H. A fast algorithm for finding crosswalks using figure-ground segmentation

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应用快速偶极子法与RACA法快速求解导体目标RCS

应用快速偶极子法与RACA法快速求解导体目标RCS胡倩倩;孙玉发【摘要】文章将快速偶极子法(fast dipole method,FDM)结合再压缩自适应交叉近似(recompressed adaptive cross approximation,RACA)算法应用于导体目标雷达散射截面(radar cross section,RCS)的计算.快速偶极子法是在等效偶极子法的基础上,将远场组相互作用的偶极子之间的距离通过泰勒级数展开,实现矩阵向量积的快速计算.为了进一步加快近场组互阻抗元素的填充,采用RACA算法对阻抗矩阵进行进一步压缩.与传统FDM相比,计算时间和内存得到了有效缩减,数值结果证明了该方法的有效性和精确性.%The fast dipole method(FDM)combined with recompressed adaptive cross approximation (RACA)algorithm is used to solve the radar cross section(RCS)for perfect conducting targets.T he FDM,which is based on the equivalent dipole-moment method(EDM),uses a simple Taylor's series to expand the distance between the interacting equivalent dipoles in far-field groups and realizes the fast calculation of matrix vector product.In order to speed up the calculation of mutual impedance ele-ments in the near-field groups,the RACA algorithm is used to further compress the impedance ma-trix.T he computational time and memory consumption of the proposed method are reduced effectively compared with the traditional FDM.Numerical results are presented to demonstrate the efficiency and accuracy of this method.【期刊名称】《合肥工业大学学报（自然科学版）》【年(卷),期】2018(041)002【总页数】4页(P207-210)【关键词】快速偶极子法(FDM);等效偶极子法;再压缩自适应交叉近似(RACA)算法;导体目标;雷达散射截面(RCS)【作者】胡倩倩;孙玉发【作者单位】安徽大学计算智能与信号处理教育部重点实验室,安徽合肥 230601;安徽大学计算智能与信号处理教育部重点实验室,安徽合肥 230601【正文语种】中文【中图分类】TN011矩量法(method of moments,MoM)在雷达散射截面(radar cross section,RCS)的计算、电磁环境的预测问题中有着广泛的应用,但是,随着目标电尺寸的不断增大,计算复杂度和内存需求迅速增加,普通计算机难以负担。

《人工智能-一种现代方法》第四版习题答案

Chapter 2
2.1 Define in your own words the following terms: agent, agent function, agent program, rationality, reflex agent, model-based agent, goal-based agent, utility-based agent, learning agent. The following are just some of the many possible definitions that can be written:
1.11 “surely computers cannot be intelligent-they can do only what their programmers tell them.” Is the latter statement true, and does it imply the former? This depends on your definition of “intelligent” and “tell.” In one sense computers only do what the programmers command them to do, but in another sense what the programmers consciously tells the computer to do often has very little to do with what the computer actually does. Anyone who has written a program with an ornery bug knows this, as does anyone who has written a successful machine learning program. So in one sense Samuel “told” the computer “learn to play checkers better than I do, and then play that way,” but in another sense he told the computer “follow this learning algorithm” and it learned to play. So we’re left in the situation where you may or may not consider learning to play checkers to be s sign of intelligence (or you may think that learning to play in the right way requires intelligence, but not in this way), and you may think the intelligence resides in the programmer or in the computer

A Sequential Algorithm for Generating Random Graphs

A Sequential Algorithm for Generating Random Graphs
Mohsen Bd Amin Saberi1
arXiv:cs/0702124v4 [] 16 Jun 2007
Stanford University {bayati,saberi}@ 2 Yonsei University jehkim@yonsei.ac.kr
(FPRAS) for generating random graphs; this we can do in almost linear time. An FPRAS provides an arbitrary close approximaiton in time that depends only polynomially on the input size and the desired error. (For precise deﬁnitions of this, see Section 2.) Recently, sequential importance sampling (SIS) has been suggested as a more suitable approach for designing fast algorithms for this and other similar problems [18, 13, 35, 6]. Chen et al. [18] used the SIS method to generate bipartite graphs with a given degree sequence. Later Blitzstein and Diaconis [13] used a similar approach to generate general graphs. Almost all existing work on SIS method are justiﬁed only through simulations and for some special cases counter examples have been proposed [11]. However the simplicity of these algorithms and their great performance in several instances, suggest further study of the SIS method is necessary. Our Result. Let d1 , . . . , dn be non-negative integers given for the degree sequence n and let i=1 di = 2m. Our algorithm is as follows: start with an empty graph and sequentially add edges between pairs of non-adjacent vertices. In every step of the procedure, the probability that an edge is added between two distinct ˆj (1 − di dj /4m) where d ˆi and d ˆj denote ˆi d vertices i and j is proportional to d the remaining degrees of vertices i and j . We will show that our algorithm produces an asymptotically uniform sample with running time of O(m dmax ) when maximum degree is of O(m1/4−τ ) and τ is any positive constant. Then we use a simple SIS method to obtain an FPRAS for any ep, δ > 0 with running time O(m dmax ǫ−2 log(1/δ )) for generating graphs with dmax = O(m1/4−τ ). Moreover, we show that for d = O(n1/2−τ ), our algorithm can generate an asymptotically uniform d-regular graph. Our results are improving the bounds of Kim and Vu [34] and Steger and Wormald [45] for regular graphs. Related Work. McKay and Wormald [37, 39] give asymptotic estimates for number of graphs within the range dmax = O(m1/3−τ ). But, the error terms in their estimates are larger than what is needed to apply Jerrum, Valiant and Vazirani’s [25] reduction to achieve asymptotic sampling. Jerrum and Sinclair [26] however, use a random walk on the self-reducibility tree and give an FPRAS for sampling graphs with maximum degree of o(m1/4 ). The running time of their algorithm is O(m3 n2 ǫ−2 log(1/δ )) [44]. A diﬀerent random walk studied by [27, 28, 10] gives an FPRAS for random generation for all degree sequences for bipartite graphs and almost all degree sequences for general graphs. However the running time of these algorithms is at least O(n4 m3 dmax log5 (n2 /ǫ)ǫ−2 log(1/δ )). For the weaker problem of generating asymptotically uniform samples (not an FPRAS) the best algorithm was given by McKay and Wormald’s switching technique on conﬁguration model [38]. Their algorithm works for graphs 2 2 3 2 with d3 max =O(m / i di ) with average running i di ) and dmax = o(m + 2 2 2 4 time of O(m + ( i di ) ). This leads to O(n d ) average running time for dregular graphs with d = o(n1/3 ). Very recently and independently from our work, Blanchet [12] have used McKay’s estimate and SIS technique to obtain an FPRAS with running time O(m2 ) for sampling bipartite graphs with given

移相相关法计算相位差的研究

移相相关法计算相位差的研究刘玉周;赵斌【摘要】为了提高相位式测距仪的测量精度，采用移相相关方法来估计两同频正弦信号的相位差。

首先将每路信号移相2π后和原信号做相关来计算自相关，以减少噪声的影响；其次用少许数据初步估算相位差，并将一路信号移相，使两路信号的相位差移到π/2（或3π/2）附近；然后用较多的采样数据计算两路信号的相位差，将结果再减去移相量得到最终的相位差。

同时分析了频率误差对相位差计算精度的影响，进行了理论分析和仿真实验验证。

结果表明，该方法计算的误差大大减小。

这对提高测距仪的测量精度是有帮助的。

%In order to improve the accuracy of a phase-shift range finder , a phase-difference algorithm based on phase-shift correlation analysis was proposed to estimate the phase-difference between two sinusoidal signals with same frequency .For reducing the influence of noise , the autocorrelation between the original and 2πshifted signal was calculated firstly.Secondly, the phase difference was estimated approximately with a few sampled data and the initial phase of one signal was shifted by Δθto make the phase difference between two signals to be near π/2(or 3π/2).Then, the phase-difference was calculated with whole set of data by correlation method and the final phase difference was obtained by subtracting Δθ.The influence of frequency error was analyzed .Theoretical analysis and simulation shows that the error of this method is greatly reduced .The proposed method can improve the accuracy of a range finder .【期刊名称】《激光技术》【年(卷),期】2014(000)005【总页数】5页(P638-642)【关键词】测量与计量;移相相关法;相位差;频率误差【作者】刘玉周;赵斌【作者单位】华中科技大学机械科学与工程学院仪器系，武汉430074;华中科技大学机械科学与工程学院仪器系，武汉430074【正文语种】中文【中图分类】TH741相位式激光测距在3-D成像［1］、机器人导航［2］、表面检测［3］等领域有着广泛的应用，它通过测量光波往返的相位差来计算时间延迟从而计算待测距离［4-5］。

a星算法预处理路径

a星算法预处理路径A*算法是一种启发式搜索算法，用于在图形空间中找到两个点之间的最短路路径。

除了起点和终点之外，它还需要一个评估函数来评估每个节点到目标节点的距离。

因为这个评估值是启发式的，所以A*算法能够在搜索空间中快速找到最优解。

与其他搜索算法不同的是，A*算法可以在不搜索整个搜索空间的情况下，找到最短路径。

A*算法的核心思想是使用一个评估函数f(n)来评估每个节点n的最小反向代价估计，这个评估函数的值是从起点到当前节点的代价g(n)和从当前节点到目标节点的最小估计代价h(n)之和，即f(n)=g(n)+h(n)。

其中h(n) 是它到目标节点的估计距离，g(n)是从起点到节点n的实际代价。

A*算法的预处理路径是将整个图形空间进行分类，将每个点都归入不同的类别中。

这样做的目的是为了让算法能够更快地搜索到目标节点。

预处理路径的过程一般包括两个步骤：建立地图和预处理路径。

建立地图时需要将地图分为不同的区域，确定每个区域的关系，并将每个区域编号。

这部分需要涵盖一定的算法和数据结构知识，如图形数据结构、二维数组、树、模拟人类思维过程的“分而治之”等。

在将地图分为不同的区域时需要考虑地图是否要求精细化处理，比如是否需要考虑建筑物的复杂形状、地形的起伏等因素。

同时，也要考虑到预处理路径的计算效率，是否需要对地图进行简化处理。

预处理路径是指在搜索之前，通过一些算法来计算出每个点到目标点的距离，这样可以加速搜索过程。

这个过程涉及到的算法有 Dijkstra 算法、BFS 算法、Floyd 算法等。

其中Dijkstra算法是一种确定路径的算法，可以用于单源最短路径问题。

BFS算法是一种广度优先搜索算法，在小型地图上表现很好，但在大型地图上会面临内存瓶颈。

Floyd算法是一种动态规划算法，可以用于求任意两点之间的最短路径，但计算量比较大。

综合考虑，一般使用 A* 算法来计算预处理路径。

在预处理路径时，需要考虑选择合适的启发式算法来评估每个节点的距离，同时需要考虑到搜索空间的大小和节点数量。

A星路径寻找算法的应用

摘
要
A*算法是一种启发式的搜索算法。使用 A*算法能够快速地在游戏地图中得到一条最优路径。本文通过对 A*算法的分析，结合面向对象的思想，用 C#语言实现可重用的 A*类库，并应用于一个小游戏中。
关键词
A*算法；代价；二叉堆；启发函数
I
Abstract
A * algorithm is a heuristic search algorithm. Using the A * algorithm can quickly get the optimal path in the game map. Based on the A* algorithm analysis, combined with object-oriented thinking, an A* reusable class librariy with the C # language is implemented and applied it to a small game.
Key words
A* Algorithm; Cost; Binary Heap; Heuristic Function
II
目
录
1 引言.................................................................................................................................................. 1 2 算法原理.......................................................................................................................................... 1 2.1 算法基本概念.......................................................................................................................1 2.1.1 地图（搜索空间）.....................................................................................................1 2.1.2 A*结点........................................................................................................................ 2 2.1.3 代价............................................................................................................................2 2.2 A* 算法................................................................................................................................. 2 2.2.1 启发函数与估价函数................................................................................................2 2.2.2 Open 表与 Closed 表..................................................................................................2 2.2.3 伪代码........................................................................................................................3 3 算法设计.......................................................................................................................................... 4 3.1 二叉堆（Binary Heap）...................................................................................................... 4 3.1.1 数据存储和方法........................................................................................................4 3.1.2 添加元素....................................................................................................................4 3.1.3 获取堆顶元素............................................................................................................5 3.1.4 删除堆顶元素............................................................................................................5 3.1.5 查找指定元素............................................................................................................7 3.1.6 获取和替换元素........................................................................................................7 3.2 AStarNode 抽象类............................................................................................................... 10 3.2.1 属性..........................................................................................................................10 3.2.2 方法..........................................................................................................................10 3.3 AStar 类................................................................................................................................ 11 3.3.1 属性.......................................................................................................................... 11 3.3.2 方法.......................................................................................................................... 11 3.4 AStarNode2D 类.................................................................................................................. 13 3.4.1 属性..........................................................................................................................14

带有模式选择的改进的方向性菱形搜索算法

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摘
星，陈祖爵
（江苏大学计算机科学与通信工程学院，江苏镇江２２１）１０３要：针对Ｈ２４ＡＣ标准的多模式运动估计算法编码模式复杂、算量大的不足，出了一种带有模式选．６／Ｖ计提
择的方向性菱形搜索算法。基于宏块最佳编码模式的统计分析，运动特征的不同进行分类，运动估计的直按将
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astar寻路算法原理 -回复

astar寻路算法原理-回复A*寻路算法原理及步骤一、简介A*（A-Star）寻路算法是一种常用的路径规划算法，用于找到两个点之间的最短路径。

它综合了Dijkstra算法和贪心算法的优点，既考虑了每个节点的代价，也考虑了每个节点到目标节点的预估代价。

本文将一步一步详细介绍A*寻路算法的原理和步骤。

二、原理A*算法的核心思想是使用一个估算函数来预测从起始节点到目标节点的代价，并在遍历过程中选择最小代价节点来进行扩展。

该算法综合了代价函数和启发函数的信息，以更快地找到最短路径。

其具体步骤如下：1. 初始化将起始节点添加到一个开放列表(open list)中，开放列表存放待扩展的节点。

同时，创建一个空的闭合列表(closed list)，用于存放已扩展过的节点。

2. 循环操作进入循环操作，直到开放列表为空或找到目标节点。

在每次循环中，选择开放列表中代价最小的节点进行扩展。

3. 节点扩展取开放列表中代价最小的节点，将其从开放列表中删除，并加入到闭合列表中。

然后，获取该节点的相邻节点，计算它们的代价和预估代价，并更新它们的代价值和路径。

4. 判断相邻节点对于每个相邻节点，判断它们是否在开放列表或闭合列表中。

若在闭合列表，则跳过该节点；若在开放列表，比较新路径与旧路径的代价，若新路径更好，则更新代价和路径；否则，不做任何操作。

5. 添加新节点对于不在开放列表中的相邻节点，将它们添加到开放列表中，并计算它们的代价和预估代价。

6. 重复操作重复步骤2至5，直到开放列表为空或找到目标节点。

若开放列表为空，则无法找到路径；若找到目标节点，则回溯路径，回到起始节点。

三、关键要点在上述步骤中，有几个关键要点需要注意：1. 代价函数代价函数用于计算节点到起始节点的实际代价，包括走过的距离、障碍物等影响因素。

根据具体情况，可以自定义代价函数。

2. 启发函数启发函数用于估算节点到目标节点的代价，即预测代价。

常见的启发函数有曼哈顿距离、欧几里得距离等，根据实际情况选择合适的启发函数。

[Archer, 88]

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一种新的矩阵平方根的迭代算法

一种新的矩阵平方根的迭代算法杨壮;彭振赟;牟继萍【摘要】For solving the constraint solution of the quadratic matrix equation X 2-A=0,an iterative algorithm is proposed. The convergence theorem of the algorithm for solving the symmetric solution of the quadratic matrix equation X 2-A=0 is proved.Numerical experiments illustrate that the algorithm is effective.%针对求解二次矩阵方程X 2-A=0的约束解问题，提出一种新的迭代算法，并给出该算法在求解二次矩阵方程对称解时的收敛性定理。

数值实验证明了算法的有效性。

【期刊名称】《桂林电子科技大学学报》【年(卷),期】2014(000)001【总页数】5页(P60-64)【关键词】二次矩阵方程;迭代算法;收敛性【作者】杨壮;彭振赟;牟继萍【作者单位】桂林电子科技大学数学与计算科学学院，广西桂林 541004;桂林电子科技大学数学与计算科学学院，广西桂林 541004;桂林电子科技大学数学与计算科学学院，广西桂林 541004【正文语种】中文【中图分类】O241.6矩阵方程X2－A＝0的解称为矩阵A 的平方根。

矩阵平方根在很多方面都有应用，如控制论、广义特征值问题、线性方程组预处理、偏微分方程边界问题。

对于矩阵平方根的存在性及个数的问题，Higham 在文献［1］和Cross等在文献［2］已经作出了详细的阐述。

求解矩阵平方根的方法大致可以分为直接法和迭代法2种。

直接法首先对矩阵A 进行Schur分解，使矩阵A 成为一个上三角矩阵，进而求解上三角矩阵的平方根，最后还原成矩阵A 的平方根。

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。

复信号fastica算法

复信号fastica算法
FastICA算法是一种用于盲源信号分离的算法，它可以从混合信号中分离出独立的源信号。

以下是FastICA控制台指令的使用方法：
1. 打开控制台：在命令行中输入“control”并按回车键，即可打开控制台。

2. 选择FastICA算法：在控制台中选择“信号处理”选项卡，然后在“盲源信号分离”中选择“FastICA”算法。

3. 导入数据：使用“import”命令导入需要分离的混合信号数据。

4. 参数设置：根据具体情况设置FastICA算法的参数，例如迭代次数、收敛阈值等。

5. 运行算法：使用“run”命令运行FastICA算法，开始分离源信号。

6. 查看结果：在控制台中查看分离出的源信号结果，可以使用绘图命令将结果可视化。

7. 保存结果：使用“save”命令将分离出的源信号结果保存到文件中，以便后续处理和分析。

需要注意的是，以上控制台指令的具体命令可能因不同版本或不同平台而有所不同，用户需根据具体的软件环境或编程环境进行调整和修改。

求解高维函数优化问题的交叉熵蝙蝠算法

【主题：求解高维函数优化问题的交叉熵蝙蝠算法】1.概述在现代科学和工程中，高维函数优化问题是一个十分重要且具有挑战性的问题。

针对这一问题，人们提出了各种各样的优化算法，其中包括了蝙蝠算法。

在本文中，我们将主要探讨一种用于求解高维函数优化问题的蝙蝠算法的变种——交叉熵蝙蝠算法，并深入讨论其原理、优势以及应用。

2.交叉熵蝙蝠算法的基本原理交叉熵蝙蝠算法是一种基于自然界中蝙蝠裙体觅食行为的优化算法。

它模拟了蝙蝠的搜索策略，通过调整蝙蝠的位置和频率来实现目标函数的最优化。

与传统蝙蝠算法相比，交叉熵蝙蝠算法在蝙蝠的位置更新和频率调整上引入了交叉熵理论，从而更好地适应了高维函数优化问题的特点。

具体来说，交叉熵蝙蝠算法通过不断调整蝙蝠裙体中每只蝙蝠的位置和频率，以逐步优化目标函数的取值。

在搜索的过程中，交叉熵蝙蝠算法引入了交叉熵概率来指导蝙蝠对目标的搜索方向和强度，使得算法具有更好的全局搜索能力和收敛速度。

通过不断迭代，蝙蝠裙体逐渐聚集到目标函数的最优解附近，从而得到最优解。

3.交叉熵蝙蝠算法的优势与传统的优化算法相比，交叉熵蝙蝠算法具有以下几点优势：（1）全局搜索能力强：交叉熵蝙蝠算法在搜索过程中引入了交叉熵概率，能够有效地引导蝙蝠裙体对目标进行全局搜索，从而避免陷入局部最优解。

（2）收敛速度快：由于引入了交叉熵概率，交叉熵蝙蝠算法能够在搜索过程中自适应调整蝙蝠的搜索方向和强度，从而加快算法的收敛速度，降低了求解高维函数优化问题的时间成本。

（3）适应性强：交叉熵蝙蝠算法能够自适应地调整蝙蝠的位置和频率，适应了不同问题的特点，具有更好的通用性和适用性。

4.交叉熵蝙蝠算法的应用交叉熵蝙蝠算法在实际问题中已经得到了广泛的应用，尤其在求解高维函数优化问题上表现出色。

在电力系统中的最优潮流计算、无线传感器网络中的能量优化以及机器学习领域的参数优化等问题中，交叉熵蝙蝠算法都取得了较好的效果。

5.个人观点与总结从个人角度来看，交叉熵蝙蝠算法作为一种新兴的优化算法，在求解高维函数优化问题方面表现出了很好的性能。

adaptive filter

Introduction to Adaptive Filters1. Basic ConceptsIn the last decades, the field of digital signal processing, and particularly adaptive signal processing, has developed enormously due to the increasingly availability of technology for the implementation of the emerging algorithms. These algorithms have been applied to an extensive number of problems including noise and echo canceling, channel equalization, signal prediction, adaptive arrays as well as many others.An adaptive filter may be understood as a self-modifying digital filter that adjusts its coefficients in order to minimize an error function. This error function, also referred to as the cost function, is a distance measurement between the reference or desired signal and the output of the adaptive filter.2. Basic configuration of an adaptive filterThe basic configuration of an adaptive filter, operating in thediscrete-time domain k, is illustrated in Figure 2.1. In such a scheme, the input signal is denoted by x(k), the reference signal d(k) represents the desired output signal (that usually includes some noise component), y(k) is the output of the adaptive filter, and the error signal is defined as e(k) = d(k).y(k).Fig. 2.1 Basic block diagram of an adaptive filter.The error signal is used by the adaptation algorithm to update the adaptive filter coefficient vector w(k) according to some performance criterion. In general, the whole adaptation process aims at minimizing some metric of the error signal, forcing the adaptive filter output signal to approximate the reference signal in a statistical sense.Fig. 2.2Channel equalization configuration of an adaptive filter: The output signal y(k) estimates the transmitted signal s(k).Fig. 2.3 Predictor configuration of an adaptive filter: The output signaly(k) estimates the presentinput sample s(k) based on past values of this same signal. Therefore, when the adaptive filter output y(k) approximates the reference, the adaptive filter operates as a predictor system.3. Adaptation algorithmSeveral optimization procedures can be employed to adjust the filter coefficients, including, for instance, the least mean-square (LMS) and its normalized version, the data-reusing (DR) including the affine projection (AP), and the recursive least-squares (RLS) algorithms. All these schemes are discussed in Section 2.3, emphasizing their main convergence and implementation characteristics. The remaining of the book focuses on the RLS algorithms, particularly, those employing QR decomposition, which achieve excellent overall convergence performance.3.1 Error MeasurementsAdaptation of the filter coefficients follows a minimization procedure of aparticular objective or cost function. This function is commonly defined as a norm of the error signal e(k). The three most commonly employed norms are the mean-square error (MSE), the instantaneous square error (ISE), and the weighted least-squares (WLS), which are introduced below.3.2 The mean-square errorThe MSE is defined as(k) = E[e2(k)] = E[|d(k)−y(k)|2].Where R and p are the input-signal correlation matrix and thecross-correlation vector between the reference signal and the input signal, respectively, and are defined asR = E[x(k)x T(k)],p = E[d(k)x T(k)].Note, from the above equations, that R and p are not represented as a function of the iteration k or not time-varying, due to the assumed stationarity of the input and reference signals.From Equation (2.5), the gradient vector of the MSE function with respect to the adaptive filter coefficient vector is given byThe so-called Wiener solution w o, that minimizes the MSE cost function, is obtained by equating the gradient vector in Equation (2.8) to zero. Assuming that R is non-singular, one gets that3.3 The instantaneous square errorThe MSE is a cost function that requires knowledge of the error function e(k) at all time k. For that purpose, the MSE cannot be determined precisely in practice and is commonly approximated by other cost functions. The simpler form to estimate the MSE function is to work with the ISE given byIn this case, the associated gradient vector with respect to the coefficient vector is determined asThis vector can be seen as a noisy estimate of the MSE gradient vector defined in Equation (2.8) or as a precise gradient of the ISE function,which, in its own turn, is a noisy estimate of the MSE cost function seen in Section 2.2.1.3.4 The weighted least-squaresAnother objective function is the WLS function given bywhere 0_⎣ < 1 is the so-called forgetting factor. The parameter ⎣ k−i emphasizes the most recent error samples (where i ≈k) in the composition of the deterministic cost function ⎩D(k), giving to this function the ability of modeling non-stationary processes. In addition, since the WLS function is based on several error samples, its stochastic nature reduces in time, being significantly smaller than the noisy ISE nature as k increases.2.3 Adaptation AlgorithmsIn this section, a number of schemes are presented to find the optimal filter solution for the error functions seen in Section 2.2. Each scheme constitutes an adaptation algorithm that adjusts the adaptive filter coefficients in order to minimize the associated error norm.The algorithms seen here can be grouped into three families, namely the LMS, the DR, and the RLS classes of algorithms. Each group presents particular characteristics of computational complexity and speed ofconvergence, which tend to determine the best possible solution to an application at hand.2.3.1 LMS and normalized-LMS algorithmsDetermining the Wiener solution for the MSE problem requires inversion of matrix R, which makes Equation (2.9) hard to implement in real time. One can then estimate the Wiener solution, in a computationally efficient manner, iteratively adjusting the coefficient vector w at each time instant k, in such a manner that the resulting sequence w(k) converges to the desired w o solution, possibly in a sufficiently small number of iterations. The LMS algorithm is summarized in Table 2.1, where the superscripts . and H denote the complex-conjugate and the Hermitian operations, respectively.The LMS algorithm is very popular and has been widely used due to its extreme simplicity. Its convergence speed, however, is highly dependent on the condition number p of the input-signal autocorrelation matrix[1–3],defined as the ratio between the maximum and minimum Eigen values of this matrix.In the NLMS algorithm, when = 0, one has w(k) = w(k−1) and theupdating halts. When υ= 1, the fastest convergence is attained at the price of a higher misadjustment then the one obtained for 0 <υ< 1.2.3.2 Data-reusing LMS algorithmsAs remarked before, the LMS algorithm estimates the MSE function with the current ISE value, yielding a noisy adaptation process. In this algorithm, information from each time sample k is disregarded in future coefficient updates. DR algorithms [9–11] employ present and past samples of the reference and input signals to improve convergence characteristics of the overall adaptation process.As a generalization of the previous idea, the AP algorithm [13–15] is among the prominent adaptation algorithms that allow trade-off between fast convergence and low computational complexity. By adjusting the number of projections, or alternatively, the number of data reuses, one obtains adaptation processes ranging from that of the NLMS algorithm to that of the sliding-window RLS algorithm [16, 17].2.3.3 RLS-type algorithmsThis subsection presents the basic versions of the RLS family of adaptive algorithms. Importance of the expressions presented here cannot be overstated for they allow an easy and smooth reading of the forthcoming chapters.The RLS-type algorithms have a high convergence speed which is independent of the Eigen value spread of the input correlation matrix. These algorithms are also very useful in applications where the environment is slowly varying.The price of all these benefits is a considerable increase in the computational complexity of the algorithms belonging to the RLS family.The main advantages associated to the QR-decomposition RLS(QRD-RLS) algorithms, as opposed to their conventional RLS counterpart, are the possibility of implementation in systolic arrays and the improved numerical behavior in limited precision environment.2.5 ConclusionIt was verified how adaptive algorithms are employed to adjust the coefficients of a digital filter to achieve a desired time-varying performance in several practical situations. Emphasis was given on the description of several adaptation algorithms. In particular, the LMS and the NLMS algorithms were seen as iterative schemes for optimizing the ISE, an instantaneous approximation of the MSE objective function. Data-reuse algorithms introduced the concept of utilizing data from past time samples, resulting in a faster convergence of the adaptive process. Finally, the RLS family of algorithms, based on the WLS function, was seen as the epitome of fast adaptation algorithms, which use all available signal samples to perform the adaptation process. In general, RLSalgorithms are used whenever fast convergence is necessary, for input signals with a high Eigen value spread, and when the increase in the computational load is tolerable. A detailed discussion on the RLS family of algorithms based on the QR decomposition, which also guarantees good numerical properties in finite-precision implementations, constitutes the main goals of this book. Practical examples of adaptive system identification and channel equalization were presented, allowing one to visualize convergence properties, such as misadjustment, speed, and stability, of several distinct algorithms discussed previously.。

Fast algorithms for frequent itemset mining using fp-trees

Fast Algorithms for FrequentItemset Mining Using FP-TreesGo¨sta Grahne,Member,IEEE,and Jianfei Zhu,Student Member,IEEE Abstract—Efficient algorithms for mining frequent itemsets are crucial for mining association rules as well as for many other data mining tasks.Methods for mining frequent itemsets have been implemented using a prefix-tree structure,known as an FP-tree,for storing compressed information about frequent itemsets.Numerous experimental results have demonstrated that these algorithms perform extremely well.In this paper,we present a novel FP-array technique that greatly reduces the need to traverse FP-trees,thus obtaining significantly improved performance for FP-tree-based algorithms.Our technique works especially well for sparse data sets.Furthermore,we present new algorithms for mining all,maximal,and closed frequent itemsets.Our algorithms use the FP-tree data structure in combination with the FP-array technique efficiently and incorporate various optimization techniques.We also present experimental results comparing our methods with existing algorithms.The results show that our methods are the fastest for many cases.Even though the algorithms consume much memory when the data sets are sparse,they are still the fastest ones when the minimum support is low.Moreover,they are always among the fastest algorithms and consume less memory than other methods when the data sets are dense.Index Terms—Data mining,association rules.æ1I NTRODUCTIONE FFICIENT mining of frequent itemsets(FIs)is a funda-mental problem for mining association rules[5],[6],[21], [32].It also plays an important role in other data mining tasks such as sequential patterns,episodes,multidimen-sional patterns,etc.[7],[22],[17].The description of the problem is as follows:Let I¼f i1;i2;...;i n g be a set of items and D be a multiset of transactions,where each transaction is a set of items such that I.For any X I,we say that a transaction contains X if X .The set X is called an itemset.The set of all X I(the powerset of I)naturally forms a lattice,called the itemset lattice.The count of an itemset X is the number of transactions in D that contain X. The support of an itemset X is the proportion of transactions in D that contain X.Thus,if the total number of transactions in D is n,then the support of X is the count of X divided by nÁ100percent.An itemset X is called frequent if its support is greater than or equal to some given percentage s,where s is called the minimum support.When a transaction database is very dense and the minimum support is very low,i.e.,when the database contains a significant number of large frequent itemsets, mining all frequent itemsets might not be a good idea.For example,if there is a frequent itemset with size l,then all 2l nonempty subsets of the itemset have to be generated. However,since frequent itemsets are downward closed in the itemset lattice,meaning that any subset of a frequent itemset is frequent,it is sufficient to discover only all the maximal frequent itemsets(MFIs).A frequent itemset X is called maximal if there does not exist frequent itemset Y such that X&Y.Mining frequent itemsets can thus be reduced to mining a“border”in the itemset lattice.All itemsets above the border are infrequent and those that are below the border are all frequent.Therefore,some existing algorithms only mine maximal frequent itemsets.However,mining only MFIs has the following deficiency: From an MFI and its support s,we know that all its subsets are frequent and the support of any of its subset is not less than s,but we do not know the exact value of the support. For generating association rules,we do need the support of all frequent itemsets.To solve this problem,another type of a frequent itemset,called closed frequent itemset(CFI),was proposed in[24].A frequent itemset X is closed if none of its proper supersets have the same support.Any frequent itemset has the support of its smallest closed superset.The set of all closed frequent itemsets thus contains complete information for generating association rules.In most cases, the number of CFIs is greater than the number of MFIs, although still far less than the number of FIs.1.1Mining FIsThe problem of mining frequent itemsets was first introduced by Agrawal et al.[5],who proposed algorithm Apriori.Apriori is a bottom-up,breadth-first search algorithm.It uses hash-trees to store frequent itemsets and candidate frequent itemsets.Because of the downward closure property of the frequency pattern,only candidate frequent itemsets,whose subsets are all frequent,are generated in each database scan.Candidate frequent item-set generation and subset testing are all based on the hash-trees.In the algorithm,transactions are not stored in the memory and,thus,Apriori needs l database scans if the size of the largest frequent itemset is l.Many algorithms,such as [28],[29],[23],are variants of Apriori.In[23],the kDCI method applies a novel counting strategy to efficiently determine the itemset supports without necessarily per-forming all the l scans..The authors are with the Department of Computer Science,ConcordiaUniversity,1455De Maisonneuve Blvd.West,Montreal,Quebec,H3G1M8,Canada.E-mail:{grahne,j_zhu}@cs.concordia.ca.Manuscript received28Apr.2004;revised27Nov.2004;accepted11Mar.2005;published online18Aug.2005.For information on obtaining reprints of this article,please send e-mail to:tkde@,and reference IEEECS Log Number TKDE-0123-0404.1041-4347/05/$20.00ß2005IEEE Published by the IEEE Computer SocietyIn[14],Han et al.introduced a novel algorithm,known as the FP-growth method,for mining frequent itemsets.The FP-growth method is a depth-first search algorithm.In the method,a data structure called the FP-tree is used for storing frequency information of the original database in a compressed form.Only two database scans are needed for the algorithm and no candidate generation is required.This makes the FP-growth method much faster than Apriori.In [27],PatriciaMine stores the FP-trees as Patricia Tries[18].A number of optimizations are used for reducing time and space of the algorithm.In[33],Zaki also proposed a depth-first search algorithm,Eclat,in which database is“verti-cally”represented.Eclat uses a linked list to organize frequent patterns,however,each itemset now corresponds to an array of transaction IDs(the“TID-array”).Each element in the array corresponds to a transaction that contains the itemset.Frequent itemset mining and candidate frequent itemset generation are done by TID-array ter,Zaki and Gouda[35]introduced a technique, called diffset,for reducing the memory requirement of TID-arrays.The diffset technique only keeps track of differences in the TID’s of candidate itemsets when it is generating frequent itemsets.The Eclat algorithm incorporating the diffset technique is called dEclat[35].1.2Mining MFIsMaximal frequent itemsets were inherent in the border notion introduced by Mannila and Toivonen in[20]. Bayardo[8]introduced MaxMiner which extends Apriori to mine only“long”patterns(maximal frequent itemsets). Since MaxMiner only looks for the maximal FIs,the search space can be reduced.MaxMiner performs not only subset infrequency pruning,where a candidate itemset with an infrequent subset will not be considered,but also a “lookahead”to do superset frequency pruning.MaxMiner still needs several passes of the database to find the maximal frequent itemsets.In[10],Burdick et al.gave an algorithm called MAFIA to mine maximal frequent itemsets.MAFIA uses a linked list to organize all frequent itemsets.Each itemset I corre-sponds to a bitvector;the length of the bitvector is the number of transactions in the database and a bit is set if its corresponding transaction contains I,otherwise,the bit is not set.Since all information contained in the database is compressed into the bitvectors,mining frequent itemsets and candidate frequent itemset generation can be done by bitvector and-operations.Pruning techniques are also used in the MAFIA algorithm.GenMax,another depth-first algorithm,proposed by Gouda and Zaki[11],takes an approach called progressive focusing to do maximality testing.This technique,instead of comparing a newly found frequent itemset with all maximal frequent itemsets found so far,maintains a set of local maximal frequent itemsets.The newly found FI is only compared with itemsets in the small set of local maximal frequent itemsets,which reduces the number of subset tests.In our earlier paper[12],we presented the FPmax algorithm for mining MFIs using the FP-tree structure. FPmax is also a depth-first algorithm.It takes advantage of the FP-tree structure so that only two database scans are needed.In FPmax,a tree structure similar to the FP-tree is used for maximality testing.The experimental results in[12]showed that FPmax outperforms GenMax and MAFIA for many,although not all,cases.Another method that uses the FP-tree structure is AFOPT [19].In the algorithm,item search order,intermediate result representation,and construction strategy,as well as tree traversal strategy,are considered dynamically;this makes the algorithm adaptive to general situations.SmartMiner [36],also a depth-first algorithm,uses a technique to quickly prune candidate frequent itemsets in the itemset lattice.The technique gathers“tail”information for a node in the lattice.The tail information is used to determine the next node to explore during the depth-first mining.Items are dynamically reordered based on the tail information. The algorithm was compared with MAFIA and GenMax on two data sets and the experiments showed that SmartMiner is about10times faster than MAFIA and GenMax.1.3Mining CFIsIn[24],Pasquier et al.introduced closed frequent itemsets. The algorithm proposed in the paper,A-close,extends Apriori to mine all CFIs.Zaki and Hsiao[34]proposed a depth-first algorithm,CHARM,for CFI mining.As in their earlier work in[11],in CHARM,each itemset corresponds to a TID-array,and the main operation of the mining is again TID-array intersections.CHARM also uses the diffset technique to reduce the memory requirement for TID-array intersections.The algorithm AFOPT[19]described in Section1.2has an option for mining CFIs in a manner similar to the way AFOPT mines MFIs.In[26],Pei et al.extended the FP-growth method to a method called CLOSET for mining CFIs.The FP-tree structure was used and some optimizations for reducing the search space were proposed.The experimental results reported in[26]showed that CLOSET is faster than CHARM and A-close.CLOSET was extended to CLOSET+by Wang et al.in[30]to find the best strategies for mining frequent closed itemsets.CLOSET+uses data structures and data traversal strategies that depend on the characteristics of the data set to be mined.Experimental results in[30]showed that CLOSET+outperformed all previous algorithms.1.4ContributionsIn this work,we use the FP-tree,the data structure that was first introduced in[14].The FP-tree has been shown to be a very efficient data structure for mining frequent patterns [14],[30],[26],[16]and its variation has been used for “iceberg”data cube computation[31].One of the important contributions of our work is a novel technique that uses a special data structure,called an FP-array,to greatly improve the performance of the algorithms operating on FP-trees.We first demonstrate that the FP-array technique drastically speeds up the FP-growth method on sparse data sets,since it now needs to scan each FP-tree only once for each recursive call emanating from it.This technique is then applied to our previous algorithm FPmax for mining maximal frequent itemsets.We call the new method FPmax*.In FPmax*,we also introduce our technique for checking if a frequent itemset is maximal,for which a variant of the FP-tree structure,called an MFI-tree, is used.For mining closed frequent itemsets,we have designed an algorithm FPclose which uses yet another variant of the FP-tree structure,called a CFI-tree,forchecking the closedness of frequent itemsets.The closednesschecking is quite different from CLOSET+.Experimentalresults in this paper show that our closedness checkingapproach is more efficient than the approach of CLOSET+.Both the experimental results in this paper and theindependent experimental results from the first IEEE ICDMWorkshop on frequent itemset mining (FIMI ’03)[3],[32]demonstrate the fact that all of our FP-algorithms have verycompetitive and robust performance.As a matter of fact,inFIMI ’03,our algorithms were considered to be the algo-rithms of choice for mining maximal and closed frequentitemsets [32].1.5Organization of the PaperIn Section 2,we briefly review the FP-growth method andintroduce our FP-array technique that results in the greatlyimproved method FPgrowth*.Section 3gives algorithmFPmax*,which is an extension of our previous algorithmFPmax,for mining MFIs.Here,we also introduce ourapproach of maximality checking.In Section 4,we givealgorithm FPclose for mining CFIs.Experimental results arepresented in Section 5.Section 6concludes and outlinesdirections for future research.2D ISCOVERING FI’S2.1The FP-Tree and FP-Growth MethodThe FP-growth method [14],[15]is a depth-first algorithm.Inthe method,Han et al.proposed a data structure called theFP-tree (frequent pattern tree).The FP-tree is a compact repre-sentation of all relevant frequency information in a database.Every branch of the FP-tree represents a frequent itemset andthe nodes along the branches are stored in decreasing order offrequency of the corresponding items with leaves represent-ing the least frequent pression is achieved bybuilding the tree in such a way that overlapping itemsetsshare prefixes of the corresponding branches.An FP-tree T has a header table,T :header ,associatedwith it.Single items and their counts are stored in theheader table in decreasing order of their frequency.Theentry for an item also contains the head of a list that links allthe corresponding nodes of the pared with breadth-first algorithms such as Apriori and its variants,which may need as many database scans as the length of the longest pattern,the FP-growth method only needs two database scans when mining all frequent itemsets.The first scan is to find all frequent items.These items are inserted into the header table in decreasing order of their count.In the second scan,as each transaction is scanned,the set of frequent items in it is inserted into the FP-tree as a branch.If an itemset shares a prefix with an itemset already in the tree,this part of the branch will be shared.In addition,a counter is associated with each node in the tree.The counter stores the number of transactions containing the itemset represented by the path from the root to the node in question.This counter is updated during the second scan,when a transaction causes the insertion of a new branch.Fig.1a shows an example of a data set and Fig.1b the FP-tree for that data set.Now,the constructed FP-tree contains all frequency information of the database.Mining the database becomes mining the FP-tree.The FP-growth method relies on the following principle:If X and Y are two itemsets,the count of itemset X [Y in the database is exactly that of Y in the restriction of the database to those transactions containing X .This restriction of the database is called the conditional pattern base of X and the FP-tree constructed from the conditional pattern base is called X 0s conditional FP-tree ,which we denote by T X .We can view the FP-tree constructed from the initial database as T ;,the conditional FP-tree for the empty itemset.Note that,for any itemset Y that is frequent in the conditional pattern base of X ,the set X [Y is a frequent itemset in the original database.Given an item i in T X :header ,by following the linked list starting at i in T X :header ,all branches that contain item i are visited.The portion of these branches from i to the root forms the conditional pattern base of X [f i g ,so the traversal obtains all frequent items in this conditional pattern base.The FP-growth method then constructs the conditional FP-tree T X [f i g by first initializing its header table based on the frequent items found,then revisiting the branches of T X along the linked list of i and inserting the corresponding itemsets in T X [f i g .Note that the order of items can be different in T X and T X [f i g .As an example,the conditionalpattern base of f f g and the conditional FP-tree T f f g for the GRAHNE ANDZHU:FAST ALGORITHMS FOR FREQUENT ITEMSET MINING USING FP-TREES 1349Fig.1.An FP-tree example.(a)A database.(b)The FP-tree for the database (minimum support =20percent ).database in Fig.1a is shown in Fig.1c.The above procedure is applied recursively,and it stops when the resulting new FP-tree contains only one branch.The complete set of frequent itemsets can be generated from all single-branch FP-trees.2.2The FP-Array TechniqueThe main work done in the FP-growth method is traversing FP-trees and constructing new conditional FP-trees after the first FP-tree is constructed from the original database.From numerous experiments,we found out that about80percent of the CPU time was used for traversing FP-trees.Thus,the question is,can we reduce the traversal time so that the method can be sped up?The answer is yes,by using a simple additional data structure.Recall that,for each item i in the header of a conditional FP-tree T X,two traversals of T X are needed for constructing the new conditional FP-tree T X[f i g. The first traversal finds all frequent items in the conditional pattern base of X[f i g and initializes the FP-tree T X[f i g by constructing its header table.The second traversal constructs the new tree T X[f i g.We can omit the first scan of T X by constructing a frequent pairs array A X while building T X. We initialize T X with an attribute A X.Definition.Let T be a conditional FP-tree and I¼f i1;i2;...;i m g be the set of items in T:header.A frequent pairs array(FP-array)of T is aðmÀ1ÞÂðmÀ1Þmatrix,where each element of the matrix corresponds to the counter of an ordered pair of items in I.Obviously,there is no need to set a counter for both item pairsði j;i kÞandði k;i jÞ.Therefore,we only store the counters for all pairsði k;i jÞsuch that k<j.We use an example to explain the construction of the FP-array.In Fig.1a,supposing that the minimum support is 20percent,after the first scan of the original database,we sort the frequent items as b:5,a:5,d:5,g:4,f:2,e:2,c:2.This order is also the order of items in the header table of T;. During the second scan of the database,we will construct T; and an FP-array A;,as shown in Fig.2a.All cells in the FP-array are initialized to0.According to the definition of an FP-array,in A;,each cell is a counter of a pair of items.Cell A;½c;b is the counter for itemset f c;b g,cell A;½c;a is the counter for itemset f c;a g, and so forth.During the second scan for constructing T;,for each transaction,all frequent items in the transaction are extracted.Suppose these items form itemset J.To insert J into T;,the items in J are sorted according to the order in T;:header.When we insert J into T;,at the same time A;½i;j is incremented by1if f i;j g is contained in J.For instance, for the second transaction,f b;a;f;g g is extracted(item h is infrequent)and sorted as b;a;g;f.This itemset is inserted into T;as usual and,at the same time,A;½f;b ;A;½f;a ,A;½f;g ;A;½g;b ,A;½g;a ;A;½a;b are all incremented by1. After the second scan,the FP-array A;contains the counts of all pairs of frequent items,as shown in Fig.2a.Next,the FP-growth method is recursively called to mine frequent itemsets for each item in T;:header.However,now for each item i,instead of traversing T;along the linked list starting at i to get all frequent items in i0s conditional pattern base,A;gives all frequent items for i.For example, by checking the third line in the table for A;,frequent items b;a;d for the conditional pattern base of g can be obtained.Sorting them according to their counts,we get b;d;a.Therefore,for each item i in T;,the FP-array A; makes the first traversal of T;unnecessary and each T f i g can be initialized directly from A;.For the same reason,from a conditional FP-tree T X,when we construct a new conditional FP-tree for X[f i g,for an item i,a new FP-array A X[f i g is calculated.During the construction of the new FP-tree T X[f i g,the FP-array A X[f i g is filled.As an example,from the FP-tree in Fig.1b,if the conditional FP-tree T f g g is constructed,the FP-array A f g g will be in Fig.2b.This FP-array is constructed as follows:From the FP-array A;,we know that the frequent items in the conditional pattern base of f g g are,in descending order of their support,b;d;a.By following the linked list of g,from the first node,we get f b;d g:2,so it is inserted asðb:2;d:2Þinto the new FP-tree T f g g.At the same time,A f g g½b;d is incremented by1.From the second node in the linked list, f b;a g:1is extracted and it is inserted asðb:1;a:1Þinto T f g g.At the same time,A f g g½b;a is incremented by1.From the third node in the linked list,f a;d g:1is extracted and it is inserted asðd:1;a:1Þinto T f g g.At the same time,A f g g½d;a is incremented by1.Since there are no other nodes in the linked list,the construction of T f g g is finished and FP-array A f g g is ready to be used for construction of FP-trees at the next level of recursion.The construction of FP-arrays and FP-trees continues until the FP-growth method terminates.Based on the foregoing discussion,we define a variant of the FP-tree structure in which,besides all attributes given in [14],an FP-tree also has an attribute,FP-array,which contains the corresponding FP-array.2.3DiscussionLet us analyze the size of an FP-array first.Suppose the number of frequent items in the first FP-tree T;is n.Then, the size of the associated FP-array is proportional to P nÀ1i¼1i¼nðnÀ1Þ=2,which is the same as the number of candidate large2-itemsets in Apriori in[6].The FP-trees constructed from the first FP-tree have fewer frequent items,so the sizes of the associated FP-arrays decrease.At any time when the space for an FP-tree is freed,so is the space for its FP-array.There are some limitations for using the FP-array technique.One potential problem is the size of the FP-array. When the number of items in T;is small,the size of the FP-array is not very big.For example,if there are5,000frequent items in the original database and the size of an integer is 4bytes,the FP-array takes only50megabytes or so. However,when n is large,nðnÀ1Þ=2becomes an extremely large number.At this case,the FP-array technique will reduce the significance of the FP-growth method,since the method mines frequent itemsets without generating any candidate frequent itemsets.Thus,one solution is to simply give up the FP-array technique until the number of items in an FP-tree is small enough.Another possible solution is to1350IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING,VOL.17,NO.10,OCTOBER2005Fig.2.Two FP-array examples.(a)A;.(b)A f g g.reduce the size of the FP-array.This can be done by generating a much smaller set of candidate large two-itemsets as in[25]and only store in memory cells of the FP-array corresponding to a two-itemset in the smaller set. However,in this paper,we suppose the main memory is big enough for all FP-arrays.The FP-array technique works very well,especially when the data set is sparse and very large.The FP-tree for a sparse data set and the recursively constructed FP-trees will be big and bushy because there are not many shared common prefixes among the FIs in the transactions.The FP-arrays save traversal time for all items and the next level FP-trees can be initialized directly.In this case,the time saved by omitting the first traversals is far greater than the time needed for accumulating counts in the associated FP-arrays.However,when a data set is dense,the FP-trees become more compact.For each item in a compact FP-tree,the traversal is fairly rapid,while accumulating counts in the associated FP-array could take more time.In this case, accumulating counts may not be a good idea.Even for the FP-trees of sparse data sets,the first levels of recursively constructed FP-trees for the first items in aheader table are always conditional FP-trees for the most common prefixes.We can therefore expect the traversal times for the first items in a header table to be fairly short,so the cells for these items are unnecessary in the FP-array.As an example,in Fig.2a,since b,a,and d are the first three items in the header table,the first two lines do not have to be calculated,thus saving counting time.Note that the data sets(the conditional pattern bases) change during the different depths of the recursion.In order to estimate whether a data set is sparse or dense, during the construction of each FP-tree,we count the number of nodes in each level of the tree.Based on experiments,we found that if the upper quarter of the tree contains less than15percent of the total number of nodes, we are most likely dealing with a dense data set.Otherwise, the data set is likely to be sparse.If the data set appears to be dense,we do not calculate the FP-array for the next level of the FP-tree.Otherwise,we calculate the FP-array of each FP-tree in the next level,but the cells for the first several(we use15based on our experience)items in its header table are not calculated. 2.4FPgrowth*:An Improved FP-Growth Method Fig.3contains the pseudo code for our new method FPgrowth*.The procedure has an FP-tree T as parameter.T has attributes:base,header,and FP-array.T:base contains the itemset X for which T is a conditional FP-tree,the attribute header contains the header table,and T:F P-array contains the FP-array A X.In FPgrowth*,line6tests if the FP-array of the current FP-tree exists.If the FP-tree corresponds to a sparse data set,its FP-array exists,and line7constructs the header table of the new conditional FP-tree from the FP-array directly.One FP-tree traversal is saved for this item compared with the FP-growth method in[14].In line9,during the construction, we also count the nodes in the different levels of the tree in order to estimate whether we shall really calculate the FP-array or just set T Y:F P-array as undefined.3FP MAX*:M INING MFI’SIn[12],we developed FPmax,another method that mines maximal frequent itemsets using the FP-tree structure.Since the FP-array technique speeds up the FP-growth method for sparse data sets,we can expect that it will be useful in FPmax too.This gives us an improved method,FPmax*. Compared to FPmax,in addition to the FP-array technique, the improved method FPmax*also has a more efficient maximality checking approach,as well as several other optimizations.It turns out that FPmax*outperforms FPmax for all cases we discussed in[12].3.1The MFI-TreeObviously,compared with FPgrowth*,the extra work that needs to be done by FPmax*is to check if a frequent itemset is maximal.The naive way to do this is during a postprocessing step.Instead,in FPmax,we introduced a global data structure,the maximal frequent itemsets tree(MFI-tree),to keep the track of MFIs.Since FPmax*is a depth-first algorithm,a newly discovered frequent itemset can only be a subset of an already discovered MFI.We therefore need to keep track of all already discovered MFIs.For this,we use the MFI-tree.A newly discovered frequent itemset is inserted into the MFI-tree,unless it is a subset of an itemset already in the tree.From experience,we learned that a further consideration for large data sets is that the MFI-tree will be quite large,and sometimes one itemset needs thousands of comparisons for maximality checking.Inspired by the way maximality checking is done in[11],in FPmax*,we still use the MFI-tree structure,but for each conditional FP-tree T X,a small local MFI-tree M X is created.The tree M X will contain all maximal itemsets in the conditional pattern base of X.To see if a local MFI Y generated from a conditional FP-tree T X is globally maximal,we only need to compare Y with the itemsets in M X.This speeds up FPmax significantly.Each MFI-tree is associated with a particular FP-tree.An MFI-tree resembles an FP-tree.There are two main differ-ences between MFI-trees and FP-trees.In an FP-tree,each node in the subtree has three fields:item-name,count,and node-link.In an MFI-tree,the count is replaced by the level of the node.The level field is used for maximality checking in a way to be explained later.Another difference is that the header table in an FP-tree is constructed from traversing the previous FP-tree or using the associated FP-array,while theGRAHNE ANDZHU:FAST ALGORITHMS FOR FREQUENT ITEMSET MINING USING FP-TREES1351Fig.3.Algorithm FPgrowth*.。

找不着钥匙？可能是你的大脑不同步啦！

找不着钥匙？可能是你的大脑不同步啦！作者：暂无来源：《华东科技》 2012年第2期1月10日发布于《认知》（Cogn ition）的一份报告中，加拿大滑铁卢大学的萨尔曼（Grayden Solma n）和同事对我们如何寻找东西进行了研究。

研究团队设定了一个简单的任务，志愿者会在电脑屏幕上看一堆有颜色的图形。

他们需要尽可能快地找出某个特定形状。

萨尔曼表示，大约有10%~20%的时间人们会漏掉特定形状的图形，尽管这些人其实已经拿起了它。

为了找出原因，研究者进行了一些后续实验。

为了确认志愿者是否只是忘记了他们的目地，研究者在志愿者进行寻找任务之前让人们记忆一个物品列表，并在任务之后进行回忆。

这是为了填充每名志愿者的记忆量，这样人们就不能再短期记忆中再放入其他信息了。

尽管这可能会对搜寻任务的成绩起到负面影响，但是实际上人们出错的几率并没有受到影响。

之后，研究者让被试执行相似的寻找任务，让他们用鼠标选择特定的图形，并分析了鼠标的移动。

研究发现，志愿者在移动并漏过目标后，他们的动作会变慢。

研究团队提出，大脑中处理动作的系统运作得太快了，视觉系统跟不上。

当你在乱糟糟的屋子里翻找钥匙时，你可能没有能够给予视觉系统足够的时间来确认哪个东西是哪个。

萨尔曼觉得，整体来说，由于时间可能会更珍贵，偶尔牺牲准确度换取速度可能是有益的。

鼠标移动慢下来这个现象显示，在一定程度上志愿者意识到了他们漏过了目标。

之前也有一些研究显示，人们在犯了错误后会试图减慢行动速度，即使他们并没有意识到犯了错误。

萨尔曼觉得这反映了大脑意图减慢运动系统的努力，以让视觉系统跟上来产生意识知觉。

美国哈佛大学的霍洛维茨（Todd Horowitz）表示，运动和知觉系统是分离的这个提法很有趣，这些系统都在试图帮助你找到钥匙，但是它们没有能够协调起来。

这个理论有其现实意义，比如医生也会通过X光寻找病灶，行李检查时人们也要寻找危险物品。

(来源：果壳科技)。

fast思考法

fast思考法Fast思考法是一种高效的思考方法。

它的目的是帮助人们在非常短的时间内获得深刻且准确的洞察。

这种方法适用于日常生活中的各种情况，从日常工作到做决定和解决问题都可以使用fast思考法。

帮助你在短时间内做出决策快速理解信息和解决问题的能力提高创造力提高生产力提高自信心1. 定义问题首先，你需要确定你要解决的问题是什么。

最好明确地描述这个问题，以便你更好地理解和思考它。

2. 收集信息收集关于问题的信息非常重要。

你需要尽可能多地获取有关该问题的信息，并且需要保持开放的思维，以便你能够更好地了解这个问题。

3. 分析问题现在你有了足够的信息，你需要开始分析问题，以便你能够更好地了解它。

这意味着你需要仔细研究所收集的信息，并尝试发现可能隐藏在其中的模式和联系。

4. 生成解决方案接下来，你需要根据你的分析结果来生成解决方案。

这一步可能需要一些创造性思维，并可能需要你激发一些脑力活动以更好地理解问题和可能的解决方案。

5. 选择最好的解决方案你现在应该有了多个解决方案，现在需要选择最好的方案来解决问题。

选择最佳的解决方法可能需要权衡不同因素，包括资源、时间和成本等。

6. 行动最后，你需要采取行动开始解决问题。

在此阶段，你需要展现出果断的行动能力，以便尽快解决问题。

总结使用fast思考法可以帮助你更快地找到问题的解决方案。

通过清晰明确的定义问题、收集关于问题的信息、分析问题、生成解决方案、选择最佳的解决方案和行动，你可以在更短的时间内做出更准确的决策。

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A Fast Algorithm for Finding Crosswalks usingFigure-Ground SegmentationJames M.Coughlan and Huiying ShenSmith-Kettlewell Eye Research InstituteSan Francisco,CA94115USA{coughlan,shen}@Abstract.Urban intersections are the most dangerous parts of a blindperson’s travel.Proper alignment is necessary to enter the crosswalk inthe right direction and avoid the danger of straying outside the cross-puter vision is a natural tool for providing such alignmentinformation.However,little work has been done on algorithms forﬁnd-ing crosswalks for the blind,and most of it focuses on fairly clean,simpleimages in which the Hough transform suﬃces for extracting the bordersof the crosswalk stripes.In real-world conditions such as cluttered sceneswith shadows,saturation eﬀects,slightly curved stripes and occlusions,the Hough transform is often unreliable as a pre-processing step.Wedemonstrate a novel,alternative approach forﬁnding zebra(i.e.multi-plely striped)crosswalks that is fast(just a few seconds per image)androbust.The approach is based onﬁgure-ground segmentation,which wecast in a graphical model framework for grouping geometric featuresinto a coherent structure.We show promising experimental results on animage database photographed by an unguided blind pedestrian,demon-strating the feasibility of the approach.21IntroductionUrban intersections are the most dangerous parts of a blind person’s travel. While most blind pedestrians have little diﬃculty walking to intersections using standard orientation and mobility skills,it is very diﬃcult for them to align themselves precisely with the crosswalk.Proper alignment would allow them order to enter the crosswalk in the right direction and avoid the danger of straying outside the crosswalk.Little work has been done on algorithms for detecting zebra crosswalks for the blind and visually impaired[10,4,5,9].This body of work focuses on fairly simple images in which the Hough transform is typically suﬃcient for extracting the borders of the crosswalk stripes.However,in real-world conditions,such as cluttered scenes with shadows,saturation eﬀects,slightly curved stripes and occlusions,the Hough transform is often inadequate as a pre-processing step. Instead of relying on a tool for grouping structures globally such as the Hough transform,we use a moreﬂexible,local grouping process based onﬁgure-ground segmentation(or segregation)using graphical models.Figure-ground segmentation has recently been successfully applied[8,3]to the detection and segmentation of speciﬁc objects or structures of interest from the background.Standard techniques such as deformable templates[12]are poorly suited toﬁnding some targets,such as printed text,stripe patterns, vegetation or buildings,particularly when the targets are regular or texture-like structures with widely varying extent,shape and scale.The cost of making the deformable templateﬂexible enough to handle such variations in structure and size is the need for many parameters to estimate,which imposes a heavy computational burden.In these cases it seems more appropriate to group target features into a common foreground class,rather than seek a detailed correspon-dence between a prototype and the target in the image,as is typically done with deformable template and shape matching techniques.Our graphical model-based approach toﬁgure-ground segmentation empha-sizes the use of the geometric relationships of features extracted from an image as a means of grouping the target features into the foreground.In contrast with related MRF techniques[2]for classifying image pixels into small numbers of categories,our approach seeks to make maximal use of geometric,rather than intensity-based,information.Geometric information is generally more intuitive to understand thanﬁlter-based feature information,and it may also be more appropriate when lighting conditions are highly variable.We formulate our approach in a generalﬁgure-ground segmentation frame-work and apply it to the problem ofﬁnding zebra crosswalks in urban scenes. Our results demonstrate a high success rate of crosswalk detections on typical images taken by an unguided blind photographer.2Graphical Model for Figure-GroundWe tackle theﬁgure-ground segmentation problem using a graphical model that assigns a label ofﬁgure or ground to each element in an image.In our application3 the elements are a sparse set of geometric features created by grouping togethersimpler features in a greedy,bottom-up fashion.The features are designed tooccur commonly on the foreground structure of interest and more rarely in thebackground.For example,in our crosswalk application,simple edges are groupedinto candidate crosswalk stripe fragment features,i.e.edge pairs that are likelyto border fragments of the crosswalk stripes.The true positive features tend tocluster into regular structures(roughly parallel stripes in this case),diﬀerentlyfrom the false positives,which are distributed more randomly.Our approach exploits this characteristic clustering of true positive features,drawing on ideas from work on object-speciﬁcﬁgure-ground segmentation[8],which uses normalized cuts to perform grouping.We use aﬃnity functions tomeasure the compatibilities of pairs of elements as potential foreground candi-dates and construct a graphical model to represent aﬁgure-ground process.Each node in the graph has two possible states,ﬁgure or ground.The graphi-cal model deﬁnes a probability distribution on all possible combinations ofﬁgure-ground labels at each node.We use belief propagation(BP)to estimate themarginal probabilities of these labels at each node;any node with a suﬃcientlyhigh marginal probability of belonging to theﬁgure is designated asﬁgure.2.1The Form of the Graphical ModelWe deﬁne the graphical model for a generalﬁgure-ground segmentation pro-cess as follows.Each of the N features extracted from the image is associatedwith a graph node(vertex)x i,where i ranges from1through N.Each nodex i can be in two possible states,0or1,representing ground andﬁgure,respec-tively.The probability of any labeling of all the nodes is given by the followingexpression:P(x1,...,x N)=1/Z N i=1ψi(x i) <ij>ψij(x i,x j).(Here we adopt the notation commonly used in the graphical model literature[11],in which theword“potential”corresponds to a factor in this expression for probability.Thisis diﬀerent from to the usage in statistical physics,in which potentials mustbe exponentiated to express a factor in the probability,e.g.e−U i where U i is apotential.)This is the expression for a pairwise MRF(graphical model),whereψi(x i)is the unary potential function,ψij(x i,x j)is the binary potential function andZ is the normalization factor.<ij>denotes the set of all pairs of features iand j that are directly connected in the graph.ψi(x i)represents a unary factorreﬂecting the likelihood of feature x i belonging to theﬁgure or ground,inde-pendent of the context of other nearby features.ψij(x i,x j)is the compatibilityfunction between features i and j,which reﬂects how the relationship between two features inﬂuences the probability of assigning them toﬁgure/ground.The unary and binary functions may be chosen by trial and error,as inthe current application,or by maximum likelihood learning.An example of thiskind of learning is in[6],in which compatibilities(binary potentials)learned fromlabeled data are used to construct graphical models for clustering.However,forthe preliminary results we show to demonstrate the feasibility of our approach,we used simple trial and error to choose unary and binary functions.4The general form of our unary and binary functions is as follows.First,ψi(x i) enforces a bias in favor of each node being assigned to the ground:ψi(x i=0)=1 (which we refer to as a“neutral”value for a potential)andψi(x i=1)<1. The magnitude of theﬁgure value for any feature will depend on one or more unary cues,or factors.In order for a node to be set to the foreground,the bi-nary functions must reward compatible pairs of nodes suﬃciently to oﬀset the unary bias.Ground-ground and ground-ﬁgure interactions are set to be neu-tral:ψij(x i=0,x j=0)=ψij(x i=0,x j=1)=ψij(x i=1,x j=0)=1. Figure-ﬁgure interactionsψij(x i=1,x j=1)are set less than1for rela-tively incompatible nodes and greater than1for compatible nodes.The value of ψij(x i=1,x j=1)will be determined by several binary cues,or compatibility factors.3Figure-Ground Process for Finding LinesA standard approach to detecting crosswalk stripes is to use the Hough trans-form toﬁnd the straight-line edges of the stripes,and then to group them into an entire zebra pattern.While this method is sound for analyzing high-quality photographs of suﬃciently well-formed crosswalks,it is inadequate under many real-world conditions because the Hough transform fails to isolate the lines cor-rectly.To illustrate the limitations of the Hough transform,consider Figure1.A straight line is speciﬁed in Hough space as a pair(d,θ):this deﬁnes a line made up of all points(u,v)such that n(θ)·(u,v)=d,where n(θ)=(cosθ,sinθ)is the unit normal vector to the line.In an image containing one straight line,each point of the line will cast votes in Hough space,and collectively the votes will concentrate on the true value of(d,θ).The lines in Figure1are not perfectly straight, however,and so the peak in Hough space corresponding to each line will be smeared.If only one such line were present in the image,this smearing could be tolerated simply by quantizing the Hough space bins coarsely enough.However, the presence of a second nearby line makes it diﬃcult for the Hough transform to resolve the two lines separately,since no choice of Hough bin quantization can group all the votes from one line without also including votes from the other.Fig.1.Two slightly curved lines(black),representing edges of crosswalk stripes(with exaggerated curvature).The straight red dashed line is tangent to both lines,which means that the Hough transform cannot resolve the two black lines separately.The global property of the Hough transform is inappropriate for such situa-tions,which is why we turn to a localﬁgure-ground process instead.5 3.1Local Figure-Ground ProcessOur localﬁgure-ground process is a graphical model with a suitable choice of unary and binary potentials.Given oriented edgelets y i=(E i,d i,θi)where E i is the edge strength,d i=u i cosθi+v i sinθi and(u i,v i)are the pixel coordinates of the point,we can deﬁne the unary potential asψi(x i)=e(αE i+β)x i whereαand βare coeﬃcients that can be learned from training data.With this deﬁntion, note thatψi(x i)always equals1whenever x i=0,and thatψi(x i=1)increases with increasing edge strength(assumingα>0).Similarly,we can deﬁne the binary potential asψij(x i,x j)=e(λC ij+τ)x i x j where C ij=|d i−d j|+sin2(θi−θj) measures how collinear edgelets y i and y j are.Again,note thatψij(x i,x j)always equals1whenever at least one of the states x i,x j is0.Unless the graph has very dense connectivity,with edgelets being connected even at very long separations,the grouping of edgelets may lack long-range coherence.Without such long-range coherence,the graph may group edgelets into lines that are very slowly curving everywhere(e.g.a large circle)rather than justﬁnding lines that are globally roughly straight.However,such dense connectivity creates great computational complexity in running BP to perform inference.The solution we adopt to this problem is to create a much smaller(and thus faster),higher-scale version of the original graph based on composite fea-ture nodes,each composed of many individual features.For our line grouping problem,we propose a greedy procedure to group the edgelets into roughly straight-line segments of varying length.We describe how this framework ap-plies to crosswalk detection in the next section.4Crosswalks and StripeletsWe have devised a bottom-up procedure for grouping edges into composite fea-tures that are characteristic of zebra crosswalks,and which are uncommon else-where in street scenes.The image is converted to grayscale,downsampled(by a factor of4with our current camera)to the size409x307and blurred slightly. We have avoided the use of color since we wish to detect crosswalks of arbitrary color.Since the crosswalk stripes are roughly horizontal under typical viewing con-ditions,our edge detectorﬁnds roughly horizontal edges byﬁnding local min-ima/maxima of a simple y derivative of the image intensity,∂I/∂y.A greedy procedure groups these individual edges into roughly straight line segments(see Figure3(a)).A candidate stripe fragment feature,or“stripelet,”is deﬁned as the com-position of any two line segments(referred to as“upper”and“lower”)with all of the following properties:(1.)The upper and lower segments have polarities consistent with a crosswalk stripe,i.e.∂I/∂y is negative on the upper segment and positive on the lower segment,since the crosswalk stripe is painted a much brighter color than the pavement.(2.)The two segments are roughly parallel6in the image.(3.)The segments have suﬃcient“overlap,”i.e.the x-coordinate range of one segment has signiﬁcant overlap with the x-coordinate range of the other.(4.)The vertical width w of the segment pair(i.e.the y-coordinate of the upper segment minus the y-coordinate of the lower,minimized across x belong-ing to both segments)must be within the range2to70pixels(i.e.spanning the typical range of stripe widths observed in our409x307images).Many stripelets are detected in a typical crosswalk scene(see Figure3(b)).4.1Crosswalk Graphical ModelOnce the stripelets(i.e.nodes)of the graph have been extracted,unary and binary cues are used to evaluate their suitability as“ﬁgure”elements.The unary cue exploits the fact that stripes lower in the image tend to be wider,which is true assuming that the camera is pointed rightside-up so that the stripes lower in the image are closer to the photographer.If we examine the empirical distribution of all stripelets(not just the ones that lie on the cross-walks)extracted from our image dataset,weﬁnd a characteristic relationship between the vertical widths w and vertical coordinates y(i.e.the vertical height of the centroid of the stripelet in the image).Figure2shows a typical scatter-plot of(y,w)values from stripelets extracted from a single crosswalk image.The envelope of points in the lower left portion of the graph,drawn as a red line, corresponds to the distribution of stripelets that lie on the crosswalk.The slope and intercept of the envelope will vary depending on the crosswalk,camera pose and presence of non-crosswalk clutter,but typically few points lie left of the envelope.In general,the closer a point is to the envelope,the more likely it is to correspond to a stripelet lying on the crosswalk.Given a crosswalk image,a line-ﬁtting procedure can be used to estimate the envelope,and thereby provide unary evidence for each stripelet belonging toﬁgure(crosswalk)or ground.Let E denote the distance(in(y,w)space) between a(y,w)point and the envelope,andˆw be the value of w along the envelope corresponding to the vertical coordinate y of the stripelet.Then E/ˆw is a normalized measure of distance from the envelope.Another source of unary evidence is the length of the stripelet:all else being equal,long(i.e.horizontally extended)stripelets are more likely to belong to ﬁgure than to ground.Denoting the lengths of the upper and lower segments of a stripelet by a and b,we choose a measure L that is the square root of the geometric mean of a and b:L=(ab)1/4.We combine these two sources of unary evidence into the unary function as follows:ψi(x i=1)=(1/10)max[1,L(1−E/ˆw)].Longer stripelets that lie close to the envelope will have larger values of unary potential forﬁgure,ψi(x i=1), but note that this value never exceeds1/10,compared to the unary potential for ground,1.One binary cue is applied in two diﬀerent ways to deﬁne the binary poten-tial between two stripelets.This cue is based on the cross ratio test[1](the application of which is inspired by crosswalk detection work of[10,9]),which is7Fig.2.Typical scatterplot of(y,w)values from stripelets extracted from single cross-walk image.Envelope of points,drawn as a red line,corresponds to distribution of stripelets lying on the crosswalk.a quantity deﬁned for four collinear points(in3-D space)that is invariant to perspective projection.Theﬁrst application of the cross ratio test is used to check for evidence that the four line segments corresponding to the two stripelets are approximately par-allel in three-dimensional space,as they should be.The cross ratio is calculated by drawing a“probe”line through the four lines in the image deﬁned by the line segments.If the four lines share a vanishing point(i.e.because they are parallel in3-D),the cross ratio should be the same no matter the choice of probe.In our algorithm,we choose two vertical probes to estimate the cost ratio twice, i.e.values of r1and r2,and the less discrepant these two values,the higher the compatibility factor.In addition,we exploit a geometric property of a zebra crosswalks:the stripe widths are equal to the separation between adjacent stripes(in3-D),and so the cross ratio from any line slicing across adjacent stripes should equal1/4,as pointed out by[9].These two properties of the cross ratio are combined into an overall error measure as follows:R=(|r1−1/4|+|r2−1/4|)/2+2|r1−r2|.This in turn is used to deﬁne the binary potentialψij(x i=1,x j=1)=(10/3)e−10R.4.2Crosswalk Implementation and ResultsSince the line-ﬁtting procedure forﬁnding the envelope is confused by noisy scatterplots,multiple envelope hypotheses can be considered if necessary,each of which gives rise to a separate version of the graph.In our experiments we chose eight diﬀerent envelope hypotheses and ran BP on each corresponding graph. The solution that yielded the highest unary belief at any node was chosen as the ﬁnal solution.For each graph,the graph connectivity was chosen according to three fac-tors describing the relationship between each possible pair of stripelets:distance between the stripe centroids,the cross ratio error measure R,and the“mono-tonicity”requirement that the higher stripe must have less vertical width than the lower stripe(i.e.the slope of the envelope is negative).If the distance is not8too long,the error measure is suﬃciently low and the monotonicity requirement is satisﬁed,then a connection is established between the stripelets.A few sweeps of BP(one sweep is a schedule of asynchronous BP message updating that updates every possible message once)are applied to each graph, and the unary beliefs,i.e.estimates of P(x i=1),are thresholded to decide if each feature belongs to theﬁgure or ground.The eﬀects of pruning out the “ground”states are shown in Figure3and Figure4.We ran our algorithm with the same exact settings and parameter values for all of the following images.The total execution time was a few seconds per image,using unoptimized Python and C++code running on a standard laptop.Note the algorithm’s ability to handle considerable amounts of scene clutter,shadows,saturation,etc.Also note that all photographs were taken by a blind photographer,and no photographs that he took were omitted from our zebra crosswalk dataset.Fig.3.Stages of crosswalk detection.Left to right:(a)Straight line segments(green).(b)Stripelets(pairs of line segments)shown as red quadrilaterals.(c)Nodes in graph-ical model.(d)Figure nodes identiﬁed after BP.5Summary and ConclusionsWe have demonstrated a novel graphical model-basedﬁgure-ground segmenta-tion approach toﬁnding zebra crosswalks in images intended for eventual use by a blind pedestrian.Our approach is fast(a few seconds per image)as well as robust,which is essential for making it feasible as an application for blind pedes-trians.We are currently investigating learning our graphical model parameters from ground truth datasets,as well as the possibility of employing additional cues.We would like to thank Roberto Manduchi for useful feedback.Both authors were supported by the National Institute on Disability and Rehabilitation Re-search grant number H133G030080and the National Eye Institute grant number EY015187-01A2.References1.R.I.Hartley and A.Zisserman.”Multiple View Geometry in Computer Vision”.2000.Cambridge University Press.9Fig.4.Crosswalk detection results for all zebra crosswalk images.2.X.He,R.S.Zemel and M.A.Carreira-Perpinan.“Multiscale Conditional Random Fields for Image Labeling.”CVPR 2004.3.S.Kumar and M.Hebert.“Man-Made Structure Detection in Natural Images using a Causal Multiscale Random Field.”CVPR 2003.4.S.Se.”Zebra-crossing Detection for the Partially Sighted.”CVPR)2000.South Carolina,June 2000.5.S.Se and M.Brady.”Road Feature Detection and Estimation.”Machine Vision and Applications Journal,Volume 14,Number 3,pages 157-165,July 2003.6.N.Shental,A.Zomet,T.Hertz and Y.Weiss.“Pairwise Clustering and Graphical Models.”NIPS 2003.7.J.Shi and J.Malik.”Normalized Cuts and Image Segmentation.”IEEE Transac-tions on Pattern Analysis and Machine Intelligence,22(8),888-905,August 2000.8.S.X.Yu and J.Shi.“Object-Speciﬁc Figure-Ground Segregation.”CVPR 2003.9.M.S.Uddin and T.Shioyama.“Bipolarity-and Projective Invariant-Based Zebra-Crossing Detection for the Visually Impaired.”1st IEEE Workshop on Computer Vision Applications for the Visually Impaired,CVPR 2005.1010.S.Utcke.”Grouping based on Projective Geometry Constraints and Uncertainty.”ICCV’98.Bombay,India.Jan.1998.11.J.S.Yedidia,W.T.Freeman,Y.Weiss.“Bethe Free Energies,Kikuchi Approx-imations,and Belief Propagation Algorithms”.2001.MERL Cambridge Research Technical Report TR2001-16.12. 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