多变点Г分布的变点估计

第二节 点估计的常用方法内容分布图示★ 矩估计法 ★ 求矩估计的方法★ 例1 ★ 例2 ★ 例3 ★ 例4★ 最大似然估计法★ 求最大似然估计的一般方法★ 例5 ★ 例6 ★ 例7 ★ 例8★ 关于有k 个未知参数的最大似然估计 ★ 内容小结 ★ 课堂练习★ 习题6-2内容要点：一、矩估计法矩估计法的基本思想是用样本矩估计总体矩. 因为由在数定理知, 当总体的k 阶矩存在时,样本的k 阶矩依概率收敛于总体的k 阶矩.例如, 可用样本均值X 作为总体均值)(X E 的估计量, 一般地, 记总体k 阶矩 );(k k X E =μ样本k 阶矩 ∑==ni kik X nA 11;总体k 阶中心矩 ;)]([k k X E X E V -= 样本k 阶中心矩 .)(11∑=-=ni kik X XnB用相应的样本矩去估计总体矩的方法就称为矩估计法. 用矩估计法确定的估计量称为矩估计量. 相应的估计值称为据估计值. 矩估计量与矩估计值统称为矩估计. 求矩估计的方法：设总体X 的分布函数),,;(1k x F θθ 中含有k 个未知参数k θθ,,1 , 则(1) 求总体X 的前k 阶矩k μμ,,1 ，一般都是这k 个未知参数的函数, 记为k i g k i i ,,2,1),,,(1 ==θθμ （*） (2) 从（*）中解得 k j h k j j ,,2,1),,,(1 ==μμθ(3) 再用),,2,1(k i i =μ的估计量i A 分别代替上式中的i μ,即可得),,2,1(k i j =θ的矩估计量:.,,2,1),,,(ˆ1k j A A h k j j ==θ注：求,,,1k V V 类似于上述步骤，最后用k B B ,,1⋅⋅⋅代替k V V ,,1 ，求出矩估计j θˆ),,2,1(k I ⋅⋅⋅=。

二、最大似然估计法引例 某同学与一位猎人一起去打猎,一只野兔从前方窜过, 只听一声枪响, 野兔应声倒下, 试猜测是谁打中的?由于只发一枪便打中,而猎人命中的概率一般大于这位同学命中的概率, 故一般会猜测这一枪是猎人射中的.最大似然估计法的思想: 在已经得到实验结果的情况下, 应该寻找使这个结果出现的可能性最大的那个θ作为θ的估计θˆ.注: 最大似然估计法首先由德国数学家高斯于1821年提出, 英国统计学家费歇于1922年重新发现并作了进一步的研究.下面分别就离散型总体和连续型总体情形作具体讨论. 离散型总体的情形: 设总体X 的概率分布为),,(}{θx p x X P ==其中θ为未知参数.如果n X X X ,,,21 是取自总体X 的样本,样本的观察值为n x x x ,,,21 ,则样本的联合分布律,),(},,,{111∏====ni i n n x p x X x X P θ对确定的样本观察值n x x x ,,,21 ,它是未知参数θ的函数,记为∏===ni i n x f x x x L L 121),(),,,,()(θθθ ,并称其为似然函数.连续型总体的情形: 设总体X 的概率密度为),(θx f ,其中θ为未知参数,此时定义似然函数∏===ni i n x f x x x L L 121),(),,,,()(θθθ .似然函数)(θL 的值的大小意味着该样本值出现的可能性的大小, 在已得到样本值nx x x ,,,21 的情况下, 则应该选择使)(θL 达到最大值的那个θ作为θ的估计θˆ. 这种求点估计的方法称为最大似然估计法.定义 若对任意给定的样本值n x x x ,,,21 , 存在),,,(ˆˆ21n x x x θθ=,使 ),(max )ˆ(θθθL L =则称),,,(ˆˆ21n x x x θθ=为θ的最大似然估计值.称相应的统计量),,,(ˆ21n X X X θ为θ最大似然估计量. 它们统称为θ的最大似然估计(MLE ).三、求最大似然估计的一般方法求未知参数θ的最大似然估计问题, 归结为求似然函数)(θL 的最大值点的问题. 当似然函数关于未知参数可微时, 可利用微分学中求最大值的方法求之. 其主要步骤:(1) 写出似然函数),,,,()(21θθn x x x L L =;(2) 令0)(=θθd dL 或)(ln =θθd L d , 求出驻点;注: 因函数L ln 是L 的单调增加函数,且函数)(ln θL 与函数)(θL 有相同的极值点,故常转化为求函数)(ln θL 的最大值点较方便.(3) 判断并求出最大值点, 在最大值点的表达式中, 用样本值代入就得参数的最大似然估计值.注：(i) 当似然函数关于未知参数不可微时，只能按最大似然估计法的基本思想求出最大值点。

第六章点估计1. 本章重点概括本章要求学生正确理解参数点估计的概念。

掌握矩估计法，明确其实质是用样本矩来替换总体矩，即皮尔逊替换原则。

掌握极大似然估计法，明确其基本思想是选取估计量，使得该样本发生的可能性最大，能熟练地求出某些常见分布中未知参数的极大似然估计量。

掌握关于判别估计量优良性的一致性、无偏性、有效性这三个准则，并能熟练地加以运用。

掌握罗-克拉美(Rao-Cramer)不等式的条件、结论，能求一些常见分布中未知参数的无偏估计量之方差的罗-克拉美下界，会求一些常见分布中未知参数的有效估计，或会证明某∧θ是θ的有效估计。

掌握充分统计量的概念和奈曼(Neyman)因子分解定理，并会加以应用。

点估计方法一般有两种，一种为矩估计法，一种为极大似然估计法。

矩估计法比较直观，对任何总体都适用，方法简单，但需要保证总体的相应的矩存在，若不存在就不能用矩估计的方法。

而极大似然估计对任何总体也都适用，从它得到估计量一般有有效性，并且常常具有无偏性，即使不具有无偏性，也可以修正偏差使估计值与待估计参数的真实值充分接近。

极大似然估计法的缺点是往往要解一个似然方程，而这个方程在有些情况下是很难解的。

在分析估计量的好坏时，应首先考虑一致性，即看估计量是否依概率收敛于所估计的参数，不具备一致性的估计量我们一般是不予考虑的。

估计量是一个随机变量，对于不同的样本值，一般给出参数不同的估计值，因而在考虑估计量的优劣时，应该从某种整体性能去衡量，而不能看它在个别样本之下表现如何。

一般来说，矩估计和极大似然估计都不一定是无偏估计。

无偏估计要111112求估计量的数学期望等于待估参数，但无偏估计不一定是有效估计，如正态总体期望的估计量∑==n i i i X k 1ˆμ，其中∑==n i i k 11是无偏估计，但只有当n n nk i ,,2,1,1 ==时，μˆ才是有效估计。

由于统计量很多，那么怎样的统计量才是最佳的呢？直观的想法是，一方面要尽可能的简单，另一方面又要能提供样本所含的“全部信息”，由此引出了充分统计量的定义。

变点问题的研究意义和价值在数学领域，变点问题是一类具有挑战性和广泛应用价值的问题。

它涉及到函数在某一点的性质发生突变的现象，这种性质的变化往往与函数的导数或某一部分的导数有关。

本文将探讨变点问题的研究意义和价值，以期为我们更深入地理解和应用这一数学工具提供启示。

一、变点问题的研究意义1.理论意义（1）完善数学理论体系：变点问题是实分析、泛函分析、微分方程等数学分支中的重要研究对象，研究变点问题有助于丰富和发展这些领域的理论体系。

（2）揭示函数性质的变化规律：通过研究变点问题，可以更深入地了解函数在某一点或某一区间上的性质变化规律，为解决实际问题提供理论依据。

2.应用意义（1）优化控制：在工程、经济、生物等领域，许多问题可以归结为最优化问题。

变点问题的研究为优化控制提供了新的方法，有助于解决实际问题。

（2）信号处理：在信号处理领域，变点检测是一个关键问题。

研究变点问题有助于提高信号处理的准确性和实时性。

二、变点问题的价值1.学术价值（1）促进数学分支的交叉融合：变点问题涉及到多个数学分支，研究此类问题可以促进不同分支之间的交叉融合，推动数学的发展。

（2）提出新的研究方法：变点问题的研究往往需要创新的研究方法，这些方法可以为其他数学问题的研究提供借鉴。

2.实践价值（1）为实际问题提供解决方案：如前所述，变点问题在优化控制、信号处理等领域具有广泛的应用。

研究变点问题可以为这些领域提供有效的解决方案。

（2）提高国家竞争力：数学是科技创新的基础，变点问题的研究有助于提高我国在相关领域的科技创新能力，从而提高国家竞争力。

综上所述，变点问题的研究具有丰富的理论意义和应用价值。

进一步深入研究变点问题，不仅有助于完善数学理论体系，还将为实际问题的解决提供有力支持。

贝叶斯估计是一种计算统计参数的重要方法，在贝叶斯方法中，一组
参数通常用来描述不同分布的变量。

贝叶斯估计可以应用于二项分布，也就是观察到一定数量的元素，并知道它们属于某一类的可能性的概率。

贝叶斯估计基于贝叶斯定理，在二项分布中，假设我们有一群由
两个元素（具有不同权重的两个参数）组成的变量，假定每个元素的
权重是比较稳定的。

在贝叶斯估计中，我们假设可变点的两个参数影响着结果，也即变量
具有不同的可能性。

此时，我们用贝叶斯定理来求解可变点二项分布
参数的不确定性。

首先，我们通过求解贝叶斯估计的概率密度来计算
不同可变点情况下二项分布参数可能性分布。

其次，把可变点看作是
未知参数，最大似然估计法也可以用来求解不同可变点情况下二项分
布参数的估计值。

贝叶斯估计一般用来推断数据的相关性，也可以用来对各种分布的参
数进行推断，例如二项分布参数多变点模型，它可以用来求解不同可
变点情况下的参数值。

使用贝叶斯估计的最大优点就是它可以更好地
发挥数据的参数推断能力，可以让我们更好地分析和推断数据，而不
需要考虑参数间的相关性问题。

因此，贝叶斯估计是对二项分布参数多变点模型的一种有效的求解方法，它可以更好地推断变化的参数值，从而更好地分析和理解数据。

Package‘beast’October12,2022Type PackageTitle Bayesian Estimation of Change-Points in the Slope ofMultivariate Time-SeriesVersion1.1Date2018-03-16Author Panagiotis PapastamoulisMaintainer Panagiotis Papastamoulis<*****************>Description Assume that a temporal process is composed of contiguous segments with differ-ing slopes and replicated noise-corrupted time series measurements are observed.The un-known mean of the data generating process is modelled as a piecewise linear func-tion of time with an unknown number of change-points.The package infers the joint poste-rior distribution of the number and position of change-points as well as the unknown mean pa-rameters per time-series by MCMC sampling.A-priori,the proposed model uses an overfit-ting number of mean parameters but,conditionally on a set of change-points,only a sub-set of them influences the likelihood.An exponentially decreasing prior distribution on the num-ber of change-points gives rise to a posterior distribution concentrating on sparse representa-tions of the underlying sequence,but also available is the Poisson distribution.See Papasta-moulis et al(2017)<arXiv:1709.06111>for a detailed presentation of the method.License GPL-2Imports RColorBrewerDepends R(>=2.10)NeedsCompilation noRepository CRANDate/Publication2018-03-1607:42:36UTCR topics documented:beast-package (2)beast (4)birthProbs (7)complexityPrior (8)computeEmpiricalPriorParameters (8)1computePosteriorParameters (9)computePosteriorParametersFree (10)FungalGrowthDataset (10)localProposal (11)logLikelihoodFullModel (11)logPrior (12)mcmcSampler (12)myUnicodeCharacters (14)normalizeTime0 (14)plot.beast.object (15)print.beast.object (16)proposeTheta (16)simMultiIndNormInvGamma (17)simulateFromPrior (18)singleLocalProposal (18)truncatedPoisson (19)updateNumberOfCutpoints (19)Index21beast-package Bayesian Estimation of Change-Points in the Slope of MultivariateTime-SeriesDescriptionAssume that a temporal process is composed of contiguous segments with differing slopes and replicated noise-corrupted time series measurements are observed.The unknown mean of the data generating process is modelled as a piecewise linear function of time with an unknown number of change-points.The package infers the joint posterior distribution of the number and position of change-points as well as the unknown mean parameters per time-series by MCMC sampling.A-priori,the proposed model uses an overfitting number of mean parameters but,conditionally on a set of change-points,only a subset of them influences the likelihood.An exponentially decreasing prior distribution on the number of change-points gives rise to a posterior distribution concentrating on sparse representations of the underlying sequence,but also available is the Poisson distribution.See Papastamoulis et al(2017)<arXiv:1709.06111>for a detailed presentation of the method. DetailsThe beast package deals with B ayesian e stimation of ch a nge-points in the s lope of multivariate t ime-series,introduced by Papastamoulis et al(2017).For a given period t=1,...,T we observe multiple time series which are assumed independent,each one consisting of multiple measurements (replicates).Each time series is assumed to have its own segmentation,which is common among its replicates.Thus,different time series have distinct mean parameters in the underlying normal distribution.The variance,which is assumed known,can be either shared between different time series or not and in practice it is estimated at a pre-processing stage.Our model infers the joint posterior distribution of the number and location of change-points by MCMC sampling.For this purpose a Metropolis-Hastings MCMC sampler is used.The main function of the package is beast.Assume that the observed data consists of N time-series,each one consisting of R variables(or replicates)measured at T consecutive time-points.The input of the main function beast should be given in the form of a list myDataList with the following attributes:•length(myDataList)should be equal to R,that is,the number of variables(or replicates)•dim(myDataList)[[1]],...,dim(myDataList)[[R]]should be all T×N,that is,rows andcolumns should correspond to time-points and different series,respectively.Then,a basic usage of the package consists of the following commands:•beastRun<-beast(myDataList=myDataList)•print(beastRun)•plot(beastRun)which correspond to running the MCMC sampler,printing and plotting output summaries,respec-tively.Author(s)Panagiotis PapastamoulisMaintainer:Panagiotis Papastamoulis<*****************>ReferencesPapastamoulis P.,Furukawa T.,van Rhijn N.,Bromley M.,Bignell E.and Rattray M.(2017).Bayesian detection of piecewise linear trends in replicated time-series with application to growth data modelling.arXiv:1709.06111[stat.AP]See Alsobeast,print.beast.object,plot.beast.objectExamples#toy-example(MCMC iterations not enough)library( beast )#load packagedata("FungalGrowthDataset")#load datasetmyIndex<-c(392,62,3,117)#run the sampler only for the#specific subset of time-seriesset.seed(1)#Run MCMC sampler with very small number of iterations(nIter):run_mcmc<-beast(myDataList=FungalGrowthDataset,subsetIndex=myIndex,zeroNormalization=TRUE,nIter=40,burn=20)#Print output:print(run_mcmc)#Plot output to file:"beast_plot.pdf"plot(run_mcmc,fileName="beast_plot_toy.pdf",timeScale=1/6,xlab="hours",ylab="growth")#Run the following commands to obtain convergence:##Not run:#This example illustrates the package using a subset of four#time-series of the fungal dataset.library( beast )#load packagedata("FungalGrowthDataset")#load datasetmyIndex<-c(392,62,3,117)#run the sampler only for the#specific subset of time-seriesset.seed(1)#optional#Run MCMC sampler with the default number of iterations(nIter=70000):run_mcmc<-beast(myDataList=FungalGrowthDataset,subsetIndex=myIndex,zeroNormalization=TRUE)#Print output:print(run_mcmc)#Plot output to file:"beast_plot.pdf"plot(run_mcmc,fileName="beast_plot.pdf",timeScale=1/6,xlab="hours",ylab="growth") #NOTE1:for a complete analysis remove the subsetIndex=myIndex argument.#NOTE2: zeroNormalization=TRUE is an optional argument that forces all#time-series to start from zero.It is not supposed to be used#for other applications.##End(Not run)beast Main functionDescriptionThis is the main function of the package,implementing the MCMC sampler described in Papasta-moulis et al(2017).Usagebeast(myDataList,burn,nIter,mhPropRange,mhSinglePropRange,startPoint, timeScale,savePlots,zeroNormalization,LRange,tau,gammaParameter,nu0,alpha0,beta0,subsetIndex,saveTheta,sameVariance,Prior)ArgumentsmyDataList Observed data in the form of a list with length R,denoting the dimensionality of the multivariate time-series data.For each r=1,...,R,myDataList[[r]]should correspond to T×N array,with myDataList[[r]][t,n]correspondingto the observed data for time-series n=1,...,N and time-point t=1,...,T.burn Number of iterations that will be discarder as burn-in period.Default value: burn=20000.nIter Number of MCMC iterations.Default value:nIter=70000.mhPropRange Positive integer corresponding to the parameter d1of MCMC Move3.a of Pa-pastamoulis et al(2017).Default value:mhPropRange=1.mhSinglePropRangePositive integer denoting the parameter d2of Papastamoulis et al(2017).Defaultvalue:mhPropRange=40.startPoint An(optional)positive integer pointing at the minimum time-point where changes are allowed to occur.Default value:startPoint=2(all possible values aretaken into account).timeScale Null.savePlots Character denoting the name of the folder where various plots will be saved to.zeroNormalizationLogical value denoting whether to normalize to zero all time time-series fort=1.Default:zeroNormalization=FALSE.LRange Range of possible values for the number of change-points.Default value:LRange =0:30.tau Positive real number corresponding to parameter c in Move2of Papastamoulis et al(2017).Default value:tau=0.05.gammaParameter Positive real number corresponding to parameterαof the exponential prior dis-tribution.Default value:gammaParameter=2.nu0Positive real number corresponding to prior parameterν0in Papastamoulis et al (2017).Default value:nu0=0.1.alpha0Positive real number corresponding to prior parameterα0in Papastamoulis et al (2017).Default value:alpha0=1.beta0Positive real number corresponding to prior parameterβ0in Papastamoulis et al (2017).Default value:beta0=1.subsetIndex Optional subset of integers corresponding to time-series indexes.If not null,the sampler will be applied only to the specified subset.saveTheta Logical value indicating whether to save the generated values of the mean per time-point across the MCMC trace.Default:FALSE.sameVariance Logical value indicating whether to assume the same variance per time-point across time-series.Default value:sameVariance=TRUE.Prior Character string specifying the prior distribution of the number of change-points.Allowed values:Prior="complexity"(default)or Prior="Poisson"(notsuggested).ValueThe output of the sampler is returned as a list,with the following features:Cutpoint_posterior_medianThe estimated medians per change-point,conditionally on the most probablenumber of change-points per time-series.Cutpoint_posterior_varianceThe estimated variances per change-points,conditionally on the most probablenumber of change-points per time-series.NumberOfCutPoints_posterior_distributionPosterior distributions of number of change-points per time-series.NumberOfCutPoints_MAPThe most probable number of change-points per time-series.Metropolis-Hastings_acceptance_rateAcceptance of the MCMC move-types.subject_ID the identifier of individual time-series.Cutpoint_mcmc_trace_mapThe sampled values of each change-point per time series,conditionally on theMAP values.theta The sampled values of the means per time-series,conditionally on the MAP values.nCutPointsTraceThe sampled values of the number of change-points,per time-series.NoteThe complexity prior distribution with parameter gammaParameter=2is the default prior assump-tion imposed on the number of change-points.Smaller(larger)values of gammaParameter will a-priori support larger(respectively:smaller)number of change-points.For completeness purposes,the Poisson distribution is also allowed in the Prior.In this latter case, the gammaParameter denotes the rate parameter of the Poisson distribution.Note that in this case the interpretation of gammaParameter is reversed:Smaller(larger)values of gammaParameter will a-priori support smaller(respectively:larger)number of change-points.Author(s)Panagiotis PapastamoulisReferencesPapastamoulis P.,Furukawa T.,van Rhijn N.,Bromley M.,Bignell E.and Rattray M.(2017).Bayesian detection of piecewise linear trends in replicated time-series with application to growth data modelling.arXiv:1709.06111[stat.AP]Examples#toy-example(MCMC iterations not enough)library( beast )#load packagedata("FungalGrowthDataset")#load datasetmyIndex<-c(392,62,3,117)#run the sampler only for the#specific subset of time-seriesset.seed(1)#Run MCMC sampler with very small number of iterations(nIter):run_mcmc<-beast(myDataList=FungalGrowthDataset,subsetIndex=myIndex,birthProbs7 zeroNormalization=TRUE,nIter=40,burn=20)#Print output:print(run_mcmc)#Plot output to file:"beast_plot.pdf"plot(run_mcmc,fileName="beast_plot_toy.pdf",timeScale=1/6,xlab="hours",ylab="growth") #Run the following commands to obtain convergence:##Not run:#This example illustrates the package using a subset of four#time-series of the fungal dataset.library( beast )#load packagedata("FungalGrowthDataset")#load datasetmyIndex<-c(392,62,3,117)#run the sampler only for the#specific subset of time-seriesset.seed(1)#optional#Run MCMC sampler with the default number of iterations(nIter=70000):run_mcmc<-beast(myDataList=FungalGrowthDataset,subsetIndex=myIndex,zeroNormalization=TRUE)#Print output:print(run_mcmc)#Plot output to file:"beast_plot.pdf"plot(run_mcmc,fileName="beast_plot.pdf",timeScale=1/6,xlab="hours",ylab="growth") #NOTE1:for a complete analysis remove the subsetIndex=myIndex argument.#NOTE2: zeroNormalization=TRUE is an optional argument that forces all#time-series to start from zero.It is not supposed to be used#for other applications.##End(Not run)birthProbs Birth ProbabilitiesDescriptionThis function defines the probability of proposing an addition of a change-point.UsagebirthProbs(LRange)ArgumentsLRange The range of possible values for the number of change-points.Valueprobs Vector of probabilitiesAuthor(s)Panagiotis Papastamoulis8computeEmpiricalPriorParameters complexityPrior Complexity prior distributionDescriptionThis function computes the complexity prior distribution on the number of change-points,defined as f( )=P( n= )∝e−α log(bT/ ),a,b>0; =0,1,2,....Note that this distribution has exponential decrease(Castillo and van der Vaart,2012)when b>1+e,so we set b=3.72. UsagecomplexityPrior(Lmax=20,gammaParameter,nTime)ArgumentsLmax maximum number of change-points(default=20).gammaParameter positive real number,corresponding toα.nTime positive integer denoting the total number of time-points.ValuelogPrior Prior distribution values in the log-scale.Author(s)Panagiotis PapastamoulisReferencesCastillo I.and van der Vaart A(2012).Needles and Straw in a Haystack:Posterior concentration for possibly sparse sequences.The Annals of Statistics,40(4),2069–2101. computeEmpiricalPriorParametersCompute the empirical mean.DescriptionThis function computes the empirical mean of the time-series.UsagecomputeEmpiricalPriorParameters(myDataList,nu0=1,alpha0=1,beta0=1)computePosteriorParameters9ArgumentsmyDataList Observed multivariate time-series.nu0positive real number.alpha0positive real number.beta0positive real number.Valuemu0Empirical meanAuthor(s)Panagiotis PapastamouliscomputePosteriorParametersCompute empirical posterior parametersDescriptionCompute empirical posterior parametersUsagecomputePosteriorParameters(myDataList,priorParameters)ArgumentsmyDataList Observed data.priorParametersPrior parameters.Valuelist of posterior parametersAuthor(s)Panagiotis Papastamoulis10FungalGrowthDataset computePosteriorParametersFreePosterior parametersDescriptionPosterior parametersUsagecomputePosteriorParametersFree(myDataList,priorParameters)ArgumentsmyDataList observed data.priorParameterslist of prior parameters.Valuelist of posterior parameters.Author(s)Panagiotis PapastamoulisFungalGrowthDataset Fungal Growth DatasetDescriptionTime-series dataset with N×R×T growth levels for R=3replicates of N=411objects (mutants)measured every10minutes for T=289time-points.See Papastamoulis et al(2017)fora detailed description.UsageFungalGrowthDatasetFormatTime-series data.ReferencesPapastamoulis P.,Furukawa T.,van Rhijn N.,Bromley M.,Bignell E.and Rattray M.(2017).Bayesian detection of piecewise linear trends in replicated time-series with application to growth data modelling.arXiv:1709.06111[stat.AP]localProposal11 localProposal Move3.bDescriptionImplements Move3.b of the Metropolis-Hastings MCMC sampler.UsagelocalProposal(cutPoints,nTime,mhPropRange,startPoint)ArgumentscutPoints Current configuration of change-points.nTime Total number of time-points.mhPropRange Parameter d2.startPoint Integer,at least equal to2.ValuenewState Candidate state of the chain.propRatio Proposal ratio.Author(s)Panagiotis PapastamoulislogLikelihoodFullModelLog-likelihood function.DescriptionLog-likelihood function.UsagelogLikelihoodFullModel(myData,cutPoints,theta,startPoint)ArgumentsmyData datacutPoints change-points.theta means.startPoint optional integer at least equal to2.Valuelog-likelihood value.Author(s)Panagiotis PapastamoulislogPrior Log-prior.DescriptionLogarithm of the prior distribution on the number of change-points.UsagelogPrior(cutPoints,nTime,startPoint)ArgumentscutPoints change-points.nTime number of time-points.startPoint optional integer,at least equal to2.Valuelogarithm of the prior distribution.Author(s)Panagiotis PapastamoulismcmcSampler MCMC samplerDescriptionThis function implements the Metropolis-Hastings MCMC sampler for individual time-series.UsagemcmcSampler(myData,nIter,finalIterationPdf,modelVariance,mhPropRange, mhSinglePropRange,movesRange,startPoint,postPar,dName,timeScale,burn,iterPerPlotPrefix,priorParameters,L=3,LRange,tau,gammaParameter,saveTheta,Prior="complexity")ArgumentsmyData observed data.nIter number of mcmc iterationsfinalIterationPdfoutput foldermodelVariance nullmhPropRange positive integermhSinglePropRangepositive integermovesRange nullstartPoint positive integerpostPar list of emprirically estimated parametersdName subject IDtimeScale nullburn burn-in period.iterPerPlotPrefixnullpriorParametersprior parameters.L nullLRange range of possible values of the number of change-points.tau real.gammaParameter real.saveTheta TRUE.Prior character.ValueMCMC output.Author(s)Panagiotis Papastamoulis14normalizeTime0 myUnicodeCharacters PrintingDescriptionPrinting various unicode symbols.UsagemyUnicodeCharacters()Valueprinted symbolnormalizeTime0Zero normalizationDescriptionZero normalization at1st time-pointUsagenormalizeTime0(myDataList)ArgumentsmyDataList dataValuenullAuthor(s)Panagiotis Papastamoulisplot.beast.object15 plot.beast.object Plot functionDescriptionThis function plots objects returned by the beast function.All output is diverted to a pdffile provided in the fileName argument.Usage##S3method for class beast.objectplot(x,fileName,width,height,pointsize,ylab,xlab,timeScale,myPal,boxwex,...) Argumentsx An object of class beast.object,which is returned by the beast function.fileName Name of the output pdffile.width Width of pdffile.Default:9height Height pdffile.Default:6pointsize Pointsize.Default:12ylab y-axis label.Default:x.xlab x-axis label.Default:t.timeScale A multiplicative-factor which will be used to scale the x-axis labels.For exam-ple,if time-points correspond to10-minute periods,then setting timeScale=1/6will make the x-axis labels correspond to hours.Default:timeScale=1.myPal Vector of colors that will be used to produce the plot with all time-series over-layed.If the distinct values of the inferred numbers of change-points is less than10,the Set1pallete of the RColorBrewer library is used.Otherwise,the userhas to manually define the colors.boxwex A scale factor to be applied to all boxes.The appearance of the plot can beimproved by making the boxes narrower or wider.Default:0.2....ignored.DetailsThe function will produce a plot with all time-series coloured according to the corresponding num-ber of change-points.In addition,it will generate individual plots per time-series displaying the observed data with boxplots which summarize the posterior distribution of change-points locations, conditionally on the most probable number of change-points.Author(s)Panagiotis Papastamoulis16proposeTheta print.beast.object Print functionDescriptionThis function prints a summary of objects returned by the beast function.Usage##S3method for class beast.objectprint(x,...)Argumentsx An object of class beast.object,which is returned by the beast function....ignored.DetailsThe function prints a summary of the most probable number(MAP)of change-points per time-series in the form of a table,as well as a list containing the MAP number of change-points and the corresponding locations(posterior medians)per time-series.Author(s)Panagiotis PapastamoulisproposeTheta Move2DescriptionProposes an update ofθaccording to Metropolis-Hastings move2.UsageproposeTheta(thetaOld,tau,alpha,beta)ArgumentsthetaOld Current valuestau Parameter c.alpha nullbeta nullsimMultiIndNormInvGamma17 Valuemean proposed values.Author(s)Panagiotis PapastamoulissimMultiIndNormInvGammaPrior random numbersDescriptionGeneration of mean values according to the priorUsagesimMultiIndNormInvGamma(mu,nu,alpha,beta)Argumentsmu meansnu precision parameteralpha prior parameterbeta prior parameterValuenullAuthor(s)Panagiotis Papastamoulis18singleLocalProposal simulateFromPrior Generate change-points according to the priorDescriptionGenerate change-points according to the prior distribution conditionally on a given number of change-points.UsagesimulateFromPrior(nTime,startPoint,L=3)ArgumentsnTime Number of time-pointsstartPoint At least equal to2.L nullValuecutPoints Change-point locationsAuthor(s)Panagiotis PapastamoulissingleLocalProposal Move3.bDescriptionImplement Metropolis-Hastings Move3.b.UsagesingleLocalProposal(cutPoints,nTime,mhSinglePropRange,startPoint)ArgumentscutPoints Current statenTime Number of time-pointsmhSinglePropRangePrior parameter.startPoint Optional.truncatedPoisson19ValuenewState Candidate statepropRatio Proposal ratioAuthor(s)Panagiotis PapastamoulistruncatedPoisson Truncated Poisson pdfDescriptionProbability density function of the truncated Poisson distribution.UsagetruncatedPoisson(Lmax=20,gammaParameter=1)ArgumentsLmax Max numbergammaParameter Location parameter.ValuelogPrior Log-pdf valuesAuthor(s)Panagiotis PapastamoulisupdateNumberOfCutpointsMove1DescriptionUpdate the number of change-points according to Metropolis-Hastings move1.UsageupdateNumberOfCutpoints(cutPoints,nTime,startPoint,LRange,birthProbs)20updateNumberOfCutpointsArgumentscutPoints Current configurationnTime Number of time-pointsstartPoint Optional integerLRange Range of possible valuesbirthProbs Birth probabilitiesValuenewState Candidate statepropRatio Proposal ratioAuthor(s)Panagiotis PapastamoulisIndex∗datasetsFungalGrowthDataset,10∗packagebeast-package,2beast,3,4beast-package,2birthProbs,7complexityPrior,8 computeEmpiricalPriorParameters,8 computePosteriorParameters,9 computePosteriorParametersFree,10 FungalGrowthDataset,10localProposal,11 logLikelihoodFullModel,11logPrior,12mcmcSampler,12 myUnicodeCharacters,14normalizeTime0,14plot.beast.object,3,15print.beast.object,3,16 proposeTheta,16 simMultiIndNormInvGamma,17 simulateFromPrior,18 singleLocalProposal,18 truncatedPoisson,19 updateNumberOfCutpoints,1921。

现代⼼理和教育统计学课后题第⼀章绪论1.名词解释随机变量：在统计学上，把取值之前不能预料取到什么值的变量称之为随机变量总体：⼜称为母全体、全域，指据有某种特征的⼀类事物的全体样本：从总体中抽取的⼀部分个体，称为总体的⼀个样本个体：构成总体的每个基本单元称为个体次数：指某⼀事件在某⼀类别中出现的数⽬，⼜成为频数，⽤f表⽰频率：⼜称相对次数，即某⼀事件发⽣的次数被总的事件数⽬除，亦即某⼀数据出现的次数被这⼀组数据总个数去除。

频率通畅⽤⽐例或百分数表⽰概率：⼜称机率。

或然率，⽤符号P表⽰，指某⼀事件在⽆限的观测中所能预料的相对出现的次数，也就是某⼀事物或某种情况在某⼀总体中出现的⽐率统计量：样本的特征值叫做统计量，⼜叫做特征值参数：总体的特性成为参数，⼜称总体参数，是描述⼀个总体情况的统计指标观测值：在⼼理学研究中，⼀旦确定了某个值，就称这个值为某⼀变量的观测值，也就是具体数据2.何谓⼼理与教育统计学学习它有何意义⼼理与教育统计学是专门研究如何运⽤统计学原理和⽅法，搜集。

整理。

分析⼼理与教育科学研究中获得的随机数据资料，并根据这些数据资料传递的信息，进⾏科学推论找出⼼理与教育活动规律的⼀门学科。

3.选⽤统计⽅法有哪⼏个步骤⾸先要分析⼀下试验设计是否合理，即所获得的数据是否适合⽤统计⽅法去处理，正确的数量化是应⽤统计⽅法的起步，如果对数量化的过程及其意义没有了解，将⼀些不着边际的数据加以统计处理是毫⽆意义的其次要分析实验数据的类型，不同数据类型所使⽤的统计⽅法有很⼤差别，了解实验数据的类型和⽔平，对选⽤恰当的统计⽅法⾄关重要第三要分析数据的分布规律，如总体⽅差的情况，确定其是否满⾜所选⽤的统计⽅法的前提条件4.什么叫随机变量⼼理与教育科学实验所获得的数据是否属于随机变量随机变量的定义：①率先⽆法确定，受随机因素影响，成随机变化，具有偶然性和规律性②有规律变化的变量5.怎样理解总体、样本与个体总体N：据有某种特征的⼀类事物的全体，⼜称为母体、样本空间，常⽤N表⽰，其构成的基本单元为个体。