[R] Empirical density estimation

Jeff Newmiller jdnewmil at dcn.davis.ca.us
Mon Mar 12 04:24:09 CET 2018 

cwhmisc package provides essentially the algorithm outlined by Dan.

If you want answers outside your original data (extrapolation) then the 
following code at least won't give broken answers, though it is not 
necessarily any more "correct" for extrapolation than the approx solution 
is.

Regarding "needing this for reporting", do thoroughly read ?density as 
Bert suggested, because the bandwidth parameter affects your answers and 
there are various historical recommendations for choosing possible 
bandwidth values, and really no "right" answer.

############
smoothed.df2 <- function ( d ) {
     F <- cumsum( d$y )
     F <- F / F[ length( F ) ] * ( length( F ) - 0.5 ) / length( F )
     eF <- splinefun( d$x, qlogis( F ), "monoH.FC" )
     function( x ) {
         efx <- eF( x )
         plogis( efx )
     }
}

set.seed( 42 )
Dat <- c( rnorm( 100, 1 ), rnorm( 100, 5 ) )

d <- density( Dat )

CDF1 <- cwhmisc::smoothed.df( d )
plot( Dat, CDF1( Dat ) )

CDF2 <- smoothed.df2( d )
plot( Dat, CDF2( Dat ) )

CDF1( -5 ) # <0
CDF2( -5 ) # >0
############

On Sun, 11 Mar 2018, Daniel Nordlund wrote:

> On 3/11/2018 3:35 PM, Christofer Bogaso wrote:
>> But for my reporting purpose, I need to generate a bell curve like
>> plot based on empirical PDF, that also contains original points.
>> 
>> Any idea would be helpful. Thanks,
>> 
>
>
> Christofer,
>
> something like the following may get you what you want:
>
> ## get the kernel density estimate
> dens <- density(Dat)
>
> ## estimate the density at your original points
> dnew <- approx(dens$x,dens$y,xout=Dat)
>
> ## plot kernel density estimate
> plot(dx)
>
> ## add your original values with the estimated density
> points(dnew, pch=1, cex=0.5, col="red")
>
>
> Hope this is helpful,
>
> Dan
>
> -- 
> Daniel Nordlund
> Port Townsend, WA  USA
>
>> On Mon, Mar 12, 2018 at 3:49 AM, Bert Gunter <bgunter.4567 at gmail.com> 
>> wrote:
>>> You need to re-read ?density and perhaps think again -- or do some study 
>>> --
>>> about how a (kernel) density estimate works. The points at which the
>>> estimate is calculated are *not* the values given, nor should they be!
>>> 
>>> Cheers,
>>> Bert
>>> 
>>> 
>>> 
>>> Bert Gunter
>>> 
>>> "The trouble with having an open mind is that people keep coming along and
>>> sticking things into it."
>>> -- Opus (aka Berkeley Breathed in his "Bloom County" comic strip )
>>> 
>>> On Sun, Mar 11, 2018 at 11:45 AM, Christofer Bogaso
>>> <bogaso.christofer at gmail.com> wrote:
>>>> 
>>>> Hi,
>>>> 
>>>> Let say I have below vector of data-points :
>>>> 
>>>> Dat = c(-0.444444444444444, -0.25, -0.237449799196787, 
>>>> -0.227467046669042,
>>>> 
>>>> -0.227454464682363, -0.22, -0.214876033057851, -0.211781206171108,
>>>> 
>>>> -0.199891067538126, -0.192920353982301, -0.192307692307692,
>>>> -0.186046511627907,
>>>> 
>>>> -0.184418145956608, -0.181818181818182, -0.181818181818182,
>>>> -0.181266261925412,
>>>> 
>>>> -0.181003118503119, -0.179064587973274, -0.178217821782178,
>>>> -0.17809021675454,
>>>> 
>>>> -0.177685950413223, -0.177570093457944, -0.176470588235294,
>>>> -0.176470588235294,
>>>> 
>>>> -0.174825741611282, -0.168021680216802, -0.166666666666667,
>>>> -0.166666666666667,
>>>> 
>>>> -0.166380789022298, -0.164209115281501, -0.164011246485473,
>>>> -0.162689804772234,
>>>> 
>>>> -0.162361623616236, -0.160161507402423, -0.16, -0.155038759689922,
>>>> 
>>>> -0.154172560113154, -0.15311004784689, -0.151515151515152,
>>>> -0.151462994836489,
>>>> 
>>>> -0.151098901098901, -0.150537634408602, -0.150442477876106,
>>>> -0.150406504065041,
>>>> 
>>>> -0.149904214559387, -0.149882903981265, -0.149797570850202,
>>>> -0.148496240601504,
>>>> 
>>>> -0.148325358851675, -0.147540983606557, -0.147239263803681,
>>>> -0.146989966555184,
>>>> 
>>>> -0.14622641509434, -0.146095717884131, -0.145994832041344,
>>>> -0.14572864321608,
>>>> 
>>>> -0.145161290322581, -0.144292237442922, -0.144144144144144,
>>>> -0.144021739130435,
>>>> 
>>>> -0.14375, -0.142212189616253, -0.141122913505311, -0.140324963072378,
>>>> 
>>>> -0.139344262295082, -0.13884007029877, -0.138356164383562,
>>>> -0.137626262626263,
>>>> 
>>>> -0.137142857142857, -0.136690647482014, -0.136577708006279,
>>>> -0.136363636363636,
>>>> 
>>>> -0.136094674556213, -0.135879774577332, -0.135586319218241,
>>>> -0.135135135135135,
>>>> 
>>>> -0.132780082987552, -0.132209405501331, -0.132023755139333,
>>>> -0.131233595800525,
>>>> 
>>>> -0.130434782608696, -0.130434782608696, -0.130268199233717,
>>>> -0.128813559322034,
>>>> 
>>>> -0.1284046692607, -0.128205128205128, -0.128182616330114,
>>>> -0.127937336814621,
>>>> 
>>>> -0.126283367556468, -0.125853658536585, -0.125448028673835,
>>>> -0.125425564840607,
>>>> 
>>>> -0.125311203319502, -0.125, -0.124401913875598, -0.124248496993988,
>>>> 
>>>> -0.124031007751938, -0.123572170301142, -0.123188405797102,
>>>> -0.122905027932961,
>>>> 
>>>> -0.121666666666667, -0.121573685907772, -0.120658135283364,
>>>> -0.120540019286403,
>>>> 
>>>> -0.119858156028369, -0.11965811965812, -0.11965811965812,
>>>> -0.119565217391304,
>>>> 
>>>> -0.118942731277533, -0.117820324005891, -0.116257947320618,
>>>> -0.115789473684211,
>>>> 
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>>>> -0.114492753623188,
>>>> 
>>>> -0.114357262103506, -0.114285714285714, -0.114035087719298,
>>>> -0.113181972212809,
>>>> 
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>>>> -0.112191473448018,
>>>> 
>>>> -0.111111111111111, -0.111111111111111, -0.110813226094727,
>>>> -0.110384300899428,
>>>> 
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>>>> -0.109792284866469,
>>>> 
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>>>> 
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>>>> 
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>>>> 
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>>>> 
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>>>> 
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>>>> 
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>>>> 
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>>>> 
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>>>> 
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>>>> 
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>>>> 
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>>>> 
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>>>> 
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>>>> 
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>>>> 
>>>> -0.09079754601227, -0.0907071455016661, -0.0906593406593406,
>>>> 
>>>> -0.0903614457831325, -0.0903323548906352, -0.09, -0.0897243107769424,
>>>> 
>>>> -0.0896358543417368, -0.0895522388059702, -0.0895052902487847,
>>>> 
>>>> -0.0891719745222929, -0.0888888888888888, -0.0887227819304518,
>>>> 
>>>> -0.0887096774193548, -0.0886956521739131, -0.0884703196347032,
>>>> 
>>>> -0.0884450784593437, -0.0884413309982488, -0.0883577310155536,
>>>> 
>>>> -0.0883054892601431, -0.0882917466410749, -0.0881628999776236,
>>>> 
>>>> -0.0881193929739248, -0.0880681818181819, -0.0878186968838525,
>>>> 
>>>> -0.087719298245614, -0.0876010781671159, -0.0873634945397815,
>>>> 
>>>> -0.0872641509433961, -0.0871512228728901, -0.0871032050299035,
>>>> 
>>>> -0.0868133772309825, -0.0865384615384615, -0.0858895705521473,
>>>> 
>>>> -0.085742525327403, -0.0855766209280403, -0.0854700854700855,
>>>> 
>>>> -0.0853994490358125, -0.0853658536585365, -0.0851063829787235,
>>>> 
>>>> -0.085048426150121, -0.0849673202614379, -0.084862385321101,
>>>> 
>>>> -0.0843335637529065, -0.0843222985633979, -0.084315503173164,
>>>> 
>>>> -0.0842105263157896, -0.0841423948220064, -0.0838323353293414,
>>>> 
>>>> -0.0836501901140684, -0.083422459893048, -0.0833333333333334,
>>>> 
>>>> -0.0833333333333333, -0.0827236916150815, -0.0826723611958512,
>>>> 
>>>> -0.0826039387308535, -0.0824242424242424, -0.0821513002364065,
>>>> 
>>>> -0.0820932539682539, -0.0820713238886174, -0.0818181818181818,
>>>> 
>>>> -0.0816618911174786, -0.0816613868356214, -0.0816326530612244,
>>>> 
>>>> -0.081619937694704, -0.0813748531139835, -0.0811965811965812,
>>>> 
>>>> -0.0810372771474878, -0.0809095716552088, -0.0808823529411764,
>>>> 
>>>> -0.0808414831492295, -0.0807511737089201, -0.0805722891566266,
>>>> 
>>>> -0.0803474484256242, -0.0800000000000001, -0.0800000000000001,
>>>> 
>>>> -0.0799584631360333, -0.0795941847087125, -0.0795930580490723,
>>>> 
>>>> -0.0794701986754966, -0.0793650793650794, -0.0791855203619909,
>>>> 
>>>> -0.0790273556231002, -0.0790273556231002, -0.0789473684210527,
>>>> 
>>>> -0.0789473684210526, -0.0787545787545788, -0.0787461773700307,
>>>> 
>>>> -0.0786724031960664, -0.0782122905027933, -0.078177727784027,
>>>> 
>>>> -0.078125, -0.0778443113772455, -0.0776310524209684, -0.0775862068965517,
>>>> 
>>>> -0.0775577557755776, -0.0773930753564156, -0.0772686433063791,
>>>> 
>>>> -0.0771513353115727, -0.0771200000000001, -0.0770370370370371,
>>>> 
>>>> -0.076923076923077, -0.076923076923077, -0.0769039940461424,
>>>> 
>>>> -0.076592082616179, -0.0763425253991292, -0.0762886597938145,
>>>> 
>>>> -0.0762886597938145, -0.0761565836298932, -0.0760869565217391,
>>>> 
>>>> -0.07597645799893, -0.0758377425044092, -0.0757314974182444,
>>>> 
>>>> -0.0755148741418765, -0.0754814305364511, -0.0754189944134078,
>>>> 
>>>> -0.0752351097178683, -0.0750988142292491, -0.074935400516796,
>>>> 
>>>> -0.0748663101604277, -0.0748031496062992, -0.0747663551401868,
>>>> 
>>>> -0.0747295968534907, -0.0747028862478777, -0.0745967741935485,
>>>> 
>>>> -0.0741784037558686, -0.0740740740740741, -0.0740740740740741,
>>>> 
>>>> -0.0740740740740741, -0.073828125, -0.0736698499317872,
>>>> -0.0735877862595421,
>>>> 
>>>> -0.0735824009102977, -0.0735632183908046, -0.0733944954128441,
>>>> 
>>>> -0.0728862973760932, -0.0728279386712096, -0.0726480203414457,
>>>> 
>>>> -0.0726329442282749, -0.072632190586868, -0.0725766664070006,
>>>> 
>>>> -0.0724137931034483, -0.0722664432341852, -0.0719257540603249,
>>>> 
>>>> -0.0717538461538461, -0.0716612377850163, -0.0715181932245923,
>>>> 
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>>>> 
>>>> -0.0708446866485014, -0.0708117443868739, -0.0708028321132845,
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>>>> -0.0706436420722135, -0.0705325930495579, -0.0705128205128205,
>>>> 
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>>>> 
>>>> -0.0701663201663202, -0.0700661736084078, -0.0699999999999999,
>>>> 
>>>> -0.0695540857691487, -0.0692979983084297, -0.0691823899371069,
>>>> 
>>>> -0.0691244239631336, -0.0690335305719921, -0.0689655172413792,
>>>> 
>>>> -0.0688536409516944, -0.068450039339103, -0.0682332955832389,
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>>>> -0.0682261208576998, -0.0681818181818182, -0.0679035250463822,
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>>>> -0.0675675675675676, -0.0675241157556271, -0.067463706233988,
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>>>> -0.0674584323040379, -0.0668523676880222, -0.0666666666666667,
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>>>> -0.066526455612132, -0.0664206642066421, -0.066321243523316,
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>>>> -0.0657534246575342, -0.0654481924472518, -0.0652900974262034,
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>>>> -0.0652225100070638, -0.0651126106388572, -0.0650887573964497,
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>>>> -0.0650684931506849, -0.0650195058517555, -0.0649954627949184,
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>>>> -0.0649081209247184, -0.064891846921797, -0.0648899188876013,
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>>>> -0.0642667310944672, -0.0641975308641976, -0.0641025641025641,
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>>>> -0.0619971604354,
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>>>> -0.0506575742815391, -0.0506024096385543, -0.0505494505494506,
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>>>> -0.046890780551147, -0.0467625899280576, -0.046753107010276,
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>>>> -0.0466592011944754, -0.0465772759350741, -0.0465753424657535,
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>>>> -0.046281851274051, -0.0459946060208471, -0.0458715596330275,
>>>> 
>>>> -0.0458563535911603, -0.0457367549668873, -0.0456500165398611,
>>>> 
>>>> -0.0454373701114679, -0.0454265548766046, -0.0454183266932271,
>>>> 
>>>> -0.045213508233324, -0.045182571340902, -0.0451018428709989,
>>>> 
>>>> -0.0451018428709989, -0.044955044955045, -0.0444136016655101,
>>>> 
>>>> -0.0443722943722945, -0.0442996742671011, -0.0442477876106195,
>>>> 
>>>> -0.0442477876106195, -0.0441176470588236, -0.043982895540623,
>>>> 
>>>> -0.0438184663536776, -0.0436363636363636, -0.043261231281198,
>>>> 
>>>> -0.0432218514782339, -0.0431654676258993, -0.0424336600525938,
>>>> 
>>>> -0.0423429781227946, -0.0423387096774193, -0.0420711974110032,
>>>> 
>>>> -0.0420083118050095, -0.0420032310177707, -0.0417256011315418,
>>>> 
>>>> -0.0416842105263158, -0.0416141235813367, -0.0414593698175788,
>>>> 
>>>> -0.0413793103448276, -0.041320942582144, -0.0407865986890023,
>>>> 
>>>> -0.0406158747013539, -0.0405803571428572, -0.0405643738977073,
>>>> 
>>>> -0.040422053796998, -0.040268456375839, -0.0400500625782228,
>>>> 
>>>> -0.0400449101796407, -0.04, -0.0399718147460123, -0.0399137001078749,
>>>> 
>>>> -0.0399044464954666, -0.0398863290368415, -0.0398230088495575,
>>>> 
>>>> -0.0394736842105263, -0.0393401015228426, -0.0392720306513411,
>>>> 
>>>> -0.0392156862745098, -0.0388349514563106, -0.0386904761904763,
>>>> 
>>>> -0.0384615384615385, -0.0382457345700217, -0.0382293762575454,
>>>> 
>>>> -0.0381840048105834, -0.0380030474141795, -0.0379746835443038,
>>>> 
>>>> -0.0379247280777747, -0.0376432078559739, -0.0375494071146245,
>>>> 
>>>> -0.0375490837696336, -0.0373345788572318, -0.0372940156114484,
>>>> 
>>>> -0.0371871275327772, -0.037037037037037, -0.0368133174791915,
>>>> 
>>>> -0.0368098159509203, -0.0367231638418078, -0.0361445783132531,
>>>> 
>>>> -0.0360915492957746, -0.0357142857142857, -0.0356663184599313,
>>>> 
>>>> -0.0356413696552528, -0.035513103110458, -0.0353194103194103,
>>>> 
>>>> -0.0352112676056338, -0.0350119904076739, -0.0349794238683129,
>>>> 
>>>> -0.0349373764007909, -0.0349091600875106, -0.0347826086956522,
>>>> 
>>>> -0.0347728547392037, -0.0347682119205298, -0.0346083788706739,
>>>> 
>>>> -0.0344706296166022, -0.0344421548425081, -0.0341176470588236,
>>>> 
>>>> -0.0338983050847458, -0.033457249070632, -0.0334208223972004,
>>>> 
>>>> -0.0333333333333334, -0.0333333333333333, -0.0329670329670331,
>>>> 
>>>> -0.0329144225014961, -0.0326797385620915, -0.0323033707865168,
>>>> 
>>>> -0.0322118826055834, -0.0321750321750322, -0.0320890635232481,
>>>> 
>>>> -0.0318302387267905, -0.0316666666666666, -0.0315586914688903,
>>>> 
>>>> -0.0312606184165817, -0.030881017257039, -0.0308764940239045,
>>>> 
>>>> -0.0305419952434597, -0.030406198638178, -0.0299651567944252,
>>>> 
>>>> -0.029945999018164, -0.0298786181139121, -0.0295741147960329,
>>>> 
>>>> -0.0293917033546227, -0.029171528588098, -0.0291327913279132,
>>>> 
>>>> -0.0290497291974397, -0.0288721376760064, -0.0287704170466792,
>>>> 
>>>> -0.0286236297198539, -0.0285714285714286, -0.0277575837684224,
>>>> 
>>>> -0.0277085471338513, -0.0275897304892068, -0.0275526742301458,
>>>> 
>>>> -0.027027027027027, -0.0270270270270269, -0.0269087523277468,
>>>> 
>>>> -0.0268918695148203, -0.02676704307821, -0.026454253142356,
>>>> -0.0263157894736842,
>>>> 
>>>> -0.0263157894736842, -0.0261660978384529, -0.025974025974026,
>>>> 
>>>> -0.0257503107796129, -0.0255319148936171, -0.0254237288135594,
>>>> 
>>>> -0.025297619047619, -0.0249174422095467, -0.0248888888888889,
>>>> 
>>>> -0.024767619719967, -0.0246053853296194, -0.0245269796776454,
>>>> 
>>>> -0.024390243902439, -0.0241935483870968, -0.0238907849829351,
>>>> 
>>>> -0.0238153295944078, -0.0234413496961306, -0.0232558139534884,
>>>> 
>>>> -0.0232558139534883, -0.0230392156862745, -0.0230360307147077,
>>>> 
>>>> -0.0228847365106026, -0.0225806451612903, -0.0224519940915805,
>>>> 
>>>> -0.0214786344110332, -0.0212360867018161, -0.0205245153933865,
>>>> 
>>>> -0.0204170602339606, -0.0200986321764215, -0.0200729927007299,
>>>> 
>>>> -0.0199828473413379, -0.0194174757281553, -0.0193536931818183,
>>>> 
>>>> -0.0192885771543086, -0.019222732971166, -0.0191414840759143,
>>>> 
>>>> -0.0189573459715641, -0.0188902007083826, -0.0186903321231681,
>>>> 
>>>> -0.0184456468273487, -0.0183066361556064, -0.0182166826462128,
>>>> 
>>>> -0.0181149908112366, -0.0179372197309418, -0.0179172441100772,
>>>> 
>>>> -0.0178571428571429, -0.0174672489082969, -0.0174216027874564,
>>>> 
>>>> -0.0171428571428572, -0.017017253604349, -0.0169252468265163,
>>>> 
>>>> -0.0165094339622642, -0.0158730158730158, -0.0158415841584159,
>>>> 
>>>> -0.0157247037374659, -0.0157089706490286, -0.0156250000000001,
>>>> 
>>>> -0.0153846153846153, -0.0151668351870576, -0.0151589242053789,
>>>> 
>>>> -0.0150489089541008, -0.0150262202501008, -0.0150081124932396,
>>>> 
>>>> -0.0148196281468128, -0.0144251166737377, -0.0142857142857142,
>>>> 
>>>> -0.0140638734251392, -0.0136986301369863, -0.0133333333333333,
>>>> 
>>>> -0.0129464285714286, -0.0129449838187702, -0.0127813811522321,
>>>> 
>>>> -0.0126030053320408, -0.0125721665147373, -0.0125, -0.0122160435399906,
>>>> 
>>>> -0.0122116689280869, -0.0121019108280255, -0.0118747750989563,
>>>> 
>>>> -0.0118277953189996, -0.0117107942973524, -0.0115172759138707,
>>>> 
>>>> -0.0114087997381585, -0.0113924050632912, -0.0111524163568772,
>>>> 
>>>> -0.0111248454882571, -0.0111111111111111, -0.0102570933795125,
>>>> 
>>>> -0.0102040816326532, -0.00963463463463469, -0.00958657878969439,
>>>> 
>>>> -0.00941028858218319, -0.00874453466583389, -0.00779588944011335,
>>>> 
>>>> -0.00713620850139625, -0.00710720027075056, -0.00709219858156026,
>>>> 
>>>> -0.00624999999999998, -0.00551977920883173, -0.00521499557217353,
>>>> 
>>>> -0.00453998797354176, -0.00444444444444444, -0.00166389351081533,
>>>> 
>>>> -0.00118483412322285, -0.00100704934541787, -0.000937500000000036,
>>>> 
>>>> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
>>>> 
>>>> 0.000218031178458569, 0.000796495420151402, 0.00167130919220062,
>>>> 
>>>> 0.00190476190476193, 0.00350058343057173, 0.00363914958820145,
>>>> 
>>>> 0.00432900432900424, 0.00456037942356804, 0.00477446915441642,
>>>> 
>>>> 0.0050825509873745, 0.00525691029336935, 0.00603318250377083,
>>>> 
>>>> 0.00652173913043472, 0.00733192494097176, 0.00747663551401869,
>>>> 
>>>> 0.00760456273764256, 0.00812529439472442, 0.00991231414410967,
>>>> 
>>>> 0.0101685066821615, 0.0106182565507793, 0.0109070034443169,
>>>> 0.0115183246073298,
>>>> 
>>>> 0.0115442244823827, 0.0127659574468085, 0.0128076846107665,
>>>> 0.0135135135135135,
>>>> 
>>>> 0.0140845070422535, 0.0150730098916627, 0.0159362549800796,
>>>> 0.0172413793103449,
>>>> 
>>>> 0.0176899063475547, 0.0186954715413378, 0.0187765706188003,
>>>> 0.0195757403189066,
>>>> 
>>>> 0.019607843137255, 0.0199999999999999, 0.0210526315789473,
>>>> 0.0221979621542939,
>>>> 
>>>> 0.0239410681399632, 0.0239999999999999, 0.025149051490515,
>>>> 0.0260869565217392,
>>>> 
>>>> 0.0262475696694749, 0.028180542563143, 0.0285714285714286,
>>>> 0.0303030303030302,
>>>> 
>>>> 0.0315904139433552, 0.0341864716636198, 0.0375782881002089,
>>>> 0.038479809976247,
>>>> 
>>>> 0.0387596899224806, 0.0416666666666667, 0.0428100987925357,
>>>> 0.0428134556574923,
>>>> 
>>>> 0.0437201907790144, 0.0451127819548872, 0.0460251046025105,
>>>> 0.0487779511180447,
>>>> 
>>>> 0.048975188781014, 0.0529872938632063, 0.0562390158172233,
>>>> 0.0589335827876521,
>>>> 
>>>> 0.0617551462621885, 0.0628115653040877, 0.0671812464265294,
>>>> 0.0721784776902887,
>>>> 
>>>> 0.0736842105263157, 0.0833333333333334, 0.083862394802894,
>>>> 0.0970464135021097,
>>>> 
>>>> 0.0971168437025795, 0.102272727272727, 0.111111111111111,
>>>> 0.117117117117117,
>>>> 
>>>> 0.123532699832309, 0.141304347826087, 0.179487179487179,
>>>> 0.191268191268191,
>>>> 
>>>> 0.205128205128205, 0.219020172910663, 0.271590909090909, 
>>>> 0.333333333333333
>>>> 
>>>> )
>>>> 
>>>> 
>>>> Now I want to estimate empirical PDF for interval of (-1, 1) including
>>>> all points of Dat
>>>> 
>>>> I have looked into the density() function, however it appears that,
>>>> empirical density is estimated for equally distant points, which not
>>>> necessarily contains actual supplied points (in my case, Dat)
>>>> 
>>>> Is there any option to achieve the same?
>>>> 
>>>> Thanks for your pointer.
>>>> 
>>>> ______________________________________________
>>>> R-help at r-project.org mailing list -- To UNSUBSCRIBE and more, see
>>>> https://stat.ethz.ch/mailman/listinfo/r-help
>>>> PLEASE do read the posting guide
>>>> http://www.R-project.org/posting-guide.html
>>>> and provide commented, minimal, self-contained, reproducible code.
>>> 
>>> 
>> 
>> ______________________________________________
>> R-help at r-project.org mailing list -- To UNSUBSCRIBE and more, see
>> https://stat.ethz.ch/mailman/listinfo/r-help
>> PLEASE do read the posting guide 
>> http://www.R-project.org/posting-guide.html
>> and provide commented, minimal, self-contained, reproducible code.
>> 
>
> ______________________________________________
> R-help at r-project.org mailing list -- To UNSUBSCRIBE and more, see
> https://stat.ethz.ch/mailman/listinfo/r-help
> PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
> and provide commented, minimal, self-contained, reproducible code.
>

---------------------------------------------------------------------------
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                                       Live:   OO#.. Dead: OO#..  Playing
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