ForImp.PCA: Imputation of missing data by alternating Nearest Neighbour...
In GenForImp: The Forward Imputation: A Sequential Distance-Based Approach for Imputing Missing Data
Description
Usage
Arguments
Details
Value
Author(s)
References
See Also
Examples
Description
This function imputes quantitative missing data by alternating Nearest Neighbour Imputation (NNI) method and Principal Component Analysis (PCA) in a forward and sequential step-by-step process that starts from the complete part of data.
Usage
1
2
Arguments
mat
a quantitative data matrix with missing entries.
stand
a logical value indicating if variables should be standardized (stand=TRUE), or not (stand=FALSE, default) before the imputation process starts.
cor
a logical value. If cor=TRUE, PCA is run on the correlation matrix; if cor=FALSE the covariance matrix is used (default).
r
a positive value (equal or greater than 1) indicating
the order of the weighted Minkowski distance to be computed for selecting donors. In particular, r=1 is Manhattan (or city-block) distance, and r=2 is Euclidean distance (default). r=Inf denotes Lagrange (or Chebyshev or sup) distance.
probs
vector of probabilities with values in [0, 1] for computing quantiles of Minkowski distances in selection of donors. Default option: probs=seq(0,1,0.1) calculates the deciles of distances. Quantiles are computed with the generic function quantile.
q
string of the form "X%", with X=integer. It gives the quantile of Minkowski distances corresponding to the first "X%" distances as computed (and named) by the function quantile with probabilities specified in the argument probs.
tol
tolerance factor introduced to prevent numerical problems occuring when distances of complete units are equal to the choosen quantile q. Default is tol=1e-6.
Details
The ForImp.PCA procedure exploits the PCA method for setting-up so-called “Pseudo-Principal Components” (PPCs). These are artificial, missing-data free variables computed for all the units (i.e. both complete and incomplete) by using common observed variables without missing values. PPCs synthesize the main relevant information in the complete part of data and transfer them to incomplete units in a way functional to the subsequent application of the NNI method. NNI is then applied to PPCs (with a weighted Minkowski distance of order r, r ≥ 1) in order to select donors for incomplete units. All is performed in four stages:
Stage 0: data preparation;
Stage 1: running PCA and computing PPC scores;
Stage 2: application of the NNI method;
Stage 3: imputation.
Steps 1 to 3 are iteratively repeated until the starting data matrix is completely imputed.
ForImp.PCA does not require a number n of units greater than the number p of variables at every step of the procedure. It can work even in the presence of a starting data matrix with n < p. This further variant is indicated as ForImp with PCO, i.e. Forward Imputation with the Principal Coordinates Analysis.
For further details, see the references below.
Value
The imputed data matrix.
Author(s)
Nadia Solaro, Alessandro Barbiero, Giancarlo Manzi, Pier Alda Ferrari
References
Gower, J.C. (2005). Principal coordinates analysis. In: Armitage, P., Colton, T. (eds), Encyclopedia of biostatistics. John Wiley & Sons, Ltd., New York
Solaro, N., Barbiero, A., Manzi. G., Ferrari, P.A. (2014). Algorithmic-type imputation techniques with different data structures: Alternative approaches in comparison. In: Vicari, D., Okada, A., Ragozini, G., Weihs, C. (eds),
Analysis and modeling of complex data in behavioural and social sciences, Studies in Classification, Data Analysis, and Knowledge Organization. Springer International Publishing, Cham (CH): 253-261
Solaro, N., Barbiero, A., Manzi, G., Ferrari, P.A. (2015). A sequential distance-based approach for imputing missing data: The Forward Imputation. Under review
See Also
Examples
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# EXAMPLE with multivariate skew-normal data (MSN)
# require('sn')
# number of variables
p <- 5
# association matrix
omega <- 0.8
Omega <- matrix(omega, p, p)
diag(Omega) <- 1
Omega
# skewness parameter
alpha <- 4
alpha <- rep(alpha,p)
alpha
# number of units
n <- 500
# percentage of missing values
percmiss <- 0.2
nummiss <- n*p*percmiss
nummiss
## computation of output parameters
## covariance matrix and univariate means and skewnesses
param <- list(xi=rep(0,p), Omega=Omega, alpha=alpha, nu=Inf)
cp <- dp2cp(param, "SN")
cp
# correlation matrix
rho <- cov2cor(cp$var.cov)
rho
# generation of a complete matrix
set.seed(1)
x0 <- rmsn(n, Omega=Omega, alpha=alpha)
x0
# generating a matrix with missing data
x <- missing.gen(x0, nummiss)
# imputing missing values
xForImpPCA <- ForImp.PCA(x)
xForImpPCA
# computing the Relative Mean Square Error
error <- sum(apply((x0-xForImpPCA)^2/diag(var(x0)),2,sum)) / n
error
# EXAMPLE with real data
data(airquality)
m0 <- airquality
m0
# selecting the first 4 columns, with quantitative data
m <- m0[, 1:4]
m
# imputation
mi <- ForImp.PCA(m)
mi
# plot of imputed values for variable "Ozone"
ozone.miss.ind <- which(is.na(m)[,1])
plot(mi[ozone.miss.ind,1], axes=FALSE, pch=19, ylab="imputed values of Ozone",
xlab="observation index")
axis(2)
axis(1, at=1:length(ozone.miss.ind), labels=ozone.miss.ind, las=2)
box()
abline(v=1:length(ozone.miss.ind), lty=3, col="grey")
# EXAMPLE with n < p: ForImp with PCO
# require('mvtnorm')
p <- 20
n <- 10
sigma <- matrix(0.4, p, p)
diag(sigma) <- 1
# complete matrix
set.seed(2)
xtrue <- rmvnorm(n=n, mean=rep(0, p), sigma=sigma)
rownames(xtrue) <- 1:n
colnames(xtrue) <- paste("V", 1:p, sep="")
xtrue
# matrix with 10 missing values
xmiss <- missing.gen(xtrue, 10)
xmiss
# number of missing values per unit
apply(is.na(xmiss),1,sum)
# imputed matrix
ximp <- ForImp.PCA(xmiss)
ximp
# computing the Relative Mean Square Error
error <- sum(apply((xtrue-ximp)^2/diag(var(xtrue)),2,sum)) / n
error
Example output
Loading required package: mvtnorm
Loading required package: sn
Loading required package: stats4
Attaching package: 'sn'
The following object is masked from 'package:stats':
sd
[,1] [,2] [,3] [,4] [,5]
[1,] 1.0 0.8 0.8 0.8 0.8
[2,] 0.8 1.0 0.8 0.8 0.8
[3,] 0.8 0.8 1.0 0.8 0.8
[4,] 0.8 0.8 0.8 1.0 0.8
[5,] 0.8 0.8 0.8 0.8 1.0
[1] 4 4 4 4 4
[1] 500
$mean
[1] 0.7301875 0.7301875 0.7301875 0.7301875 0.7301875
$var.cov
[,1] [,2] [,3] [,4] [,5]
[1,] 0.4668262 0.2668262 0.2668262 0.2668262 0.2668262
[2,] 0.2668262 0.4668262 0.2668262 0.2668262 0.2668262
[3,] 0.2668262 0.2668262 0.4668262 0.2668262 0.2668262
[4,] 0.2668262 0.2668262 0.2668262 0.4668262 0.2668262
[5,] 0.2668262 0.2668262 0.2668262 0.2668262 0.4668262
$gamma1
[1] 0.5238823 0.5238823 0.5238823 0.5238823 0.5238823
[,1] [,2] [,3] [,4] [,5]
[1,] 1.000000 0.571575 0.571575 0.571575 0.571575
[2,] 0.571575 1.000000 0.571575 0.571575 0.571575
[3,] 0.571575 0.571575 1.000000 0.571575 0.571575
[4,] 0.571575 0.571575 0.571575 1.000000 0.571575
[5,] 0.571575 0.571575 0.571575 0.571575 1.000000
[,1] [,2] [,3] [,4] [,5]
[1,] 1.3997702246 1.740905120 2.126122910 1.806735795 1.105844380
[2,] 0.6945391557 0.486987263 1.054385270 0.147232551 0.265429499
[3,] 0.0963506889 0.046859206 0.324425715 1.091941861 0.924948103
[4,] 1.5833087432 0.796238948 0.869323764 0.438609958 1.111387571
[5,] 0.6795276080 0.904965960 0.425291902 0.570364009 0.238987974
[6,] 0.7307073314 1.762965575 0.328104738 1.155380084 1.418594511
[7,] 1.4165927789 0.636352733 1.639694414 1.507674146 0.991960068
[8,] 1.1448306975 0.680608813 0.092404825 1.563944164 0.484800455
[9,] 1.2154284344 1.384501470 0.326826275 0.958317223 0.545371140
[10,] 1.2048923363 0.919392866 1.861007627 1.535329959 1.124918601
[11,] 1.1481853408 -0.593608627 0.493363578 0.174160918 1.169243172
[12,] 0.2627416734 -0.297756929 0.098524127 0.125394215 0.044491540
[13,] 0.4088261404 0.337686146 0.224069250 0.682056785 1.491290997
[14,] -0.8897334430 0.227716368 -0.339363218 0.505376462 0.313932281
[15,] 1.3570517897 0.640564203 0.977727393 0.785415459 0.978074917
[16,] 0.8544953271 0.967586085 0.784236396 0.798043074 1.225607456
[17,] 1.5571990226 1.399569788 1.393668911 1.646169646 1.617987748
[18,] 0.6489690193 0.327420382 -0.362463720 0.672190044 -0.086148301
[19,] 1.3825513260 0.831548181 0.607638404 1.142151245 1.265300573
[20,] 0.3998247806 0.571939627 0.726870943 0.265196677 -0.513671886
[21,] 0.4026152344 -0.342528717 -0.088703174 -0.557792174 0.181478740
[22,] 0.9254945018 0.651442044 0.792811727 0.500505337 0.100877853
[23,] 0.7765367291 0.392487414 1.137392990 1.432482983 0.021438053
[24,] -0.6903784017 0.378222840 0.557717218 -0.234170759 0.521574309
[25,] 0.3096630891 -0.016584249 0.118054202 -0.308676393 0.694769944
[26,] 1.5572532510 0.924402315 1.314817160 1.849894765 1.981529107
[27,] 0.7934043980 0.814134136 0.509982131 1.080624317 0.931095378
[28,] 0.5956033763 1.788218537 1.467529075 0.985186703 0.746052865
[29,] 0.3471820345 0.851673588 0.014705896 0.557952417 0.598690264
[30,] 0.6766525482 0.525475024 0.649804150 1.018914501 0.050772781
[31,] 0.7260920983 -0.415665024 -0.007297023 0.319166282 0.080484776
[32,] 0.5883068508 1.097897468 0.352635176 0.934708414 0.179883910
[33,] 0.2319538768 0.070674836 -0.265121541 -0.847686270 0.832175071
[34,] 0.1321912885 -0.018307099 0.125956546 0.630414722 -0.051235932
[35,] 0.1821017809 1.509363173 1.000249452 0.974804386 0.560209539
[36,] 0.0269107026 -0.069037554 0.493153299 0.087482661 0.651376020
[37,] 0.6211647108 0.935775388 0.694111173 0.742373595 1.324025177
[38,] 1.1627481017 1.694896876 1.366816770 1.465459653 0.064460958
[39,] 2.7721817242 2.462606433 2.775098467 1.903799231 1.776066754
[40,] 0.7983977899 0.299328300 0.453502396 0.807814587 0.426485798
[41,] -0.0448453630 -0.127328026 0.262061066 0.209182687 -0.206389938
[42,] 0.8559723794 1.120968371 1.043970282 0.841740123 1.006447789
[43,] 0.7443870987 0.599985044 -0.027722497 0.396297929 0.424235006
[44,] 0.8173781838 0.548487747 0.192326181 0.207708868 0.630471830
[45,] 0.1668676881 1.025713353 1.071672701 -0.677318205 0.318370502
[46,] 0.8193987355 1.144602152 1.507892830 1.029192947 1.466242098
[47,] 0.1496825249 -0.305873600 0.328864518 0.023361975 -0.194686549
[48,] 1.7620200106 1.591461148 1.372680664 1.123788161 1.820729107
[49,] 0.8636662906 0.055828357 1.360165917 1.313912862 1.134362694
[50,] 0.4101076728 0.126684978 -0.016375026 0.062854109 -0.061034903
[51,] 0.7621136819 0.759510348 0.521813162 0.384218112 0.948390355
[52,] -0.0333803441 0.182076814 0.380336361 0.277669554 -0.106827856
[53,] 0.8199176619 0.212507441 0.404105959 1.382063513 0.479409383
[54,] 0.0399193520 0.445329178 1.545139148 0.367889225 0.111754913
[55,] 1.3419565727 1.238832757 0.748588647 0.590827959 -0.089418633
[56,] 0.9216801580 -0.352515495 -0.067369198 0.231206239 -0.113681520
[57,] 0.1122608235 -0.620081478 1.022697369 0.957583908 -0.528631959
[58,] 0.0061941965 0.842819104 0.193955877 0.563884926 0.186765856
[59,] 1.2422034905 0.762913912 1.658755985 0.409471993 1.131579336
[60,] 0.0442327068 -0.089138795 0.023821715 0.079006411 0.405394831
[61,] 1.0140762553 -0.654174616 -0.902998902 0.418304960 0.568907575
[62,] 0.2686845246 0.279305563 0.801180715 -0.019078581 0.065612056
[63,] 0.4417461372 0.374570408 0.440606672 -0.221946328 -0.207503323
[64,] 0.9811509803 0.736836709 0.746702898 1.385349070 0.897876390
[65,] 0.5914096642 0.722215277 0.503920145 0.995708827 1.240568689
[66,] 0.5898124013 0.926265023 -0.319305190 0.012638943 1.203919673
[67,] 0.4464647666 1.244917729 1.270479185 2.096825926 0.638677828
[68,] 2.4088506883 1.618998936 1.014389418 2.141835029 1.995003672
[69,] 0.1269219986 -0.194556145 -0.180184294 0.610158352 -0.050923979
[70,] 1.2437658548 0.518990624 0.121039409 -0.071895677 0.285815188
[71,] 0.3708355214 -0.255029238 0.572014816 0.146332697 0.369656055
[72,] 0.0736765837 0.759007933 0.019176391 -0.270291995 0.910808867
[73,] 0.3286740357 0.341672890 -0.115416065 0.071660090 -0.236141063
[74,] -0.1996279844 0.080423324 0.414249086 0.436104623 0.148874116
[75,] 0.0733043831 1.028060597 1.238553336 0.013221449 0.402626256
[76,] 0.4576312546 0.700058976 -0.161745099 0.171730148 0.842873759
[77,] 0.4917420262 0.937450012 0.043692170 1.355425039 0.546198901
[78,] 0.7447294802 1.538360445 -0.184665144 0.960592272 0.876869120
[79,] 0.5678512175 -0.238703385 0.969864592 0.587239796 0.821510975
[80,] 0.2493369910 0.084925071 0.656449207 0.813345111 0.911124678
[81,] 1.1161737417 0.858423347 1.849207462 0.981289074 1.731462696
[82,] 0.0608606010 0.193696586 0.055499439 0.526905648 -0.764655677
[83,] 1.2780445956 1.315056382 0.654093712 -0.201199290 1.064270953
[84,] -0.1772016552 1.236814571 0.465113687 0.005767215 0.369219332
[85,] 2.2920500300 2.188882200 1.948697673 2.164836429 1.875522731
[86,] 0.1485071042 0.327945111 -0.353183937 0.369315974 -0.530032577
[87,] 1.4646475076 0.880367718 1.426411037 0.166112479 1.170301524
[88,] -0.0536103020 0.281249772 -0.109705023 0.310450055 -0.169075162
[89,] 0.5552235556 0.519204931 0.576817955 0.335872846 0.078827296
[90,] 0.3456803662 -0.085334422 0.364333826 0.264399391 0.139034937
[91,] -0.2064820386 -0.052346118 0.763751206 -0.400224790 -0.249441842
[92,] 1.3689990377 0.716827944 1.051680126 0.382172480 0.932523641
[93,] 1.9844547930 1.955412070 1.118317520 1.350134853 0.988626345
[94,] 0.4882287511 -0.265908008 0.259450869 0.672100751 -0.105381817
[95,] 2.0293888176 0.412800795 1.444060559 1.007899878 1.821624220
[96,] 1.4473929775 1.471632294 0.297601518 1.060250553 1.539135498
[97,] -0.0299984447 0.283270275 1.334510926 0.980762766 0.400868504
[98,] 1.0439635421 1.157340367 1.777761136 0.800969943 1.097845717
[99,] 0.5373924396 1.499825859 0.751675927 0.273825687 1.834708342
[100,] -0.1804339883 -0.244852264 0.287185568 0.088882746 -0.191495262
[101,] 0.6104811867 0.784503687 1.048923778 1.035955289 0.532781052
[102,] 0.7379861401 1.306368201 0.116440008 0.726265818 0.708531083
[103,] -0.2876305844 0.371535197 0.659574715 -0.432098276 0.573397741
[104,] 0.0937199304 0.227971092 -0.141382696 0.107187956 0.221962569
[105,] 1.1572386246 1.428405225 1.566783856 1.408689745 1.554578146
[106,] 2.9132345268 1.590547067 2.321669632 2.416690750 1.725644283
[107,] 0.4420395422 -0.500559087 -0.109340191 0.523798259 0.801914335
[108,] 0.7358177083 0.196618667 -0.095575314 0.719018172 0.256363930
[109,] 1.7525829237 2.347079458 1.308199909 1.444159679 1.052483472
[110,] 1.3130649042 0.565409014 0.667894548 0.497865977 -0.046760094
[111,] -0.1284516001 0.251439214 0.195245067 0.412608305 -0.167434136
[112,] 0.2332677621 0.592773865 0.687284971 -0.463152269 1.262729708
[113,] 1.4778671461 0.616154446 0.937909506 0.933874552 0.869956419
[114,] 0.3666192583 0.140845996 1.058151043 0.311237493 1.232927404
[115,] 0.9327784412 2.085144007 0.195845004 1.068530241 0.488073523
[116,] 0.3372931429 0.366007614 0.835367052 0.917834072 0.079238414
[117,] 0.5849929379 0.626640613 0.483407447 0.754072830 1.263182640
[118,] 0.9785252378 0.414018789 1.153722741 1.129519078 1.537642748
[119,] 0.4254967260 0.083031779 0.170266887 0.197157346 0.095879188
[120,] 1.4422484926 1.433234024 0.942811786 2.359565313 1.433088298
[121,] 0.2564058123 0.407220985 0.500270193 0.587636400 0.369424830
[122,] 1.3742231655 0.007004910 0.753632921 1.039044944 0.871800166
[123,] 1.5207142471 2.327154659 1.148294237 1.347683518 2.205922789
[124,] 0.0438786122 0.824675844 -0.282110464 -0.164002117 0.271291082
[125,] 0.2030135619 0.523196806 0.637543229 -0.075521100 0.286147251
[126,] 1.2756801054 0.490360296 0.580637316 0.716376747 1.667893461
[127,] -0.0210801382 0.295662252 -0.149264798 0.383003449 -0.448969045
[128,] 0.3364503404 0.258452463 -0.540727887 0.667295664 1.012720822
[129,] 0.2129725498 0.425752703 0.336584299 0.020619835 0.872863828
[130,] 0.6220435245 1.317383849 0.434987979 0.697059758 1.035519295
[131,] 0.0669473856 -0.444498818 0.758670099 -0.147241501 -0.054187755
[132,] 0.2776721280 0.369342926 0.270094349 0.178649291 0.563979926
[133,] 0.9343747667 0.383884991 0.430418803 1.589454669 0.182530367
[134,] -0.0194467519 0.741651964 0.208670365 0.578161077 1.146789076
[135,] 0.8374246983 0.068420483 0.937728525 0.384106454 1.629287157
[136,] 0.2056860835 0.692136139 1.482572633 1.018184816 0.733503813
[137,] 0.9114676577 1.341598455 0.528110159 1.518032157 0.575262918
[138,] 0.1383091754 0.582881024 0.186377691 0.347742175 0.399579286
[139,] 0.2800814771 0.221636514 1.229440529 0.575900214 0.240163145
[140,] 0.0914826882 0.339047268 -0.202132630 -0.208883794 0.091754324
[141,] -0.3299474246 -0.334262452 1.162668390 1.119109758 0.404601994
[142,] 2.0035494765 1.286243058 1.605592322 1.503117852 0.903455691
[143,] -0.0433631399 1.062377787 0.210386193 0.942337373 1.482001675
[144,] 0.4790645276 0.450558969 1.241902608 0.844270750 0.463748541
[145,] -0.1733715834 0.366014709 0.682083846 0.594669758 0.143179980
[146,] -0.0668627174 -0.319259150 0.310534056 0.323913610 0.500186859
[147,] 1.8708708235 1.167842140 0.397314715 1.043097010 1.047923012
[148,] 0.2908074965 -0.443579056 0.353901005 0.228164067 0.930242805
[149,] 0.2361163367 0.747373098 1.216983191 1.535419151 0.006536608
[150,] 0.4138741722 1.127748150 1.046795988 1.502799337 1.208482190
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[384,] 0.6539619225 0.644135969 0.270791926 0.009444134 1.275304204
[385,] 1.3235583861 1.183459377 0.836223965 0.576451735 0.279578676
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[414,] 0.0872480022 0.530217368 0.702879339 -0.148937750 0.665879256
[415,] 2.2177118959 3.165500531 2.129520056 2.406724427 2.256465810
[416,] 0.8276579974 0.313678454 0.601124349 0.017892828 1.554676801
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[418,] 0.5517635433 0.946712680 1.002607641 0.382882066 0.369320818
[419,] -0.5653612880 0.158307255 0.623478045 -0.393729323 0.385962636
[420,] 1.8419281348 1.756237102 1.245027180 0.504389005 0.358376832
[421,] 1.0149892091 0.592127518 0.582511472 0.349894422 -0.132325963
[422,] 0.5763820055 1.839301660 0.778873668 0.608870178 0.703524479
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[429,] 0.0788832234 0.763302216 0.079342586 0.119264015 0.476940537
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[449,] 1.0672963694 0.756186826 1.095294795 1.123375088 1.014033367
[450,] 0.7876668073 2.135957126 1.793550352 1.956823382 1.639250350
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[500,] 0.6343152373 0.794084956 0.808473583 1.190503178 1.077501296
attr(,"family")
[1] "SN"
attr(,"parameters")
attr(,"parameters")$xi
[1] 0 0 0 0 0
attr(,"parameters")$Omega
[,1] [,2] [,3] [,4] [,5]
[1,] 1.0 0.8 0.8 0.8 0.8
[2,] 0.8 1.0 0.8 0.8 0.8
[3,] 0.8 0.8 1.0 0.8 0.8
[4,] 0.8 0.8 0.8 1.0 0.8
[5,] 0.8 0.8 0.8 0.8 1.0
attr(,"parameters")$alpha
[1] 4 4 4 4 4
attr(,"parameters")$tau
[1] 0
x.1 x.2 x.3 x.4 x.5
1 1.3997702246 1.740905120 2.126122910 1.806735795 1.105844380
2 0.6945391557 0.486987263 1.054385270 0.147232551 0.265429499
3 0.0963506889 0.046859206 0.324425715 1.091941861 0.924948103
4 1.5833087432 0.796238948 0.869323764 0.438609958 1.111387571
5 0.4171984786 0.904965960 0.425291902 0.570364009 0.238987974
6 0.7307073314 0.725446609 0.328104738 0.717063392 1.418594511
7 1.0493284286 0.636352733 0.999543004 1.507674146 0.991960068
8 1.1448306975 0.680608813 0.092404825 1.563944164 0.484800455
9 1.4622091863 1.384501470 1.375850324 1.329629732 1.597098228
10 1.2048923363 1.218718595 1.279922379 1.535329959 1.124918601
11 1.1481853408 -0.593608627 0.493363578 0.174160918 1.169243172
12 0.2627416734 -0.297756929 0.098524127 0.125394215 0.281881536
13 0.4088261404 0.337686146 0.224069250 0.682056785 1.491290997
14 -0.8897334430 0.227716368 -0.339363218 0.505376462 0.313932281
15 1.3570517897 0.931834409 1.007377643 0.785415459 0.978074917
16 0.8544953271 0.967586085 0.784236396 0.798043074 1.225607456
17 1.5571990226 1.128904622 1.393668911 1.646169646 1.065821927
18 0.6489690193 0.327420382 -0.362463720 0.672190044 -0.086148301
19 1.0333958224 0.831548181 0.607638404 1.142151245 1.265300573
20 0.3998247806 0.571939627 0.726870943 0.265196677 -0.513671886
21 0.4856701500 -0.342528717 -0.088703174 0.318120679 0.202561311
22 0.5067897519 0.449178573 0.792811727 0.500505337 0.100877853
23 0.7765367291 0.392487414 1.137392990 1.432482983 0.021438053
24 -0.6903784017 0.378222840 0.557717218 -0.234170759 0.521574309
25 0.3096630891 -0.016584249 0.118054202 -0.308676393 0.694769944
26 1.5572532510 1.471452187 1.556592468 1.479667651 1.981529107
27 0.7934043980 0.814134136 0.509982131 1.080624317 0.931095378
28 0.5956033763 1.788218537 0.832453788 0.985186703 0.746052865
29 0.3471820345 0.851673588 0.487966344 0.557952417 0.598690264
30 0.5356690695 0.525475024 0.497669907 1.018914501 0.050772781
31 0.7260920983 -0.415665024 -0.007297023 0.365567377 0.358388418
32 0.5883068508 1.097897468 0.352635176 0.934708414 0.179883910
33 0.2319538768 0.070674836 -0.265121541 -0.847686270 0.832175071
34 0.1321912885 -0.018307099 0.432558974 0.630414722 -0.051235932
35 1.0487730796 1.509363173 1.000249452 0.974804386 0.560209539
36 0.0269107026 0.638840032 0.493153299 0.087482661 0.651376020
37 0.6211647108 0.935775388 0.694111173 0.742373595 1.324025177
38 1.1627481017 1.694896876 1.366816770 1.465459653 0.064460958
39 2.7721817242 2.462606433 2.775098467 1.903799231 1.757605529
40 0.7983977899 0.648291848 0.453502396 0.807814587 0.426485798
41 -0.0448453630 -0.127328026 0.262061066 0.209182687 -0.206389938
42 0.8559723794 1.120968371 1.043970282 0.841740123 0.771219357
43 0.7443870987 0.599985044 -0.027722497 0.396297929 0.424235006
44 0.8173781838 0.548487747 0.192326181 0.207708868 0.630471830
45 0.1668676881 1.025713353 1.071672701 -0.677318205 0.327030686
46 0.8193987355 1.144602152 1.507892830 1.097872111 1.466242098
47 0.1496825249 -0.305873600 0.328864518 0.023361975 0.251777211
48 1.7620200106 1.591461148 1.232348192 1.123788161 1.820729107
49 0.8636662906 0.939725035 1.360165917 1.313912862 1.134362694
50 0.4101076728 0.126684978 -0.016375026 0.298393172 0.233936135
51 0.7621136819 0.759510348 0.521813162 0.384218112 0.948390355
52 -0.0333803441 0.182076814 0.380336361 0.277669554 -0.106827856
53 0.8199176619 0.212507441 0.404105959 1.382063513 0.504051826
54 0.8233698576 0.445329178 1.545139148 0.367889225 0.772095308
55 0.3616123128 1.238832757 0.748588647 0.590827959 -0.089418633
56 0.9216801580 -0.352515495 -0.067369198 0.231206239 -0.113681520
57 0.1122608235 -0.620081478 1.022697369 0.957583908 -0.528631959
58 0.0061941965 0.842819104 0.193955877 0.434682639 0.186765856
59 1.2422034905 0.762913912 1.658755985 0.409471993 1.131579336
60 0.0442327068 -0.089138795 0.023821715 0.447778336 0.405394831
61 0.3949430320 -0.654174616 -0.902998902 0.418304960 0.568907575
62 0.2686845246 0.279305563 0.801180715 -0.019078581 0.065612056
63 0.4417461372 0.413105040 0.440606672 -0.221946328 -0.207503323
64 0.9811509803 0.736836709 0.746702898 1.385349070 0.897876390
65 0.5914096642 0.722215277 0.850367511 0.995708827 1.240568689
66 0.5898124013 0.926265023 0.610096508 0.012638943 1.203919673
67 0.4464647666 1.244917729 1.270479185 2.096825926 0.638677828
68 2.4088506883 1.618998936 1.014389418 1.550608292 1.995003672
69 0.2941132767 -0.194556145 -0.180184294 0.326396516 -0.050923979
70 1.2437658548 0.518990624 0.121039409 0.685908084 0.285815188
71 0.3708355214 0.374369520 0.494927586 0.146332697 0.369656055
72 0.0736765837 0.759007933 0.019176391 -0.270291995 0.910808867
73 0.3286740357 0.294844308 -0.115416065 0.071660090 -0.236141063
74 -0.1996279844 0.080423324 0.414249086 0.436104623 0.310248426
75 0.0733043831 1.028060597 0.561443051 0.013221449 0.402626256
76 0.4576312546 0.700058976 -0.161745099 0.171730148 0.842873759
77 0.4917420262 0.937450012 0.043692170 1.355425039 0.546198901
78 0.7447294802 0.628874508 -0.184665144 0.960592272 0.876869120
79 0.5678512175 0.974949209 0.969864592 0.587239796 0.821510975
80 0.2493369910 0.084925071 0.656449207 0.813345111 0.911124678
81 1.1161737417 0.932789289 1.849207462 0.981289074 1.731462696
82 0.0608606010 0.193696586 0.055499439 0.526905648 0.212588974
83 0.8316726325 1.315056382 0.654093712 -0.201199290 1.064270953
84 -0.1772016552 1.236814571 0.465113687 0.005767215 0.369219332
85 2.2920500300 1.701409313 1.948697673 1.712982823 1.875522731
86 0.1485071042 0.404808527 0.242668751 0.369315974 -0.530032577
87 1.4646475076 0.880367718 1.426411037 0.166112479 1.170301524
88 -0.0536103020 0.281249772 0.273409553 0.310450055 0.244193409
89 0.5552235556 0.519204931 0.576817955 0.335872846 0.078827296
90 0.3456803662 -0.085334422 0.364333826 0.264399391 0.139034937
91 -0.2064820386 -0.052346118 0.287864038 0.433494277 -0.249441842
92 1.3689990377 0.716827944 1.051680126 0.933349696 0.932523641
93 1.3735519011 1.955412070 1.118317520 1.350134853 0.988626345
94 0.4882287511 0.384266837 0.259450869 0.672100751 -0.105381817
95 2.0293888176 0.412800795 1.444060559 1.007899878 1.821624220
96 1.4473929775 1.471632294 0.297601518 1.060250553 1.539135498
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343 0.3835012871 1.219849413 0.347409074 0.149134565 -0.612855802
344 0.4304634629 0.471592257 0.921867712 0.762868065 0.139435683
345 1.2992532243 2.244757485 2.996824040 2.568087044 1.956396777
346 -0.4676296157 0.365683142 0.673485516 0.361658805 0.002410185
347 0.7810262911 0.692634868 -0.199863725 0.602468794 0.252859318
348 0.3590790164 0.585706415 0.133452158 0.764560628 0.462458038
349 0.3697650508 0.381754451 0.167966900 0.297595665 0.806710593
350 1.3599896395 0.562182129 -0.014553920 0.749667114 0.631377418
351 1.0807901534 0.526188194 0.646292113 0.427214251 0.920750657
352 0.9783358856 0.900996631 0.860754293 1.397725834 0.101399964
353 0.9145155592 0.468967292 0.801761463 1.231528563 0.745433071
354 0.8772550067 0.042954276 1.481857967 1.437526274 0.917362169
355 0.6889716817 0.428968617 -0.505812461 -0.612954164 0.737776142
356 1.3964090331 1.114439303 1.222729963 0.817660441 1.705111032
357 1.9554043795 2.237279087 2.189610774 2.485596401 1.464107663
358 0.3485623532 0.051695094 0.227729230 0.311517932 0.870728653
359 0.6151404546 0.908882771 1.699811714 1.385293604 1.563124644
360 0.4868226706 0.187178323 0.465775193 0.694167608 0.392268137
361 0.4698391646 0.775517930 0.340980865 0.448729967 0.377598449
362 0.7975822120 0.532534588 -0.459155854 0.138147245 0.262924385
363 1.2738687665 1.277564018 1.146111816 1.094326935 1.921782385
364 1.4566200006 1.472739008 1.307385413 1.290875140 2.226795912
365 0.7724753239 0.998247782 1.451399701 0.515849299 0.403447897
366 0.4593913357 0.704270099 0.343946126 0.388007193 0.168751894
367 0.6598277392 0.301277257 -0.086612589 0.871697242 -0.433145648
368 0.2727471773 0.345437546 -0.334936773 0.218460227 0.491391694
369 1.2275158773 1.096359183 1.173776533 1.361896038 1.052881605
370 1.9806620090 1.782287106 1.868807091 1.926231311 2.557661316
371 0.9292858181 1.817353966 1.007832695 0.727380365 0.824733759
372 1.1865567842 1.183365950 0.902476034 0.910079334 0.982155400
373 0.1311338631 0.295134474 0.306988497 0.016053584 0.226001756
374 0.5014455113 0.363212428 0.519916926 0.405368082 0.603715478
375 -0.0202566836 0.360892819 1.013672306 0.528273339 0.388434759
376 1.5619774963 1.508064984 1.504305788 1.514537327 1.629872963
377 0.2057060386 0.210968623 0.537937602 0.481593788 0.398060703
378 0.2867649189 0.718353252 0.222701947 -0.287538013 0.406849289
379 0.3028227898 0.509726312 0.695506524 0.065717114 0.467381395
380 1.2851530626 0.266344934 0.394218987 1.060926639 0.061510237
381 2.0303247550 1.283042627 1.507699467 1.683230802 1.540479586
382 2.1461350130 1.843640069 1.063502927 1.500524495 1.583386649
383 0.2650744378 0.698614707 0.465245910 1.784322527 -0.039141314
384 0.6539619225 0.644135969 0.270791926 0.009444134 0.466377909
385 1.3235583861 1.183459377 0.836223965 0.576451735 0.279578676
386 1.1663769983 1.881599927 0.968708404 1.221162119 0.939539674
387 0.8083644096 0.423081802 0.673617866 0.848791936 1.409423706
388 1.8212707100 2.295013252 1.120559023 2.000864663 1.512032153
389 -0.3125561538 0.292639549 1.300561643 0.764183494 -0.009164656
390 1.0123255035 1.519255898 1.294859970 1.439438010 0.823164390
391 1.6302179620 1.060530567 1.029264747 0.752902132 1.445845259
392 0.8714798596 0.457422988 0.437740488 0.629272948 0.787685683
393 0.8130905083 0.818309730 0.142258502 -0.016041588 0.531361468
394 1.1653296409 0.821775283 1.402437524 0.657673073 0.889671067
395 0.4147485544 0.728476075 0.359880916 0.456414997 0.291332893
396 -0.3112394607 0.142159442 0.190767532 0.326802891 0.235736173
397 1.0448263042 -0.110841563 -0.024755527 0.504125315 0.759732571
398 1.4109732234 1.865064082 1.543643011 1.404795391 1.311419755
399 0.2497186175 -0.652223581 0.450042025 -0.452102464 0.814659118
400 0.3647744987 -0.113867554 0.260800271 0.285815004 0.348583411
401 1.1478843333 0.010234422 1.374978041 0.355080537 0.609339596
402 0.8067086235 -0.057604782 -0.089751537 -0.088543871 -0.035788005
403 -0.1554298312 0.280554357 0.362144895 0.545850394 -0.110393010
404 1.2578851565 0.819547901 1.327516689 1.560274266 1.808898461
405 0.3344431550 0.796267256 0.248676862 0.586856371 0.659319629
406 1.5921629388 1.373423460 1.557836053 1.157874928 2.745977294
407 0.0391522837 0.479606981 0.281865192 0.385154845 0.417847798
408 0.2542299672 0.356362702 -0.176182684 0.231909323 0.534097355
409 1.1489241034 1.452084975 0.434340699 0.888747708 1.881090975
410 0.5655852822 0.543579201 0.772436546 1.091205497 1.163959047
411 0.7809429737 1.010931938 1.367981501 0.891134818 0.879614334
412 1.9086173673 1.403335153 1.685292572 1.499973973 1.548409561
413 1.9197716984 0.785593613 1.140856545 1.326169866 1.543571812
414 0.0872480022 0.530217368 0.702879339 -0.148937750 0.665879256
415 2.2177118959 3.165500531 2.129520056 2.406724427 1.470814677
416 0.8276579974 0.313678454 0.601124349 0.017892828 1.554676801
417 1.1525635739 0.905725684 1.034924086 0.982576839 1.493830920
418 0.5517635433 0.946712680 0.588591894 0.438469146 0.369320818
419 0.1911741196 0.158307255 0.623478045 -0.393729323 0.370166003
420 0.6321223873 0.576737195 1.245027180 0.737155550 0.358376832
421 1.0149892091 0.509564073 0.582511472 0.349894422 0.624780111
422 0.5763820055 1.839301660 0.778873668 0.608870178 0.703524479
423 0.6699065309 0.839038331 1.364369855 0.405989451 -0.054326428
424 0.5165206224 -0.052285794 1.290252615 0.657239098 0.058939492
425 0.6179220867 1.099570783 0.624929639 0.633975007 0.546218631
426 1.2382446106 0.075650767 0.464914121 0.620437551 0.105569331
427 -0.4517096344 -0.034360341 0.090037763 -0.024565980 0.871784432
428 -0.1532259748 0.171677233 0.328049076 0.235296410 -0.223043417
429 0.0788832234 0.763302216 0.079342586 0.346397178 0.476940537
430 1.2393608495 1.268033743 1.195552147 0.634831558 1.543526359
431 -0.2893547342 1.038733810 0.417678840 1.050764894 0.332237042
432 0.4740520223 0.392763472 0.462441281 -0.037307347 -0.298559796
433 1.0778025969 1.770401158 2.677195942 1.826325826 1.192468301
434 0.7281908433 0.810244174 1.384011706 0.800686433 0.761729554
435 0.6858011849 0.703760609 0.255215145 -0.633044821 0.500048781
436 0.7679983965 1.717099275 0.624999061 0.660103553 0.663652049
437 0.7789961486 0.359325006 0.348850970 0.567309859 -0.314310202
438 0.5862629371 0.067599716 0.728722435 1.631352938 0.108618293
439 0.8515743861 1.028751284 1.731100495 1.295264353 1.021129158
440 1.2548726772 0.567646719 1.633566005 1.149898947 1.012845297
441 0.7174907920 0.916405371 0.997641566 0.703785179 0.883232905
442 1.0010134828 0.386715237 0.496782963 0.535667381 0.584015186
443 1.5406531575 1.889544523 1.324290594 1.339994842 1.229747204
444 0.8675957798 0.536711261 0.587806427 0.486121956 0.531409666
445 1.0447698861 0.576658412 0.564991154 0.847925590 1.770143629
446 -1.0466793862 0.764736812 -0.133037292 0.427461730 0.894892830
447 -0.1062427750 1.100388752 0.751966615 0.550767480 0.570713672
448 1.1431429304 2.236790212 0.042426819 1.552504040 1.758753133
449 1.0672963694 0.756186826 1.095294795 0.958391001 1.014033367
450 1.7518669621 2.135957126 1.602162493 1.956823382 1.639250350
451 0.2816609918 0.350999803 -0.453958746 -0.317809034 0.404032095
452 -0.1815496123 0.495494997 0.155736646 0.041199272 0.489211905
453 2.0506704709 2.630069675 2.382815483 1.846620963 1.614818645
454 0.4896877989 1.345604853 1.094698989 1.121410915 1.546959130
455 -0.4637147532 0.204901488 0.319866776 -0.346587388 0.851198467
456 0.4387578297 0.433208593 0.190713677 0.399225237 0.241567340
457 0.5363220588 0.661983797 0.266755477 0.720442825 -0.436044030
458 -0.4874575908 0.713738018 1.024491130 0.520704087 -0.185740840
459 -0.0369443317 0.278424691 0.765886523 0.909835714 0.075991173
460 1.9410352661 0.796728910 2.498946950 1.371291030 1.391352511
461 -0.0583528385 -0.145416418 0.157158807 0.378976935 0.658364989
462 1.0276564101 -0.054518498 0.173809231 -0.015014901 0.523886783
463 1.4010812965 0.382105425 1.852589269 0.965756595 2.229171688
464 0.5814140612 0.512498112 0.323716733 0.447054053 -0.131459487
465 1.2341913025 0.647848620 1.101372242 1.294973937 2.040942822
466 2.0240765499 1.430203910 2.721843778 2.039963597 1.525803771
467 0.0926748400 0.730429704 -0.322409760 -0.512653833 0.375296934
468 0.3157234402 -0.373379426 1.098983365 0.978463498 0.282284551
469 0.7942551823 0.349720282 -0.126728805 -0.154457229 0.344339430
470 0.5056091025 -0.166861345 -0.402655196 0.365967524 -0.499861896
471 1.6552493616 2.237543752 1.431070123 1.662822254 2.240142381
472 -0.0619300866 1.085686691 0.186590999 0.320754430 0.815504764
473 0.1363872100 -0.185133836 -0.425976512 0.784844323 0.144853101
474 1.3433356642 0.847743272 0.876913030 1.329796709 1.105238385
475 0.3889798126 -1.019899844 0.872496732 0.140490049 0.548402999
476 -0.0538060912 0.544544943 0.496175924 0.504353504 1.148527419
477 1.2108732416 1.179200385 1.168545991 1.521336006 1.056961059
478 0.2234005579 0.261341265 -0.060807896 -0.289487705 -0.080731082
479 2.2218638505 2.424530372 1.745101182 1.730359704 1.706807004
480 0.4401640797 0.479359500 0.423558707 0.361232325 0.846691735
481 1.5615376428 1.920666806 1.463862278 1.432632347 1.395026206
482 0.4914551246 0.330789605 0.412741592 0.355418115 0.601084853
483 0.8316452179 0.784454447 1.168112572 0.632877683 1.852258521
484 0.1950071551 0.398717128 0.381487187 -0.137720371 1.029064321
485 1.0114904994 0.027343484 0.074367840 0.346627092 0.230489210
486 3.3892940964 2.226207936 2.362123485 2.305562227 2.271571692
487 0.5884305043 0.351302080 1.057697472 0.667220497 0.929285248
488 0.1326068343 0.634674982 0.704599277 1.309664614 0.244490977
489 -0.4561553423 0.231741203 0.437037313 0.774215478 0.333403692
490 0.6958549364 0.705817376 0.576192923 0.845894141 0.651655393
491 1.2188160937 1.570599483 1.971099759 1.920439752 1.287808861
492 2.0157620014 0.619789366 0.809975786 1.311502877 0.624184690
493 2.1063871198 2.746916392 2.340228960 2.705616600 1.461522687
494 0.5449314664 0.901207278 0.596426007 0.054761905 0.070165841
495 2.2396228948 0.396634299 0.836431933 0.432022654 0.157280977
496 -0.2760935019 0.349602463 0.353771615 0.289892288 0.256280999
497 0.3338808096 -0.164133316 0.138176785 0.220501398 0.337010077
498 1.4023172187 1.430412316 2.262429082 1.109038342 1.506119262
499 2.1211488748 1.367098610 1.657866732 2.662131305 1.396535987
500 0.6343152373 0.828641146 0.808473583 1.190503178 1.077501296
[1] 0.6102628
Ozone Solar.R Wind Temp Month Day
1 41 190 7.4 67 5 1
2 36 118 8.0 72 5 2
3 12 149 12.6 74 5 3
4 18 313 11.5 62 5 4
5 NA NA 14.3 56 5 5
6 28 NA 14.9 66 5 6
7 23 299 8.6 65 5 7
8 19 99 13.8 59 5 8
9 8 19 20.1 61 5 9
10 NA 194 8.6 69 5 10
11 7 NA 6.9 74 5 11
12 16 256 9.7 69 5 12
13 11 290 9.2 66 5 13
14 14 274 10.9 68 5 14
15 18 65 13.2 58 5 15
16 14 334 11.5 64 5 16
17 34 307 12.0 66 5 17
18 6 78 18.4 57 5 18
19 30 322 11.5 68 5 19
20 11 44 9.7 62 5 20
21 1 8 9.7 59 5 21
22 11 320 16.6 73 5 22
23 4 25 9.7 61 5 23
24 32 92 12.0 61 5 24
25 NA 66 16.6 57 5 25
26 NA 266 14.9 58 5 26
27 NA NA 8.0 57 5 27
28 23 13 12.0 67 5 28
29 45 252 14.9 81 5 29
30 115 223 5.7 79 5 30
31 37 279 7.4 76 5 31
32 NA 286 8.6 78 6 1
33 NA 287 9.7 74 6 2
34 NA 242 16.1 67 6 3
35 NA 186 9.2 84 6 4
36 NA 220 8.6 85 6 5
37 NA 264 14.3 79 6 6
38 29 127 9.7 82 6 7
39 NA 273 6.9 87 6 8
40 71 291 13.8 90 6 9
41 39 323 11.5 87 6 10
42 NA 259 10.9 93 6 11
43 NA 250 9.2 92 6 12
44 23 148 8.0 82 6 13
45 NA 332 13.8 80 6 14
46 NA 322 11.5 79 6 15
47 21 191 14.9 77 6 16
48 37 284 20.7 72 6 17
49 20 37 9.2 65 6 18
50 12 120 11.5 73 6 19
51 13 137 10.3 76 6 20
52 NA 150 6.3 77 6 21
53 NA 59 1.7 76 6 22
54 NA 91 4.6 76 6 23
55 NA 250 6.3 76 6 24
56 NA 135 8.0 75 6 25
57 NA 127 8.0 78 6 26
58 NA 47 10.3 73 6 27
59 NA 98 11.5 80 6 28
60 NA 31 14.9 77 6 29
61 NA 138 8.0 83 6 30
62 135 269 4.1 84 7 1
63 49 248 9.2 85 7 2
64 32 236 9.2 81 7 3
65 NA 101 10.9 84 7 4
66 64 175 4.6 83 7 5
67 40 314 10.9 83 7 6
68 77 276 5.1 88 7 7
69 97 267 6.3 92 7 8
70 97 272 5.7 92 7 9
71 85 175 7.4 89 7 10
72 NA 139 8.6 82 7 11
73 10 264 14.3 73 7 12
74 27 175 14.9 81 7 13
75 NA 291 14.9 91 7 14
76 7 48 14.3 80 7 15
77 48 260 6.9 81 7 16
78 35 274 10.3 82 7 17
79 61 285 6.3 84 7 18
80 79 187 5.1 87 7 19
81 63 220 11.5 85 7 20
82 16 7 6.9 74 7 21
83 NA 258 9.7 81 7 22
84 NA 295 11.5 82 7 23
85 80 294 8.6 86 7 24
86 108 223 8.0 85 7 25
87 20 81 8.6 82 7 26
88 52 82 12.0 86 7 27
89 82 213 7.4 88 7 28
90 50 275 7.4 86 7 29
91 64 253 7.4 83 7 30
92 59 254 9.2 81 7 31
93 39 83 6.9 81 8 1
94 9 24 13.8 81 8 2
95 16 77 7.4 82 8 3
96 78 NA 6.9 86 8 4
97 35 NA 7.4 85 8 5
98 66 NA 4.6 87 8 6
99 122 255 4.0 89 8 7
100 89 229 10.3 90 8 8
101 110 207 8.0 90 8 9
102 NA 222 8.6 92 8 10
103 NA 137 11.5 86 8 11
104 44 192 11.5 86 8 12
105 28 273 11.5 82 8 13
106 65 157 9.7 80 8 14
107 NA 64 11.5 79 8 15
108 22 71 10.3 77 8 16
109 59 51 6.3 79 8 17
110 23 115 7.4 76 8 18
111 31 244 10.9 78 8 19
112 44 190 10.3 78 8 20
113 21 259 15.5 77 8 21
114 9 36 14.3 72 8 22
115 NA 255 12.6 75 8 23
116 45 212 9.7 79 8 24
117 168 238 3.4 81 8 25
118 73 215 8.0 86 8 26
119 NA 153 5.7 88 8 27
120 76 203 9.7 97 8 28
121 118 225 2.3 94 8 29
122 84 237 6.3 96 8 30
123 85 188 6.3 94 8 31
124 96 167 6.9 91 9 1
125 78 197 5.1 92 9 2
126 73 183 2.8 93 9 3
127 91 189 4.6 93 9 4
128 47 95 7.4 87 9 5
129 32 92 15.5 84 9 6
130 20 252 10.9 80 9 7
131 23 220 10.3 78 9 8
132 21 230 10.9 75 9 9
133 24 259 9.7 73 9 10
134 44 236 14.9 81 9 11
135 21 259 15.5 76 9 12
136 28 238 6.3 77 9 13
137 9 24 10.9 71 9 14
138 13 112 11.5 71 9 15
139 46 237 6.9 78 9 16
140 18 224 13.8 67 9 17
141 13 27 10.3 76 9 18
142 24 238 10.3 68 9 19
143 16 201 8.0 82 9 20
144 13 238 12.6 64 9 21
145 23 14 9.2 71 9 22
146 36 139 10.3 81 9 23
147 7 49 10.3 69 9 24
148 14 20 16.6 63 9 25
149 30 193 6.9 70 9 26
150 NA 145 13.2 77 9 27
151 14 191 14.3 75 9 28
152 18 131 8.0 76 9 29
153 20 223 11.5 68 9 30
Ozone Solar.R Wind Temp
1 41 190 7.4 67
2 36 118 8.0 72
3 12 149 12.6 74
4 18 313 11.5 62
5 NA NA 14.3 56
6 28 NA 14.9 66
7 23 299 8.6 65
8 19 99 13.8 59
9 8 19 20.1 61
10 NA 194 8.6 69
11 7 NA 6.9 74
12 16 256 9.7 69
13 11 290 9.2 66
14 14 274 10.9 68
15 18 65 13.2 58
16 14 334 11.5 64
17 34 307 12.0 66
18 6 78 18.4 57
19 30 322 11.5 68
20 11 44 9.7 62
21 1 8 9.7 59
22 11 320 16.6 73
23 4 25 9.7 61
24 32 92 12.0 61
25 NA 66 16.6 57
26 NA 266 14.9 58
27 NA NA 8.0 57
28 23 13 12.0 67
29 45 252 14.9 81
30 115 223 5.7 79
31 37 279 7.4 76
32 NA 286 8.6 78
33 NA 287 9.7 74
34 NA 242 16.1 67
35 NA 186 9.2 84
36 NA 220 8.6 85
37 NA 264 14.3 79
38 29 127 9.7 82
39 NA 273 6.9 87
40 71 291 13.8 90
41 39 323 11.5 87
42 NA 259 10.9 93
43 NA 250 9.2 92
44 23 148 8.0 82
45 NA 332 13.8 80
46 NA 322 11.5 79
47 21 191 14.9 77
48 37 284 20.7 72
49 20 37 9.2 65
50 12 120 11.5 73
51 13 137 10.3 76
52 NA 150 6.3 77
53 NA 59 1.7 76
54 NA 91 4.6 76
55 NA 250 6.3 76
56 NA 135 8.0 75
57 NA 127 8.0 78
58 NA 47 10.3 73
59 NA 98 11.5 80
60 NA 31 14.9 77
61 NA 138 8.0 83
62 135 269 4.1 84
63 49 248 9.2 85
64 32 236 9.2 81
65 NA 101 10.9 84
66 64 175 4.6 83
67 40 314 10.9 83
68 77 276 5.1 88
69 97 267 6.3 92
70 97 272 5.7 92
71 85 175 7.4 89
72 NA 139 8.6 82
73 10 264 14.3 73
74 27 175 14.9 81
75 NA 291 14.9 91
76 7 48 14.3 80
77 48 260 6.9 81
78 35 274 10.3 82
79 61 285 6.3 84
80 79 187 5.1 87
81 63 220 11.5 85
82 16 7 6.9 74
83 NA 258 9.7 81
84 NA 295 11.5 82
85 80 294 8.6 86
86 108 223 8.0 85
87 20 81 8.6 82
88 52 82 12.0 86
89 82 213 7.4 88
90 50 275 7.4 86
91 64 253 7.4 83
92 59 254 9.2 81
93 39 83 6.9 81
94 9 24 13.8 81
95 16 77 7.4 82
96 78 NA 6.9 86
97 35 NA 7.4 85
98 66 NA 4.6 87
99 122 255 4.0 89
100 89 229 10.3 90
101 110 207 8.0 90
102 NA 222 8.6 92
103 NA 137 11.5 86
104 44 192 11.5 86
105 28 273 11.5 82
106 65 157 9.7 80
107 NA 64 11.5 79
108 22 71 10.3 77
109 59 51 6.3 79
110 23 115 7.4 76
111 31 244 10.9 78
112 44 190 10.3 78
113 21 259 15.5 77
114 9 36 14.3 72
115 NA 255 12.6 75
116 45 212 9.7 79
117 168 238 3.4 81
118 73 215 8.0 86
119 NA 153 5.7 88
120 76 203 9.7 97
121 118 225 2.3 94
122 84 237 6.3 96
123 85 188 6.3 94
124 96 167 6.9 91
125 78 197 5.1 92
126 73 183 2.8 93
127 91 189 4.6 93
128 47 95 7.4 87
129 32 92 15.5 84
130 20 252 10.9 80
131 23 220 10.3 78
132 21 230 10.9 75
133 24 259 9.7 73
134 44 236 14.9 81
135 21 259 15.5 76
136 28 238 6.3 77
137 9 24 10.9 71
138 13 112 11.5 71
139 46 237 6.9 78
140 18 224 13.8 67
141 13 27 10.3 76
142 24 238 10.3 68
143 16 201 8.0 82
144 13 238 12.6 64
145 23 14 9.2 71
146 36 139 10.3 81
147 7 49 10.3 69
148 14 20 16.6 63
149 30 193 6.9 70
150 NA 145 13.2 77
151 14 191 14.3 75
152 18 131 8.0 76
153 20 223 11.5 68
Ozone Solar.R Wind Temp
1 41.00000 190.0000 7.4 67
2 36.00000 118.0000 8.0 72
3 12.00000 149.0000 12.6 74
4 18.00000 313.0000 11.5 62
5 20.47224 178.3231 14.3 56
6 28.00000 193.3949 14.9 66
7 23.00000 299.0000 8.6 65
8 19.00000 99.0000 13.8 59
9 8.00000 19.0000 20.1 61
10 26.61911 194.0000 8.6 69
11 7.00000 158.5913 6.9 74
12 16.00000 256.0000 9.7 69
13 11.00000 290.0000 9.2 66
14 14.00000 274.0000 10.9 68
15 18.00000 65.0000 13.2 58
16 14.00000 334.0000 11.5 64
17 34.00000 307.0000 12.0 66
18 6.00000 78.0000 18.4 57
19 30.00000 322.0000 11.5 68
20 11.00000 44.0000 9.7 62
21 1.00000 8.0000 9.7 59
22 11.00000 320.0000 16.6 73
23 4.00000 25.0000 9.7 61
24 32.00000 92.0000 12.0 61
25 15.42723 66.0000 16.6 57
26 30.66671 266.0000 14.9 58
27 21.32364 142.1746 8.0 57
28 23.00000 13.0000 12.0 67
29 45.00000 252.0000 14.9 81
30 115.00000 223.0000 5.7 79
31 37.00000 279.0000 7.4 76
32 40.18203 286.0000 8.6 78
33 31.27023 287.0000 9.7 74
34 39.31116 242.0000 16.1 67
35 60.98784 186.0000 9.2 84
36 60.05528 220.0000 8.6 85
37 48.35681 264.0000 14.3 79
38 29.00000 127.0000 9.7 82
39 83.45562 273.0000 6.9 87
40 71.00000 291.0000 13.8 90
41 39.00000 323.0000 11.5 87
42 69.39093 259.0000 10.9 93
43 69.71556 250.0000 9.2 92
44 23.00000 148.0000 8.0 82
45 35.76567 332.0000 13.8 80
46 35.94283 322.0000 11.5 79
47 21.00000 191.0000 14.9 77
48 37.00000 284.0000 20.7 72
49 20.00000 37.0000 9.2 65
50 12.00000 120.0000 11.5 73
51 13.00000 137.0000 10.3 76
52 37.53622 150.0000 6.3 77
53 34.19981 59.0000 1.7 76
54 26.77555 91.0000 4.6 76
55 37.54133 250.0000 6.3 76
56 26.60216 135.0000 8.0 75
57 30.62680 127.0000 8.0 78
58 18.99009 47.0000 10.3 73
59 27.43092 98.0000 11.5 80
60 23.79767 31.0000 14.9 77
61 36.90639 138.0000 8.0 83
62 135.00000 269.0000 4.1 84
63 49.00000 248.0000 9.2 85
64 32.00000 236.0000 9.2 81
65 38.95093 101.0000 10.9 84
66 64.00000 175.0000 4.6 83
67 40.00000 314.0000 10.9 83
68 77.00000 276.0000 5.1 88
69 97.00000 267.0000 6.3 92
70 97.00000 272.0000 5.7 92
71 85.00000 175.0000 7.4 89
72 37.49463 139.0000 8.6 82
73 10.00000 264.0000 14.3 73
74 27.00000 175.0000 14.9 81
75 66.05723 291.0000 14.9 91
76 7.00000 48.0000 14.3 80
77 48.00000 260.0000 6.9 81
78 35.00000 274.0000 10.3 82
79 61.00000 285.0000 6.3 84
80 79.00000 187.0000 5.1 87
81 63.00000 220.0000 11.5 85
82 16.00000 7.0000 6.9 74
83 44.40482 258.0000 9.7 81
84 45.08041 295.0000 11.5 82
85 80.00000 294.0000 8.6 86
86 108.00000 223.0000 8.0 85
87 20.00000 81.0000 8.6 82
88 52.00000 82.0000 12.0 86
89 82.00000 213.0000 7.4 88
90 50.00000 275.0000 7.4 86
91 64.00000 253.0000 7.4 83
92 59.00000 254.0000 9.2 81
93 39.00000 83.0000 6.9 81
94 9.00000 24.0000 13.8 81
95 16.00000 77.0000 7.4 82
96 78.00000 218.6714 6.9 86
97 35.00000 189.7878 7.4 85
98 66.00000 202.1405 4.6 87
99 122.00000 255.0000 4.0 89
100 89.00000 229.0000 10.3 90
101 110.00000 207.0000 8.0 90
102 77.91793 222.0000 8.6 92
103 32.70190 137.0000 11.5 86
104 44.00000 192.0000 11.5 86
105 28.00000 273.0000 11.5 82
106 65.00000 157.0000 9.7 80
107 25.13334 64.0000 11.5 79
108 22.00000 71.0000 10.3 77
109 59.00000 51.0000 6.3 79
110 23.00000 115.0000 7.4 76
111 31.00000 244.0000 10.9 78
112 44.00000 190.0000 10.3 78
113 21.00000 259.0000 15.5 77
114 9.00000 36.0000 14.3 72
115 31.08529 255.0000 12.6 75
116 45.00000 212.0000 9.7 79
117 168.00000 238.0000 3.4 81
118 73.00000 215.0000 8.0 86
119 69.07124 153.0000 5.7 88
120 76.00000 203.0000 9.7 97
121 118.00000 225.0000 2.3 94
122 84.00000 237.0000 6.3 96
123 85.00000 188.0000 6.3 94
124 96.00000 167.0000 6.9 91
125 78.00000 197.0000 5.1 92
126 73.00000 183.0000 2.8 93
127 91.00000 189.0000 4.6 93
128 47.00000 95.0000 7.4 87
129 32.00000 92.0000 15.5 84
130 20.00000 252.0000 10.9 80
131 23.00000 220.0000 10.3 78
132 21.00000 230.0000 10.9 75
133 24.00000 259.0000 9.7 73
134 44.00000 236.0000 14.9 81
135 21.00000 259.0000 15.5 76
136 28.00000 238.0000 6.3 77
137 9.00000 24.0000 10.9 71
138 13.00000 112.0000 11.5 71
139 46.00000 237.0000 6.9 78
140 18.00000 224.0000 13.8 67
141 13.00000 27.0000 10.3 76
142 24.00000 238.0000 10.3 68
143 16.00000 201.0000 8.0 82
144 13.00000 238.0000 12.6 64
145 23.00000 14.0000 9.2 71
146 36.00000 139.0000 10.3 81
147 7.00000 49.0000 10.3 69
148 14.00000 20.0000 16.6 63
149 30.00000 193.0000 6.9 70
150 26.27180 145.0000 13.2 77
151 14.00000 191.0000 14.3 75
152 18.00000 131.0000 8.0 76
153 20.00000 223.0000 11.5 68
V1 V2 V3 V4 V5 V6
1 -0.27294622 0.5649844 1.6517405 -0.45378443 0.359638060 0.5243731
2 1.61893357 -0.9300666 1.2307204 1.51345863 0.003216759 -1.8996916
3 -0.09156827 -1.3119584 -0.4464256 1.68003787 0.687738071 1.7477167
4 -1.75697208 1.2015881 -0.9164589 -0.24929173 0.020322163 -1.0069711
5 0.18531297 -1.8997072 -0.9993217 -1.64912872 -2.296420684 0.8252367
6 1.57761882 0.9472044 0.5019121 0.16468517 0.077490188 2.3317513
7 1.33040094 0.4765310 1.6940212 0.03539862 -0.439885071 0.7853790
8 -0.30232217 0.4667190 -0.4757372 0.21561522 0.727385605 -0.3569848
9 -0.10808600 -0.9109851 -0.6241070 -0.13797592 -1.433679131 -1.1014573
10 -1.32985277 -1.5091918 -0.2279206 -1.40811383 -0.110213013 0.3796918
V7 V8 V9 V10 V11 V12
1 0.97018018 0.2361315 1.9589677 0.3142968 0.74531168 1.1822632
2 0.36905840 -0.4627000 0.6130300 0.2237436 0.57177136 0.2464576
3 -0.03107033 0.1351887 0.0629055 -0.7230052 -0.44377809 1.8061065
4 -1.92010585 -0.7430641 -0.3066002 -1.0654466 -1.08542269 -0.1158405
5 -1.09228974 -0.8066865 -0.8859033 -0.2866407 0.65348405 0.7160437
6 1.47340510 2.3012685 0.4189516 0.4735910 -0.05045945 0.5512949
7 -0.14968739 0.8637884 -0.9664251 -0.9807089 0.90329170 0.6253347
8 0.65834024 0.5753060 -0.4755417 -0.6232189 0.90526977 0.0181135
9 1.19217328 -2.0427906 -1.3760800 0.3504487 0.42653643 0.2338093
10 -0.51216355 -0.1326851 -1.0383768 -0.1415861 -1.51708483 -0.5911722
V13 V14 V15 V16 V17 V18
1 0.1176203 -0.3835233 1.8023094 -1.36834561 1.10236499 0.4495366
2 0.8329853 -0.2207156 -0.6022181 -0.46200466 -1.33754621 -0.6997470
3 -0.2299583 1.1937213 -0.6058899 -1.31720639 -0.04461599 0.9304723
4 -0.4815356 -0.0349740 -0.4134190 -1.07445030 0.63789061 0.2260202
5 -1.5029909 -1.6384298 -1.7578831 -1.55682426 0.93153809 -0.5802509
6 1.1108469 1.7979764 1.1823493 1.09854680 1.69883875 1.6339145
7 1.0436978 -1.1928054 1.3082314 1.29887088 1.16840837 0.9781224
8 0.7087555 -0.5047357 1.1387356 -0.89102083 0.74217847 0.5089179
9 -0.3726949 -0.1658626 -1.1174908 -0.92191686 -0.62003392 -1.1407242
10 -1.5737854 -0.7379942 0.3107172 0.01095196 -0.87466556 -1.1424905
V19 V20
1 1.2063345 0.7566320
2 -0.4336555 -0.1915558
3 1.0880000 1.5003867
4 0.4434772 -1.4640166
5 -1.2388603 -1.0518300
6 0.8279166 0.1385917
7 2.0034276 -0.7571446
8 0.3569348 0.6664090
9 0.2193313 -1.6748832
10 -1.1953611 -1.1098792
V1 V2 V3 V4 V5 V6
1 -0.27294622 0.5649844 1.6517405 NA 0.359638060 0.5243731
2 1.61893357 -0.9300666 1.2307204 1.51345863 0.003216759 -1.8996916
3 -0.09156827 -1.3119584 -0.4464256 NA 0.687738071 1.7477167
4 -1.75697208 1.2015881 -0.9164589 -0.24929173 0.020322163 -1.0069711
5 0.18531297 -1.8997072 -0.9993217 -1.64912872 -2.296420684 0.8252367
6 1.57761882 0.9472044 0.5019121 0.16468517 0.077490188 2.3317513
7 NA 0.4765310 1.6940212 0.03539862 -0.439885071 0.7853790
8 -0.30232217 0.4667190 -0.4757372 0.21561522 0.727385605 -0.3569848
9 -0.10808600 -0.9109851 -0.6241070 -0.13797592 -1.433679131 -1.1014573
10 -1.32985277 -1.5091918 -0.2279206 -1.40811383 -0.110213013 0.3796918
V7 V8 V9 V10 V11 V12
1 0.97018018 0.2361315 1.9589677 0.3142968 0.74531168 1.1822632
2 0.36905840 -0.4627000 0.6130300 0.2237436 0.57177136 0.2464576
3 -0.03107033 0.1351887 0.0629055 -0.7230052 -0.44377809 1.8061065
4 -1.92010585 -0.7430641 -0.3066002 -1.0654466 -1.08542269 NA
5 -1.09228974 -0.8066865 -0.8859033 -0.2866407 0.65348405 0.7160437
6 NA 2.3012685 0.4189516 0.4735910 -0.05045945 0.5512949
7 -0.14968739 0.8637884 -0.9664251 -0.9807089 0.90329170 0.6253347
8 0.65834024 0.5753060 -0.4755417 -0.6232189 0.90526977 0.0181135
9 1.19217328 -2.0427906 -1.3760800 0.3504487 0.42653643 0.2338093
10 -0.51216355 -0.1326851 -1.0383768 -0.1415861 -1.51708483 -0.5911722
V13 V14 V15 V16 V17 V18
1 0.1176203 -0.3835233 1.8023094 -1.36834561 1.1023650 0.4495366
2 0.8329853 -0.2207156 -0.6022181 -0.46200466 -1.3375462 -0.6997470
3 -0.2299583 1.1937213 -0.6058899 -1.31720639 NA 0.9304723
4 NA -0.0349740 -0.4134190 -1.07445030 0.6378906 0.2260202
5 -1.5029909 -1.6384298 -1.7578831 NA 0.9315381 -0.5802509
6 1.1108469 1.7979764 1.1823493 1.09854680 1.6988387 1.6339145
7 1.0436978 -1.1928054 1.3082314 1.29887088 1.1684084 0.9781224
8 0.7087555 -0.5047357 1.1387356 -0.89102083 0.7421785 0.5089179
9 -0.3726949 -0.1658626 -1.1174908 -0.92191686 -0.6200339 -1.1407242
10 -1.5737854 -0.7379942 0.3107172 0.01095196 -0.8746656 -1.1424905
V19 V20
1 1.2063345 0.7566320
2 -0.4336555 -0.1915558
3 1.0880000 1.5003867
4 0.4434772 -1.4640166
5 -1.2388603 -1.0518300
6 0.8279166 NA
7 2.0034276 -0.7571446
8 0.3569348 0.6664090
9 0.2193313 NA
10 -1.1953611 -1.1098792
1 2 3 4 5 6 7 8 9 10
1 0 2 2 1 2 1 0 1 0
V1 V2 V3 V4 V5 V6
1 -0.27294622 0.5649844 1.6517405 1.51345863 0.359638060 0.5243731
2 1.61893357 -0.9300666 1.2307204 1.51345863 0.003216759 -1.8996916
3 -0.09156827 -1.3119584 -0.4464256 0.21561522 0.687738071 1.7477167
4 -1.75697208 1.2015881 -0.9164589 -0.24929173 0.020322163 -1.0069711
5 0.18531297 -1.8997072 -0.9993217 -1.64912872 -2.296420684 0.8252367
6 1.57761882 0.9472044 0.5019121 0.16468517 0.077490188 2.3317513
7 -0.30232217 0.4765310 1.6940212 0.03539862 -0.439885071 0.7853790
8 -0.30232217 0.4667190 -0.4757372 0.21561522 0.727385605 -0.3569848
9 -0.10808600 -0.9109851 -0.6241070 -0.13797592 -1.433679131 -1.1014573
10 -1.32985277 -1.5091918 -0.2279206 -1.40811383 -0.110213013 0.3796918
V7 V8 V9 V10 V11 V12
1 0.97018018 0.2361315 1.9589677 0.3142968 0.74531168 1.1822632
2 0.36905840 -0.4627000 0.6130300 0.2237436 0.57177136 0.2464576
3 -0.03107033 0.1351887 0.0629055 -0.7230052 -0.44377809 1.8061065
4 -1.92010585 -0.7430641 -0.3066002 -1.0654466 -1.08542269 -0.5911722
5 -1.09228974 -0.8066865 -0.8859033 -0.2866407 0.65348405 0.7160437
6 0.97018018 2.3012685 0.4189516 0.4735910 -0.05045945 0.5512949
7 -0.14968739 0.8637884 -0.9664251 -0.9807089 0.90329170 0.6253347
8 0.65834024 0.5753060 -0.4755417 -0.6232189 0.90526977 0.0181135
9 1.19217328 -2.0427906 -1.3760800 0.3504487 0.42653643 0.2338093
10 -0.51216355 -0.1326851 -1.0383768 -0.1415861 -1.51708483 -0.5911722
V13 V14 V15 V16 V17 V18
1 0.1176203 -0.3835233 1.8023094 -1.36834561 1.1023650 0.4495366
2 0.8329853 -0.2207156 -0.6022181 -0.46200466 -1.3375462 -0.6997470
3 -0.2299583 1.1937213 -0.6058899 -1.31720639 0.7421785 0.9304723
4 -1.5737854 -0.0349740 -0.4134190 -1.07445030 0.6378906 0.2260202
5 -1.5029909 -1.6384298 -1.7578831 0.01095196 0.9315381 -0.5802509
6 1.1108469 1.7979764 1.1823493 1.09854680 1.6988387 1.6339145
7 1.0436978 -1.1928054 1.3082314 1.29887088 1.1684084 0.9781224
8 0.7087555 -0.5047357 1.1387356 -0.89102083 0.7421785 0.5089179
9 -0.3726949 -0.1658626 -1.1174908 -0.92191686 -0.6200339 -1.1407242
10 -1.5737854 -0.7379942 0.3107172 0.01095196 -0.8746656 -1.1424905
V19 V20
1 1.2063345 0.7566320
2 -0.4336555 -0.1915558
3 1.0880000 1.5003867
4 0.4434772 -1.4640166
5 -1.2388603 -1.0518300
6 0.8279166 0.7566320
7 2.0034276 -0.7571446
8 0.3569348 0.6664090
9 0.2193313 0.6664090
10 -1.1953611 -1.1098792
[1] 1.890908
GenForImp documentation built on May 2, 2019, 2:15 p.m.
R Package Documentation
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Description Usage Arguments Details Value Author(s) References See Also Examples
Description
This function imputes quantitative missing data by alternating Nearest Neighbour Imputation (NNI) method and Principal Component Analysis (PCA) in a forward and sequential step-by-step process that starts from the complete part of data.
Usage
1 2 |
Arguments
mat |
a quantitative data matrix with missing entries. |
stand |
a logical value indicating if variables should be standardized ( |
cor |
a logical value. If |
r |
a positive value (equal or greater than 1) indicating
the order of the weighted Minkowski distance to be computed for selecting donors. In particular, |
probs |
vector of probabilities with values in [0, 1] for computing quantiles of Minkowski distances in selection of donors. Default option: |
q |
string of the form |
tol |
tolerance factor introduced to prevent numerical problems occuring when distances of complete units are equal to the choosen quantile |
Details
The ForImp.PCA procedure exploits the PCA method for setting-up so-called “Pseudo-Principal Components” (PPCs). These are artificial, missing-data free variables computed for all the units (i.e. both complete and incomplete) by using common observed variables without missing values. PPCs synthesize the main relevant information in the complete part of data and transfer them to incomplete units in a way functional to the subsequent application of the NNI method. NNI is then applied to PPCs (with a weighted Minkowski distance of order r, r ≥ 1) in order to select donors for incomplete units. All is performed in four stages:
Stage 0: data preparation;
Stage 1: running PCA and computing PPC scores;
Stage 2: application of the NNI method;
Stage 3: imputation.
Steps 1 to 3 are iteratively repeated until the starting data matrix is completely imputed.
ForImp.PCA does not require a number n of units greater than the number p of variables at every step of the procedure. It can work even in the presence of a starting data matrix with n < p. This further variant is indicated as ForImp with PCO, i.e. Forward Imputation with the Principal Coordinates Analysis.
For further details, see the references below.
Value
The imputed data matrix.
Author(s)
Nadia Solaro, Alessandro Barbiero, Giancarlo Manzi, Pier Alda Ferrari
References
Gower, J.C. (2005). Principal coordinates analysis. In: Armitage, P., Colton, T. (eds), Encyclopedia of biostatistics. John Wiley & Sons, Ltd., New York
Solaro, N., Barbiero, A., Manzi. G., Ferrari, P.A. (2014). Algorithmic-type imputation techniques with different data structures: Alternative approaches in comparison. In: Vicari, D., Okada, A., Ragozini, G., Weihs, C. (eds), Analysis and modeling of complex data in behavioural and social sciences, Studies in Classification, Data Analysis, and Knowledge Organization. Springer International Publishing, Cham (CH): 253-261
Solaro, N., Barbiero, A., Manzi, G., Ferrari, P.A. (2015). A sequential distance-based approach for imputing missing data: The Forward Imputation. Under review
See Also
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 | # EXAMPLE with multivariate skew-normal data (MSN)
# require('sn')
# number of variables
p <- 5
# association matrix
omega <- 0.8
Omega <- matrix(omega, p, p)
diag(Omega) <- 1
Omega
# skewness parameter
alpha <- 4
alpha <- rep(alpha,p)
alpha
# number of units
n <- 500
# percentage of missing values
percmiss <- 0.2
nummiss <- n*p*percmiss
nummiss
## computation of output parameters
## covariance matrix and univariate means and skewnesses
param <- list(xi=rep(0,p), Omega=Omega, alpha=alpha, nu=Inf)
cp <- dp2cp(param, "SN")
cp
# correlation matrix
rho <- cov2cor(cp$var.cov)
rho
# generation of a complete matrix
set.seed(1)
x0 <- rmsn(n, Omega=Omega, alpha=alpha)
x0
# generating a matrix with missing data
x <- missing.gen(x0, nummiss)
# imputing missing values
xForImpPCA <- ForImp.PCA(x)
xForImpPCA
# computing the Relative Mean Square Error
error <- sum(apply((x0-xForImpPCA)^2/diag(var(x0)),2,sum)) / n
error
# EXAMPLE with real data
data(airquality)
m0 <- airquality
m0
# selecting the first 4 columns, with quantitative data
m <- m0[, 1:4]
m
# imputation
mi <- ForImp.PCA(m)
mi
# plot of imputed values for variable "Ozone"
ozone.miss.ind <- which(is.na(m)[,1])
plot(mi[ozone.miss.ind,1], axes=FALSE, pch=19, ylab="imputed values of Ozone",
xlab="observation index")
axis(2)
axis(1, at=1:length(ozone.miss.ind), labels=ozone.miss.ind, las=2)
box()
abline(v=1:length(ozone.miss.ind), lty=3, col="grey")
# EXAMPLE with n < p: ForImp with PCO
# require('mvtnorm')
p <- 20
n <- 10
sigma <- matrix(0.4, p, p)
diag(sigma) <- 1
# complete matrix
set.seed(2)
xtrue <- rmvnorm(n=n, mean=rep(0, p), sigma=sigma)
rownames(xtrue) <- 1:n
colnames(xtrue) <- paste("V", 1:p, sep="")
xtrue
# matrix with 10 missing values
xmiss <- missing.gen(xtrue, 10)
xmiss
# number of missing values per unit
apply(is.na(xmiss),1,sum)
# imputed matrix
ximp <- ForImp.PCA(xmiss)
ximp
# computing the Relative Mean Square Error
error <- sum(apply((xtrue-ximp)^2/diag(var(xtrue)),2,sum)) / n
error
|
Example output
Loading required package: mvtnorm
Loading required package: sn
Loading required package: stats4
Attaching package: 'sn'
The following object is masked from 'package:stats':
sd
[,1] [,2] [,3] [,4] [,5]
[1,] 1.0 0.8 0.8 0.8 0.8
[2,] 0.8 1.0 0.8 0.8 0.8
[3,] 0.8 0.8 1.0 0.8 0.8
[4,] 0.8 0.8 0.8 1.0 0.8
[5,] 0.8 0.8 0.8 0.8 1.0
[1] 4 4 4 4 4
[1] 500
$mean
[1] 0.7301875 0.7301875 0.7301875 0.7301875 0.7301875
$var.cov
[,1] [,2] [,3] [,4] [,5]
[1,] 0.4668262 0.2668262 0.2668262 0.2668262 0.2668262
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[4,] 0.2668262 0.2668262 0.2668262 0.4668262 0.2668262
[5,] 0.2668262 0.2668262 0.2668262 0.2668262 0.4668262
$gamma1
[1] 0.5238823 0.5238823 0.5238823 0.5238823 0.5238823
[,1] [,2] [,3] [,4] [,5]
[1,] 1.000000 0.571575 0.571575 0.571575 0.571575
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[4,] 0.571575 0.571575 0.571575 1.000000 0.571575
[5,] 0.571575 0.571575 0.571575 0.571575 1.000000
[,1] [,2] [,3] [,4] [,5]
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attr(,"family")
[1] "SN"
attr(,"parameters")
attr(,"parameters")$xi
[1] 0 0 0 0 0
attr(,"parameters")$Omega
[,1] [,2] [,3] [,4] [,5]
[1,] 1.0 0.8 0.8 0.8 0.8
[2,] 0.8 1.0 0.8 0.8 0.8
[3,] 0.8 0.8 1.0 0.8 0.8
[4,] 0.8 0.8 0.8 1.0 0.8
[5,] 0.8 0.8 0.8 0.8 1.0
attr(,"parameters")$alpha
[1] 4 4 4 4 4
attr(,"parameters")$tau
[1] 0
x.1 x.2 x.3 x.4 x.5
1 1.3997702246 1.740905120 2.126122910 1.806735795 1.105844380
2 0.6945391557 0.486987263 1.054385270 0.147232551 0.265429499
3 0.0963506889 0.046859206 0.324425715 1.091941861 0.924948103
4 1.5833087432 0.796238948 0.869323764 0.438609958 1.111387571
5 0.4171984786 0.904965960 0.425291902 0.570364009 0.238987974
6 0.7307073314 0.725446609 0.328104738 0.717063392 1.418594511
7 1.0493284286 0.636352733 0.999543004 1.507674146 0.991960068
8 1.1448306975 0.680608813 0.092404825 1.563944164 0.484800455
9 1.4622091863 1.384501470 1.375850324 1.329629732 1.597098228
10 1.2048923363 1.218718595 1.279922379 1.535329959 1.124918601
11 1.1481853408 -0.593608627 0.493363578 0.174160918 1.169243172
12 0.2627416734 -0.297756929 0.098524127 0.125394215 0.281881536
13 0.4088261404 0.337686146 0.224069250 0.682056785 1.491290997
14 -0.8897334430 0.227716368 -0.339363218 0.505376462 0.313932281
15 1.3570517897 0.931834409 1.007377643 0.785415459 0.978074917
16 0.8544953271 0.967586085 0.784236396 0.798043074 1.225607456
17 1.5571990226 1.128904622 1.393668911 1.646169646 1.065821927
18 0.6489690193 0.327420382 -0.362463720 0.672190044 -0.086148301
19 1.0333958224 0.831548181 0.607638404 1.142151245 1.265300573
20 0.3998247806 0.571939627 0.726870943 0.265196677 -0.513671886
21 0.4856701500 -0.342528717 -0.088703174 0.318120679 0.202561311
22 0.5067897519 0.449178573 0.792811727 0.500505337 0.100877853
23 0.7765367291 0.392487414 1.137392990 1.432482983 0.021438053
24 -0.6903784017 0.378222840 0.557717218 -0.234170759 0.521574309
25 0.3096630891 -0.016584249 0.118054202 -0.308676393 0.694769944
26 1.5572532510 1.471452187 1.556592468 1.479667651 1.981529107
27 0.7934043980 0.814134136 0.509982131 1.080624317 0.931095378
28 0.5956033763 1.788218537 0.832453788 0.985186703 0.746052865
29 0.3471820345 0.851673588 0.487966344 0.557952417 0.598690264
30 0.5356690695 0.525475024 0.497669907 1.018914501 0.050772781
31 0.7260920983 -0.415665024 -0.007297023 0.365567377 0.358388418
32 0.5883068508 1.097897468 0.352635176 0.934708414 0.179883910
33 0.2319538768 0.070674836 -0.265121541 -0.847686270 0.832175071
34 0.1321912885 -0.018307099 0.432558974 0.630414722 -0.051235932
35 1.0487730796 1.509363173 1.000249452 0.974804386 0.560209539
36 0.0269107026 0.638840032 0.493153299 0.087482661 0.651376020
37 0.6211647108 0.935775388 0.694111173 0.742373595 1.324025177
38 1.1627481017 1.694896876 1.366816770 1.465459653 0.064460958
39 2.7721817242 2.462606433 2.775098467 1.903799231 1.757605529
40 0.7983977899 0.648291848 0.453502396 0.807814587 0.426485798
41 -0.0448453630 -0.127328026 0.262061066 0.209182687 -0.206389938
42 0.8559723794 1.120968371 1.043970282 0.841740123 0.771219357
43 0.7443870987 0.599985044 -0.027722497 0.396297929 0.424235006
44 0.8173781838 0.548487747 0.192326181 0.207708868 0.630471830
45 0.1668676881 1.025713353 1.071672701 -0.677318205 0.327030686
46 0.8193987355 1.144602152 1.507892830 1.097872111 1.466242098
47 0.1496825249 -0.305873600 0.328864518 0.023361975 0.251777211
48 1.7620200106 1.591461148 1.232348192 1.123788161 1.820729107
49 0.8636662906 0.939725035 1.360165917 1.313912862 1.134362694
50 0.4101076728 0.126684978 -0.016375026 0.298393172 0.233936135
51 0.7621136819 0.759510348 0.521813162 0.384218112 0.948390355
52 -0.0333803441 0.182076814 0.380336361 0.277669554 -0.106827856
53 0.8199176619 0.212507441 0.404105959 1.382063513 0.504051826
54 0.8233698576 0.445329178 1.545139148 0.367889225 0.772095308
55 0.3616123128 1.238832757 0.748588647 0.590827959 -0.089418633
56 0.9216801580 -0.352515495 -0.067369198 0.231206239 -0.113681520
57 0.1122608235 -0.620081478 1.022697369 0.957583908 -0.528631959
58 0.0061941965 0.842819104 0.193955877 0.434682639 0.186765856
59 1.2422034905 0.762913912 1.658755985 0.409471993 1.131579336
60 0.0442327068 -0.089138795 0.023821715 0.447778336 0.405394831
61 0.3949430320 -0.654174616 -0.902998902 0.418304960 0.568907575
62 0.2686845246 0.279305563 0.801180715 -0.019078581 0.065612056
63 0.4417461372 0.413105040 0.440606672 -0.221946328 -0.207503323
64 0.9811509803 0.736836709 0.746702898 1.385349070 0.897876390
65 0.5914096642 0.722215277 0.850367511 0.995708827 1.240568689
66 0.5898124013 0.926265023 0.610096508 0.012638943 1.203919673
67 0.4464647666 1.244917729 1.270479185 2.096825926 0.638677828
68 2.4088506883 1.618998936 1.014389418 1.550608292 1.995003672
69 0.2941132767 -0.194556145 -0.180184294 0.326396516 -0.050923979
70 1.2437658548 0.518990624 0.121039409 0.685908084 0.285815188
71 0.3708355214 0.374369520 0.494927586 0.146332697 0.369656055
72 0.0736765837 0.759007933 0.019176391 -0.270291995 0.910808867
73 0.3286740357 0.294844308 -0.115416065 0.071660090 -0.236141063
74 -0.1996279844 0.080423324 0.414249086 0.436104623 0.310248426
75 0.0733043831 1.028060597 0.561443051 0.013221449 0.402626256
76 0.4576312546 0.700058976 -0.161745099 0.171730148 0.842873759
77 0.4917420262 0.937450012 0.043692170 1.355425039 0.546198901
78 0.7447294802 0.628874508 -0.184665144 0.960592272 0.876869120
79 0.5678512175 0.974949209 0.969864592 0.587239796 0.821510975
80 0.2493369910 0.084925071 0.656449207 0.813345111 0.911124678
81 1.1161737417 0.932789289 1.849207462 0.981289074 1.731462696
82 0.0608606010 0.193696586 0.055499439 0.526905648 0.212588974
83 0.8316726325 1.315056382 0.654093712 -0.201199290 1.064270953
84 -0.1772016552 1.236814571 0.465113687 0.005767215 0.369219332
85 2.2920500300 1.701409313 1.948697673 1.712982823 1.875522731
86 0.1485071042 0.404808527 0.242668751 0.369315974 -0.530032577
87 1.4646475076 0.880367718 1.426411037 0.166112479 1.170301524
88 -0.0536103020 0.281249772 0.273409553 0.310450055 0.244193409
89 0.5552235556 0.519204931 0.576817955 0.335872846 0.078827296
90 0.3456803662 -0.085334422 0.364333826 0.264399391 0.139034937
91 -0.2064820386 -0.052346118 0.287864038 0.433494277 -0.249441842
92 1.3689990377 0.716827944 1.051680126 0.933349696 0.932523641
93 1.3735519011 1.955412070 1.118317520 1.350134853 0.988626345
94 0.4882287511 0.384266837 0.259450869 0.672100751 -0.105381817
95 2.0293888176 0.412800795 1.444060559 1.007899878 1.821624220
96 1.4473929775 1.471632294 0.297601518 1.060250553 1.539135498
97 0.8542947551 0.678520481 1.334510926 0.980762766 0.400868504
98 1.0439635421 1.157340367 1.777761136 0.800969943 1.097845717
99 0.5373924396 1.499825859 1.081441559 0.273825687 1.834708342
100 -0.1804339883 -0.244852264 0.287185568 0.088882746 -0.191495262
101 0.6104811867 0.768302077 1.048923778 1.035955289 0.818302191
102 0.7379861401 1.306368201 0.116440008 0.726265818 0.708531083
103 0.1237864538 0.561328445 0.659574715 -0.432098276 0.573397741
104 0.0937199304 0.227971092 0.319106879 0.107187956 0.236875526
105 1.1572386246 1.428405225 1.335022214 1.183110016 1.554578146
106 1.5433365072 1.590547067 2.321669632 1.601007052 1.725644283
107 0.4420395422 -0.500559087 -0.109340191 0.358810803 0.801914335
108 0.7358177083 0.196618667 -0.095575314 0.719018172 0.256363930
109 1.7525829237 2.347079458 1.308199909 1.509087731 1.052483472
110 0.3309263402 0.475652414 0.667894548 0.497865977 -0.046760094
111 -0.1284516001 0.251439214 0.195245067 0.412608305 -0.167434136
112 0.2803784739 0.592773865 0.291821222 -0.463152269 1.262729708
113 1.4778671461 0.616154446 0.937909506 0.933874552 0.869956419
114 0.3666192583 0.506696987 0.518110538 0.311237493 1.232927404
115 0.9327784412 0.712838305 0.195845004 1.068530241 0.599000116
116 0.3372931429 0.477344988 0.835367052 0.917834072 0.079238414
117 0.6718968374 0.626640613 0.483407447 0.588322139 1.263182640
118 0.9785252378 1.063981635 1.153722741 1.129519078 0.970523972
119 0.4254967260 0.083031779 0.170266887 0.197157346 0.095879188
120 1.4422484926 1.433234024 0.942811786 2.359565313 1.433088298
121 0.2564058123 0.407220985 0.500270193 0.587636400 0.369424830
122 1.3742231655 1.163535446 0.753632921 1.039044944 0.871800166
123 1.8253730986 2.327154659 1.609789589 1.347683518 2.205922789
124 0.0438786122 0.824675844 -0.282110464 -0.164002117 0.461385497
125 0.2030135619 0.523196806 0.637543229 0.449371287 0.286147251
126 1.2756801054 0.838944329 0.867893662 0.716376747 0.935064731
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128 0.3364503404 0.258452463 0.549070564 0.667295664 1.012720822
129 0.2129725498 0.425752703 0.336584299 0.020619835 0.872863828
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131 0.2221278368 -0.444498818 0.758670099 -0.147241501 0.478970328
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411 0.7809429737 1.010931938 1.367981501 0.891134818 0.879614334
412 1.9086173673 1.403335153 1.685292572 1.499973973 1.548409561
413 1.9197716984 0.785593613 1.140856545 1.326169866 1.543571812
414 0.0872480022 0.530217368 0.702879339 -0.148937750 0.665879256
415 2.2177118959 3.165500531 2.129520056 2.406724427 1.470814677
416 0.8276579974 0.313678454 0.601124349 0.017892828 1.554676801
417 1.1525635739 0.905725684 1.034924086 0.982576839 1.493830920
418 0.5517635433 0.946712680 0.588591894 0.438469146 0.369320818
419 0.1911741196 0.158307255 0.623478045 -0.393729323 0.370166003
420 0.6321223873 0.576737195 1.245027180 0.737155550 0.358376832
421 1.0149892091 0.509564073 0.582511472 0.349894422 0.624780111
422 0.5763820055 1.839301660 0.778873668 0.608870178 0.703524479
423 0.6699065309 0.839038331 1.364369855 0.405989451 -0.054326428
424 0.5165206224 -0.052285794 1.290252615 0.657239098 0.058939492
425 0.6179220867 1.099570783 0.624929639 0.633975007 0.546218631
426 1.2382446106 0.075650767 0.464914121 0.620437551 0.105569331
427 -0.4517096344 -0.034360341 0.090037763 -0.024565980 0.871784432
428 -0.1532259748 0.171677233 0.328049076 0.235296410 -0.223043417
429 0.0788832234 0.763302216 0.079342586 0.346397178 0.476940537
430 1.2393608495 1.268033743 1.195552147 0.634831558 1.543526359
431 -0.2893547342 1.038733810 0.417678840 1.050764894 0.332237042
432 0.4740520223 0.392763472 0.462441281 -0.037307347 -0.298559796
433 1.0778025969 1.770401158 2.677195942 1.826325826 1.192468301
434 0.7281908433 0.810244174 1.384011706 0.800686433 0.761729554
435 0.6858011849 0.703760609 0.255215145 -0.633044821 0.500048781
436 0.7679983965 1.717099275 0.624999061 0.660103553 0.663652049
437 0.7789961486 0.359325006 0.348850970 0.567309859 -0.314310202
438 0.5862629371 0.067599716 0.728722435 1.631352938 0.108618293
439 0.8515743861 1.028751284 1.731100495 1.295264353 1.021129158
440 1.2548726772 0.567646719 1.633566005 1.149898947 1.012845297
441 0.7174907920 0.916405371 0.997641566 0.703785179 0.883232905
442 1.0010134828 0.386715237 0.496782963 0.535667381 0.584015186
443 1.5406531575 1.889544523 1.324290594 1.339994842 1.229747204
444 0.8675957798 0.536711261 0.587806427 0.486121956 0.531409666
445 1.0447698861 0.576658412 0.564991154 0.847925590 1.770143629
446 -1.0466793862 0.764736812 -0.133037292 0.427461730 0.894892830
447 -0.1062427750 1.100388752 0.751966615 0.550767480 0.570713672
448 1.1431429304 2.236790212 0.042426819 1.552504040 1.758753133
449 1.0672963694 0.756186826 1.095294795 0.958391001 1.014033367
450 1.7518669621 2.135957126 1.602162493 1.956823382 1.639250350
451 0.2816609918 0.350999803 -0.453958746 -0.317809034 0.404032095
452 -0.1815496123 0.495494997 0.155736646 0.041199272 0.489211905
453 2.0506704709 2.630069675 2.382815483 1.846620963 1.614818645
454 0.4896877989 1.345604853 1.094698989 1.121410915 1.546959130
455 -0.4637147532 0.204901488 0.319866776 -0.346587388 0.851198467
456 0.4387578297 0.433208593 0.190713677 0.399225237 0.241567340
457 0.5363220588 0.661983797 0.266755477 0.720442825 -0.436044030
458 -0.4874575908 0.713738018 1.024491130 0.520704087 -0.185740840
459 -0.0369443317 0.278424691 0.765886523 0.909835714 0.075991173
460 1.9410352661 0.796728910 2.498946950 1.371291030 1.391352511
461 -0.0583528385 -0.145416418 0.157158807 0.378976935 0.658364989
462 1.0276564101 -0.054518498 0.173809231 -0.015014901 0.523886783
463 1.4010812965 0.382105425 1.852589269 0.965756595 2.229171688
464 0.5814140612 0.512498112 0.323716733 0.447054053 -0.131459487
465 1.2341913025 0.647848620 1.101372242 1.294973937 2.040942822
466 2.0240765499 1.430203910 2.721843778 2.039963597 1.525803771
467 0.0926748400 0.730429704 -0.322409760 -0.512653833 0.375296934
468 0.3157234402 -0.373379426 1.098983365 0.978463498 0.282284551
469 0.7942551823 0.349720282 -0.126728805 -0.154457229 0.344339430
470 0.5056091025 -0.166861345 -0.402655196 0.365967524 -0.499861896
471 1.6552493616 2.237543752 1.431070123 1.662822254 2.240142381
472 -0.0619300866 1.085686691 0.186590999 0.320754430 0.815504764
473 0.1363872100 -0.185133836 -0.425976512 0.784844323 0.144853101
474 1.3433356642 0.847743272 0.876913030 1.329796709 1.105238385
475 0.3889798126 -1.019899844 0.872496732 0.140490049 0.548402999
476 -0.0538060912 0.544544943 0.496175924 0.504353504 1.148527419
477 1.2108732416 1.179200385 1.168545991 1.521336006 1.056961059
478 0.2234005579 0.261341265 -0.060807896 -0.289487705 -0.080731082
479 2.2218638505 2.424530372 1.745101182 1.730359704 1.706807004
480 0.4401640797 0.479359500 0.423558707 0.361232325 0.846691735
481 1.5615376428 1.920666806 1.463862278 1.432632347 1.395026206
482 0.4914551246 0.330789605 0.412741592 0.355418115 0.601084853
483 0.8316452179 0.784454447 1.168112572 0.632877683 1.852258521
484 0.1950071551 0.398717128 0.381487187 -0.137720371 1.029064321
485 1.0114904994 0.027343484 0.074367840 0.346627092 0.230489210
486 3.3892940964 2.226207936 2.362123485 2.305562227 2.271571692
487 0.5884305043 0.351302080 1.057697472 0.667220497 0.929285248
488 0.1326068343 0.634674982 0.704599277 1.309664614 0.244490977
489 -0.4561553423 0.231741203 0.437037313 0.774215478 0.333403692
490 0.6958549364 0.705817376 0.576192923 0.845894141 0.651655393
491 1.2188160937 1.570599483 1.971099759 1.920439752 1.287808861
492 2.0157620014 0.619789366 0.809975786 1.311502877 0.624184690
493 2.1063871198 2.746916392 2.340228960 2.705616600 1.461522687
494 0.5449314664 0.901207278 0.596426007 0.054761905 0.070165841
495 2.2396228948 0.396634299 0.836431933 0.432022654 0.157280977
496 -0.2760935019 0.349602463 0.353771615 0.289892288 0.256280999
497 0.3338808096 -0.164133316 0.138176785 0.220501398 0.337010077
498 1.4023172187 1.430412316 2.262429082 1.109038342 1.506119262
499 2.1211488748 1.367098610 1.657866732 2.662131305 1.396535987
500 0.6343152373 0.828641146 0.808473583 1.190503178 1.077501296
[1] 0.6102628
Ozone Solar.R Wind Temp Month Day
1 41 190 7.4 67 5 1
2 36 118 8.0 72 5 2
3 12 149 12.6 74 5 3
4 18 313 11.5 62 5 4
5 NA NA 14.3 56 5 5
6 28 NA 14.9 66 5 6
7 23 299 8.6 65 5 7
8 19 99 13.8 59 5 8
9 8 19 20.1 61 5 9
10 NA 194 8.6 69 5 10
11 7 NA 6.9 74 5 11
12 16 256 9.7 69 5 12
13 11 290 9.2 66 5 13
14 14 274 10.9 68 5 14
15 18 65 13.2 58 5 15
16 14 334 11.5 64 5 16
17 34 307 12.0 66 5 17
18 6 78 18.4 57 5 18
19 30 322 11.5 68 5 19
20 11 44 9.7 62 5 20
21 1 8 9.7 59 5 21
22 11 320 16.6 73 5 22
23 4 25 9.7 61 5 23
24 32 92 12.0 61 5 24
25 NA 66 16.6 57 5 25
26 NA 266 14.9 58 5 26
27 NA NA 8.0 57 5 27
28 23 13 12.0 67 5 28
29 45 252 14.9 81 5 29
30 115 223 5.7 79 5 30
31 37 279 7.4 76 5 31
32 NA 286 8.6 78 6 1
33 NA 287 9.7 74 6 2
34 NA 242 16.1 67 6 3
35 NA 186 9.2 84 6 4
36 NA 220 8.6 85 6 5
37 NA 264 14.3 79 6 6
38 29 127 9.7 82 6 7
39 NA 273 6.9 87 6 8
40 71 291 13.8 90 6 9
41 39 323 11.5 87 6 10
42 NA 259 10.9 93 6 11
43 NA 250 9.2 92 6 12
44 23 148 8.0 82 6 13
45 NA 332 13.8 80 6 14
46 NA 322 11.5 79 6 15
47 21 191 14.9 77 6 16
48 37 284 20.7 72 6 17
49 20 37 9.2 65 6 18
50 12 120 11.5 73 6 19
51 13 137 10.3 76 6 20
52 NA 150 6.3 77 6 21
53 NA 59 1.7 76 6 22
54 NA 91 4.6 76 6 23
55 NA 250 6.3 76 6 24
56 NA 135 8.0 75 6 25
57 NA 127 8.0 78 6 26
58 NA 47 10.3 73 6 27
59 NA 98 11.5 80 6 28
60 NA 31 14.9 77 6 29
61 NA 138 8.0 83 6 30
62 135 269 4.1 84 7 1
63 49 248 9.2 85 7 2
64 32 236 9.2 81 7 3
65 NA 101 10.9 84 7 4
66 64 175 4.6 83 7 5
67 40 314 10.9 83 7 6
68 77 276 5.1 88 7 7
69 97 267 6.3 92 7 8
70 97 272 5.7 92 7 9
71 85 175 7.4 89 7 10
72 NA 139 8.6 82 7 11
73 10 264 14.3 73 7 12
74 27 175 14.9 81 7 13
75 NA 291 14.9 91 7 14
76 7 48 14.3 80 7 15
77 48 260 6.9 81 7 16
78 35 274 10.3 82 7 17
79 61 285 6.3 84 7 18
80 79 187 5.1 87 7 19
81 63 220 11.5 85 7 20
82 16 7 6.9 74 7 21
83 NA 258 9.7 81 7 22
84 NA 295 11.5 82 7 23
85 80 294 8.6 86 7 24
86 108 223 8.0 85 7 25
87 20 81 8.6 82 7 26
88 52 82 12.0 86 7 27
89 82 213 7.4 88 7 28
90 50 275 7.4 86 7 29
91 64 253 7.4 83 7 30
92 59 254 9.2 81 7 31
93 39 83 6.9 81 8 1
94 9 24 13.8 81 8 2
95 16 77 7.4 82 8 3
96 78 NA 6.9 86 8 4
97 35 NA 7.4 85 8 5
98 66 NA 4.6 87 8 6
99 122 255 4.0 89 8 7
100 89 229 10.3 90 8 8
101 110 207 8.0 90 8 9
102 NA 222 8.6 92 8 10
103 NA 137 11.5 86 8 11
104 44 192 11.5 86 8 12
105 28 273 11.5 82 8 13
106 65 157 9.7 80 8 14
107 NA 64 11.5 79 8 15
108 22 71 10.3 77 8 16
109 59 51 6.3 79 8 17
110 23 115 7.4 76 8 18
111 31 244 10.9 78 8 19
112 44 190 10.3 78 8 20
113 21 259 15.5 77 8 21
114 9 36 14.3 72 8 22
115 NA 255 12.6 75 8 23
116 45 212 9.7 79 8 24
117 168 238 3.4 81 8 25
118 73 215 8.0 86 8 26
119 NA 153 5.7 88 8 27
120 76 203 9.7 97 8 28
121 118 225 2.3 94 8 29
122 84 237 6.3 96 8 30
123 85 188 6.3 94 8 31
124 96 167 6.9 91 9 1
125 78 197 5.1 92 9 2
126 73 183 2.8 93 9 3
127 91 189 4.6 93 9 4
128 47 95 7.4 87 9 5
129 32 92 15.5 84 9 6
130 20 252 10.9 80 9 7
131 23 220 10.3 78 9 8
132 21 230 10.9 75 9 9
133 24 259 9.7 73 9 10
134 44 236 14.9 81 9 11
135 21 259 15.5 76 9 12
136 28 238 6.3 77 9 13
137 9 24 10.9 71 9 14
138 13 112 11.5 71 9 15
139 46 237 6.9 78 9 16
140 18 224 13.8 67 9 17
141 13 27 10.3 76 9 18
142 24 238 10.3 68 9 19
143 16 201 8.0 82 9 20
144 13 238 12.6 64 9 21
145 23 14 9.2 71 9 22
146 36 139 10.3 81 9 23
147 7 49 10.3 69 9 24
148 14 20 16.6 63 9 25
149 30 193 6.9 70 9 26
150 NA 145 13.2 77 9 27
151 14 191 14.3 75 9 28
152 18 131 8.0 76 9 29
153 20 223 11.5 68 9 30
Ozone Solar.R Wind Temp
1 41 190 7.4 67
2 36 118 8.0 72
3 12 149 12.6 74
4 18 313 11.5 62
5 NA NA 14.3 56
6 28 NA 14.9 66
7 23 299 8.6 65
8 19 99 13.8 59
9 8 19 20.1 61
10 NA 194 8.6 69
11 7 NA 6.9 74
12 16 256 9.7 69
13 11 290 9.2 66
14 14 274 10.9 68
15 18 65 13.2 58
16 14 334 11.5 64
17 34 307 12.0 66
18 6 78 18.4 57
19 30 322 11.5 68
20 11 44 9.7 62
21 1 8 9.7 59
22 11 320 16.6 73
23 4 25 9.7 61
24 32 92 12.0 61
25 NA 66 16.6 57
26 NA 266 14.9 58
27 NA NA 8.0 57
28 23 13 12.0 67
29 45 252 14.9 81
30 115 223 5.7 79
31 37 279 7.4 76
32 NA 286 8.6 78
33 NA 287 9.7 74
34 NA 242 16.1 67
35 NA 186 9.2 84
36 NA 220 8.6 85
37 NA 264 14.3 79
38 29 127 9.7 82
39 NA 273 6.9 87
40 71 291 13.8 90
41 39 323 11.5 87
42 NA 259 10.9 93
43 NA 250 9.2 92
44 23 148 8.0 82
45 NA 332 13.8 80
46 NA 322 11.5 79
47 21 191 14.9 77
48 37 284 20.7 72
49 20 37 9.2 65
50 12 120 11.5 73
51 13 137 10.3 76
52 NA 150 6.3 77
53 NA 59 1.7 76
54 NA 91 4.6 76
55 NA 250 6.3 76
56 NA 135 8.0 75
57 NA 127 8.0 78
58 NA 47 10.3 73
59 NA 98 11.5 80
60 NA 31 14.9 77
61 NA 138 8.0 83
62 135 269 4.1 84
63 49 248 9.2 85
64 32 236 9.2 81
65 NA 101 10.9 84
66 64 175 4.6 83
67 40 314 10.9 83
68 77 276 5.1 88
69 97 267 6.3 92
70 97 272 5.7 92
71 85 175 7.4 89
72 NA 139 8.6 82
73 10 264 14.3 73
74 27 175 14.9 81
75 NA 291 14.9 91
76 7 48 14.3 80
77 48 260 6.9 81
78 35 274 10.3 82
79 61 285 6.3 84
80 79 187 5.1 87
81 63 220 11.5 85
82 16 7 6.9 74
83 NA 258 9.7 81
84 NA 295 11.5 82
85 80 294 8.6 86
86 108 223 8.0 85
87 20 81 8.6 82
88 52 82 12.0 86
89 82 213 7.4 88
90 50 275 7.4 86
91 64 253 7.4 83
92 59 254 9.2 81
93 39 83 6.9 81
94 9 24 13.8 81
95 16 77 7.4 82
96 78 NA 6.9 86
97 35 NA 7.4 85
98 66 NA 4.6 87
99 122 255 4.0 89
100 89 229 10.3 90
101 110 207 8.0 90
102 NA 222 8.6 92
103 NA 137 11.5 86
104 44 192 11.5 86
105 28 273 11.5 82
106 65 157 9.7 80
107 NA 64 11.5 79
108 22 71 10.3 77
109 59 51 6.3 79
110 23 115 7.4 76
111 31 244 10.9 78
112 44 190 10.3 78
113 21 259 15.5 77
114 9 36 14.3 72
115 NA 255 12.6 75
116 45 212 9.7 79
117 168 238 3.4 81
118 73 215 8.0 86
119 NA 153 5.7 88
120 76 203 9.7 97
121 118 225 2.3 94
122 84 237 6.3 96
123 85 188 6.3 94
124 96 167 6.9 91
125 78 197 5.1 92
126 73 183 2.8 93
127 91 189 4.6 93
128 47 95 7.4 87
129 32 92 15.5 84
130 20 252 10.9 80
131 23 220 10.3 78
132 21 230 10.9 75
133 24 259 9.7 73
134 44 236 14.9 81
135 21 259 15.5 76
136 28 238 6.3 77
137 9 24 10.9 71
138 13 112 11.5 71
139 46 237 6.9 78
140 18 224 13.8 67
141 13 27 10.3 76
142 24 238 10.3 68
143 16 201 8.0 82
144 13 238 12.6 64
145 23 14 9.2 71
146 36 139 10.3 81
147 7 49 10.3 69
148 14 20 16.6 63
149 30 193 6.9 70
150 NA 145 13.2 77
151 14 191 14.3 75
152 18 131 8.0 76
153 20 223 11.5 68
Ozone Solar.R Wind Temp
1 41.00000 190.0000 7.4 67
2 36.00000 118.0000 8.0 72
3 12.00000 149.0000 12.6 74
4 18.00000 313.0000 11.5 62
5 20.47224 178.3231 14.3 56
6 28.00000 193.3949 14.9 66
7 23.00000 299.0000 8.6 65
8 19.00000 99.0000 13.8 59
9 8.00000 19.0000 20.1 61
10 26.61911 194.0000 8.6 69
11 7.00000 158.5913 6.9 74
12 16.00000 256.0000 9.7 69
13 11.00000 290.0000 9.2 66
14 14.00000 274.0000 10.9 68
15 18.00000 65.0000 13.2 58
16 14.00000 334.0000 11.5 64
17 34.00000 307.0000 12.0 66
18 6.00000 78.0000 18.4 57
19 30.00000 322.0000 11.5 68
20 11.00000 44.0000 9.7 62
21 1.00000 8.0000 9.7 59
22 11.00000 320.0000 16.6 73
23 4.00000 25.0000 9.7 61
24 32.00000 92.0000 12.0 61
25 15.42723 66.0000 16.6 57
26 30.66671 266.0000 14.9 58
27 21.32364 142.1746 8.0 57
28 23.00000 13.0000 12.0 67
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30 115.00000 223.0000 5.7 79
31 37.00000 279.0000 7.4 76
32 40.18203 286.0000 8.6 78
33 31.27023 287.0000 9.7 74
34 39.31116 242.0000 16.1 67
35 60.98784 186.0000 9.2 84
36 60.05528 220.0000 8.6 85
37 48.35681 264.0000 14.3 79
38 29.00000 127.0000 9.7 82
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41 39.00000 323.0000 11.5 87
42 69.39093 259.0000 10.9 93
43 69.71556 250.0000 9.2 92
44 23.00000 148.0000 8.0 82
45 35.76567 332.0000 13.8 80
46 35.94283 322.0000 11.5 79
47 21.00000 191.0000 14.9 77
48 37.00000 284.0000 20.7 72
49 20.00000 37.0000 9.2 65
50 12.00000 120.0000 11.5 73
51 13.00000 137.0000 10.3 76
52 37.53622 150.0000 6.3 77
53 34.19981 59.0000 1.7 76
54 26.77555 91.0000 4.6 76
55 37.54133 250.0000 6.3 76
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63 49.00000 248.0000 9.2 85
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72 37.49463 139.0000 8.6 82
73 10.00000 264.0000 14.3 73
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101 110.00000 207.0000 8.0 90
102 77.91793 222.0000 8.6 92
103 32.70190 137.0000 11.5 86
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128 47.00000 95.0000 7.4 87
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130 20.00000 252.0000 10.9 80
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148 14.00000 20.0000 16.6 63
149 30.00000 193.0000 6.9 70
150 26.27180 145.0000 13.2 77
151 14.00000 191.0000 14.3 75
152 18.00000 131.0000 8.0 76
153 20.00000 223.0000 11.5 68
V1 V2 V3 V4 V5 V6
1 -0.27294622 0.5649844 1.6517405 -0.45378443 0.359638060 0.5243731
2 1.61893357 -0.9300666 1.2307204 1.51345863 0.003216759 -1.8996916
3 -0.09156827 -1.3119584 -0.4464256 1.68003787 0.687738071 1.7477167
4 -1.75697208 1.2015881 -0.9164589 -0.24929173 0.020322163 -1.0069711
5 0.18531297 -1.8997072 -0.9993217 -1.64912872 -2.296420684 0.8252367
6 1.57761882 0.9472044 0.5019121 0.16468517 0.077490188 2.3317513
7 1.33040094 0.4765310 1.6940212 0.03539862 -0.439885071 0.7853790
8 -0.30232217 0.4667190 -0.4757372 0.21561522 0.727385605 -0.3569848
9 -0.10808600 -0.9109851 -0.6241070 -0.13797592 -1.433679131 -1.1014573
10 -1.32985277 -1.5091918 -0.2279206 -1.40811383 -0.110213013 0.3796918
V7 V8 V9 V10 V11 V12
1 0.97018018 0.2361315 1.9589677 0.3142968 0.74531168 1.1822632
2 0.36905840 -0.4627000 0.6130300 0.2237436 0.57177136 0.2464576
3 -0.03107033 0.1351887 0.0629055 -0.7230052 -0.44377809 1.8061065
4 -1.92010585 -0.7430641 -0.3066002 -1.0654466 -1.08542269 -0.1158405
5 -1.09228974 -0.8066865 -0.8859033 -0.2866407 0.65348405 0.7160437
6 1.47340510 2.3012685 0.4189516 0.4735910 -0.05045945 0.5512949
7 -0.14968739 0.8637884 -0.9664251 -0.9807089 0.90329170 0.6253347
8 0.65834024 0.5753060 -0.4755417 -0.6232189 0.90526977 0.0181135
9 1.19217328 -2.0427906 -1.3760800 0.3504487 0.42653643 0.2338093
10 -0.51216355 -0.1326851 -1.0383768 -0.1415861 -1.51708483 -0.5911722
V13 V14 V15 V16 V17 V18
1 0.1176203 -0.3835233 1.8023094 -1.36834561 1.10236499 0.4495366
2 0.8329853 -0.2207156 -0.6022181 -0.46200466 -1.33754621 -0.6997470
3 -0.2299583 1.1937213 -0.6058899 -1.31720639 -0.04461599 0.9304723
4 -0.4815356 -0.0349740 -0.4134190 -1.07445030 0.63789061 0.2260202
5 -1.5029909 -1.6384298 -1.7578831 -1.55682426 0.93153809 -0.5802509
6 1.1108469 1.7979764 1.1823493 1.09854680 1.69883875 1.6339145
7 1.0436978 -1.1928054 1.3082314 1.29887088 1.16840837 0.9781224
8 0.7087555 -0.5047357 1.1387356 -0.89102083 0.74217847 0.5089179
9 -0.3726949 -0.1658626 -1.1174908 -0.92191686 -0.62003392 -1.1407242
10 -1.5737854 -0.7379942 0.3107172 0.01095196 -0.87466556 -1.1424905
V19 V20
1 1.2063345 0.7566320
2 -0.4336555 -0.1915558
3 1.0880000 1.5003867
4 0.4434772 -1.4640166
5 -1.2388603 -1.0518300
6 0.8279166 0.1385917
7 2.0034276 -0.7571446
8 0.3569348 0.6664090
9 0.2193313 -1.6748832
10 -1.1953611 -1.1098792
V1 V2 V3 V4 V5 V6
1 -0.27294622 0.5649844 1.6517405 NA 0.359638060 0.5243731
2 1.61893357 -0.9300666 1.2307204 1.51345863 0.003216759 -1.8996916
3 -0.09156827 -1.3119584 -0.4464256 NA 0.687738071 1.7477167
4 -1.75697208 1.2015881 -0.9164589 -0.24929173 0.020322163 -1.0069711
5 0.18531297 -1.8997072 -0.9993217 -1.64912872 -2.296420684 0.8252367
6 1.57761882 0.9472044 0.5019121 0.16468517 0.077490188 2.3317513
7 NA 0.4765310 1.6940212 0.03539862 -0.439885071 0.7853790
8 -0.30232217 0.4667190 -0.4757372 0.21561522 0.727385605 -0.3569848
9 -0.10808600 -0.9109851 -0.6241070 -0.13797592 -1.433679131 -1.1014573
10 -1.32985277 -1.5091918 -0.2279206 -1.40811383 -0.110213013 0.3796918
V7 V8 V9 V10 V11 V12
1 0.97018018 0.2361315 1.9589677 0.3142968 0.74531168 1.1822632
2 0.36905840 -0.4627000 0.6130300 0.2237436 0.57177136 0.2464576
3 -0.03107033 0.1351887 0.0629055 -0.7230052 -0.44377809 1.8061065
4 -1.92010585 -0.7430641 -0.3066002 -1.0654466 -1.08542269 NA
5 -1.09228974 -0.8066865 -0.8859033 -0.2866407 0.65348405 0.7160437
6 NA 2.3012685 0.4189516 0.4735910 -0.05045945 0.5512949
7 -0.14968739 0.8637884 -0.9664251 -0.9807089 0.90329170 0.6253347
8 0.65834024 0.5753060 -0.4755417 -0.6232189 0.90526977 0.0181135
9 1.19217328 -2.0427906 -1.3760800 0.3504487 0.42653643 0.2338093
10 -0.51216355 -0.1326851 -1.0383768 -0.1415861 -1.51708483 -0.5911722
V13 V14 V15 V16 V17 V18
1 0.1176203 -0.3835233 1.8023094 -1.36834561 1.1023650 0.4495366
2 0.8329853 -0.2207156 -0.6022181 -0.46200466 -1.3375462 -0.6997470
3 -0.2299583 1.1937213 -0.6058899 -1.31720639 NA 0.9304723
4 NA -0.0349740 -0.4134190 -1.07445030 0.6378906 0.2260202
5 -1.5029909 -1.6384298 -1.7578831 NA 0.9315381 -0.5802509
6 1.1108469 1.7979764 1.1823493 1.09854680 1.6988387 1.6339145
7 1.0436978 -1.1928054 1.3082314 1.29887088 1.1684084 0.9781224
8 0.7087555 -0.5047357 1.1387356 -0.89102083 0.7421785 0.5089179
9 -0.3726949 -0.1658626 -1.1174908 -0.92191686 -0.6200339 -1.1407242
10 -1.5737854 -0.7379942 0.3107172 0.01095196 -0.8746656 -1.1424905
V19 V20
1 1.2063345 0.7566320
2 -0.4336555 -0.1915558
3 1.0880000 1.5003867
4 0.4434772 -1.4640166
5 -1.2388603 -1.0518300
6 0.8279166 NA
7 2.0034276 -0.7571446
8 0.3569348 0.6664090
9 0.2193313 NA
10 -1.1953611 -1.1098792
1 2 3 4 5 6 7 8 9 10
1 0 2 2 1 2 1 0 1 0
V1 V2 V3 V4 V5 V6
1 -0.27294622 0.5649844 1.6517405 1.51345863 0.359638060 0.5243731
2 1.61893357 -0.9300666 1.2307204 1.51345863 0.003216759 -1.8996916
3 -0.09156827 -1.3119584 -0.4464256 0.21561522 0.687738071 1.7477167
4 -1.75697208 1.2015881 -0.9164589 -0.24929173 0.020322163 -1.0069711
5 0.18531297 -1.8997072 -0.9993217 -1.64912872 -2.296420684 0.8252367
6 1.57761882 0.9472044 0.5019121 0.16468517 0.077490188 2.3317513
7 -0.30232217 0.4765310 1.6940212 0.03539862 -0.439885071 0.7853790
8 -0.30232217 0.4667190 -0.4757372 0.21561522 0.727385605 -0.3569848
9 -0.10808600 -0.9109851 -0.6241070 -0.13797592 -1.433679131 -1.1014573
10 -1.32985277 -1.5091918 -0.2279206 -1.40811383 -0.110213013 0.3796918
V7 V8 V9 V10 V11 V12
1 0.97018018 0.2361315 1.9589677 0.3142968 0.74531168 1.1822632
2 0.36905840 -0.4627000 0.6130300 0.2237436 0.57177136 0.2464576
3 -0.03107033 0.1351887 0.0629055 -0.7230052 -0.44377809 1.8061065
4 -1.92010585 -0.7430641 -0.3066002 -1.0654466 -1.08542269 -0.5911722
5 -1.09228974 -0.8066865 -0.8859033 -0.2866407 0.65348405 0.7160437
6 0.97018018 2.3012685 0.4189516 0.4735910 -0.05045945 0.5512949
7 -0.14968739 0.8637884 -0.9664251 -0.9807089 0.90329170 0.6253347
8 0.65834024 0.5753060 -0.4755417 -0.6232189 0.90526977 0.0181135
9 1.19217328 -2.0427906 -1.3760800 0.3504487 0.42653643 0.2338093
10 -0.51216355 -0.1326851 -1.0383768 -0.1415861 -1.51708483 -0.5911722
V13 V14 V15 V16 V17 V18
1 0.1176203 -0.3835233 1.8023094 -1.36834561 1.1023650 0.4495366
2 0.8329853 -0.2207156 -0.6022181 -0.46200466 -1.3375462 -0.6997470
3 -0.2299583 1.1937213 -0.6058899 -1.31720639 0.7421785 0.9304723
4 -1.5737854 -0.0349740 -0.4134190 -1.07445030 0.6378906 0.2260202
5 -1.5029909 -1.6384298 -1.7578831 0.01095196 0.9315381 -0.5802509
6 1.1108469 1.7979764 1.1823493 1.09854680 1.6988387 1.6339145
7 1.0436978 -1.1928054 1.3082314 1.29887088 1.1684084 0.9781224
8 0.7087555 -0.5047357 1.1387356 -0.89102083 0.7421785 0.5089179
9 -0.3726949 -0.1658626 -1.1174908 -0.92191686 -0.6200339 -1.1407242
10 -1.5737854 -0.7379942 0.3107172 0.01095196 -0.8746656 -1.1424905
V19 V20
1 1.2063345 0.7566320
2 -0.4336555 -0.1915558
3 1.0880000 1.5003867
4 0.4434772 -1.4640166
5 -1.2388603 -1.0518300
6 0.8279166 0.7566320
7 2.0034276 -0.7571446
8 0.3569348 0.6664090
9 0.2193313 0.6664090
10 -1.1953611 -1.1098792
[1] 1.890908
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