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Multinomial Randomness Models for Retrieval withDocument FieldsVassilis Plachouras1and Iadh Ounis21Yahoo!Research,Barcelona,Spain2University of Glasgow,Glasgow,UKvassilis@,ounis@ Abstract.Documentﬁelds,such as the title or the headings of a document,offer a way to consider the structure of documents for retrieval.Most of the pro-posed approaches in the literature employ either a linear combination of scoresassigned to differentﬁelds,or a linear combination of frequencies in the termfrequency normalisation component.In the context of the Divergence From Ran-domness framework,we have a sound opportunity to integrate documentﬁeldsin the probabilistic randomness model.This paper introduces novel probabilis-tic models for incorporatingﬁelds in the retrieval process using a multinomialrandomness model and its information theoretic approximation.The evaluationresults from experiments conducted with a standard TREC Web test collectionshow that the proposed models perform as well as a state-of-the-artﬁeld-basedweighting model,while at the same time,they are theoretically founded and moreextensible than currentﬁeld-based models.1IntroductionDocumentﬁelds provide a way to incorporate the structure of a document in Information Retrieval(IR)models.In the context of HTML documents,the documentﬁelds may correspond to the contents of particular HTML tags,such as the title,or the heading tags.The anchor text of the incoming hyperlinks can also be seen as a documentﬁeld. In the case of email documents,theﬁelds may correspond to the contents of the email’s subject,date,or to the email address of the sender[9].It has been shown that using documentﬁelds for Web retrieval improves the retrieval effectiveness[17,7].The text and the distribution of terms in a particularﬁeld depend on the function of thatﬁeld.For example,the titleﬁeld provides a concise and short description for the whole document,and terms are likely to appear once or twice in a given title[6].The anchor textﬁeld also provides a concise description of the document,but the number of terms depends on the number of incoming hyperlinks of the document.In addition, anchor texts are not always written by the author of a document,and hence,they may enrich the document representation with alternative terms.The combination of evidence from the differentﬁelds in a retrieval model requires special attention.Robertson et al.[14]pointed out that the linear combination of scores, which has been the approach mostly used for the combination ofﬁelds,is difﬁcult to interpret due to the non-linear relation between the assigned scores and the term frequencies in each of theﬁelds.Hawking et al.[5]showed that the term frequency G.Amati,C.Carpineto,and G.Romano(Eds.):ECIR2007,LNCS4425,pp.28–39,2007.c Springer-Verlag Berlin Heidelberg2007Multinomial Randomness Models for Retrieval with Document Fields 29normalisation applied to each ﬁeld depends on the nature of the corresponding ﬁeld.Zaragoza et al.[17]introduced a ﬁeld-based version of BM25,called BM25F,which applies term frequency normalisation and weighting of the ﬁelds independently.Mac-donald et al.[7]also introduced normalisation 2F in the Divergence From Randomness (DFR)framework [1]for performing independent term frequency normalisation and weighting of ﬁelds.In both cases of BM25F and the DFR models that employ normali-sation 2F,there is the assumption that the occurrences of terms in the ﬁelds follow the same distribution,because the combination of ﬁelds takes place in the term frequency normalisation component,and not in the probabilistic weighting model.In this work,we introduce weighting models,where the combination of evidence from the different ﬁelds does not take place in the term frequency normalisation part of the model,but instead,it constitutes an integral part of the probabilistic randomness model.We propose two DFR weighting models that combine the evidence from the different ﬁelds using a multinomial distribution,and its information theoretic approx-imation.We evaluate the performance of the introduced weighting models using the standard .Gov TREC Web test collection.We show that the models perform as well as the state-of-the-art model ﬁeld-based PL2F,while at the same time,they employ a theoretically founded and more extensible combination of evidence from ﬁelds.The remainder of this paper is structured as follows.Section 2provides a description of the DFR framework,as well as the related ﬁeld-based weighting models.Section 3introduces the proposed multinomial DFR weighting models.Section 4presents the evaluation of the proposed weighting models with a standard Web test collection.Sec-tions 5and 6close the paper with a discussion related to the proposed models and the obtained results,and some concluding remarks drawn from this work,respectively.2Divergence from Randomness Framework and Document Fields The Divergence From Randomness (DFR)framework [1]generates a family of prob-abilistic weighting models for IR.It provides a great extent of ﬂexibility in the sense that the generated models are modular,allowing for the evaluation of new assumptions in a principled way.The remainder of this section provides a description of the DFR framework (Section 2.1),as well as a brief description of the combination of evidence from different document ﬁelds in the context of the DFR framework (Section 2.2).2.1DFR ModelsThe weighting models of the Divergence From Randomness framework are based on combinations of three components:a randomness model RM ;an information gain model GM ;and a term frequency normalisation model.Given a collection D of documents,the randomness model RM estimates the probability P RM (t ∈d |D )of having tf occurrences of a term t in a document d ,and the importance of t in d corresponds to the informative content −log 2(P RM (t ∈d |D )).Assuming that the sampling of terms corresponds to a sequence of independent Bernoulli trials,the randomness model RM is the binomial distribution:P B (t ∈d |D )= T F tfp tf (1−p )T F −tf (1)30V .Plachouras and I.Ouniswhere TF is the frequency of t in the collection D ,p =1N is a uniform prior probabilitythat the term t appears in the document d ,and N is the number of documents in the collection D .A limiting form of the binomial distribution is the Poisson distribution P :P B (t ∈d |D )≈P P (t ∈d |D )=λtf tf !e −λwhere λ=T F ·p =T FN (2)The information gain model GM estimates the informative content 1−P risk of the probability P risk that a term t is a good descriptor for a document.When a term t appears many times in a document,then there is very low risk in assuming that t describes the document.The information gain,however,from any future occurrences of t in d is lower.For example,the term ‘evaluation’is likely to have a high frequency in a document about the evaluation of IR systems.After the ﬁrst few occurrences of the term,however,each additional occurrence of the term ‘evaluation’provides a diminishing additional amount of information.One model to compute the probability P risk is the Laplace after-effect model:P risk =tf tf +1(3)P risk estimates the probability of having one more occurrence of a term in a document,after having seen tf occurrences already.The third component of the DFR framework is the term frequency normalisation model,which adjusts the frequency tf of the term t in d ,given the length l of d and the average document length l in D .Normalisation 2assumes a decreasing density function of the normalised term frequency with respect to the document length l .The normalised term frequency tfn is given as follows:tfn =tf ·log 2(1+c ·l l )(4)where c is a hyperparameter,i.e.a tunable parameter.Normalisation 2is employed in the framework by replacing tf in Equations (2)and (3)with tfn .The relevance score w d,q of a document d for a query q is given by:w d,q =t ∈qqtw ·w d,t where w d,t =(1−P risk )·(−log 2P RM )(5)where w d,t is the weight of the term t in document d ,qtw =qtf qtf max ,qtf is the frequency of t in the query q ,and qtf max is the maximum qtf in q .If P RM is estimatedusing the Poisson randomness model,P risk is estimated using the Laplace after-effect model,and tfn is computed according to normalisation 2,then the resulting weight-ing model is denotedby PL2.The factorial is approximated using Stirling’s formula:tf !=√2π·tftf +0.5e −tf .The DFR framework generates a wide range of weighting models by using different randomness models,information gain models,or term frequency normalisation models.For example,the next section describes how normalisation 2is extended to handle the normalisation and weighting of term frequencies for different document ﬁelds.Multinomial Randomness Models for Retrieval with Document Fields31 2.2DFR Models for Document FieldsThe DFR framework has been extended to handle multiple documentﬁelds,and to apply per-ﬁeld term frequency normalisation and weighting.This is achieved by ex-tending normalisation2,and introducing normalisation2F[7],which is explained below.Suppose that a document has kﬁelds.Each occurrence of a term can be assigned to exactly oneﬁeld.The frequency tf i of term t in the i-thﬁeld is normalised and weighted independently of the otherﬁelds.Then,the normalised and weighted term frequencies are combined into one pseudo-frequency tfn2F:tfn2F=ki=1w i·tf i log21+c i·l il i(6)where w i is the relative importance or weight of the i-thﬁeld,tf i is the frequency of t in the i-thﬁeld of document d,l i is the length of the i-thﬁeld in d,l i is the average length of the i-thﬁeld in the collection D,and c i is a hyperparameter for the i-thﬁeld.The above formula corresponds to normalisation2F.The weighting model PL2F corresponds to PL2using tfn2F as given in Equation(6).The well-known BM25 weighting model has also been extended in a similar way to BM25F[17].3Multinomial Randomness ModelsThis section introduces DFR models which,instead of extending the term frequency normalisation component,as described in the previous section,use documentﬁelds as part of the randomness model.While the weighting model PL2F has been shown to perform particularly well[7,8],the documentﬁelds are not an integral part of the ran-domness weighting model.Indeed,the combination of evidence from the differentﬁelds takes place as a linear combination of normalised frequencies in the term frequency nor-malisation component.This implies that the term frequencies are drawn from the same distribution,even though the nature of eachﬁeld may be different.We propose two weighting models,which,instead of assuming that term frequen-cies inﬁelds are drawn from the same distribution,use multinomial distributions to incorporate documentﬁelds in a theoretically driven way.Theﬁrst one is based on the multinomial distribution(Section3.1),and the second one is based on an information theoretic approximation of the multinomial distribution(Section3.2).3.1Multinomial DistributionWe employ the multinomial distribution to compute the probability that a term appears a given number of times in each of theﬁelds of a document.The formula of the weighting model is derived as follows.Suppose that a document d has kﬁelds.The probability that a term occurs tf i times in the i-thﬁeld f i,is given as follows:P M(t∈d|D)=T Ftf1tf2...tf k tfp tf11p tf22...p tf kkp tf (7)32V .Plachouras and I.OunisIn the above equation,T F is the frequency of term t in the collection,p i =1k ·N is the prior probability that a term occurs in a particular ﬁeld of document d ,and N is the number of documents in the collection D .The frequency tf =T F − ki =1tf i cor-responds to the number of occurrences of t in other documents than d .The probability p =1−k 1k ·N =N −1N corresponds to the probability that t does not appear in any of the ﬁelds of d .The DFR weighting model is generated using the multinomial distribution from Equation (7)as a randomness model,the Laplace after-effect from Equation (3),and replacing tf i with the normalised term frequency tfn i ,obtained by applying normal-isation 2from Equation (4).The relevance score of a document d for a query q is computed as follows:w d,q = t ∈q qtw ·w d,t = t ∈qqtw ·(1−P risk )· −log 2(P M (t ∈d |D )=t ∈q qtw k i =1tfn i +1· −log 2(T F !)+k i =1 log 2(tfn i !)−tfn i log 2(p i ) +log 2(tfn !)−tfn log 2(p ) (8)where qtw is the weight of a term t in query q ,tfn =T F − k i =1tfn i ,tfn i =tf i ·log 2(1+c i ·li l i )for the i -th ﬁeld,and c i is the hyperparameter of normalisation 2for the i -th ﬁeld.The weighting model introduced in the above equation is denoted by ML2,where M stands for the multinomial randomness model,L stands for the Laplace after-effect model,and 2stands for normalisation 2.Before continuing,it is interesting to note two issues related to the introduced weight-ing model ML2,namely setting the relative importance,or weight,of ﬁelds in the do-cument representation,and the computation of factorials.Weights of ﬁelds.In Equation (8),there are two different ways to incorporate weights for the ﬁelds of documents.The ﬁrst one is to multiply each of the normalised term frequencies tfn i with a constant w i ,in a similar way to normalisation 2F (see Equa-tion (6)):tfn i :=w i ·tfn i .The second way is to adjust the prior probabilities p i of ﬁelds,in order to increase the scores assigned to terms occurring in ﬁelds with low prior probabilities:p i :=p i w i .Indeed,the assigned score to a query term occurring in a ﬁeld with low probability is high,due to the factor −tfn i log 2(p i )in Equation (8).Computing factorials.As mentioned in Section 2.1,the factorial in the weighting model PL2is approximated using Stirling’s formula.A different method to approximate the factorial is to use the approximation of Lanczos to the Γfunction [12,p.213],which has a lower approximation error than Stirling’s formula.Indeed,preliminary experi-mentation with ML2has shown that using Stirling’s formula affects the performance of the weighting model,due to the accumulation of the approximation error from com-puting the factorial k +2times (k is the number of ﬁelds).This is not the case for the Poisson-based weighting models PL2and PL2F,where there is only one factorial com-putation for each query term (see Equation (2)).Hence,the computation of factorials in Equation (8)is performed using the approximation of Lanczos to the Γfunction.Multinomial Randomness Models for Retrieval with Document Fields33 3.2Approximation to the Multinomial DistributionThe DFR framework generates different models by replacing the binomial randomness model with its limiting forms,such as the Poisson randomness model.In this section, we introduce a new weighting model by replacing the multinomial randomness model in ML2with the following information theoretic approximation[13]:T F!tf1!tf2!···tf k!tf !p1tf1p2tf2···p k tf k p tf ≈1√2πT F k2−T F·Dtf iT F,p ip t1p t2···p tk p t(9)Dtf iT F,p icorresponds to the information theoretic divergence of the probability p ti=tf iT Fthat a term occurs in aﬁeld,from the prior probability p i of theﬁeld:D tfiT F,p i=ki=1tfiT Flog2tf iT F·p i+tfT Flog2tfT F·p(10)where tf =T F− ki=1tf i.Hence,the multinomial randomness model M in theweighting model ML2can be replaced by its approximation from Equation(9):w d,q=t∈q qtw·k2log2(2πT F)ki=1tfn i+1·ki=1tfn i log2tfn i/T Fp i+12log2tfn iT F+tfn log2tfn /T Fp+12log2tfnT F(11)The above model is denoted by M D L2.The deﬁnitions of the variables involved in theabove equation have been introduced in Section3.1.It should be noted that the information theoretic divergence Dtf iT F,p iis deﬁnedonly when tf i>0for1≤i≤k.In other words,Dtf iT F,p iis deﬁned only whenthere is at least one occurrence of a query term in all theﬁelds.This is not always the case,because a Web document may contain all the query terms in its body,but it may contain only some of the query terms in its title.To overcome this issue,the weight of a query term t in a document is computed by considering only theﬁelds in which the term t appears.The weights of differentﬁelds can be deﬁned in the same way as in the case of the weighting model ML2,as described in Section3.1.In more detail,the weighting of ﬁelds can be achieved by either multiplying the frequency of a term in aﬁeld by a constant,or by adjusting the prior probability of the correspondingﬁeld.An advantage of the weighting model M D L2is that,because it approximates the multinomial distribution,there is no need to compute factorials.Hence,it is likely to provide a sufﬁciently accurate approximation to the multinomial distribution,and it may lead to improved retrieval effectiveness compared to ML2,due to the lower accu-mulated numerical errors.The experimental results in Section4.2will indeed conﬁrm this advantage of M D L2.34V.Plachouras and I.Ounis4Experimental EvaluationIn this section,we evaluate the proposed multinomial DFR models ML2and M D L2, and compare their performance to that of PL2F,which has been shown to be particu-larly effective[7,8].A comparison of the retrieval effectiveness of PL2F and BM25F has shown that the two models perform equally well on various search tasks and test collections[11],including those employed in this work.Hence,we experiment only with the multinomial models and PL2F.Section4.1describes the experimental setting, and Section4.2presents the evaluation results.4.1Experimental SettingThe evaluation of the proposed models is conducted with TREC Web test collection,a crawl of approximately1.25million documents from domain.The .Gov collection has been used in the TREC Web tracks between2002and2004[2,3,4]. In this work,we employ the tasks from the Web tracks of TREC2003and2004,because they include both informational tasks,such as the topic distillation(td2003and td2004, respectively),as well as navigational tasks,such as named pageﬁnding(np2003and np2004,respectively)and home pageﬁnding(hp2003and hp2004,respectively).More speciﬁcally,we train and test for each type of task independently,in order to get insight on the performance of the proposed models[15].We employ each of the tasks from the TREC2003Web track for training the hyperparameters of the proposed models.Then, we evaluate the models on the corresponding tasks from the TREC2004Web track.In the reported set of experiments,we employ k=3documentﬁelds:the contents of the<BODY>tag of Web documents(b),the anchor text associated with incoming hyperlinks(a),and the contents of the<TITLE>tag(t).Moreﬁelds can be deﬁned for other types ofﬁelds,such as the contents of the heading tags<H1>for example. It has been shown,however,that the body,title and anchor textﬁelds are particularly effective for the considered search tasks[11].The collection of documents is indexed after removing stopwords and applying Porter’s stemming algorithm.We perform the experiments in this work using the Terrier IR platform[10].The proposed models ML2and M D L2,as well as PL2F,have a range of hyperpa-rameters,the setting of which can affect the retrieval effectiveness.More speciﬁcally,all three weighting models have two hyperparameters for each employed documentﬁeld: one related to the term frequency normalisation,and a second one related to the weight of thatﬁeld.As described in Sections3.1and3.2,there are two ways to deﬁne the weights ofﬁelds for the weighting models ML2and M D L2:(i)multiplying the nor-malised frequency of a term in aﬁeld;(ii)adjusting the prior probability p i of the i-th ﬁeld.Theﬁeld weights in the case of PL2F are only deﬁned in terms of multiplying the normalised term frequency by a constant w i,as shown in Equation(6).In this work,we consider only the term frequency normalisation hyperparameters, and we set all the weights ofﬁelds to1,in order to avoid having one extra parameter in the discussion of the performance of the weighting models.We set the involved hyperparameters c b,c a,and c t,for the body,anchor text,and titleﬁelds,respectively, by directly optimising mean average precision(MAP)on the training tasks from the Web track of TREC2003.We perform a3-dimensional optimisation to set the valuesMultinomial Randomness Models for Retrieval with Document Fields 35of the hyperparameters.The optimisation process is the following.Initially,we apply a simulated annealing algorithm,and then,we use the resulting hyperparameter values as a starting point for a second optimisation algorithm [16],to increase the likelihood of detecting a global maximum.For each of the three training tasks,we apply the above optimisation process three times,and we select the hyperparameter values that result in the highest MAP.We employ the above optimisation process to increase the likelihood that the hyperparameters values result in a global maximum for MAP.Figure 1shows the MAP obtained by ML2on the TREC 2003home page ﬁnding topics,for each iteration of the optimisation process.Table 1reports the hyperparameter values that resulted in the highest MAP for each of the training tasks,and that are used for the experiments in this work.0 0.20.40.60.80 40 80 120 160 200M A PiterationML2Fig.1.The MAP obtained by ML2on the TREC 2003home page ﬁnding topics,during the optimisation of the term frequency normalisation hyperparametersThe evaluation results from the Web tracks of TREC 2003[3]and 2004[4]have shown that employing evidence from the URLs of Web documents results in important improvements in retrieval effectiveness for the topic distillation and home page ﬁnd-ing tasks,where relevant documents are home pages of relevant Web sites.In order to provide a more complete evaluation of the proposed models for these two types of Web search tasks,we also employ the length in characters of the URL path,denoted by URLpathlen ,using the following formula to transform it to a relevance score [17]:w d,q :=w d,q +ω·κκ+URLpathlen (12)where w d,q is the relevance score of a document.The parameters ωand κare set by per-forming a 2-dimensional optimisation as described for the case of the hyperparameters c i .The resulting values for ωand κare shown in Table 2.4.2Evaluation ResultsAfter setting the hyperparameter values of the proposed models,we evaluate the models with the search tasks from TREC 2004Web track [4].We report the ofﬁcial TREC evaluation measures for each search task:mean average precision (MAP)for the topic distillation task (td2004),and mean reciprocal rank (MRR)of the ﬁrst correct answer for both named page ﬁnding (np2004)and home page ﬁnding (hp2004)tasks.36V.Plachouras and I.OunisTable1.The values of the hyperparameters c b,c a,and c t,for the body,anchor text and titleﬁelds,respectively,which resulted in the highest MAP on the training tasks of TREC2003Web trackML2Task c b c a c ttd20030.0738 4.326810.8220np20030.1802 4.70578.4074hp20030.1926310.3289624.3673M D L2Task c b c a c ttd20030.256210.038324.6762np20031.02169.232121.3330hp20030.4093355.2554966.3637PL2FTask c b c a c ttd20030.1400 5.0527 4.3749np20031.015311.96529.1145hp20030.2785406.1059414.7778Table2.The values of the hyperparameters ωandκ,which resulted in the high-est MAP on the training topic distillation (td2003)and home pageﬁnding(hp2003) tasks of TREC2003Web trackML2Taskωκtd20038.809514.8852hp200310.66849.8822M D L2Taskωκtd20037.697412.4616hp200327.067867.3153PL2FTaskωκtd20037.36388.2178hp200313.347628.3669Table3presents the evaluation results for the proposed models ML2,M D L2,and the weighting model PL2F,as well as their combination with evidence from the URLs of documents(denoted by appending U to the weighting model’s name).When only the documentﬁelds are employed,the multinomial weighting models have similar perfor-mance compared to the weighting model PL2F.The weighting models PL2F and M D L2 outperform ML2for both topic distillation and home pageﬁnding tasks.For the named pageﬁnding task,ML2results in higher MRR than M D L2and PL2F.Using the Wilcoxon signed rank test,we tested the signiﬁcance of the differences in MAP and MRR between the proposed new multinomial models and PL2F.In the case of the topic distillation task td2004,PL2F and M D L2were found to perform statistically signiﬁcantly better than ML2,with p<0.001in both cases.There was no statistically signiﬁcant difference between PL2F and M D L2.Regarding the named pageﬁnding task np2004,there is no statistically signiﬁcant difference between any of the three proposed models.For the home pageﬁnding task hp2004,only the difference between ML2and PL2F was found to be statistically signiﬁcant(p=0.020).Regarding the combination of the weighting models with the evidence from the URLs of Web documents,Table3shows that PL2FU and M D L2U outperform ML2U for td2004.The differences in performance are statistically signiﬁcant,with p=0.002 and p=0.012,respectively,but there is no signiﬁcant difference in the retrieval ef-fectiveness between PL2FU and M D L2U.When considering hp2004,we can see that PL2F outperforms the multinomial weighting models.The only statistically signiﬁcant difference in MRR was found between PL2FU and M D L2FU(p=0.012).Multinomial Randomness Models for Retrieval with Document Fields37 Table3.Evaluation results for the weighting models ML2,M D L2,and PL2F on the TREC 2004Web track topic distillation(td2004),named pageﬁnding(np2004),and home pageﬁnding (hp2004)tasks.ML2U,M D L2U,and PL2FU correspond to the combination of each weighting model with evidence from the URL of documents.The table reports mean average precision (MAP)for the topic distillation task,and mean reciprocal rank(MRR)of theﬁrst correct answer for the named pageﬁnding and home pageﬁnding tasks.ML2U,M D L2U and PL2FU are evalu-ated only for td2004and hp2004,where the relevant documents are home pages(see Section4.1).Task ML2M D L2PL2FMAPtd20040.12410.13910.1390MRRnp20040.69860.68560.6878hp20040.60750.62130.6270Task ML2U M D L2U PL2FUMAPtd20040.19160.20120.2045MRRhp20040.63640.62200.6464A comparison of the evaluation results with the best performing runs submitted to the Web track of TREC2004[4]shows that the combination of the proposed mod-els with the evidence from the URLs performs better than the best performing run of the topic distillation task in TREC2004,which achieved MAP0.179.The performance of the proposed models is comparable to that of the most effective method for the named pageﬁnding task(MRR0.731).Regarding the home pageﬁnding task,the dif-ference is greater between the performance of the proposed models with evidence from the URLs,and the best performing methods in the same track(MRR0.749).This can be explained in two ways.First,the over-ﬁtting of the parametersωandκon the training task may result in lower performance for the test task.Second,usingﬁeld weights may be more effective for the home pageﬁnding task,which is a high precision task,where the correct answers to the queries are documents of a very speciﬁc type.From the results in Table3,it can be seen that the model M D L2,which employs the information theoretic approximation to the multinomial distribution,signiﬁcantly outperforms the model ML2,which employs the multinomial distribution,for the topic distillation task.As discussed in Section3.2,this may suggest that approximating the multinomial distribution is more effective than directly computing it,because of the number of computations involved,and the accumulated small approximation errors from the computation of the factorial.The difference in performance may be greater if more documentﬁelds are considered.Overall,the evaluation results show that the proposed multinomial models ML2and M D L2have a very similar performance to that of PL2F for the tested search tasks. None of the models outperforms the others consistently for all three tested tasks,and the weighting models M D L2and PL2F achieve similar levels of retrieval effectiveness. The next section discusses some points related to the new multinomial models.。

基于RF-NSGA-Ⅱ的建筑能耗多目标优化

第38卷第2期2021年4月Vol.38No.2Apr.2021土木工程与管理学报Journal of Civil Engineering and Management基于RF-NSGA-II的建筑能耗多目标优化吴贤国1,杨赛1,王成龙2,王洪涛2,陈虹宇3,朱海军2,王雷1(1.华中科技大学土木与水利工程学院，湖北武汉430074；2.中建三局集团有限公司，湖北武汉430064；3.南洋理工大学土木工程与环境学院，新加坡639798)摘要：本文提出一种随机森林-带精英策略的非支配排序遗传算法(RF-NSGA-II)相结合的方法，基于建筑外围护结构设计优选进行建筑能耗多目标优化。

首先通过建立BIM模型导入DesignBuilder软件，利用正交试验获取建筑外维护结构主要参数与对应能耗的模拟样本集,再将样本数据集用于RF模型预测中进行训练，将训练好的RF回归函数作为适应度函数并作为优化目标之一。

在此基础上引入室内热舒适度函数为另一个适应度函数，采用NSGA-II进行多目标优化，得到Pareto前沿解集并从中利用理想点法得到一组多目标最优解。

通过本研究可以得出：利用RF对建筑能耗进行预测精度高，所得预测效果好，通过NSGA-II算法和理想点法可以获得满足建筑能耗和舒适度要求的外围护设计参数最优取值，结果与实际相符，效果良好。

对实现低建筑能耗和高热舒适度优化具有参考价值。

关键词：建筑能耗模拟；围护结构；随机森林；遗传算法；热舒适度；多目标优化中图分类号：TU201.5文献标识码：A文章编号：2095-0985(2021)02-0001-08Multi-objective Optimization of Building Energy Consumption Based on RF-NSGA-II WU Xianguo1,YANG Sai1,WANG Chenglong2,WANG Hongtao2,CHEN Hongyu3,ZHU Haijun2,WANG Lei(1.School of Civil and Hydraulic Engineering,Huazhong University of Science and Technology,Wuhan430074,China； 2.China Construction Third Engineering Bureau Group Co Ltd, Wuhan430064,China； 3.School of Civil and Environmental Engineering,Nanyang Technological University,Singapore639798,Singapore)Abstract:This paper proposes a combination method of random forest and non-dominated sorting genetic algorithm with elite strategy(RF-NSGA-II),which is based on the optimization of building envelope structure design for multi-objective optimization of building energy consumption.Firstly,by establishing a BIM model and importing it into DesignBuilder software,we use orthogonal experiments to obtain a simulation sample set of the main parameters of the maintenance structure outside the building and the corresponding energy consumption.And then we use the sample data set in the RF model prediction for training and take the trained RF regression function as a fitness function and one of the optimization goals.On this basis,the indoor thermal comfort function is introduced as another fitness function,and NSGA-II is used for multi-objective optimization,and the Pareto frontier solution set is obtained,and a set of multi-objective optimal solutions are obtained by using the ideal point method.Through this research,it can be concluded that the prediction accuracy of building energy consumption by RF is used,and the obtained prediction effect is good.The optimal envelope design parameters can be obtained to meet the requirements of building energy consumption and comfort through NS-GA-II algorithm and ideal point method.The result is consistent with the actual value and the effect is good.It has reference value to realize the optimization of low building energy consumption and high收稿日期：2020-10-05修回日期：2020-12-09作者简介：吴贤国(1964—)，女，湖北武汉人，博士，教授，研究方向为土木工程施工及管理(Email：wxg0220@)通讯作者：王成龙(1990—)，男，湖北武汉人，工程师，研究方向为土木工程施工及管理(Email：*****************)基金项目：国家重点研发计划(2016YFC0800208)；国家自然科学基金(51378235；71571078；51308240)・2・土木工程与管理学报2021年thermal comfort.Key words:building energy consumption simulation；envelope structure;random forest;genetic algorithm；thermal comfort;multi-objective optimization能源消耗一直是世界各国备受关注的问题,其中建筑能耗问题同样需要引起重视。

机器学习题库

机器学习题库一、极大似然1、 ML estimation of exponential model (10)A Gaussian distribution is often used to model data on the real line, but is sometimesinappropriate when the data are often close to zero but constrained to be nonnegative. In such cases one can fit an exponential distribution, whose probability density function is given by()1xb p x e b-=Given N observations x i drawn from such a distribution:(a) Write down the likelihood as a function of the scale parameter b.(b) Write down the derivative of the log likelihood.(c) Give a simple expression for the ML estimate for b.2、换成Poisson 分布：()|,0,1,2,...!x e p x y x θθθ-==()()()()()1111log |log log !log log !N Ni i i i N N i i i i l p x x x x N x θθθθθθ======--⎡⎤=--⎢⎥⎣⎦∑∑∑∑3、二、贝叶斯假设在考试的多项选择中，考生知道正确答案的概率为p ，猜测答案的概率为1-p ，并且假设考生知道正确答案答对题的概率为1，猜中正确答案的概率为1，其中m 为多选项的数目。

最近鲁棒优化进展Recent Advances in Robust Optimization and Robustness An Overview

Recent Advances in Robust Optimization and Robustness:An OverviewVirginie Gabrel∗and C´e cile Murat†and Aur´e lie Thiele‡July2012AbstractThis paper provides an overview of developments in robust optimization and robustness published in the aca-demic literature over the pastﬁve years.1IntroductionThis review focuses on papers identiﬁed by Web of Science as having been published since2007(included),be-longing to the area of Operations Research and Management Science,and having‘robust’and‘optimization’in their title.There were exactly100such papers as of June20,2012.We have completed this list by considering 726works indexed by Web of Science that had either robustness(for80of them)or robust(for646)in their title and belonged to the Operations Research and Management Science topic area.We also identiﬁed34PhD disserta-tions dated from the lastﬁve years with‘robust’in their title and belonging to the areas of operations research or management.Among those we have chosen to focus on the works with a primary focus on management science rather than system design or optimal control,which are broadﬁelds that would deserve a review paper of their own, and papers that could be of interest to a large segment of the robust optimization research community.We feel it is important to include PhD dissertations to identify these recent graduates as the new generation trained in robust optimization and robustness analysis,whether they have remained in academia or joined industry.We have also added a few not-yet-published preprints to capture ongoing research efforts.While many additional works would have deserved inclusion,we feel that the works selected give an informative and comprehensive view of the state of robustness and robust optimization to date in the context of operations research and management science.∗Universit´e Paris-Dauphine,LAMSADE,Place du Mar´e chal de Lattre de Tassigny,F-75775Paris Cedex16,France gabrel@lamsade.dauphine.fr Corresponding author†Universit´e Paris-Dauphine,LAMSADE,Place du Mar´e chal de Lattre de Tassigny,F-75775Paris Cedex16,France mu-rat@lamsade.dauphine.fr‡Lehigh University,Industrial and Systems Engineering Department,200W Packer Ave Bethlehem PA18015,USA aure-lie.thiele@2Theory of Robust Optimization and Robustness2.1Deﬁnitions and BasicsThe term“robust optimization”has come to encompass several approaches to protecting the decision-maker against parameter ambiguity and stochastic uncertainty.At a high level,the manager must determine what it means for him to have a robust solution:is it a solution whose feasibility must be guaranteed for any realization of the uncertain parameters?or whose objective value must be guaranteed?or whose distance to optimality must be guaranteed? The main paradigm relies on worst-case analysis:a solution is evaluated using the realization of the uncertainty that is most unfavorable.The way to compute the worst case is also open to debate:should it use aﬁnite number of scenarios,such as historical data,or continuous,convex uncertainty sets,such as polyhedra or ellipsoids?The answers to these questions will determine the formulation and the type of the robust counterpart.Issues of over-conservatism are paramount in robust optimization,where the uncertain parameter set over which the worst case is computed should be chosen to achieve a trade-off between system performance and protection against uncertainty,i.e.,neither too small nor too large.2.2Static Robust OptimizationIn this framework,the manager must take a decision in the presence of uncertainty and no recourse action will be possible once uncertainty has been realized.It is then necessary to distinguish between two types of uncertainty: uncertainty on the feasibility of the solution and uncertainty on its objective value.Indeed,the decision maker generally has different attitudes with respect to infeasibility and sub-optimality,which justiﬁes analyzing these two settings separately.2.2.1Uncertainty on feasibilityWhen uncertainty affects the feasibility of a solution,robust optimization seeks to obtain a solution that will be feasible for any realization taken by the unknown coefﬁcients;however,complete protection from adverse realiza-tions often comes at the expense of a severe deterioration in the objective.This extreme approach can be justiﬁed in some engineering applications of robustness,such as robust control theory,but is less advisable in operations research,where adverse events such as low customer demand do not produce the high-proﬁle repercussions that engineering failures–such as a doomed satellite launch or a destroyed unmanned robot–can have.To make the robust methodology appealing to business practitioners,robust optimization thus focuses on obtaining a solution that will be feasible for any realization taken by the unknown coefﬁcients within a smaller,“realistic”set,called the uncertainty set,which is centered around the nominal values of the uncertain parameters.The goal becomes to optimize the objective,over the set of solutions that are feasible for all coefﬁcient values in the uncertainty set.The speciﬁc choice of the set plays an important role in ensuring computational tractability of the robust problem and limiting deterioration of the objective at optimality,and must be thought through carefully by the decision maker.A large branch of robust optimization focuses on worst-case optimization over a convex uncertainty set.The reader is referred to Bertsimas et al.(2011a)and Ben-Tal and Nemirovski(2008)for comprehensive surveys of robust optimization and to Ben-Tal et al.(2009)for a book treatment of the topic.2.2.2Uncertainty on objective valueWhen uncertainty affects the optimality of a solution,robust optimization seeks to obtain a solution that performs well for any realization taken by the unknown coefﬁcients.While a common criterion is to optimize the worst-case objective,some studies have investigated other robustness measures.Roy(2010)proposes a new robustness criterion that holds great appeal for the manager due to its simplicity of use and practical relevance.This framework,called bw-robustness,allows the decision-maker to identify a solution which guarantees an objective value,in a maximization problem,of at least w in all scenarios,and maximizes the probability of reaching a target value of b(b>w).Gabrel et al.(2011)extend this criterion from aﬁnite set of scenarios to the case of an uncertainty set modeled using intervals.Kalai et al.(2012)suggest another criterion called lexicographicα-robustness,also deﬁned over aﬁnite set of scenarios for the uncertain parameters,which mitigates the primary role of the worst-case scenario in deﬁning the solution.Thiele(2010)discusses over-conservatism in robust linear optimization with cost uncertainty.Gancarova and Todd(2012)studies the loss in objective value when an inaccurate objective is optimized instead of the true one, and shows that on average this loss is very small,for an arbitrary compact feasible region.In combinatorial optimization,Morrison(2010)develops a framework of robustness based on persistence(of decisions)using the Dempster-Shafer theory as an evidence of robustness and applies it to portfolio tracking and sensor placement.2.2.3DualitySince duality has been shown to play a key role in the tractability of robust optimization(see for instance Bertsimas et al.(2011a)),it is natural to ask how duality and robust optimization are connected.Beck and Ben-Tal(2009) shows that primal worst is equal to dual best.The relationship between robustness and duality is also explored in Gabrel and Murat(2010)when the right-hand sides of the constraints are uncertain and the uncertainty sets are represented using intervals,with a focus on establishing the relationships between linear programs with uncertain right hand sides and linear programs with uncertain objective coefﬁcients using duality theory.This avenue of research is further explored in Gabrel et al.(2010)and Remli(2011).2.3Multi-Stage Decision-MakingMost early work on robust optimization focused on static decision-making:the manager decided at once of the values taken by all decision variables and,if the problem allowed for multiple decision stages as uncertainty was realized,the stages were incorporated by re-solving the multi-stage problem as time went by and implementing only the decisions related to the current stage.As theﬁeld of static robust optimization matured,incorporating–ina tractable manner–the information revealed over time directly into the modeling framework became a major area of research.2.3.1Optimal and Approximate PoliciesA work going in that direction is Bertsimas et al.(2010a),which establishes the optimality of policies afﬁne in the uncertainty for one-dimensional robust optimization problems with convex state costs and linear control costs.Chen et al.(2007)also suggests a tractable approximation for a class of multistage chance-constrained linear program-ming problems,which converts the original formulation into a second-order cone programming problem.Chen and Zhang(2009)propose an extension of the Afﬁnely Adjustable Robust Counterpart framework described in Ben-Tal et al.(2009)and argue that its potential is well beyond what has been in the literature so far.2.3.2Two stagesBecause of the difﬁculty in incorporating multiple stages in robust optimization,many theoretical works have focused on two stages.Regarding two-stage problems,Thiele et al.(2009)presents a cutting-plane method based on Kelley’s algorithm for solving convex adjustable robust optimization problems,while Terry(2009)provides in addition preliminary results on the conditioning of a robust linear program and of an equivalent second-order cone program.Assavapokee et al.(2008a)and Assavapokee et al.(2008b)develop tractable algorithms in the case of robust two-stage problems where the worst-case regret is minimized,in the case of interval-based uncertainty and scenario-based uncertainty,respectively,while Minoux(2011)provides complexity results for the two-stage robust linear problem with right-hand-side uncertainty.2.4Connection with Stochastic OptimizationAn early stream in robust optimization modeled stochastic variables as uncertain parameters belonging to a known uncertainty set,to which robust optimization techniques were then applied.An advantage of this method was to yield approaches to decision-making under uncertainty that were of a level of complexity similar to that of their deterministic counterparts,and did not suffer from the curse of dimensionality that afﬂicts stochastic and dynamic programming.Researchers are now making renewed efforts to connect the robust optimization and stochastic opti-mization paradigms,for instance quantifying the performance of the robust optimization solution in the stochastic world.The topic of robust optimization in the context of uncertain probability distributions,i.e.,in the stochastic framework itself,is also being revisited.2.4.1Bridging the Robust and Stochastic WorldsBertsimas and Goyal(2010)investigates the performance of static robust solutions in two-stage stochastic and adaptive optimization problems.The authors show that static robust solutions are good-quality solutions to the adaptive problem under a broad set of assumptions.They provide bounds on the ratio of the cost of the optimal static robust solution to the optimal expected cost in the stochastic problem,called the stochasticity gap,and onthe ratio of the cost of the optimal static robust solution to the optimal cost in the two-stage adaptable problem, called the adaptability gap.Chen et al.(2007),mentioned earlier,also provides a robust optimization perspective to stochastic programming.Bertsimas et al.(2011a)investigates the role of geometric properties of uncertainty sets, such as symmetry,in the power ofﬁnite adaptability in multistage stochastic and adaptive optimization.Duzgun(2012)bridges descriptions of uncertainty based on stochastic and robust optimization by considering multiple ranges for each uncertain parameter and setting the maximum number of parameters that can fall within each range.The corresponding optimization problem can be reformulated in a tractable manner using the total unimodularity of the feasible set and allows for aﬁner description of uncertainty while preserving tractability.It also studies the formulations that arise in robust binary optimization with uncertain objective coefﬁcients using the Bernstein approximation to chance constraints described in Ben-Tal et al.(2009),and shows that the robust optimization problems are deterministic problems for modiﬁed values of the coefﬁcients.While many results bridging the robust and stochastic worlds focus on giving probabilistic guarantees for the solutions generated by the robust optimization models,Manuja(2008)proposes a formulation for robust linear programming problems that allows the decision-maker to control both the probability and the expected value of constraint violation.Bandi and Bertsimas(2012)propose a new approach to analyze stochastic systems based on robust optimiza-tion.The key idea is to replace the Kolmogorov axioms and the concept of random variables as primitives of probability theory,with uncertainty sets that are derived from some of the asymptotic implications of probability theory like the central limit theorem.The authors show that the performance analysis questions become highly structured optimization problems for which there exist efﬁcient algorithms that are capable of solving problems in high dimensions.They also demonstrate that the proposed approach achieves computationally tractable methods for(a)analyzing queueing networks,(b)designing multi-item,multi-bidder auctions with budget constraints,and (c)pricing multi-dimensional options.2.4.2Distributionally Robust OptimizationBen-Tal et al.(2010)considers the optimization of a worst-case expected-value criterion,where the worst case is computed over all probability distributions within a set.The contribution of the work is to deﬁne a notion of robustness that allows for different guarantees for different subsets of probability measures.The concept of distributional robustness is also explored in Goh and Sim(2010),with an emphasis on linear and piecewise-linear decision rules to reformulate the original problem in aﬂexible manner using expected-value terms.Xu et al.(2012) also investigates probabilistic interpretations of robust optimization.A related area of study is worst-case optimization with partial information on the moments of distributions.In particular,Popescu(2007)analyzes robust solutions to a certain class of stochastic optimization problems,using mean-covariance information about the distributions underlying the uncertain parameters.The author connects the problem for a broad class of objective functions to a univariate mean-variance robust objective and,subsequently, to a(deterministic)parametric quadratic programming problem.The reader is referred to Doan(2010)for a moment-based uncertainty model for stochastic optimization prob-lems,which addresses the ambiguity of probability distributions of random parameters with a minimax decision rule,and a comparison with data-driven approaches.Distributionally robust optimization in the context of data-driven problems is the focus of Delage(2009),which uses observed data to deﬁne a”well structured”set of dis-tributions that is guaranteed with high probability to contain the distribution from which the samples were drawn. Zymler et al.(2012a)develop tractable semideﬁnite programming(SDP)based approximations for distributionally robust individual and joint chance constraints,assuming that only theﬁrst-and second-order moments as well as the support of the uncertain parameters are given.Becker(2011)studies the distributionally robust optimization problem with known mean,covariance and support and develops a decomposition method for this family of prob-lems which recursively derives sub-policies along projected dimensions of uncertainty while providing a sequence of bounds on the value of the derived policy.Robust linear optimization using distributional information is further studied in Kang(2008).Further,Delage and Ye(2010)investigates distributional robustness with moment uncertainty.Speciﬁcally,uncertainty affects the problem both in terms of the distribution and of its moments.The authors show that the resulting problems can be solved efﬁciently and prove that the solutions exhibit,with high probability,best worst-case performance over a set of distributions.Bertsimas et al.(2010)proposes a semideﬁnite optimization model to address minimax two-stage stochastic linear problems with risk aversion,when the distribution of the second-stage random variables belongs to a set of multivariate distributions with knownﬁrst and second moments.The minimax solutions provide a natural distribu-tion to stress-test stochastic optimization problems under distributional ambiguity.Cromvik and Patriksson(2010a) show that,under certain assumptions,global optima and stationary solutions of stochastic mathematical programs with equilibrium constraints are robust with respect to changes in the underlying probability distribution.Works such as Zhu and Fukushima(2009)and Zymler(2010)also study distributional robustness in the context of speciﬁc applications,such as portfolio management.2.5Connection with Risk TheoryBertsimas and Brown(2009)describe how to connect uncertainty sets in robust linear optimization to coherent risk measures,an example of which is Conditional Value-at-Risk.In particular,the authors show the link between polyhedral uncertainty sets of a special structure and a subclass of coherent risk measures called distortion risk measures.Independently,Chen et al.(2007)present an approach for constructing uncertainty sets for robust opti-mization using new deviation measures that capture the asymmetry of the distributions.These deviation measures lead to improved approximations of chance constraints.Dentcheva and Ruszczynski(2010)proposes the concept of robust stochastic dominance and shows its applica-tion to risk-averse optimization.They consider stochastic optimization problems where risk-aversion is expressed by a robust stochastic dominance constraint and develop necessary and sufﬁcient conditions of optimality for such optimization problems in the convex case.In the nonconvex case,they derive necessary conditions of optimality under additional smoothness assumptions of some mappings involved in the problem.2.6Nonlinear OptimizationRobust nonlinear optimization remains much less widely studied to date than its linear counterpart.Bertsimas et al.(2010c)presents a robust optimization approach for unconstrained non-convex problems and problems based on simulations.Such problems arise for instance in the partial differential equations literature and in engineering applications such as nanophotonic design.An appealing feature of the approach is that it does not assume any speciﬁc structure for the problem.The case of robust nonlinear optimization with constraints is investigated in Bertsimas et al.(2010b)with an application to radiation therapy for cancer treatment.Bertsimas and Nohadani (2010)further explore robust nonconvex optimization in contexts where solutions are not known explicitly,e.g., have to be found using simulation.They present a robust simulated annealing algorithm that improves performance and robustness of the solution.Further,Boni et al.(2008)analyzes problems with uncertain conic quadratic constraints,formulating an approx-imate robust counterpart,and Zhang(2007)provide formulations to nonlinear programming problems that are valid in the neighborhood of the nominal parameters and robust to theﬁrst order.Hsiung et al.(2008)present tractable approximations to robust geometric programming,by using piecewise-linear convex approximations of each non-linear constraint.Geometric programming is also investigated in Shen et al.(2008),where the robustness is injected at the level of the algorithm and seeks to avoid obtaining infeasible solutions because of the approximations used in the traditional approach.Interval uncertainty-based robust optimization for convex and non-convex quadratic programs are considered in Li et al.(2011).Takeda et al.(2010)studies robustness for uncertain convex quadratic programming problems with ellipsoidal uncertainties and proposes a relaxation technique based on random sampling for robust deviation optimization sserre(2011)considers minimax and robust models of polynomial optimization.A special case of nonlinear problems that are linear in the decision variables but convex in the uncertainty when the worst-case objective is to be maximized is investigated in Kawas and Thiele(2011a).In that setting,exact and tractable robust counterparts can be derived.A special class of nonconvex robust optimization is examined in Kawas and Thiele(2011b).Robust nonconvex optimization is examined in detail in Teo(2007),which presents a method that is applicable to arbitrary objective functions by iteratively moving along descent directions and terminates at a robust local minimum.3Applications of Robust OptimizationWe describe below examples to which robust optimization has been applied.While an appealing feature of robust optimization is that it leads to models that can be solved using off-the-shelf software,it is worth pointing the existence of algebraic modeling tools that facilitate the formulation and subsequent analysis of robust optimization problems on the computer(Goh and Sim,2011).3.1Production,Inventory and Logistics3.1.1Classical logistics problemsThe capacitated vehicle routing problem with demand uncertainty is studied in Sungur et al.(2008),with a more extensive treatment in Sungur(2007),and the robust traveling salesman problem with interval data in Montemanni et al.(2007).Remli and Rekik(2012)considers the problem of combinatorial auctions in transportation services when shipment volumes are uncertain and proposes a two-stage robust formulation solved using a constraint gener-ation algorithm.Zhang(2011)investigates two-stage minimax regret robust uncapacitated lot-sizing problems with demand uncertainty,in particular showing that it is polynomially solvable under the interval uncertain demand set.3.1.2SchedulingGoren and Sabuncuoglu(2008)analyzes robustness and stability measures for scheduling in a single-machine environment subject to machine breakdowns and embeds them in a tabu-search-based scheduling algorithm.Mittal (2011)investigates efﬁcient algorithms that give optimal or near-optimal solutions for problems with non-linear objective functions,with a focus on robust scheduling and service operations.Examples considered include parallel machine scheduling problems with the makespan objective,appointment scheduling and assortment optimization problems with logit choice models.Hazir et al.(2010)considers robust scheduling and robustness measures for the discrete time/cost trade-off problem.3.1.3Facility locationAn important question in logistics is not only how to operate a system most efﬁciently but also how to design it. Baron et al.(2011)applies robust optimization to the problem of locating facilities in a network facing uncertain demand over multiple periods.They consider a multi-periodﬁxed-charge network location problem for which they ﬁnd the number of facilities,their location and capacities,the production in each period,and allocation of demand to facilities.The authors show that different models of uncertainty lead to very different solution network topologies, with the model with box uncertainty set opening fewer,larger facilities.?investigate a robust version of the location transportation problem with an uncertain demand using a2-stage formulation.The resulting robust formulation is a convex(nonlinear)program,and the authors apply a cutting plane algorithm to solve the problem exactly.Atamt¨u rk and Zhang(2007)study the networkﬂow and design problem under uncertainty from a complexity standpoint,with applications to lot-sizing and location-transportation problems,while Bardossy(2011)presents a dual-based local search approach for deterministic,stochastic,and robust variants of the connected facility location problem.The robust capacity expansion problem of networkﬂows is investigated in Ordonez and Zhao(2007),which provides tractable reformulations under a broad set of assumptions.Mudchanatongsuk et al.(2008)analyze the network design problem under transportation cost and demand uncertainty.They present a tractable approximation when each commodity only has a single origin and destination,and an efﬁcient column generation for networks with path constraints.Atamt¨u rk and Zhang(2007)provides complexity results for the two-stage networkﬂow anddesign plexity results for the robust networkﬂow and network design problem are also provided in Minoux(2009)and Minoux(2010).The problem of designing an uncapacitated network in the presence of link failures and a competing mode is investigated in Laporte et al.(2010)in a railway application using a game theoretic perspective.Torres Soto(2009)also takes a comprehensive view of the facility location problem by determining not only the optimal location but also the optimal time for establishing capacitated facilities when demand and cost parameters are time varying.The models are solved using Benders’decomposition or heuristics such as local search and simulated annealing.In addition,the robust networkﬂow problem is also analyzed in Boyko(2010),which proposes a stochastic formulation of minimum costﬂow problem aimed atﬁnding network design andﬂow assignments subject to uncertain factors,such as network component disruptions/failures when the risk measure is Conditional Value at Risk.Nagurney and Qiang(2009)suggests a relative total cost index for the evaluation of transportation network robustness in the presence of degradable links and alternative travel behavior.Further,the problem of locating a competitive facility in the plane is studied in Blanquero et al.(2011)with a robustness criterion.Supply chain design problems are also studied in Pan and Nagi(2010)and Poojari et al.(2008).3.1.4Inventory managementThe topic of robust multi-stage inventory management has been investigated in detail in Bienstock and Ozbay (2008)through the computation of robust basestock levels and Ben-Tal et al.(2009)through an extension of the Afﬁnely Adjustable Robust Counterpart framework to control inventories under demand uncertainty.See and Sim (2010)studies a multi-period inventory control problem under ambiguous demand for which only mean,support and some measures of deviations are known,using a factor-based model.The parameters of the replenishment policies are obtained using a second-order conic programming problem.Song(2010)considers stochastic inventory control in robust supply chain systems.The work proposes an inte-grated approach that combines in a single step dataﬁtting and inventory optimization–using histograms directly as the inputs for the optimization model–for the single-item multi-period periodic-review stochastic lot-sizing problem.Operation and planning issues for dynamic supply chain and transportation networks in uncertain envi-ronments are considered in Chung(2010),with examples drawn from emergency logistics planning,network design and congestion pricing problems.3.1.5Industry-speciﬁc applicationsAng et al.(2012)proposes a robust storage assignment approach in unit-load warehouses facing variable supply and uncertain demand in a multi-period setting.The authors assume a factor-based demand model and minimize the worst-case expected total travel in the warehouse with distributional ambiguity of demand.A related problem is considered in Werners and Wuelﬁng(2010),which optimizes internal transports at a parcel sorting center.Galli(2011)describes the models and algorithms that arise from implementing recoverable robust optimization to train platforming and rolling stock planning,where the concept of recoverable robustness has been deﬁned in。

The effect of distribution-free newsboy problem

The effect of advertising on the distribution-free newsboy problemChih-Ming Lee a ,n ,Shu-Lu Hsu ba Department of Business Administration,Soochow University,56Kuei-Yang St.Sec.1,Taipei 100,TaiwanbDepartment of Management Information Systems,National Chiayi University,580Sinmin Road,Chiayi City 600,Taiwana r t i c l e i n f oArticle history:Received 30November 2009Accepted 4October 2010Available online 20October 2010Keywords:Newsboy problems AdvertisingDistribution-free.a b s t r a c tAdvertising is very important for the newsboy problem because the shelf-life of the newsboy product is short and advertising may increase sales to avoid overstocking.In this paper,models to study the effect of advertising are developed for the distribution-free newsboy problem where only the mean and variance of the demand are known.As in Khouja and Robbins (2003),it is assumed that the mean demand is an increasing and concave function of advertising expenditure.Three cases are considered:(1)demand has constant variance,(2)demand has constant coefﬁcient of variation,and (3)demand has an increasing coefﬁcient of variation.This paper provides closed-form solutions or steps to solve the problem.Numerical results of the model are also compared with those from other papers.The effects of model parameters on optimal expenditure on advertising,optimal order quantity,and the lower bound on expected proﬁt are derived or discussed.&2010Elsevier B.V.All rights reserved.1.IntroductionThe newsboy problem is that of ﬁnding a product’s order quantity that maximizes the expected proﬁt under probabilistic demand.Researchers’interest in the newsboy problem has increased in recent decades.Khouja (1999)classiﬁed the newsboy problem literature into eleven categories and provided useful suggestions for future research.The aim in the classical newsboy problem is maximizing the expected proﬁt.However,Ozler et al.(2009)proposed the multi-product newsboy problem under a Value at Risk (VaR)constraint.A mathematical programming approach was used to ﬁnd the solution.Wang (2010)studied a game setting where multiple newsvendors with loss aversion preferences are competing for inventory from a risk-neutral supplier.Moreover,the classical newsboy problem assumes that demand follows a speciﬁc distribution with the known parameters.However,several authors have analyzed the distribution-free newsboy problem in which only the ﬁrst two moments of demand,mean and variance,are assumed to be known.Scarf (1958)ﬁrst addressed the distribution-free newsboy problem and derived a closed-form expression for the optimal ordering rule that max-imizes the expected proﬁt against the worst possible distribution of the demand with the mean m and the variance s 2.Gallego and Moon (1993)proved the optimality of Scarf’s rule and extended Scarf’s ideas to four cases.Alfares and Elmorra (2005)extended the models proposed in Gallego and Moon (1993)to incorporate a shortage penalty cost beyond the lost proﬁt.Moon and Choi (1995)derived an ordering rule for the distribution-free newsboy problem with balking.Balking means that once the inventory level falls to the balking level or lower,the per-unit probability of a sale declines from one to less than one.Mostard et al.(2005)studied the case that the returned goods arriving before the end of the selling season can be resold if there is sufﬁcient demand.They found that the distribution-free ordering rule performs well when the coefﬁcient of variation is less than 0.5.Yue et al.(2006)deﬁned the expected value of distribution information,EVDI ,as the difference in cost functions between a distribution-free decision and the optimal decision under the true demand distribution.The maximum EVDI can be used as a robustness measurement for the distribution-free decision.Advertising is one of the major tools used by companies to target buyers and populations.It consists of impersonal forms of com-munication conducted through paid media under clear sponsor-ship.Advertising can be used to build up a long-term image for a product or to trigger quick sales (Kotler,2001).Vidale and Wolfe (1957)published one of the earliest advertising response models.The model is based on three parameters:a sales decay constant,the saturation level,and the response constant.The response function may be S-shaped or concave.The S-shaped function indicates ﬁrst increase in returns and then,after an inﬂection point,decrease in returns.The concave response function indicates that as advertising expenditures increase,so do sales,but at monotonically diminishing returns from the begin-ning.Rao and Miller (1975)illustrated a procedure for estimating an S-shaped sales response function from historical data.The procedure uses,as building blocks,distributed lag models that relate market share to advertising expenditures on a market-by-market basis.Little (1979)suggested that several phenomenaContents lists available at ScienceDirectjournal homepage:/locate/ijpeInt.J.Production Economics0925-5273/$-see front matter &2010Elsevier B.V.All rights reserved.doi:10.1016/j.ijpe.2010.10.009nCorresponding author.Tel.:+886223111531x3414;fax:+886223822326.E-mail address:cmlee@.tw (C.-M.Lee).Int.J.Production Economics 129(2011)217–224should be considered in building dynamic models of advertising response.These include sales which respond upward and down-ward at different rates,a steady-state response that can be concave or S-shaped and can have positive sales at zero advertising,the effect on sales of competitive advertising,and advertising-dollar effectiveness that can change over time.Simon and Arndt(1980) reviewed over100studies.Evidence consistently indicates that only the concave downward function exists.However,Mahajan and Muller(1986)presented an analytical model that can be used to evaluate the impact of various pulsed and uniform advertising policies.They found that pulsing is optimal only with an S-shaped response ing a modiﬁed Lanchester model,Mesak and Darrat(1993)demonstrated that the policy of constant advertising spending is superior to a cyclic counterpart,provided that the response functions of the competingﬁrms are concave.Vakratsas et al.(2004)formulated a switching regression model with two regimes,in only one of which advertising is effective.They also estimated a variety of model types,including both standard concave and S-shaped response functions.A sequence of compar-isons among these models strongly suggested that the threshold effect exists.However,the debate about the shape of the function continues in the advertising literature.Khouja and Robbins(2003)assumed that mean demand is increasing and concave in advertising expenditure and studied three cases of demand variation as a function of advertising expenditure:(1)demand has constant variance,(2)demand has a constant coefﬁcient of variation,and(3)demand has an increas-ing coefﬁcient of variation.They provided closed-or near-closed-form solutions of this problem under several different demand distributions.The present paper extends some of their concepts to the distribution-free newsboy problem.The rest of this paper is organized as follows.In Section2,the models are developed.Section3illustrates the results of these models using numerical examples and provides managerial insights.Section4presents the mainﬁndings of this research and points out the direction of future research.2.The modelWe introduce the following notations:c40unit cost,p¼(1+m)c4c unit selling price,with m being the markup rate, s¼(1Àd)c o c unit salvage value,with d being the discount rate, l¼kc unit shortage penalty cost,with k being the shortage penalty rate,D random demand with mean m and variance s2,s o m, m0expected demand without advertising(the original market size),s0standard deviation of the demand without advertising, s0o m0,Q order quantity,B expenditure on advertising,x+¼max{x,0}the positive part of x.Consider a distribution-free newsboy problem.Let D be a random variable representing the demand in the period under consideration,with distribution G.As in Gallego and Moon(1993) and Alfares and Elmorra(2005),no assumptions are made on G other than to say that it belongs to the class W of cumulative distribution functions with known mean m and variance s2.In each period,the decision-maker needs to decide the expenditure on advertising B and the order quantity Q to maximize the expected proﬁt against the worst possible distribution of demand.2.1.Constant variance case(CVC)Because of diminishing returns from advertising,assume that the mean m is a concave increasing function of B,i.e.,m¼m0+m0o B a, where o and a are empirically determined positive constants that represent the effectiveness of advertising and0o a o1.For any o40,the larger the values of a and o,the more effective is theadvertising.In the CVC,it is assumed that advertising shifts the mean,but preserves the standard deviation,i.e.,s¼s0.This situation may occur when a company steps up its advertising efforts to attract more buyers but its competitors remain unaware or have no resources to do the same.The decision-maker would like to maximize the expected proﬁt p G(Q,B),wheremaxQ,B Z0p GðQ,BÞ¼pEðmin f Q,D gÞþsEðQÀDÞþÀcQÀlEðDÀQÞþÀB, such that m¼m0+m0o B a,s¼s0:ð1ÞNote that if o¼0so that demand is not affected by advertising, the decision-maker will set the expenditure on advertising B to be zero,and(1)reduces to the model presented in Alfares and Elmorra (2005).Next,with the deﬁnitions of m,d,k,and the following relationships:min{Q,D}¼D–(D–Q)+and(Q–D)+¼(Q–D)+(D–Q)+,the expected proﬁt p G(Q,B)can be rewritten asp GðQ,BÞ¼c fðdþmÞmÀdQÀðdþmþkÞEðDÀQÞþgÀB:ð2ÞThen the following lemmas are required:Lemma1.EðDÀQÞþr½s2þðQÀmÞ2 1=2ÀðQÀmÞ2:ð3ÞLemma2.For every Q,there exists a distribution G*A W where the bound(3)is tight.For the proofs of Lemmas1and2,please refer to Gallego and Moon(1993).Combining(2)and(3)we getp GðQ,BÞZ cðdþmÞmÀdQÀðdþmþkÞ½s2þðQÀmÞ2 1=2ÀðQÀmÞ()ÀB:ð4ÞThe lower bound is maximized to derive the optimal values of Q and B.Rearranging(4)yields the lower bound of expected proﬁt:p GL¼c2ðmþdÀkÞmÀðdÀmÀkÞQÀðdþmþkÞ½s2þðQÀmÞ2 1=2n oÀB:ð5ÞFrom(5),itsﬁrst partial derivative with respect to Q is@p G L@Q¼Àc2ðdÀmÀkÞþðdþmþkÞðQÀmÞ½s2þðQÀmÞ2 1=2():ð6ÞBy setting(6)to zero and solving for Q,Scarf’s ordering rule is obtained:Q S¼mþsðmþkÀdÞ2ðkdþmdÞ1=2¼mþs2kþmd1=2Àdkþm1=2():ð7ÞNote that(7)is the same as the result presented by Alfares and Elmorra(2005).Substituting the constraints of(1)into(7),the result is the optimal order quantity:QÃ1¼m0þm0o B aþs02kþmd1=2Àdkþm1=2():ð8ÞC.-M.Lee,S.-L.Hsu/Int.J.Production Economics129(2011)217–224 218To verify that (5)is strictly concave in Q ,the second derivative of (5)with respect to Q is calculated from (6):@2p G L@Q 2¼Àc ðd þm þk Þs 22½s 2þðQ Àm Þ2 3=2o 0:ð9ÞNext,(7)is substituted into (5)to yield the lower bound on the expected proﬁt:p GL ¼cm m 1Às m m ðkd þmd Þ1=2ÀB :ð10ÞLemma 3.cm m 1Às m mðkd þmd Þ1=2ÀB r p G ðQ ,B Þr cm m ÀB :For the proof of Lemma 3,please refer to Gallego and Moon (1993).From Lemma 3,when demand is probabilistic (s 40),ðs =m m Þðkd þmd Þ1=2can be regarded as the maximal fractional cost of demand randomness.Lemma 4.If m 2=ðkd þmd Þr ðs =m Þ2,then the lower bound on expected proﬁt is not positive,and the decision-maker will set B ¼0and Q ¼0;otherwise B and Q are positive.From (1)and (10),it is clear that Lemma 4is true.From Lemma 4,it follows that if a newsboy product has a higher value of the CV of demand,a higher discount rate (a lower salvage value),a higher penalty rate,or a lower markup rate,the lower bound on the expected proﬁt is less likely be positive;therefore,the decision-maker may not order or advertise the product.In other words,the decision-maker may not want to be in the business of selling this kind of newsboy product.As for the rest of this case,consider only the situation that the lower bound is positive,i.e.,m 2=ðkd þmd Þ4ðs 0=m Þ2.Substituting the constraints of (1)into (10)givesp G L ¼cm ðm 0þm 0o B a ÞÀc s 0ðkd þmd Þ1=2ÀB :ð11ÞDifferentiating (11)with respect to B and setting the ﬁrstderivative to zero yield the optimal expenditure on advertising:B Ã1¼ðcm m 0oa Þ11Àa:ð12ÞTo verify that (11)is strictly concave in B ,the second derivatives of (11)are computed.Because 0o a o 1,the second derivatives are @2p G L¼cm m 0oa ða À1ÞB a À2o 0;@2p G L¼0:ð13ÞFrom (9)and (13),the Hessian matrix of the lower bound on theexpected proﬁt (5)is negative deﬁnite,and therefore the lower bound has a maximum value.Theorem 1.B Ã1is an increasing function of c,m,m 0,a ,and o .However,d,k,and s 0do not affect B Ã1.Proof.Theorem 1can be proved taking the ﬁrst derivatives of (13)with respect to each parameter and determining their signs:dB Ã1dc ¼m m 0oa B Ãa 1=ð1Àa Þ40,dB Ã1¼c m 0oa B Ãa1=ð1Àa Þ40,dB Ã1dd ¼dB Ã1dk ¼dB Ã1d s 0¼0,dB Ã1d m 0¼cm oa B Ãa 1=ð1Àa Þ40,dB Ã1d o ¼cm m 0a B Ãa 1=ð1Àa Þ40,dB Ã1a¼B Ã1(ð1Àa ÞÀ2ln ðcm m 0oa Þþ1=a ð1Àa Þ)40:&The results of Theorem 1can be described and explained asfollows.From (10),the lower bound on the expected proﬁt increases with the mean m (the average market size).Therefore,when the unit cost c or the markup rate m increases (i.e.,the unit proﬁt increases),enlarging the average market size by increasing the expenditure on advertising can increase proﬁts.When the mean m 0(the original market size)increases,enlarging average market size by increasing the expenditure on advertising can increase proﬁts.When the advertising effectiveness constants a and o increase,advertising becomes more effective in increasing market size,and therefore the decision-maker should increase the expenditure on advertising.Furthermore,in the CVC,advertising does not affect the standard deviation of demand,and therefore there is no point in increasing or decreasing the expenditure on advertising when the standard deviation s 0increases.However,it is interesting to note that the discount rate d and the penalty rate k do not affect the optimal cost of advertising.The reason may be that discount and penalty costs occur when true demand deviates from the mean,but that advertising cannot affect the standard deviation of demand in the CVC.In summary,advertising is more suitable for a product with a higher unit cost,a higher marginal proﬁt (markup),a higher advertising effect,or a lower coefﬁcient of variation of demand.Now (13)can be back-substituted into (8)to obtain the optimal order quantity:Q Ã1¼m 0þm 0o B Ãa 1þs 02k þm d 1=2Àd k þm 1=2():ð14ÞTheorem 2.(i)Q Ã1is an increasing function of c,m,k,m 0,a ,and o ,but Q Ã1is a decreasing function of d.(ii)When k+m 4d,the optimal order quantity Q Ã1is an increasingfunction of s 0.When k+m o d,Q Ã1is a decreasing function of s 0.When k+m ¼d,Q Ã1is not affected by s 0.Proof.Theorem 2can be proved taking the ﬁrst derivatives of (14)with respect to each parameter and determining their signs as in the proof of Theorem 1:dQ Ã1¼m 0oa B Ãa À11dB Ã140,dQ Ã1o ¼m 0B Ãa 1þm 0oa B Ãa À11dB Ã1o40,dQ Ã1dd ¼Às 0ðm þk Þðm þk þd Þ4ðkd þmd Þ3=2o 0,dQ Ã1dk ¼s 0d4ðkd þmd Þ3=240,dQ Ã1m 0¼1þo B Ãa 1þm 0oa B Ãa À11dB Ã1m 040,dQ Ã1d s 0¼12k þm d 1=2Àdk þm 1=2(),dQ Ã1a ¼m 0o B Ãa 1ln B Ã1þm 0oa B Ãa À11dB Ã1a 40,dQ Ã1dm ¼m 0oa B Ãa À11dB Ã1dm þs 0d ðm þk þd Þ4ðkd þmd Þ3=240:&Note that Scarf’s ordering rules,as stated in Gallego and Moon(1993)and Alfares and Elmorra (2005),do not appear to be affected by the unit cost c .However,in this case,the optimal order quantity increases with the unit cost c .From Theorem 2,when the unit cost c ,the markup rate m ,or the penalty rate k increases,the decision-maker should increase the order quantity because the unit proﬁt has increased and a large order quantity may increase sales and prevent shortages.When the mean m 0increases,the original market size has increased and a large order quantity may increase sales.When the advertising effectiveness constants a or o increase,a larger order quantity is optimal.When the discount rate dC.-M.Lee,S.-L.Hsu /Int.J.Production Economics 129(2011)217–224219increases,the salvage value has decreased and a smaller order quantity is preferred.Next,consider a product with k +m 4d .If one more unit of product is ordered,the proﬁt k +m when the unit is sold is greater than the loss d when the unit is not sold.Therefore,when the standard deviation s 0increases,a larger order quantity may increase proﬁt.Similar explanations can be given for the situations that k +m o d or k +m ¼d .In summary,a larger order quantity is more suitable for a product with a higher mean,a higher unit cost,a higher marginal proﬁt (markup),a higher penalty cost,a higher advertising effect,or a lower discount rate.For a product with k +m 4d ,a larger order quantity is suitable when the demand uncertainty (standard deviation)increases;for a product with k +m o d ,a smaller order quantity is suitable;and for a product with k +m ¼d ,the increase of demand uncertainty has no effect on order quantity.Next,(12)and the constraints in (1)are substituted into (11)to yield the optimal lower bound on expected proﬁt:p G ÃL 1¼cm f m 0þm 0o B Ãa 1gÀc s 0ðkd þmd Þ1=2ÀB Ã1:ð15ÞTheorem 3.p G ÃL 1is an increasing function of c,m,m 0,a ,and o ,but p G ÃL 1is a decreasing function of d,k,and s 0.Proof.Theorem 3can be proved taking the ﬁrst derivatives of (15)with respect to each parameter and determining their signs using Lemma 4and Theorem 1:d p G ÃL 1d a ¼cm m 0o B Ãa 1ln B Ã140,d p G ÃL 1dd ¼Àc s 0ðk þm Þðkd þmd ÞÀ1=2=2o 0,d p G ÃL 1d m 0¼cm ð1þo B Ãa 1Þ40,d p G ÃL 1d s 0¼Àc ðkd þmd Þ1=2o 0,d p G ÃL 1d o ¼cm m 0B Ãa 140,d p G ÃL 1¼Àc s 0d ðkd þmd ÞÀ1=2=2o 0,d p G ÃL 1¼c (m Às 0d ðkd þmd ÞÀ1=2=2)40,d p G ÃL 1dc¼m m 0o B Ãa 1þðm m 0Às 0ðkd þmd Þ1=2Þ40:&From Theorem 3,when the unit cost c or the markup rate m increases,the optimal lower bound on expected proﬁt increases.When the mean m 0(the original market size)increases,demand increases,and therefore the optimal lower bound also increases.When the advertising effectiveness constants a or o increase,this means that advertising is more effective,and therefore the optimal lower bound increases.However,when the discount rate d increases (implying a lower salvage value),the penalty rate k increases (meaning a higher shortage cost),or the standard deviation s 0increases (meaning a higher demand uncertainty),and the optimal lower bound decreases.2.2.Constant coefﬁcient of variation case (CCVC)In this case,advertising increases both the mean m and the standard deviation s of demand in a proportional fashion,i.e.,CV is a constant.Therefore,it can be assumed that both the mean and the standard deviation are concave increasing functions of B ,i.e.,m ¼m 0+m 0o B a and s ¼s 0+s 0o B a ,according to Khouja and Robbins (2003).This situation may occur when a company steps up its advertising efforts to attract more buyers and its competitors are likely to do the same,with the result that the demand becomesmore uncertain at a higher level of advertising expenditure.Thedecision-maker’s objective ismax Q ,B Z 0p G ðQ ,B Þ¼pE ðmin f Q ,D gÞþsE ðQ ÀD ÞþÀcQ ÀlE ðD ÀQ ÞþÀB ,such that m ¼m 0+m 0o B a ,s ¼s 0þs 0o B a :ð16ÞThe process to derive the optimal values of Q and B is similar to that in the previous case.The optimal order quantity in this case isQ Ã2¼m 0þm 0o B a þs 0þs 0o B a 2k þm d 1=2Àd k þm 1=2():ð17ÞThe lower bound on expected proﬁt in the CCVC isp G L ¼cm ðm 0þm 0o B a ÞÀc ðs 0þs 0o B a Þðkd þmd Þ1=2ÀB :ð18ÞAs in Lemma 4,only the situation where the lower boundin (18)is positive will be considered,i.e.,m 2=ðkd þmd Þ4ðs =m Þ2¼ðs 0=m 0Þ2.The optimal expenditure on advertising in the CCVC isB Ã2¼f c oa ðm m 0Às 0ðkd þmd Þ1=2Þg 11Àa:ð19ÞTo verify that (18)is strictly concave in B ,the second derivatives of (18)are computed.Because 0o a o 1and m 2=ðkd þmd Þ4ðs 0=m 0Þ2,we have@2p G L@B2¼c oa ða À1ÞB a À2ðm m 0Às 0ðkd þmd Þ1=2Þo 0;@2p G L@B @Q¼0:ð20ÞFrom (9)and (20),the Hessian matrix of the lower bound on theexpected proﬁt (5)is negative deﬁnite,and therefore the lower bound has a maximum value.Theorem 4.B Ã2is an increasing function of c,m,m 0,a ,and o ,but B Ã2is a decreasing function of d,k,and s 0.Proof.Theorem 4can be proved taking the ﬁrst derivatives of (19)with respect to each parameter and determining their signs:dB Ã2d m 0¼cm oa B Ãa 2=ð1Àa Þ40,dB Ã2d s 0¼Àc oa ðkd þmd Þ1=2B Ãa 2=ð1Àa Þo 0,dB Ã2dc ¼oa ðm m 0Às 0ðkd þmd Þ1=2ÞB Ãa 2=ð1Àa Þ40,dB Ã2dm ¼c oa B Ãa 2ðm 0Às 0d ðkd þmd ÞÀ1=2=2Þ=ð1Àa Þ40,dB Ã2dd ¼Àc oas 0ðm þk Þðkd þmd ÞÀ1=2B Ãa 2=2ð1Àa Þo 0,dB Ã2dk ¼Àc oas 0d ðkd þmd ÞÀ1=2B Ãa 2=2ð1Àa Þo 0,dB Ã2o¼c a ðm m 0Às 0ðkd þmd Þ1=2ÞB Ãa 2=ð1Àa Þ40,dB Ã2a¼B Ã2ð1Àa ÞÀ2ln ðc oa ðm m 0Às 0ðkd þmd Þ1=2ÞÞþ1=a ð1Àa Þn o 40:&The results for Theorems 2and 4are similar.However,in Theorem 4,the effects of the standard deviation s 0,the discount rate d ,and the penalty rate k on the optimal expenditure on advertising B Ã2are negative.In the CCVC,advertising increases the uncertainty of demand (the standard deviation of demand),but it does not do so in the CVC.This means that when the standard deviation increases,it is better not to increase advertising expen-diture to cause the demand to become more uncertain.When the discount rate d increases,it is also better not to increase advertising expenditure to increase the uncertainty of demand because the product is more likely to remain unsold and its salvage value is low.When the penalty rate k increases,it is also better not to increaseC.-M.Lee,S.-L.Hsu /Int.J.Production Economics 129(2011)217–224220advertising expenditure to increase the uncertainty of demand because the demand is more likely to be unsatisﬁed and its shortage penalty is high.In summary,advertising is more suitable for a product with a higher unit cost,a higher markup rate,a lower discount rate,a lower penalty rate,higher advertising effects,or a lower coefﬁcient of variation of demand.The optimal order quantity QÃ2is as follows:QÃ2¼m0þm0o BÃa2þs0þs0o BÃa2kþm1=2Àd1=2():ð21ÞTheorem5.QÃ2is an increasing function of c,m,k,m0,a,and o,but QÃ2is a decreasing function of d.The effect of s0on QÃ2depends on the values of the model parameters.Proof.Theorem5can be proved taking theﬁrst derivatives of(21) with respect to each parameter and determining their signs using Theorem4:dQÃ2¼oa BÃaÀ12mþs0ðmþkÀdÞ2ðkdþmdÞ()dBÃ240,dQÃ2 dm ¼oa BÃaÀ12mþs0ðmþkÀdÞ2ðkdþmdÞ1=2()dBÃ2dm þðs0þs0o BÃa2ÞdðmþkþdÞ4ðkdþmdÞ40,dQÃ2 dd ¼oa BÃaÀ12mþs0ðmþkÀdÞ2ðkdþmdÞ()dBÃ2ddÀðs0þs0o BÃa2ÞðmþkÞðmþkþdÞ4ðkdþmdÞ3=2o0,dQÃ2 dk ¼oa BÃaÀ12mþs0ðmþkÀdÞ2ðkdþmdÞ1=2()dBÃ2dk þðs0þs0o BÃa2ÞdðmþkþdÞ4ðkdþmdÞ40,dQÃ2 m0¼1þo BÃa2þoa BÃaÀ12mþs0ðmþkÀdÞ2ðkdþmdÞ()dBÃ2m40,dQÃ2 d s0¼oa BÃaÀ12mþs0ðmþkÀdÞ2ðkdþmdÞ1=2()dBÃ2d s0þð1þo BÃa2ÞðmþkÀdÞ2ðkdþmdÞ1=2,dQÃ2o¼mþs0ðmþkÀdÞ2ðkdþmdÞ()BÃa2þoa BÃaÀ12dBÃ2o40,dQÃ2 d a ¼oðBÃa2ln BÃ2þa BÃaÀ12dBÃ2d aÞm0þs0ðmþkÀdÞ2ðkdþmdÞ()40:&The results for Theorems2and5are similar.However,in Theorem5,the effect of the standard deviation on the optimal order quantity is more complicated.Next,the optimal lower bound on expected proﬁt isp GÃL2¼cm f m0þm0o BÃa2gÀc f s0þs0o BÃa2gðkdþmdÞ1=2ÀBÃ2:ð22ÞTheorem6.p GÃL2is an increasing function of c,m,m0,a,and o,but p GÃL2 is a decreasing function of d,k,and s0.Proof.Theorem6can be proved taking theﬁrst derivatives of(22) with respect to each parameter and determining their signs using Lemma4and Theorem4:d p GÃL2m¼cmð1þo BÃa2Þ40,d p GÃL2d s0¼Àcð1þo BÃa2ÞðkdþmdÞ1=2o0,d p GÃL2 dc ¼ð1þo BÃa2Þðm m0Às0ðkdþmdÞ1=2Þ40,d p GÃL2¼cð1þo BÃa2Þðm0Às0dðkdþmdÞÀ1=2=2Þ40,d p GÃL2¼Àc s0ð1þo BÃa2ÞðkþmÞðkdþmdÞÀ1=2=2o0,d p GÃL2dk¼Àcd s0ð1þo BÃa2ÞðkdþmdÞÀ1=2=2o0,d p GÃL2d o¼cBÃa2ðm m0Às0ðkdþmdÞ1=2Þ40,d p GÃL2d a¼c o BÃa2ln BÃ2ðm m0Às0ðkdþmdÞ1=2Þ40:&The results for Theorems3and6are the same.Theorem7.(i)BÃ1Z BÃ2.(ii)p GÃL1Z p GÃL2.Proof.From(12)and(19),part(i)is true.From(1),only oneconstraint is effective,and s¼s0is not an effective constraint.From(16),two constraints are effective.Moreover,(1)and(16)have thesame objective function,and thus part(ii)is true.&From Theorem7,it follows that the decision-maker in the CVCspends more on advertising and earns more proﬁts than in theCCVC.The reason may be that advertising in the CVC shifts not onlythe mean demand(positive effect)but also both the mean demand(positive effect)and the standard deviation(negative effect)inthe CCVC.2.3.Increase in coefﬁcient of variation case(ICVC)In this case,advertising increases the standard deviation s at afaster rate than it increases the mean m.Khouja and Robbins(2003)assumed that s/m¼(1+gB g)s0/m0,where g and g are empiricallydetermined positive constants and0o g o1.Therefore,inthe present model,it is assumed that m¼m0+m0o B a ands¼s0(1+o B a)(1+gB g).The larger the values of g or g,the morequickly the demand uncertainty increases.In other words,g and grepresent the negative effect of advertising.This situation mayoccur when a company steps up its advertising efforts to attractmore buyers and its competitors are likely to do the same,in whichcase the effects of unanticipated economic conditions may lead tovery large deviations from the expected demand at higher adver-tising expenditure.The decision-maker’s objective function ismaxQ,B Z0p GðQ,BÞ¼pEðmin f Q,D gÞþsEðQÀDÞþÀcQÀlEðDÀQÞþÀB,such that m¼m0+m0o B a,s¼s0ð1þo B aÞð1þgB gÞ:ð23ÞThe process of deriving optimal values of Q and B is similar tothat in the CVC.The optimal order quantity is as follows:QÃ3¼m0þm0o B aþs0ð1þo B aÞð1þgB gÞkþm1=2Àd1=2():ð24ÞThe lower bound on expected proﬁt isp GL¼cm m0ð1þo B aÞÀc s0ð1þo B aÞð1þgB gÞðkdþmdÞ1=2ÀB:ð25ÞAs in the previous cases,the following discussion of this caseconsiders only the situation that the lower bound is positive,i.e.,m2=ðkdþmdÞ4ðs=mÞ24ðs0=m0Þ2.Differentiating(25)with respect to B and setting the derivativeto zero yield the optimal expenditure on advertising BÃ3from(26),and BÃ3can be solved for using a computer-based spreadsheetsoftware package:cm aom0B aÀ1Àc s0ðkdþmdÞ1=2f oa B aÀ1ð1þgB gÞþð1þo B aÞg g B gÀ1g¼1:ð26ÞC.-M.Lee,S.-L.Hsu/Int.J.Production Economics129(2011)217–224221。

Optimal stimulus coding by neural populations using rate codes

Optimal Stimulus Coding by Neural Populations using Rate CodesDon H.Johnson and Will RayDepartment of Electrical&Computer Engineering,MS366Rice University6100Main StreetHouston,Texas77251–1892Revised May19,2003AbstractWe create a framework based on Fisher information for determining the most effective population coding scheme for representing a continuous-valued stimulus attribute over its entire ing this scheme,we derive optimal single-and multi-neuron rate codes for homogeneous populations using several statistical models frequently used to describe neural data.We show that each neuron’s discharge rate should increase quadratically with the stimulus and that statistically independent neural outputs provides optimal coding.Only cooperative populations can achieve this condition in an informationally effective way.Contact:Don H.Johnson713.348.4956,713.348.5686(FAX)e-mail:dhj@ web:/˜dhj1IntroductionRecordings from populations have,in some cases,shown statistical dependence among the constituent neuron’s responses(Dan et al.,1998;deCharms and Merzenich,1996;Gerstein et al.,1989;Perkel and Bullock,1968;Nirenberg et al.,2001)and little or no dependence in others(Gawne and Rich-mond,1993;Johnson and Kiang,1976;Reich et al.,2001;Zohary et al.,1994).Because a population code presumes that neurons jointly represent information(ﬁgure1),statistical dependence would be expected.Statistical dependence among individual responses is usually categorized into two classes. Stimulus-induced dependence arises because each neuron receives the same input as the others(ﬁg-ure1),while connection-induced correlation arises when neurons are interconnected somehow.Con-sequently,studies thatﬁnd little dependence are ing a variety of theoretical approaches, many researchers have shown that the effectiveness of neural population coding can be enhanced by statistical dependence among the individual responses(Abbott and Dayan,1999;Jenison,2000;John-son,2003;Panzeri et al.,1999;Pola et al.,2003;Seung and Sompolinsky,1993;Shamir and Som-polinsky,2001;Sompolinksy et al.,2001;Wilke and Eurich,2002;Wu et al.,2002),while some of these(Abbott and Dayan,1999;Panzeri et al.,1999;Sompolinksy et al.,2001)and others(Gawne and Richmond,1993;Zohary et al.,1994)have shown that dependence can hinder population coding. These studies judged the efﬁcacy of population coding using either Fisher information or mutual in-formation,and assumed a speciﬁc parametric form for individual and population coding(Seung and Sompolinsky,1993;Shamir and Sompolinsky,2001)or examined performance for a speciﬁc value of the stimulus(Johnson,2003;Abbott and Dayan,1999).One recent study explicitly decomposed the mutual information between stimulus and population response into independent and stimulus-induced and connection-induced dependence terms(Pola et al.,2003).This report takes a different approach:regardless of how dependence might arise,what is the op-timal population code?Here,we develop an approach to deriving the optimal code for populations faced with effectively representing a stimulus attribute over its entire range.A single stimulus attribute, denoted here by,is to be encoded by neural discharge patterns as the attribute varies over the nor-malized range of.We ask,in the case of single neurons,how should discharge rate vary with the stimulus attribute and,in the case of a population,how should individual discharge rates and sta-tistical dependence vary to achieve the most effective neural code.In both cases,we seek the coding function the way discharge rate or dependence vary with the stimulus that maximizes the ability of the optimal processing system to extract the value of the stimulus attribute throughout its range.We could not solve this problem without focusing on a particular coding mechanism the rate code and on speciﬁc statistical models that describe small(three or fewer neurons)populations.2MethodsThe notation we use here is the same as in Johnson(2003).We use the symbols and to denote a neuron’s input and output,respectively(ﬁgure1).The population output is designated by,where denotes the number of neurons in the population.How stimuli are encoded in the population’s output is expressed by the joint probability function,with denoting the stimulus attribute being coded.(1)The quantity describes how the population processes its input to produce its output.Ideally, we would seek the optimal form this conditional probability function should take for a speciﬁc input probability function.We found this problem too difﬁcult.Our approach is toﬁnd the optimal form for the output’s joint probability function when we assume a speciﬁc model for the individual neuron’s output probability function.For the various models considered here,we found a similarity of results.Based on this similarity,we infer what form the optimal population code might be regardless of model.Optimal response criterion.We assume the stimulus attribute is a scalar,and that one population code is better than another if it could lead to a smaller error in estimating the attribute from the popula-tion response.Our optimization criterion is to minimize the mean-squared estimation error, where denotes the estimate of.To approach this problem,we use Fisher information.Fisher information is important because of the Cram´e r-Rao bound,which states that the mean-squared er-ror for any unbiased estimator of a parameter’s value cannot be smaller than the reciprocal of the Fisher information evaluated at that value(Johnson et al.,2001):,where.denotes expected value with respect to the probability function .By using the Cram´e r-Rao bound,we need not specify how the attribute is decoded from the population response to determine how well the decoder could rmation theoretic meth-ods boil down to Fisher information when attribute values must be estimated(Sinanovi´c and Johnson, 2003).Note that,in general,the Fisher information varies with the value of,which means that the best possible mean-squared error that any estimator can achieve depends on the stimulus attribute’s value.By maximizing the Fisher information,we minimize the smallest value the mean-squared er-ror can achieve for a given parameter value.As the amount of data increases,the Cram´e r-Rao bound can be achieved with equality with a maximum-likelihood estimator(Cram´e r,1946).Consequently,by calculating the Fisher information,we can assess how well a stimulus attribute can be measured from a population response.Even though reliance on mean-squared error is limiting,it is well-understoodand gives us more analytic freedom.An important property of Fisher information exploited here is additivity:when consists of statistically independent components,the population’s Fisher infor-mation equals the sum of the individual neuron’s Fisher information.Stated mathematically,when,.We assume that the probability function governing the homogeneous population’s collective re-sponse can be described by the parameter vector.For example,consider a Bernoulli model for the number of spikes occurring in a time bin.This model merely says that no more than one spike can occur in a bin and that the probability of a spike in a bin is,which equals the product of discharge rate and binwidth.spikes in a bin(2)This model is parameterized by the spike probability;thus,.How a probability function’s parameters vary with the stimulus deﬁne what we call coding functions:an analytic description of how a one-dimensional stimulus attribute is encoded by a single neuron or by a population as a whole. Using its deﬁnition and the chain rule,the Fisher information that results when all of the components of varying in concert with a single stimulus attribute is given by the quadratic form(3)where denotes the transpose of.The Fisher information is a matrix that depends solely on the population’s statistical model and how this model depends on its parameters.does not depend on the stimulus;rather,it depends only on the probabilistic model and how the parameters inﬂuence the model.For example,in a two-neuron Bernoulli model,which will be detailed later,this matrix varies with two parameters:spike discharge probability and inter-neuron correlation coefﬁcient.The quantity indicates how the population’s Fisher information varies with the stimulus attribute, which means,in the context of the Bernoulli model,the result of how spike probability and inter-neuron correlation vary together with the stimulus according to some coding function.From an information coding viewpoint,it is that determines how well the stimulus attribute is coded.We seek the“best”coding function by establishing a criterion for how the mean-squared error should vary with the stimulus and solving this differential equation.Many choices could be made for how the mean-squared error varies with the stimulus.The most general speciﬁcation is to demand the mean-squared error vary as,with a speciﬁcation of how the optimal estimator’s mean-squared error should vary with the attribute’s value and with the proportionality constant chosen as small as possible for optimality.We equate with thereciprocal of the Fisher information.The Fisher information transformation equation becomeswith as small as possible.(4)The population size directly affects the Fisher information because it depends on the population’s joint response statistics.We want to apply this criterion for a given population and investigate the optimal coding strategy for it.By merging the square-root of the mean-squared error criterion with the derivative ,we obtain a much simpler way ofﬁnding the optimal coding function.with as small as possible.For each value of,the left side is a quadratic form we seek to maximize.The maximum value of occurs when is proportional to,the eigenvector of having the largest eigenvalue .This optimality criterion means that is proportional to the eigenvector corre-sponding to largest eigenvalue for each value of.The differential equation that would need to be solved is(5)Although complicated,this equation can be solved numerically.We solved this equation for the plus and minus signs separately,and took the one yielding the smallest meaningful value of.1 Perhaps the simplest choice for the mean-squared error criterion derives from the assumptions that the population strives to represent the attribute uniformly well over the attribute’s entire range.With this choice,optimal decoders observing the population’s response would yield constant mean-squared error for any attribute value.We thus seek the coding function that yields the largest possible con-stant Fisher information:.Estimators achieving constant mean-squared error are termed equivariance estimators and have the property that they minimize the maximal estimation er-ror(Lehmann and Casella,1998).Assuming that the stimulus attribute is deﬁned over the normalized range,the differential equation(5)becomesas large as possible(6)Solving this differential equation yields equivariance coding functions for each assumed value of; boundary conditions described below determine.The standard statistical model for a single neuron’s response is the point process(Johnson,1996). Because correlating point processes in general presents analytic difﬁculties,we focus on two simpler 1We found that if is chosen too small,negative probabilities or probabilities greater than one resulted.models frequently used to describe neural data:Bernoulli and Poisson ing these essen-tially means we focus on rate codes.For each model,we use to denote all of the parameters governing the joint probability distribution that describes the population’s output.Because sequences of Bernoulli random variables are statistically independent from bin to bin,this model does not take into account temporal statistical correlations that may be present in the response.Because of this independence,the Fisher information accumulated over a time interval is simply the sum of each bin’s contribution and we only need to consider a single bin.We also derived results for the Poisson and deadtime-modiﬁed Poisson counting models,the latter of which incorporates a simple description of refractory phenomena. 3ResultsOptimal single-neuron rate codes.Weﬁrst derive the optimal coding function for the single-neuron Bernoulli model(equation(2)).With the probability function’s parameter vector consisting only of the discharge probability,the differential equation(6)becomesbecause in a one-dimensional problem the largest eigenvalue corresponds to the Fisher information of the neuron’s statistical model,which in this case equals.This differential equation’s solution describes how the discharge probability should vary with the stimulus attribute so that the smallest possible constant estimation error would result.In other words,the solution is the optimal way(according to the equivariance criterion)to encode the stimulus in a single neuron’s discharge probability.With the initial condition that,this equation has the closed form solutionSolving for,weﬁnd that this quantity depends on the discharge probability at the largest value of the stimulus attribute:.The speciﬁcation of the discharge probability at the extreme values of the stimulus attribute comprise the boundary conditions mentioned after equation(6). Requiring that,for example,results in a Fisher information value of, which is equivalent to a root-mean-squared error bound of/bin.Figure2compares the Fisher information for this equivariance coding with several suboptimal(i.e.,non-constant Fisher information) choices.By considering the mean-squared error of these alternative rate coding schemes,we found that the equivariance criterion yields a well-behaved variation of the error with the stimulus.For the Poisson case,wherein the discharge rate is constant and the number of spikes occurring in an interval of duration has a Poisson probability function,the Fisher information is. The optimal coding function found by solving(6)iswhere,are the maximum and minimum rates,respectively,achieved as varies.This coding function differs mathematically from the Bernoulli coding function found previously,but both have similar forms(seeﬁgure3).The Fisher information for the optimal coding function is.We were also able toﬁnd the optimal coding function when an absolute refractory interval was included in the Poisson model with rate being the only parameter optimized.The Fisher information is ,and solving(6)for the optimal coding function yieldsDespite its complexity,this coding function differed little from that found in the Poisson case when .The expression for the equivariance Fisher information is also complicated,but in the small refractory-interval case,. Thus,refractoriness reduces the Fisher information.Optimal population rate codes.We elaborated these statistical models to describe small populations by correlating the occurrence of events in each model neuron.In addition toﬁnding the optimal value of Fisher information according to the equivariance principle,we need compare that value with that of an independent neural population,which equals the sum of Fisher information values contributed by each neuron.For example,this baseline value for two independent neurons described by the Bernoulli model described above will be.Bernoulli models for the two-neuron population are speciﬁed by the correlation coefﬁcient and the spike probabilities for each neuron.2(7)We assume that the spike probabilities and the correlation coefﬁcient,deﬁned in the usual way to be,could vary with stimulus attribute.An independent population occurs when(which corresponds to the baseline situation).A noncooperative population,such 2Note that some combinations of correlation coefﬁcient and spike probabilities,are not permitted as they would yield negative probabilities for the population’s probability function.as shown inﬁgure1and expresses stimulus-induced dependence,can only yield positive values for (Johnson,2003).Negative and positive values of can arise from a cooperative population,which expresses both stimulus-and connection-induced dependence.Consequently,we can only generally determine population structure from the correlation coefﬁcient.If is negative,the structure must be cooperative;if is zero,the optimal structure could be the independent structure or a cooperative one;and if positive,the structure could be cooperative or noncooperative.To make the population homogeneous,we set the spike probabilities equal:.Thus,the two-neuron Bernoulli model is parameterized as.For three-neuron populations,,with being a correlation-like quantity related to the joint probability of all neurons responding during a bin.No analytic results for the population models could be derived;we relied on numerical solution of the differential equation(6).Before considering the general case wherein both spike probability and correlation parameters vary with the stimulus,we can examine the special case of pure correlation coding(deCharms and Merzenich,1996)for a two-neuron population,in which the population encodes by varying the cor-relation between neurons while keeping spike probabilityﬁxed.Allowing the correlation to range between0and1in solving(6)yields a Fisher information of with and with. Thus,the Fisher information resulting from pure correlation coding depends on the sustained discharge probability.Negative correlation coding generally yields smaller Fisher information:forand for.Thus,correlation coding is maximally efﬁcient when high spike probabilities occur,and negative and positive correlations are equally effective at this extreme.Even so,correlation coding yields a Fisher information roughly a factor of two smaller than the baseline value of.Figure3shows optimal coding functions for the general case when both spike discharge proba-bility and inter-neuron correlation vary with the stimulus for both two-and three-neuron populations described by a Bernoulli model.The most striking aspect of these results is that optimal population encoding required very small correlation values.The greatest value of achieved was in the two-neuron case and zero in the three-neuron case.Furthermore,positive correlation resulted in a larger Fisher information()than negative correlation()for the two-neuron case.Only positive cor-relation coding resulted in a Fisher information greater than the baseline,but the Fisher information increase is small(or).Thus,statistically independent population outputs are essentially op-timal.For the three-neuron model,optimal choices for and were identically zero.Based on these two-and three-neuron Bernoulli models,we infer that independent populations are optimal according to the equivariance criterion.Furthermore,pure correlation coding is inferior,at least in the context of Bernoulli models.In the Poisson model for correlated activity between two neurons,the output spike count oversome time interval equals,where is the Poisson-distributed spike count in the th neuron and is a common spike count that each neuron produces(Holgate,1964).These spike counts are statistically independent of each other and of.All counts are Poisson random variables having the same rate parameter and the common count is also Poisson with rate.The common count among all neurons means the outputs are statistically dependent with positive correlation unless the common-count rate equals zero.For the two-neuron case,solving equation(6)resulted in a decreasing rate for the common count component(ﬁgure3).Larger values of the Fisher information were found here(61.8compared to the baseline value of48.2).However,if the common rate was constrained to be zero at,which corresponds to a nominally independent population,the common-count rate was zero over the entire range.Again,the independent population is found to be optimal.Population structures exhibiting optimal rate coding.These results create a quandary.From the viewpoint of a population endeavoring to code a continuous stimulus attribute as well as possible,the most effective rate coding scheme,as quantiﬁed by the smallest Fisher information,occurs when the population outputs are statistically independent.A population having a common input should result in stimulus-induced correlation,which is always positive(Johnson,2003).An independent population, wherein each neuron in the population receives input statistically independent of the others,results in statistically independent outputs but provides no information gain beyond that of a single neuron(John-son,2003).One possible role for cooperation among neurons is to create connection-induced correlation that decorrelates the stimulus-induced correlation produced by a noncooperative structure.We envision the structure shown inﬁgure4,where a lateral connection network interacts the outputs of a noncoop-erative population to produce a cooperative one.If this kind of cooperation has the right properties, it will be lossless from an information processing viewpoint,and the output would have theoretically ideal properties:statistically independent outputs(thereby producing optimal coding properties from an equivariance viewpoint)while exhibiting the information processing properties of a noncoopera-tive population.We showed in our companion paper(Johnson,2003)that noncooperative populations always yield maximal processingﬁdelity as the population size increases.Complicating the creation of the decorrelation transformation is that it would need to apply over the range of possible joint probability functions produced by the cooperative population without depending on.An adaptive cooperative structure could be envisioned wherein the network changes the way the population’s outputs interact as stimulus conditions vary.In this paper,we take the analytically simpler tact of exploring aﬁxed structure that would make the noncooperative outputs as uncorrelated aspossible and would exploit population size to create a transformation that would not depend on discharge rates or inter-neuron correlation.We constructed an interconnection network function that has the following properties.The transformation is invertible.Mathematically,must be deﬁned for all possible inputs and outputs of the interconnection network.Invertibility requires that the number of outputs be no smaller than the population’s size.The reason for this condition is that invertible transformations are,from an information processing viewpoint,lossless(Sinanovi´c and Johnson,2003).Hence the cooperative population’s information transfer ratio equals that of the noncooperative population and would thus have all of its asymptotic properties.The transformation decorrelates the population’s output regardless of.A transformation that yields statistically independent outputs would depend greatly on the jointprobability function.However,by decorrelating,which makes the pairwise correlation coefﬁcients zero,we canﬁnd a interconnection network without needing to know the details of the output probability function.In addition,we need the transformation to not depend on the stimulus-induced correlation that the noncooperative structure yields.As shown in the appendix,if the population outputs are scalar quantities(spike counts,for example),a transformation having these properties has the form(8)Here,is the average of the population outputs:.In words,each output equals an input minus a constant times the average value of all inputs.Note that as the population size grows,the weighting constant tends to be independent of and.The threshold population size needed to achieve this insensitivity is.Curiously,insensitivity is easier to achieve when the correlation among population outputs is strong(when);weaker correlation demands larger populations.In this way,uncorrelated population outputs can be obtained so that optimal encoding can be achieved while maintaining the information transfer properties of a noncooperative population.4ConclusionsOur theory forﬁnding the optimal coding strategy can be generalized to other statistical models,includ-ing ones other than rate codes,and to mean-squared error criteria other than the equivariance principle. To use this approach,the statistical model and the mean-squared error criterion would need to be de-Johnson & RayOptimal Stimulus Codingﬁned, model parameterized, the Fisher information with respect to that parameter set computed, and equation (5) solved. Our choice of constant mean-squared error for the performance criterion was not without consideration. Figure 2 shows the optimal mean-square error proﬁles that various rate functions would yield. For example, a linear increase in discharge rate will result in an error in determining the stimulus attribute that increases as the attribute’s value increases. While choosing the equivariance principle is arbitrary, it seems reasonable. We do not know how robust our results are to other choices for the variation of mean-squared error with the stimulus. We found that the discharge rate should increase as the square of the stimulus attribute and that the population outputs should be statistically independent. The attribute not refer to any speciﬁc stimulus parameter, such as amplitude. to stimulus parameters. Instead, is trying to code efﬁciently.« is intentionally abstract, does Note that « need not be proportional« could be proportional to that stimulus aspect the sensory system For example, « could be related to the logarithm of stimulus amplitude.Unfortunately, because these results are tied to speciﬁc statistical models, our optimal coding results may not be a general principle, even for all rate codes. That said, the parabolic behavior we found mimics what was found from other coding viewpoints (Miller and Troyer, 2002; Stein, 1967). Although our derivations were limited to three models Bernoulli, Poisson, and Poisson with deadtime and tosmall populations, this parabolic input-output behavior seems to be an efﬁcient way to encode a scalar stimulus feature in a neuron’s discharge rate from an equivariance viewpoint (constant mean-squared error regardless of the stimulus attribute’s value). The similarity of the results for the various models is striking, and leads to the speculation that the square-law/independent-output population behavior applies generally for individual and population rate codes. Our conclusion that independent coding is optimal would seem to mean that no collective population coding should occur because stimulus-induced dependence is not present. We have showed that regardless of the neural code employed, noncooperative populations can represent the stimulus increasingly well as the population size increases, with no information loss occurring asymptotically (Johnson, 2003). Connection-induced dependence resulting from cooperation among the neurons can enhance, diminish, or leave unchanged the efﬁcacy of the population’s code relative to the baseline performance of the noncooperative population. We have shown here that cooperation, which causes connection-induced dependence, can cancel stimulus-induced correlation to yield a population output having optimal coding properties from the equivariance viewpoint. One consequence of this structure is that inferring population structure from population response statistics can’t be done uniquely: an uncorrelated output results from both the structure shown in ﬁgure 4 and the independent population. Also, the cooperative decorrelation structure is relatively insensitive to neuron loss. Two factors make it less robust than the11Johnson & RayOptimal Stimulus Codingnoncooperative structure. First of all, achieving perfect decorrelation without knowing the stimulusinduced correlation requires a sufﬁciently large population. Losing neurons would eventually comprise the large-population assumption. Secondly, the average of responses also depends on the population size; a robust average would need to be obtained. Many researchers have recorded population responses, looking for population codes that collectively express the stimulus better than the individual neurons do. Several have concluded that dependence can enhance coding efﬁcacy (Abbott and Dayan, 1999; deCharms and Merzenich, 1996; Jenison, 2000; Wu et al., 2002) while others have concluded it impairs population coding (Gawne and Richmond, 1993). This study ﬁnds that independent rate codes are optimal for a population that must represent a stimulus parameter well over its entire range. In the companion paper (Johnson, 2003), we found using the Bernoulli model that if a binary-valued stimulus attribute is to be encoded (for example, coding whether the light is on or off), correlated population responses provide optimal coding. Thus, the ﬁndings that dependence can be detrimental or beneﬁcial are both correct: no single population coding strategy exists that applies universally. Optimal population coding strategies depend at least on what is to be encoded.12。

NUMERICAL STRATEGIES FOR OPTIMAL EXPERIMENTAL DESIGN FOR PARAMETER IDENTIFICATION OF NON-LI

∂y F = ∫ (t ) ∂p 0
tf
T
Байду номын сангаас
∂y Q (t ) dt ∂p
(1)
with between the brackets the sensitivity matrix of the model outputs (in the vector y) with respect to the parameters (in the vector p) and Q the inverse of the measurement error covariance matrix under the assumption that the outputs are corrupted with zero-mean gaussian noise. Thus, the mathematical formulation of the OED problem can be written as. Find u(t) to minimize J = φ(F) subject to
Abstract
The problem of optimal experimental design (OED) for parameter estimation of non-linear dynamic systems is considered. It is shown how this problem can be formulated as a dynamic optimization (optimal control) problem where the performance index is usually a scalar function of the Fisher information matrix. Numerical solutions can be obtained using direct methods, which transform the original problem into a nonlinear programming (NLP) problem via discretizations. However, due to the frequent non-smoothness of the cost functions, the use of gradient-based methods to solve this NLP might lead to local solutions. Stochastic methods of global optimization are suggested as robust alternatives. A case study considering the OED for parameter estimation in a fed-batch bioreactor is used to illustrate the performance and advantages of two selected stochastic algorithms.

并联混合动力汽车ECMS的时变等效因子提取算法的研究

2021年（第43卷）第2期汽车工程Automotive Engineering2021（Vol.43）No.2并联混合动力汽车ECMS的时变等效因子提取算法的研究*李跃娟1，齐巍2，王成2，张博2，卢强2（1.北京工业大学机械工程与应用电子技术学院，北京100124；2.中国汽车技术研究中心有限公司，天津300300）［摘要］为解决当前等效燃油消耗最小控制策略（ECMS）未能根据实际工况选取最优等效因子的问题，利用动态规划算法（DP）和ECMS各自的优点，构建并联混合动力汽车能量算法模型，即采用动态规划算法的等效燃油消耗最小控制策略（ECMSwDP），将等效因子作为全局最优算法的控制变量，通过对等效因子的离散全局优化，获得基于工况的最佳时变等效因子。

在标准工况下对时变等效因子实时控制策略与全局最优控制策略DP的各项性能参数进行了数值仿真，验证了时变等效因子提取算法的有效性和等效因子初始值选取方法的可行性。

关键词：混合动力汽车；动态规划；等效消耗最小化策略；时变等效因子Study on Extraction Algorithm for Time‑varying Equivalent Factor of ECMS forParallel Hybrid Electric VehicleLi Yuejuan1，Qi Wei2，Wang Cheng2，Zhang Bo2&Lu Qiang21.College of Mechanical Engineering and Applied Electronics Technology，Beijing University of Technology，Beijing100124；2.China Automotive Technology&Research Center Co.，Ltd.，Tianjin300300［Abstract］For solving the problem that the current equivalent consumption minimization strategy（ECMS）cannot select the optimal equivalent factor according to the actual driving condition，an energy algorithm model for parallel hybrid electric vehicle，i.e.ECMS with dynamic programming（DP），is constucted by taking the respective advantages of DP and ECMS.With equivalent factor as the control variable of global optimum alroithm，the condition ‑based optimal time‑varying equivalent factor is abtained through the discrete global optimizaion of equivalent factor.A numerical simulation is conducted on all performance parameters of real‑time control strategy for time‑varying equivalent factor and the DP for globally optimal control strategy in normal condition，verifying the effectiveness of the extraction algorithm for time‑varying equivalent factor and the feasibility of the selecting method for the initial value of equivalent factor.Keywords：hybrid electric vehicle；dynamic programing；equivalent consumption minimization strategy；time⁃varying equivalent factor前言面对大气污染加剧和石油资源消耗过度的紧迫形势，汽车的环保与节能引起了广泛的社会关注，我国三部委联合颁布的《汽车产业中长期发展规划》的八大重点工程中的智能网联汽车推进工程和先进节能环保汽车技术提升工程为混合动力电动汽车节能研究带来了新的机遇［1］。

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