# [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,
>>>>
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>>>> -0.148496240601504,
>>>>
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>>>> -0.146989966555184,
>>>>
>>>> -0.14622641509434, -0.146095717884131, -0.145994832041344,
>>>> -0.14572864321608,
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>>>> -0.144021739130435,
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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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>>>> -0.0458563535911603, -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.
>

---------------------------------------------------------------------------
Jeff Newmiller                        The     .....       .....  Go Live...
DCN:<jdnewmil at dcn.davis.ca.us>        Basics: ##.#.       ##.#.  Live Go...
Live:   OO#.. Dead: OO#..  Playing
Research Engineer (Solar/Batteries            O.O#.       #.O#.  with
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```