Possible biases induced by MCMC convergence diagnostics

Possible biases induced by MCMC convergence diagnostics

by

Mary Kathryn Cowles*,Gareth O.Roberts**and Je?rey S.Rosenthal***

(December,1997;revised October,1998.)

Abstract

Convergence diagnostics are widely used to determine how many initial“burn-in”iterations should be discarded from the output of a Markov chain Monte Carlo(MCMC)sampler in the hope that the remaining samples are representative of the target distribution of interest.This paper demonstrates that some ways of applying convergence diagnostics may actually introduce bias into estimation based on the sampler output.To avoid this possibility,we recommend choosing the number of burn-in iterations r by applying con-vergence diagnostics to one or more pilot chains,and then basing estimation and inference on a separate long chain from which the?rst r iterations have been discarded.

1.Introduction.

Markov chain Monte Carlo(MCMC)methods(Metropolis et al.,1953;Hastings, 1970;Geman and Geman,1984;Tanner and Wong,1987;Gelfand and Smith,1990;Smith and Roberts,1993;Neal,1993)are very commonly used to estimate the expected value μ=E(g)of a functional g with respect to a complicated probability measureπ(·).The procedure involves setting up a Markov chain P(·,·)having stationary distributionπ(·), running this chain on a computer to obtain output X0,X1,...,X n,and then using this

sample to estimateμ.The estimator used is typically of the form μ=1

n?r

n

i=r+1

g(X i),

corresponding to throwing away the?rst r runs and then averaging the functional over the remaining n?r runs.

*Department of Statistics and Actuarial Science,University of Iowa,Iowa City,IA 52242-1419,U.S.A.Internet:kcowles@https://www.wendangku.net/doc/c66992014.html,.

**Department of Mathematics and Statistics,Fylde College,Lancaster University,Lan-caster,LA14YF,England.Internet:g.o.roberts@https://www.wendangku.net/doc/c66992014.html,.

***Department of Statistics,University of Toronto,Toronto,Ontario,Canada M5S3G3. Internet:jeff@https://www.wendangku.net/doc/c66992014.html,.Partially supported by NSERC of Canada.

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However,it is not at all clear how to choose the burn-in time“r”.Theoretical analysis has sometimes succeeded in providing useful values of r(Meyn and Tweedie,1994;Rosen-thal,1995b,1996;Cowles and Rosenthal,1996),but only in certain restricted cases.More often,convergence diagnostics(Roberts,1992,1994;Raftery and Lewis,1992;Geweke, 1992;Gelman and Rubin,1992;Geyer,1992;Cowles and Carlin,1996;Brooks and Roberts, 1995)are used.That is,the value of r is itself random(i.e.output dependent),and is ob-tained by a statistical analysis of the initial sample values of the chain.Very roughly, convergence diagnostics instruct the user to wait until the values of one or several runs of the Markov chain have“settled down”in some sense.

It is well known(see e.g.Cowles and Carlin,1996for a review)that for certain target distributions(e.g.the“witch’s hat”example,cf.Matthews,1993),convergence diagnostics will prematurely claim convergence,i.e.will provide a value of r which is too small and therefore leads to severely biased estimators μ.However,another possible problem with convergence diagnostics is less well studied.Speci?cally,the dependence of r on the actual sample values can lead to further biasing of μ.Most dramatically,even if the chain was in fact started in the stationary distributionπ(·),so that for any non-random(i.e.independent of the output)choice of r the estimator μwould be perfectly unbiased,it is still possible that μmay have bias for natural choices of r which depend on the actual sample values. Such“diagnostic biases”are the subject of the present paper.

This paper is organised as follows.In Section2,we present an over-simpli?ed version of a convergence diagnostic,and study analytically its performance on certain simple Markov chains.We restrict ourselves primarily to chains which in fact produce i.i.d.samples from π(·),i.e.which mix extremely rapidly.For such chains,any non-random choice of r would result in a perfectly unbiased estimator;however,the convergence diagnostic introduces biases of its own,which we are able to compute explicitly.

In Section3,we present numerical simulations of variants of the Geweke(1992)diag-nostic to two simple Bayesian inference problems.We demonstrate that in certain cases, signi?cant biases can be caused by the naive use of these diagnostics.Diagnostic bias is thus an important practical issue.

In Section4,we investigate the option of choosing r to be non-random,https://www.wendangku.net/doc/c66992014.html,pletely

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independent of the sample values.This avoids the biases of dependent r;however it raises the question of what r to choose.We concentrate on a simple two-state Markov chain in which we can explicitly compute the standard error of the estimator μ.We show that the optimal choice of r depends very strongly on the relative sizes of the transition probabilities: it is sometimes very small,but can be a substantial fraction of n.

Finally,in Section5,we discuss the practical implications of our results.we argue that Our results suggest that an e?ective approach combines diagnostics with non-random choices of r,by?rst analysing some initial runs to determine a reasonable choice of r,and then creating an estimator based on a fresh new run of the Markov chain,which has r ?xed in advance.Such an approach avoids the e?ects of bias due to the use of convergence diagnostics,although the increased computational expense of such a procedure needs to be taken into account when choosing an implementation strategy.

2.Analytic study of biases from simple diagnostics.

In this section,we consider an over-simpli?ed convergence diagnostic,applied to i.i.d. chains.For this simple case,we are able to analytically compute the biases involved.

Speci?cally,suppose that X0,X1,X2,...are the output of a Markov chain,and that in fact the X n are independent and each distributed according toπ(·)(so that the true burn-in time for the chain is0).

Suppose the convergence is diagnosed by way of batch means.Speci?cally,suppose we choose a?xed positive integer m and a?xed >0.We then compute the quantities

A m,i=1

m m(i+1)

t=mi+1

X t,for i=0,1,2,...,and let i?be the smallest value of i such that

|A m,i??A m,i??1|< .We then take our estimator to be μ=1

m m(i?+1)

t=mi?+1

g(X t).In words,

we assess“convergence”as the?rst time that two consecutive batches(each of length m) give the same mean to within a tolerance of ;we then use the most recent m sample values to estimateμ.

We show that even in this simple case,biases are introduced through the use of this

sampler.Indeed,suppose the law of1

m

m

t=1

X t has density f with respect to Lebesgue

measure.Assume that f is piecewise continuous.Then,as 0,it is easily seen that the

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law of μwill converge weakly to the distribution having density proportional to f2.That is,for small >0,the probability of having two independent batch means both very close to a particular value is approximately proportional to the square of having just one batch mean close to that value.

This result can be demonstrated formally by making the following argument rigorous. Given that the Markov chain is currently at state x,by piece-wise continuity of f,for almost all x,f is approximatly constant on a small enough interval(x? ,x+ ),so that the probability of obtaining a value less than from x is approximately2 f(x).Therefore, given that the probability of being at x in the?rst place is proportional to f(x),the probability of stopping at x is approximately proportional to2 f2.

So,the very act of demanding that two(independent)consecutive batch means be very close to each other,in fact replaces their probability density by(approximately)its square. Is this change signi?cant?Well,as m→∞the distribution of the batch mean will(by the weak law of large numbers)be close to a point mass at the true meanμ,so squaring the density will have little impact,at least on the mean of the distribution sampled.However, for?nite values of m this squaring will have a noticeable e?ect.Moreover,if the bias is calculated in other ways(for instance total variation distance between the true target density and the density actually simulated)it is possible as we shall see that the bias does not decrease at all with sample size

We consider some examples here.

Example1.Suppose that each X i has the standard normal distribution N(0,1).Then

the law of1

m

m

t=1

X t is the normal distribution N(0,1/m),so f is the density function for

N(0,1/m).But then it is easily seen that f2is proportional to the density function for N(0,1/2m).That is,in this case,demanding that two consecutive batch means be very close to each other in fact means the sampled batch mean will have distribution N(0,1/2m) instead of N(0,1/m).Of course,by symmetry the mean of each of these distributions is0, so there is no bias in the mean.However,the variance is o?by1/m.Moreover the error in total variation distance is constant as a function of m;it does not go to0as m→∞.

Example 2.Suppose that each X i has the gamma distribution Gamma(a,b),with

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density proportional to x a ?1e ?bx .Then the law of 1

m m t =1X t is the gamma distribution

Gamma(ma,mb ).Now,the square of the density of this distribution is proportional to the density of Gamma(2ma ?1,2mb ).Hence,in this case the diagnostic replaces the Gamma(ma,mb )distribution by the Gamma(2ma ?1,2mb )distribution.Among other things,this leads to a bias in the mean:It replaces the mean ma mb by the mean

2ma ?12mb ,giving a bias of ?12mb .Example 3.Suppose that each X i has the standard Cauchy distribution,with den-

sity 1π(1+x 2).Then 1m m t =1X t again has the standard Cauchy distribution.Hence,in

this case the diagnostic replaces the standard Cauchy distribution by a distribution hav-ing density 2

π(1+x 2)2,regardless of the choice of m .(Note that here the mean μis

not de?ned,so the weak law of large numbers does not apply.)Furthermore,we com-pute the total variation distance (between the target and the sampled densities)to be 12∞ ?∞

1π(1+x 2)?2π(1+x 2)2 dx =1/π.=0.318,again regardless of the choice of m .We thus see that,in all three of these examples,demanding that two consecutive batch means be very similar signi?cantly a?ects the distribution that is actually sampled.In cases where μis ?nite (i.e.the ?rst two examples above),the errors for the mean and variance decrease to 0as m →∞,i.e.as the batch size gets large.

We close this section with a number of remarks.

Remarks.

1.We have stated our diagnostic in terms of requiring two consecutive batch means (of the same Markov chain run)to be similar.However,our results are clearly identical if we instead imagine running two independent copies of the chain,and taking the smallest i such that the i th batch mean of the ?rst chain is similar to the i th batch mean of the second chain.This version is closer in spirit to the multiple independent chain approach to convergence diagnostics (e.g.Gelman and Rubin,1992).

2.More generally,we might require that k (instead of just 2)consecutive batch means are similar.In this case,the density f would be approximately replaced by its k th power,f k .This makes the resulting errors even worse .For example,for the Gamma(a,b )example above,this “k equal”diagnostic would replace the Gamma(ma,mb )distri-

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bution by the Gamma(kma?(k?1),kmb)distribution.Hence,the bias in the mean

would be kma?(k?1)

kmb ?ma

mb

=?k?1

kmb

,which is increasing as a function of k(but is never

more than twice the bias of the k=2case).

3.Of course,if the Markov chain is really providing i.i.d.samples,then we can avoid all

diagnostic-related problems by simply using the following batch mean for our estima-tor,rather than using one of the batch means we actually used to diagnose conver-gence.However,in a more realistic situation in which the Markov chain only slowly changes values,this“solution”would not be a solution at all,and the diagnostic biasing problems would be virtually unchanged.

4.There may be times when we actually want to sample from the distribution with

density proportional to f2(or to f k),but it is easier(for some reason)to construct and run a Markov chain having stationary distribution with density f.In that case, our analysis above suggests that one way to proceed is to use our simple diagnostic rule to convert a sampler for f into one for f2(or f k as in Remark2above).However, this is likely not a practical approach in general,since as 0the waiting time until “convergence”goes to in?nity.

5.For a Markov chain which does not converge immediately to the stationary distri-

bution,there will be some interplay between diagnostic bias as discussed here,and more traditional convergence bias(i.e.because the chain hasn’t converged yet).It is even possible in rare cases that these biases will cancel each other out.For example, consider the Markov chain on the two-point space{1,2},with initial probabilities

(1 2,1

2

),with transition probabilities p12=2δand p21=δ,and hence with stationary

distribution(1

3,2

3

).For very small values ofδ,and with a small batch size m,our

diagnostic would likely claim convergence immediately,and hence the sampled distri-

bution would be(1

2,1

2

).(Note that for the multiple-chain diagnostic as in Remark1

above,the situation is slightly more complicated.)On the other hand,ifδ=α/k for the k equal rule(as in Remark2above),then the probability that our sampled point will be1is

e?α

2?e?α

2(1+e?α?e?2α)

;

hence,for an appropriate choice ofαthis can be made equal to2

3

,as desired.Thus,

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in this example,it is indeed possible for the two types of bias(diagnostic and conver-gence)to cancel each other out.

6.On the other hand,it is not possible in general for the diagnostic bias to cancel out

from every choice of initial state.To see this,consider the following simple example together with the“two equal”stopping rule.Let K(·,·)be the transition probabilities for a discrete state space Markov chain.Suppose that K is irreducible and aperiodic with stationary distribution f.Also let p(x,y)denote the probability that starting at x our terminal value(ie the?rst repeated value)is y.Suppose we assume that X0has the stationary distribution f(so as to eliminate the e?ect of the convergence bias).

Now if we want the bias to be non-existent,we will also require that the probability of terminating in the state y should be f(y).Therefore f solves

f(x)=

y

f(y)p(y,x)

also.That is,p becomes a transition matrix with stationary distribution f.However because of the particular construction of the stopping time,for y=x,

p(y,x)=

K(y,z)p(z,x)?K(y,y)p(y,x)

whereas

p(x,x)=

K(y,z)p(z,x)+(1?p(x,x))K(x,x).

Summing these two equations over y gives

y

f(y)K(y,y)P(y,x)=f(x)K(x,x)

so that by uniqueness of the invariant measure f for p(p is irreducible and strongly aperiodic by inspection)f(y)K(y,y)must be a constant multiple of f and so K(y,y) is necessarily constant.To summarise,it is impossible for the bias to be completely removed unless K(y,y)is constant,and it is easy to check that under this condition, the“k equal”stopping rule introduces no bias.

7.If the bias is measured in terms of theχ2distance,it has a natural interpretation in

terms of the coe?cient of variation of the random variable f(X).Speci?cally,recall

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theχ2distance with respect to the density f of two densities g and h:

g?h =E f

g(X)

f(X)

?

h(X)

f(X)

2

.

Then a simple calculation yields that

f?Cf2 =(cv f(f(X)))2,

where C is chosen so that Cf2is a density,and where cv(Y)is the coe?cient of variation of a non-negative random variable:cv(Y)=Var(Y)1/2/E(Y).

3.Simulation study of biases from realistic diagnostics.

To investigate whether bias may be introduced by convergence diagnostics that are used in common practice,we performed a simulation study of two variants of the conver-gence diagnostic proposed by Geweke(1992).One of these variants is built into the BUGS (“Bayesian inference Using Gibbs Sampling”)software package(Spiegelhalter,Thomas, Best and Gilks,1995),and another is included in CODA(Best,Cowles,and Vines,1995), a suite of Splus routines distributed with BUGS that performs convergence diagnosis and output analysis on Gibbs sampler output.

To apply Geweke’s diagnostic,for each individual parameter of interest,one computes the sample meanˉx early of a sequence of iterates from the beginning of the MCMC chain(he suggests the?rst10%)and the sample meanˉx late from a longer sequence at the end of the chain(he suggests the last50%of the chain).If many iterations separate the“early”and “late”segments of the chain,thenˉx early andˉx late should be approximately independent, and

Z=

ˉx early?ˉx late

Var(ˉx early)+Var(ˉx late)

should have approximately a standard normal distribution.Geweke suggests that if|Z|> 1.96,iterates from the“early”segment were not yet drawn from the target distribution and should be discarded.He recommends using time-series methods based on spectral densities to compute Var(ˉx early)and Var(ˉx late).

Geweke’s diagnostic with this method of variance estimation is included in CODA, with user-modi?able default values of.1for the proportion of iterates in the“early”segment

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and.5for the proportion“late.”The“diag”command in BUGS invokes a variant of the Geweke diagnostic that uses the less computationally-intensive“batch means”method of variance estimation.The“diag”command assigns the?rst25%of iterates to the“early”segment and the last50%to the“late”segment.The|Z|statistic is then computed as above.

For our simulation study,we used two simple two-parameter models for which com-puting the true posterior joint and marginal distributions is straightforward;thus estimates from MCMC runs could be compared to the“truth.”The?rst is the normal mean model with likelihood f(x)=N(μ,σ2)and noninformative priorπ(μ,σ2)∝1

,which we?t to

σ2

the data on observed breaking strength of20samples of yarn given in Box and Tiao(1973), Table2.2.1.

We ran300parallel Gibbs samplers,all started in stationarity(i.e.initial values were random draws from the true joint posterior distribution)and run for1000iterations.We then applied the Geweke diagnostic with both variance-estimation methods(designated “Batch”and“Spectral”in the tables)and two choices of proportion of iterates in the “early”segment(.1and.25)to the output from each sampler for each parameter in the following sequential manner.[This type of procedure is often used in applied practice, especially when samplers are run in one software package(e.g.BUGS)and convergence is assessed in another(e.g.CODA),so that it is easy to remove burn-in iterations but less convenient to extend the length of the chain.]

If the value of the|Z|statistic was greater than1.96for either parameter,the?rst 100(or250when the second de?nition of“early”was being used)iterates were discarded for all parameters and the diagnostic was reapplied to the remaining portion of the chain. This process was continued until either the|Z|statistic was≤1.96for both parameters or the next discarding would have resulted in fewer than250iterates being retained(in which case the sampler run was deemed to have failed to converge and no statistics were stored).

For each parameter,the?rst section of Table1presents the analytically computed posterior mean and width of the95%HPD region.The next section shows the mean across all300chains of the sample mean,the mean squared error,and the width of the

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interval between the.025and.975quantiles for each parameter.The“t”statistics are for univariate tests of the di?erence between the true value and the corresponding value estimated from the samplers.

Because the samplers were started in stationarity and mixing is rapid for this model (i.e.,autocorrelations within the sampler output for each parameter,as well as cross-correlations between the parameters,are negligible),no burn-in iterations should have been discarded.However,application of the four versions of the diagnostic resulted in discarding initial iterations from41to49of the300chains.The section of Table1for each version of the diagnostic presents the means,MSEs,interval widths,and number of iterations retained averaged over only those chains for which some initial iterations were discarded.There is no evidence of systematic upward or downward bias in the estimates of the means for either parameter.However,when diagnostics result in discarding iterates, the estimation ofμis a?ected in two ways:the mean squared errors are in?ated by33%–125%,and the widths of the95%HPD region are slightly(by about1.2%),but statistically signi?cantly,underestimated.Even for the“Batch”diagnostic with“proportion early= .25,”the estimated interval width forμis smaller than for the full run with no diagnostics; the t-statistic is less extreme only because of the small number of chains on which it is based.

To investigate the e?ect of these diagnostics when Gibbs samplers mix slowly due to high correlations,we next tried a bivariate normal model with both marginal distributions N(0,1)and known correlation coe?cientρ=.95.We ran200parallel chains,each started in stationarity and run for1500iterations.The four versions of the diagnostic were applied in the same manner as for the?rst model.The results are presented in Table2.Here the interval widths for both parameters are signi?cantly though slightly underestimated(by 1%)even in the complete chains;for a slow-mixing model like this,many more than 1500iterations are needed,or a di?erent MCMC algorithm should be used.Discarding iterations in order to obtain a more homogeneous chain makes the problem worse.When diagnostics result in discarding iterations,interval widths are underestimated by up to 3.5%and mean squared errors are two to four times as large as when the full run is used.The“batch”means method appears to introduce less bias in interval widths than

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the“spectral”method,which suggests that the former may be providing more accurate estimates of Var(ˉx early)and Var(ˉx late).For each method of variance estimation,the magnitudes of the mean squared errors and of the bias in estimated interval widths are greater when the proportion early is.25than.10.

4.Choosing r=cn,with c non-random.

One way to avoid the bias problems of the previous sections is to consider simply setting r=cn,where c is some?xed constant,independent of the sample run.In this case,if the Markov chain is started in stationarity,then the resulting estimator will clearly be unbiased.Indeed,if starting in stationarity,then it is optimal to take c=0,i.e.to have no burn-in period at all.However,if the chain is not started in stationarity,then it may be desirable to choose some c>0.We investigate this question here.For concreteness,we consider the very simple case of a two-state Markov chain.This will allow us to compute the standard error of μexplicitly.

Consider the two-state Markov chain with X={0,1},and with

P=

1?p p

q1?q

.

Furthermore let g be the identity function,i.e.suppose that we are interested in estimating the mean of the stationary distributionπ(·).To?x ideas,imagine that the chain is in fact started at the point massδ0(·).

We are interested in the estimator μ=e/(n?r),where e=

n

i=r+1

X i.Speci?cally,we

shall consider the squared-error loss function

E

( μ?μ)2

=

1

(n?r)2

E

(e?π(e))2

=

1

(n?r)2

E(e2)?2π(e)E(e)+π(e)2

.

We set a=p

q+p

,a =q

q+p

=1?a,andλ=1?p?q.We recall(cf.Hoel,Port,and

Stone,1972,Section1.2;Rosenthal,1995c)that

P i=

a +aλi a?aλi

a ?a λi a+a λi

,

and that furthermoreπ(0)=a andπ(1)=a.We then have that

π(e)=(n?r)a,

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that

E(e)=

n

i=r+1

P i(0,1)=

n

i=r+1

(a?aλi)=a

(n?r)?

λr+1?λn+1

1?λ

,

and

E

( μ?μ)2

=

[1?2π(e)]E(e)+2s+[π(e)]2

/(n?r)2,

with

s=

n?1

i=r+1

n?i

k=1

[a?aλi][a+a λk].

Now,the explicit evaluation of this double sum is possible but messy.In particular,if p,q, and p/q are all O(1),then most of the terms are negligible,and we get the approximation

E

(e?π(e))2

≈an(1?c)

1+2a

λ

1?λ

+o(n),

valid as n→∞,with r=cn and with c,p,and q?xed.(Note that the O(n2)terms cancel,as expected.)It then follows that

E

(μ? μ)2

≈an?1(1?c)?1

1+2a

λ

1?λ

.

Hence,this squared error is minimised when(1?c)is maximised,i.e.when c is minimised. Hence,as expected,we should simply take c=0,i.e.r=0,i.e.no burn-in period,in this case.

However,if p and q are both very small(say,O(n?1)),which corresponds more closely to a typical diagnostic situation on a more complicated state space,then the question gets more interesting.In this case,terms likeλn cannot necessarily be neglected.

We have investigated this question for di?erent values of n,p,and q,using the Math-ematica computation system(Wolfram,1988).If p and q are at all large(i.e.,if the chain is mixing quickly),or if q≥p(i.e.,if the stationary distribution is not so far from the starting distribution),then the expected squared error is minimised when c is very small, i.e.it is optimal to let μaverage over nearly all of the available sample values.However, if p and q are very small and q p(i.e.,if the chain is mixing slowly and our starting distribution is far from the stationary distribution),then this is not necessarily the case.

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For a speci?c example,suppose that n=1000,with p=0.01and q=0.001(so that the stationary distribution puts mass1/11at0,and mass10/11at1).Then the squared error,as a function of the burn-in fraction c,is near0.0257for c close to0;dips down to0.0186for c near0.2;then climbs up steadily to0.0693when c=0.95.Hence,in this example it is optimal to choose c close to0.2,i.e.to discard the?rst20%of the sample values.

Keeping n=1000?xed,we can vary the values of p and q to see how the optimal c value changes.If p=0.1and q=0.01(or,if q=0.001),then the optimal c value is very close to0;convergence is so quick(due to the large value of p)that hardly any sample values should be discarded.On the other hand,if p=0.01and q=0.0001,then the optimal c value is near0.45,i.e.in this case it is optimal to discard the?rst45%of the sample values.For p=0.001and q=0.0001,the optimal c value is very close to1; convergence is so slow that later sample values are much more helpful than earlier ones.

To summarise:in the case of slow convergence when starting far from stationary(i.e. p,q≤O(1/n)and q p),the optimal choice of burn-in fraction c is not close to0;indeed, it increases towards1as p,q,and q/p all go to0.

5.Discussion and conclusion.

In this paper,we have considered biases that may result from the stochastic depen-dence of convergence diagnostics on the actual sample values used for estimating.We have argued that such dependence can signi?cantly a?ect the distribution of the sample values used.

The results of Section2indicate that,if the diagnostic makes use of batch means of size m,then diagnostic biases may indeed occur(and,in simple cases,they can be computed explicitly).As m→∞the biasing errors for the mean and variance go to0,at least when the diagnostic random variable has?nite mean.(However,the total variation distance errors may not decrease with m.)This suggests that,for our over-simpli?ed diagnostic at least,it is better to choose a su?ciently large batch size.

The results of Section3show that,when certain standard diagnostics are applied to the output of MCMC samplers,the diagnostic biases can sometimes be signi?cant.

13

Geweke’s diagnostic was selected for illustration because of its similarity to the batch means approach explored analytically in Section2.Similar results might be expected from other standard convergence diagnostics that seek to detect an initial transient.

The results of Section4suggest one way of avoiding this type of bias,albeit at the expense of a more computationally expensive implementation strategy.By choosing our burn-in time non-randomly,the optimal choice of burn-in time turns out to depend heavily on the nature of the Markov chain and of the starting distribution.In particular,the farther we start from the stationary distribution,and the more slowly the Markov chain is mixing,the larger a burn-in time we require.Furthermore,the optimal choice cannot even be computed without detailed knowledge of the target density which will typically not be available.Thus,as a result of the investigations of Section4,the proposed strategy cannot be strongly endorsed.

There is a way to combine random and non-random burn-in times,which avoids the biases of diagnostics but also avoids the uncertainty of?xed burn-in times.Speci?cally, it is possible to?rst apply a diagnostic procedure to one or more initial runs of the chain to estimate a burn-in time r.This diagnostic procedure should include inspection of the sample paths of quantities of interest and assessment of autocorrelations and cross-correlations,as well as use of a convergence diagnostic.The choice of r from the initial run(s)is then?xed and used for a fresh?nal run on which all estimation and inference are based.Starting values for the?nal run must be generated from the same distribution as was used to start the initial runs.In fact,the same starting values may be used as long as the random number seed is di?erent for the?nal run so that the sample paths are di?erent.

This combined approach achieves the best of both of these alternative approaches, is simple to implement,and appears to be a sensible procedure in the absence of any theoretical knowledge about convergence rate of the Markov chain being used. Acknowledgements.We thank Radford Neal for suggestions related to Section4herein.

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Table1:Normal Means Model

Diagnostics applied to all parameters

300parallel chains,1000iterations each

mu tau

True posterior values

mean500.0546

width 4.00590.0688

Full run,no diagnostics applied

mean50.00010.0547

t0.0282 1.7071

MSE0.00120

width 3.99440.0688

t?1.444?0.436

Batch,proportion early=.1

mean50.00340.0546

t0.59910.2458

MSE0.00160

width 3.95820.0682

t?2.8612?1.5381

Chains with initial iterations discarded:49

Mean iterations retained:873

Failures to converge:0

Batch,proportion early=.25

mean49.98890.0546

t?1.61560.3522

MSE0.00210

width 3.98930.0685

t?0.7665?0.7971

Chains with initial iterations discarded:42

Mean iterations retained:720

Failures to converge:0

17

Spectral,proportion early=.1

mean50.00110.0546

t0.1756?0.1669

MSE0.00180

width 3.96310.0683

t?2.3912?1.3365

Chains with initial iterations discarded:48

Mean iterations retained:873

Failures to converge:0

Spectral,proportion early=.25

mean49.99930.0546

t?0.08840.0235

MSE0.00270

width 3.95590.0689

t?2.08450.2221

Chains with initial iterations discarded:41

Mean iterations retained:701

Failures to converge:1

Table2:Bivariate Normal,rho=0.95 Diagnostics applied to all parameters

200parallel chains,1500iterations each

mu1mu2

True posterior values

mean00

width 3.9199 3.9199

Full run,no diagnostics applied

mean0.00360.0034

t0.43790.4148

MSE0.01340.0135

width 3.8801 3.8761

t?2.528?2.799

Batch,proportion early=.1

mean0.03370.0321

t 1.2661 1.2014

MSE0.02590.0261

width 3.8903 3.892

t?0.767?0.6757

Chains with initial iterations discarded:36

18

Mean iterations retained:1267

Failures to converge:0

Batch,proportion early=.25

mean0.02730.028

t0.4460.4576 MSE0.04950.0494 width 3.828 3.8124 t?1.6536?1.6729 Chains with initial iterations discarded:14

Mean iterations retained:1018

Failures to converge:0

Spectral,proportion early=.1

mean0.01060.0101 t0.64620.6128 MSE0.02270.0228 width 3.8369 3.8416 t?2.9355?2.7723 Chains with initial iterations discarded:85

Mean iterations retained:1196

Failures to converge:0

Spectral,proportion early=.25

mean0.05680.0555 t 2.0729 2.0048 MSE0.04830.0491 width 3.7822 3.7849 t?2.8077?2.6567 Chains with initial iterations discarded:61

Mean iterations retained:928

Failures to converge:5

19

MCMC算法

第7章MCMC算法 本章将介绍的马氏链蒙特卡罗（MCMC）方法是用来生成近似服从f分布的随机变量X的样本，从而估计关于X的函数的期望。 7.1 Metropolis-Hastings 算法 Metropolis-Hastings 算法是一种非常通用的构造马氏链的方法。这个方法从t=0开始，取X(0)=x(0)，其中x(0)是从某个初始分布g中随机抽取的样本使得满足f(x(0))>0。给定X(t)=x(t)，下面的算法用于产生X(t+1)。 (1)由某提案密度g(?|x(0))产生一个候选值X?. (2)计算Metropolis-Hastings比率R(x(t),X?)，其中 R(u,v)=f(v)g(u|v) f(u)g(u|v) (7.1) 注意R(x(t),X?)总是有定义的，因为只有f(x(t))>0且g(x?|x(t))>0时才有X?=x?。 (3)根据下式抽取X(t+1)： X(t+1)={X?,以概率min{R(x(t),X?),1}, x(t),否则. (7.2) (4)增加t，返回第1步。 我们将第t步迭代称作产生X(t)=x(t)的过程。 通过Metropolis-Hastings算法构造得到的链满足马氏性，因为X(t+1)仅依赖于X(t)。而是否是非周期不可约的则取决于提案分布的选取，需要自己去检验是否满足这些条件。如果满足了，那么这样生成的链具有唯一的极限平稳分布。 7.1.1独立链

假设选取Metropolis-Hastings算法的提案分布为某个固定的密度函数使得g(x?|x(t))=g(x?)。此时Metropolis-Hastings比率为 R(x(t),X?)=f(X?)g(x(t)) （7.4） f(x)g(X) 如果g(x)>0，则有f(x)>0，那么得到的马氏链就是非周期不可约的。 注：选择重要抽样包络的准则同样适用于选择提案密度。提案密度g应 与目标分布f近似，并在尾部包含f。 Code 7.1.2 随机游动链

五种最优化方法

五种最优化方法 1.最优化方法概述 1.1最优化问题的分类 1）无约束和有约束条件； 2）确定性和随机性最优问题（变量是否确定）； 3）线性优化与非线性优化（目标函数和约束条件是否线性）； 4）静态规划和动态规划（解是否随时间变化）。 1.2最优化问题的一般形式（有约束条件）： 式中f（X）称为目标函数（或求它的极小，或求它的极大），si（X）称为不等式约束，hj（X）称为等式约束。化过程就是优选X，使目标函数达到最优值。 2.牛顿法 2.1简介 1）解决的是无约束非线性规划问题； 2）是求解函数极值的一种方法； 3）是一种函数逼近法。 2.2原理和步骤

3.最速下降法（梯度法） 3.1最速下降法简介 1）解决的是无约束非线性规划问题； 2）是求解函数极值的一种方法； 3）沿函数在该点处目标函数下降最快的方向作为搜索方向； 3.2最速下降法算法原理和步骤

4.模式搜索法（步长加速法） 4.1简介 1）解决的是无约束非线性规划问题； 2）不需要求目标函数的导数，所以在解决不可导的函数或者求导异常麻烦的函数的优化问题时非常有效。 3）模式搜索法每一次迭代都是交替进行轴向移动和模式移动。轴向移动的目的是探测有利的下降方向，而模式移动的目的则是沿着有利方向加速移动。 4.2模式搜索法步骤

5.评价函数法 5.1简介 评价函数法是求解多目标优化问题中的一种主要方法。在许多实际问题中，衡量一个方案的好坏标准往往不止一个，多目标最优化的数学描述如下：min (f_1(x),f_2(x),...,f_k(x)) s.t. g(x)<=0 传统的多目标优化方法本质是将多目标优化中的各分目标函数，经处理或数学变换，转变成一个单目标函数，然后采用单目标优化技术求解。常用的方法有“线性加权和法”、“极大极小法”、“理想点法”。选取其中一种线性加权求合法介绍。 5.2线性加权求合法 6.遗传算法 智能优化方法是通过计算机学习和存贮大量的输入-输出模式映射关系，进

MCMC及R实现

贝叶斯集锦（3）：从MC、MC到MCMC 2013-07-31 23:03:39 #####一份草稿 贝叶斯计算基础 一、从MC、MC到MCMC 斯坦福统计学教授Persi Diaconis是一位传奇式的人物。Diaconis14岁就成了一名魔术师,为了看懂数学家Feller的概率论著作，24岁时进入大学读书。他向《科学美国人》投稿介绍他的洗牌方法，在《科学美国人》上常年开设数学游戏专栏的著名数学科普作家马丁?加德纳给他写了推荐信去哈佛大学，当时哈佛的统计学家Mosteller 正在研究魔术，于是Diaconis成了Mosteller的学生。（对他这段传奇经历有兴趣的读者可以看一看统计学史话《女士品茶》）。下面要讲的这个故事，是Diaconis 在他的文章The Markov Chain Monte Carlo Revolution中给出的破译犯人密码的例子。一天，一位研究犯罪心理学的心理医生来到斯坦福拜访Diaconis。他带来了一个囚犯所写的密码信息。他希望Diaconis帮助他把这个密码中的信息找出来。这个密码里的每个符号应该对应着某个字母，但是如何把这些字母准确地找出来呢？Diaconis和他的学生Marc采用了一种叫做MCMC（马尔科夫链蒙特卡洛）的方法解决了这个问题。这个MCMC方法就是这一节我们所要讨论的内容。 (1)贝叶斯推断的计算问题 在上节我们看到，贝叶斯统计学是利用后验分布对θ进行推断。这种推断的计算很多情况下要用积分计算来完成。比如，我们要计算θ的函数g(θ)的期望： E(g(θ∣x))=∫g(θ)fθ∣x(θ∣x)dθ 其中函数f表示后验分布。当g(θ)=θ时，得到的就是关于θ的点估计。

最优化方法及其应用 - 更多gbj149 相关pdf电子书下载

最优化方法及其应用 作者：郭科 出版社：高等教育出版社 类别：不限 出版日期：20070701 最优化方法及其应用 的图书简介 系统地介绍了最优化的理论和计算方法,由浅入深,突出方法的原则,对最优化技术的理论作丁适当深度的讨论,着重强调方法与应用的有机结合,包括最优化问题总论,线性规划及其对偶问题,常用无约束最优化方法,动态规划,现代优化算法简介,其中前八章为传统优化算法,最后一章还给出了部分优化问题的设计实例,也可供一般工科研究生以及数学建模竞赛参赛人员和工程技术人员参考, 最优化方法及其应用 的pdf电子书下载 最优化方法及其应用 的电子版预览 第一章 最优化问题总论1.1 最优化问题数学模型1.2 最优化问题的算法1.3 最优化算法分类1.4

组合优化问題简卉习题一第二章 最优化问题的数学基础2.1 二次型与正定矩阵2.2 方向导数与梯度2.3 Hesse矩阵及泰勒展式2.4 极小点的判定条件2.5 锥、凸集、凸锥2.6 凸函数2.7 约束问题的最优性条件习题二第三章 线性规划及其对偶问题3.1线性规划数学模型基本原理3.2 线性规划迭代算法3.3 对偶问题的基本原理3.4 线性规划问题的灵敏度习题三第四章 一维搜索法4.1 搜索区间及其确定方法4.2 对分法4.3 Newton切线法4.4 黄金分割法4.5 抛物线插值法习题四第五章 常用无约束最优化方法5.1 最速下降法5.2 Newton法5.3 修正Newton法5.4 共轭方向法5.5 共轭梯度法5.6 变尺度法5.7 坐标轮换法5.8 单纯形法习題五第六章 常用约束最优化方法6.1外点罚函数法6.2 內点罚函数法6.3 混合罚函数法6.4 约束坐标轮换法6.5 复合形法习题六第七章 动态规划7.1 动态规划基本原理7.2 动态规划迭代算法7.3 动态规划有关说明习题七第八章 多目标优化8.1 多目标最优化问题的基本原理8.2 评价函数法8.3 分层求解法8.4目标规划法习题八第九章 现代优化算法简介9.1 模拟退火算法9.2遗传算法9.3 禁忌搜索算法9.4 人工神经网络第十章 最优化问题程序设计方法10.1 最优化问题建模的一般步骤10.2 常用最优化方法的特点及选用标准10.3 最优化问题编程的一般过程10.4 优化问题设计实例参考文献 更多 最优化方法及其应用 相关pdf电子书下载

蒙特卡罗马尔科夫链模拟方法MCMC

Monte Carlo Simulation Methods (蒙特卡罗模拟方法) 主要内容: 1.各种随机数的生成方法. 2.MCMC方法. 1

2 从Buffon 投针问题谈起 Buffon 投针问题:平面上画很多平行线，间距为a ．向此平面投掷长 为l (l < a) 的针, 求此针与任一平行线相交的概率p 。 22[0,/2] [0,] sin ,{:sin }. l l a X A X 随机投针可以理解成针的中心 点与最近的平行线的距离X 是均匀 地分布在区间 上的r.v.，针 与平行线的夹角是均匀地分布 在区间 上的r.v.，且X 与相互独立， 于是针与平行线相交的充要条件为 即相交

3Buffon 投针问题 2sin 0022(sin ) 2l l l p P X dxd a a 于是有： 2l ap 若我们独立重复地作n 次投针试验，记 ()n A 为A 发生的次数。()n f A 为A 在n 次中出现的频率。假如我们取 ()n f A 作为()p P A 的估计，即?()n p f A 。 然后取2?() n l af A 作为的估计。根据大数定律，当n 时，..?().a s n p f A p 从而有2?()P n l af A 。这样可以用随机试验的方法求得的估计。历史上 有如下的试验结果。

4 3.14159292 180834080.831925Lazzarini 3.1595148910300.751884Fox 3.15665121832040.601855Smith 3.15956253250000.801850Wolf π的估计值相交次数投针次数针长时间(年)试验者

《最优化方法与应用》实验指导书

《最优化方法与应用》 实验指导书 信息与计算科学系编制

1 实验目的 基于单纯形法求解线性规划问题，编写算法步骤，绘制算法流程图，编写单纯形法程序，并针对实例完成计算求解。 2实验要求 程序设计语言：C++ 输入：线性规划模型（包括线性规划模型的价值系数、系数矩阵、右侧常数等） 输出：线性规划问题的最优解及目标函数值 备注：可将线性规划模型先转化成标准形式，也可以在程序中将线性规划模型从一般形式转化成标准形式。 3实验数据 123()-5-4-6=Min f x x x x 121231212320 324423230,,0３-+≤??++≤??+≤??≥? x x x x x x st x x x x x

1 实验目的 基于线性搜索的对分法、Newton 切线法、黄金分割法、抛物线法等的原理及方法，编写算法步骤和算法流程图，编写程序求解一维最优化问题，并针对实例具体计算。 2实验要求 程序设计语言：C++ 输入：线性搜索模型（目标函数系数，搜索区间，误差限等） 输出：最优解及对应目标函数值 备注：可从对分法、Newton 切线法、黄金分割法、抛物线法中选择2种具体的算法进行算法编程。 3实验数据 2211 ()+-6(0.3)0.01(0.9)0.04 = -+-+Min f x x x 区间[0.3,1],ε=10-4

实验三 无约束最优化方法 1实验目的 了解最速下降法、牛顿法、共轭梯度法、DFP 法和BFGS 法等的基本原理及方法，掌握其迭代步骤和算法流程图，运用Matlab 软件求解无约束非线性多元函数的最小值问题。 2实验要求 程序设计语言：Matlab 针对实验数据，对比最速下降法、牛顿法、共轭梯度法、DFP 法和BFGS 法等算法，比较不同算法的计算速度和收敛特性。 3实验数据 Rosenbrock's function 222211()(100)+(1-)=-Min f x x x x 初始点x=[-1.9, 2]，,ε=10-4

MATLAB教程2012a第3章习题解答 张志涌

第3章 数值阵列及其运算 习题3及解答 1 在MATLAB 中，先运行指令A=magic(3), B=[1,2,1;3,4,3;5,6,7]生成阵列33?A ，33?B ，然后根据运行结果回答以下问题： 〖目的〗 ● 体验矩阵乘法次序不可交换； ● 体验矩阵左除、右除的不同； ● 体验数组乘法次序可交换； ● 体验数组左除、右除的相同性； ● 体验矩阵乘法与数组乘法的根本性差别 ● 体验矩阵求逆的两种方法； ● 体验数组“逆”概念 〖解答〗 A=magic(3), B=[1,2,1;3,4,3;5,6,7] %创建阵列 A = 8 1 6 3 5 7 4 9 2 B = 1 2 1 3 4 3 5 6 7 （1） C1amb=A*B %相乘矩阵的次序不可交换 C1bma=B*A C1amb = 41 56 53 53 68 67 41 56 45 C1bma = 18 20 22 48 50 52 86 98 86 （2） C2adb=A\B %矩阵左除和右除根本不同 C2bda=B/A C2adb = 0.0333 0.1000 0.1611 0.5333 0.6000 0.7444 0.0333 0.1000 -0.1722 C2bda = 0.0056 0.0889 0.1722 0.1389 0.2222 0.3056

0.2333 0.7333 0.2333 （3） C3amb=A.*B %数组乘法不分左、右乘，因为是“元素对元素的运算”C3bma=B.*A C3amb = 8 2 6 9 20 21 20 54 14 C3bma = 8 2 6 9 20 21 20 54 14 （4） C4adb=A.\B %数组除法不分左、右除，因为是“元素对元素的运算”C4bda=B./A C4adb = 0.1250 2.0000 0.1667 1.0000 0.8000 0.4286 1.2500 0.6667 3.5000 C4bda = 0.1250 2.0000 0.1667 1.0000 0.8000 0.4286 1.2500 0.6667 3.5000 （5） C5ada=A\A %相当于inv(A)*A，所以得到“单位阵” C5adda=A.\A %相当于“数组逆”乘数组，得到“单位数组” C5ada = 1 0 0 0 1 0 0 0 1 C5adda = 1 1 1 1 1 1 1 1 1 （6） C6ade=A\eye(3) %矩阵求逆的代数方程法 C6inv=inv(A) %直接利用求逆指令。两者结果相同 C6ade = 0.1472 -0.1444 0.0639 -0.0611 0.0222 0.1056 -0.0194 0.1889 -0.1028 C6inv = 0.1472 -0.1444 0.0639 -0.0611 0.0222 0.1056 -0.0194 0.1889 -0.1028 （7） A C7add1=A.\1 %求“数组逆” C7ade=A\eye(3) %求“矩阵逆” A = 8 1 6 3 5 7 4 9 2 C7add1 = 0.1250 1.0000 0.1667 0.3333 0.2000 0.1429 0.2500 0.1111 0.5000

最优化方法及应用

陆吾生教授是加拿大维多利亚大学电气与计算机工程系 (Dept. of Elect. and Comp. Eng. University of Victoria) 的正教授, 且为我校兼职教授，曾多次来我校数学系电子系讲学。陆吾生教授的研究方向是：最优化理论和小波理论及其在1维和2维的数字信号处理、数字图像处理、控制系统优化方面的应用。 现陆吾生教授计划在 2007 年 10－11 月来校开设一门为期一个月的短期课程“最优化理论及其应用”（每周两次，每次两节课），对象是数学系、计算机系、电子系的教师、高年级本科生及研究生，以他在2006年出版的最优化理论的专著作为教材。欢迎数学系、计算机系、电子系的研究生及高年级本科生选修该短期课程，修毕的研究生及本科生可给学分。 上课地点及时间：每周二及周四下午2:00开始，在闵行新校区第三教学楼326教室。（自10月11日至11月8日） 下面是此课程的内容介绍。 ----------------------------------- 最优化方法及应用 I. 函数的最优化及应用 1.1 无约束和有约束的函数优化问题 1.2 有约束优化问题的Karush-Kuhn-Tucker条件 1.3 凸集、凸函数和凸规划 1.4 Wolfe对偶 1.5 线性规划与二次规划 1.6 半正定规划 1.7 二次凸锥规划 1.8 多项式规划 1.9解最优化问题的计算机软件 II 泛函的最优化及应用 2.1 有界变差函数 2.2 泛函的变分与泛函的极值问题 2.3 Euler-Lagrange方程 2.4 二维图像的Osher模型 2.5 泛函最优化方法在图像处理中的应用 2.5.1 噪声的消减 2.5.2 De-Blurring 2.5.3 Segmentation ----------------------------------------------- 注：这是一门约二十学时左右的短期课程，旨在介绍函数及泛函的最优化理论和方法，及其在信息处理中的应用。只要学过一元及多元微积分和线性代数的学生就能修读并听懂本课程。课程中涉及到的算法实现和应用举例都使用数学软件MATLAB 华东师大数学系

MCMC方法分析

龙源期刊网 https://www.wendangku.net/doc/c66992014.html, MCMC方法分析 作者：黄小艳 来源：《中国市场》2015年第14期 [摘要]本文主要介绍了Monte Carlo integration和 Markov Chain的构成原理，阐述了Metropolis-Hastings算法和Gibbs抽样法两种算法的基本原理。 [关键词]MCMC方法；M-H算法；Gibbs抽样 1949年，Metropolis和Ulam共同发表了第一篇关于蒙特卡罗方法的论文，其基本思想是：当所求解问题是某种随机事件出现的概率或者是某个随机变量的期望值时，通过某种“实验”的方法，以这种事件出现的频率估计这一随机事件的概率，或者得到这个随机变量的某些数字特征，并将其作为问题的解。 1953年6月，Nicolas Metropolis与Marshall N等在《The Journal of Chemical Physics》上 发表了一篇题为“Equations o f State Calculations by Fast Computing Machines”的文章，标志着Markov Chain Monte Carlo（MCMC）方法的诞生。MCMC方法的基本思想是构造一个概率转移矩阵，建立一个以分布π（x）为平稳分布的Markov链来得到π（x）的样本，通过随机抽样得到的这些样本然后进行各种统计推断。 1 MCMC的构成 1.1 Monte Carlo integration 但是如果后验概率函数难以得到时，该方法则不适用。 1.2 Markov Chain 3 结论 近年来，统计学家逐渐把研究重点转向了MCMC方法，这种方法应用甚广，可靠性强，主要用来模拟复杂的、非标准化的多元（多变量）分布，特别是当信息不全或者数据缺失时，可以通过Gibbs抽样，根据条件分布来推导联合分布，可以有效改善以往方法不能考虑总体情况的缺陷，在大样本情况下，能够很好地将已知的总体分布信息纳入到对缺失数据的处理当中；当然仿真导致的伪随机的算法再好也不是真正随机，因此，应用MCMC方法时应加以注意。 参考文献： [1]茆诗松.贝叶斯统计[M].北京：中国统计出版社，1999.

差分码ASK信号抽样仿真

长沙理工大学 《通信原理》课程设计报告 李秉坤 学 院 城南学院 专 业 通信工程 班 级 通信1104 学 号 201185250429 学生姓名 李秉坤 指导教师 黄红兵 课程成绩 完成日期 2014年1月9日

课程设计成绩评定 学院城南学院专业通信工程 班级通信1104 学号20118525042 学生姓名李秉坤指导教师黄红兵 课程成绩完成日期2014年1月9日指导教师对学生在课程设计中的评价 指导教师对课程设计的评定意见

课程设计任务书 城南学院通信工程专业

差分码ASK信号PAM调制仿真 学生姓名：李秉坤指导老师：黄红兵 摘要本课程设计主要用matlab/Simulink平台仿真一个差分码ASK信号抽样仿真系统，利用图形输入法设计相关电路，用示波器和频谱模块分析系统性能。首先根据原理画出图形，构建调制解调电路，在Simulink中调出各模块组成电路，设置调制解调电路各模块的参数值并运行，把运行仿真结果输入显示器，根据显示结果分析所设计的系统性能。通过波形分析，达到仿真的目的。 关键词抽样仿真；差分码；Matlab/Simulink 1 引言 MATLAB的名称源自Matrix Laboratory，它是一种科学计算软件，专门以矩阵的形式处理数据[1]。MATLAB将高性能的数值计算和可视化集成在一起，并提供了大量的内置函数，从而被广泛地应用于科学计算、控制系统、信息处理等领域的分析、仿真和设计工作。本课程设计主要用matlab中的Simulink平台仿真一个差分码ASK信号抽样仿真系统分别在理想信道和非理想信道中运行，并把运行仿真结果输入显示器，根据显示结果分析所设计的系统性能。 1.1 课程设计目的 通信原理课程设计是《通信原理》理论课程的辅助实践环节。着重体现学生对通信原理教学知识的应用，培养学生理论与实际工程相结合的能力。以小课题的方式来加深、扩展通信原理知识[3]。通过设计差分码ASK信号抽样仿真系统，并使其在不同的噪声信道中运行，让学生进一步理解通信系统的基本组成、模拟通信和数字通信的基础理论、通信系统发射端信号的形成原理、通信系统信号传输质量的检测等方面的相关知识，并学会运用这些知识。 1.2 课程设计的步骤 学习MATLAB的基本知识，熟悉MATLAB集成环境下的Simulink仿真平台。 利用通信原理中所学到的相关知识，在Simulink仿真平台中设计差分码ASK信号

_马尔可夫链蒙特卡洛_MCMC_方法在估计IRT模型参数中的应用

ＩＲＴ自２０世纪６０年代出现以来，由于其理论模型的科学性和精确性见长，一开始就受到心理和教育测量学的研究者和实际工作者的关注和兴趣。至今已成为考试技术学研究领域中最有影响的一种现代测量理论。但理论的严谨性又导致了计算的复杂性，因而也影响了ＩＲＴ的普及和应用乃至它的考试研究２００６年１０月第２卷第４期ＥｘａｍｉｎａｔｉｏｎｓＲｅｓｅａｒｃｈＯｃｔ．２００６Ｖｏｌ．２，Ｎｏ．４ “马尔可夫链蒙特卡洛”(M CM C)方法在估计IRT 模型参数中的应用[1][2] 王权编译【摘要】本文介绍和阐述怎样运用“马尔可夫链蒙特卡洛”（ＭＣＭＣ）技术，并结合Ｂａｙｅｓ方法来估计ＩＲＴ的模型参数。首先简要地概述了ＭＣＭＣ方法估计模型参数的基本原理；其次介绍ＭＣＭＣ方法估计模型参数的一般方法，涉及Ｇｉｂｂｓ抽样、取舍抽样、Ｍｅｔｒｏｐｏｌｉｓ－Ｈａｓｔｉｎｇｓ算法等概念和方法；最后以ＩＲＴ的“二参数逻辑斯蒂”（２ＰＬ）模型为例，重点介绍了用“Ｇｉｂｂｓ范围内的Ｍ－Ｈ算法”估计项目参数（β１ｊβ２ｊ）的算法过程。结束本文时还解说了ＭＣＭＣ方法的特点。 阅读本文需具有随机过程、Ｍａｒｋｏｖ链、Ｂａｙｅｓ方法等概率论的基本知识。 【关键词】项目反应理论 马尔可夫链蒙特卡洛Ｇｉｂｂｓ抽样取舍抽样作者简介王权，教授，浙江大学教育系。浙江杭州，３１００２８。４５

《考试研究》第２卷第４期 发展速度。令我们欣喜的是在２０世纪９０年代，国外统计学家又推陈出新地提出了参数估计的新方法，使ＩＲＴ的应用和发展又迈出了新的一步。 模型参数的估计是ＩＲＴ的核心内容。以往的参数估计方法主要有“条件极大似然估计”（ＣＭＬＥ）、“联合极大似然估计”（ＪＭＬＥ）、“边际极大似然估计” （ＭＭＬＥ）和“条件期望—极大化算法”（Ｅ－ＭＡｌｇｏｒｉｔｈｍ）等，大致上后一种算法均是前一种算法的改进［３］。Ｅ－Ｍ算法是由Ｒ．Ｄ．Ｂｏｃｋ和Ｍ．Ａｉｔｋｉｎ于１９８１年创立，它是以ＭＭＬＥ方法为基础发展而成。在Ｅ－Ｍ算法中，Ｅ步要涉及精确的数字积分计算，或者在Ｍ步要涉及偏导计算，当模型较复杂时，计算就十分困难。加之，它还难以将项目参数估计中的“不可靠性”（ｕｎｃｅｒｔａｉｎｔｙ）结合进能力参数估计时不可靠性的计算；反之亦然。 “马尔可夫链蒙特卡洛”（ＭａｒｋｏｖＣｈａｉｎＭｏｎｔｅＣａｒｌｏ，ＭＣＭＣ）方法是一种动态的计算机模拟技术，它是根据任一多元理论分布，特别是根据以贝叶斯（Ｂａｙｅｓ）推断为中心的多元后验分布来模拟随机样本的一种方法。它在估计ＩＲＴ模型参数的应用中，一方面继承了以往估计能力参数和项目参数时所采用的“分而治之”（ｄｉｖｉｄｅ－ａｎｄ－ｃｏｎｑｕｅｒ）的策略，采用能力参数与项目参数交替迭代计算的方法生成Ｍａｒｋｏｖ链；然后采取迥然不同于极大似然方法的思路，充分发挥计算机模拟技术的优势，采集充分大的状态样本，用初等的方法来估计模型参数，绕开了Ｅ－Ｍ算法中的复杂计算，从而提高了估计的成功率。 —“Ｇｉｂｂｓ采样１９９２年统计学家Ｊ．Ｈ．Ａｌｂｅｒｔ首先将一种特殊的ＭＣＭＣ方法—— 法”应用于ＩＲＴ问题的研究。现在它已被推广应用于多种复杂的ＩＲＴ模型，在应用于大范围的教育测验评价中尤显它的长处。本文主要介绍ＭＣＭＣ方法的基本原理和基本方法，为说明方便，只列举应用于较为简单状况的二参数逻辑斯蒂模型，它是进一步推广应用的基础。 一、ＭＣＭＣ方法的基本原理 用ＭＣＭＣ方法估计ＩＲＴ的模型参数的基本思路是：首先定义一Ｍａｒｋｏｖ链，Ｍ０，Ｍ１，Ｍ２，…，Ｍｋ，…状态Ｍｋ＝（θｋ，βｋ），ｋ＝１，２，…其中θ为能力参数，β为项目参数，θ和β可以为多维；然后根据Ｍａｒｋｏｖ链模拟观测（即模拟状态）；最后用所得的模拟观测推断参数θ和β。在一定的规则条件下，随着ｋ的增长，状态Ｍｋ的４６

最优化方法及其应用课后答案

1 2 ( ( 最优化方法部分课后习题解答 1.一直优化问题的数学模型为： 习题一 min f (x ) = (x ? 3)2 + (x ? 4)2 ? g (x ) = x ? x ? 5 ≥ ? 1 1 2 2 ? 试用图解法求出： s .t . ?g 2 (x ) = ?x 1 ? x 2 + 5 ≥ 0 ?g (x ) = x ≥ 0 ? 3 1 ??g 4 (x ) = x 2 ≥ 0 （1） 无约束最优点，并求出最优值。 （2） 约束最优点，并求出其最优值。 （3） 如果加一个等式约束 h (x ) = x 1 ? x 2 = 0 ，其约束最优解是什么？ * 解 ：（1）在无约束条件下， f (x ) 的可行域在整个 x 1 0x 2 平面上，不难看出，当 x =（3，4） 时， f (x ) 取最小值，即，最优点为 x * =（3，4）：且最优值为： f (x * ) =0 （2）在约束条件下， f (x ) 的可行域为图中阴影部分所示，此时，求该问题的最优点就是 在约束集合即可行域中找一点 (x 1 , x 2 ) ，使其落在半径最小的同心圆上，显然，从图示中可 以看出，当 x * = 15 , 5 ) 时， f (x ) 所在的圆的半径最小。 4 4 ?g (x ) = x ? x ? 5 = 0 ? 15 ?x 1 = 其中：点为 g 1 (x ) 和 g 2 (x ) 的交点，令 ? 1 1 2 ? 2 求解得到： ? 4 5 即最优点为 x * = ? ?g 2 (x ) = ?x 1 ? x 2 + 5 = 0 15 , 5 ) ：最优值为： f (x * ) = 65 ?x = ?? 2 4 4 4 8 （3）.若增加一个等式约束，则由图可知，可行域为空集，即此时最优解不存在。 2.一个矩形无盖油箱的外部总面积限定为 S ，怎样设计可使油箱的容量最大？试列出这个优 化问题的数学模型，并回答这属于几维的优化问题. 解：列出这个优化问题的数学模型为： max f (x ) = x 1x 2 x 3 ?x 1x 2 + 2x 2 x 3 + 2x 1x 3 ≤ S

LMS算法

N=31; % 滤波器阶数 sample_N=500; % 总采样次数 d=sin(2*pi*0.02* [0:sample_N-1]); % 期望的输入信号figure ; % 滤波器的权值矩阵 subplot(2,1,1); plot(d,'r'); grid on; xlabel('样本'); ylabel('幅值'); title('自适应滤波器的理想输入'); x=awgn(d,10); % 经过信道加入高斯噪声信噪比为10dB M=length(d); % 接受信号长度 wn=randn(M,1); hn=zeros(N,1); rxx=zeros(N,N); rxd=zeros(1,N); y=zeros(M,1); xt=zeros(M+N,1); % 生成输入信号与理想信号的互相关矩阵 for i=1:N, for m=i:1:M, rxd(1,i)=rxd(1,i)+x(1,m)*d(1,m-i+1); end end % 生成输入信号的自相关矩阵 for i=1:N, for m=i:1:M, rxx(1,i)=rxx(1,i)+x(1,m)*x(1,m-i+1); end end for i=2:N, for m=2:N,

rxx(i,m)=rxx(i-1,m-1); end end rxt=rxx'; for i=1:N rxt(i,i)=0; end rxx=rxx+rxt; irxx=inv(rxx); % RLS算法 randn('seed', 0) ; rand('seed', 0) ; NoOfData = 8000 ; % Set no of data points used for training Order = 32 ; % 自适应滤波权数 Lambda = 0.98 ; % 遗忘因子 Delta = 0.001 ; % 相关矩阵R的初始化 x = randn(NoOfData, 1) ;%高斯随机系列 h = rand(Order, 1) ; % 系统随机抽样 d = filter(h, 1, x) ; % 期望输出 % RLS算法的初始化 P = Delta * eye ( Order, Order ) ;%相关矩阵 w = zeros ( Order, 1 ) ;%滤波系数矢量的初始化 % RLS Adaptation for n = Order : NoOfData ; u = x(n:-1:n-Order+1) ;%延时函数 pi_ = u' * P ;%互相关函数 k = Lambda + pi_ * u ; K = pi_'/k;%增益矢量 e(n) = d(n) - w' * u ;%误差函数 w = w + K * e(n) ;%递归公式 PPrime = K * pi_ ; P = ( P - PPrime ) / Lambda ;%误差相关矩阵

最优化求解法在实际问题中的应用

本科毕业论文 （2014届） 题目：最优化求解法在实际问题中的应用学院：计算机与科学技术学院 专业：数学与应用数学 班级：10数本班 学号：1006131084 姓名：严慧 指导老师：孙钢钢

目录 1．摘要 (3) 2．关键字 (3) 3．引言 (3) 4．最优化求解法在实际问题中的应用 (4) 4.1．无约束最优化问题的求解............................................... ....... 4.2．有约束最优化问题的求解............................................... ....... 4.3．线性规划问题的求解............................................... ........... ... 4.4．非线性规划问题的求解............................................... ........... 5．结束语................................................................................................参考书目

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基于矩阵分解的对角加载波束形成算法

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最优化方法在化学工程中到的应用

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最优化方法及其应用

最优化方法及其应用

第一章 最优化问题总论 无论做任何一件事，人们总希望以最少的代价取得最大的效益，也就是力求最好，这就是优化问题．最优化就是在一切可能的方案中选择一个最好的方案以达到最优目标的学科．例如，从甲地到乙地有公路、水路、铁路、航空四种走法，如果我们追求的目标是省钱，那么只要比较一下这四种走法的票价，从中选择最便宜的那一种走法就达到目标．这是最简单的最优化问题，实际优化问题一般都比较复杂． 概括地说，凡是追求最优目标的数学问题都属于最优化问题．作为最优化问题，一般要有三个要素：第一是目标；第二是方案；第三是限制条件．而且目标应是方案的“函数”．如果方案与时间无关，则该问题属于静态最优化问题；否则称为动态最优化问题． §1.1 最优化问题数学模型 最简单的最优化问题实际上在高等数学中已遇到，这就是所谓函数极值，我们习惯上又称之为经典极值问题． 例1.1 对边长为a 的正方形铁板，在四个角处剪去相等的正方形以制成方形无盖水槽，问如何剪法使水槽的容积最大？ 解 设剪去的正方形边长为x ，由题意易知，与此相应的水槽容积为 x x a x f 2 )2()(-=． 令 0)6)(2()2()2)(2(2)('2 =--=-+--=x a x a x a x x a x f ， 得两个驻点： a x a x 6 121== ，．

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