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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.。

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3 level carhart style model -回复

3 level carhart style model -回复什么是3级Carhart风格模型？3级Carhart风格模型是一种用于分析和预测股票价格和股票回报的经济模型。

该模型由风险、公司基本面因素和技术因素构成，被广泛应用于金融市场和投资决策中。

第一级：风险因素在3级Carhart风格模型中，风险因素被认为是影响股票价格和回报的主要因素之一。

风险因素包括市场风险、规模因素和价值因素。

市场风险是指整个市场的波动和变化对股票的影响。

市场风险通常通过市场指数（如S&P 500指数）来衡量。

规模因素是指公司的市值对股票价格和回报的影响。

通常认为，市值较小的公司具有较高的风险和回报。

价值因素是指公司的估值水平对股票价格和回报的影响。

价值较低的公司可能具有较高的风险和回报。

第二级：公司基本面因素公司基本面因素指的是公司内部的经营状况和财务状况对股票价格和回报的影响。

这些因素包括公司的利润、收入、负债、现金流等。

公司利润是指公司经营活动所获得的收入减去成本和费用后的剩余金额。

利润水平对股票价格和回报有直接影响。

公司收入是指公司从销售产品或提供服务中获得的总额。

收入越高，通常意味着公司的前景较好，可能会推动股票价格上涨。

负债是指公司的债务和负债占总资产的比例。

高负债率可能意味着公司面临较高的风险，这可能会对股票价格和回报产生负面影响。

现金流是指公司从经营活动中产生的现金流入和流出情况。

良好的现金流有助于提高公司的稳定性和回报。

第三级：技术因素技术因素指的是基于股票价格和交易量的技术分析方法对股票价格和回报进行预测和分析。

技术因素包括股票的趋势、交易信号和移动平均线等指标。

趋势分析是通过观察股票价格的变化趋势来预测未来的股票价格走势。

趋势分析可以帮助投资者判断股票价格的上升或下降趋势。

交易信号是通过股票价格和交易量的变化来触发买或卖的信号。

交易信号可以帮助投资者做出买入或卖出决策。

移动平均线是一种平滑股票价格的指标，可以帮助投资者识别趋势和价格的转折点。

从乘数-加速数原理看2008年美国经济危

３７７《商场现代化》２００９年３月（上旬刊）总第５６８期２００８年，由美国次贷危机引起的世界性金融海啸深受全世界关注，广大的经济学家竞相以自己的学识从各个方面对这次危机做出了各种分析解释。

而本文将从经济周期的角度，利用乘数—加速数模型对这次危机进行一定的分析。

一、一些经济周期理论对此次危机的解释２００８年９月１５日，美国第四大投资银行雷曼兄弟公司宣布申请破产保护，标志着这场席卷全球的经济危机全面爆发。

时至今日，经济学者们总是试图用各种理论来对这次危机进行分析，随着“这次美国经济危机是经济周期的一部分”这一认识已经越来越毋庸置疑，从经济周期理论出发对这次危机的分析也随之出现了，以下就是其中的一些：１．政治经济周期理论的解释政治经济周期理论将经济周期归因于政府政策，以及政治家为重新当选而对财政政策和货币政策操纵等等，认为格林斯潘时代的美联储应该对造成当前信贷市场的泡沫负责，谴责美联储在２００２年至２００６年间采取了过分宽松的政策。

２．货币主义经济周期理论的解释货币主义理论认为经济周期纯粹是一种货币现象，是由于银行体系周期性的扩张和紧缩信用造成的。

２０００年后美联储连续降息、扩张信贷规模催生了次贷，而在２００３年后的升息就促使了次贷危机的形成。

３．长波理论的解释长波又叫长周期或康德拉季耶夫周期，主要指资本主义经济增长过程中长达５０年～６０年的明显规则性波动。

根据近一个半世纪以来世界主要国家平均利润率的变化情况可以发现，在２０世界８０年代初以来的这个长波中，不管是从经济增长速度还是利润率来看，现如今的美国经济都已经越过了本次长波的顶峰，进入了下降时期。

由于经济周期理论的多样性，利用经济周期理论对此次危机的解释也多种多样，各有千秋。

在本文接下来的部分，将从凯恩斯主义的乘数—加速数原理出发，对这次经济危机进行分析。

二、乘数—加速数原理简介乘数—加速数原理是基于凯恩斯主义的乘数原理，结合加速数原理，以说明经济周期的原因的理论。

3 level carhart style model -回复

3 level carhart style model -回复什么是3级Carhartt风格模型？Carhartt风格模型是一个模型框架，用于解释和预测经济现象和宏观经济变量之间的关系。

它以美国经济学家Thomas Carhartt 的名字命名，是一个被广泛使用的经济学工具。

该模型的最大特点是将经济领域划分为三个层次：宏观经济层面、产业层面和公司层面。

这三个层次相互作用，并对经济运行产生影响。

下面将详细介绍这三个层次及其相互关系。

1. 宏观经济层面：宏观经济层面是模型的第一层，对整个经济系统进行宏观分析。

在这个层面，我们考虑国家或地区的整体经济变量，如国内生产总值（GDP）、物价指数、消费支出、投资等。

宏观经济政策如货币政策和财政政策的变化对整个经济体的发展有着重大影响。

在Carhartt风格模型中，宏观经济层面提供了基本的经济环境，这些环境将影响下一层次：产业层面。

2. 产业层面：产业层面是模型的第二层，考虑不同产业之间的相互作用和产业对经济的影响。

在这个层面，我们分析不同产业的生产、销售和就业情况。

每个产业都有其特定的市场结构和竞争力，其行为对经济体产生影响。

例如，一个动荡的能源市场可能导致油价上涨，从而影响到其他与能源相关的产业。

此外，技术创新和市场竞争也会对产业结构和发展产生影响。

在Carhartt 风格模型中，产业层面的变化将影响到下一层次：公司层面。

3. 公司层面：公司层面是模型的第三层，涉及个别公司的经营和决策。

在这个层面，我们研究企业的生产过程、市场竞争、利润和就业等。

不同公司在其所属产业中扮演着不同的角色，并通过他们的经营决策影响着经济体的发展。

公司的投资、生产能力、市场份额和技术创新等都会对经济体产生影响。

在Carhartt风格模型中，公司层面的变化将反过来影响到产业层面和宏观经济层面。

这三个层次相互关联，构成了一个完整的模型，以解释经济领域中的变化和相互作用。

例如，当宏观经济政策发生变化时，比如货币政策有所调整，这将引起产业层面的变化，影响公司的经营决策和市场竞争。

3 level carhart style model -回复

3 level carhart style model -回复什么是3级Carhart风格模型？3级Carhart风格模型是用于分析股票投资组合的金融模型。

它主要通过评估投资组合的风险、收益和相关性，来为投资者提供决策建议。

该模型以马科维茨新兴资产组合理论（Modern Portfolio Theory，简称MPT）为基础，为投资者提供了一种科学且系统的方法来构建股票投资组合。

第一级：收益率模型在3级Carhart风格模型中，首先需要建立一个基于收益率的模型。

收益率模型可以用来度量投资组合中不同股票或资产的预期收益。

它主要基于历史数据分析，使用数学和统计模型来预测未来收益。

为了建立收益率模型，首先需要收集和整理足够的历史数据。

这些数据可以包括股票价格、交易量等信息。

然后，可以利用回归分析技术来确定各个因素对投资组合收益的影响程度。

例如，可以考虑宏观经济因素、行业因素和公司特定因素等。

第二级：风险模型在3级Carhart风格模型中，风险模型用于评估投资组合的风险。

风险主要分为系统性风险和非系统性风险。

系统性风险是所有股票所面临的共同风险，而非系统性风险则是各个股票独有的风险。

为了评估投资组合的系统性风险，可以使用Beta系数。

Beta系数被认为是一个股票与整个市场的相关性度量。

它可以帮助投资者了解一个股票在市场上的表现，并预测其未来可能的波动。

对于非系统性风险，可以使用特异风险指标，如波动率或标准差来度量。

这些指标可以帮助投资者了解个别股票的风险特征，并避免投资太多的同类风险股票。

第三级：相关性模型在3级Carhart风格模型中，相关性模型用于评估投资组合中各个股票之间的相关性。

相关性可以帮助投资者了解股票之间的关系，并帮助他们构建一个具有多样性的投资组合，以降低整体风险。

为了评估股票之间的相关性，可以使用相关系数。

相关系数是一个度量两个变量之间关联关系的统计指标。

它可以帮助投资者了解股票之间是正相关、负相关还是没有相关性。

Decay Constants $f_{D_s^}$ and $f_{D_s}$ from ${bar{B}}^0to D^+ l^- {bar{nu}}$ and ${bar{B}

$Decay Constants $f_{D_s^}$ and $f_{D_s}$ from ${bar{B}}^0to D^+ l^- {bar{nu}}$ and ${bar{B}$

∗ = 336 ± 79 M eV , and fD ∗ /fD we predict fDs s = 1.41 ± 0.41 for (pole/constant)-type s
form factor.
PACS index : 12.15.-y, 13.20.-v, 13.25.Hw, 14.40.Nd, 14.65.Fy Keywards : Factorization, Non-leptonic Decays, Decay Constant, Penguin Eﬀects
∗ experimentally from leptonic B and Ds decays. For instance, determine fB , fBs fDs and fDs
+ the decay rate for Ds is given by [1]
+ Γ(Ds
m2 G2 2 2 l 1 − m M → ℓ ν ) = F fD D s 2 8π s ℓ MD s
1/2
(4)
.
(5)
In the zero lepton-mass limit, 0 ≤ q 2 ≤ (mB − mD )2 .
2
For the q 2 dependence of the form factors, Wirbel et al. [8] assumed a simple pole formula for both F1 (q 2 ) and F0 (q 2 ) (we designate this scenario ’pole/pole’): q2 F1 (q ) = F1 (0) /(1 − 2 ), mF1
∗ amount to about 11 % for B → DDs and 5 % for B → DDs , which have been mentioned in

交通流

Network impacts of a road capacity reduction:Empirical analysisand model predictionsDavid Watling a ,⇑,David Milne a ,Stephen Clark baInstitute for Transport Studies,University of Leeds,Woodhouse Lane,Leeds LS29JT,UK b Leeds City Council,Leonardo Building,2Rossington Street,Leeds LS28HD,UKa r t i c l e i n f o Article history:Received 24May 2010Received in revised form 15July 2011Accepted 7September 2011Keywords:Trafﬁc assignment Network models Equilibrium Route choice Day-to-day variabilitya b s t r a c tIn spite of their widespread use in policy design and evaluation,relatively little evidencehas been reported on how well trafﬁc equilibrium models predict real network impacts.Here we present what we believe to be the ﬁrst paper that together analyses the explicitimpacts on observed route choice of an actual network intervention and compares thiswith the before-and-after predictions of a network equilibrium model.The analysis isbased on the ﬁndings of an empirical study of the travel time and route choice impactsof a road capacity reduction.Time-stamped,partial licence plates were recorded across aseries of locations,over a period of days both with and without the capacity reduction,and the data were ‘matched’between locations using special-purpose statistical methods.Hypothesis tests were used to identify statistically signiﬁcant changes in travel times androute choice,between the periods of days with and without the capacity reduction.A trafﬁcnetwork equilibrium model was then independently applied to the same scenarios,and itspredictions compared with the empirical ﬁndings.From a comparison of route choice pat-terns,a particularly inﬂuential spatial effect was revealed of the parameter specifying therelative values of distance and travel time assumed in the generalised cost equations.When this parameter was ‘ﬁtted’to the data without the capacity reduction,the networkmodel broadly predicted the route choice impacts of the capacity reduction,but with othervalues it was seen to perform poorly.The paper concludes by discussing the wider practicaland research implications of the study’s ﬁndings.Ó2011Elsevier Ltd.All rights reserved.1.IntroductionIt is well known that altering the localised characteristics of a road network,such as a planned change in road capacity,will tend to have both direct and indirect effects.The direct effects are imparted on the road itself,in terms of how it can deal with a given demand ﬂow entering the link,with an impact on travel times to traverse the link at a given demand ﬂow level.The indirect effects arise due to drivers changing their travel decisions,such as choice of route,in response to the altered travel times.There are many practical circumstances in which it is desirable to forecast these direct and indirect impacts in the context of a systematic change in road capacity.For example,in the case of proposed road widening or junction improvements,there is typically a need to justify econom-ically the required investment in terms of the beneﬁts that will likely accrue.There are also several examples in which it is relevant to examine the impacts of road capacity reduction .For example,if one proposes to reallocate road space between alternative modes,such as increased bus and cycle lane provision or a pedestrianisation scheme,then typically a range of alternative designs exist which may differ in their ability to accommodate efﬁciently the new trafﬁc and routing patterns.0965-8564/$-see front matter Ó2011Elsevier Ltd.All rights reserved.doi:10.1016/j.tra.2011.09.010⇑Corresponding author.Tel.:+441133436612;fax:+441133435334.E-mail address:d.p.watling@ (D.Watling).168 D.Watling et al./Transportation Research Part A46(2012)167–189Through mathematical modelling,the alternative designs may be tested in a simulated environment and the most efﬁcient selected for implementation.Even after a particular design is selected,mathematical models may be used to adjust signal timings to optimise the use of the transport system.Road capacity may also be affected periodically by maintenance to essential services(e.g.water,electricity)or to the road itself,and often this can lead to restricted access over a period of days and weeks.In such cases,planning authorities may use modelling to devise suitable diversionary advice for drivers,and to plan any temporary changes to trafﬁc signals or priorities.Berdica(2002)and Taylor et al.(2006)suggest more of a pro-ac-tive approach,proposing that models should be used to test networks for potential vulnerability,before any reduction mate-rialises,identifying links which if reduced in capacity over an extended period1would have a substantial impact on system performance.There are therefore practical requirements for a suitable network model of travel time and route choice impacts of capac-ity changes.The dominant method that has emerged for this purpose over the last decades is clearly the network equilibrium approach,as proposed by Beckmann et al.(1956)and developed in several directions since.The basis of using this approach is the proposition of what are believed to be‘rational’models of behaviour and other system components(e.g.link perfor-mance functions),with site-speciﬁc data used to tailor such models to particular case studies.Cross-sectional forecasts of network performance at speciﬁc road capacity states may then be made,such that at the time of any‘snapshot’forecast, drivers’route choices are in some kind of individually-optimum state.In this state,drivers cannot improve their route selec-tion by a unilateral change of route,at the snapshot travel time levels.The accepted practice is to‘validate’such models on a case-by-case basis,by ensuring that the model—when supplied with a particular set of parameters,input network data and input origin–destination demand data—reproduces current mea-sured mean link trafﬁcﬂows and mean journey times,on a sample of links,to some degree of accuracy(see for example,the practical guidelines in TMIP(1997)and Highways Agency(2002)).This kind of aggregate level,cross-sectional validation to existing conditions persists across a range of network modelling paradigms,ranging from static and dynamic equilibrium (Florian and Nguyen,1976;Leonard and Tough,1979;Stephenson and Teply,1984;Matzoros et al.,1987;Janson et al., 1986;Janson,1991)to micro-simulation approaches(Laird et al.,1999;Ben-Akiva et al.,2000;Keenan,2005).While such an approach is plausible,it leaves many questions unanswered,and we would particularly highlight two: 1.The process of calibration and validation of a network equilibrium model may typically occur in a cycle.That is to say,having initially calibrated a model using the base data sources,if the subsequent validation reveals substantial discrep-ancies in some part of the network,it is then natural to adjust the model parameters(including perhaps even the OD matrix elements)until the model outputs better reﬂect the validation data.2In this process,then,we allow the adjustment of potentially a large number of network parameters and input data in order to replicate the validation data,yet these data themselves are highly aggregate,existing only at the link level.To be clear here,we are talking about a level of coarseness even greater than that in aggregate choice models,since we cannot even infer from link-level data the aggregate shares on alternative routes or OD movements.The question that arises is then:how many different combinations of parameters and input data values might lead to a similar link-level validation,and even if we knew the answer to this question,how might we choose between these alternative combinations?In practice,this issue is typically neglected,meaning that the‘valida-tion’is a rather weak test of the model.2.Since the data are cross-sectional in time(i.e.the aim is to reproduce current base conditions in equilibrium),then in spiteof the large efforts required in data collection,no empirical evidence is routinely collected regarding the model’s main purpose,namely its ability to predict changes in behaviour and network performance under changes to the network/ demand.This issue is exacerbated by the aggregation concerns in point1:the‘ambiguity’in choosing appropriate param-eter values to satisfy the aggregate,link-level,base validation strengthens the need to independently verify that,with the selected parameter values,the model responds reliably to changes.Although such problems–ofﬁtting equilibrium models to cross-sectional data–have long been recognised by practitioners and academics(see,e.g.,Goodwin,1998), the approach described above remains the state-of-practice.Having identiﬁed these two problems,how might we go about addressing them?One approach to theﬁrst problem would be to return to the underlying formulation of the network model,and instead require a model deﬁnition that permits analysis by statistical inference techniques(see for example,Nakayama et al.,2009).In this way,we may potentially exploit more information in the variability of the link-level data,with well-deﬁned notions(such as maximum likelihood)allowing a systematic basis for selection between alternative parameter value combinations.However,this approach is still using rather limited data and it is natural not just to question the model but also the data that we use to calibrate and validate it.Yet this is not altogether straightforward to resolve.As Mahmassani and Jou(2000) remarked:‘A major difﬁculty...is obtaining observations of actual trip-maker behaviour,at the desired level of richness, simultaneously with measurements of prevailing conditions’.For this reason,several authors have turned to simulated gaming environments and/or stated preference techniques to elicit information on drivers’route choice behaviour(e.g. 1Clearly,more sporadic and less predictable reductions in capacity may also occur,such as in the case of breakdowns and accidents,and environmental factors such as severe weather,ﬂoods or landslides(see for example,Iida,1999),but the responses to such cases are outside the scope of the present paper. 2Some authors have suggested more systematic,bi-level type optimization processes for thisﬁtting process(e.g.Xu et al.,2004),but this has no material effect on the essential points above.D.Watling et al./Transportation Research Part A46(2012)167–189169 Mahmassani and Herman,1990;Iida et al.,1992;Khattak et al.,1993;Vaughn et al.,1995;Wardman et al.,1997;Jou,2001; Chen et al.,2001).This provides potentially rich information for calibrating complex behavioural models,but has the obvious limitation that it is based on imagined rather than real route choice situations.Aside from its common focus on hypothetical decision situations,this latter body of work also signiﬁes a subtle change of emphasis in the treatment of the overall network calibration problem.Rather than viewing the network equilibrium calibra-tion process as a whole,the focus is on particular components of the model;in the cases above,the focus is on that compo-nent concerned with how drivers make route decisions.If we are prepared to make such a component-wise analysis,then certainly there exists abundant empirical evidence in the literature,with a history across a number of decades of research into issues such as the factors affecting drivers’route choice(e.g.Wachs,1967;Huchingson et al.,1977;Abu-Eisheh and Mannering,1987;Duffell and Kalombaris,1988;Antonisse et al.,1989;Bekhor et al.,2002;Liu et al.,2004),the nature of travel time variability(e.g.Smeed and Jeffcoate,1971;Montgomery and May,1987;May et al.,1989;McLeod et al., 1993),and the factors affecting trafﬁcﬂow variability(Bonsall et al.,1984;Huff and Hanson,1986;Ribeiro,1994;Rakha and Van Aerde,1995;Fox et al.,1998).While these works provide useful evidence for the network equilibrium calibration problem,they do not provide a frame-work in which we can judge the overall‘ﬁt’of a particular network model in the light of uncertainty,ambient variation and systematic changes in network attributes,be they related to the OD demand,the route choice process,travel times or the network data.Moreover,such data does nothing to address the second point made above,namely the question of how to validate the model forecasts under systematic changes to its inputs.The studies of Mannering et al.(1994)and Emmerink et al.(1996)are distinctive in this context in that they address some of the empirical concerns expressed in the context of travel information impacts,but their work stops at the stage of the empirical analysis,without a link being made to net-work prediction models.The focus of the present paper therefore is both to present theﬁndings of an empirical study and to link this empirical evidence to network forecasting models.More recently,Zhu et al.(2010)analysed several sources of data for evidence of the trafﬁc and behavioural impacts of the I-35W bridge collapse in Minneapolis.Most pertinent to the present paper is their location-speciﬁc analysis of linkﬂows at 24locations;by computing the root mean square difference inﬂows between successive weeks,and comparing the trend for 2006with that for2007(the latter with the bridge collapse),they observed an apparent transient impact of the bridge col-lapse.They also showed there was no statistically-signiﬁcant evidence of a difference in the pattern ofﬂows in the period September–November2007(a period starting6weeks after the bridge collapse),when compared with the corresponding period in2006.They suggested that this was indicative of the length of a‘re-equilibration process’in a conceptual sense, though did not explicitly compare their empiricalﬁndings with those of a network equilibrium model.The structure of the remainder of the paper is as follows.In Section2we describe the process of selecting the real-life problem to analyse,together with the details and rationale behind the survey design.Following this,Section3describes the statistical techniques used to extract information on travel times and routing patterns from the survey data.Statistical inference is then considered in Section4,with the aim of detecting statistically signiﬁcant explanatory factors.In Section5 comparisons are made between the observed network data and those predicted by a network equilibrium model.Finally,in Section6the conclusions of the study are highlighted,and recommendations made for both practice and future research.2.Experimental designThe ultimate objective of the study was to compare actual data with the output of a trafﬁc network equilibrium model, speciﬁcally in terms of how well the equilibrium model was able to correctly forecast the impact of a systematic change ap-plied to the network.While a wealth of surveillance data on linkﬂows and travel times is routinely collected by many local and national agencies,we did not believe that such data would be sufﬁciently informative for our purposes.The reason is that while such data can often be disaggregated down to small time step resolutions,the data remains aggregate in terms of what it informs about driver response,since it does not provide the opportunity to explicitly trace vehicles(even in aggre-gate form)across more than one location.This has the effect that observed differences in linkﬂows might be attributed to many potential causes:it is especially difﬁcult to separate out,say,ambient daily variation in the trip demand matrix from systematic changes in route choice,since both may give rise to similar impacts on observed linkﬂow patterns across re-corded sites.While methods do exist for reconstructing OD and network route patterns from observed link data(e.g.Yang et al.,1994),these are typically based on the premise of a valid network equilibrium model:in this case then,the data would not be able to give independent information on the validity of the network equilibrium approach.For these reasons it was decided to design and implement a purpose-built survey.However,it would not be efﬁcient to extensively monitor a network in order to wait for something to happen,and therefore we required advance notiﬁcation of some planned intervention.For this reason we chose to study the impact of urban maintenance work affecting the roads,which UK local government authorities organise on an annual basis as part of their‘Local Transport Plan’.The city council of York,a historic city in the north of England,agreed to inform us of their plans and to assist in the subsequent data collection exercise.Based on the interventions planned by York CC,the list of candidate studies was narrowed by considering factors such as its propensity to induce signiﬁcant re-routing and its impact on the peak periods.Effectively the motivation here was to identify interventions that were likely to have a large impact on delays,since route choice impacts would then likely be more signiﬁcant and more easily distinguished from ambient variability.This was notably at odds with the objectives of York CC,170 D.Watling et al./Transportation Research Part A46(2012)167–189in that they wished to minimise disruption,and so where possible York CC planned interventions to take place at times of day and of the year where impacts were minimised;therefore our own requirement greatly reduced the candidate set of studies to monitor.A further consideration in study selection was its timing in the year for scheduling before/after surveys so to avoid confounding effects of known signiﬁcant‘seasonal’demand changes,e.g.the impact of the change between school semesters and holidays.A further consideration was York’s role as a major tourist attraction,which is also known to have a seasonal trend.However,the impact on car trafﬁc is relatively small due to the strong promotion of public trans-port and restrictions on car travel and parking in the historic centre.We felt that we further mitigated such impacts by sub-sequently choosing to survey in the morning peak,at a time before most tourist attractions are open.Aside from the question of which intervention to survey was the issue of what data to collect.Within the resources of the project,we considered several options.We rejected stated preference survey methods as,although they provide a link to personal/socio-economic drivers,we wanted to compare actual behaviour with a network model;if the stated preference data conﬂicted with the network model,it would not be clear which we should question most.For revealed preference data, options considered included(i)self-completion diaries(Mahmassani and Jou,2000),(ii)automatic tracking through GPS(Jan et al.,2000;Quiroga et al.,2000;Taylor et al.,2000),and(iii)licence plate surveys(Schaefer,1988).Regarding self-comple-tion surveys,from our own interview experiments with self-completion questionnaires it was evident that travellersﬁnd it relatively difﬁcult to recall and describe complex choice options such as a route through an urban network,giving the po-tential for signiﬁcant errors to be introduced.The automatic tracking option was believed to be the most attractive in this respect,in its potential to accurately map a given individual’s journey,but the negative side would be the potential sample size,as we would need to purchase/hire and distribute the devices;even with a large budget,it is not straightforward to identify in advance the target users,nor to guarantee their cooperation.Licence plate surveys,it was believed,offered the potential for compromise between sample size and data resolution: while we could not track routes to the same resolution as GPS,by judicious location of surveyors we had the opportunity to track vehicles across more than one location,thus providing route-like information.With time-stamped licence plates, the matched data would also provide journey time information.The negative side of this approach is the well-known poten-tial for signiﬁcant recording errors if large sample rates are required.Our aim was to avoid this by recording only partial licence plates,and employing statistical methods to remove the impact of‘spurious matches’,i.e.where two different vehi-cles with the same partial licence plate occur at different locations.Moreover,extensive simulation experiments(Watling,1994)had previously shown that these latter statistical methods were effective in recovering the underlying movements and travel times,even if only a relatively small part of the licence plate were recorded,in spite of giving a large potential for spurious matching.We believed that such an approach reduced the opportunity for recorder error to such a level to suggest that a100%sample rate of vehicles passing may be feasible.This was tested in a pilot study conducted by the project team,with dictaphones used to record a100%sample of time-stamped, partial licence plates.Independent,duplicate observers were employed at the same location to compare error rates;the same study was also conducted with full licence plates.The study indicated that100%surveys with dictaphones would be feasible in moderate trafﬁcﬂow,but only if partial licence plate data were used in order to control observation errors; for higherﬂow rates or to obtain full number plate data,video surveys should be considered.Other important practical les-sons learned from the pilot included the need for clarity in terms of vehicle types to survey(e.g.whether to include motor-cycles and taxis),and of the phonetic alphabet used by surveyors to avoid transcription ambiguities.Based on the twin considerations above of planned interventions and survey approach,several candidate studies were identiﬁed.For a candidate study,detailed design issues involved identifying:likely affected movements and alternative routes(using local knowledge of York CC,together with an existing network model of the city),in order to determine the number and location of survey sites;feasible viewpoints,based on site visits;the timing of surveys,e.g.visibility issues in the dark,winter evening peak period;the peak duration from automatic trafﬁcﬂow data;and speciﬁc survey days,in view of public/school holidays.Our budget led us to survey the majority of licence plate sites manually(partial plates by audio-tape or,in lowﬂows,pen and paper),with video surveys limited to a small number of high-ﬂow sites.From this combination of techniques,100%sampling rate was feasible at each site.Surveys took place in the morning peak due both to visibility considerations and to minimise conﬂicts with tourist/special event trafﬁc.From automatic trafﬁc count data it was decided to survey the period7:45–9:15as the main morning peak period.This design process led to the identiﬁcation of two studies:2.1.Lendal Bridge study(Fig.1)Lendal Bridge,a critical part of York’s inner ring road,was scheduled to be closed for maintenance from September2000 for a duration of several weeks.To avoid school holidays,the‘before’surveys were scheduled for June and early September.It was decided to focus on investigating a signiﬁcant southwest-to-northeast movement of trafﬁc,the river providing a natural barrier which suggested surveying the six river crossing points(C,J,H,K,L,M in Fig.1).In total,13locations were identiﬁed for survey,in an attempt to capture trafﬁc on both sides of the river as well as a crossing.2.2.Fishergate study(Fig.2)The partial closure(capacity reduction)of the street known as Fishergate,again part of York’s inner ring road,was scheduled for July2001to allow repairs to a collapsed sewer.Survey locations were chosen in order to intercept clockwiseFig.1.Intervention and survey locations for Lendal Bridge study.around the inner ring road,this being the direction of the partial closure.A particular aim wasFulford Road(site E in Fig.2),the main radial affected,with F and K monitoring local diversion I,J to capture wider-area diversion.studies,the plan was to survey the selected locations in the morning peak over a period of approximately covering the three periods before,during and after the intervention,with the days selected so holidays or special events.Fig.2.Intervention and survey locations for Fishergate study.In the Lendal Bridge study,while the‘before’surveys proceeded as planned,the bridge’s actualﬁrst day of closure on Sep-tember11th2000also marked the beginning of the UK fuel protests(BBC,2000a;Lyons and Chaterjee,2002).Trafﬁcﬂows were considerably affected by the scarcity of fuel,with congestion extremely low in theﬁrst week of closure,to the extent that any changes could not be attributed to the bridge closure;neither had our design anticipated how to survey the impacts of the fuel shortages.We thus re-arranged our surveys to monitor more closely the planned re-opening of the bridge.Unfor-tunately these surveys were hampered by a second unanticipated event,namely the wettest autumn in the UK for270years and the highest level ofﬂooding in York since records began(BBC,2000b).Theﬂooding closed much of the centre of York to road trafﬁc,including our study area,as the roads were impassable,and therefore we abandoned the planned‘after’surveys. As a result of these events,the useable data we had(not affected by the fuel protests orﬂooding)consisted ofﬁve‘before’days and one‘during’day.In the Fishergate study,fortunately no extreme events occurred,allowing six‘before’and seven‘during’days to be sur-veyed,together with one additional day in the‘during’period when the works were temporarily removed.However,the works over-ran into the long summer school holidays,when it is well-known that there is a substantial seasonal effect of much lowerﬂows and congestion levels.We did not believe it possible to meaningfully isolate the impact of the link fully re-opening while controlling for such an effect,and so our plans for‘after re-opening’surveys were abandoned.3.Estimation of vehicle movements and travel timesThe data resulting from the surveys described in Section2is in the form of(for each day and each study)a set of time-stamped,partial licence plates,observed at a number of locations across the network.Since the data include only partial plates,they cannot simply be matched across observation points to yield reliable estimates of vehicle movements,since there is ambiguity in whether the same partial plate observed at different locations was truly caused by the same vehicle. Indeed,since the observed system is‘open’—in the sense that not all points of entry,exit,generation and attraction are mon-itored—the question is not just which of several potential matches to accept,but also whether there is any match at all.That is to say,an apparent match between data at two observation points could be caused by two separate vehicles that passed no other observation point.Theﬁrst stage of analysis therefore applied a series of specially-designed statistical techniques to reconstruct the vehicle movements and point-to-point travel time distributions from the observed data,allowing for all such ambiguities in the data.Although the detailed derivations of each method are not given here,since they may be found in the references provided,it is necessary to understand some of the characteristics of each method in order to interpret the results subsequently provided.Furthermore,since some of the basic techniques required modiﬁcation relative to the published descriptions,then in order to explain these adaptations it is necessary to understand some of the theoretical basis.3.1.Graphical method for estimating point-to-point travel time distributionsThe preliminary technique applied to each data set was the graphical method described in Watling and Maher(1988).This method is derived for analysing partial registration plate data for unidirectional movement between a pair of observation stations(referred to as an‘origin’and a‘destination’).Thus in the data study here,it must be independently applied to given pairs of observation stations,without regard for the interdependencies between observation station pairs.On the other hand, it makes no assumption that the system is‘closed’;there may be vehicles that pass the origin that do not pass the destina-tion,and vice versa.While limited in considering only two-point surveys,the attraction of the graphical technique is that it is a non-parametric method,with no assumptions made about the arrival time distributions at the observation points(they may be non-uniform in particular),and no assumptions made about the journey time probability density.It is therefore very suitable as aﬁrst means of investigative analysis for such data.The method begins by forming all pairs of possible matches in the data,of which some will be genuine matches(the pair of observations were due to a single vehicle)and the remainder spurious matches.Thus, for example,if there are three origin observations and two destination observations of a particular partial registration num-ber,then six possible matches may be formed,of which clearly no more than two can be genuine(and possibly only one or zero are genuine).A scatter plot may then be drawn for each possible match of the observation time at the origin versus that at the destination.The characteristic pattern of such a plot is as that shown in Fig.4a,with a dense‘line’of points(which will primarily be the genuine matches)superimposed upon a scatter of points over the whole region(which will primarily be the spurious matches).If we were to assume uniform arrival rates at the observation stations,then the spurious matches would be uniformly distributed over this plot;however,we shall avoid making such a restrictive assumption.The method begins by making a coarse estimate of the total number of genuine matches across the whole of this plot.As part of this analysis we then assume knowledge of,for any randomly selected vehicle,the probabilities:h k¼Prðvehicle is of the k th type of partial registration plateÞðk¼1;2;...;mÞwhereX m k¼1h k¼1172 D.Watling et al./Transportation Research Part A46(2012)167–189。

1999.Multilevel Hypergraph Partitioning__Applications in VLSI Domain

Multilevel Hypergraph Partitioning:Applications in VLSI DomainGeorge Karypis,Rajat Aggarwal,Vipin Kumar,Senior Member,IEEE,and Shashi Shekhar,Senior Member,IEEE Abstract—In this paper,we present a new hypergraph-partitioning algorithm that is based on the multilevel paradigm.In the multilevel paradigm,a sequence of successivelycoarser hypergraphs is constructed.A bisection of the smallesthypergraph is computed and it is used to obtain a bisection of theoriginal hypergraph by successively projecting and reﬁning thebisection to the next levelﬁner hypergraph.We have developednew hypergraph coarsening strategies within the multilevelframework.We evaluate their performance both in terms of thesize of the hyperedge cut on the bisection,as well as on the runtime for a number of very large scale integration circuits.Ourexperiments show that our multilevel hypergraph-partitioningalgorithm produces high-quality partitioning in a relatively smallamount of time.The quality of the partitionings produced by ourscheme are on the average6%–23%better than those producedby other state-of-the-art schemes.Furthermore,our partitioningalgorithm is signiﬁcantly faster,often requiring4–10times lesstime than that required by the other schemes.Our multilevelhypergraph-partitioning algorithm scales very well for largehypergraphs.Hypergraphs with over100000vertices can bebisected in a few minutes on today’s workstations.Also,on thelarge hypergraphs,our scheme outperforms other schemes(inhyperedge cut)quite consistently with larger margins(9%–30%).Index Terms—Circuit partitioning,hypergraph partitioning,multilevel algorithms.I.I NTRODUCTIONH YPERGRAPH partitioning is an important problem withextensive application to many areas,including very largescale integration(VLSI)design[1],efﬁcient storage of largedatabases on disks[2],and data mining[3].The problemis to partition the vertices of a hypergraphintois deﬁnedas a set ofvertices[4],and the size ofa hyperedge is the cardinality of this subset.Manuscript received April29,1997;revised March23,1998.This workwas supported under IBM Partnership Award NSF CCR-9423082,by theArmy Research Ofﬁce under Contract DA/DAAH04-95-1-0538,and by theArmy High Performance Computing Research Center,the Department of theArmy,Army Research Laboratory Cooperative Agreement DAAH04-95-2-0003/Contract DAAH04-95-C-0008.G.Karypis,V.Kumar,and S.Shekhar are with the Department of ComputerScience and Engineering,Minneapolis,University of Minnesota,Minneapolis,MN55455-0159USA.R.Aggarwal is with the Lattice Semiconductor Corporation,Milpitas,CA95131USA.Publisher Item Identiﬁer S1063-8210(99)00695-2.During the course of VLSI circuit design and synthesis,itis important to be able to divide the system speciﬁcation intoclusters so that the inter-cluster connections are minimized.This step has many applications including design packaging,HDL-based synthesis,design optimization,rapid prototyping,simulation,and testing.In particular,many rapid prototyp-ing systems use partitioning to map a complex circuit ontohundreds of interconnectedﬁeld-programmable gate arrays(FPGA’s).Such partitioning instances are challenging becausethe timing,area,and input/output(I/O)resource utilizationmust satisfy hard device-speciﬁc constraints.For example,ifthe number of signal nets leaving any one of the clustersis greater than the number of signal p-i-n’s available in theFPGA,then this cluster cannot be implemented using a singleFPGA.In this case,the circuit needs to be further partitioned,and thus implemented using multiple FPGA’s.Hypergraphscan be used to naturally represent a VLSI circuit.The verticesof the hypergraph can be used to represent the cells of thecircuit,and the hyperedges can be used to represent the netsconnecting these cells.A high quality hypergraph-partitioningalgorithm greatly affects the feasibility,quality,and cost ofthe resulting system.A.Related WorkThe problem of computing an optimal bisection of a hy-pergraph is at least NP-hard[5].However,because of theimportance of the problem in many application areas,manyheuristic algorithms have been developed.The survey byAlpert and Khang[1]provides a detailed description andcomparison of such various schemes.In a widely used class ofiterative reﬁnement partitioning algorithms,an initial bisectionis computed(often obtained randomly)and then the partitionis reﬁned by repeatedly moving vertices between the twoparts to reduce the hyperedge cut.These algorithms oftenuse the Schweikert–Kernighan heuristic[6](an extension ofthe Kernighan–Lin(KL)heuristic[7]for hypergraphs),or thefaster Fiduccia–Mattheyses(FM)[8]reﬁnement heuristic,toiteratively improve the quality of the partition.In all of thesemethods(sometimes also called KLFM schemes),a vertex ismoved(or a vertex pair is swapped)if it produces the greatestreduction in the edge cuts,which is also called the gain formoving the vertex.The partition produced by these methodsis often poor,especially for larger hypergraphs.Hence,thesealgorithms have been extended in a number of ways[9]–[12].Krishnamurthy[9]tried to introduce intelligence in the tie-breaking process from among the many possible moves withthe same high gain.He used a Look Ahead()algorithm,which looks ahead uptoa move.PROP [11],introduced by Dutt and Deng,used a probabilistic gain computation model for deciding which vertices need to move across the partition line.These schemes tend to enhance the performance of the basic KLFM family of reﬁnement algorithms,at the expense of increased run time.Dutt and Deng [12]proposed two new methods,namely,CLIP and CDIP,for computing the gains of hyperedges that contain more than one node on either side of the partition boundary.CDIP in conjunctionwithand CLIP in conjunction with PROP are two schemes that have shown the best results in their experiments.Another class of hypergraph-partitioning algorithms [13]–[16]performs partitioning in two phases.In the ﬁrst phase,the hypergraph is coarsened to form a small hypergraph,and then the FM algorithm is used to bisect the small hypergraph.In the second phase,these algorithms use the bisection of this contracted hypergraph to obtain a bisection of the original hypergraph.Since FM reﬁnement is done only on the small coarse hypergraph,this step is usually fast,but the overall performance of such a scheme depends upon the quality of the coarsening method.In many schemes,the projected partition is further improved using the FM reﬁnement scheme [15].Recently,a new class of partitioning algorithms was devel-oped [17]–[20]based upon the multilevel paradigm.In these algorithms,a sequence of successively smaller (coarser)graphs is constructed.A bisection of the smallest graph is computed.This bisection is now successively projected to the next-level ﬁner graph and,at each level,an iterative reﬁnement algorithm such as KLFM is used to further improve the bisection.The various phases of multilevel bisection are illustrated in Fig.1.Iterative reﬁnement schemes such as KLFM become quite powerful in this multilevel context for the following reason.First,the movement of a single node across a partition bound-ary in a coarse graph can lead to the movement of a large num-ber of related nodes in the original graph.Second,the reﬁned partitioning projected to the next level serves as an excellent initial partitioning for the KL or FM reﬁnement algorithms.This paradigm was independently studied by Bui and Jones [17]in the context of computing ﬁll-reducing matrix reorder-ing,by Hendrickson and Leland [18]in the context of ﬁnite-element mesh-partitioning,and by Hauck and Borriello (called Optimized KLFM)[20],and by Cong and Smith [19]for hy-pergraph partitioning.Karypis and Kumar extensively studied this paradigm in [21]and [22]for the partitioning of graphs.They presented new graph coarsening schemes for which even a good bisection of the coarsest graph is a pretty good bisec-tion of the original graph.This makes the overall multilevel paradigm even more robust.Furthermore,it allows the use of simpliﬁed variants of KLFM reﬁnement schemes during the uncoarsening phase,which signiﬁcantly speeds up the reﬁne-ment process without compromising overall quality.METIS [21],a multilevel graph partitioning algorithm based upon this work,routinely ﬁnds substantially better bisections and is often two orders of magnitude faster than the hitherto state-of-the-art spectral-based bisection techniques [23],[24]for graphs.The improved coarsening schemes of METIS work only for graphs and are not directly applicable to hypergraphs.IftheFig.1.The various phases of the multilevel graph bisection.During the coarsening phase,the size of the graph is successively decreased;during the initial partitioning phase,a bisection of the smaller graph is computed,and during the uncoarsening and reﬁnement phase,the bisection is successively reﬁned as it is projected to the larger graphs.During the uncoarsening and reﬁnement phase,the dashed lines indicate projected partitionings and dark solid lines indicate partitionings that were produced after reﬁnement.G 0is the given graph,which is the ﬁnest graph.G i +1is the next level coarser graph of G i ,and vice versa,G i is the next level ﬁner graph of G i +1.G 4is the coarsest graph.hypergraph is ﬁrst converted into a graph (by replacing each hyperedge by a set of regular edges),then METIS [21]can be used to compute a partitioning of this graph.This technique was investigated by Alpert and Khang [25]in their algorithm called GMetis.They converted hypergraphs to graphs by simply replacing each hyperedge with a clique,and then they dropped many edges from each clique randomly.They used METIS to compute a partitioning of each such random graph and then selected the best of these partitionings.Their results show that reasonably good partitionings can be obtained in a reasonable amount of time for a variety of benchmark problems.In particular,the performance of their resulting scheme is comparable to other state-of-the art schemes such as PARABOLI [26],PROP [11],and the multilevel hypergraph partitioner from Hauck and Borriello [20].The conversion of a hypergraph into a graph by replacing each hyperedge with a clique does not result in an equivalent representation since high-quality partitionings of the resulting graph do not necessarily lead to high-quality partitionings of the hypergraph.The standard hyperedge-to-edge conversion [27]assigns a uniform weightofisthe of the hyperedge,i.e.,thenumber of vertices in the hyperedge.However,the fundamen-tal problem associated with replacing a hyperedge by its clique is that there exists no scheme to assign weight to the edges of the clique that can correctly capture the cost of cutting this hyperedge [28].This hinders the partitioning reﬁnement algo-rithm since vertices are moved between partitions depending on how much this reduces the number of edges they cut in the converted graph,whereas the real objective is to minimize the number of hyperedges cut in the original hypergraph.Furthermore,the hyperedge-to-clique conversion destroys the natural sparsity of the hypergraph,signiﬁcantly increasing theKARYPIS et al.:MULTILEVEL HYPERGRAPH PARTITIONING:APPLICATIONS IN VLSI DOMAIN 71run time of the partitioning algorithm.Alpert and Khang [25]solved this problem by dropping many edges of the clique randomly,but this makes the graph representation even less accurate.A better approach is to develop coarsening and reﬁnement schemes that operate directly on the hypergraph.Note that the multilevel scheme by Hauck and Borriello [20]operates directly on hypergraphs and,thus,is able to perform accurate reﬁnement during the uncoarsening phase.However,all coarsening schemes studied in [20]are edge-oriented;i.e.,they only merge pairs of nodes to construct coarser graphs.Hence,despite a powerful reﬁnement scheme (FM with theuse oflook-ahead)during the uncoarsening phase,their performance is only as good as that of GMetis [25].B.Our ContributionsIn this paper,we present a multilevel hypergraph-partitioning algorithm hMETIS that operates directly on the hypergraphs.A key contribution of our work is the development of new hypergraph coarsening schemes that allow the multilevel paradigm to provide high-quality partitions quite consistently.The use of these powerful coarsening schemes also allows the reﬁnement process to be simpliﬁed considerably (even beyond plain FM reﬁnement),making the multilevel scheme quite fast.We investigate various algorithms for the coarsening and uncoarsening phases which operate on the hypergraphs without converting them into graphs.We have also developed new multiphase reﬁnement schemes(-cycles)based on the multilevel paradigm.These schemes take an initial partition as input and try to improve them using the multilevel scheme.These multiphase schemes further reduce the run times,as well as improve the solution quality.We evaluate their performance both in terms of the size of the hyperedge cut on the bisection,as well as on run time on a number of VLSI circuits.Our experiments show that our multilevel hypergraph-partitioning algorithm produces high-quality partitioning in a relatively small amount of time.The quality of the partitionings produced by our scheme are on the average 6%–23%better than those produced by other state-of-the-art schemes [11],[12],[25],[26],[29].The difference in quality over other schemes becomes even greater for larger hypergraphs.Furthermore,our partitioning algorithm is signiﬁcantly faster,often requiring 4–10times less time than that required by the other schemes.For many circuits in the well-known ACM/SIGDA benchmark set [30],our scheme is able to ﬁnd better partitionings than those reported in the literature for any other hypergraph-partitioning algorithm.The remainder of this paper is organized as follows.Section II describes the different algorithms used in the three phases of our multilevel hypergraph-partitioning algorithm.Section III describes a new partitioning reﬁnement algorithm based on the multilevel paradigm.Section IV compares the results produced by our algorithm to those produced by earlier hypergraph-partitioning algorithms.II.M ULTILEVEL H YPERGRAPH B ISECTIONWe now present the framework of hMETIS ,in which the coarsening and reﬁnement scheme work directly with hyper-edges without using the clique representation to transform them into edges.We have developed new algorithms for both the phases,which,in conjunction,are capable of delivering very good quality solutions.A.Coarsening PhaseDuring the coarsening phase,a sequence of successively smaller hypergraphs are constructed.As in the case of mul-tilevel graph bisection,the purpose of coarsening is to create a small hypergraph,such that a good bisection of the small hypergraph is not signiﬁcantly worse than the bisection di-rectly obtained for the original hypergraph.In addition to that,hypergraph coarsening also helps in successively reducing the sizes of the hyperedges.That is,after several levels of coarsening,large hyperedges are contracted to hyperedges that connect just a few vertices.This is particularly helpful,since reﬁnement heuristics based on the KLFM family of algorithms [6]–[8]are very effective in reﬁning small hyperedges,but are quite ineffective in reﬁning hyperedges with a large number of vertices belonging to different partitions.Groups of vertices that are merged together to form single vertices in the next-level coarse hypergraph can be selected in different ways.One possibility is to select pairs of vertices with common hyperedges and to merge them together,as illustrated in Fig.2(a).A second possibility is to merge together all the vertices that belong to a hyperedge,as illustrated in Fig.2(b).Finally,a third possibility is to merge together a subset of the vertices belonging to a hyperedge,as illustrated in Fig.2(c).These three different schemes for grouping vertices together for contraction are described below.1)Edge Coarsening (EC):The heavy-edge matching scheme used in the multilevel-graph bisection algorithm can also be used to obtain successively coarser hypergraphs by merging the pairs of vertices connected by many hyperedges.In this EC scheme,a heavy-edge maximal 1matching of the vertices of the hypergraph is computed as follows.The vertices are visited in a random order.For eachvertex are considered,and the one that is connected via the edge with the largest weight is matchedwithandandofsize72IEEE TRANSACTIONS ON VERY LARGE SCALE INTEGRATION(VLSI)SYSTEMS,VOL.7,NO.1,MARCH1999Fig.2.Various ways of matching the vertices in the hypergraph and the coarsening they induce.(a)In edge-coarsening,connected pairs of vertices are matched together.(b)In hyperedge-coarsening,all the vertices belonging to a hyperedge are matched together.(c)In MHEC,we match together all the vertices in a hyperedge,as well as all the groups of vertices belonging to a hyperedge.weight of successively coarser graphs does not decrease very fast.In order to ensure that for every group of vertices that are contracted together,there is a decrease in the hyperedge weight in the coarser graph,each such group of vertices must be connected by a hyperedge.This is the motivation behind the HEC scheme.In this scheme,an independent set of hyperedges is selected and the vertices that belong to individual hyperedges are contracted together.This is implemented as follows.The hyperedges are initially sorted in a nonincreasing hyperedge-weight order and the hyperedges of the same weight are sorted in a nondecreasing hyperedge size order.Then,the hyperedges are visited in that order,and for each hyperedge that connects vertices that have not yet been matched,the vertices are matched together.Thus,this scheme gives preference to the hyperedges that have large weight and those that are of small size.After all of the hyperedges have been visited,the groups of vertices that have been matched are contracted together to form the next level coarser graph.The vertices that are not part of any contracted hyperedges are simply copied to the next level coarser graph.3)Modiﬁed Hyperedge Coarsening(MHEC):The HEC algorithm is able to signiﬁcantly reduce the amount of hyperedge weight that is left exposed in successively coarser graphs.However,during each coarsening phase,a majority of the hyperedges do not get contracted because vertices that belong to them have been contracted via other hyperedges. This leads to two problems.First,the size of many hyperedges does not decrease sufﬁciently,making FM-based reﬁnement difﬁcult.Second,the weight of the vertices(i.e.,the number of vertices that have been collapsed together)in successively coarser graphs becomes signiﬁcantly different,which distorts the shape of the contracted hypergraph.To correct this problem,we implemented a MHEC scheme as follows.After the hyperedges to be contracted have been selected using the HEC scheme,the list of hyperedges is traversed again,and for each hyperedge that has not yet been contracted,the vertices that do not belong to any other contracted hyperedge are contracted together.B.Initial Partitioning PhaseDuring the initial partitioning phase,a bisection of the coarsest hypergraph is computed,such that it has a small cut, and satisﬁes a user-speciﬁed balance constraint.The balance constraint puts an upper bound on the difference between the relative size of the two partitions.Since this hypergraph has a very small number of vertices(usually less than200),the time toﬁnd a partitioning using any of the heuristic algorithms tends to be small.Note that it is not useful toﬁnd an optimal partition of this coarsest graph,as the initial partition will be sub-stantially modiﬁed during the reﬁnement phase.We used the following two algorithms for computing the initial partitioning. Theﬁrst algorithm simply creates a random bisection such that each part has roughly equal vertex weight.The second algorithm starts from a randomly selected vertex and grows a region around it in a breadth-ﬁrst fashion[22]until half of the vertices are in this region.The vertices belonging to the grown region are then assigned to theﬁrst part,and the rest of the vertices are assigned to the second part.After a partitioning is constructed using either of these algorithms,the partitioning is reﬁned using the FM reﬁnement algorithm.Since both algorithms are randomized,different runs give solutions of different quality.For this reason,we perform a small number of initial partitionings.At this point,we can select the best initial partitioning and project it to the original hypergraph,as described in Section II-C.However,the parti-tioning of the coarsest hypergraph that has the smallest cut may not necessarily be the one that will lead to the smallest cut in the original hypergraph.It is possible that another partitioning of the coarsest hypergraph(with a higher cut)will lead to a bet-KARYPIS et al.:MULTILEVEL HYPERGRAPH PARTITIONING:APPLICATIONS IN VLSI DOMAIN 73ter partitioning of the original hypergraph after the reﬁnement is performed during the uncoarsening phase.For this reason,instead of selecting a single initial partitioning (i.e.,the one with the smallest cut),we propagate all initial partitionings.Note that propagation of.Thus,by increasing the value ofis to drop unpromising partitionings as thehypergraph is uncoarsened.For example,one possibility is to propagate only those partitionings whose cuts arewithinissufﬁciently large,then all partitionings will be maintained and propagated in the entire reﬁnement phase.On the other hand,if the valueof,many partitionings may be available at the coarsest graph,but the number of such available partitionings will decrease as the graph is uncoarsened.This is useful for two reasons.First,it is more important to have many alternate partitionings at the coarser levels,as the size of the cut of a partitioning at a coarse level is a less accurate reﬂection of the size of the cut of the original ﬁnest level hypergraph.Second,reﬁnement is more expensive at the ﬁne levels,as these levels contain far more nodes than the coarse levels.Hence,by choosing an appropriate valueof(from 10%to a higher value such as 20%)did not signiﬁcantly improve the quality of the partitionings,although it did increase the run time.C.Uncoarsening and Reﬁnement PhaseDuring the uncoarsening phase,a partitioning of the coarser hypergraph is successively projected to the next-level ﬁner hypergraph,and a partitioning reﬁnement algorithm is used to reduce the cut set (and thus to improve the quality of the partitioning)without violating the user speciﬁed balance con-straints.Since the next-level ﬁner hypergraph has more degrees of freedom,such reﬁnement algorithms tend to improve the solution quality.We have implemented two different partitioning reﬁnement algorithms.The ﬁrst is the FM algorithm [8],which repeatedly moves vertices between partitions in order to improve the cut.The second algorithm,called hyperedge reﬁnement (HER),moves groups of vertices between partitions so that an entire hyperedge is removed from the cut.These algorithms are further described in the remainder of this section.1)FM:The partitioning reﬁnement algorithm by Fiduccia and Mattheyses [8]is iterative in nature.It starts with an initial partitioning of the hypergraph.In each iteration,it tries to ﬁnd subsets of vertices in each partition,such that moving them to other partitions improves the quality of the partitioning (i.e.,the number of hyperedges being cut decreases)and this does not violate the balance constraint.If such subsets exist,then the movement is performed and this becomes the partitioning for the next iteration.The algorithm continues by repeating the entire process.If it cannot ﬁnd such a subset,then the algorithm terminates since the partitioning is at a local minima and no further improvement can be made by this algorithm.In particular,for eachvertexto the other partition.Initially allvertices are unlocked ,i.e.,they are free to move to the other partition.The algorithm iteratively selects an unlockedvertex is moved,it is locked ,and the gain of the vertices adjacentto[8].For reﬁnement in the context of multilevel schemes,the initial partitioning obtained from the next level coarser graph is actually a very good partition.For this reason,we can make a number of optimizations to the original FM algorithm.The ﬁrst optimization limits the maximum number of passes performed by the FM algorithm to only two.This is because the greatest reduction in the cut is obtained during the ﬁrst or second pass and any subsequent passes only marginally improve the quality.Our experience has shown that this optimization signiﬁcantly improves the run time of FM without affecting the overall quality of the produced partitionings.The second optimization aborts each pass of the FM algorithm before actually moving all the vertices.The motivation behind this is that only a small fraction of the vertices being moved actually lead to a reduction in the cut and,after some point,the cut tends to increase as we move more vertices.When FM is applied to a random initial partitioning,it is quite likely that after a long sequence of bad moves,the algorithm will climb74IEEE TRANSACTIONS ON VERY LARGE SCALE INTEGRATION (VLSI)SYSTEMS,VOL.7,NO.1,MARCH1999Fig.3.Effect of restricted coarsening .(a)Example hypergraph with a given partitioning with the required balance of 40/60.(b)Possible condensed version of (a).(c)Another condensed version of a hypergraph.out of a local minima and reach to a better cut.However,in the context of a multilevel scheme,a long sequence of cut-increasing moves rarely leads to a better local minima.For this reason,we stop each pass of the FM algorithm as soon as we haveperformedto be equal to 1%of the number ofvertices in the graph we are reﬁning.This modiﬁcation to FM,called early-exit FM (FM-EE),does not signiﬁcantly affect the quality of the ﬁnal partitioning,but it dramatically improves the run time (see Section IV).2)HER:One of the drawbacks of FM (and other similar vertex-based reﬁnement schemes)is that it is often unable to reﬁne hyperedges that have many nodes on both sides of the partitioning boundary.However,a reﬁnement scheme that moves all the vertices that belong to a hyperedge can potentially solve this problem.Our HER works as follows.It randomly visits all the hyperedges and,for each one that straddles the bisection,it determines if it can move a subset of the vertices incident on it,so that this hyperedge will become completely interior to a partition.In particular,consider ahyperedgebe the verticesofto partition 0.Now,depending on these gains and subject to balance constraints,it may move one of the twosets .In particular,if.III.M ULTIPHASE R EFINEMENT WITHR ESTRICTED C OARSENINGAlthough the multilevel paradigm is quite robust,random-ization is inherent in all three phases of the algorithm.In particular,the random choice of vertices to be matched in the coarsening phase can disallow certain hyperedge cuts,reducing reﬁnement in the uncoarsening phase.For example,consider the example hypergraph in Fig.3(a)and its two possible con-densed versions [Fig.3(b)and (c)]with the same partitioning.The version in Fig.3(b)is obtained by selectinghyperedgesto be compressed in the HEC phase and then selecting pairs ofnodesto be compressed inthe HEC phase and then selecting pairs ofnodesand apartitioningfor ,be the sequence of hypergraphsand partitionings.Given ahypergraphandorpartition,,are collapsedtogether to formvertexof,thenvertex belong。

VLSI题库

VLSI系统结构设计典型习题及解答1、填空题1）通用数字系统设计的性能指标是：（1）所需的硬件电路和资源；（2）执行的速度；（3）功耗；（4）有限字长的性能。

2）DSP区别于其它通用计算的两个重要特性是：（1）实时吞吐率的需要;（2）数据驱动的性质。

3）DSP程序的迭代边界是指关键环路的环路边界。

计算迭代边界的算法有最长矩阵路径(LPM)算法和最小环均值（MCM）算法。

4）在DFG中，关键路径是指具有零延时的所有路径中具有最长运算时间的路径。

5）流水线技术就是采用沿着数据通路引入流水锁存器的方法来减小有效关键路径，从而可以提高时钟速度或采样速度，或者可以在同样的速度下降低功耗。

6）并行处理技术就是多个输出在一个时钟周期内并行的计算，从而使有效采样速度提高到与并行级数相当的倍数。

7）重定时在同步电路设计中有很多应用，包括缩短电路的时钟周期，减少电路中寄存器的数目、降低电路的功耗和逻辑综合的规模。

8）展开变换可以揭示隐藏在用DFG描述的数字信号处理系统中的并发性。

因此，展开技术可以用来缩短DSP算法的迭代周期。

展开可以用来生成高速低功耗应用中的字并行架构。

9）折叠变换是一种设计分时复用架构的系统变换技术。

在DSP架构中，折叠技术提供了一种以时间换取面积的方法。

10）重定时在同步电路设计中有很多应用，包括缩短电路的时钟周期，减少电路中寄存器的数目、降低电路的功耗和逻辑综合的规模。

11）脉动阵列是一种有节奏地计算并通过系统传输数据的处理单元网络，这种系统的特征是模块化和规则化。

12）对于3阶FIR滤波器y(n)=b0x(n)+b1x(n-1)+b2x(n-2),请说明下列参数的含义：y(n)表示 n时刻在输出端Y的输出 ; x(n)表示： n时刻在输入端X的输入; x(n-2)表示n-2时刻在输入端X的输入在N时刻在电路中某点的值。

13）利用折叠变换进行VLSI系统的结构设计中，画折叠图的六个步骤分别是：（1）定义折叠集；（2）计算折叠方程；（3）画出折叠运算节点；（4）添加各节点的输入；（5）根据折叠方程添加延时及连线；(6)图形整理。

MOST包（多阶段优化策略）说明说明书

Package‘MOST’October12,2022Type PackageTitle Multiphase Optimization StrategyVersion0.1.2Depends R(>=2.15.0)Encoding UTF-8Copyright2022by The Pennsylvania State UniversityDescription Provides functions similar to the'SAS'macros previously provided to accompany Collins,Dziak,and Li(2009)<DOI:10.1037/a0015826>and Dziak,Nahum-Shani,and Collins(2012)<DOI:10.1037/a0026972>,paperswhich outline practical beneﬁts and challenges of factorialand fractional factorial experiments for scientists interestedin developing biological and/or behavioral interventions,especiallyin the context of the multiphase optimization strategy(see Collins,Kugler&Gwadz2016)<DOI:10.1007/s10461-015-1145-4>.The package currently contains three functions.First,RelativeCosts1()draws a graphof the relative cost of complete and reduced factorial designs versusother alternatives.Second,RandomAssignmentGenerator()returns a dataframewhich contains a list of random numbers that can be used to convenientlyassign participants to conditions in an experiment withmany conditions.Third,FactorialPowerPlan()estimates the power,detectable effectsize,or required sample size of a factorial or fractional factorialexperiment,for main effects or interactions,given several possible choicesof effect size metric,and allowing pretests and clustering.License GPL(>=2)NeedsCompilation noRoxygenNote7.2.0Author Linda Collins[aut],Liying Huang[aut],John Dziak[aut,cre]Maintainer John Dziak<*****************>Repository CRANDate/Publication2022-06-2322:20:08UTC1R topics documented:FactorialPowerPlan (2)RandomAssignmentGenerator (4)RelativeCosts1 (5)Index7 FactorialPowerPlan sample size,power and effect size calculations for a factorial or frac-tional factorial experimentDescriptionThere are three ways to use this function:1.Estimate power available from a given sample size and a given effect size.2.Estimate sample size needed for a given power and a given effect size.3.Estimate effect size detectable from a given power at a given sample size.That is,there are three main pieces of information:power,sample size,and effect size.The user provides two of them,and this function calculates the third.UsageFactorialPowerPlan(alpha=0.05,assignment="unclustered",change_score_icc=NULL,cluster_size=NULL,cluster_size_sd=NULL,d_main=NULL,effect_size_ratio=NULL,icc=NULL,model_order=1,nclusters=NULL,nfactors=1,ntotal=NULL,power=NULL,pre_post_corr=NULL,pretest="none",raw_coef=NULL,raw_main=NULL,sigma_y=NULL,std_coef=NULL)Argumentsalpha Two sided Type I error level for the test to be performed(default=0.05).assignment One of three options:(default=unclustered)1.“independent”or equivalently“unclustered”2.“within”or equivalently“within_clusters”3.“between”or equivalently“between_clusters”Clusters in this context are preexisting units within which responses may bedependent(e.g.,clinics or schools).A within-cluster experiment involves ran-domizing individual members,while a between-cluster experiment involves ran-domizing clusters as whole units(see Dziak,Nahum-Shani,and Collins,2012)<DOI:10.1037/a0026972>change_score_iccThe intraclass correlation of the change scores(posttest minus pretest).Relevantonly if assignment is between clusters and there is a pretest.cluster_size The mean number of members in each cluster.Relevant only if assignment isbetween clusters or within clusters.cluster_size_sdRelevant only if assignment is between clusters.The standard deviation of thenumber of members in each cluster(the default is0which means that the clustersare expected to be of equal size).d_main Effect size measure:standardized mean difference raw_main/sigma_y.effect_size_ratioEffect size measure:signal to noise ratio raw_coef^2/sigma_y^2.icc Relevant only if assignment is between clusters or within clusters.The intraclasscorrelation of the variable of interest in the absence of treatment.model_order The highest order term to be included in the regression model in the plannedanalysis(1=main effects,2=two-way interactions,3=three-way interactions,etc.);must be>=1and<=nfactors(default=1).nclusters The total number of clusters available(for between clusters or within clustersassignment).nfactors The number of factors(independent variables)in the planned experiment(default=1).ntotal The total sample size available(for unclustered assignment.For clustered as-signment,use“cluster_size”and“nclusters.”power If speciﬁed:The desired power of the test.If returned in the output list:Theexpected power of the test.pre_post_corr Relevant only if there is a pretest.The correlation between the pretest and theposttest.pretest One of three options:1.“no”or“none”for no pretest.2.“covariate”for pretest to be entered as a covariate in the model.3.“repeated”for pretest to be considered as a repeated measure.4RandomAssignmentGeneratorThe option“yes”is also allowed and is interpreted as“repeated.”The option“covariate”is not allowed if assignment is between clusters.This is becausepredicting power for covariate-adjusted cluster-level randomization is somewhatcomplicated,although it can be approximated in practice by using the formulafor the repeated-measures cluster-level randomization(see simulations in Dziak,Nahum-Shani,and Collins,2012).raw_coef Effect size measure:unstandardized effect-coded regression coefﬁcient.raw_main Effect size measure:unstandardized mean difference.sigma_y The assumed standard deviation of the response variable after treatment,withineach treatment condition(i.e.,adjusting for treatment but not adjusting for post-test).This statement must be used if the effect size argument used is either“raw_main”or“raw_coef”.std_coef Effect size measure:standardized effect-coded regression coefﬁcient raw_coef/sigma_y. ValueA list with power,sample size and effect size.ExamplesFactorialPowerPlan(assignment="independent",model_order=2,nfactors=5,ntotal=300,raw_main=3,sigma_y=10)FactorialPowerPlan(assignment="independent",model_order=2,nfactors=5,ntotal=300,pre_post_corr=.6,pretest="covariate",raw_main=3,sigma_y=10)RandomAssignmentGeneratorRandom Assignment Generator for a Factorial Experiment with ManyConditionsDescriptionThis function provides a list of random numbers that can be used to assign participants to conditions(cells)in an experiment with many conditions,such as a factorial experiment.The randomizationis restricted as follows:if the number of participants available is a multiple of the number of condi-tions,then cell sizes will be balanced;otherwise,they will be as near balanced as possible.UsageRandomAssignmentGenerator(N,C)ArgumentsN The total number of participants to be randomized.C The total number of conditions for the experiment you are planning.Note that for a complete factorial experiment having k factors,there will be2^k conditions.ValueA dataframe with1variable ranList with N observations,each observation of ranList provides arandom number for each participant.This will be a number from1to C.For example,if the4thnumber in the list is7,the4th subject is randomly assigned to experiment condition7.Randomnumbers will be generated so that the experiment is approximately balanced.Examplesresult<-RandomAssignmentGenerator(35,17)print(result)RelativeCosts1The relative cost of reduced factorial designsDescriptionDraw a graph of the relative cost of complete factorial,fractional factorial,and unbalanced reducedfactorial designs,as presented by Collins,Dziak and Li(2009;https:///pubmed/19719358).For purposes of illustration,a normally distributed response variable,dichotomous factors,and neg-ligible interactions are assumed in this function.UsageRelativeCosts1(number_of_factors,desired_fract_resolution=4,min_target_d_per_factor=0.2,condition_costlier_than_subject=1,max_graph_ratio=5)Argumentsnumber_of_factorsThe number of factors to be tested.desired_fract_resolutionThe desired resolution of the fractional factorial experiment to be compared.The default value is set to be4.min_target_d_per_factorThe minimum Cohen’s d(standardized difference,i.e.,response difference be-tween levels on a given factor,divided by response standard deviation)that isdesired to be detected with80The default value is set to be0.2.condition_costlier_than_subjectThe default value is set to be1.max_graph_ratioThe default value is set to be5.ExamplesRelativeCosts1(number_of_factors=9,desired_fract_resolution=4,min_target_d_per_factor=.2,condition_costlier_than_subject=1,max_graph_ratio=4)IndexFactorialPowerPlan,2 RandomAssignmentGenerator,4RelativeCosts1,57。

cost-sensitive methods -回复

cost-sensitive methods -回复什么是“costsensitive methods”，以及这些方法在数据分析和机器学习中的作用？【引言】在数据分析和机器学习领域中，我们经常会遇到一类特殊的问题：不同的错误类型会给我们带来不同的损失。

这就意味着在处理这些问题时，我们需要更加关注某些错误的影响，而不仅仅是简单地预测准确性。

这时候就需要利用“costsensitive methods”（成本敏感方法）来解决这些问题。

【什么是“costsensitive methods”】“Costsensitive methods”是一种特殊的数据分析和机器学习方法，旨在解决在不同错误类型带来不同损失的问题。

传统的分类算法通常只关注分类的准确性，它们通过最小化错误率来指导模型的训练。

然而，在某些应用场景中，错误的后果可能会对我们的决策产生重大影响。

这时候，我们就需要引入成本因素，将不同错误类型的损失考虑进来。

【常见的错误类型】在成本敏感方法中，我们通常将错误划分为以下两类：假阳性（False Positive，FP）和假阴性（False Negative，FN）。

假阳性是指模型错误地将负类样本误分类为正类样本，而假阴性是指模型错误地将正类样本误分类为负类样本。

【成本矩阵】成本矩阵是成本敏感方法中的关键组成部分。

它描述了在不同错误类型下的损失。

通常，成本矩阵是一个二维矩阵，行代表实际标签，列代表预测标签。

对于分类问题，成本矩阵的主对角线上的元素表示正确分类的样本，而非主对角线上的元素则表示错误分类的样本。

通过设定不同错误分类的损失权重，我们可以更准确地度量不同错误类型的成本。

【成本敏感的算法】在成本敏感方法中，我们需要选择合适的算法来训练我们的模型。

常见的成本敏感算法有：1. 加权支持向量机（Weighted Support Vector Machine）：在传统的支持向量机算法中，所有样本的权重都是相同的，而在成本敏感的支持向量机中，根据成本矩阵的设定，我们可以为不同类的样本设置不同的权重。

自适应均衡rls算法仿真流程

自适应均衡rls算法仿真流程自适应均衡RLS算法是一种自适应信号处理算法，它利用递归最小二乘法（RLS）来对信号进行均衡处理。

该算法在通信系统中广泛应用，能够实现对信号的实时均衡处理，提高系统的性能和稳定性。

本文将对自适应均衡RLS算法的原理、流程和仿真结果进行详细介绍，旨在帮助读者全面理解该算法的工作原理和应用场景。

一、自适应均衡RLS算法原理1.1 RLS算法简介递归最小二乘法（RLS）是一种基于最小二乘准则的自适应滤波算法，主要用于对信号进行滤波和均衡等处理。

该算法利用衰减因子对历史数据进行加权衰减处理，实现对信号的实时自适应处理。

RLS算法采用递归更新方式，能够快速收敛并适用于实时信号处理场景。

1.2自适应均衡RLS原理自适应均衡RLS算法是在RLS算法基础上进行改进，主要用于通信系统中信号均衡处理。

其主要原理是通过对接收信号进行均衡处理，使得输出信号与目标信号尽可能接近，从而实现对信号的高效处理和恢复。

自适应均衡RLS算法采用递归最小二乘准则，并结合均衡器来实现对信号的实时均衡处理。

1.3算法工作流程自适应均衡RLS算法主要包括初始化阶段和递归更新阶段两部分。

初始化阶段主要用于初始化算法参数和权值矩阵，递归更新阶段则是通过递归最小二乘法对信号进行实时处理。

二、自适应均衡RLS算法仿真流程2.1仿真环境搭建在进行自适应均衡RLS算法仿真之前，需要先搭建仿真环境。

通常情况下，可以使用MATLAB等工具进行仿真实验，通过编写相应的算法代码和仿真模型来模拟算法的工作过程。

在搭建仿真环境时，需要考虑信号源、传输通道和接收端等因素，以便进行真实的信号处理和均衡实验。

2.2信号仿真模型设计在进行自适应均衡RLS算法仿真时，需要设计相应的信号仿真模型。

该模型主要包括信号发生器、传输通道和接收端等组成部分，通过对信号的产生、传输和接收进行模拟，可以得到真实的信号处理和均衡效果。

通常情况下，可以选择合适的信号波形和参数设置，以便对算法的性能进行全面测试。

lsi 2008 参数

lsi 2008 参数LSI 2008 参数LSI 2008 是一种常用的文本分析方法，它是基于词频-逆文档频率（Term Frequency-Inverse Document Frequency，TF-IDF）的基础上，通过奇异值分解（Singular Value Decomposition，SVD）将文本数据降维，从而实现对文本的主题建模和相似度计算。

LSI 2008 参数是指在进行 LSI 分析时所使用的参数集合。

我们需要了解 LSI 的基本原理。

LSI 是一种无监督学习方法，通过对文本进行向量化表示，从而实现对文本的语义分析。

在 LSI 中，文本被表示为一个矩阵，其中每一行表示一个文档，每一列表示一个词语，矩阵中的元素表示词语在文档中的出现频率。

LSI 2008 参数主要包括以下几个关键参数：1. 词频计算方法（Term Frequency，TF）：TF 是指词语在文档中的出现频率。

常见的计算方法有原始频率（Raw Frequency）、对数频率（Logarithmic Frequency）和双重词频（Double Frequency）等。

不同的计算方法会对结果产生不同的影响，需要根据具体情况选择合适的计算方法。

2. 逆文档频率计算方法（Inverse Document Frequency，IDF）：IDF 是指词语在整个文档集合中的重要程度。

常见的计算方法有平滑逆文档频率（Smooth IDF）和逆文档频率平滑化（IDF Smoothing）等。

IDF 用于降低常见词语的权重，提高关键词语的权重，从而更好地体现词语的重要性。

3. 停用词过滤（Stop Words Filtering）：停用词是指在文本中频繁出现但对文本主题没有明显贡献的词语，如“的”、“是”、“在”等。

通过过滤停用词可以减少噪声，提高文本分析的准确性和效率。

4. 文档预处理方法（Document Preprocessing）：文档预处理是指在进行 LSI 分析之前对文本进行的一系列处理操作，如分词、去除标点符号、转换为小写等。

基于二次指数平滑的干预模型在CCI中的应用

２００８年金融危机对中国的影响是多重的，有直接影响、接影响和心理影响，至会产生思想间甚认识方面的影响．因此．费者信心指数是反映消消
费者信心强弱的指标，综合反映并量化消费者对是
经济发展过程产生影响。至改变经济结构．０９甚２０年底经济复苏对中国的影响是明显的，特别对我国消费者信心影响是很明显的．几年，间序列分近时析方法的研究和应用飞速发展，特别在经济界，越
第２３卷第２期
２１００年６月
海南师范大学学报（自然科学版）
ＪｕｎｌｆＨｉａｏｍｌｎｖｒｉ（ａｒｌｃｅｃ）ｏｒａｏａｎｎＮｒａＵｉｅｓｙＮｔａＳｉｈｔｕｎ
Ｖｏ１２．．３Ｎｏ２
Ｊｎ２ＩＣ中的应用
朱冬和，胡晓华，杨帅国
（海南师范大学数学与统计学院，海南海口５１５）７１８
摘要：用Ｅｉ软件和二次指数平滑的干预模型方法对我国消费者信心指数建立了动态模利ｖｅｗｓ型．并预测了未来４个月消费者信心的发展趋势．结果表明，型比较好地解释了我国消费者此模
Ｋｅｒｓ：ｏｂｅｅｐｎｎｉｌｓｏｈｎｃｎｕｒｃｎｄｎｅｉｄｘ；ｔｒｅｔｎｙｗｏｄｄｕｌｘｏｅｔｍｏｔｉｇ；ｏｓｍｅｏｆｅｃｅｉｅｖｎｉａｉｎｎ０

参数分解法critical parameter management案例

参数分解法在Critical Parameter Management中的应用案例参数分解法是一种在Critical Parameter Management（CPM）中常用的方法，它通过对参数进行分解和优化，提高产品的质量和性能。

下面是一个参数分解法在CPM中的应用案例。

某汽车制造公司生产一款新型汽车，该汽车在市场上受到广泛关注。

然而，在生产过程中，公司发现汽车的刹车性能存在一些问题，导致刹车距离过长，影响了汽车的安全性能。

为了解决这个问题，公司决定采用参数分解法对刹车系统进行优化。

首先，他们对刹车系统进行了详细的分析，包括刹车盘的材料、尺寸、摩擦系数等参数。

然后，他们将这些参数进行分解，并针对每个参数进行优化。

在优化过程中，他们采用了多种方法，包括实验设计、回归分析、敏感性分析等。

通过这些方法，他们找到了影响刹车性能的关键参数，并针对这些参数进行了优化。

经过优化后，汽车的刹车性能得到了显著提高。

实验结果表明，汽车的刹车距离缩短了20%，大大提高了汽车的安全性能。

同时，优化后的刹车系统还具有更高的稳定性和可靠性，减少了故障率。

这个案例充分展示了参数分解法在CPM中的应用价值。

通过对参数进行分解和优化，公司成功地提高了产品的质量和性能，增强了市场竞争力。

此外，参数分解法还可以应用于其他领域，如机械制造、电子工程、生物医学等。

在应用过程中，需要结合具体问题和实际情况，选择合适的分解方法和优化方法，以达到最佳的优化效果。

总之，参数分解法是一种有效的CPM方法，可以帮助企业提高产品质量和性能，增强市场竞争力。

在未来的发展中，随着技术的不断进步和应用领域的不断扩展，参数分解法将会发挥更大的作用。

数据方体的Benifit算法

数据方体的Benifit算法
Greedy算法
固件需要维护一张Block属性表，记录每个Block当前的Valid Page数量。

假设每次GC处理8个Block，查表挑出Valid Page最少的8个Block进行GC，这样做的好处是复制Valid Page的开销最小。

Cost-Benefit算法
u代表valid page在该Block中的比例，age代表该Block距离最近一次修改的时间。

1-u是对这个Block进行GC以后能够获得Free Page的数量
2u是对这个Block进行GC的开销，读取Valid Page（1个u）然后写入到新的Block（再1个u）
（1-u）/2u可以理解为投入产出比
固件需要维护的Block属性表里，需要记录每个Block最后一次被写入的时间，GC时选择更久没有被修改的Block（冷数据）该策略就是选择投入产出比更高，未修改时间更长的Block进行GC，两者相乘数字更大的优先被GC。

西蒙信号测试仪signaltek ii-汉语-说明书

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1058IEEE TRANSACTIONS ON VERY LARGE SCALE INTEGRATION (VLSI) SYSTEMS, VOL. 16, NO. 8, AUGUST 2008Cost-Effective Triple-Mode Reconﬁgurable Pipeline FFT/IFFT/2-D DCT ProcessorChin-Teng Lin, Fellow, IEEE, Yuan-Chu Yu, Member, IEEE, and Lan-Da Van, Member, IEEEAbstract—This investigation proposes a novel radix-42 algorithm with the low computational complexity of a radix-16 algorithm but the lower hardware requirement of a radix-4 algorithm. The proposed pipeline radix-42 single delay feedback path (R42 SDF) architecture adopts a multiplierless radix-4 butterﬂy structure, based on the speciﬁc linear mapping of common factor algorithm (CFA), to support both 256-point fast Fourier trans8 2-D form/inverse fast Fourier transform (FFT/IFFT) and 8 discrete cosine transform (DCT) modes following with the high efﬁcient feedback shift registers architecture. The segment shift register (SSR) and overturn shift register (OSR) structure are adopted to minimize the register cost for the input re-ordering and post computation operations in the 8 8 2-D DCT mode, respectively. Moreover, the retrenched constant multiplier and eight-folded complex multiplier structures are adopted to decrease the multiplier cost and the coefﬁcient ROM size with the complex conjugate symmetry rule and subexpression elimination technology. To further decrease the chip cost, a ﬁnite wordlength analysis is provided to indicate that the proposed architecture only requires a 13-bit internal wordlength to achieve 40-dB signal-to-noise ratio (SNR) performance in 256-point FFT/IFFT modes and high digital video (DV) compression quality in 8 8 2-D DCT mode. The comprehensive comparison results indicate that the proposed cost effective reconﬁgurable design has the smallest hardware requirement and largest hardware utilization among the tested architectures for the FFT/IFFT computation, and thus has the highest cost efﬁciency. The derivation and chip implementation results show that the proposed pipeline 256-point FFT/IFFT/2-D DCT triple-mode chip consumes 22.37 mW at 100 MHz at 1.2-V supply voltage in TSMC 0.13- m CMOS process, which is very appropriate for the RSoCs IP of next-generation handheld devices. Index Terms—Computation complexity, cost effective, hardware utilization, next-generation wireless communications, pipeline architecture, R42 SDF, triple modes.I. INTRODUCTIONNEXT-GENERATION mobile multimedia applications, including mobile phones and personal digital assistants (PDAs), require much sufﬁciently high processing power forManuscript received March 16, 2007; revised July 23, 2007. This work was supported in part by the National Science Council of the Republic of China, Taiwan under Contract NSC93-2218-E-009-061 and Contract MOEA-96-EC-17-A-01–S1-048. C.-T. Lin is with the University Provost and the Chair Professor of Electrical and Control Engineering, National Chiao-Tung University, Hsinchu 300, Taiwan, R.O.C. (e-mail: ctlin@.tw). Y.-C. Yu is with the Department of Electrical and Control Engineering, National Chiao-Tung University, Hsinchu 300, Taiwan, R.O.C. (e-mail: Vincent_yu@). L.-D. Van is with the Department of Computer Science, National Chiao-Tung University, Hsinchu 300, Taiwan, R.O.C. (e-mail: ldvan@.tw). Digital Object Identiﬁer 10.1109/TVLSI.2008.2000676multimedia applications. Multimedia applications include video/audio codecs, speech recognition, and echo cancellers. The speech recognition requires the speech extraction and autocorrelation coefﬁcient computations [2] in the voice command application. The video codec is the most challenging element of a multimedia application, since it requires much processing power and bandwidth. Hence, a ﬂexible and low cost pipeline processor with the superiority of high processing rate is required to realize necessary computation-intensive algorithms, such as 256-point fast Fourier transform/inverse fast Fourier transform (FFT/IFFT) and 8 8 2-D discrete cosine transform (DCT) [3]–[5]. Additionally, a major integration challenge is to design the digital baseband and accompanying control logic. The WiMAX baseband is constructed around orthogonal frequency division multiplexing (OFDM) technology requiring high processing throughput. The ﬁxed, IEEE 802.16e [1], version of WiMAX also needs a 256-point FFT computation. Many researchers have recently concentrated on designing an optimized reconﬁgurable DSP processor to achieve a high processing rate and low power consumption in next-generation mobile multimedia applications [3], [4]. The software-based architecture such as the coprocessor and dual-MAC designs have been proposed by Chai et al. [3] and Kolagotla et al. [4], respectively. However, they induce the large chip size because of the high ﬂexibility. Vorbach et al. have also presented hardware-based concepts such as the processing element (PE) array [5], which achieves a high processing rate with reasonable ﬂexibility. However, the processing kernel has the ﬂaw of a low utilization rate with a large array memory and muti-MACs, leading to poor cost efﬁciency. The speciﬁc application-speciﬁc integrated circuit (ASIC)-based design on a fast computation algorithm provides high cost efﬁciency [6]–[8]. Tell et al. [6] presented the FFT/WALSH/1-D DCT processor for multiple radio standards of the upcoming fourth generation wireless systems. Conversely, some designs [6]–[8] only support 1-D DCT computation, and have no 2-D DCT support. However, 2-D DCT is desirable for the video compression among wireless communication applications. This study not only presents a single reconﬁgurable architecture for the 256-point FFT/IFFT modes and the 8 8 2-D DCT mode, but also achieves high cost-efﬁciency in portable multimedia applications. He et al. [9] has presented several reliable architectures and the detailed comparisons of the corresponding hardware cost for efﬁcient pipeline FFT processors. The comparison results of these architectures indicate that the has the highest butterﬂy utilization and lowest hardware resource usage in the pipeline FFT/IFFT architecture. However, the radix-21063-8210/$25.00 © 2008 IEEELIN et al.: COST-EFFECTIVE TRIPLE-MODE RECONFIGURABLE PIPELINE FFT/IFFT/2-D DCT PROCESSOR1059algorithm has a higher multiplicative complexity than the high-radix FFT and other mixed-radix algorithms. Results of comprehensive comparison further indicate that the proposed -based pipeline processor achieves a higher utilization with a smaller hardware requirement than -based pipeline processor [9] in the 256-point FFT/IFFT mode, and thus has higher cost efﬁciency. The proposed -based design also achieves satisfactory performance for the DV encoding standard with the lowest cost in the 8 8 2-D DCT mode. Finally, the chip implementation has been applied in the TSMC 0.13- m 1P8M CMOS process, which only consumes 22.37 mW under 1.2 V at 100 MHz. Thus, the proposed design is very appropriate for reconﬁgurable system-on-chips (RSoCs) IP in next-generation mobile multimedia applications. The remainder of this study is structured as follows. A new FFT/IFFT and 8 8 2-D DCT algorithm is given in Section II. Section III demonstrates the proposed FFT/IFFT/2-D DCT algorithm. The ﬁnite pipeline architecture using the wordlength analysis is given in Section IV, and indicates that the proposed architecture achieves the required system performance in both 256-point FFT/IFFT and 8 8 2-D DCT modes with the lowest hardwire cost. Section V tabulates the comparison results in terms of hardware utilization and cost to demonstrate the high cost-efﬁciency of the proposed architecture, and also discusses the chip implementation. The ﬁnal section draws conclusions. II. NEW RADIX-4 -BASED FFT/IFFT AND 8 ALGORITHM A. Radix-4 -Based FFT Formula The FFT of the -point input is given by (1) where . Applying a 3-D linear index map 8 2-D DCTwhere the butterﬂy structure of the ﬁrst stage takes the form(4) Notably, the radix-4 butterﬂy structure only requires trivial multiplication, which involves real-imaginary swapping and sign inversion, and which does not require any complex multiplication of the type shown in (4). The structure has only four four-input complex adders and some shufﬂe circuits. Decomposing the FFT computation with a higher radix, induces increasingly complex multiplications in the single butterﬂy structure. The radix-4 based is more cost-efﬁcient than higher-radix-based butterﬂy structures. Following a similar decomposition procedure, (3) can be decomposed as(5) Meanwhile, the butterﬂy structure of the second stage can be obtained as(6) Clearly, the decomposition creates three multipliers: , , and , as written in (6). The three full complex multipliers from the second butterﬂy stage can be simpliﬁed as one architecture. single constant multiplier in the proposed The constant multiplier cost can be further reduced by applying the subexpression elimination algorithm. The detailed hardware structure is described in Section III. The second radix-4 butterﬂy structure in (6) is the same as the ﬁrst radix-4 butterﬂy structure in (4) after simpliﬁcation of the common factor of the constant multiplier. The complete radix-4 DIF FFT algorithm is obtained by applying the CFA procedure recursively to the remaining FFTs of length N/16 in (5), as illustrated in Fig. 1. Fig. 1 indicate that the proposed radix-4 algorithm decomposes the -points FFT computation by cascading the number of radix-16-based butterﬂy (R16-BF) computations, which can be split into two cascading radix-4-based butterﬂy (R4-BF) computations as depicted in (4) and (6). When the variables of , , and were treated as constants for each single output as depicted in (3) and (5), the summation rages indicate that the required computation results of ﬁrst and second(2) The common factor algorithm (CFA) [10] form can be written as(3)1060IEEE TRANSACTIONS ON VERY LARGE SCALE INTEGRATION (VLSI) SYSTEMS, VOL. 16, NO. 8, AUGUST 2008where the butterﬂy structure of the ﬁrst and second stage has the formFig. 1. CFA decomposition procedure of the proposed radix-4 -based FFT algorithm.N -point(9a) andradix-4 butterﬂy stage were N/4 and N/16, respectively, as depicted in Fig. 1. The radix-4 algorithm has the same multiplicative complexity as the radix-16 algorithm, but still retains the radix-4 butterﬂy structure. Signiﬁcantly, the radix-16 algorithm clearly has a lower multiplicative complexity than other low-radix algorithm, such as a radix-2 algorithm. For instance, the number of complex multiplications of the 256-point FFT algorithms are computation adopting the radix-2 and radix 1539 and 224, respectively. Thus, the proposed design based on the new radix-4 algorithm has a lower multiplication comdesign [9], [12]. Furthermore, plexity (85.4%) than the as mentioned before, the radix-4 algorithm does not require any multiplication in the single butterﬂy structure. B. Radix-4 -Based IFFT Formula Following the similar procedure, the radix-4 IFFT algorithm can be obtained as follows. The IFFT of the -point input is given by (7) can be implemented with the In (7), the coefﬁcient simple right-shift circuit. This study focuses on the 256-point pipeline FFT/IFFT computations, where the eight right-shift is induced. Thus, the IFFT derivation results can be written as(9b) Notably, the only difference between FFT and IFFT algorithm are the sign bits as given in (4), (6), (9a), and (9b). Therefore, the pipeline FFT/IFFT processor can be easily implemented with a single module by controlling the sign coefﬁcient. Additionally, the proposed pipeline IFFT processor has a similar butterﬂy structure and a single constant multiplier structure with the proposed pipeline FFT processor, which could replace the three multipliers: , , and . C. 8 8 2-D FFT and 8 8 2-D DCT FormulaTwo concurrent 2-D DCTs can be calculated by the single 2-D shifted FFT (SFFT) algorithm [11] from the input reordering and post computation. This study presented a high-speed pipeline processor to support the triple-mode 256-point FFT/IFFT/8 8 2-D DCT with the radix-4 algorithm. Two concurrent 2-D DCTs results can be obtained by the proposed radix-4 -based architecture in the 8 8 2-D DCT mode. The 8 8 2-D DCT of the input signal is given by(10) neglects the post-scaling factor in (10). The input data could then be reordered as This study of(8) (11)LIN et al.: COST-EFFECTIVE TRIPLE-MODE RECONFIGURABLE PIPELINE FFT/IFFT/2-D DCT PROCESSOR1061where . After the scaling and input data reordering of (11), (10) can be recast asof tively, can then be created as,respec-(12) The value of is then calculated with the 8 SFFT with a time-domain shift of 1/4 samples 8 2D(17a)(17b) To help understand the serial pipeline operation, the 2-D locaand can be substituted as tion and , respectively. Then, the speciﬁc 2-D linear index map is applied as follows: (13) where of the input signal . In (13), the 8 8 2-D FFT is given bywhere(14) (18) where . Since the input data form a real-valued sequence, the second half output can be derived as The word numbers of the shift registers in the post-computation of the fourth stage can be minimized by following the speciﬁc mapping in (18). The 8 8 2-D FFT CFA form can then be written as(15) where . By combining (13) and (15), the 8 8 2-D DCT output can be recast as (16) . Equation (16) adopts only the real value and the imaginary value of to calcu. By combining two reordered input sequences for two independent sequences , and forming a complex input sequence , the double throughput of 2-D 8 8 DCT of can be derived by single 2-D 8 8 SFFT computation. Consequently, two independent 8 8 2-D DCTs where of late the (19) The butterﬂy structures for 8 8 2-D DCT, corresponding to (10)–(19), are summarized as follows. Butterﬂy stage I:1062IEEE TRANSACTIONS ON VERY LARGE SCALE INTEGRATION (VLSI) SYSTEMS, VOL. 16, NO. 8, AUGUST 2008Butterﬂy stage II:Butterﬂy stage III:Fig. 2. Block diagram of the R4 SDF-based 256-point FFT/IFFT and 8 2-D-DCT architecture.28The additional stage of 2-D DCT:The time-domain shift stage of 2-D DCT:Butterﬂy stage IV:(20) 8 2-D DCT computation results and are calculated concurrently in the post-computation of the butterﬂy stage IV. The 8 8 2-D IDCT computation can also be obtained following a similar decomposition procedure. Because of the cost-effective constraint in the physical design, this study only considers the triple-mode FFT/ IFFT and 2-D DCT computations. The derivation results of the radix-4 -based FFT/IFFT/2-D DCT algorithm indicate that all butterﬂy computation can be easily implemented with four fourinput complex adders and some shufﬂe circuits. The radix-4 butterﬂy structure has no multipliers. Additionally, the regular structure can be easily derived in both the 8 8 2-D DCT and 256-point FFT/IFFT pipeline processor architecture. Two 8 III. PIPELINE 256-POINT FFT/IFFT/8 8 2-D-DCT PROCESSOR ARCHITECTURE He et al. presented several pipeline FFT/IFFT architectures [9]. The serial delay feedback (SDF)-based architecture isknown to have a low hardware cost and high cost-efﬁciency advantages with the feedback type shift registers architecture [9], [12], [13]. The delay-feedback type shift register approaches are always more efﬁcient than other corresponding approaches in terms of memory utilization since the butterﬂy output share the same storage with its input [9]. The pipeline architecture has the same computation complexity as the radix-4 algorithm, and few hardware requirements as the radix-2 algorithm. This study presents a reconﬁgurable pipeline architecture with a low computation complexity as the radix-16 algorithm, and low hardware requirements as the radix-4 algorithm. Signiﬁcantly, the proposed triple-mode radix-4 butterﬂy structure, like the radix-2 butterﬂy structure, does not require a complex multiplier or constant multiplier. Section V presents detailed comparisons between the and architectures. This section describes a novel radix-4 single-path delay feedback architecture to support the three modes, 256-point FFT, 256-point IFFT, and 8 8 2-D DCT, based on the radix-4 DIF FFT algorithm obtained in Section II. Fig. 2 shows a block diagram of the purposed -based 256-point FFT/IFFT and 8 8 2-D DCT pipeline processor. Based on the proposed pipeline architecture, the cost-effective 256-point FFT/IFFT processor is ﬁrst constructed. This processor only requires four butterﬂy stages with 255-word shift registers, two constant multipliers and one complex multiplier with one coefﬁcient ROM, represented by black solid color in Fig. 2. Fig. 2 indicates that a single data stream passes through two constant multipliers and one complex multiplier to realize different combinations of , , and of , as illustrated in (3). Using some control circuits, one additional radix-2 butterﬂy (R2-BF) with an one-word shift register and additional eight-word shift register, two concurrent 2-D-DCT operations are calculated from the single 2-D-SFFT computation as depicted in (11), (17a), and (17b). These extra circuits were embedded at the ﬁrst butterﬂy stage, the additional stage, and the fourth butterﬂy stage, which is represented by the gray color in Fig. 2. These circuits complete the input reordering, time domain shift, and post-computation in the 8 8 2-D DCT computation, respectively. Notably, only one radix-2 butterﬂy and nine-word shift register are needed to support additional two 8 8 2-D DCT computations in the original pipeline SDF-based FFT/IFFT architecture. The proposed architecture has, in total, four radix-4 butterﬂies, one radix-2 butterﬂy, two constant multipliers, oneLIN et al.: COST-EFFECTIVE TRIPLE-MODE RECONFIGURABLE PIPELINE FFT/IFFT/2-D DCT PROCESSOR1063TABLE I CORRESPONDING EQUATION NUMBERS FOR EACH BUILDING BLOCKB. Memory Structure The memory structure of butterﬂy stage is well known to be an important issue for the high cost-effective FFT/IFFT pipeline processor design. From the exiting researches, there are mainly two different approaches: delay commutator (DC) [9] and delay feedback (DF) [9], [12], [13]. In this study, the DF-based memory structure is adopted and depicted in Fig. 2. In order to compute the radix-4-based butterﬂy computations, the input data and the intermediate results have to be reordered as four concurrently data streams using memory as shown in Fig. 3. Each radix-4 butterﬂy unit applies the three parallel memories to store the serial data input and butterﬂy output in the feedback paths as presented in Fig. 2. The timing sequence of N-point FFT computation can be divided into four stages, each containing N/4 clock cycles. In the ﬁrst N/4 cycles (i.e., ﬁrst stage), the butterﬂy units simply store the input samples into the ﬁrst feedback memory. Similarly, the second and third feedback memory are ﬁlled in the second and third stages. , After the 3N/4 cycles, the butterﬂy units retrieves the , and samples from the feedback memory, performs corresponding operations with the sample and then feeds the output into the next butterﬂy units as depicted in Fig. 2. The required number of memory cells for the th stage is . Thus, the 256-points FFT/IFFT computations require the 64 3, 16 3, 4 3, and 1 3 word shift registers in the ﬁrst, second, third, and fourth butterﬂy stages, respectively. Signiﬁcantly, the SDF-based pipeline FFT/IFFT structure is highly regular, which has the high effective memory structure with the simpler routing complexity [9], [12], [13]. In this study, the shift registers were all realized by the cascaded ﬂip-ﬂops, which are composed of two latch circuits. C. Input Reordering and First Butterﬂy Computation Consider the new radix-4 algorithm presented in Section II. The proposed SDF architecture is estimated to need 64 3-word shift registers at the ﬁrst butterﬂy stage in the 256-point 8 2-D DCT mode only FFT/IFFT mode. Although the 8 requires 16 3-word shift registers at the ﬁrst butterﬂy stage, the 8 8 2-D DCT mode needs a swapping buffer to complete the input reordering and post-computation in (11) and (17) from the 8 8 2-D SFFT computation. Notably, the number of shift registers at the fourth butterﬂy stage for the post computation in the 8 8 2-D DCT mode depends on the sequential order of the input data at the ﬁrst butterﬂy stage. Following the speciﬁc linear mapping in (18), the number of shift registers can be reduced to only eight words at the fourth butterﬂy stage, as revealed in Fig. 2. Comparing with the other linear mapping, the proposed architecture could reduce at least 96% shift registers cost. The segmented shift registers (SSR) structure is also proposed to realize both the input reordering and butterﬂy computation operation at the ﬁrst stage to support the 256-point FFT/IFFT and 8 8 2-D DCT modes. Fig. 4(a) shows the 12 proposed operation mechanisms of the ﬁrst butterﬂy stage to ﬁnish the input reordering and the ﬁrst-stage computation. Operation mechanisms 0–3 are adoptedFig. 3. Block diagram of the radix-4 butterﬂy architecture.complex multiplier, one coefﬁcient ROM, and a 264-word shift register. To help understand the corresponding functions of each building block, the respective equation numbers related to each element are shown in Table I. The detailed operations of each element are described as follows. A. Radix-4 Butterﬂy and Radix-2 Butterﬂy The derivation results of the radix-4 -based algorithm reveal that both the FFT/IFFT butterﬂy computation in (4), (6), (9a), (9b), and the 8 8 2-D-DCT butterﬂy computation in (20), can be easily completed with the radix-4-based butterﬂy architecture. The only difference between the 8 8 2-D-DCT and the 256-point FFT/IFFT butterﬂy computation is the summands and minuends at the butterﬂy stage one and four, which can be easily realized with the multiplex circuits in the radix-4 butterﬂy structure. Signiﬁcantly, the number of the two ﬁrst stages in the 8 8 2-D-DCT computation can be completed concurrently at the ﬁrst butterﬂy stage in parallel. Additionally, the input reordering operation at the ﬁrst stage and the post-computation operation at the fourth stage of the 8 8 2-D-DCT mode are described in detail. Fig. 3 illustrates the proposed radix-4 butterﬂy structure, which only includes four four-input complex adders with no complex multipliers inside. This conﬁguration means that the proposed radix-4 butterﬂy structure has a low hardware cost of higher-radix butterﬂy structures. Moreover, the proposed radix-4 algorithm has the same complex multiplication complexity as the radix-16 algorithm, so has the high cost efﬁciency of lower radix architectures. To obtain the additional stage of the 8 8 2-D DCT mode in (20), one additional SDF-based radix-2 butterﬂy structure with one word shift register is required, as illustrated in Fig. 2. The additional stage of the 2-D DCT mode only requires two 2-input complex adders and one shift register, giving it a small hardware penalty.1064IEEE TRANSACTIONS ON VERY LARGE SCALE INTEGRATION (VLSI) SYSTEMS, VOL. 16, NO. 8, AUGUST 2008Fig. 4. Block diagram of the proposed ﬁrst radix-4 butterﬂy stage in the R4 SDF-based 256-point FFT/IFFT and 8 8 2-D-DCT architecture. (a) The proposed 12 reconﬁgurable operation mechanisms of the ﬁrst butterﬂy stage. (b) The timing sequences of operation mechanism in the ﬁrst butterﬂy stage. (c) The storage content in SSR in the 8 8 2-D-DCT mode. (d) The content of the 8 8 2-D DCT computation result in SSR.222in the FFT/IFFT computation, and operation mechanisms 4–11 are adopted in the 8 8 2-D DCT computation. In the 8 8 2-D DCT mode, reconﬁgurable operation mechanisms 5 and 6 are adopted for the butterﬂy computation, and reconﬁgurable operation mechanisms 4 and 7–11 are adopted for the input reordering. Fig. 4(b) lists the corresponding timing sequence of the ﬁrst butterﬂy stage, which discusses the relationships among the input data, output data, and respective operation mechanisms during each clock (block number) in the 8 8 2-D DCT mode. Additionally, Fig. 4(c) and (d) illustrate the data content before and after the butterﬂy computation in SSR, respectively. The ﬁrst stage butterﬂy computation is completed by applying operation mechanisms 5 and 6. Most results of 8 8 2-D DCT computation are then pushed back into the SSR, as shown in Fig. 4(d), where denotes the computation results from (20). Fig. 4(d) presents the complete computation results. The 64 3-word shift register is segmented as , which is easily realized by three dependent clock domains with a simple 3-bit counter controller, as depicted in Fig. 4(c). These three segments in the SSR are called the power-saving, swapping and storage segments, and their sizes are 40 3, 8 3,and 16 3, respectively. Since 2-D DCT mode as depicted in (19) has a low computation complexity, the ﬁrst, second, and third butterﬂy stages have shift registers comprising 40 3, 8 3, and 2 3 words, respectively. These shift registers are set as power-saving segments, and gated to reduce power consumption. To perform the input reordering operation, 64 serial input data words are split into eight blocks of eight words, as shown in Fig. 4(b) and (c). The block numbers, written inside the brackets in Fig. 4(b) and (c), denote the serial input sequential order of eight-word blocks. In Fig. 4(c), the terms and , respectively, represent the 2-D DCT image data in the previous and current frame, which both contain 64 points in each frame. Following the operation mechanisms 4, 7, 8, 9, 10, and 11 in Fig. 4(a), the serial input data of each block adopt the swapping segment as the swapping space to achieve the required storage ordering in the storage segment. The detail timing sequence of the proposed 8 8 2-D DCT computations is given as follows. Operation mechanism 4 pushes the input data into the swapping segment from the clock number 0 to 7 (block number 0). At the same time, the original data in the swapping segment areLIN et al.: COST-EFFECTIVE TRIPLE-MODE RECONFIGURABLE PIPELINE FFT/IFFT/2-D DCT PROCESSOR1065Fig. 5. Block diagram of the proposed constant multiplier architecture.simultaneously pushed into the storage segment as illustrated in Fig. 4(a) and (c). In the following 16 clock cycles, operation mechanisms 5 and 6 replace the swapping segments with the and (i.e., block number 1 input data and 2), as presented in Fig. 4(a)–(c). Operation mechanisms 5 and 6 provide the original swapping segment data and for the ﬁrst butterﬂy stage computation in (20), along with the data in the storage segment. At the same time, 48 new 8 8 2-D DCT results of , as listed in the top three rows of Fig. 4(d), are pushed into the storage segment by the feedback path. Furthermore, the other 16 new 8 8 2-D DCT results, which are listed in the ﬁnal row in Fig. 4(d), are pushed directly to the second butterﬂy stage, as shown in Fig. 4(b). Notably, the 48 different 2-D DCT results in the storage segment are pushed out one-by-one due to the swapping operation by the swapping segment data in the following 48 clock cycles. Following a similar procedure, the serial input data of block numbers 3, 4, 5, and 7 complete their respective swapping operations by operation mechanisms 7, 8, 9, and 11. The block number 6 is stored into the storage segment directly by operation mechanism 10 as illustrated in Figs. 4(a) and (c). The input reordering operation is ﬁnished after a period of 64 clock cycles, which includes 7 clock cycles of input swapping latency. D. Constant Multiplier Based on the derivation results in Section II, the radix-4 algorithm requires some complex multiplications, namely , , in the 256-point FFT/IFFT mode in (6), (9), and , , , in the 8 8 2-D DCT mode in (20). Due to the ﬁnite range of and in (6) and (9b), namely 0–3, the three complex multiplications, , , and can be written as ,, and . Follow -ing the similar procedure, , , , and in (20) can be expanded as , , , and . The system has in total 38 different twiddle factor values, which could be implemented as 38 different constant multipliers by only shifters and adders. Based on the SDF-based architecture, the proposed design only has to calculate one complex multiplication in (6), (9), and (19) during each clock cycle. The 38 twiddle factor values can thus be reduced to the extension of two different values of and using the complex conjugate symmetry rule. Accordingly, the other 36 twiddle factor values can be expressed as the real-imaginary swapping or sign inversion of these two constant values. Moreover, the repeated shifters and adders of two constant multipliers could be simpliﬁed using the subexpression elimination algorithm [14] as illustrated in Fig. 5. According to our implementation results, the small cost penalty for the multiplexer control (i.e., S0, S1, and S2) could be neglected as shown in Fig. 5. Following the three steps to reduce the complex multipliers to the most economical constant multipliers are summarized as follows. First, the twiddle factors from (6), (9), and (20) are realized as the constant multipliers, which only contain shifters and adders as shown in Fig. 2. Second, the complex conjugate symmetry rule is applied to decrease the number of complex multiplications (12) to only two constant multiplications per stage with some shufﬂe circuits as shown in Fig. 5, thus achieving a constant multiplier cost reduction of 94.7%. Finally, the subexpression elimination algorithm [14] is adopted to reduce the number of shift circuits by more than 20%, and the number of complex adders by 50% in the one constant multiplier, as depicted in Fig. 5. The strictest constant multiplier is obtained in the purposed architecture by following these three steps. The cost penalty of the constant multiplier is thus minimized.。