Sangyeol_lee
Wednesday, April 08, 2015
###Reference : http://www.unt.edu/rss/class/Jon/R_SC/
####Why not use MLE? : Overfitting ####Why not use MAP? : No representive of our uncertainty ####Why use Bayesian? : Optimize Loss Function
####Bayesian methods focus on five essential elements.
####'First', the incorporation of prior information (e.g.,expert opinion, a thorough literature review of the same or similar variables, and/or prior data). Prior information is generally specified quantitatively in the form of a distribution (e.g., normal/Gaussian, Poisson, binomial, etc.)
####'Second', the prior is combined with a likelihood function. The likelihood function represents the data Second, the prior is combined with a likelihood function.
####'Third', the combination of the prior with the likelihood function results in the creation of a posterior distribution of coefficient values.
####'Fourth', simulates are drawn from the posterior distribution to create an empirical distribution of likely values for the population parameter.
####'Fifth', basic statistics are used to summarize the empirical distribution of simulates from the posterior
#traditional Ordinary Least Squares (OLS) regression, stepwise regression
library(ISLR)## Warning: package 'ISLR' was built under R version 3.1.2
data(Hitters)
head(Hitters)## AtBat Hits HmRun Runs RBI Walks Years CAtBat CHits
## -Andy Allanson 293 66 1 30 29 14 1 293 66
## -Alan Ashby 315 81 7 24 38 39 14 3449 835
## -Alvin Davis 479 130 18 66 72 76 3 1624 457
## -Andre Dawson 496 141 20 65 78 37 11 5628 1575
## -Andres Galarraga 321 87 10 39 42 30 2 396 101
## -Alfredo Griffin 594 169 4 74 51 35 11 4408 1133
## CHmRun CRuns CRBI CWalks League Division PutOuts Assists
## -Andy Allanson 1 30 29 14 A E 446 33
## -Alan Ashby 69 321 414 375 N W 632 43
## -Alvin Davis 63 224 266 263 A W 880 82
## -Andre Dawson 225 828 838 354 N E 200 11
## -Andres Galarraga 12 48 46 33 N E 805 40
## -Alfredo Griffin 19 501 336 194 A W 282 421
## Errors Salary NewLeague
## -Andy Allanson 20 NA A
## -Alan Ashby 10 475.0 N
## -Alvin Davis 14 480.0 A
## -Andre Dawson 3 500.0 N
## -Andres Galarraga 4 91.5 N
## -Alfredo Griffin 25 750.0 A
Hitters=na.omit(Hitters)
model.1 <- step(lm(Salary ~ ., data = Hitters, x = TRUE, y = TRUE),direction="backward")## Start: AIC=3046.02
## Salary ~ AtBat + Hits + HmRun + Runs + RBI + Walks + Years +
## CAtBat + CHits + CHmRun + CRuns + CRBI + CWalks + League +
## Division + PutOuts + Assists + Errors + NewLeague
##
## Df Sum of Sq RSS AIC
## - CHmRun 1 1138 24201837 3044.0
## - CHits 1 3930 24204629 3044.1
## - Years 1 7869 24208569 3044.1
## - NewLeague 1 9784 24210484 3044.1
## - RBI 1 16076 24216776 3044.2
## - HmRun 1 48572 24249272 3044.6
## - Errors 1 58324 24259023 3044.7
## - League 1 62121 24262821 3044.7
## - Runs 1 63291 24263990 3044.7
## - CRBI 1 135439 24336138 3045.5
## - CAtBat 1 159864 24360564 3045.8
## <none> 24200700 3046.0
## - Assists 1 280263 24480963 3047.1
## - CRuns 1 374007 24574707 3048.1
## - CWalks 1 609408 24810108 3050.6
## - Division 1 834491 25035190 3052.9
## - AtBat 1 971288 25171987 3054.4
## - Hits 1 991242 25191941 3054.6
## - Walks 1 1156606 25357305 3056.3
## - PutOuts 1 1319628 25520328 3058.0
##
## Step: AIC=3044.03
## Salary ~ AtBat + Hits + HmRun + Runs + RBI + Walks + Years +
## CAtBat + CHits + CRuns + CRBI + CWalks + League + Division +
## PutOuts + Assists + Errors + NewLeague
##
## Df Sum of Sq RSS AIC
## - Years 1 7609 24209447 3042.1
## - NewLeague 1 10268 24212106 3042.2
## - CHits 1 14003 24215840 3042.2
## - RBI 1 14955 24216793 3042.2
## - HmRun 1 52777 24254614 3042.6
## - Errors 1 59530 24261367 3042.7
## - League 1 63407 24265244 3042.7
## - Runs 1 64860 24266698 3042.7
## - CAtBat 1 174992 24376830 3043.9
## <none> 24201837 3044.0
## - Assists 1 285766 24487603 3045.1
## - CRuns 1 611358 24813196 3048.6
## - CWalks 1 645627 24847464 3049.0
## - Division 1 834637 25036474 3050.9
## - CRBI 1 864220 25066057 3051.3
## - AtBat 1 970861 25172699 3052.4
## - Hits 1 1025981 25227819 3052.9
## - Walks 1 1167378 25369216 3054.4
## - PutOuts 1 1325273 25527110 3056.1
##
## Step: AIC=3042.12
## Salary ~ AtBat + Hits + HmRun + Runs + RBI + Walks + CAtBat +
## CHits + CRuns + CRBI + CWalks + League + Division + PutOuts +
## Assists + Errors + NewLeague
##
## Df Sum of Sq RSS AIC
## - NewLeague 1 9931 24219377 3040.2
## - RBI 1 15989 24225436 3040.3
## - CHits 1 18291 24227738 3040.3
## - HmRun 1 54144 24263591 3040.7
## - Errors 1 57312 24266759 3040.7
## - Runs 1 63172 24272619 3040.8
## - League 1 65732 24275178 3040.8
## <none> 24209447 3042.1
## - CAtBat 1 266205 24475652 3043.0
## - Assists 1 293479 24502926 3043.3
## - CRuns 1 646350 24855797 3047.1
## - CWalks 1 649269 24858716 3047.1
## - Division 1 827511 25036958 3049.0
## - CRBI 1 872121 25081568 3049.4
## - AtBat 1 968713 25178160 3050.4
## - Hits 1 1018379 25227825 3050.9
## - Walks 1 1164536 25373983 3052.5
## - PutOuts 1 1334525 25543972 3054.2
##
## Step: AIC=3040.22
## Salary ~ AtBat + Hits + HmRun + Runs + RBI + Walks + CAtBat +
## CHits + CRuns + CRBI + CWalks + League + Division + PutOuts +
## Assists + Errors
##
## Df Sum of Sq RSS AIC
## - RBI 1 15800 24235177 3038.4
## - CHits 1 15859 24235237 3038.4
## - Errors 1 54505 24273883 3038.8
## - HmRun 1 54938 24274316 3038.8
## - Runs 1 62294 24281671 3038.9
## - League 1 107479 24326856 3039.4
## <none> 24219377 3040.2
## - CAtBat 1 261336 24480713 3041.1
## - Assists 1 295536 24514914 3041.4
## - CWalks 1 648860 24868237 3045.2
## - CRuns 1 661449 24880826 3045.3
## - Division 1 824672 25044049 3047.0
## - CRBI 1 880429 25099806 3047.6
## - AtBat 1 999057 25218434 3048.9
## - Hits 1 1034463 25253840 3049.2
## - Walks 1 1157205 25376583 3050.5
## - PutOuts 1 1335173 25554550 3052.3
##
## Step: AIC=3038.4
## Salary ~ AtBat + Hits + HmRun + Runs + Walks + CAtBat + CHits +
## CRuns + CRBI + CWalks + League + Division + PutOuts + Assists +
## Errors
##
## Df Sum of Sq RSS AIC
## - CHits 1 13483 24248660 3036.5
## - HmRun 1 44586 24279763 3036.9
## - Runs 1 54057 24289234 3037.0
## - Errors 1 57656 24292833 3037.0
## - League 1 108644 24343821 3037.6
## <none> 24235177 3038.4
## - CAtBat 1 252756 24487934 3039.1
## - Assists 1 294674 24529851 3039.6
## - CWalks 1 639690 24874868 3043.2
## - CRuns 1 693535 24928712 3043.8
## - Division 1 808984 25044161 3045.0
## - CRBI 1 893830 25129008 3045.9
## - Hits 1 1034884 25270061 3047.4
## - AtBat 1 1042798 25277975 3047.5
## - Walks 1 1145013 25380191 3048.5
## - PutOuts 1 1340713 25575890 3050.6
##
## Step: AIC=3036.54
## Salary ~ AtBat + Hits + HmRun + Runs + Walks + CAtBat + CRuns +
## CRBI + CWalks + League + Division + PutOuts + Assists + Errors
##
## Df Sum of Sq RSS AIC
## - HmRun 1 40487 24289148 3035.0
## - Errors 1 51930 24300590 3035.1
## - Runs 1 79343 24328003 3035.4
## - League 1 114742 24363402 3035.8
## <none> 24248660 3036.5
## - Assists 1 283442 24532103 3037.6
## - CAtBat 1 613356 24862016 3041.1
## - Division 1 801474 25050134 3043.1
## - CRBI 1 903248 25151908 3044.2
## - CWalks 1 1011953 25260613 3045.3
## - Walks 1 1246164 25494824 3047.7
## - AtBat 1 1339620 25588280 3048.7
## - CRuns 1 1390808 25639469 3049.2
## - PutOuts 1 1406023 25654684 3049.4
## - Hits 1 1607990 25856650 3051.4
##
## Step: AIC=3034.98
## Salary ~ AtBat + Hits + Runs + Walks + CAtBat + CRuns + CRBI +
## CWalks + League + Division + PutOuts + Assists + Errors
##
## Df Sum of Sq RSS AIC
## - Errors 1 44085 24333232 3033.5
## - Runs 1 49068 24338215 3033.5
## - League 1 103837 24392985 3034.1
## <none> 24289148 3035.0
## - Assists 1 247002 24536150 3035.6
## - CAtBat 1 652746 24941894 3040.0
## - Division 1 795643 25084791 3041.5
## - CWalks 1 982896 25272044 3043.4
## - Walks 1 1205823 25494971 3045.7
## - AtBat 1 1300972 25590120 3046.7
## - CRuns 1 1351200 25640348 3047.2
## - CRBI 1 1353507 25642655 3047.2
## - PutOuts 1 1429006 25718154 3048.0
## - Hits 1 1574140 25863288 3049.5
##
## Step: AIC=3033.46
## Salary ~ AtBat + Hits + Runs + Walks + CAtBat + CRuns + CRBI +
## CWalks + League + Division + PutOuts + Assists
##
## Df Sum of Sq RSS AIC
## - Runs 1 54113 24387345 3032.0
## - League 1 91269 24424501 3032.4
## <none> 24333232 3033.5
## - Assists 1 220010 24553242 3033.8
## - CAtBat 1 650513 24983746 3038.4
## - Division 1 799455 25132687 3040.0
## - CWalks 1 971260 25304493 3041.8
## - Walks 1 1239533 25572765 3044.5
## - CRBI 1 1331672 25664904 3045.5
## - CRuns 1 1361070 25694302 3045.8
## - AtBat 1 1378592 25711824 3045.9
## - PutOuts 1 1391660 25724892 3046.1
## - Hits 1 1649291 25982523 3048.7
##
## Step: AIC=3032.04
## Salary ~ AtBat + Hits + Walks + CAtBat + CRuns + CRBI + CWalks +
## League + Division + PutOuts + Assists
##
## Df Sum of Sq RSS AIC
## - League 1 113056 24500402 3031.3
## <none> 24387345 3032.0
## - Assists 1 280689 24668034 3033.1
## - CAtBat 1 596622 24983967 3036.4
## - Division 1 780369 25167714 3038.3
## - CWalks 1 946687 25334032 3040.1
## - Walks 1 1212997 25600342 3042.8
## - CRuns 1 1334397 25721742 3044.1
## - CRBI 1 1361339 25748684 3044.3
## - PutOuts 1 1455210 25842555 3045.3
## - AtBat 1 1522760 25910105 3046.0
## - Hits 1 1718870 26106215 3047.9
##
## Step: AIC=3031.26
## Salary ~ AtBat + Hits + Walks + CAtBat + CRuns + CRBI + CWalks +
## Division + PutOuts + Assists
##
## Df Sum of Sq RSS AIC
## <none> 24500402 3031.3
## - Assists 1 313650 24814051 3032.6
## - CAtBat 1 534156 25034558 3034.9
## - Division 1 798473 25298875 3037.7
## - CWalks 1 965875 25466276 3039.4
## - CRuns 1 1265082 25765484 3042.5
## - Walks 1 1290168 25790569 3042.8
## - CRBI 1 1326770 25827172 3043.1
## - PutOuts 1 1551523 26051925 3045.4
## - AtBat 1 1589780 26090181 3045.8
## - Hits 1 1716068 26216469 3047.1
summary(model.1)##
## Call:
## lm(formula = Salary ~ AtBat + Hits + Walks + CAtBat + CRuns +
## CRBI + CWalks + Division + PutOuts + Assists, data = Hitters,
## x = TRUE, y = TRUE)
##
## Residuals:
## Min 1Q Median 3Q Max
## -939.11 -176.87 -34.08 130.90 1910.55
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 162.53544 66.90784 2.429 0.015830 *
## AtBat -2.16865 0.53630 -4.044 7.00e-05 ***
## Hits 6.91802 1.64665 4.201 3.69e-05 ***
## Walks 5.77322 1.58483 3.643 0.000327 ***
## CAtBat -0.13008 0.05550 -2.344 0.019858 *
## CRuns 1.40825 0.39040 3.607 0.000373 ***
## CRBI 0.77431 0.20961 3.694 0.000271 ***
## CWalks -0.83083 0.26359 -3.152 0.001818 **
## DivisionW -112.38006 39.21438 -2.866 0.004511 **
## PutOuts 0.29737 0.07444 3.995 8.50e-05 ***
## Assists 0.28317 0.15766 1.796 0.073673 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 311.8 on 252 degrees of freedom
## Multiple R-squared: 0.5405, Adjusted R-squared: 0.5223
## F-statistic: 29.64 on 10 and 252 DF, p-value: < 2.2e-16
confint(model.1)## 2.5 % 97.5 %
## (Intercept) 30.76564039 294.30524369
## AtBat -3.22485010 -1.11245000
## Hits 3.67507200 10.16096299
## Walks 2.65203223 8.89441706
## CAtBat -0.23937493 -0.02078463
## CRuns 0.63939259 2.17710543
## CRBI 0.36150857 1.18711573
## CWalks -1.34995509 -0.31169761
## DivisionW -189.60973671 -35.15037828
## PutOuts 0.15076820 0.44397699
## Assists -0.02732144 0.59365750
#But what does this really tell us?#
library(LearnBayes)## Warning: package 'LearnBayes' was built under R version 3.1.2
#Gives a simulated sample from the joint posterior distribution of the regression vector and the error standard deviation for a linear regression model with a noninformative
joint.posterior.samples <- blinreg(model.1$y, model.1$x, 5000)
oldpar <- par(oma=c(0,0,3,0), mfrow=c(2,2))
hist(joint.posterior.samples$beta[,1], main = "Intercept", xlab = "beta 0")
hist(joint.posterior.samples$beta[,2], main = "AtBat", xlab = "beta 1")
hist(joint.posterior.samples$beta[,3], main = "Hits", xlab = "beta 2")
hist(joint.posterior.samples$beta[,4], main = "Walks", xlab = "beta 3")
par(oldpar)# To display the 95% credible intervals[Bayesian confidence interval:a posterior probability distribution used for interval estimation] (and medians) from the distributions, use an 'apply' function.
apply(joint.posterior.samples$beta, 2, quantile, c(.025, .500, .975))## X(Intercept) XAtBat XHits XWalks XCAtBat XCRuns
## 2.5% 28.76785 -3.236407 3.608064 2.607493 -0.23725174 0.6251287
## 50% 161.27018 -2.184659 6.949457 5.780419 -0.13173403 1.4100760
## 97.5% 292.03988 -1.098075 10.209252 8.886494 -0.01961774 2.2020981
## XCRBI XCWalks XDivisionW XPutOuts XAssists
## 2.5% 0.3604256 -1.3642696 -189.17936 0.1542088 -0.02701138
## 50% 0.7779286 -0.8313523 -112.29701 0.2976190 0.28687689
## 97.5% 1.1929659 -0.3216264 -33.07594 0.4476204 0.58842072
library(arm)## Warning: package 'arm' was built under R version 3.1.3
## Loading required package: MASS
## Warning: package 'MASS' was built under R version 3.1.2
## Loading required package: Matrix
## Warning: package 'Matrix' was built under R version 3.1.2
## Loading required package: lme4
## Warning: package 'lme4' was built under R version 3.1.2
## Loading required package: Rcpp
## Warning: package 'Rcpp' was built under R version 3.1.3
##
## arm (Version 1.8-4, built: 2015-04-07)
##
## Working directory is /Users/moodern/Google 드라이브/베이지안통계/part1/R/bayesianregression
# Conduct the Bayesian Generalized linear model (here family = Gaussian, as is default).
model.2 <- bayesglm(Salary~AtBat+Hits+Walks+CAtBat+CRuns+CRBI+CWalks+Division+PutOuts+Assists, family = "gaussian", data = Hitters, prior.scale=Inf, prior.df=Inf)
summary(model.2)##
## Call:
## bayesglm(formula = Salary ~ AtBat + Hits + Walks + CAtBat + CRuns +
## CRBI + CWalks + Division + PutOuts + Assists, family = "gaussian",
## data = Hitters, prior.scale = Inf, prior.df = Inf)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -939.11 -176.87 -34.08 130.91 1910.55
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 162.53058 66.90782 2.429 0.015833 *
## AtBat -2.16865 0.53630 -4.044 7.00e-05 ***
## Hits 6.91802 1.64665 4.201 3.69e-05 ***
## Walks 5.77322 1.58483 3.643 0.000327 ***
## CAtBat -0.13008 0.05550 -2.344 0.019858 *
## CRuns 1.40825 0.39040 3.607 0.000373 ***
## CRBI 0.77431 0.20961 3.694 0.000271 ***
## CWalks -0.83083 0.26359 -3.152 0.001818 **
## DivisionW -112.38006 39.21438 -2.866 0.004511 **
## PutOuts 0.29737 0.07444 3.995 8.50e-05 ***
## Assists 0.28317 0.15766 1.796 0.073673 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for gaussian family taken to be 97223.82)
##
## Null deviance: 53319113 on 262 degrees of freedom
## Residual deviance: 24500402 on 252 degrees of freedom
## AIC: 3779.6
##
## Number of Fisher Scoring iterations: 7
simulates <- coef(sim(model.2))
head(simulates)## (Intercept) AtBat Hits Walks CAtBat CRuns
## [1,] 245.9365 -2.564983 8.406301 6.018244 -0.24366964 1.9861266
## [2,] 188.7658 -1.515596 5.963264 2.648046 -0.07708377 0.6727034
## [3,] 119.9218 -2.021112 7.147965 4.939572 -0.14827582 2.1962228
## [4,] 157.3588 -2.249339 6.244926 7.303011 -0.06406493 1.0822331
## [5,] 196.0815 -2.519319 7.709565 6.215528 -0.09313694 1.4661667
## [6,] 221.3343 -2.291642 7.244021 4.692335 -0.15535993 1.2913480
## CRBI CWalks DivisionW PutOuts Assists
## [1,] 1.2601220 -1.2663429 -171.8177 0.3053617 0.2984496540
## [2,] 0.9139821 -0.5559680 -191.2886 0.3095893 0.1591645637
## [3,] 0.2485923 -0.9565678 -148.0399 0.3211596 0.0005705172
## [4,] 0.9285909 -1.0882823 -114.2606 0.2530632 0.3171362772
## [5,] 0.7367792 -1.2145861 -191.4720 0.4424243 0.3083065025
## [6,] 0.8667205 -0.4926049 -145.2815 0.3220155 0.5428532727
posterior.open <- simulates[,2]
head(posterior.open)## [1] -2.564983 -1.515596 -2.021112 -2.249339 -2.519319 -2.291642
par(mfrow=c(1,1))
hist(posterior.open)
plot(density(posterior.open), main = "", xlab = "Posterior.open", ylab = "Density")
# Retrieve the 95% credible interval for the open variable's coefficient.
quantile(posterior.open, c(.025, .975))## 2.5% 97.5%
## -3.139237 -1.306707
library(MCMCpack)## Warning: package 'MCMCpack' was built under R version 3.1.2
## Loading required package: coda
## Warning: package 'coda' was built under R version 3.1.3
##
## Attaching package: 'coda'
##
## The following object is masked from 'package:arm':
##
## traceplot
##
## ##
## ## Markov Chain Monte Carlo Package (MCMCpack)
## ## Copyright (C) 2003-2015 Andrew D. Martin, Kevin M. Quinn, and Jong Hee Park
## ##
## ## Support provided by the U.S. National Science Foundation
## ## (Grants SES-0350646 and SES-0350613)
## ##
##
## Attaching package: 'MCMCpack'
##
## The following object is masked from 'package:LearnBayes':
##
## rdirichlet
model.3 <- MCMCregress(Salary~AtBat+Hits+Walks+CAtBat+CRuns+CRBI+CWalks+Division+PutOuts+Assists, data = Hitters, burnin = 3000, mcmc = 10000, verbose = 1000, seed = NA, beta.start = NA)##
##
## MCMCregress iteration 1 of 13000
## beta =
## 257.80280
## -2.15326
## 6.72180
## 4.89756
## -0.20469
## 1.59195
## 0.92860
## -0.54330
## -109.88147
## 0.29914
## 0.15717
## sigma2 = 98250.91761
##
##
## MCMCregress iteration 1001 of 13000
## beta =
## 227.84376
## -2.86991
## 8.68673
## 6.48707
## -0.16508
## 1.86055
## 0.75150
## -1.09328
## -131.84219
## 0.30023
## 0.42341
## sigma2 = 105370.41062
##
##
## MCMCregress iteration 2001 of 13000
## beta =
## 170.70258
## -1.58901
## 5.95458
## 3.42729
## -0.16681
## 1.26187
## 0.81976
## -0.43564
## -56.47505
## 0.27070
## 0.08646
## sigma2 = 93211.05921
##
##
## MCMCregress iteration 3001 of 13000
## beta =
## 91.53963
## -1.05203
## 3.34177
## 5.24995
## -0.22319
## 2.16926
## 0.81533
## -1.03503
## -30.24605
## 0.30729
## 0.30309
## sigma2 = 116917.77875
##
##
## MCMCregress iteration 4001 of 13000
## beta =
## 187.03759
## -2.70954
## 9.10087
## 5.38630
## -0.17708
## 1.66874
## 0.68773
## -0.62454
## -96.54137
## 0.25939
## 0.08316
## sigma2 = 76510.77109
##
##
## MCMCregress iteration 5001 of 13000
## beta =
## 200.90334
## -2.68242
## 7.70420
## 6.58449
## -0.13567
## 1.68241
## 0.72978
## -0.97267
## -95.74549
## 0.35683
## 0.33232
## sigma2 = 85954.26307
##
##
## MCMCregress iteration 6001 of 13000
## beta =
## 182.54547
## -2.71216
## 8.32391
## 6.39576
## -0.14158
## 1.68089
## 0.44301
## -0.59778
## -119.13613
## 0.40282
## -0.02833
## sigma2 = 87004.65096
##
##
## MCMCregress iteration 7001 of 13000
## beta =
## 227.88812
## -2.02647
## 5.12662
## 8.00250
## -0.18917
## 1.91685
## 1.06716
## -1.27613
## -94.87868
## 0.28557
## 0.40533
## sigma2 = 89519.89504
##
##
## MCMCregress iteration 8001 of 13000
## beta =
## 195.50564
## -2.95633
## 8.86822
## 8.51649
## -0.11171
## 1.03846
## 0.66465
## -0.34423
## -115.99559
## 0.23310
## 0.26118
## sigma2 = 108311.15708
##
##
## MCMCregress iteration 9001 of 13000
## beta =
## 181.43587
## -1.95807
## 6.44895
## 5.02445
## -0.12647
## 1.41957
## 0.89163
## -0.95369
## -107.23072
## 0.23844
## 0.22974
## sigma2 = 85083.55117
##
##
## MCMCregress iteration 10001 of 13000
## beta =
## 144.65261
## -1.88796
## 5.93995
## 7.85979
## -0.17399
## 1.71799
## 0.85562
## -0.93492
## -162.87604
## 0.27071
## 0.09325
## sigma2 = 99865.68958
##
##
## MCMCregress iteration 11001 of 13000
## beta =
## 11.69946
## -1.22992
## 3.79550
## 7.50951
## -0.08459
## 1.27727
## 0.80287
## -1.08464
## -119.93612
## 0.40165
## 0.19863
## sigma2 = 92937.41009
##
##
## MCMCregress iteration 12001 of 13000
## beta =
## 235.66299
## -2.62503
## 8.51615
## 4.92052
## -0.11200
## 1.07168
## 0.71490
## -0.45159
## -124.84402
## 0.26825
## 0.34124
## sigma2 = 91372.95941
summary(model.3)##
## Iterations = 3001:13000
## Thinning interval = 1
## Number of chains = 1
## Sample size per chain = 10000
##
## 1. Empirical mean and standard deviation for each variable,
## plus standard error of the mean:
##
## Mean SD Naive SE Time-series SE
## (Intercept) 163.5554 6.781e+01 6.781e-01 6.781e-01
## AtBat -2.1803 5.439e-01 5.439e-03 5.439e-03
## Hits 6.9395 1.655e+00 1.655e-02 1.655e-02
## Walks 5.7946 1.600e+00 1.600e-02 1.559e-02
## CAtBat -0.1301 5.550e-02 5.550e-04 5.550e-04
## CRuns 1.4119 3.915e-01 3.915e-03 3.985e-03
## CRBI 0.7739 2.106e-01 2.106e-03 2.106e-03
## CWalks -0.8347 2.647e-01 2.647e-03 2.647e-03
## DivisionW -112.5032 3.890e+01 3.890e-01 3.890e-01
## PutOuts 0.2980 7.443e-02 7.443e-04 7.342e-04
## Assists 0.2848 1.594e-01 1.594e-03 1.567e-03
## sigma2 98054.3746 8.868e+03 8.868e+01 9.302e+01
##
## 2. Quantiles for each variable:
##
## 2.5% 25% 50% 75% 97.5%
## (Intercept) 3.148e+01 117.5466 163.8727 2.092e+02 2.964e+02
## AtBat -3.263e+00 -2.5411 -2.1748 -1.810e+00 -1.109e+00
## Hits 3.668e+00 5.8411 6.9523 8.039e+00 1.018e+01
## Walks 2.647e+00 4.7077 5.8131 6.860e+00 8.937e+00
## CAtBat -2.381e-01 -0.1670 -0.1303 -9.382e-02 -1.966e-02
## CRuns 6.348e-01 1.1560 1.4076 1.678e+00 2.168e+00
## CRBI 3.610e-01 0.6321 0.7752 9.136e-01 1.182e+00
## CWalks -1.347e+00 -1.0120 -0.8364 -6.574e-01 -3.169e-01
## DivisionW -1.877e+02 -138.8548 -112.6978 -8.599e+01 -3.653e+01
## PutOuts 1.511e-01 0.2481 0.2983 3.487e-01 4.435e-01
## Assists -3.096e-02 0.1777 0.2854 3.930e-01 5.929e-01
## sigma2 8.177e+04 91844.4297 97528.6826 1.037e+05 1.167e+05
plot(model.3)
