Introduction to R a computing software for statistical ...nunez/mastertecnologiastelecomu... · It is the top choice of statistical software among academic statisticians but also

The R Project for Statistical Computing Statistical Tables using R Monte Carlo method Resampling from data

Introduction to R – a computing software forstatistical analysis

Krzysztof PodgorskiDepartment of Mathematics and Statistics

University of Limerick

September 8, 2009


Quotation of the lecture

“I was still a couple of miles above the clouds when it broke,and with such violence I fell to the ground that I found myselfstunned, and in a hole nine fathoms under the grass, when Irecovered, hardly knowing how to get out again. Looking down,I observed that I had on a pair of boots with exceptionallysturdy straps. Grasping them firmly, I pulled with all my might.Soon I had hoist myself to the top and stepped out on terrafirma without further ado.”

R. E. Raspe, Singular Travels, Campaigns and Adventures ofBaron Munchausen, 1786.


Outline

1 The R Project for Statistical Computing

2 Statistical Tables using R

3 Monte Carlo method

4 Resampling from data


Downloading and installing the R-package

can be downloaded from the following webside:http://www.r-project.org/index.html

http://www.r-project.org/index.html


The highlights

The package is available for all popular operating systems:WindowsMac OS 10Linux

It is free!

Everyone (knowledgeable enough) can contribute to the software bywriting a package

Packages are available for download through a convienient facility

It is fairly well documented and the documentation is available eitherfrom the program help menu or from the website

It is the top choice of statistical software among academic statisticiansbut also very popular in industry specially among biostatisticians andmedical researchers (mostly due to the huge package calledBioconductor that is built on the top of R)

It is a powerful tool not only for doing statistics but also all kind ofscientific programming


The highlights


It is free!







The highlights

The package is available for all popular operating systems:

WindowsMac OS 10Linux

It is free!







The highlights

The package is available for all popular operating systems:Windows

Mac OS 10Linux

It is free!







The highlights

The package is available for all popular operating systems:WindowsMac OS 10

Linux

It is free!







The highlights


It is free!







The highlights


It is free!







The highlights


It is free!







The highlights


It is free!







The highlights


It is free!







The highlights


It is free!







The highlights


It is free!







What is R? – only some basic information

R is a language and environment for statistical computing and graphics.

R provides a wide variety of statistical and graphical techniques, and is highly extensible. Among its tools

one can find implementedlinear and nonlinear modelling,classical statistical tests,time-series analysis,classification,clustering,...

One of R’s strengths is the ease with which well-designed publication-quality plots can be produced,including mathematical symbols and formulae where needed.

R is an integrated suite of software facilities for data manipulation, calculation and graphical display. It

includesan effective data handling and storage facility,a suite of operators for calculations on arrays, in particular matrices,a large, coherent, integrated collection of intermediate tools for data analysis,graphical facilities for data analysis and display either on-screen or on hardcopy, anda well-developed, simple and effective programming language which includes conditionals, loops,user-defined recursive functions and input and output facilities.













one can find implemented

linear and nonlinear modelling,classical statistical tests,time-series analysis,classification,clustering,...








one can find implementedlinear and nonlinear modelling,

classical statistical tests,time-series analysis,classification,clustering,...








one can find implementedlinear and nonlinear modelling,classical statistical tests,

time-series analysis,classification,clustering,...








one can find implementedlinear and nonlinear modelling,classical statistical tests,time-series analysis,

classification,clustering,...








one can find implementedlinear and nonlinear modelling,classical statistical tests,time-series analysis,classification,

clustering,...








one can find implementedlinear and nonlinear modelling,classical statistical tests,time-series analysis,classification,clustering,

...



























includes

an effective data handling and storage facility,a suite of operators for calculations on arrays, in particular matrices,a large, coherent, integrated collection of intermediate tools for data analysis,graphical facilities for data analysis and display either on-screen or on hardcopy, anda well-developed, simple and effective programming language which includes conditionals, loops,user-defined recursive functions and input and output facilities.








includesan effective data handling and storage facility,

a suite of operators for calculations on arrays, in particular matrices,a large, coherent, integrated collection of intermediate tools for data analysis,graphical facilities for data analysis and display either on-screen or on hardcopy, anda well-developed, simple and effective programming language which includes conditionals, loops,user-defined recursive functions and input and output facilities.








includesan effective data handling and storage facility,a suite of operators for calculations on arrays, in particular matrices,

a large, coherent, integrated collection of intermediate tools for data analysis,graphical facilities for data analysis and display either on-screen or on hardcopy, anda well-developed, simple and effective programming language which includes conditionals, loops,user-defined recursive functions and input and output facilities.








includesan effective data handling and storage facility,a suite of operators for calculations on arrays, in particular matrices,a large, coherent, integrated collection of intermediate tools for data analysis,

graphical facilities for data analysis and display either on-screen or on hardcopy, anda well-developed, simple and effective programming language which includes conditionals, loops,user-defined recursive functions and input and output facilities.








includesan effective data handling and storage facility,a suite of operators for calculations on arrays, in particular matrices,a large, coherent, integrated collection of intermediate tools for data analysis,graphical facilities for data analysis and display either on-screen or on hardcopy, and

a well-developed, simple and effective programming language which includes conditionals, loops,user-defined recursive functions and input and output facilities.










Outline






Table a normal distribution

The following is a fragment of the table of values of F (x) for thestandard normal cumulative distribution function


The same “table” in RHere is a simple code in R that produce the same values as in the table

#Preceding line with the symbol ’#’ makes it a comment in R#The following line produce a single value of the standard normal cumulative#function. It is the value corresponding to the first value in the table

pnorm(-3.4)

#[1] 0.0003369293

#Then the first row of the table

z=seq(-3.4,-3.31,by=0.01)pnorm(z)

# [1] 0.0003369293 0.0003494631 0.0003624291 0.0003758409 0.0003897124# [6] 0.0004040578 0.0004188919 0.0004342299 0.0004500872 0.0004664799

#And all values from the table

z=seq(-3.4,3.4,by=0.01)pnorm(z)

[1] 0.0003369293 0.0003494631 0.0003624291 0.0003758409 0.0003897124 0.0004040578 0.0004188919 0.0004342299 0.0004500872 0.0004664799[11] 0.0004834241 0.0005009369 0.0005190354 0.0005377374 0.0005570611 0.0005770250 0.0005976485 0.0006189511 0.0006409530 0.0006636749[21] 0.0006871379 0.0007113640 0.0007363753 0.0007621947 0.0007888457 0.0008163523 0.0008447392 0.0008740315 0.0009042552 0.0009354367[31] 0.0009676032 0.0010007825 0.0010350030 0.0010702939 0.0011066850 0.0011442068 0.0011828907 0.0012227687 0.0012638734 0.0013062384[41] 0.0013498980 0.0013948872 0.0014412419 0.0014889987 0.0015381952 0.0015888696 0.0016410612 0.0016948100 0.0017501569 0.0018071438[51] 0.0018658133 0.0019262091 0.0019883759 0.0020523590 0.0021182050 0.0021859615 0.0022556767 0.0023274002 0.0024011825 0.0024770750[61] 0.0025551303 0.0026354021 0.0027179449 0.0028028146 0.0028900681 0.0029797632 0.0030719592 0.0031667163 0.0032640958 0.0033641604[71] 0.0034669738 0.0035726010 0.0036811080 0.0037925623 0.0039070326 0.0040245885 0.0041453014 0.0042692434 0.0043964883 0.0045271111[81] 0.0046611880 0.0047987966 0.0049400158 0.0050849257 0.0052336082 0.0053861460 0.0055426234 0.0057031263 0.0058677417 0.0060365581[91] 0.0062096653 0.0063871548 0.0065691191 0.0067556526 0.0069468508 0.0071428107 0.0073436310 0.0075494114 0.0077602536 0.0079762603

[101] 0.0081975359 0.0084241864 0.0086563190 0.0088940426 0.0091374675 0.0093867055 0.0096418699 0.0099030756 0.0101704387 0.0104440771[111] 0.0107241100 0.0110106583 0.0113038442 0.0116037915 0.0119106254 0.0122244727 0.0125454614 0.0128737214 0.0132093838 0.0135525811[121] 0.0139034475 0.0142621184 0.0146287308 0.0150034230 0.0153863348 0.0157776074 0.0161773834 0.0165858067 0.0170030226 0.0174291779[131] 0.0178644206 0.0183088999 0.0187627664 0.0192261722 0.0196992704 0.0201822154 0.0206751629 0.0211782696 0.0216916938 0.0222155944[141] 0.0227501319 0.0232954678 0.0238517643 0.0244191853 0.0249978951 0.0255880595 0.0261898449 0.0268034189 0.0274289497 0.0280666067[151] 0.0287165598 0.0293789800 0.0300540390 0.0307419089 0.0314427630 0.0321567748 0.0328841187 0.0336249694 0.0343795024 0.0351478936[161] 0.0359303191 0.0367269557 0.0375379803 0.0383635704 0.0392039033 0.0400591569 0.0409295090 0.0418151376 0.0427162208 0.0436329365[171] 0.0445654628 0.0455139773 0.0464786579 0.0474596818 0.0484572263 0.0494714680 0.0505025835 0.0515507485 0.0526161385 0.0536989281[181] 0.0547992917 0.0559174025 0.0570534332 0.0582075556 0.0593799406 0.0605707580 0.0617801767 0.0630083645 0.0642554878 0.0655217121[191] 0.0668072013 0.0681121180 0.0694366233 0.0707808770 0.0721450370 0.0735292596 0.0749336995 0.0763585095 0.0778038405 0.0792698415[201] 0.0807566592 0.0822644387 0.0837933224 0.0853434508 0.0869149619 0.0885079914 0.0901226725 0.0917591357 0.0934175090 0.0950979178[211] 0.0968004846 0.0985253290 0.1002725680 0.1020423151 0.1038346811 0.1056497737 0.1074876971 0.1093485524 0.1112324374 0.1131394464[221] 0.1150696702 0.1170231960 0.1190001075 0.1210004844 0.1230244031 0.1250719356 0.1271431506 0.1292381122 0.1313568810 0.1334995132[231] 0.1356660609 0.1378565720 0.1400710901 0.1423096544 0.1445722997 0.1468590564 0.1491699503 0.1515050028 0.1538642304 0.1562476450[241] 0.1586552539 0.1610870595 0.1635430593 0.1660232461 0.1685276075 0.1710561263 0.1736087803 0.1761855422 0.1787863796 0.1814112549[251] 0.1840601253 0.1867329430 0.1894296548 0.1921502021 0.1948945213 0.1976625431 0.2004541933 0.2032693918 0.2061080536 0.2089700879[261] 0.2118553986 0.2147638842 0.2176954376 0.2206499463 0.2236272924 0.2266273524 0.2296499972 0.2326950923 0.2357624978 0.2388520681[271] 0.2419636522 0.2450970937 0.2482522305 0.2514288951 0.2546269147 0.2578461108 0.2610862997 0.2643472921 0.2676288935 0.2709309038[281] 0.2742531178 0.2775953248 0.2809573089 0.2843388490 0.2877397188 0.2911596868 0.2945985162 0.2980559654 0.3015317875 0.3050257309[291] 0.3085375387 0.3120669494 0.3156136965 0.3191775088 0.3227581103 0.3263552203 0.3299685537 0.3335978206 0.3372427268 0.3409029738[301] 0.3445782584 0.3482682735 0.3519727076 0.3556912452 0.3594235668 0.3631693488 0.3669282640 0.3706999811 0.3744841653 0.3782804782[311] 0.3820885778 0.3859081188 0.3897387524 0.3935801268 0.3974318868 0.4012936743 0.4051651283 0.4090458849 0.4129355774 0.4168338365[321] 0.4207402906 0.4246545653 0.4285762841 0.4325050683 0.4364405371 0.4403823076 0.4443299952 0.4482832133 0.4522415740 0.4562046875[331] 0.4601721627 0.4641436074 0.4681186280 0.4720968298 0.4760778173 0.4800611942 0.4840465631 0.4880335266 0.4920216863 0.4960106437[341] 0.5000000000 0.5039893563 0.5079783137 0.5119664734 0.5159534369 0.5199388058 0.5239221827 0.5279031702 0.5318813720 0.5358563926[351] 0.5398278373 0.5437953125 0.5477584260 0.5517167867 0.5556700048 0.5596176924 0.5635594629 0.5674949317 0.5714237159 0.5753454347[361] 0.5792597094 0.5831661635 0.5870644226 0.5909541151 0.5948348717 0.5987063257 0.6025681132 0.6064198732 0.6102612476 0.6140918812[371] 0.6179114222 0.6217195218 0.6255158347 0.6293000189 0.6330717360 0.6368306512 0.6405764332 0.6443087548 0.6480272924 0.6517317265[381] 0.6554217416 0.6590970262 0.6627572732 0.6664021794 0.6700314463 0.6736447797 0.6772418897 0.6808224912 0.6843863035 0.6879330506[391] 0.6914624613 0.6949742691 0.6984682125 0.7019440346 0.7054014838 0.7088403132 0.7122602812 0.7156611510 0.7190426911 0.7224046752[401] 0.7257468822 0.7290690962 0.7323711065 0.7356527079 0.7389137003 0.7421538892 0.7453730853 0.7485711049 0.7517477695 0.7549029063[411] 0.7580363478 0.7611479319 0.7642375022 0.7673049077 0.7703500028 0.7733726476 0.7763727076 0.7793500537 0.7823045624 0.7852361158[421] 0.7881446014 0.7910299121 0.7938919464 0.7967306082 0.7995458067 0.8023374569 0.8051054787 0.8078497979 0.8105703452 0.8132670570[431] 0.8159398747 0.8185887451 0.8212136204 0.8238144578 0.8263912197 0.8289438737 0.8314723925 0.8339767539 0.8364569407 0.8389129405[441] 0.8413447461 0.8437523550 0.8461357696 0.8484949972 0.8508300497 0.8531409436 0.8554277003 0.8576903456 0.8599289099 0.8621434280[451] 0.8643339391 0.8665004868 0.8686431190 0.8707618878 0.8728568494 0.8749280644 0.8769755969 0.8789995156 0.8809998925 0.8829768040[461] 0.8849303298 0.8868605536 0.8887675626 0.8906514476 0.8925123029 0.8943502263 0.8961653189 0.8979576849 0.8997274320 0.9014746710[471] 0.9031995154 0.9049020822 0.9065824910 0.9082408643 0.9098773275 0.9114920086 0.9130850381 0.9146565492 0.9162066776 0.9177355613[481] 0.9192433408 0.9207301585 0.9221961595 0.9236414905 0.9250663005 0.9264707404 0.9278549630 0.9292191230 0.9305633767 0.9318878820[491] 0.9331927987 0.9344782879 0.9357445122 0.9369916355 0.9382198233 0.9394292420 0.9406200594 0.9417924444 0.9429465668 0.9440825975[501] 0.9452007083 0.9463010719 0.9473838615 0.9484492515 0.9494974165 0.9505285320 0.9515427737 0.9525403182 0.9535213421 0.9544860227[511] 0.9554345372 0.9563670635 0.9572837792 0.9581848624 0.9590704910 0.9599408431 0.9607960967 0.9616364296 0.9624620197 0.9632730443[521] 0.9640696809 0.9648521064 0.9656204976 0.9663750306 0.9671158813 0.9678432252 0.9685572370 0.9692580911 0.9699459610 0.9706210200[531] 0.9712834402 0.9719333933 0.9725710503 0.9731965811 0.9738101551 0.9744119405 0.9750021049 0.9755808147 0.9761482357 0.9767045322[541] 0.9772498681 0.9777844056 0.9783083062 0.9788217304 0.9793248371 0.9798177846 0.9803007296 0.9807738278 0.9812372336 0.9816911001[551] 0.9821355794 0.9825708221 0.9829969774 0.9834141933 0.9838226166 0.9842223926 0.9846136652 0.9849965770 0.9853712692 0.9857378816[561] 0.9860965525 0.9864474189 0.9867906162 0.9871262786 0.9874545386 0.9877755273 0.9880893746 0.9883962085 0.9886961558 0.9889893417[571] 0.9892758900 0.9895559229 0.9898295613 0.9900969244 0.9903581301 0.9906132945 0.9908625325 0.9911059574 0.9913436810 0.9915758136[581] 0.9918024641 0.9920237397 0.9922397464 0.9924505886 0.9926563690 0.9928571893 0.9930531492 0.9932443474 0.9934308809 0.9936128452[591] 0.9937903347 0.9939634419 0.9941322583 0.9942968737 0.9944573766 0.9946138540 0.9947663918 0.9949150743 0.9950599842 0.9952012034[601] 0.9953388120 0.9954728889 0.9956035117 0.9957307566 0.9958546986 0.9959754115 0.9960929674 0.9962074377 0.9963188920 0.9964273990[611] 0.9965330262 0.9966358396 0.9967359042 0.9968332837 0.9969280408 0.9970202368 0.9971099319 0.9971971854 0.9972820551 0.9973645979[621] 0.9974448697 0.9975229250 0.9975988175 0.9976725998 0.9977443233 0.9978140385 0.9978817950 0.9979476410 0.9980116241 0.9980737909[631] 0.9981341867 0.9981928562 0.9982498431 0.9983051900 0.9983589388 0.9984111304 0.9984618048 0.9985110013 0.9985587581 0.9986051128[641] 0.9986501020 0.9986937616 0.9987361266 0.9987772313 0.9988171093 0.9988557932 0.9988933150 0.9989297061 0.9989649970 0.9989992175[651] 0.9990323968 0.9990645633 0.9990957448 0.9991259685 0.9991552608 0.9991836477 0.9992111543 0.9992378053 0.9992636247 0.9992886360[661] 0.9993128621 0.9993363251 0.9993590470 0.9993810489 0.9994023515 0.9994229750 0.9994429389 0.9994622626 0.9994809646 0.9994990631[671] 0.9995165759 0.9995335201 0.9995499128 0.9995657701 0.9995811081 0.9995959422 0.9996102876 0.9996241591 0.9996375709 0.9996505369[681] 0.9996630707


There is more than meets the eye in the table

It is not only the table values that can be explored for thestandard normal distribution using R. Recall that the normaldistribution equally often referred to as a Gaussian distribution,is defined by the density

f (z) =1√2π

e−z2/2.

The density represents distribution of probability for arandom variable associated with it.The area under the density represents the probability sothe that the total area under it is equal to one.The area accumulated up to certain value z representsprobability that a corresponding random variable takesvalue smaller than z and this probability defines thecumulative distribution function F (z) which is tabularized.




f (z) =1√2π

e−z2/2.





f (z) =1√2π

e−z2/2.

The density represents distribution of probability for arandom variable associated with it.

The area under the density represents the probability sothe that the total area under it is equal to one.The area accumulated up to certain value z representsprobability that a corresponding random variable takesvalue smaller than z and this probability defines thecumulative distribution function F (z) which is tabularized.




f (z) =1√2π

e−z2/2.

The density represents distribution of probability for arandom variable associated with it.The area under the density represents the probability sothe that the total area under it is equal to one.

The area accumulated up to certain value z representsprobability that a corresponding random variable takesvalue smaller than z and this probability defines thecumulative distribution function F (z) which is tabularized.




f (z) =1√2π

e−z2/2.



All this can be seen in RThe following code explores various aspects of the standard normal distribution

#Plotting the density function of the standard normal variable

z=seq(-3,3,by=0.01)plot(z,dnorm(z),type=’l’,col="red",lwd=4)

#Plotting the cumulative distribution function (that one from the table)

plot(z,pnorm(z),type=’l’,col="red",lwd=4)

#And plotting them one at the top of the other

par(mfrow=c(2, 1))

plot(z,dnorm(z),type=’l’,col="red",lwd=4)


#Side by side

par(mfrow=c(1, 2))

plot(z,dnorm(z),type=’l’,col="red",lwd=4)



Outline






Deciding for Poisson model

Recall that the Poisson model is given by

P(X = x |θ) =θxe−θ

x!.

It is relatively easy to demonstrate that the mean value of thisdistribution is equal to θ and standard deviation is also equal to θ.

How?Thus for a sample of observations x = (x1, . . . , xn) it is reasonable toconsider both

θ1 = x,

θ2 = x2 − x2

as estimators of θ.Important Question

Which one is better?





x!.

It is relatively easy to demonstrate that the mean value of thisdistribution is equal to θ and standard deviation is also equal to θ.How?

Thus for a sample of observations x = (x1, . . . , xn) it is reasonable toconsider both

θ1 = x,

θ2 = x2 − x2







x!.

It is relatively easy to demonstrate that the mean value of thisdistribution is equal to θ and standard deviation is also equal to θ.How?Thus for a sample of observations x = (x1, . . . , xn) it is reasonable toconsider both

θ1 = x,

θ2 = x2 − x2

as estimators of θ.

Important Question






x!.

It is relatively easy to demonstrate that the mean value of thisdistribution is equal to θ and standard deviation is also equal to θ.How?Thus for a sample of observations x = (x1, . . . , xn) it is reasonable toconsider both

θ1 = x,

θ2 = x2 − x2




Monte Carlo study

Computational approach to the problem is to run many samples from thePoisson distribution and check which of the estimates performs better. This iswhat is done in the following R code:

%Generating in a loop samples of size n=20 from Poisson distribution with parameter theta=4%and evaluating theis means and standard deviationsn=20theta=4N=1000means=vector(’numeric’,N)vars=vector(’numeric’,N)for(i in 1:N){x=rpois(n,theta)means[i]=mean(x)vars[i]=var(x)

}

%Plotting histogramssplit.screen(c(2,1),1)hist(means)hist(vars)

It is quite clear from the graphs that the estimator based on the mean isbetter than the one based on the variance.


Monte Carlo study

Computational approach to the problem is to run many samples from thePoisson distribution and check which of the estimates performs better. This iswhat is done in the following R code:

%Generating in a loop samples of size n=20 from Poisson distribution with parameter theta=4%and evaluating theis means and standard deviationsn=20theta=4N=1000means=vector(’numeric’,N)vars=vector(’numeric’,N)for(i in 1:N){x=rpois(n,theta)means[i]=mean(x)vars[i]=var(x)

}

%Plotting histogramssplit.screen(c(2,1),1)hist(means)hist(vars)

It is quite clear from the graphs that the estimator based on the mean isbetter than the one based on the variance.


Outline






Nitrate ion concentration measurements in a certainchemical lab

Also in the file Table2_1.txt

0.51 0.51 0.51 0.50 0.51 0.49 0.52 0.53 0.50 0.470.51 0.52 0.53 0.48 0.49 0.50 0.52 0.49 0.49 0.500.49 0.48 0.46 0.49 0.49 0.48 0.49 0.49 0.51 0.470.51 0.51 0.51 0.48 0.50 0.47 0.50 0.51 0.49 0.480.51 0.50 0.50 0.53 0.52 0.52 0.50 0.50 0.51 0.51


The mean concentration

#Getting data in a vectorx=scan(’Table2_1.txt’)

mean(x)#[1] 0.4998

sd(x)#[1] 0.01647385

QuestionWhat is the error of this determination of the nitrate concentration?


Pulling yourselves up by bootstraps, i.e. something fornothing

If we would repeat our experiment of collecting 50 samples ofnitrate concentrations many times we would see the range oferror.

But it would be a waste of resources and not a viable method.

Instead we resample ‘new’ data from our data and use soobtained new samples for assessment of the error.

The following R code does the job.#Getting data in a vectorm=mean(x)bootstrap=vector(’numeric’,500)for(i in 1:500){bootstrap[i]=mean(sample(x,replace=T))-mean(x)}

#The distribution of estimation errorhist(boostrap)

We can safely say that the nitrate concentration is 49.99± 0.005.


Gaussian distribution or is it?

Abraham de Moivre French mathematician residing in England, anarticle in 1733 proved that coins count will follow Normal law,

Anecdote: He noted shortly before his death that it was necessary orhim to sleep a quarter of hour longer each day, he computed that the

day he will have to sleep 24 hours was on 27 November 1754. Hedeclared this date to be the day of his death and indeed he died on this

day.

Pierre-Simon, marquis de Laplace a great French mathematician andphysicist (when Gauss was two years old), an article in 1778.

Anecdote: His brain was removed by his physician and kept for manyyears, it was reportedly smaller than the average brain.

Carl Friedrich Gauss a great German mathematician and physicist, anarticle in 1809.

Anecdote: While his wife was lying sick, Gauss became engrossed in atheoretical problem. When the doctor told him that his wife was dying,

Gauss wave him away, Tell her, he ordered without looking up, to wait aminute until I’ve finished.






day.











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Stigler’s law of eponymy (Stephen Stigler, statisticsprofessor, University of Chicago)

No scientific discovery is named after its originaldiscoverer.

attributed to Robert K. Merton









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