Imputation of missing data by using Nearest Neighbour Imputation with the Mahalanobis distance

Description

This function imputes quantitative missing data by using Nearest Neighbour Imputation (NNI) with the Mahalanobis distance in a forward and sequential step-by-step process that starts from the complete part of data.

Usage

ForImp.Mahala(mat, probs=seq(0, 1, 0.1), q="10%", add.unit=TRUE, squared=FALSE, 
tol=1e-6)

Arguments

Details

ForImp.Mahala is a forward imputation method alternative to the ForImp.PCA procedure for imputing quantitative missing data (see ForImp.PCA). It does not embrace Stage 1 since it works directly on the original variables. Regarding Stage 2, the basic metric for the NNI method is the Mahalanobis distance. Steps 2 to 3 are therefore iteratively repeated until the starting data matrix is completely imputed.

Unlike ForImp.PCA, the ForImp.Mahala procedure requires that the number n of units is equal or greater than the number p of variables at every step of the procedure, otherwise the covariance matrix involved in the Mahalanobis distance is not invertible. For further details, see the references below.

Value

The imputed data matrix.

Author(s)

Nadia Solaro, Alessandro Barbiero, Giancarlo Manzi, Pier Alda Ferrari

References

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

ForImp.PCA

Examples

# EXAMPLE with multivariate normal data (MVN)
# require('mvtnorm')
# number of variables
p <- 5
# correlation matrix
rho <- 0.8
Rho <- matrix(rho, p, p)
diag(Rho) <- 1
Rho
# mean vector
vmean <- rep(0,p)
vmean
# number of units
n <- 1000
# percentage of missing values
percmiss <- 0.2
nummiss <- n*p*percmiss
nummiss
# generation of a complete matrix
set.seed(1)
x0 <- rmvnorm(n, mean=vmean, sigma=Rho)
x0
# generating a matrix with missing data
x <- missing.gen(x0, nummiss) 
# imputing missing values
xForImpMahala <- ForImp.Mahala(x)
xForImpMahala
# computing the Relative Mean Square Error
error <- sum(apply((x0-xForImpMahala)^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.Mahala(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")

Results

R version 3.3.1 (2016-06-21) -- "Bug in Your Hair"
Copyright (C) 2016 The R Foundation for Statistical Computing
Platform: x86_64-pc-linux-gnu (64-bit)

R is free software and comes with ABSOLUTELY NO WARRANTY.
You are welcome to redistribute it under certain conditions.
Type 'license()' or 'licence()' for distribution details.

R is a collaborative project with many contributors.
Type 'contributors()' for more information and
'citation()' on how to cite R or R packages in publications.

Type 'demo()' for some demos, 'help()' for on-line help, or
'help.start()' for an HTML browser interface to help.
Type 'q()' to quit R.

> library(GenForImp)
Loading required package: mvtnorm
Loading required package: sn
Loading required package: stats4

Attaching package: 'sn'

The following object is masked from 'package:stats':

    sd

> png(filename="/home/ddbj/snapshot/RGM3/R_CC/result/GenForImp/ForImp.Mahala.Rd_%03d_medium.png", width=480, height=480)
> ### Name: ForImp.Mahala
> ### Title: Imputation of missing data by using Nearest Neighbour Imputation
> ###   with the Mahalanobis distance
> ### Aliases: ForImp.Mahala
> ### Keywords: multivariate NA nonparametric
> 
> ### ** Examples
> 
> # EXAMPLE with multivariate normal data (MVN)
> # require('mvtnorm')
> # number of variables
> p <- 5
> # correlation matrix
> rho <- 0.8
> Rho <- matrix(rho, p, p)
> diag(Rho) <- 1
> Rho
     [,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
> # mean vector
> vmean <- rep(0,p)
> vmean
[1] 0 0 0 0 0
> # number of units
> n <- 1000
> # percentage of missing values
> percmiss <- 0.2
> nummiss <- n*p*percmiss
> nummiss
[1] 1000
> # generation of a complete matrix
> set.seed(1)
> x0 <- rmvnorm(n, mean=vmean, sigma=Rho)
> x0
                [,1]         [,2]         [,3]          [,4]         [,5]
   [1,] -0.073045466  0.289240987 -0.166591281  0.9205444587  0.354473551
   [2,] -0.150413417  0.434496098  0.546700045  0.4740084472  0.079937360
   [3,]  0.737168822  0.235422926 -0.216747503 -0.9293641690  0.564164131
   [4,]  0.716212507  0.729066922  1.158403813  1.1035687109  1.001908173
   [5,]  0.541124844  0.479927657  0.163492144 -0.7595194552  0.407340171
   [6,] -0.583583473 -0.628155807 -1.216222400 -0.7723171434 -0.371572790
   [7,]  0.675776186  0.022188149  0.241528234  0.0440938729 -0.547683537
   [8,]  0.133114778  0.142374161  0.292180231  0.8106522901  0.660008559
   [9,] -0.026477811 -0.066207410  0.358790875  0.2960467281 -0.260921533
  [10,]  0.066320473  0.545767936  0.726420298  0.3324791725  0.776765280
  [11,]  0.316101051 -0.135643835  0.290616053 -0.3670038417  0.778930371
  [12,]  1.207285507  0.157397311 -0.145327452  0.5764101112  0.261225493
  [13,]  1.822844112  0.731259336  1.057268819  0.7613309446  0.416406114
  [14,]  0.781458529 -0.110173992  1.352444105  0.7655650224  1.668649523
  [15,] -0.367795521 -0.897947543 -0.307324719 -0.9981910075 -1.141091748
  [16,] -0.083045534 -0.411630406 -0.212889925 -0.1801378023 -0.477026035
  [17,] -0.400236530 -0.206373856  0.380936380 -0.8272799279  0.119700669
  [18,]  0.702927423  1.029460191  0.417992304  0.7195049324  0.673477701
  [19,]  1.075263213  1.858060451  1.836833373  1.6310306103  2.027539041
  [20,] -0.708141615 -1.528813729 -1.214276424 -1.5055677477 -1.169615538
  [21,] -0.913733886 -0.617462683 -1.043674020 -0.5656248587 -0.929036626
  [22,]  2.540101658  2.070268091  2.156789054  1.9215596793  2.502038777
  [23,] -0.451954064 -0.374097878  0.472892011 -0.4586443340 -0.260387566
  [24,] -0.391980907 -0.359417022 -0.341135124  0.0046958801 -0.295616463
  [25,] -0.116440352  0.710455926  0.013867875  0.0295305823  0.065024042
  [26,]  0.189109828 -0.162503235 -0.146434747 -0.4344520673 -0.274622308
  [27,] -0.360525461 -0.650791648 -0.149737704 -1.0664765043 -0.250333184
  [28,] -1.672362873 -1.119842239 -1.221495579 -1.2768672750 -1.010686636
  [29,] -2.132164757 -0.749853142 -2.020635505 -1.4833342927 -1.775091838
  [30,] -0.839873409  0.429312311 -0.496317387 -1.0793480293 -1.237798044
  [31,] -0.536717185 -0.746347187 -0.880291479 -1.1536703650 -1.403259451
  [32,] -0.548633962  0.379433128 -0.345632262 -0.6869278575  0.768178831
  [33,]  0.674646007  0.377809111  0.957903349  0.8809555996  0.207601428
  [34,]  1.175565974  0.074915403 -0.448086416  0.1243894934  0.281780925
  [35,]  1.820310276  0.835467215  0.992527037  0.7536471162  0.638781240
  [36,]  1.606981246  1.974754338  2.550588980  2.0819750642  2.162704256
  [37,] -0.862682158  0.127993892 -0.213664221 -0.9681917440 -0.079009129
  [38,] -0.332655254  0.393325321 -0.604260322 -0.4540545819 -0.675826795
  [39,] -0.362644346 -0.103655913 -0.610688778  0.0879131238 -0.823712093
  [40,] -0.658298647  0.454879550 -0.643926567 -0.0053850776 -0.360048161
  [41,]  0.525545766  1.097742792  1.051999616  0.1944692304 -0.679532703
  [42,]  2.463702209  1.645035048  1.588802933  1.3407215403  1.574841411
  [43,] -0.242155424  0.019496047 -0.347640102 -0.7814199085  0.273130386
  [44,]  0.857571341  0.039847724 -0.382567514  0.4651397471  0.157926170
  [45,] -2.011655575 -1.235583343 -1.518415618 -1.3890225239 -1.753771624
  [46,]  0.911128838 -0.043347486 -0.613268163  0.1929268493  0.222434990
  [47,] -1.724040593 -2.575130090 -1.569597615 -1.0280267186 -1.309874551
  [48,]  0.073890394  0.368603918 -0.412803170  0.6082908689  0.115407341
  [49,]  1.037027873  1.183175965  0.820652407  0.3277389358  1.240802487
  [50,] -1.518158923 -0.867295792 -0.737997384 -0.6979495528 -0.167292633
  [51,]  0.356240242  0.477410848  0.264169380  0.1845610992  0.606379737
  [52,]  0.277194176 -1.310112027  0.020722187 -0.0678328793 -0.425600235
  [53,]  1.339982699  0.740604683  0.787520305  1.2981221518  1.683717550
  [54,] -0.716238363 -1.025817015 -1.368798218 -0.9850530349 -1.257315038
  [55,]  0.553516869  0.845686767  0.288352968  1.8540584868  0.739085583
  [56,]  0.244495305 -1.284676117  0.070437041 -0.8495914820  0.150399959
  [57,]  0.188628144 -0.171673395  0.602805389 -0.3030214101 -0.249079685
  [58,] -0.218647360 -0.069772828  0.651745291  0.4230032269  0.678566597
  [59,]  0.012215556  0.354999838  0.295875800 -0.4511597476  0.982019683
  [60,]  0.540395977  0.822655436  0.907419262  0.4576554858  0.343504342
  [61,]  1.389239734  0.521210733  1.871185584  0.8180112103  1.729332972
  [62,]  1.144392361  0.505213689  0.721779190  0.0099182759  0.612563185
  [63,]  0.465708232  0.043299799 -0.204105648 -0.2942840079 -0.017074444
  [64,]  0.370589189 -0.323994521 -0.161671719 -0.6863113564  0.113509068
  [65,]  1.300086419  1.384672170  1.079521215 -0.1210198902  0.930745225
  [66,]  0.232322808 -0.129712648  0.104577974 -0.3580758610  0.332094001
  [67,] -0.541534694 -1.096965559 -0.559579647  0.2149431769 -0.544745845
  [68,]  0.584594081  0.680224267  0.258865460  0.0993349745  0.020011266
  [69,] -0.512646145 -1.318993117 -0.733291666 -0.9728685549 -1.856289815
  [70,]  0.012943671  1.052657047  0.557458202  1.0022399247  1.487104991
  [71,]  1.914030305  1.557774280  1.424815082  1.1686685251  1.960545266
  [72,] -0.645684690 -1.437168253 -0.981934147 -1.7704912497 -0.999651100
  [73,] -1.964631204 -0.217667392 -1.089530497 -0.9975994084 -0.881028486
  [74,]  0.249015099  0.278585758  0.229218652 -0.2808260280 -0.634459576
  [75,] -0.869312268 -0.571183417 -1.063563463 -0.7262569968 -1.217664862
  [76,]  0.500749190  0.562090981  0.439227529  0.4311618854  1.293149526
  [77,]  1.068420057  1.224428090  0.314510300  0.8008769050  1.243834525
  [78,] -1.485552746 -2.412554628 -1.306055883 -2.3121975703 -1.823041917
  [79,]  1.626399805  1.309608240  1.522496270  1.1714049341  0.985022575
  [80,] -0.136603248  0.989134677  0.296891342 -0.0430531531  0.663701524
  [81,]  0.980386438  1.347644422  0.230213242  0.3250804308  0.313741683
  [82,] -0.262105736 -0.258068972 -0.226337606  0.5506950609 -0.406055236
  [83,] -0.138684162  0.326732797  0.537923154 -0.3103888240 -0.192313439
  [84,]  0.737041498  0.958024778  0.390547033 -0.1366412599  1.267213413
  [85,]  0.471915087 -0.487555171 -0.198966640 -0.9567258759  0.084920322
  [86,]  0.039558163 -1.413719411 -1.030138884 -0.9682482877 -0.986116465
  [87,] -1.709162148 -0.575811471 -1.484890140 -0.8110284699 -0.534658357
  [88,]  0.125029574  0.612583541  0.202165132 -0.0759772978  0.502725979
  [89,]  0.568138409  1.554890130  0.753988318  1.1435447684  0.853905948
  [90,] -3.557584019 -2.823287618 -2.402008086 -2.1064419713 -3.260041419
  [91,] -0.661034223 -1.361425437 -1.427814987 -1.7868086852 -1.741319078
  [92,] -0.615925628 -0.413321067 -1.579319772 -1.0902823229 -0.401513593
  [93,]  0.937652079  2.108638656  1.697371271  1.4065712394  1.142440815
  [94,]  1.010974644  0.570349888  1.166898513  1.5480125115  1.370565290
  [95,] -0.603986231 -0.964276079 -0.289331862 -0.0431206220  0.363890532
  [96,] -0.447003761  0.305654071  0.247120628  0.6766493526  0.130214608
  [97,] -0.170396250 -0.523478218 -0.242004820 -0.1725540192  1.040592507
  [98,]  1.022952563 -0.145721503 -0.537911798 -0.9534928553  0.158998622
  [99,]  0.867674379  2.377124424  0.959333494  1.4050820185  3.059401247
 [100,] -1.275810704 -0.642343101 -1.274909333 -0.6244156384 -1.170426849
 [101,] -0.093575571 -0.260910271 -0.657308595 -0.1230963347  0.315310886
 [102,]  0.718242427 -0.608496632 -0.106230892  0.5239089225 -0.492896871
 [103,] -2.664989199 -1.952757662 -1.966771950 -1.5129717701 -1.714765897
 [104,]  0.426314813  0.133895914  0.490097430  0.1591038388  0.855016144
 [105,] -0.705047671 -0.245314701 -0.771021534 -0.2822814518 -0.396650359
 [106,] -0.696907385 -0.008066927  0.584240747  0.3611644595  0.120495438
 [107,] -0.133839684  0.922275389  0.434800521  0.1973410752  1.133717780
 [108,]  0.011737463  0.491576759  0.929120013  0.6252318340  0.218605082
 [109,]  0.385577321  0.731521022  0.802335321  0.5810210675  1.162327890
 [110,] -0.682683764 -0.966424726 -0.412943248 -1.6379097876 -0.477946811
 [111,]  0.321511764 -0.001302960 -0.400337542 -0.0773828871  0.792369269
 [112,] -1.381992588 -2.092256164 -0.686235599 -1.2736757455 -1.277958222
 [113,] -1.098166668 -0.564727871 -0.241035597 -0.8173692943 -0.825976330
 [114,]  0.799295271  0.198173165  0.237352637  0.0290063159  0.711549468
 [115,] -0.110858401  0.713554825  0.694079524  0.1246410107  0.713183606
 [116,]  0.588683708  0.404812028  1.052923701 -0.7302042170 -0.373581669
 [117,]  0.198940589  0.889527442  1.523206364  1.5649134472  1.032911618
 [118,]  0.655059611  0.197314969  0.471835529  0.4303626025 -0.078242576
 [119,] -0.831513455 -0.760650820 -0.077001174 -1.1641034137 -1.645907397
 [120,]  0.233876848 -0.474705742 -0.304740562  0.2948887732 -0.450970169
 [121,]  0.518840913  1.343275381  0.907637094  0.9138460483  0.610248520
 [122,] -0.711302081 -0.872251258 -0.302840344  0.6922568056 -0.103790287
 [123,]  0.518522442  0.593766442  0.272742388 -0.1805291753  1.641640304
 [124,] -1.153969712 -1.098030189 -1.766051121 -1.0754041830 -1.075089207
 [125,]  0.589216259 -0.095659039  1.501609448  1.4921837652  1.011882568
 [126,] -0.360837548  0.450966932  0.021203756 -0.0115840880  0.740821687
 [127,] -1.838368018 -1.517866421 -1.616242574 -1.2804088637 -1.992648356
 [128,] -0.294311392  0.340403372  0.228279240 -0.4153135434  0.270318581
 [129,] -1.970879157 -1.035239079 -0.564091451 -1.1777000695 -0.899249338
 [130,] -1.729851265 -0.638167505 -1.844831020 -1.1620529326 -1.131462396
 [131,] -0.007560559 -0.403082012  0.783474989 -0.6742833319  0.078848226
 [132,] -2.214354325 -0.898567178 -0.671718188 -1.3847490921 -0.584436670
 [133,] -0.581981720 -0.293371924 -1.107151903 -1.1640553723 -1.522251747
 [134,]  0.268481270 -0.186035502  0.720447123  0.4445720681  1.391981417
 [135,] -0.962201908  0.084007638 -0.680305631 -1.0958831805 -0.188220532
 [136,] -1.925871641 -1.756738730 -1.612703242 -1.1905250478 -1.731576574
 [137,]  0.216981223 -0.801245740 -0.168165068 -0.2010576302 -0.741666025
 [138,] -0.066765266 -0.168531433  0.553975137  0.8208088447 -0.340537365
 [139,] -0.034631247 -0.484536326 -1.330552385 -0.7888462655 -0.244563705
 [140,] -0.372930836 -1.530059322 -0.178298089 -0.1448763352 -0.574004987
 [141,] -0.304728339  0.893419604 -0.028542839  0.0054370886 -0.490719422
 [142,] -0.657040377 -0.737103080 -0.244637032  0.5526968112  0.728393787
 [143,]  1.046041832  0.924574521  0.238391526 -0.2028087835  0.198045513
 [144,] -1.027457956 -0.088909833  0.793742929  0.1099866128  0.541518924
 [145,]  0.347029734 -0.220149468  0.907933253  0.0931339459 -0.308091114
 [146,] -0.108549916 -0.819288306  0.246143776  0.0841582502 -0.298812985
 [147,] -0.594660368  0.042495053 -0.851351804 -0.8162063986 -0.186498911
 [148,]  0.427996894  1.043540916  0.541004492  1.1641840374  0.851444747
 [149,] -1.828868211 -1.422001684 -0.251130944 -0.5158721360 -1.257289029
 [150,]  0.183005062  1.247131510  0.421824825  1.5906184240  1.577119944
 [151,] -1.988153109 -1.715783457 -1.896826856 -1.0391750556 -1.914845105
 [152,] -0.543037897  1.271017209  0.816928373  0.7324721359  0.070098765
 [153,] -0.204151571  0.078021643 -0.772093835 -0.2546778052 -0.120512393
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