0.207171446
## [101,] 1.03125734 -0.876430074
## [102,] 0.17139057 -0.269182237
## [103,] -1.04440475 -0.222904296We calculate factor scores without rotation using WLS
stock.fa = factanal(stock, factors = 2, rotation = "none", scores = "Bartlett")
stock.fa$scores## Factor1 Factor2
## [1,] -1.81015485 0.47157713
## [2,] 0.29926209 0.41007876
## [3,] -0.15535353 -0.46550636
## [4,] -1.20954479 0.95126882
## [5,] 0.94005033 -0.21296820
## [6,] 0.35335600 -0.51696625
## [7,] 0.99041738 0.84578680
## [8,] 0.08906855 1.03310622
## [9,] -1.26871680 -0.92502622
## [10,] -0.90478765 0.66356384
## [11,] -0.49160291 -0.32792290
## [12,] 1.20958875 0.81131426
## [13,] 0.83134309 -0.59458588
## [14,] 1.23181833 -2.48831492
## [15,] -1.12380072 -0.66313639
## [16,] 0.11266891 -0.89301528
## [17,] 0.99179302 -2.16010181
## [18,] -0.38358357 -0.47438699
## [19,] -0.87114383 0.56610828
## [20,] 1.17845980 0.95157115
## [21,] -0.07109786 0.19910860
## [22,] 0.41384808 0.61912543
## [23,] 1.00332367 -0.58768675
## [24,] -0.26483129 -0.38790239
## [25,] -1.07056403 -0.28228641
## [26,] 0.37615550 -1.39600492
## [27,] -0.06737044 -1.25341267
## [28,] -1.99713174 1.01724899
## [29,] 0.42348738 0.61919199
## [30,] -0.85692391 -1.28285237
## [31,] 0.70553538 1.09320798
## [32,] 0.46117536 2.26886837
## [33,] -0.12705572 1.23115796
## [34,] 0.70475345 0.04604807
## [35,] -0.29805832 0.23101129
## [36,] 0.27461166 -0.34875098
## [37,] -1.10493980 -1.74861330
## [38,] 0.60500412 1.02704826
## [39,] 0.33144936 -0.47401027
## [40,] -0.07271269 -1.06984629
## [41,] 0.03863536 -2.37629790
## [42,] 0.86392417 2.45445757
## [43,] 0.67287776 2.03179918
## [44,] 0.79635836 0.53759118
## [45,] -0.29800298 -2.42692822
## [46,] 1.04474627 0.20812352
## [47,] -0.34172923 0.34330063
## [48,] -0.79099064 0.32040016
## [49,] -0.14199506 0.62567946
## [50,] 0.92676597 1.51683459
## [51,] -0.01334738 -0.59902316
## [52,] -1.12301723 0.44990606
## [53,] 0.26309467 -1.60953176
## [54,] 0.12601062 -0.51711613
## [55,] 0.75139739 0.18572776
## [56,] 1.77140211 1.22636910
## [57,] -0.31688840 -0.50784066
## [58,] 1.45746350 -1.70280609
## [59,] 0.41472230 -0.08407779
## [60,] 1.19684621 0.09353796
## [61,] -1.17764281 -1.05308615
## [62,] -0.52290087 -0.68641947
## [63,] -2.06302981 -1.87787363
## [64,] 0.42983459 -0.19020210
## [65,] 0.57594640 0.66308326
## [66,] -1.49482154 0.16564917
## [67,] 0.59590525 0.55254434
## [68,] -0.62314821 1.58117765
## [69,] 0.11354071 -0.19063767
## [70,] -1.95963088 -0.71567504
## [71,] 0.43257854 2.62569827
## [72,] 0.97914080 -1.00011633
## [73,] -0.38654012 0.26235840
## [74,] 0.09420306 -0.11433981
## [75,] 2.15569638 -0.42456477
## [76,] -0.16176366 -0.91326535
## [77,] 1.29222040 -0.94412618
## [78,] -0.32846626 0.13490297
## [79,] -1.28919058 1.34485438
## [80,] -2.08218913 -1.19182470
## [81,] 0.45977808 -0.94297505
## [82,] 1.48732067 -0.54226258
## [83,] 2.13947360 -1.33341520
## [84,] -1.71072997 1.12287594
## [85,] -0.94056662 -1.58590755
## [86,] 1.97110254 0.30133030
## [87,] -0.98705755 1.35637979
## [88,] 0.31220460 0.55476359
## [89,] -0.51281133 -0.86673306
## [90,] 0.34720309 -0.20372222
## [91,] -1.75777921 0.57481309
## [92,] -0.85370440 0.16978744
## [93,] -2.16472564 0.76622676
## [94,] 2.02366777 1.59573776
## [95,] 0.31086136 0.28195408
## [96,] -1.30712576 3.22458864
## [97,] 1.50896270 -0.29191830
## [98,] 0.18010412 1.56056119
## [99,] -0.22426808 -0.85235425
## [100,] -0.19809233 0.08757334
## [101,] -0.74631464 1.12899771
## [102,] -0.24666460 0.20246081
## [103,] -0.34665021 -1.01009961We calculate factor scores without rotation using regression
stock.fa = factanal(stock, factors = 2, rotation = "none", scores = "regression")
stock.fa$scores## Factor1 Factor2
## [1,] -1.80116560 0.38432667
## [2,] 0.29777596 0.33420663
## [3,] -0.15458204 -0.37937911
## [4,] -1.20353818 0.77526656
## [5,] 0.93538203 -0.17356516
## [6,] 0.35160124 -0.42131797
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