DPdensity之学习

DPdensity {DPpackage} R Documentation

Semiparametric Bayesian density estimation using a DPM of normals Description

This function generates a posterior density sample for a Dirichlet process mixture of normals model.

Usage

DPdensity(y,ngrid=1000,grid=NULL,prior,mcmc,state,status, method=\ na.action=na.fail)

Arguments

y

a vector or matrix giving the data from which the density estimate is to be computed.

number of grid points where the density estimate is evaluated. This is only used if dimension of y is lower or equal than低于或等于 2. The default value is 1000.

ngrid

grid

matrix of dimension ngrid*nvar of grid points where the density estimate is evaluated. This is only used if dimension of y is lower or equal than 2. The default value缺省值 is NULL and the grid is chosen according to the range of the data.由数据范围决定

a list giving the prior information. The list includes the following

parameter: a0 and b0 giving the hyperparameters for prior distribution of the precision parameter of the Dirichlet process prior, alpha giving the value of the precision parameter (it must be specified if a0 is missing, see details below), nu2 and psiinv2 giving the hyperparameters of the inverted Wishart prior distribution for the scale matrix, Psi1, of the inverted Wishart part of the baseline distribution, tau1 and tau2 giving the hyperparameters for the gamma prior distribution of the scale

parameter k0 of the normal part of the baseline distribution, m2 and s2 giving the mean and the covariance of the normal prior for the mean, m1,

prior

of the normal component of the baseline distribution, respectively, nu1 and psiinv1 (it must be specified if nu2 is missing, see details below) giving the hyperparameters of the inverted Wishart part of the baseline distribution and, m1 giving the mean of the normal part of the baseline distribution (it must be specified if m2 is missing, see details below) and, k0 giving the scale parameter of the normal part of the baseline

distribution (it must be specified if tau1 is missing, see details below).超参数的给定

a list giving the MCMC parameters. The list must include the following integers: nburn giving the number of burn-in scans, nskip giving the thinning interval, nsave giving the total number of scans to be saved, and ndisplay giving the number of saved scans to be displayed on screen (the function reports on the screen when every ndisplay iterations have been carried out).

a list giving the current value现值 of the parameters. This list is used if the current analysis is the continuation of a previous analysis.

a logical variable indicating whether this run is new (TRUE) or the

continuation of a previous analysis (FALSE). In the latter case the current value of the parameters must be specified in the object state. the method to be used. See Details. data frame.

mcmc

state

status method data

a function that indicates what should happen when the data contain NAs.

na.action The default action (na.fail) causes DPdensity to print an error message

and terminate if there are any incomplete observations.默认出现NA结束

Details

This generic function fits a Dirichlet process mixture of normal model for density estimation (Escobar and West, 1995):

yi | mui, Sigmai ~ N(mui,Sigmai), i=1,…,n

(mui,Sigmai) | G ~ G G | alpha, G0 ~ DP(alpha G0)

where, the baseline distribution is the conjugate normal-inverted-Wishart,

G0 = N(mu| m1, (1/k0) Sigma) IW (Sigma | nu1, psi1)

To complete the model specification, independent hyperpriors are assumed (optional),

alpha | a0, b0 ~ Gamma(a0,b0)

m1 | m2, s2 ~ N(m2,s2)

k0 | tau1, tau2 ~ Gamma(tau1/2,tau2/2)

psi1 | nu2, psi2 ~ IW(nu2,psi2)

Note that the inverted-Wishart prior is parametrized such that if A ~ IWq(nu, psi) then E(A)= psiinv/(nu-q-1).

To let part of the baseline distribution fixed at a particular value, set the

corresponding hyperparameters of the prior distributions to NULL in the hyperprior specification of the model.

Although the baseline distribution, G0, is a conjugate prior in this model specification, the algorithms with auxiliary parameters described in MacEachern and Muller (1998) and Neal (2000) are adopted. Specifically, the no-gaps algorithm of MacEachern and Muller (1998), \, and the algorithm 8 with m=1 of Neal (2000), \, are considered in the DPdensity function. The default method is the algorithm 8 of Neal.

Value

An object of class DPdensity representing the DP mixture of normals model fit. Generic functions such as print, summary, and plot have methods to show the results of the fit. The results include the baseline parameters, alpha, and the number of clusters.

The function DPrandom can be used to extract the posterior mean of the subject-specific means and covariance matrices.

The MCMC samples of the parameters and the errors in the model are stored in the object thetasave and randsave, respectively. Both objects are included in the list save.state and are matrices which can be analyzed directly by functions provided by the coda package.

The list state in the output object contains the current value of the parameters

necessary to restart the analysis. If you want to specify different starting values to run multiple chains set status=TRUE and create the list state based on this starting values. In this case the list state must include the following objects: ncluster an integer giving the number of clusters.整数代表聚类数 muclus

a matrix of dimension (nobservations+100)*(nvariables) giving the means of the clusters (only the first ncluster are considered to start the chain).聚类均值？

sigmaclus a matrix of dimension

(nobservations+100)*( (nvariables)*((nvariables)+1)/2) giving the lower matrix of the covariance matrix of the clusters (only the first ncluster are considered to start the chain).

ss alpha m1 k0 psi1

an interger vector defining to which of the ncluster clusters each observation belongs.每一个观测值属于哪个类 giving the value of the precision parameter.

giving the mean of the normal components of the baseline distribution. giving the scale parameter of the normal part of the baseline distribution. giving the scale matrix of the inverted-Wishart part of the baseline distribution.

Author(s)

Alejandro Jara

References

Escobar, M.D. and West, M. (1995) Bayesian Density Estimation and Inference Using Mixtures. Journal of the American Statistical Association, 90: 577-588. MacEachern, S. N. and Muller, P. (1998) Estimating mixture of Dirichlet Process Models. Journal of Computational and Graphical Statistics, 7 (2): 223-338. Neal, R. M. (2000). Markov Chain sampling methods for Dirichlet process mixture models. Journal of Computational and Graphical Statistics, 9: 249-265.

See Also

DPrandom, PTdensity, BDPdensity

Examples

## Not run:

#################################### # Univariate example

####################################

# Data

先加载DPpackage

data(galaxy) speed 1 9172 2 9350 3 9483 4 9558 5 9775 6 10227 7 10406 8 16084 9 16170 10 18419 11 18552 12 18600 13 18927 14 19052 15 19070 16 19330 17 19343 18 19349 19 19440 20 19473 21 19529 22 19541 23 19547 24 19663 25 19846 26 19856 27 19863 28 19914 29 19918 30 19973 31 19989 32 20166 33 20175 34 20179 35 20196 36 20215 37 20221 38 20415 39 20629 40 20795 41 20821

42 20846 43 20875 44 20986 45 21137 46 21492 47 21701 48 21814 49 21921 50 21960 51 22185 52 22209 53 22242 54 22249 55 22314 56 22374 57 22495 58 22746 59 22747 60 22888 61 22914 62 23206 63 23241 64 23263 65 23484 66 23538 67 23542 68 23666 69 23706 70 23711 71 24129 72 24285 73 24289 74 24368 75 24717 76 24990 77 25633 78 26960 79 26995 80 32065 81 32789 82 34279

galaxy <- data.frame(galaxy,speeds=galaxy$speed/1000) speed speeds 1 9172 9.172

2 9350 9.350 3 9483 9.483 4 9558 9.558 5 9775 9.775 6 10227 10.227 7 10406 10.406 8 16084 16.084 9 16170 16.170 10 18419 18.419 11 18552 18.552 12 18600 18.600 13 18927 18.927 14 19052 19.052 15 19070 19.070 16 19330 19.330 17 19343 19.343 18 19349 19.349 19 19440 19.440 20 19473 19.473 21 19529 19.529 22 19541 19.541 23 19547 19.547 24 19663 19.663 25 19846 19.846 26 19856 19.856 27 19863 19.863 28 19914 19.914 29 19918 19.918 30 19973 19.973 31 19989 19.989 32 20166 20.166 33 20175 20.175 34 20179 20.179 35 20196 20.196 36 20215 20.215 37 20221 20.221 38 20415 20.415 39 20629 20.629 40 20795 20.795 41 20821 20.821 42 20846 20.846 43 20875 20.875 44 20986 20.986 45 21137 21.137

46 21492 21.492 47 21701 21.701 48 21814 21.814 49 21921 21.921 50 21960 21.960 51 22185 22.185 52 22209 22.209 53 22242 22.242 54 22249 22.249 55 22314 22.314 56 22374 22.374 57 22495 22.495 58 22746 22.746 59 22747 22.747 60 22888 22.888 61 22914 22.914 62 23206 23.206 63 23241 23.241 64 23263 23.263 65 23484 23.484 66 23538 23.538 67 23542 23.542 68 23666 23.666 69 23706 23.706 70 23711 23.711 71 24129 24.129 72 24285 24.285 73 24289 24.289 74 24368 24.368 75 24717 24.717 76 24990 24.990 77 25633 25.633 78 26960 26.960 79 26995 26.995 80 32065 32.065 81 32789 32.789 82 34279 34.279

attach(galaxy)

# Initial state state <- NULL

# MCMC parameters

nburn <- 1000 nsave <- 10000 nskip <- 10 ndisplay <- 100 mcmc <-

list(nburn=nburn,nsave=nsave,nskip=nskip,ndisplay=ndisplay)

# Example of Prior information 1 # Fixing alpha, m1, and Psi1

prior1 <- list(alpha=1,m1=rep(0,1),psiinv1=diag(0.5,1),nu1=4, tau1=1,tau2=100)

# Example of Prior information 2 # Fixing alpha and m1

prior2 <- list(alpha=1,m1=rep(0,1),psiinv2=solve(diag(0.5,1)), nu1=4,nu2=4,tau1=1,tau2=100)

# Example of Prior information 3 # Fixing only alpha

prior3 <- list(alpha=1,m2=rep(0,1),s2=diag(100000,1), psiinv2=solve(diag(0.5,1)), nu1=4,nu2=4,tau1=1,tau2=100)

# Example of Prior information 4

# Everything is random一般拿到数据应该所有的都未知

prior4 <- list(a0=2,b0=1,m2=rep(0,1),s2=diag(100000,1), psiinv2=solve(diag(0.5,1)), nu1=4,nu2=4,tau1=1,tau2=100) 不同的先验给出四组 # Fit the models

fit1.1 <- DPdensity(y=speeds,prior=prior1,mcmc=mcmc, state=state,status=TRUE)

fit1.2 <- DPdensity(y=speeds,prior=prior2,mcmc=mcmc, state=state,status=TRUE)

fit1.3 <- DPdensity(y=speeds,prior=prior3,mcmc=mcmc,

state=state,status=TRUE)

fit1.4 <- DPdensity(y=speeds,prior=prior4,mcmc=mcmc, state=state,status=TRUE) 花点时间运行

MCMC scan 10000 of 10000 (CPU time: 75.177 s) # Posterior means fit1.1

DPM of normals model for density estimation

Call:

DPdensity.default(y = speeds, prior = prior1, mcmc = mcmc, state = state, status = TRUE)

Posterior Predictive Distributions (log):

Min. 1st Qu. Median Mean 3rd Qu. Max. -15.670 -2.744 -2.145 -2.698 -1.986 -1.750

Posterior Inference of Parameters:仅估ko，n k0 ncluster 0.001293 4.553300

Number of Observations: 82 Number of Variables: 1 fit1.2

DPM of normals model for density estimation

Call:

DPdensity.default(y = speeds, prior = prior2, mcmc = mcmc, state = state, status = TRUE)

Posterior Predictive Distributions (log):

Min. 1st Qu. Median Mean 3rd Qu. Max. -7.549 -2.751 -2.130 -2.606 -1.999 -1.828

Posterior Inference of Parameters:

k0 psi1-speeds ncluster 0.002715 0.442537 4.507900

Number of Observations: 82 Number of Variables: 1 fit1.3

f normals model for density estimation

Call:

DPdensity.default(y = speeds, prior = prior3, mcmc = mcmc, state = state, status = TRUE)

Posterior Predictive Distributions (log):

Min. 1st Qu. Median Mean 3rd Qu. Max. -6.393 -2.747 -2.201 -2.563 -1.815 -1.581

Posterior Inference of Parameters:

m1-speeds k0 psi1-speeds ncluster 19.28765 0.01193 0.52103 6.99910

Number of Observations: 82 Number of Variables: 1

fit1.4

DPM of normals model for density estimation

Call:

DPdensity.default(y = speeds, prior = prior4, mcmc = mcmc, state = state, status = TRUE)

Posterior Predictive Distributions (log):

Min. 1st Qu. Median Mean 3rd Qu. Max. -7.845 -2.801 -2.218 -2.583 -1.806 -1.555

Posterior Inference of Parameters:估的所有参数

m1-speeds k0 psi1-speeds ncluster alpha 20.52637 0.01475 0.62578 11.08530 2.97309 信息越少，分的类越多

Number of Observations: 82 Number of Variables: 1

# Plot the estimated density plot(fit1.1,ask=FALSE) plot(fit1.2,ask=FALSE) plot(fit1.3,ask=FALSE) plot(fit1.4,ask=FALSE) > plot(fit1.1,ask=FALSE)

错误于plot.window(xlim, ylim, \值不能是无限的 此外: 警告信息：

1: In max(aa$intensities + aa$density) : max里所有的参数都不存在；回覆-Inf

2: In max(aa$intensities + aa$density) : max里所有的参数都不存在；回覆-Inf

> plot(fit1.2,ask=FALSE)

错误于plot.window(xlim, ylim, \值不能是无限的 此外: 警告信息：

1: In max(aa$intensities + aa$density) : max里所有的参数都不存在；回覆-Inf

2: In max(aa$intensities + aa$density) : max里所有的参数都不存在；回覆-Inf

> plot(fit1.3,ask=FALSE)

错误于plot.window(xlim, ylim, \值不能是无限的 此外: 警告信息：

1: In max(aa$intensities + aa$density) : max里所有的参数都不存在；回覆-Inf

2: In max(aa$intensities + aa$density) : max里所有的参数都不存在；回覆-Inf

> plot(fit1.4,ask=FALSE)

错误于plot.window(xlim, ylim, \值不能是无限的 此外: 警告信息：

1: In max(aa$intensities + aa$density) : max里所有的参数都不存在；回覆-Inf

2: In max(aa$intensities + aa$density) : max里所有的参数都不存在；回覆-Inf

# Extracting the density estimate cbind(fit1.1$x1,fit1.1$dens) > cbind(fit1.1$x1,fit1.1$dens) [,1] [,2] [1,] 6.887923 0.0002934141 [2,] 6.917628 0.0002919231 [3,] 6.947333 0.0002904421 [4,] 6.977038 0.0002890151 [5,] 7.006743 0.0002876935 [6,] 7.036447 0.0002865322 [7,] 7.066152 0.0002855868 [8,] 7.095857 0.0002849112 [9,] 7.125562 0.0002845553 [10,] 7.155267 0.0002845648 [11,] 7.184972 0.0002849808 [12,] 7.214677 0.0002858407 [13,] 7.244381 0.0002871791 [14,] 7.274086 0.0002890286 [15,] 7.303791 0.0002914206 [16,] 7.333496 0.0002943859

[17,] 7.363201 0.0002979553 [18,] 7.392906 0.0003021604 [19,] 7.422611 0.0003070352 [20,] 7.452315 0.0003126188 [21,] 7.482020 0.0003189580 [22,] 7.511725 0.0003261125 [23,] 7.541430 0.0003341582 [24,] 7.571135 0.0003431921 [25,] 7.600840 0.0003533360 [26,] 7.630545 0.0003647379 [27,] 7.660249 0.0003775738 [28,] 7.689954 0.0003920466 [29,] 7.719659 0.0004083850 [30,] 7.749364 0.0004268419 [31,] 7.779069 0.0004476934 [32,] 7.808774 0.0004712393 [33,] 7.838479 0.0004978052 [34,] 7.868183 0.0005277478 [35,] 7.897888 0.0005614616 [36,] 7.927593 0.0005993886 [37,] 7.957298 0.0006420295 [38,] 7.987003 0.0006899566 [39,] 8.016708 0.0007438270 [40,] 8.046413 0.0008043965 [41,] 8.076117 0.0008725331 [42,] 8.105822 0.0009492306 [43,] 8.135527 0.0010356217 [44,] 8.165232 0.0011329913 [45,] 8.194937 0.0012427898 [46,] 8.224642 0.0013666471 [47,] 8.254347 0.0015063881 [48,] 8.284051 0.0016640494 [49,] 8.313756 0.0018418972 [50,] 8.343461 0.0020424482 [51,] 8.373166 0.0022684916 [52,] 8.402871 0.0025231125 [53,] 8.432576 0.0028097164 [54,] 8.462281 0.0031320533 [55,] 8.491985 0.0034942409 [56,] 8.521690 0.0039007845 [57,] 8.551395 0.0043565927 [58,] 8.581100 0.0048669863 [59,] 8.610805 0.0054376986 [60,] 8.640510 0.0060748647

[61,] 8.670215 0.0067849971 [62,] 8.699919 0.0075749451 [63,] 8.729624 0.0084518356 [64,] 8.759329 0.0094229914 [65,] 8.789034 0.0104958265 [66,] 8.818739 0.0116777135 [67,] 8.848444 0.0129758223 [68,] 8.878149 0.0143969277 [69,] 8.907853 0.0159471860 [70,] 8.937558 0.0176318777 [71,] 8.967263 0.0194551220 [72,] 8.996968 0.0214195633 [73,] 9.026673 0.0235260388 [74,] 9.056378 0.0257732355 [75,] 9.086083 0.0281573511 [76,] 9.115787 0.0306717744 [77,] 9.145492 0.0333068052 [78,] 9.175197 0.0360494324 [79,] 9.204902 0.0388831933 [80,] 9.234607 0.0417881299 [81,] 9.264312 0.0447408604 [82,] 9.294017 0.0477147750 [83,] 9.323721 0.0506803625 [84,] 9.353426 0.0536056662 [85,] 9.383131 0.0564568610 [86,] 9.412836 0.0591989387 [87,] 9.442541 0.0617964785 [88,] 9.472246 0.0642144794 [89,] 9.501951 0.0664192220 [90,] 9.531656 0.0683791292 [91,] 9.561360 0.0700655904 [92,] 9.591065 0.0714537190 [93,] 9.620770 0.0725230120 [94,] 9.650475 0.0732578886 [95,] 9.680180 0.0736480870 [96,] 9.709885 0.0736889070 [97,] 9.739590 0.0733812917 [98,] 9.769294 0.0727317486 [99,] 9.798999 0.0717521150 [100,] 9.828704 0.0704591798 [101,] 9.858409 0.0688741778 [102,] 9.888114 0.0670221774 [103,] 9.917819 0.0649313849 [104,] 9.947524 0.0626323928

[105,] 9.977228 0.0601573992 [106,] 10.006933 0.0575394267 [107,] 10.036638 0.0548115680 [108,] 10.066343 0.0520062827 [109,] 10.096048 0.0491547654 [110,] 10.125753 0.0462864027 [111,] 10.155458 0.0434283302 [112,] 10.185162 0.0406050936 [113,] 10.214867 0.0378384170 [114,] 10.244572 0.0351470719 [115,] 10.274277 0.0325468406 [116,] 10.303982 0.0300505614 [117,] 10.333687 0.0276682443 [118,] 10.363392 0.0254072419 [119,] 10.393096 0.0232724623 [120,] 10.422801 0.0212666105 [121,] 10.452506 0.0193904463 [122,] 10.482211 0.0176430473 [123,] 10.511916 0.0160220693 [124,] 10.541621 0.0145239958 [125,] 10.571326 0.0131443713 [126,] 10.601030 0.0118780154 [127,] 10.630735 0.0107192151 [128,] 10.660440 0.0096618943 [129,] 10.690145 0.0086997608 [130,] 10.719850 0.0078264322 [131,] 10.749555 0.0070355417 [132,] 10.779260 0.0063208260 [133,] 10.808964 0.0056761973 [134,] 10.838669 0.0050957995 [135,] 10.868374 0.0045740532 [136,] 10.898079 0.0041056873 [137,] 10.927784 0.0036857611 [138,] 10.957489 0.0033096760 [139,] 10.987194 0.0029731795 [140,] 11.016898 0.0026723612 [141,] 11.046603 0.0024036438 [142,] 11.076308 0.0021637679 [143,] 11.106013 0.0019497750 [144,] 11.135718 0.0017589872 [145,] 11.165423 0.0015889867 [146,] 11.195128 0.0014375940 [147,] 11.224832 0.0013028479 [148,] 11.254537 0.0011829853

[149,] 11.284242 0.0010764226 [150,] 11.313947 0.0009817382 [151,] 11.343652 0.0008976568 [152,] 11.373357 0.0008230341 [153,] 11.403062 0.0007568440 [154,] 11.432766 0.0006981655 [155,] 11.462471 0.0006461728 [156,] 11.492176 0.0006001245 [157,] 11.521881 0.0005593554 [158,] 11.551586 0.0005232685 [159,] 11.581291 0.0004913280 [160,] 11.610996 0.0004630537 [161,] 11.640700 0.0004380150 [162,] 11.670405 0.0004158266 [163,] 11.700110 0.0003961446 [164,] 11.729815 0.0003786623 [165,] 11.759520 0.0003631079 [166,] 11.789225 0.0003492405 [167,] 11.818930 0.0003368486 [168,] 11.848634 0.0003257465 [169,] 11.878339 0.0003157729 [170,] 11.908044 0.0003067875 [171,] 11.937749 0.0002986696 [172,] 11.967454 0.0002913155 [173,] 11.997159 0.0002846366 [174,] 12.026864 0.0002785580 [175,] 12.056568 0.0002730167 [176,] 12.086273 0.0002679610 [177,] 12.115978 0.0002633491 [178,] 12.145683 0.0002591490 [179,] 12.175388 0.0002553372 [180,] 12.205093 0.0002518984 [181,] 12.234798 0.0002488238 [182,] 12.264503 0.0002461105 [183,] 12.294207 0.0002437592 [184,] 12.323912 0.0002417729 [185,] 12.353617 0.0002401545 [186,] 12.383322 0.0002389055 [187,] 12.413027 0.0002380238 [188,] 12.442732 0.0002375026 [189,] 12.472437 0.0002373297 [190,] 12.502141 0.0002374869 [191,] 12.531846 0.0002379500 [192,] 12.561551 0.0002386899

[193,] 12.591256 0.0002396727 [194,] 12.620961 0.0002408619 [195,] 12.650666 0.0002422194 [196,] 12.680371 0.0002437074 [197,] 12.710075 0.0002452900 [198,] 12.739780 0.0002469347 [199,] 12.769485 0.0002486141 [200,] 12.799190 0.0002503060 [201,] 12.828895 0.0002519951 [202,] 12.858600 0.0002536728 [203,] 12.888305 0.0002553371 [204,] 12.918009 0.0002569931 [205,] 12.947714 0.0002586520 [206,] 12.977419 0.0002603312 [207,] 13.007124 0.0002620532 [208,] 13.036829 0.0002638455 [209,] 13.066534 0.0002657391 [210,] 13.096239 0.0002677682 [211,] 13.125943 0.0002699688 [212,] 13.155648 0.0002723776 [213,] 13.185353 0.0002750307 [214,] 13.215058 0.0002779628 [215,] 13.244763 0.0002812055 [216,] 13.274468 0.0002847870 [217,] 13.304173 0.0002887308 [218,] 13.333877 0.0002930559 [219,] 13.363582 0.0002977762 [220,] 13.393287 0.0003029005 [221,] 13.422992 0.0003084331 [222,] 13.452697 0.0003143742 [223,] 13.482402 0.0003207203 [224,] 13.512107 0.0003274652 [225,] 13.541811 0.0003346011 [226,] 13.571516 0.0003421194 [227,] 13.601221 0.0003500124 [228,] 13.630926 0.0003582738 [229,] 13.660631 0.0003669007 [230,] 13.690336 0.0003758944 [231,] 13.720041 0.0003852616 [232,] 13.749745 0.0003950151 [233,] 13.779450 0.0004051744 [234,] 13.809155 0.0004157654 [235,] 13.838860 0.0004268206 [236,] 13.868565 0.0004383776

[237,] 13.898270 0.0004504784 [238,] 13.927975 0.0004631680 [239,] 13.957679 0.0004764922 [240,] 13.987384 0.0004904966 [241,] 14.017089 0.0005052247 [242,] 14.046794 0.0005207171 [243,] 14.076499 0.0005370113 [244,] 14.106204 0.0005541419 [245,] 14.135909 0.0005721420 [246,] 14.165613 0.0005910450 [247,] 14.195318 0.0006108877 [248,] 14.225023 0.0006317123 [249,] 14.254728 0.0006535700 [250,] 14.284433 0.0006765226 [251,] 14.314138 0.0007006444 [252,] 14.343843 0.0007260230 [253,] 14.373547 0.0007527587 [254,] 14.403252 0.0007809642 [255,] 14.432957 0.0008107629 [256,] 14.462662 0.0008422878 [257,] 14.492367 0.0008756804 [258,] 14.522072 0.0009110903 [259,] 14.551777 0.0009486749 [260,] 14.581481 0.0009886006 [261,] 14.611186 0.0010310438 [262,] 14.640891 0.0010761928 [263,] 14.670596 0.0011242496 [264,] 14.700301 0.0011754322 [265,] 14.730006 0.0012299761 [266,] 14.759711 0.0012881367 [267,] 14.789415 0.0013501905 [268,] 14.819120 0.0014164367 [269,] 14.848825 0.0014871987 [270,] 14.878530 0.0015628257 [271,] 14.908235 0.0016436941 [272,] 14.937940 0.0017302098 [273,] 14.967645 0.0018228101 [274,] 14.997350 0.0019219660 [275,] 15.027054 0.0020281851 [276,] 15.056759 0.0021420135 [277,] 15.086464 0.0022640385 [278,] 15.116169 0.0023948896 [279,] 15.145874 0.0025352392 [280,] 15.175579 0.0026858017

[281,] 15.205284 0.0028473314 [282,] 15.234988 0.0030206171 [283,] 15.264693 0.0032064759 [284,] 15.294398 0.0034057426 [285,] 15.324103 0.0036192561 [286,] 15.353808 0.0038478424 [287,] 15.383513 0.0040922920 [288,] 15.413218 0.0043533335 [289,] 15.442922 0.0046316008 [290,] 15.472627 0.0049275963 [291,] 15.502332 0.0052416473 [292,] 15.532037 0.0055738595 [293,] 15.561742 0.0059240657 [294,] 15.591447 0.0062917733 [295,] 15.621152 0.0066761124 [296,] 15.650856 0.0070757867 [297,] 15.680561 0.0074890307 [298,] 15.710266 0.0079135776 [299,] 15.739971 0.0083466403 [300,] 15.769676 0.0087849102 [301,] 15.799381 0.0092245769 [302,] 15.829086 0.0096613715 [303,] 15.858790 0.0100906362 [304,] 15.888495 0.0105074212 [305,] 15.918200 0.0109066094 [306,] 15.947905 0.0112830651 [307,] 15.977610 0.0116318052 [308,] 16.007315 0.0119481837 [309,] 16.037020 0.0122280820 [310,] 16.066724 0.0124680927 [311,] 16.096429 0.0126656837 [312,] 16.126134 0.0128193312 [313,] 16.155839 0.0129286079 [314,] 16.185544 0.0129942202 [315,] 16.215249 0.0130179887 [316,] 16.244954 0.0130027719 [317,] 16.274658 0.0129523415 [318,] 16.304363 0.0128712155 [319,] 16.334068 0.0127644647 [320,] 16.363773 0.0126375057 [321,] 16.393478 0.0124958949 [322,] 16.423183 0.0123451362 [323,] 16.452888 0.0121905126 [324,] 16.482592 0.0120369480

[325,] 16.512297 0.0118889052 [326,] 16.542002 0.0117503169 [327,] 16.571707 0.0116245518 [328,] 16.601412 0.0115144087 [329,] 16.631117 0.0114221347 [330,] 16.660822 0.0113494618 [331,] 16.690526 0.0112976549 [332,] 16.720231 0.0112675674 [333,] 16.749936 0.0112596988 [334,] 16.779641 0.0112742520 [335,] 16.809346 0.0113111870 [336,] 16.839051 0.0113702708 [337,] 16.868756 0.0114511219 [338,] 16.898460 0.0115532493 [339,] 16.928165 0.0116760870 [340,] 16.957870 0.0118190225 [341,] 16.987575 0.0119814223 [342,] 17.017280 0.0121626520 [343,] 17.046985 0.0123620930 [344,] 17.076690 0.0125791560 [345,] 17.106394 0.0128132916 [346,] 17.136099 0.0130639978 [347,] 17.165804 0.0133308264 [348,] 17.195509 0.0136133866 [349,] 17.225214 0.0139113477 [350,] 17.254919 0.0142244416 [351,] 17.284624 0.0145524631 [352,] 17.314328 0.0148952714 [353,] 17.344033 0.0152527904 [354,] 17.373738 0.0156250092 [355,] 17.403443 0.0160119835 [356,] 17.433148 0.0164138366 [357,] 17.462853 0.0168307608 [358,] 17.492558 0.0172630200 [359,] 17.522262 0.0177109521 [360,] 17.551967 0.0181749721 [361,] 17.581672 0.0186555763 [362,] 17.611377 0.0191533461 [363,] 17.641082 0.0196689527 [364,] 17.670787 0.0202031620 [365,] 17.700492 0.0207568401 [366,] 17.730197 0.0213309579 [367,] 17.759901 0.0219265975 [368,] 17.789606 0.0225449566

[369,] 17.819311 0.0231873547 [370,] 17.849016 0.0238552384 [371,] 17.878721 0.0245501868 [372,] 17.908426 0.0252739175 [373,] 17.938131 0.0260282924 [374,] 17.967835 0.0268153242 [375,] 17.997540 0.0276371821 [376,] 18.027245 0.0284961990 [377,] 18.056950 0.0293948774 [378,] 18.086655 0.0303358959 [379,] 18.116360 0.0313221143 [380,] 18.146065 0.0323565786 [381,] 18.175769 0.0334425241 [382,] 18.205474 0.0345833775 [383,] 18.235179 0.0357827573 [384,] 18.264884 0.0370444718 [385,] 18.294589 0.0383725158 [386,] 18.324294 0.0397710649 [387,] 18.353999 0.0412444664 [388,] 18.383703 0.0427972288 [389,] 18.413408 0.0444340070 [390,] 18.443113 0.0461595840 [391,] 18.472818 0.0479788476 [392,] 18.502523 0.0498967627 [393,] 18.532228 0.0519183363 [394,] 18.561933 0.0540485772 [395,] 18.591637 0.0562924469 [396,] 18.621342 0.0586548034 [397,] 18.651047 0.0611403361 [398,] 18.680752 0.0637534924 [399,] 18.710457 0.0664983946 [400,] 18.740162 0.0693787479 [401,] 18.769867 0.0723977398 [402,] 18.799571 0.0755579301 [403,] 18.829276 0.0788611325 [404,] 18.858981 0.0823082890 [405,] 18.888686 0.0858993375 [406,] 18.918391 0.0896330745 [407,] 18.948096 0.0935070146 [408,] 18.977801 0.0975172495 [409,] 19.007505 0.1016583092 [410,] 19.037210 0.1059230287 [411,] 19.066915 0.1103024245 [412,] 19.096620 0.1147855849

[413,] 19.126325 0.1193595796 [414,] 19.156030 0.1240093923 [415,] 19.185735 0.1287178831 [416,] 19.215439 0.1334657862 [417,] 19.245144 0.1382317471 [418,] 19.274849 0.1429924055 [419,] 19.304554 0.1477225289 [420,] 19.334259 0.1523951996 [421,] 19.363964 0.1569820593 [422,] 19.393669 0.1614536111 [423,] 19.423373 0.1657795793 [424,] 19.453078 0.1699293240 [425,] 19.482783 0.1738723040 [426,] 19.512488 0.1775785824 [427,] 19.542193 0.1810193623 [428,] 19.571898 0.1841675417 [429,] 19.601603 0.1869982716 [430,] 19.631307 0.1894895020 [431,] 19.661012 0.1916224985 [432,] 19.690717 0.1933823104 [433,] 19.720422 0.1947581736 [434,] 19.750127 0.1957438297 [435,] 19.779832 0.1963377453 [436,] 19.809537 0.1965432179 [437,] 19.839241 0.1963683592 [438,] 19.868946 0.1958259484 [439,] 19.898651 0.1949331588 [440,] 19.928356 0.1937111637 [441,] 19.958061 0.1921846368 [442,] 19.987766 0.1903811661 [443,] 20.017471 0.1883306069 [444,] 20.047175 0.1860644010 [445,] 20.076880 0.1836148885 [446,] 20.106585 0.1810146406 [447,] 20.136290 0.1782958335 [448,] 20.165995 0.1754896853 [449,] 20.195700 0.1726259678 [450,] 20.225405 0.1697326047 [451,] 20.255109 0.1668353596 [452,] 20.284814 0.1639576151 [453,] 20.314519 0.1611202391 [454,] 20.344224 0.1583415312 [455,] 20.373929 0.1556372401 [456,] 20.403634 0.1530206388

[457,] 20.433339 0.1505026471 [458,] 20.463044 0.1480919870 [459,] 20.492748 0.1457953606 [460,] 20.522453 0.1436176400 [461,] 20.552158 0.1415620599 [462,] 20.581863 0.1396304090 [463,] 20.611568 0.1378232116 [464,] 20.641273 0.1361399006 [465,] 20.670978 0.1345789761 [466,] 20.700682 0.1331381513 [467,] 20.730387 0.1318144841 [468,] 20.760092 0.1306044950 [469,] 20.789797 0.1295042721 [470,] 20.819502 0.1285095632 [471,] 20.849207 0.1276158578 [472,] 20.878912 0.1268184575 [473,] 20.908616 0.1261125384 [474,] 20.938321 0.1254932038 [475,] 20.968026 0.1249555314 [476,] 20.997731 0.1244946124 [477,] 21.027436 0.1241055855 [478,] 21.057141 0.1237836659 [479,] 21.086846 0.1235241693 [480,] 21.116550 0.1233225316 [481,] 21.146255 0.1231743249 [482,] 21.175960 0.1230752701 [483,] 21.205665 0.1230212457 [484,] 21.235370 0.1230082941 [485,] 21.265075 0.1230326243 [486,] 21.294780 0.1230906135 [487,] 21.324484 0.1231788051 [488,] 21.354189 0.1232939059 [489,] 21.383894 0.1234327812 [490,] 21.413599 0.1235924493 [491,] 21.443304 0.1237700748 [492,] 21.473009 0.1239629615 [493,] 21.502714 0.1241685455 [494,] 21.532418 0.1243843870 [495,] 21.562123 0.1246081638 [496,] 21.591828 0.1248376637 [497,] 21.621533 0.1250707776 [498,] 21.651238 0.1253054933 [499,] 21.680943 0.1255398890 [500,] 21.710648 0.1257721278

[501,] 21.740352 0.1260004525 [502,] 21.770057 0.1262231809 [503,] 21.799762 0.1264387020 [504,] 21.829467 0.1266454728 [505,] 21.859172 0.1268420153 [506,] 21.888877 0.1270269147 [507,] 21.918582 0.1271988177 [508,] 21.948286 0.1273564318 [509,] 21.977991 0.1274985243 [510,] 22.007696 0.1276239226 [511,] 22.037401 0.1277315140 [512,] 22.067106 0.1278202458 [513,] 22.096811 0.1278891265 [514,] 22.126516 0.1279372261 [515,] 22.156220 0.1279636772 [516,] 22.185925 0.1279676759 [517,] 22.215630 0.1279484825 [518,] 22.245335 0.1279054223 [519,] 22.275040 0.1278378858 [520,] 22.304745 0.1277453281 [521,] 22.334450 0.1276272683 [522,] 22.364154 0.1274832877 [523,] 22.393859 0.1273130273 [524,] 22.423564 0.1271161847 [525,] 22.453269 0.1268925106 [526,] 22.482974 0.1266418045 [527,] 22.512679 0.1263639105 [528,] 22.542384 0.1260587132 [529,] 22.572088 0.1257261336 [530,] 22.601793 0.1253661251 [531,] 22.631498 0.1249786709 [532,] 22.661203 0.1245637808 [533,] 22.690908 0.1241214892 [534,] 22.720613 0.1236518537 [535,] 22.750318 0.1231549536 [536,] 22.780022 0.1226308892 [537,] 22.809727 0.1220797814 [538,] 22.839432 0.1215017715 [539,] 22.869137 0.1208970209 [540,] 22.898842 0.1202657113 [541,] 22.928547 0.1196080455 [542,] 22.958252 0.1189242476 [543,] 22.987956 0.1182145643 [544,] 23.017661 0.1174792659

[545,] 23.047366 0.1167186483 [546,] 23.077071 0.1159330345 [547,] 23.106776 0.1151227770 [548,] 23.136481 0.1142882598 [549,] 23.166186 0.1134299007 [550,] 23.195891 0.1125481529 [551,] 23.225595 0.1116435066 [552,] 23.255300 0.1107164894 [553,] 23.285005 0.1097676664 [554,] 23.314710 0.1087976390 [555,] 23.344415 0.1078070430 [556,] 23.374120 0.1067965459 [557,] 23.403825 0.1057668431 [558,] 23.433529 0.1047186543 [559,] 23.463234 0.1036527191 [560,] 23.492939 0.1025697926 [561,] 23.522644 0.1014706418 [562,] 23.552349 0.1003560418 [563,] 23.582054 0.0992267735 [564,] 23.611759 0.0980836212 [565,] 23.641463 0.0969273716 [566,] 23.671168 0.0957588137 [567,] 23.700873 0.0945787388 [568,] 23.730578 0.0933879420 [569,] 23.760283 0.0921872230 [570,] 23.789988 0.0909773878 [571,] 23.819693 0.0897592505 [572,] 23.849397 0.0885336340 [573,] 23.879102 0.0873013715 [574,] 23.908807 0.0860633059 [575,] 23.938512 0.0848202897 [576,] 23.968217 0.0835731834 [577,] 23.997922 0.0823228530 [578,] 24.027627 0.0810701673 [579,] 24.057331 0.0798159938 [580,] 24.087036 0.0785611943 [581,] 24.116741 0.0773066203 [582,] 24.146446 0.0760531083 [583,] 24.176151 0.0748014751 [584,] 24.205856 0.0735525134 [585,] 24.235561 0.0723069886 [586,] 24.265265 0.0710656357 [587,] 24.294970 0.0698291573 [588,] 24.324675 0.0685982230

[589,] 24.354380 0.0673734686 [590,] 24.384085 0.0661554972 [591,] 24.413790 0.0649448805 [592,] 24.443495 0.0637421603 [593,] 24.473199 0.0625478511 [594,] 24.502904 0.0613624418 [595,] 24.532609 0.0601863980 [596,] 24.562314 0.0590201642 [597,] 24.592019 0.0578641653 [598,] 24.621724 0.0567188074 [599,] 24.651429 0.0555844792 [600,] 24.681133 0.0544615521 [601,] 24.710838 0.0533503799 [602,] 24.740543 0.0522512993 [603,] 24.770248 0.0511646286 [604,] 24.799953 0.0500906678 [605,] 24.829658 0.0490296977 [606,] 24.859363 0.0479819796 [607,] 24.889067 0.0469477550 [608,] 24.918772 0.0459272456 [609,] 24.948477 0.0449206531 [610,] 24.978182 0.0439281600 [611,] 25.007887 0.0429499295 [612,] 25.037592 0.0419861068 [613,] 25.067297 0.0410368191 [614,] 25.097001 0.0401021768 [615,] 25.126706 0.0391822743 [616,] 25.156411 0.0382771905 [617,] 25.186116 0.0373869898 [618,] 25.215821 0.0365117226 [619,] 25.245526 0.0356514261 [620,] 25.275231 0.0348061246 [621,] 25.304935 0.0339758306 [622,] 25.334640 0.0331605448 [623,] 25.364345 0.0323602571 [624,] 25.394050 0.0315749469 [625,] 25.423755 0.0308045837 [626,] 25.453460 0.0300491279 [627,] 25.483165 0.0293085307 [628,] 25.512869 0.0285827354 [629,] 25.542574 0.0278716772 [630,] 25.572279 0.0271752838 [631,] 25.601984 0.0264934759 [632,] 25.631689 0.0258261674

[633,] 25.661394 0.0251732661 [634,] 25.691099 0.0245346741 [635,] 25.720803 0.0239102882 [636,] 25.750508 0.0233000009 [637,] 25.780213 0.0227037006 [638,] 25.809918 0.0221212725 [639,] 25.839623 0.0215525991 [640,] 25.869328 0.0209975599 [641,] 25.899033 0.0204560321 [642,] 25.928738 0.0199278901 [643,] 25.958442 0.0194130046 [644,] 25.988147 0.0189112423 [645,] 26.017852 0.0184224649 [646,] 26.047557 0.0179465279 [647,] 26.077262 0.0174832802 [648,] 26.106967 0.0170325627 [649,] 26.136672 0.0165942082 [650,] 26.166376 0.0161680405 [651,] 26.196081 0.0157538742 [652,] 26.225786 0.0153515142 [653,] 26.255491 0.0149607553 [654,] 26.285196 0.0145813814 [655,] 26.314901 0.0142131640 [656,] 26.344606 0.0138558613 [657,] 26.374310 0.0135092152 [658,] 26.404015 0.0131729487 [659,] 26.433720 0.0128467621 [660,] 26.463425 0.0125303292 [661,] 26.493130 0.0122232927 [662,] 26.522835 0.0119252599 [663,] 26.552540 0.0116357992 [664,] 26.582244 0.0113544370 [665,] 26.611949 0.0110806570 [666,] 26.641654 0.0108139015 [667,] 26.671359 0.0105535755 [668,] 26.701064 0.0102990534 [669,] 26.730769 0.0100496904 [670,] 26.760474 0.0098048347 [671,] 26.790178 0.0095638446 [672,] 26.819883 0.0093261048 [673,] 26.849588 0.0090910451 [674,] 26.879293 0.0088581572 [675,] 26.908998 0.0086270101 [676,] 26.938703 0.0083972621

[677,] 26.968408 0.0081686692 [678,] 26.998112 0.0079410886 [679,] 27.027817 0.0077144786 [680,] 27.057522 0.0074888933 [681,] 27.087227 0.0072644744 [682,] 27.116932 0.0070414399 [683,] 27.146637 0.0068200705 [684,] 27.176342 0.0066006958 [685,] 27.206046 0.0063836790 [686,] 27.235751 0.0061694036 [687,] 27.265456 0.0059582593 [688,] 27.295161 0.0057506304 [689,] 27.324866 0.0055468851 [690,] 27.354571 0.0053473667 [691,] 27.384276 0.0051523860 [692,] 27.413980 0.0049622162 [693,] 27.443685 0.0047770887 [694,] 27.473390 0.0045971909 [695,] 27.503095 0.0044226659 [696,] 27.532800 0.0042536132 [697,] 27.562505 0.0040900906 [698,] 27.592210 0.0039321174 [699,] 27.621914 0.0037796784 [700,] 27.651619 0.0036327278 [701,] 27.681324 0.0034911939 [702,] 27.711029 0.0033549835 [703,] 27.740734 0.0032239858 [704,] 27.770439 0.0030980762 [705,] 27.800144 0.0029771199 [706,] 27.829848 0.0028609745 [707,] 27.859553 0.0027494923 [708,] 27.889258 0.0026425226 [709,] 27.918963 0.0025399131 [710,] 27.948668 0.0024415114 [711,] 27.978373 0.0023471661 [712,] 28.008078 0.0022567277 [713,] 28.037782 0.0021700490 [714,] 28.067487 0.0020869862 [715,] 28.097192 0.0020073991 [716,] 28.126897 0.0019311513 [717,] 28.156602 0.0018581107 [718,] 28.186307 0.0017881494 [719,] 28.216012 0.0017211442 [720,] 28.245716 0.0016569762

[721,] 28.275421 0.0015955309 [722,] 28.305126 0.0015366988 [723,] 28.334831 0.0014803745 [724,] 28.364536 0.0014264571 [725,] 28.394241 0.0013748506 [726,] 28.423946 0.0013254629 [727,] 28.453650 0.0012782067 [728,] 28.483355 0.0012329990 [729,] 28.513060 0.0011897611 [730,] 28.542765 0.0011484187 [731,] 28.572470 0.0011089016 [732,] 28.602175 0.0010711438 [733,] 28.631880 0.0010350833 [734,] 28.661585 0.0010006618 [735,] 28.691289 0.0009678249 [736,] 28.720994 0.0009365214 [737,] 28.750699 0.0009067035 [738,] 28.780404 0.0008783263 [739,] 28.810109 0.0008513475 [740,] 28.839814 0.0008257274 [741,] 28.869519 0.0008014281 [742,] 28.899223 0.0007784135 [743,] 28.928928 0.0007566486 [744,] 28.958633 0.0007360996 [745,] 28.988338 0.0007167333 [746,] 29.018043 0.0006985165 [747,] 29.047748 0.0006814163 [748,] 29.077453 0.0006653994 [749,] 29.107157 0.0006504322 [750,] 29.136862 0.0006364810 [751,] 29.166567 0.0006235119 [752,] 29.196272 0.0006114911 [753,] 29.225977 0.0006003856 [754,] 29.255682 0.0005901634 [755,] 29.285387 0.0005807946 [756,] 29.315091 0.0005722516 [757,] 29.344796 0.0005645101 [758,] 29.374501 0.0005575493 [759,] 29.404206 0.0005513528 [760,] 29.433911 0.0005459084 [761,] 29.463616 0.0005412086 [762,] 29.493321 0.0005372500 [763,] 29.523025 0.0005340337 [764,] 29.552730 0.0005315643

[765,] 29.582435 0.0005298496 [766,] 29.612140 0.0005289001 [767,] 29.641845 0.0005287285 [768,] 29.671550 0.0005293490 [769,] 29.701255 0.0005307771 [770,] 29.730959 0.0005330289 [771,] 29.760664 0.0005361213 [772,] 29.790369 0.0005400715 [773,] 29.820074 0.0005448971 [774,] 29.849779 0.0005506160 [775,] 29.879484 0.0005572467 [776,] 29.909189 0.0005648081 [777,] 29.938893 0.0005733201 [778,] 29.968598 0.0005828034 [779,] 29.998303 0.0005932802 [780,] 30.028008 0.0006047741 [781,] 30.057713 0.0006173108 [782,] 30.087418 0.0006309181 [783,] 30.117123 0.0006456266 [784,] 30.146827 0.0006614698 [785,] 30.176532 0.0006784847 [786,] 30.206237 0.0006967120 [787,] 30.235942 0.0007161966 [788,] 30.265647 0.0007369874 [789,] 30.295352 0.0007591384 [790,] 30.325057 0.0007827080 [791,] 30.354761 0.0008077597 [792,] 30.384466 0.0008343620 [793,] 30.414171 0.0008625885 [794,] 30.443876 0.0008925177 [795,] 30.473581 0.0009242331 [796,] 30.503286 0.0009578230 [797,] 30.532991 0.0009933802 [798,] 30.562695 0.0010310017 [799,] 30.592400 0.0010707885 [800,] 30.622105 0.0011128453 [801,] 30.651810 0.0011572797 [802,] 30.681515 0.0012042021 [803,] 30.711220 0.0012537254 [804,] 30.740925 0.0013059646 [805,] 30.770629 0.0013610366 [806,] 30.800334 0.0014190605 [807,] 30.830039 0.0014801576 [808,] 30.859744 0.0015444520

[809,] 30.889449 0.0016120713 [810,] 30.919154 0.0016831471 [811,] 30.948859 0.0017578164 [812,] 30.978563 0.0018362215 [813,] 31.008268 0.0019185115 [814,] 31.037973 0.0020048422 [815,] 31.067678 0.0020953765 [816,] 31.097383 0.0021902840 [817,] 31.127088 0.0022897405 [818,] 31.156793 0.0023939269 [819,] 31.186497 0.0025030283 [820,] 31.216202 0.0026172325 [821,] 31.245907 0.0027367285 [822,] 31.275612 0.0028617049 [823,] 31.305317 0.0029923484 [824,] 31.335022 0.0031288424 [825,] 31.364727 0.0032713657 [826,] 31.394432 0.0034200910 [827,] 31.424136 0.0035751838 [828,] 31.453841 0.0037368009 [829,] 31.483546 0.0039050892 [830,] 31.513251 0.0040801840 [831,] 31.542956 0.0042622075 [832,] 31.572661 0.0044512665 [833,] 31.602366 0.0046474502 [834,] 31.632070 0.0048508278 [835,] 31.661775 0.0050614454 [836,] 31.691480 0.0052793223 [837,] 31.721185 0.0055044479 [838,] 31.750890 0.0057367770 [839,] 31.780595 0.0059762259 [840,] 31.810300 0.0062226675 [841,] 31.840004 0.0064759268 [842,] 31.869709 0.0067357762 [843,] 31.899414 0.0070019310 [844,] 31.929119 0.0072740460 [845,] 31.958824 0.0075517120 [846,] 31.988529 0.0078344542 [847,] 32.018234 0.0081217315 [848,] 32.047938 0.0084129375 [849,] 32.077643 0.0087074028 [850,] 32.107348 0.0090043996 [851,] 32.137053 0.0093031468 [852,] 32.166758 0.0096028176

[853,] 32.196463 0.0099025471 [854,] 32.226168 0.0102014414 [855,] 32.255872 0.0104985864 [856,] 32.285577 0.0107930566 [857,] 32.315282 0.0110839238 [858,] 32.344987 0.0113702649 [859,] 32.374692 0.0116511684 [860,] 32.404397 0.0119257402 [861,] 32.434102 0.0121931081 [862,] 32.463806 0.0124524252 [863,] 32.493511 0.0127028716 [864,] 32.523216 0.0129436562 [865,] 32.552921 0.0131740173 [866,] 32.582626 0.0133932230 [867,] 32.612331 0.0136005719 [868,] 32.642036 0.0137953945 [869,] 32.671740 0.0139770545 [870,] 32.701445 0.0141449520 [871,] 32.731150 0.0142985270 [872,] 32.760855 0.0144372631 [873,] 32.790560 0.0145606923 [874,] 32.820265 0.0146683988 [875,] 32.849970 0.0147600228 [876,] 32.879674 0.0148352631 [877,] 32.909379 0.0148938792 [878,] 32.939084 0.0149356917 [879,] 32.968789 0.0149605826 [880,] 32.998494 0.0149684952 [881,] 33.028199 0.0149594333 [882,] 33.057904 0.0149334611 [883,] 33.087608 0.0148907031 [884,] 33.117313 0.0148313446 [885,] 33.147018 0.0147556317 [886,] 33.176723 0.0146638724 [887,] 33.206428 0.0145564359 [888,] 33.236133 0.0144337530 [889,] 33.265838 0.0142963143 [890,] 33.295542 0.0141446686 [891,] 33.325247 0.0139794196 [892,] 33.354952 0.0138012222 [893,] 33.384657 0.0136107781 [894,] 33.414362 0.0134088301 [895,] 33.444067 0.0131961564 [896,] 33.473772 0.0129735647

[897,] 33.503476 0.0127418854 [898,] 33.533181 0.0125019653 [899,] 33.562886 0.0122546612 [900,] 33.592591 0.0120008332 [901,] 33.622296 0.0117413388 [902,] 33.652001 0.0114770265 [903,] 33.681706 0.0112087304 [904,] 33.711410 0.0109372647 [905,] 33.741115 0.0106634183 [906,] 33.770820 0.0103879505 [907,] 33.800525 0.0101115864 [908,] 33.830230 0.0098350132 [909,] 33.859935 0.0095588768 [910,] 33.889640 0.0092837791 [911,] 33.919344 0.0090102758 [912,] 33.949049 0.0087388750 [913,] 33.978754 0.0084700363 [914,] 34.008459 0.0082041716 [915,] 34.038164 0.0079416456 [916,] 34.067869 0.0076827778 [917,] 34.097574 0.0074278459 [918,] 34.127279 0.0071770889 [919,] 34.156983 0.0069307112 [920,] 34.186688 0.0066888875 [921,] 34.216393 0.0064517671 [922,] 34.246098 0.0062194788 [923,] 34.275803 0.0059921348 [924,] 34.305508 0.0057698340 [925,] 34.335213 0.0055526651 [926,] 34.364917 0.0053407083 [927,] 34.394622 0.0051340363 [928,] 34.424327 0.0049327145 [929,] 34.454032 0.0047368003 [930,] 34.483737 0.0045463429 [931,] 34.513442 0.0043613814 [932,] 34.543147 0.0041819438 [933,] 34.572851 0.0040080463 [934,] 34.602556 0.0038396915 [935,] 34.632261 0.0036768687 [936,] 34.661966 0.0035195533 [937,] 34.691671 0.0033677067 [938,] 34.721376 0.0032212771 [939,] 34.751081 0.0030802003 [940,] 34.780785 0.0029444002

[941,] 34.810490 0.0028137901 [942,] 34.840195 0.0026882739 [943,] 34.869900 0.0025677466 [944,] 34.899605 0.0024520964 [945,] 34.929310 0.0023412045 [946,] 34.959015 0.0022349469 [947,] 34.988719 0.0021331950 [948,] 35.018424 0.0020358165 [949,] 35.048129 0.0019426760 [950,] 35.077834 0.0018536362 [951,] 35.107539 0.0017685583 [952,] 35.137244 0.0016873029 [953,] 35.166949 0.0016097309 [954,] 35.196653 0.0015357036 [955,] 35.226358 0.0014650838 [956,] 35.256063 0.0013977364 [957,] 35.285768 0.0013335284 [958,] 35.315473 0.0012723295 [959,] 35.345178 0.0012140125 [960,] 35.374883 0.0011584533 [961,] 35.404587 0.0011055314 [962,] 35.434292 0.0010551298 [963,] 35.463997 0.0010071349 [964,] 35.493702 0.0009614369 [965,] 35.523407 0.0009179298 [966,] 35.553112 0.0008765113 [967,] 35.582817 0.0008370827 [968,] 35.612521 0.0007995490 [969,] 35.642226 0.0007638193 [970,] 35.671931 0.0007298060 [971,] 35.701636 0.0006974255 [972,] 35.731341 0.0006665976 [973,] 35.761046 0.0006372460 [974,] 35.790751 0.0006092978 [975,] 35.820455 0.0005826837 [976,] 35.850160 0.0005573378 [977,] 35.879865 0.0005331975 [978,] 35.909570 0.0005102033 [979,] 35.939275 0.0004882990 [980,] 35.968980 0.0004674310 [981,] 35.998685 0.0004475488 [982,] 36.028389 0.0004286043 [983,] 36.058094 0.0004105518 [984,] 36.087799 0.0003933479

[985,] 36.117504 0.0003769512 [986,] 36.147209 0.0003613224 [987,] 36.176914 0.0003464236 [988,] 36.206619 0.0003322188 [989,] 36.236323 0.0003186733 [990,] 36.266028 0.0003057538 [991,] 36.295733 0.0002934284 [992,] 36.325438 0.0002816665 [993,] 36.355143 0.0002704387 [994,] 36.384848 0.0002597170 [995,] 36.414553 0.0002494746 [996,] 36.444257 0.0002396861 [997,] 36.473962 0.0002303277 [998,] 36.503667 0.0002213766 [999,] 36.533372 0.0002128120 [1000,] 36.563077 0.0002046143

cbind(fit1.2$x1,fit1.2$dens) cbind(fit1.3$x1,fit1.3$dens) cbind(fit1.4$x1,fit1.4$dens)

# Plot the parameters (only prior 2 for illustration)

# (to see the plots gradually set ask=TRUE)一张张的参数的图 plot(fit1.2,ask=FALSE,output=\

# Plot the a specific parameters

# (to see the plots gradually set ask=TRUE)只显示要求的 plot(fit1.2,ask=FALSE,output=\ nfigr=1,nfigc=2)

# Extracting the posterior mean of the specific # means and covariance matrices # (only prior 2 for illustration) DPrandom(fit1.2) > DPrandom(fit1.2)

Random effect information for the DP object:

Call:

DPdensity.default(y = speeds, prior = prior2, mcmc = mcmc, state = state, status = TRUE)

Posterior mean of subject-specific components:

mu-speeds var-speeds 1 9.6953 0.4884 2 9.6998 0.4867 3 9.7026 0.4938 4 9.7016 0.4895 5 9.7058 0.4855 6 9.7168 0.4919 7 9.7227 0.4975 8 19.5718 3.4006 9 19.5863 3.3918 10 20.8024 3.0366 11 20.7697 2.9610 12 20.7691 2.9496 13 20.7142 2.7965 14 20.6886 2.7431 15 20.6887 2.7316 16 20.6806 2.6852 17 20.6807 2.6947 18 20.6676 2.6764 19 20.6843 2.6769 20 20.6752 2.6628 21 20.6908 2.6862 22 20.6869 2.6767 23 20.6933 2.6766 24 20.6976 2.6831 25 20.7211 2.7078 26 20.7164 2.6991 27 20.7330 2.7112 28 20.7455 2.7291 29 20.7391 2.7308 30 20.7573 2.7422 31 20.7613 2.7550 32 20.8112 2.8155 33 20.8296 2.8421 34 20.8202 2.8203 35 20.8192 2.8179 36 20.8216 2.8283 37 20.8415 2.8422 38 20.9339 2.9486 39 21.0917 3.1361 40 21.2113 3.2582 41 21.2295 3.2784 42 21.2574 3.3021

43 21.2882 3.3141 44 21.3720 3.4144 45 21.5041 3.5108 46 21.7668 3.7045 47 21.8760 3.7680 48 21.9104 3.7926 49 21.9370 3.7906 50 21.9496 3.8151 51 22.0083 3.8418 52 21.9943 3.8050 53 22.0078 3.8331 54 22.0035 3.8280 55 22.0123 3.8286 56 22.0222 3.8222 57 22.0396 3.8228 58 22.0606 3.8216 59 22.0669 3.8422 60 22.0734 3.8424 61 22.0714 3.8427 62 22.0943 3.8413 63 22.0908 3.8424 64 22.0942 3.8489 65 22.1127 3.8536 66 22.1068 3.8523 67 22.1105 3.8351 68 22.1245 3.8448 69 22.1188 3.8523 70 22.1202 3.8454 71 22.1362 3.8604 72 22.1576 3.8924 73 22.1560 3.8577 74 22.1613 3.8890 75 22.1841 3.9122 76 22.2227 3.9596 77 22.3448 4.0127 78 22.7167 4.1157 79 22.7200 4.1273 80 32.8417 2.0389 81 32.8634 2.0264 82 32.9155 2.0338

# Ploting predictive information about the specific # means and covariance matrices # with HPD and Credibility intervals # (only prior 2 for illustration)

# (to see the plots gradually set ask=TRUE)

plot(DPrandom(fit1.2,predictive=TRUE),ask=FALSE)

plot(DPrandom(fit1.2,predictive=TRUE),ask=FALSE,hpd=FALSE)

# Ploting information about all the specific means # and covariance matrices

# with HPD and Credibility intervals # (only prior 2 for illustration)

# (to see the plots gradually set ask=TRUE) plot(DPrandom(fit1.2),ask=FALSE,hpd=FALSE)

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