Package‘coda’
October16,2015
Version0.18-1
Date2015-10-16
Title Output Analysis and Diagnostics for MCMC
Depends R(>=2.14.0)
Imports lattice
Description Provides functions for summarizing and plotting the
output from Markov Chain Monte Carlo(MCMC)simulations,as
well as diagnostic tests of convergence to the equilibrium
distribution of the Markov chain.
License GPL(>=2)
NeedsCompilation no
Author Martyn Plummer[aut,cre,trl],
Nicky Best[aut],
Kate Cowles[aut],
Karen Vines[aut],
Deepayan Sarkar[aut],
Douglas Bates[aut],
Russell Almond[aut],
Arni Magnusson[aut]
Maintainer Martyn Plummer
Repository CRAN
Date/Publication2015-10-1620:00:43
R topics documented:
as.ts.mcmc (3)
autocorr (3)
autocorr.diag (4)
autocorr.plot (5)
batchSE (5)
bugs2jags (6)
coda.options (7)
1
2R topics documented:
codamenu (9)
Cramer (9)
crosscorr (10)
crosscorr.plot (10)
cumuplot (11)
densplot (11)
effectiveSize (12)
gelman.diag (13)
gelman.plot (15)
geweke.diag (17)
geweke.plot (18)
heidel.diag (19)
HPDinterval (20)
line (21)
mcmc (21)
mcmc.convert (22)
mcmc.list (23)
mcmc.subset (24)
mcmcUpgrade (25)
mcpar (25)
multi.menu (26)
nchain (26)
plot.mcmc (27)
raftery.diag (28)
read.and.check (29)
read.coda (30)
read.coda.interactive (31)
read.openbugs (32)
rejectionRate (33)
spectrum0 (33)
spectrum0.ar (35)
summary.mcmc (36)
thin (36)
time.mcmc (37)
traceplot (38)
trellisplots (38)
varnames (42)
window.mcmc (43)
Index44
as.ts.mcmc3 as.ts.mcmc Coerce mcmc object to time series
Description
the as.ts method for mcmc objects coerces an mcmc object to a time series.
Usage
##S3method for class mcmc
as.ts(x,...)
Arguments
x an mcmc object
...unused arguments for compatibility with generic as.ts
Author(s)
Martyn Plummer
See Also
as.ts
autocorr Autocorrelation function for Markov chains
Description
autocorr calculates the autocorrelation function for the Markov chain mcmc.obj at the lags given by lags.The lag values are taken to be relative to the thinning interval if relative=TRUE.
High autocorrelations within chains indicate slow mixing and,usually,slow convergence.It may be useful to thin out a chain with high autocorrelations before calculating summary statistics:a thinned chain may contain most of the information,but take up less space in memory.Re-running the MCMC sampler with a different parameterization may help to reduce autocorrelation.
Usage
autocorr(x,lags=c(0,1,5,10,50),relative=TRUE)
4autocorr.diag
Arguments
x an mcmc object
lags a vector of lags at which to calculate the autocorrelation
relative a logical?ag.TRUE if lags are relative to the thinning interval of the chain,or FALSE if they are absolute difference in iteration numbers
Value
A vector or array containing the autocorrelations.
Author(s)
Martyn Plummer
See Also
acf,autocorr.plot.
autocorr.diag Autocorrelation function for Markov chains
Description
autocorr.diag calculates the autocorrelation function for the Markov chain mcmc.obj at the lags given by lags.The lag values are taken to be relative to the thinning interval if relative=TRUE.
Unlike autocorr,if mcmc.obj has many parmeters it only computes the autocorrelations with itself and not the cross correlations.In cases where autocorr would return a matrix,this function returns the diagonal of the matrix.Hence it is more useful for chains with many parameters,but may not be as helpful at spotting parameters.
If mcmc.obj is of class mcmc.list then the returned vector is the average autocorrelation across all chains.
Usage
autocorr.diag(mcmc.obj,...)
Arguments
mcmc.obj an object of class mcmc or mcmc.list
...optional arguments to be passed to autocorr
Value
A vector containing the autocorrelations.
autocorr.plot5
Author(s)
Russell Almond
See Also
autocorr,acf,autocorr.plot.
autocorr.plot Plot autocorrelations for Markov Chains
Description
Plots the autocorrelation function for each variable in each chain in x.
Usage
autocorr.plot(x,lag.max,https://www.doczj.com/doc/ee3825955.html,yout=TRUE,ask,...)
Arguments
x A Markov Chain
lag.max Maximum value at which to calculate acf
https://www.doczj.com/doc/ee3825955.html,yout If TRUE then,set up own layout for plots,otherwise use existing one.
ask If TRUE then prompt user before displaying each page of plots.Default is dev.interactive() in R and interactive()in S-PLUS.
...graphical parameters
See Also
autocorr.
batchSE Batch Standard Error
Description
Effective standard deviation of population to produce the correct standard errors.
Usage
batchSE(x,batchSize=100)
6bugs2jags
Arguments
x An mcmc or mcmc.list object.
batchSize Number of observations to include in each batch.
Details
Because of the autocorrelation,the usual method of taking var(x)/n overstates the precision of the estimate.This method works around the problem by looking at the means of batches of the parameter.If the batch size is large enough,the batch means should be approximately uncorrelated and the normal formula for computing the standard error should work.
The batch standard error procedure is usually thought to be not as accurate as the time series meth-ods used in summary and effectiveSize.It is included here for completeness.
Value
A vector giving the standard error for each column of x.
Author(s)
Russell Almond
References
Roberts,GO(1996)Markov chain concepts related to sampling algorithms,in Gilks,WR,Richard-son,S and Spiegelhalter,DJ,Markov Chain Monte Carlo in Practice,Chapman and Hall,45-58. See Also
spectrum0.ar,effectiveSize,summary.mcmc
bugs2jags Convert WinBUGS data?le to JAGS data?le
Description
bugs2jags converts a WinBUGS data in the format called"S-Plus"(i.e.the format created by the dput function)and writes it in dump format used by JAGS.
NB WinBUGS stores its arrays in row order.This is different from R and JAGS which both store ar-rays in column order.This difference is taken into account by bugs2jags which will automatically reorder the data in arrays,without changing the dimension.
Not yet available in S-PLUS.
Usage
bugs2jags(infile,outfile)
Arguments
infile name of the input?le
outfile name of the output?le
Note
If the input?le is saved from WinBUGS,it must be saved in plain text format.The default format for?les saved from WinBUGS is a binary compound document format(with extension odc)that cannot be read by bugs2jags.
Author(s)
Martyn Plummer
References
Spiegelhalter DJ,Thomas A,Best NG and Lunn D(2003).WinBUGS version1.4user manual MRC Biostatistics Unit,Cambridge,UK.
See Also
dput,dump.
coda.options Options settings for the codamenu driver
Description
coda.options is a utility function that queries and sets options for the codamenu()function.These settings affect the behaviour of the functions in the coda library only when they are called via the codamenu()interface.
The coda.options()function behaves just like the options()function in the base library,with the additional feature that coda.options(default=TRUE)will reset all options to the default values.
Options can be pretty-printed using the display.coda.options()function,which groups the op-tions into sections.
Available options are
bandwidth Bandwidth function used when smoothing samples to produce density estimates.De-faults to Silverman’s“Rule of thumb”.
combine.corr Logical option that determines whether to combine multiple chains when calculating cross-correlations.
combine.plots Logical option that determines whether to combine multiple chains when plotting.
combine.plots Logical option that determines whether to combine multiple chains when calculat-ing summary statistics.
data.saved For internal use only.
densplot Logical option that determines whether to plot a density plot when plot methods are called for mcmc objects.
digits Number of signi?cant digits to use when printing.
frac1For Geweke diagnostic,fraction to use from start of chain.Defaults to0.1
frac2For Geweke diagnostic,fraction to use from end of chain.Default to0.5.
gr.bin For Geweke-Brooks plot,number of iterations to use per bin.
gr.max For Geweke-Brooks plot,maximum number of bins to use.This option overrides gr.bin.
halfwidth For Heidelberger and Welch diagnostic,the target value for the ratio of half width to sample mean.
lowess Logical option that controls whether to plot a smooth line through a trace plot when plotting MCMC output.
q For Raftery and Lewis diagnostic,the target quantile to be estimated
r For Raftery and Lewis diagnostic,the required precision.
s For Raftery and Lewis diagnostic,the probability of obtaining an estimate in the interval(q-r, q+r).
quantiles Vector of quantiles to print when calculating summary statistics for MCMC output.
trace Logical option that determines whether to plot a trace of the sampled output when plotting MCMC output.
https://www.doczj.com/doc/ee3825955.html,yout Logical option that determines whether current value of par("mfrow")should be used for plots(TRUE)or whether the optimal layout should be calculated(FALSE).
Usage
coda.options(...)
display.coda.options(stats=FALSE,plots=FALSE,diags=FALSE)
.Coda.Options
.Coda.Options.Default
Arguments
stats logical?ag:show summary statistic options?
plots logical?ag:show plotting options?
diags logical?ag:show plotting options?
...list of options
See Also
options
codamenu9 codamenu Main menu driver for the coda package
Description
codamenu presents a simple menu-based interface to the functions in the coda package.It is de-signed for users who know nothing about the R/S language.
Usage
codamenu()
Author(s)
Kate Cowles,Nicky Best,Karen Vines,Martyn Plummer
Cramer The Cramer-von Mises Distribution
Description
Distribution function of the Cramer-von Mises distribution.
Usage
pcramer(q,eps)
Arguments
q vector of quantiles.
eps accuracy required
Value
pcramer gives the distribution function,
References
Anderson TW.and Darling DA.Asymptotic theory of certain‘goodness of?t’criteria based on stochastic processes.Ann.Math.Statist.,23,192-212(1952).
Csorgo S.and Faraway,JJ.The exact and asymptotic distributions of the Cramer-von Mises statistic.
J.Roy.Stat.Soc.(B),58,221-234(1996).
10crosscorr.plot crosscorr Cross correlations for MCMC output
Description
crosscorr calculates cross-correlations between variables in Markov Chain Monte Carlo output.
If x is an mcmc.list then all chains in x are combined before calculating the correlation.
Usage
crosscorr(x)
Arguments
x an mcmc or mcmc.list object.
Value
A matrix or3-d array containing the correlations.
See Also
crosscorr.plot,autocorr.
crosscorr.plot Plot image of correlation matrix
Description
crosscorr.plot provides an image of the correlation matrix for x.If x is an mcmc.list object, then all chains are combined.
The range[-1,1]is divided into a number of equal-length categories given by the length of col and assigned the corresponding color.By default,topographic colours are used as this makes it easier to distinguish positive and negative correlations.
Usage
crosscorr.plot(x,col=topo.colors(10),...)
Arguments
x an mcmc or mcmc.list object
col color palette to use
...graphical parameters
cumuplot11
See Also
crosscorr,image,topo.colors.
cumuplot Cumulative quantile plot
Description
Plots the evolution of the sample quantiles as a function of the number of iterations.
Usage
cumuplot(x,probs=c(0.025,0.5,0.975),ylab="",
lty=c(2,1),lwd=c(1,2),type="l",ask,
https://www.doczj.com/doc/ee3825955.html,yout=TRUE,col=1,...)
Arguments
x an mcmc object
probs vector of desired quantiles
ylab,lty,lwd,type,col
graphical parameters
https://www.doczj.com/doc/ee3825955.html,yout If TRUE,then set up own layout for plots,otherwise use existing one.
ask If TRUE then prompt user before displaying each page of plots.Default is dev.interactive() in R and interactive()in S-PLUS.
...further graphical parameters
Author(s)
Arni Magnusson
densplot Probability density function estimate from MCMC output
Description
Displays a plot of the density estimate for each variable in x,calculated by the density function.
Usage
densplot(x,show.obs=TRUE,bwf,
ylim,xlab,ylab="",type="l",main,...)
12effectiveSize
Arguments
x An mcmc or mcmc.list object
show.obs Show observations along the x-axis
bwf Function for calculating the bandwidth.If omitted,the bandwidth is calculate by1.06times the minimum of the standard deviation and the interquartile range
divided by1.34times the sample size to the negative one?fth power ylim Limits on y axis.See plot.window
xlab X-axis label.By default this will show the sample size and the bandwidth used for smoothing.See plot
ylab Y-axis label.By default,this is blank.See plot
type Plot type.See plot
main An overall title for the plot.See title
...Further graphical parameters
Note
You can call this function directly,but it is more usually called by the plot.mcmc function.
If a variable is bounded below at0,or bounded in the interval[0,1],then the data are re?ected at the boundary before being passed to the density()function.This allows correct estimation of a non-zero density at the boundary.
See Also
density,plot.mcmc.
effectiveSize Effective sample size for estimating the mean
Description
Sample size adjusted for autocorrelation.
Usage
effectiveSize(x)
Arguments
x An mcmc or mcmc.list object.
Details
For a time series x of length N,the standard error of the mean is the square root of var(x)/n where n is the effective sample size.n=N only when there is no autocorrelation.
Estimation of the effective sample size requires estimating the spectral density at frequency zero.
This is done by the function spectrum0.ar
For a mcmc.list object,the effective sizes are summed across chains.To get the size for each chain individually use lapply(x,effectiveSize).
Value
A vector giving the effective sample size for each column of x.
See Also
spectrum0.ar.
gelman.diag Gelman and Rubin’s convergence diagnostic
Description
The‘potential scale reduction factor’is calculated for each variable in x,together with upper and lower con?dence limits.Approximate convergence is diagnosed when the upper limit is close to
1.For multivariate chains,a multivariate value is calculated that bounds above the potential scale
reduction factor for any linear combination of the(possibly transformed)variables.
The con?dence limits are based on the assumption that the stationary distribution of the variable under examination is normal.Hence the‘transform’parameter may be used to improve the normal approximation.
Usage
gelman.diag(x,confidence=0.95,transform=FALSE,autoburnin=TRUE,
multivariate=TRUE)
Arguments
x An mcmc.list object with more than one chain,and with starting values that are overdispersed with respect to the posterior distribution.
confidence the coverage probability of the con?dence interval for the potential scale reduc-tion factor
transform a logical?ag indicating whether variables in x should be transformed to im-prove the normality of the distribution.If set to TRUE,a log transform or logit
transform,as appropriate,will be applied.
autoburnin a logical?ag indicating whether only the second half of the series should be used in the computation.If set to TRUE(default)and start(x)is less than
end(x)/2then start of series will be adjusted so that only second half of series
is used.
multivariate a logical?ag indicating whether the multivariate potential scale reduction factor should be calculated for multivariate chains
Value
An object of class gelman.diag.This is a list with the following elements:
psrf A list containing the point estimates of the potential scale reduction factor(la-belled Point est.)and their upper con?dence limits(labelled Upper C.I.).
mpsrf The point estimate of the multivariate potential scale reduction factor.This is NULL if there is only one variable in x.
The gelman.diag class has its own print method.
Theory
Gelman and Rubin(1992)propose a general approach to monitoring convergence of MCMC out-put in which m>1parallel chains are run with starting values that are overdispersed relative to the posterior distribution.Convergence is diagnosed when the chains have‘forgotten’their initial values,and the output from all chains is indistinguishable.The gelman.diag diagnostic is applied to a single variable from the chain.It is based a comparison of within-chain and between-chain variances,and is similar to a classical analysis of variance.
There are two ways to estimate the variance of the stationary distribution:the mean of the empirical variance within each chain,W,and the empirical variance from all chains combined,which can be expressed as
σ2=(n?1)W
n +
B
n
where n is the number of iterations and B/n is the empirical between-chain variance.
If the chains have converged,then both estimates are unbiased.Otherwise the?rst method will underestimate the variance,since the individual chains have not had time to range all over the stationary distribution,and the second method will overestimate the variance,since the starting points were chosen to be overdispersed.
The convergence diagnostic is based on the assumption that the target distribution is normal.A Bayesian credible interval can be constructed using a t-distribution with mean
μ=Sample mean of all chains combined
and variance
V= σ2+B
mn
and degrees of freedom estimated by the method of moments
d=2? V2 Var( V)
Use of the t-distribution accounts for the fact that the mean and variance of the posterior distribution are estimated.
The convergence diagnostic itself is
R=
(d+3) V
(d+1)W
Values substantially above1indicate lack of convergence.If the chains have not converged, Bayesian credible intervals based on the t-distribution are too wide,and have the potential to shrink by this factor if the MCMC run is continued.
Note
The multivariate a version of Gelman and Rubin’s diagnostic was proposed by Brooks and Gelman (1998).Unlike the univariate proportional scale reduction factor,the multivariate version does not include an adjustment for the estimated number of degrees of freedom.
References
Gelman,A and Rubin,DB(1992)Inference from iterative simulation using multiple sequences, Statistical Science,7,457-511.
Brooks,SP.and Gelman,A.(1998)General methods for monitoring convergence of iterative simu-lations.Journal of Computational and Graphical Statistics,7,434-455.
See Also
gelman.plot.
gelman.plot Gelman-Rubin-Brooks plot
Description
This plot shows the evolution of Gelman and Rubin’s shrink factor as the number of iterations increases.
Usage
gelman.plot(x,bin.width=10,max.bins=50,
confidence=0.95,transform=FALSE,autoburnin=TRUE,https://www.doczj.com/doc/ee3825955.html,yout=TRUE,
ask,col,lty,xlab,ylab,type,...)
Arguments
x an mcmc object
bin.width Number of observations per segment,excluding the?rst segment which always has at least50iterations.
max.bins Maximum number of bins,excluding the last one.
confidence Coverage probability of con?dence interval.
transform Automatic variable transformation(see gelman.diag)
autoburnin Remove?rst half of sequence(see gelman.diag)
https://www.doczj.com/doc/ee3825955.html,yout If TRUE then,set up own layout for plots,otherwise use existing one.
ask Prompt user before displaying each page of plots.Default is dev.interactive() in R and interactive()in S-PLUS.
col graphical parameter(see par)
lty graphical parameter(see par)
xlab graphical parameter(see par)
ylab graphical parameter(see par)
type graphical parameter(see par)
...further graphical parameters.
Details
The Markov chain is divided into bins according to the arguments bin.width and max.bins.Then the Gelman-Rubin shrink factor is repeatedly calculated.The?rst shrink factor is calculated with observations1:50,the second with observations1:(50+n)where n is the bin width,the third contains samples1:(50+2n)and so on.
Theory
A potential problem with gelman.diag is that it may mis-diagnose convergence if the shrink factor
happens to be close to1by chance.By calculating the shrink factor at several points in time, gelman.plot shows if the shrink factor has really converged,or whether it is still?uctuating.
References
Brooks,S P.and Gelman,A.(1998)General Methods for Monitoring Convergence of Iterative Simulations.Journal of Computational and Graphical Statistics,7,434-455.
See Also
gelman.diag.
geweke.diag17 geweke.diag Geweke’s convergence diagnostic
Description
Geweke(1992)proposed a convergence diagnostic for Markov chains based on a test for equality of the means of the?rst and last part of a Markov chain(by default the?rst10%and the last50%).
If the samples are drawn from the stationary distribution of the chain,the two means are equal and Geweke’s statistic has an asymptotically standard normal distribution.
The test statistic is a standard Z-score:the difference between the two sample means divided by its estimated standard error.The standard error is estimated from the spectral density at zero and so takes into account any autocorrelation.
The Z-score is calculated under the assumption that the two parts of the chain are asymptotically independent,which requires that the sum of frac1and frac2be strictly less than1.
Usage
geweke.diag(x,frac1=0.1,frac2=0.5)
Arguments
x an mcmc object
frac1fraction to use from beginning of chain
frac2fraction to use from end of chain
Value
Z-scores for a test of equality of means between the?rst and last parts of the chain.A separate statistic is calculated for each variable in each chain.
References
Geweke,J.Evaluating the accuracy of sampling-based approaches to calculating posterior mo-ments.In Bayesian Statistics4(ed JM Bernado,JO Berger,AP Dawid and AFM Smith).Clarendon Press,Oxford,UK.
See Also
geweke.plot.
18geweke.plot geweke.plot Geweke-Brooks plot
Description
If geweke.diag indicates that the?rst and last part of a sample from a Markov chain are not drawn
from the same distribution,it may be useful to discard the?rst few iterations to see if the rest of the
chain has"converged".This plot shows what happens to Geweke’s Z-score when successively larger
numbers of iterations are discarded from the beginning of the chain.To preserve the asymptotic
conditions required for Geweke’s diagnostic,the plot never discards more than half the chain.
The?rst half of the Markov chain is divided into nbins-1segments,then Geweke’s Z-score is
repeatedly calculated.The?rst Z-score is calculated with all iterations in the chain,the second after
discarding the?rst segment,the third after discarding the?rst two segments,and so on.The last
Z-score is calculated using only the samples in the second half of the chain.
Usage
geweke.plot(x,frac1=0.1,frac2=0.5,nbins=20,
pvalue=0.05,https://www.doczj.com/doc/ee3825955.html,yout=TRUE,ask,...)
Arguments
x an mcmc object
frac1fraction to use from beginning of chain.
frac2fraction to use from end of chain.
nbins Number of segments.
pvalue p-value used to plot con?dence limits for the null hypothesis.
https://www.doczj.com/doc/ee3825955.html,yout If TRUE then,set up own layout for plots,otherwise use existing one.
ask If TRUE then prompt user before displaying each page of plots.Default is dev.interactive() in R and interactive()in S-PLUS.
...Graphical parameters.
Note
The graphical implementation of Geweke’s diagnostic was suggested by Steve Brooks.
See Also
geweke.diag.
heidel.diag19 heidel.diag Heidelberger and Welch’s convergence diagnostic
Description
heidel.diag is a run length control diagnostic based on a criterion of relative accuracy for the estimate of the mean.The default setting corresponds to a relative accuracy of two signi?cant digits.
heidel.diag also implements a convergence diagnostic,and removes up to half the chain in order to ensure that the means are estimated from a chain that has converged.
Usage
heidel.diag(x,eps=0.1,pvalue=0.05)
Arguments
x An mcmc object
eps Target value for ratio of halfwidth to sample mean
pvalue signi?cance level to use
Details
The convergence test uses the Cramer-von-Mises statistic to test the null hypothesis that the sampled values come from a stationary distribution.The test is successively applied,?rstly to the whole chain,then after discarding the?rst10%,20%,...of the chain until either the null hypothesis is accepted,or50%of the chain has been discarded.The latter outcome constitutes‘failure’of the stationarity test and indicates that a longer MCMC run is needed.If the stationarity test is passed, the number of iterations to keep and the number to discard are reported.
The half-width test calculates a95%con?dence interval for the mean,using the portion of the chain which passed the stationarity test.Half the width of this interval is compared with the estimate of the mean.If the ratio between the half-width and the mean is lower than eps,the halfwidth test is passed.Otherwise the length of the sample is deemed not long enough to estimate the mean with suf?cient accuracy.
Theory
The heidel.diag diagnostic is based on the work of Heidelberger and Welch(1983),who com-bined their earlier work on simulation run length control(Heidelberger and Welch,1981)with the work of Schruben(1982)on detecting initial transients using Brownian bridge theory.
Note
If the half-width test fails then the run should be extended.In order to avoid problems caused by sequential testing,the test should not be repeated too frequently.Heidelberger and Welch(1981) suggest increasing the run length by a factor I>1.5,each time,so that estimate has the same, reasonably large,proportion of new data.
20HPDinterval
References
Heidelberger P and Welch PD.A spectral method for con?dence interval generation and run length control in https://www.doczj.com/doc/ee3825955.html,m.ACM.24,233-245(1981)
Heidelberger P and Welch PD.Simulation run length control in the presence of an initial transient.
Opns Res.,31,1109-44(1983)
Schruben LW.Detecting initialization bias in simulation experiments.Opns.Res.,30,569-590 (1982).
HPDinterval Highest Posterior Density intervals
Description
Create Highest Posterior Density(HPD)intervals for the parameters in an MCMC sample.
Usage
HPDinterval(obj,prob=0.95,...)
##S3method for class mcmc
HPDinterval(obj,prob=0.95,...)
##S3method for class mcmc.list
HPDinterval(obj,prob=0.95,...)
Arguments
obj The object containing the MCMC sample-usually of class"mcmc"or"mcmc.list".
prob A numeric scalar in the interval(0,1)giving the target probability content of
the intervals.The nominal probability content of the intervals is the multiple of
1/nrow(obj)nearest to prob.
...Optional additional arguments for methods.None are used at present.
Details
For each parameter the interval is constructed from the empirical cdf of the sample as the shortest interval for which the difference in the ecdf values of the endpoints is the nominal probability.
Assuming that the distribution is not severely multimodal,this is the HPD interval.
Value
For an"mcmc"object,a matrix with columns"lower"and"upper"and rows corresponding to the parameters.The attribute"Probability"is the nominal probability content of the intervals.A list of such matrices is returned for an"mcmc.list"object.
Author(s)
Douglas Bates
马尔柯夫链预测法 如果事物的发展过程及状态只与事物当时的状态有关，而与以前状态无关时，则此事物的发展变化称为马尔柯夫链。如果系统的安全状况具有马尔柯夫性质，且一种状态转变为另一种状态的规律又是可知的，那么可以利用马尔柯夫链的概念进行计算和分析．来预测未来特定时刻的系统安全状态。 马尔柯夫链是表征一个系统在变化过程中的特性状态，可用一组随时间进程而变化的变量来描述。如果系统在任何时刻上的状态是随机性的，则变化过程是一个随机过程，当时刻 t 变到时刻1+t ，状态变量从某个取值变到另一个取值，系统就实现了状态转移。系统从某 种状态转移到各种状态的可能性大小，可用转移概率来描述。 假定系统的初始状态可用状态向量表示为： ()()()()()[] 00302010,,,,n s s s s s = (5-19) 状态转移概率矩阵为： ????? ???????=nn n n n n p p p p p p p p p p 21 22221 11211 (5-20) 状态转移矩阵是一个n 阶方阵，满足概率矩阵的一般性质，即满足10≤≤ij p 且 11 =∑=n j ij p 。也就是说，状态转移矩阵的所有行变量都是概率向量。 一次转移向量() 1s 为： ()p s s 0) 1(= 二次转移向量() 2s 为： ()()20)1(2p s p s s ==
类似地 ()()10)1(++=k k p s s 【例5-4】某单位对1250名接触硅尘人员进行使康检查时，发现职工的健康状况分布如表5-6所示。 表5-6 接尘职工健康状况 根据统计资料，一年后接尘人员的健康变化规律为： 健康人员继续保持健康者剩70％。有20％变为疑似硅肺，10％的人被定为硅肺，即 7.011=p , 2.012=p ，1.013=p 原有疑似硅肺者一般不可能恢复为健康者，仍保持原状者为80％，有20％被正式定为硅肺，即 021=p ，8.022=p ，2.023=p 硅肺患者一般不可能恢复为健康或返回疑似硅肺，即 ,031=p 032=p ，133=p 。 状态转移矩阵为： ???? ? ?????=3332 31 232221 131211 p p p p p p p p p p 预测一年后接尘人员的健康状况为： () ()()() () [] ???? ??????=?=3332 31 2322 21131211 03020101p p p p p p p p p S S S p S S
信息论与编码课程习题1——预备知识 概率论与马尔可夫链 1、某同学下周一上午是否上课，取决于当天情绪及天气情况，且当天是否下雨与心情好坏没有关系。若下雨且心情好，则50%的可能会上课；若不下雨且心情好，则有10%的可能性不上课；若不下雨且心情不好则有40%的可能性上课；若下雨且心情不好，则有90%的可能不会上课。假设当天下雨的概率为30%，该同学当天心情好的概率为20%，试计算该同学周一上课的可能性是多大？ 分析： 天气情况用随机变量X 表示，“0”表示下雨，“1”表示不下雨；心情好坏用Y 表示，“0”表示心情好用“0”表示，心情不好用“1”表示；是否上课用随机变量Z 表示，“0”表示上课，“1”表示不上课。由题意可知 已知[]5.00,0|0====Y X Z P ，[]5.00,0|1====Y X Z P []1.00,1|1====Y X Z P ，[]9.00,1|0====Y X Z P []4.01,1|0====Y X Z P ，[]6.01,1|1====Y X Z P []9.01,0|1====Y X Z P ，[]1.01,0|0====Y X Z P []3.00==X P ，[]7.01==X P []2.00==Y P ，[]8.01==Y P 即题目实际上给出了八个个条件概率和四个概率 [][][][]0,0|00|000===?==?===X Y Z P X Y P X P Z P [][][]0,1|00|10===?==?=+X Y Z P X Y P X P [][][]1,0|01|01===?==?=+X Y Z P X Y P X P [][][]1,1|01|11===?==?=+X Y Z P X Y P X P 由于X ，Y 相互独立，则有 [][][][]0,0|0000===?=?===X Y Z P Y P X P Z P [][][]0,1|010===?=?=+X Y Z P Y P X P [][][]1,0|001===?=?=+X Y Z P Y P X P [][][]1,1|011===?=?=+X Y Z P Y P X P []5.02.03.00??==Z P 1.08.03.0??+9.02.07.0??+1.08.07.0??+ =? 注意：全概率公式的应用 2、已知随机变量X 和Y 的联合分布律如又表所示， 且()Y X Y X g Z +==2 11,，()Y X Y X g Z /,22==，求： 1）1Z 的分布律与数学期望
文献综述 数学与应用数学 马尔柯夫链在班级成绩预测中的应用 马尔柯夫链是数学中具有马尔科夫性质的离散随机过程. 在该过程中, 给定当前信息的情况下, 过去(即当期以前的历史状态)对于预测将来(即当期以后的未来状态)是无关的. 考虑只取有限个或可数个值的随机过程{}n ,0,1,2,X n =K , 若不另外说明, 过程可能取得值的集合将以非负整数集{}0,1,2,K 来表示. 若n X i =, 就说过程在时刻n 处于状态i , 假设每当过程处于状态i , 则在下一时刻处于状态j 的概率是固定的ij P , 也即假设对一切状态011,,,,,n i i i i j -K 及一切0n ≥又 {}1111100,,,,/n n n n ij P X j P X i X i X i X i +--======K , 这样的随机过程称为马尔柯夫链. 式(2.1)解释为, 对马尔柯夫链, 给定现在的状态n X 及过去的状态011,,,n X X X -K , 将来的状态1n X +的条件分布于过去的状态无关, 只依赖于现在的状态, 这称为马尔科夫性. 马尔柯夫链是一个有着广泛应用的随机过程模型, 它对一个系统由一种状态转移到另一种状态给出了定量分析. 马尔柯夫在1906年首先做出了这类过程. 而将此一般化到可数无限状态空间是由柯尔莫果洛夫在1936年给出的. 马尔柯夫链与布朗运动以及遍历假说被列为二十世纪初期重要课题, 但马尔柯夫寻求的不仅在于数学动机, 名义上是对于纵属事件大数法则的扩张. 其中, 马尔柯夫链用在基于观察数据的二到三维离散变量的随机模拟. 这一应用类似于“克里金”地理统计学, 被称为是“马尔柯夫链地理统计学”. 经过近百年的发展已形成完整的理论体系, 并且广泛被应用于社会、经济、科技、生态、农业、环境、医学、水利水电等众多科学领域. 自从我国著名的数学家、教育家中科院王梓坤院士在上世纪50年代将马尔柯夫理论引入国内以后, 我国学者对马尔柯夫过程的研究也取得了丰硕的成果, 在生灭过程的构造和它的积分型泛函的分布、马尔科夫过程的零壹律、Martin 边界与过份函数、马尔柯夫过程与位势理论的关系、多参数马尔柯夫过程等方面做了很多开创性工作, 近年来也不断有新的研究成果推出, 这些都标志着我国数学界对马尔柯夫理论的研究理论研究达到了世界领先水平.
1、对单支股票走势、收益的预侧 现以上海A股精伦电子的股价时间序列为例(原始资料如表1)，应用马尔可夫链对股价分别进行中短期和长期预测分析，这里不妨将时间序列的单位以天记。 表1:上海A股精伦电子2002年6月13日一7月17日23个交易日的收盘价格资料 将表1中这23个收盘价格划分成4个价格区间(由低到高每区间1.5个价格单位)，得到区间状态为: S1：(26.00以下)、S2：(26.00--27.50)、S3：(27.50--28.00)、S4：(28.00及以上)。则到达个区间的频数分别为5, 3, 9, 6。综合这些资料于是得到这23个交易日的收盘价格状态转移情况如表2， 由此得到各状态之间的转移概率和转移概率矩阵: 表1知，第23个交易日的收盘价格是27.53(即为k状态区间)，所以用马尔可夫链进行预测时初始状态向量，P(0) =( 0,0,1,0)，第24, 25日的收盘价格状态向量分别为即
P(1)=P(0)P=(0,0.125,0.625,0.25); P(2)=P(1)P=(0.042,0.078,0.451,0.323) 预测这两日的收盘价格处于k状态区间的概率最大，与实际情况27.21和27.39一致. 随着交易日的增加，即n足够大时，只要状态转移概率不变(即稳定条件)，则状态向量趋向于一个和初始状态无关的值，并稳定下来.按马尔可夫系统平稳定条件，可得一个线性方程组: 解得的数值即为较长时间后股价处于各区间的平稳分布。对照资料可以看出，由上述公式计算出的各收盘价格状态区间基本上是准确的。 2、用马氏链对沪市的走势进行预铡及相应分析 我们利用沪市1998年1月5日至2001年11月2日的上证综合指数每周收盘资料，将上证指数划分为六个区间，即六种状态:区间1(1000点一1300点);区间2 (1300点一1600点);区间3 (1600点一1800点):区间4 (1800点~2000点);区间 5 (2000点~2200点);区间6 (2200点以上)。即可得到上证综合指数以周为单位的转移概率矩阵 因为11月2日上证综合指数周收盘为1691点，处于状态3，所以在对沪市进行预测时，初始状态向量P(0)=(0,0,1,0,0,0)，然后按上例中的马尔可夫方法进行中短期和长期预测分析。通过对比可以发现，马尔可夫链对整个证券市场的预测结果是比较准确的，而且长期预测所得的结论与股票价格根本上是由股票内在投资价值决定的这一基本原理也是惊人的一致。
马尔可夫链模型 在考察随机因素影响的动态系统时，常常碰到这样的情况，系统在每个时期所处的状态是随机的，从这个时期到下个时期的状态按照一定的概率进行转移，并且下个时期的状态只取决于这个时期的状态和转移概率，与以前各时期的状态无关。这种性质称为无后效性或马尔可夫性。通俗的说就是已知现在，将来与历史无关。 具有马氏性的，时间、状态无为离散的随机转移过程通常用马氏链(Markov Chain)模型描述。 马氏链模型在经济、社会、生态、遗传等许多领域中有着广泛的应用。值得提出的是，虽然它是解决随机转移过程的工具，但是一些确定性系统的状态转移问题也能用马氏链模型处理。 马氏链简介： 马氏链及其基本方程：按照系统的发展，时间离散化为 0,1,2,n = ，对每个n ，系统的状态用随机变量n X 表示，设n X 可以 取k 个离散值1,2,,n X k = ，且n X i =的概率记作() i a n ，称为状态概 率，从n X i =到1 n X j +=的概率记作ij p ，称为转移概率。如果1 n X +的 取值只取决于n X 的取值及转移概率，而与1 2,,n n X X -- 的取值无关， 那么这种离散状态按照离散时间的随机转移过程称为马氏链。 由状态转移的无后效性和全概率公式可以写出马氏链的基本方程为 1 (1)()1,2,,k i j ij j a n a n p i k =+= =∑
并且() i a n 和ij p 应满足 1 1 ()10,1,2,;0 ;1 1,2,,k k j ij ij j j a n n p p i k ====≥==∑∑ 引入状态概率向量和转移概率矩阵 12()((),(),,()) {}k ij k a n a n a n a n P p == 则基本方程可以表为1 (1)()(0)n a n a n P a P ++== 例1：某商店每月考察一次经营情况，其结果用经营状况好与孬表示。若本月经营状况好，则下月保持好的概率为0.5，若本月经营状况不好，则下月保持好的概率为0.4，试分析该商店若干时间后的经营状况。 解：商店的经营状况是随机的，每月转变一次。用随机变量n X 表示第n 个月的经营状况，称为经营系统的状态.1,2 n X =分别表示 好与不好，0,1,n = 。用() i a n 表示第n 月处于状态i 的概率（1,2i =） 即()()i n a n P X i ==，ij p 表示本月处于状态i ，下月转为状态j 的概率。 这里1 n X +无后效性，只取决于n X 和ij p 。 112112220.5,0.4,0.5,0.6p p p p ==∴== 根据全概率公式可以得到： 11112212112222 (1)()()0.50.5(1)()(1)()()0.4 0.6a n a n p a n p a n a n P P a n a n p a n p +=+??? ?+==? ?+=+?? ? 假设这个递推公式存在极限w ，有w w P = ，即()0w P E -=。于 是当经营状况好或孬时，经计算可以得到下面的结果
3．5 马尔可夫链预测方法 一、基于绝对分布的马尔可夫链预测方法 对于一列相依的随机变量，用步长为一的马尔可夫链模型和初始分布推算出未来时段的绝对分布来做预测分析方法，称为“基于绝对分布的马尔可夫链预测方法”，不妨记其为“ADMCP 法”。其具体方法步骤如下: 1．计算指标值序列均值x ，均方差s ，建立指标值的分级标准，即确定马尔可夫链的状态空间I ，这可根据资料序列的长短及具体间题的要求进行。例如，可用样本均方差为标准，将指标值分级，确定马尔可夫链的状态空间 I =[1, 2，…，m ]； 2．按步骤1所建立的分级标准，确定资料序列中各时段指标值所对应的状态； 3．对步骤2所得的结果进行统计计算，可得马尔可夫链的一步转移概率矩阵1P ，它决定了指标值状态转移过程的概率法则； 4．进行“马氏性” 检验； 5．若以第1时段作为基期，该时段的指标值属于状态i ，则可认为初始分布为 (0)(0,,0,1,0,0)P = 这里P (0)是一个单位行向量，它的第i 个分量为1，其余分量全为0。于是第2时段的绝对分布为 1(1)(0)P P P =12((1),(1),,(1))m p p p = 则第2时段的预测状态j 满足：(1)max{(1),}j i p p i I =∈； 同样预测第k ＋1时段的状态，则有 1()(0)k P k P P =12((),(),,())m p k p k p k = 得到所预测的状态j 满足: ()max{(),}j i p k p k i I =∈ 6．进一步对该马尔可夫链的特征(遍历性、平稳分布等)进行分析。 二、叠加马尔可夫链预测方法 对于一列相依的随机变量，利用各种步长的马尔可夫链求得的绝对分布叠加来做预测分析，的方法，称为“叠加马尔可夫链预测方法”，不妨记其为“SPMCP 法’。其具体方法步骤如下: 1) 计算指标值序列均值x ，均方差s ，建立指标值的分级标准(相当于确定马尔可夫链的状态空间)，可根据资料序列的长短及具体问题的要求进行； 2) 按1)所建立的分级标准，确定资料序列中各时段指标值所对应的状态； 3) 对2)所得的结果进行统计，可得不同滞时(步长)的马尔可夫链的转移概率矩阵，它决定了指标值状态转移过程的概率法则； 4) 马氏性检验； 5) 分别以前面若干时段的指标值为初始状态，结合其相应的各步转移概率矩阵即可预测出该时段指标值的状态概率 (6)将同一状态的各预测概率求和作为指标值处于该状态的预测概率，即 ,所对应的i 即为该时段指标值的预测状态。待该时段的指标值确定之后，将其加 入到原序列之中，再重复步骤"(1)一(6)"，可进行下时段指标值状态的预测。 (7)可进一步对该马尔可夫链的特征(遍历性、平稳分布等)进行分析。
利用马尔柯夫链对天津市恩格尔系数的实证分析 把天津市恩格尔系数的变化过程看成是一个马尔可夫链，并针对恩格尔系数的特点引入恩格尔系数增减率，建立天津市恩格尔系数变化对马尔可夫链模型，并进行预测分析，以供有关方面参考。 标签：马尔柯夫链天津市恩格尔系数 1 分析背景 恩格尔系数是从一个方面反映一个国家或地区消费结构状况，衡量居民生活水平高低，且被世界各国广泛采用的消费结构指标。联合国粮农组织(FAO)根据各国的消费习惯，利用恩格尔系数对一个国家或地区的居民生活质量提出了一个相对标准，即60%以上为绝对贫困，50%-60%为勉强度日，40%-50%为小康，30%-40%为富裕，30%以下为最富裕。联合国粮农组织的这一举措，使恩格尔系数成为评价国家或地区生活水平高低的重要标准之一，恩格尔系数和恩格尔定律得到了广泛的认同。 中国从改革开放以来，随着经济发展，居民收入差距扩大，消费档次逐步拉开，引起人们对恩格尔系数普遍关注。另外，中国宣布“总体达到小康”，其衡量标准之一就是恩格尔系数。我国劳动和社会保障部确定最低工资标准的方法之一就是恩格尔系数法。因此研究恩格尔系数具有和重要的现实意义。 2 马尔可夫链 马尔可夫链的数学定义为：设随机过程的状态空间S为R中的可列集。如果对T中任意n个参数t1＜t2＜…tn，以及使 成立的S中任意状态i1，…in-1与in均有则称为马尔可夫链。设I为离散的马尔可夫链的状态空间。称条件概率 ，为的h步转移概率。转移概率表示已知过程在m的马尔可夫链称为齐次马尔可夫链。此时，k步转移概率可以记为p(k)。当时k=1，称为一步转移概率，简记为p；并且p(k)=pk，k≥1。概率转移矩阵中的元素具有非负性以及行和为1两个性质。 应用马尔可夫链的方法预测的基本思路是:如果某种事物或某种现象的各状态的时间序列为马尔可夫链，则根据T(u-1)时刻的状态估计或预报T(u)时刻的状态。对于一个符合马尔可夫过程的时间序列，先根据具体情况，将其划分成若干离散的状态，再计算一阶转移概率矩阵。由T(u-1)时刻的S(u-1)某状态，经一步转移到T(u)时刻的S(u)某状态的概率，称为一步转移概率。一步转移概率为：，其中ωu为状态S(u)出现的次数，ωuk为从状态S(u)转移到状态S(k)的次数，puk 为由状态S(u)经过一阶转移到状态S(k)的转移概率。
6.2 Suppose that a one-celled organism can be in one of two states-either A or B. An individual in state A will change to state B at an exponential rate α; an individual in state B divides into two new individuals of type A at an exponential rate β. Define an appropriate continuous-time Markov chain for a population of such organisms and determine the appropriate parameters for this model. 6.3 Consider two machines that are maintained by a single repairman. Machine i functions for an exponential time with rate μbefore breaking down, i = 1,2. The repair times (for either i machine) are exponential with rate μ. Can we analyze this as a birth and death process? If so, what are the parameters? If not, how can we analyze it?
1、4.23 Trials are performed in sequence. If the last two trials were successes, then the next trial is a success with probability 0.8; otherwise the next trial is a success with probability 0.5. In the long run, what proportion of trials are successes? 2、4.32 Each of two switches is either on or off during a day. On day n, each switch will independently be on with probability [1+#of on switches during day n-1]/4. For instance, if both switches are on during day n-1, then each will independently be on during day n with probability3/4. What fraction of days are both switches on? What fractions are both off?
3、Let ri denote the long-run proportion of time a given irreducible Markov chain is in state i. Explain why ri is also the proportion of transitions that are into state i as well as being the proportion of transition that are from state i. 4、4.44 Suppose that a population consists of a fixed number, say, m, of genes in any generation. Each gene is one of two possible genetic types. If any generation has exactly i (of its m) genes being type 1, then the next generation will have j type 1 genes with probability j m j m i m m i j m- ? ? ? ? ?- ? ? ? ? ? ?? ? ? ? ? . Let Xn denote the number of type 1 genes in the nth generation, and assume that X0 = i. (a) Find E[Xn] (b) What is the probability that eventually all the genes will be type 1?
马尔可夫链模型（Markov Chain Model） 目录 [隐藏] 1 马尔可夫链模型概述 2 马尔可夫链模型的性质 3 离散状态空间中的马尔可夫链模 型 4 马尔可夫链模型的应用 o 4.1 科学中的应用 o 4.2 人力资源中的应用 5 马尔可夫模型案例分析[1] o 5.1 马尔可夫模型的建立 o 5.2 马尔可夫模型的应用 6 参考文献 [编辑] 马尔可夫链模型概述 马尔可夫链因安德烈·马尔可夫（Andrey Markov，1856－1922）得名，是数学中具有马尔可夫性质的离散时间随机过程。该过程中，在给定当前知识或信息的情况下，过去（即当期以前的历史状态）对于预测将来（即当期以后的未来状态）是无关的。 时间和状态都是离散的马尔可夫过程称为马尔可夫链, 简记为 。 马尔可夫链是随机变量的一个数列。这些变量的范围，即他们所有可能取值的集合，被称为“状态空间”，而Xn的值则是在时间n的状态。如果Xn + 1对于过去状态的条件概率分布仅是Xn的一个函数，则 这里x为过程中的某个状态。上面这个恒等式可以被看作是马尔可夫性质。
马尔可夫在1906年首先做出了这类过程。而将此一般化到可数无限状态空间是由柯尔莫果洛夫在1936年给出的。 马尔可夫链与布朗运动以及遍历假说这两个二十世纪初期物理学重要课题是相联系的，但马尔可夫寻求的似乎不仅于数学动机，名义上是对于纵属事件大数法则的扩张。 马尔可夫链是满足下面两个假设的一种随机过程： 1、t+l时刻系统状态的概率分布只与t时刻的状态有关，与t时刻以前的状态无关； 2、从t时刻到t+l时刻的状态转移与t的值无关。一个马尔可夫链模型可表示为=(S，P，Q)，其中各元的含义如下： 1）S是系统所有可能的状态所组成的非空的状态集，有时也称之为系统的状态空间，它可以是有限的、可列的集合或任意非空集。本文中假定S是可数集(即有限或可列)。用小写字母i,j(或S i,S j)等来表示状态。 2）是系统的状态转移概率矩阵，其中P ij表示系统在时刻t处于状态i，在下一时刻t+l处于状态i的概率，N是系统所有可能的状态 的个数。对于任意i∈s，有。 3）是系统的初始概率分布，q i是系统在初始时刻处 于状态i的概率，满足。 [编辑] 马尔可夫链模型的性质 马尔可夫链是由一个条件分布来表示的 P(X | X n) n+ 1 这被称为是随机过程中的“转移概率”。这有时也被称作是“一步转移概率”。二、三，以及更多步的转移概率可以导自一步转移概率和马尔可夫性质：
马尔可夫链预测方法 一、基于绝对分布的马尔可夫链预测方法 对于一列相依的随机变量，用步长为一的马尔可夫链模型和初始分布推算出未来时段的绝对分布来做预测分析方法，称为“基于绝对分布的马尔可夫链预测方法”，不妨记其为“ADMCP 法”。其具体方法步骤如下: 1．计算指标值序列均值x ，均方差s ，建立指标值的分级标准，即确定马尔可夫链的状态空间I ，这可根据资料序列的长短及具体间题的要求进行。例如，可用样本均方差为标准，将指标值分级，确定马尔可夫链的状态空间 I =[1, 2，…，m ]； 2．按步骤1所建立的分级标准，确定资料序列中各时段指标值所对应的状态； 3．对步骤2所得的结果进行统计计算，可得马尔可夫链的一步转移概率矩阵1P ，它决定了指标值状态转移过程的概率法则； 4．进行“马氏性” 检验； 5．若以第1时段作为基期，该时段的指标值属于状态i ，则可认为初始分布为 (0)(0,,0,1,0,0)P = 这里P (0)是一个单位行向量，它的第i 个分量为1，其余分量全为0。于是第2时段的绝对分布为 1(1)(0)P P P =12((1),(1),,(1))m p p p = 则第2时段的预测状态j 满足：(1)max{(1),}j i p p i I =∈； 同样预测第k ＋1时段的状态，则有 1()(0)k P k P P =12((),(),,())m p k p k p k = 得到所预测的状态j 满足: ()max{(),}j i p k p k i I =∈ 6．进一步对该马尔可夫链的特征(遍历性、平稳分布等)进行分析。 二、叠加马尔可夫链预测方法 对于一列相依的随机变量，利用各种步长的马尔可夫链求得的绝对分布叠加来做预测分析，的方法，称为“叠加马尔可夫链预测方法”，不妨记其为“SPMCP 法’。其具体方法步骤如下: 1) 计算指标值序列均值x ，均方差s ，建立指标值的分级标准(相当于确定马尔可夫链的状态空间)，可根据资料序列的长短及具体问题的要求进行； 2) 按1)所建立的分级标准，确定资料序列中各时段指标值所对应的状态； 3) 对2)所得的结果进行统计，可得不同滞时(步长)的马尔可夫链的转移概率矩阵，它决定了指标值状态转移过程的概率法则； 4) 马氏性检验； 5) 分别以前面若干时段的指标值为初始状态，结合其相应的各步转移概率矩阵即可预测出该时段指标值的状态概率 (6)将同一状态的各预测概率求和作为指标值处于该状态的预测概率，即 ,所对应的i 即为该时段指标值的预测状态。待该时段的指标值确定之后，将其加 入到原序列之中，再重复步骤"(1)一(6)"，可进行下时段指标值状态的预测。 (7)可进一步对该马尔可夫链的特征(遍历性、平稳分布等)进行分析。
第十三章马尔可夫链(习题课) 习题十三 1. 已知齐次马尔可夫链的转移概率矩阵 ? ?=03131P 323132???????? ?31310 问此马尔可夫链有几个状态？求二步转移概率矩阵. 解 因为转移概率矩阵是三阶的, 故此马尔可夫链的状态有三个; 二步转移概率矩阵 2) 2()2()(P p P ij == ? ?=0313*******???????? ?31310 ? ?031313 23132?????????31310 ??=929293949594???????? ? 939292 . 2. 在一串贝努利试验中,事件A 在 每次试验中发生的概率为p ,令
? ??=发生次试验第不发生次试验第A n A n X n ,1,0 ,Λ,3,2,1=n (1) },2,1,{Λ=n X n 是否齐次马尔可夫链？ (2) 写出状态空间和转移概率矩阵; (3) 求n 步转移概率矩阵. 解 (1) 根据题设条件 知道ΛΛ,,,,2 1 n X X X 是相互独立的, 所以 },2,1,{Λ=n X n 是马尔可夫链, 又转移概率 ???=======++1 ,0,}{}|{1 1j p j q j X P i X j X P n n n 与n 无关, 故},2,1,{Λ=n X n 是齐次马尔可夫链; (2) 状态空间}1,0{=S , 一步转移概率矩阵 )(ij p P = ? ?=q q ?? ?? p p , ? ??========++1,0,}{}|{1 1j p j q j X P i X j X P p n n n ij . (3) n 步移概率矩阵
马尔可夫性与马尔可夫链 【教学目标】 1．掌握马尔可夫性与马尔可夫链。 2．熟练运用马尔可夫性与马尔可夫链解决具体问题。 3．亲历马尔可夫性与马尔可夫链的探索过程，体验分析归纳得出马尔可夫性与马尔可夫链，进一步发展学生的探究、交流能力。 【教学重难点】 重点：掌握马尔可夫性与马尔可夫链。 难点：马尔可夫性与马尔可夫链的实际应用。 【教学过程】 一、直接引入 师：今天这节课我们主要学习马尔可夫性与马尔可夫链，这节课的主要内容有马尔可夫性与马尔可夫链，并且我们要掌握这些知识的具体应用，能熟练解决相关问题。 二、讲授新课 （1）教师引导学生在预习的基础上了解马尔可夫性与马尔可夫链内容，形成初步感知。 （2）首先，我们先来学习马尔可夫性，它的具体内容是： 1n X +的随机变化规律与0X ，1X ，…1n X -的取值都没有关系，随机变量序列{}n X 的所具有的这类性质称为马尔可夫性 它是如何在题目中应用的呢？我们通过一道例题来具体说明。 例： 马尔可夫性描述了一种_____。 解析：状态序列 可以给学生一定的提示。 根据例题的解题方法，让学生自己动手练习。 练习： 序列所有可能取值的集合，被称为_____。 （3）接着，我们再来看下马尔可夫链内容，它的具体内容是：
一般地，我们称具有马尔可夫性的随机变量序列{}n X为马尔可夫链。 它是如何在题目中应用的呢？我们也通过一道例题来具体说明。 例：请同学们查询资料，判断马尔可夫链与布朗运动是否有联系 解析：马尔可夫链与布朗运动以及遍历假说这两个二十世纪初期物理学重要课题是相联系的，但马尔可夫寻求的似乎不仅于数学动机，名义上是对于纵属事件大数法则的扩张。 根据例题的解题方法，让学生自己动手练习。 练习： 请写出马尔科夫链满足的两个假设。 三、课堂总结 （1）这节课我们主要讲了马尔可夫性与马尔可夫链 （2）它们在解题中具体怎么应用？ 四、习题检测 1．请同学们写出马尔可夫性的定义。 2．请同学们写出马尔科夫链的定义。 3．请同学们写出马尔科夫性和马尔科夫链之间的联系。