A Distributed Stochastic Approximation Approach for Max-Min Fair Rate Control of Flows in P

模型可视化算法全文共四篇示例，供读者参考第一篇示例：模型可视化算法是指通过将模型的数据或结果以图形化的方式展示出来，帮助用户更直观地理解模型的建立和预测过程。

随着大数据和人工智能的发展，模型可视化算法在各个领域的应用逐渐增加，成为提高模型效果和效率的重要工具之一。

一般来说，模型可视化算法通常分为静态可视化和动态可视化两种方式。

静态可视化主要通过图表、表格、热图等方式将模型的数据或输出结果以静态的形式展示出来，便于用户观察和分析。

而动态可视化则是通过动画、交互式界面等方式展示模型的过程和结果，提供更丰富的信息与体验。

在静态可视化方面，常用的算法包括matplotlib、seaborn、ggplot等。

这些算法能够快速绘制各种图表，如折线图、散点图、直方图等，帮助用户直观地了解模型的数据分布、特征相关性等。

通过使用matplotlib库可以绘制一个数据点分布的散点图，进而观察数据的聚类情况和异常点。

在动态可视化方面，常用的算法包括D3.js、Plotly、Bokeh等。

这些算法支持用户交互式操作，用户可以根据需要调整展示方式、筛选数据、查看特定区域的详细信息等。

这种交互式的可视化方式不仅使用户更加深入地理解模型，还提高了用户的参与感和学习效果。

除了静态和动态可视化，还有一些特殊的可视化算法，如深度学习模型的可视化算法。

深度学习模型由于其复杂性和黑盒特性，常常难以解释其预测结果。

为了解决这一问题，研究人员提出了一系列的深度学习可视化算法，如t-SNE、CAM、Grad-CAM等，通过可视化神经网络的中间层、梯度信息等，帮助用户理解模型预测的原理和依据。

模型可视化算法是数据分析和机器学习领域的重要工具，可以帮助用户更加直观地了解模型的内在机制和预测结果。

通过合理选择和应用可视化算法，可以提高模型的解释性、可信度和效率，为用户提供更好的数据分析和决策支持。

我们应该进一步探索模型可视化算法的研究和应用，不断完善其理论和方法，促进其在各个领域的广泛应用。

A Distributed Algorithm for Joins in Sensor NetworksAlexandru Coman Mario A.NascimentoDepartment of Computing ScienceUniversity of Alberta,Canada{acoman|mn}@cs.ualberta.caAbstractGiven their autonomy,flexibility and large range of func-tionality,wireless sensor networks can be used as an effec-tive and discrete means for monitoring data in many do-mains.Typical sensor nodes are very constrained,in par-ticular regarding their energy and memory resources.Thus, any query processing solution over these devices should consider their limitations.We investigate the problem of processing join queries within a sensor network.Due to the limited memory at nodes,joins are typically processed in a distributed manner over a set of nodes.Previous approaches have either assumed that the join processing nodes have sufficient memory to buffer the subset of the join relations assigned to them,or that the amount of available memory at nodes is known in advance.These assumptions are not realistic for most scenarios.In this context we pro-pose and investigate DIJ,a distributed algorithm for join processing that considers the memory limitations at nodes and does not make a priori assumptions on the available memory at the processing nodes.At the same time,our al-gorithm still aims at minimizing the energy cost of query processing.1.IntroductionRecent technological advances,decreasing production costs and increasing capabilities have made sensor net-works suitable for many applications,including environ-mental monitoring,warehouse management and battlefield surveillance.Despite the relative novelty and small num-ber of real-life deployments,sensor networks are consid-ered a highly promising technology that will change the way we interact with our environment[13].Typical sen-sor networks will be typically be formed by a large number of small,radio-enabled,sensing nodes.Each node is ca-pable of observing the environment,storing the observed values,processing them and exchanging them with other nodes over the wireless network.While these capabilitiesare expected to rapidly grow in the near future,the energy source,be it either a battery or some sort of energy har-vesting[8],is likely to remain the main limitation of these devices.Hence,energy efficient data processing and net-working protocols must be developed in order to make the long-term use of such devices practical.Our focus is on en-ergy efficient processing of queries,joins in particular,over sensor networks.We study this problem in an environment where each sensor node is only aware of the existence of the other sensor nodes located within its wireless communica-tion range,and the query can be introduced in the network at any node.Users query the sensor network to retrieve the collected data on the monitored environment.The most popular form for expressing queries in a sensor network is using an SQL-like declarative language[6].The data collected in the sen-sor network can be seen as one relation distributed over the sensor nodes,called the sensor relation in the following.The queries typically accept one or more of the following operators[6,9]:selection,projection,union,grouping and aggregations.We note that the join operation in sensor net-works has been mostly neglected in the literature.A scenario where join queries are important is as fol-lows.National Parks administration is interested in long-term monitoring of the animals in the managed park.A sensor network is deployed over the park,with the task of monitoring the animals(e.g.,using RFID sensing).Park rangers patrol the park and,upon observing certain patterns, query the sensor network through mobile devices tofind in-formation of interest.For instance,uponfinding two ani-mals killed in region A,respectively B,the rangers need to find what animals,possibly ill of rabies,have killed them.The ranger would issue the query“What animals have been in both region A and B between times T1and T2?”.If joins cannot be processed in-network,then two,possibly long, lists of animals IDs appearing in each region will be re-trieved and joined at the user’s device.On the other hand, if the join is processed in-network,only possibly very few animal IDs are retrieved,substantially reducing the commu-nication cost.1In this paper we focus on the processing of the join op-erator in sensor networks.Since the energy required for communication is three to four orders of magnitude higher than the energy required by sensing and computation[9], it is important to minimize the energy cost of communica-tion during query processing.Recently,a few works ad-dressed in-network processing of join queries.Bonfils and Bonnet[3]investigate placing a correlation operator at a node in the network.Pandit and Gupta[11]propose two algorithms for processing a range-join operator in the net-work and Yu at al.[16]propose an algorithm for processing equi-joins.These works study the self-join problem where subsets of the sensor relation are joined.Abadi et al.[1]pro-pose several solutions for the join with an external relation, where the sensor relation is joined with a relation stored at the user’s an et al.[5]study the cost of several join processing solutions with respect to the location of the network region where the join is performed.Most previous solutions either assume that nodes have sufficient memory to buffer the partition of the join relations assigned to them for processing,or that the amount of memory available at each node is known in advance and the assigned data par-titions can be set accordingly.These assumptions are un-realistic for most scenarios.It is well known that sensor networks are very constrained on main memory and the en-ergy cost of using theirflash storage(for those devices that have it)is rather prohibitive to be used for data buffering during query processing.In addition,in large scale sensor networks,it is not feasible for the sensor nodes or the user station to be aware of up-to-date information on memory availability of all network nodes.In this paper our contributions are three-fold.First we analyze the requirements of a distributed in-network join processing algorithm.Second,to our knowledge,this is the first work to develop and discuss in details a distributed al-gorithm for in-network join processing.Third,based on the present algorithm,we develop a cost model that can be used to select the most efficient join plan during the execution of the query.Our join algorithm is general in the sense that it can be used with different types of joins,including semi-joins,with minor modifications to the presented algorithm and cost model.As well,our algorithm can be used within the core of other previously proposed join solutions for re-laxing their assumptions on memory availability.2.BackgroundIn our work we consider a sensor network formed by thousands offixed nodes.Each node has several sensing units(e.g.,temperature,RFID reader),a processor,a few kilobytes of main memory for buffer and data processing,a few megabytes offlash storage for long-term storage of sen-sor observations,fixed-range wireless radio and it is battery operated.These characteristics encompass a wide range ofsensor node hardware,making our work independent of aparticular sensor platform.Further on,we consider thateach node is aware of its location,which is periodicallyrefreshed through GPS or a localization algorithm[14]toaccount for any variation in a node’s position due to envi-ronmental hazards.Each node is aware of the nodes locatedwithin its wireless range,which form its1-hop neighbour-hood.A node communicates with nodes other than its1-hop neighbours using multi-hop routing over the wirelessnetwork.As sensor nodes are not designed for user inter-action,users query the sensor network through personal de-vices,which introduce the query in the network through oneof the nodes in their vicinity.We consider a sensor network deployment where nodesacquire observations periodically and the observations arestored locally for future querying.The data stored at thesensor nodes forms a virtual relation over all nodes,denotedR∗.As nodes store the acquired data locally,each node holds the values of the observations recorded by its sensingunits and the time when each recording was performed.We analyze the self-join processing problem in sensornetworks,i.e.,the joined relations are spatially and tempo-rally constrained subsets of the sensor relation R∗.We im-pose no restrictions on the join condition,that is,any tuplefrom a relation could match any tuple of the other relation.For instance,the query“What animals have been in bothregions R A and R B between times T1and T2?”(from ourexample in Section1)can be expressed in pseudo-SQL as:SELECT S.animalIDFROM R∗as S,R∗as TWHERE S.location IN Region R AAND T.location IN Region R BAND S.time IN TimeRange[T1,T2]AND T.time IN TimeRange[T1,T2]AND S.animalID=T.animalIDLet us denote by A the subset of R∗restricted to Region R A and by B the subset of R∗restricted to Region R B. The query may also contain other operators ops(selection, projection,etc.)on each tuple of R∗or on the result of the join.As our focus is on join processing,we consider the relations A and B as the resulting relations after the query operators that can be applied individually on each node’s relation have been applied.We assume operators that can be processed locally by each sensor node on its stored relation and thus they do not involve any communication.We denote with J the result of the join of relations A and B,including any operators on the join result required by the query:J= ops J(A B).We assume operators on the join result can be processed in a pipelined fashion immediately following the join of two tuples.A general query tree and the notations we use are shown in Figure1.A UU opsJopsAopsopsAR iR jkR RmR oR nR pops BopsBopsBopsBJBAFigure1.Query tree and notations3.DIJ:A Distributed Join Processing Algo-rithm for Sensor NetworksJoin processing in sensor networks is a highly complex operation due to the distributed nature of the processing and the limited memory available at nodes.We discuss some of the requirements of an effective and efficient join pro-cessing algorithm for sensor networks,namely:distributed processing,memory management and synchronized com-munication.•Distributed processing.In large scale sensor net-works the join operation must be processed in a dis-tributed manner using localized knowledge.For most queries no single node can buffer all the data required for the join.In addition,no node(or user station) has global network knowledge tofind the optimal join strategy.As nodes have information only about their neighbourhood,the challenge is to take correct and consistent decisions among nodes with respect to pro-cessing the join.For instance,when the join operation is evaluated over a group of nodes,each node in the group must route and buffer tuples such that each pair of join tuples is evaluated exactly once in the join.•Memory management.Each node participating in the join must have sufficient memory to buffer the tu-ples that it joins and the resulting tuples.For some join queries the join relations are larger than the avail-able memory of a single node.Typically,several nodes must collaborate to process the join operator,pooling their memory and processing resources together.A join processing algorithm should pool these resources together and allocate tasks and data among the partici-pating nodes such that the efficiency of the processing is maximized.•Synchronized dataflow.Inter-node communication must be synchronized such that a node does not re-ceive new tuples to process when its memory is full.Otherwise,the node would have to drop some of the buffered or new tuples,which is unacceptable as it may invalidate the result of the join.Thus,each node mustfully process the join tuples it holds before receivingany new tuples.A similar problem occurs also for thenodes routing the data.A parent node routing data formultiple children may not be able to buffer all receiveddata before it can forward it.Thus,a join processingalgorithm should carefully consider theflow of dataduring its execution.In this work we propose a distributed join processing al-gorithm which considers the above requirements.In ourpresentation we focus on the join between two restrictions(A and B)of the R∗relation,where the join condition isgeneral(theta-join).Thus,every pair of tuples from re-lations A and B must be verified against the join condi-tion.Relations A and B are located within regions R A andR B and they are joined in-network in a join region R J. Technique forfinding the location of the join region havebeen presented elsewhere[4,5,16]and are orthogonal toour problem.In fact,our algorithm is general with respectto the join relations and their locations and could be usedwithin the core of other previously proposed join solutions(e.g.[5]),including solutions using semi-joins(e.g.[16]).For clarity of presentation we describe our join algorithm inthe context of the Mediated Join[5]solution.The Mediated Join solution works as follows:relationsA andB are sent to the join region(R J)where they are joined and the resulting relation J is transmitted to the query originator node.(Recall that a query can be posed at any node of the network.)Figure2shows in overview the query processing steps and the dataflow.The Mediated Join seems straightforward based on this description,but there are several issues that must be carefully addressed in the low-level sensor implementation to ensure the correct-ness of the query result,e.g.:•How to ensure that both relation A and B are transmit-ted to the same region R J?•How large should region R J be to have sufficient re-sources,i.e.,memory at nodes,to process the join?•How should A and B be transmitted such that the join is processed correctly at the nodes in R J?•How to process the join in R J such that the join is processed correctly using minimum resources?We now describe in details DIJ,our join processing al-gorithm addressing these questions.The steps of DIJ are: 1.Multi-cast the query from originator node O to nodesin R A and R B.Designate the nodes closest to the cen-tres C A and C B of the regions R A,respectively R B,as regional coordinators.Designate the coordinator lo-cation C J for join region R J.Disseminate the infor-mation about the coordinators along with the query.Figure2.Mediated Join-dataflow2.Construct routing trees in regions R A and R B rootedat their respective coordinators C A and C B.3.Collect information on the number of query relevanttuples for each region at the corresponding coordina-tors.Each coordinator sends this information to coor-dinator C J of the join region R J.4.Construct the join region.C J constructs R J so that ithas sufficient memory space at its nodes to buffer A.5.Distribute A over R J.(a)C J asks C A to start sending packets with tuples.Once C J receives A’s tuples,it forwards them toa node in R J with available memory.(b)Upon receiving a request for data from C J,C Aasks for relevant tuples from its children in therouting tree.The process is repeated by all inter-nal tree nodes until all relevant tuples have beenforwarded up in the tree.6.Broadcast B over R J(a)Once C J receives a signal from C A that it hasno more packets(i.e.,tuples)to send,C J asksfor one packet with tuples from C B.When thepacket is received,it is broadcast to nodes in R J.(b)Each node in R J joins the tuples in the packet re-ceived from B with its local partition of A,send-ing the resulting tuples to O.Once the join iscomplete,each node asks for another packet ofB’s tuples from C J.(c)Upon receiving a request for tuples from C J,C Basks for a number of join tuples from its childrenin the routing tree.The process is repeated byall internal tree nodes if they cannot satisfy therequest alone.(d)Once C J receives requests for B’s tuples from allnodes in R J,Step6is repeated unless C B signalsthat it has no more packets(i.e.,tuples)to send.In the steps above we chose,only for the sake of presen-tation,that relation A is distributed over the nodes in R J and relation B is broadcast over the nodes in R J.The steps above are symmetrical if the roles of A and B are switched, however the actual order does matter in terms of query cost. In Section4we explore this issue and show how to deter-mine which relation should be distributed and which should be broadcast in order to minimize the cost of the processing the join operator.Steps1-3of DIJ are typical to in-network query pro-cessing and do not present particular challenges.In Step4, the join coordinator C J must request and pool together the memory of other nodes in its vicinity for allocating relation A to these nodes(in Step5a).This is a non-trivial task as C J does not have information about the nodes in its vicin-ity(except its1-hop neighbours).Steps5and6also pose a challenge,that is,how to control theflow of tuples effi-ciently without buffer overflows,ensuring correct execution of the join.We detail these steps in the following.3.1.Constructing the join region(Step4)Once node C J receives the size of the join relations A and B from C A and C B(in Step1),it mustfind the nodes in its vicinity where to buffer relation A.DIJ uses the fol-lowing heuristic for this task,called k-hop-pooling: If C J alone does not have sufficient memory tobuffer relation A,C J asks its1-hop neighbours toreport how much memory they have available forprocessing the query.If relation A is smaller thanthe total memory available at the1-hop neigh-bours,C J stops the memory search.Otherwise,C J asks its2-hop neighbours to report their avail-able memory.This process is repeated for k-hops,where k represents the number of hops such thatthe total memory available at the nodes up to khops away from C J plus the memory available atC J is sufficient to buffer relation A.An interesting question is how much memory should a node allocate for processing a particular query.If the sensor network processes only one join query at a time(e.g.,there is a central point that controls the insertion of join queries in the network),then nodes can allocate all the memory they have available for processing the join.However,if nodes al-locate all their memory for a query,but several join queries are processed simultaneously in the network,it may happen that a coordinator C J will notfind any nodes with available memory in its immediate vicinity,forcing it to use farther away nodes during processing,and,thus,consuming more energy.For networks where multiple queries may coexist in the network,nodes should allocate only a part of their avail-able memory for a certain query,reserving the rest for other queries.How to actually best allocate the memory of an individual node is orthogonal to our problem.In this work we assume that nodes report as available only the memorythey are willing to use for processing the requested query.Figure3shows a possible memory allocation scheme at anode.3.2Distributing A over R J(Step5)In this step two tasks are carried out concurrently:C Arequests and gathers relevant tuples(grouped in data pack-ets)from R A,and C J distributes the packets received fromC A over R J.Once the set of k-hop neighbours that will buffer A hasbeen constructed,C J asks for relation A from C A,packetby packet,and distributes each packet of A’s tuples in around-robin fashion to its neighbours,ordered by their hopdistance to C J.When deciding to which node to send a newpacket with A’s tuples,a straightforward packet allocationstrategy would be for C J to pick a node from its list andsend to it all new packets with A’s tuples until its allocatedmemory is full.This strategy has two disadvantages.As allpackets use the same route(for most routing algorithms)toget to their destination node,their delivery will be delayed ifthere is a delay on one of the links in the route.Also,con-secutive packets may contain tuples with values such thatthey all(or many of them)will join with the same tuple inB.In this case,the node holding all these tuples will gener-ate many result tuples that have to be transmitted,delayingthe processing of the join.The hop-based round-robin al-location also ensures that all k-hop neighbours have a fairchance of having some free memory at the end of the allo-cation process,memory that can be used for other queries.Once node C A receives a request for tuples from C J,ithas to gather relevant tuples from R A.If C A would simplybroadcast the tuple request in the routing tree constructedover R A,nodes in R A will start sending these tuples to-ward C A.As each internal tree node has(likely)severalchildren,it should receive and buffer many packages beforebeing able to send these packages out.Some nodes maynot be able to handle such a dataflow due to lack of bufferspace,possibly dropping some of the packets.To ensurethat no packages are lost due to lack of buffer space,wepropose aflow synchronization scheme where each nodewill only buffer one package.In this scheme,the requestfor A’s tuples is transmitted one link at a time.Each nodein the routing tree is in one of the following states duringthe synchronized tupleflow(Figure4):•Wait for a tuple request from the parent node(or C J in the case of C A)in the routing tree constructed in Step2.•Send local tuples(from the local storage or receive buffer)to the parent node.•If buffer space has been freed and there are relevant tu-ples available at the children nodes in the routing tree,Figure3.Memory allocation schemeFigure4.A node’s states during tuple routingrequest tuples from a child node that still has tuples to send.Figure5shows the routing tree for a region and the information maintained in each node of the tree as tuples are routed from either R A or R B to R J.Note that the number of tuples that each child node will pro-vide has been collected as part of Step3.•Receive tuples from child,buffer the tuples and update the number of tuples that the child still has available.Once a node has forwarded to its parent all of A’s tuples from its routing sub-tree,it can free all buffers used for pro-cessing the query.{local: 2 tuples}{local: 2 tuples{local: 0 tuples}{local: 3 tuples}{local: 3 tuples}{local: 2 tuples{local: 3 tuplesN5: 8 tuplesN6: 5 tuples}N3: 0 tuplesN1: 2 tuplesN2: 3 tuples}N4: 3 tuples}N7N5N1N2N3N6N4Figure5.Join tuples information at nodes3.3.Broadcasting B over R J(Step6)The collection of B’s tuples proceeds much like the collection of A’s tuples,with one important difference. Whereas C A gathers and sends all of the relevant tuples of A as a a result of a single tuple request from C J,C B only sends one packet with tuples for each request it re-ceives from C J.This way,C J can broadcast such a packet of tuples to all nodes in R J,wait until all nodes fully pro-cess the local joins and send the results,and then request a new packet of tuples from R B when each node in the join region R J is ready to receive and join a new set of tuples.4.Selecting the relation to be distributedIn the previous discussions we have assumed for clarity of presentation that relation A is distributed over the nodes in region R J and B is broadcast over the nodes in the re-gion.An interesting question is which of the two join rela-tion should be distributed and whether the choice makes a major difference in cost.Let us focusfirst on which of the two join relation should be distributed and,subsequently,which should be incre-mentally broadcast.To decide on this matter,the query optimizer has to estimate the cost of the two options(i.e., distribute A or B)and compare their costs to decide which alternative is more energy efficient.For generality,we de-rive in the following a cost model for processing the join by distributing relation R d and broadcasting relation R b.The actual relations A and B can then be substituted into R d and R b(or vice-versa)to estimate the processing costs.Considering the steps of DIJ,the cost of query process-ing can be decomposed into a sum of components,with one component associated to each step.Several of these com-ponents are independent of the choice of the relation that is distributed.Thus,they do not affect the decision of which relation to distribute and do not need to be derived.For instance,we have the cost for disseminating the query in regions A and B(Step1)and the cost for constructing the routing tree over regions R A and R B(Step2).These costs are identical when processing the join by distributing A or B and do not affect the decision.The steps that have differ-ent costs when A or B are the distributed relation R d are the construction of the join region R J(Step4),the distribution of the relation R d(Step5a)and the broadcast of the relation R b(Step6a).Note that we are only interested in differences in the communication cost between the two alternatives. 4.1.Constructing the join region(Step4)As discussed in Section3.1,we use the k-hop-pooling strategy to construct the join region R J.In each round of memory allocation,C J broadcasts its request for memory in a hop-wise increasing fashion,until sufficient nodes with the required buffer space are located.During a round h,each node within h-hops from C J broadcast the memory request and its1-hop neighbours re-ceive the request message.Thus,the total energy cost is:E memreq4=k−1h=0(E t N h n M r+E r N h n N1n M r),where N h n represents the average number of nodes within h hops from a node,E t and E r represents the energy required to transmit,respectively receive,one bit of information and M r represents the size of the memory request message(in bits).N h n is a network-dependent value independent of our technique and it is derived in the Appendix.When a node receives a memory request message for the first time,it allocates buffer space in its memory and sends the memory information to C J.The nodes located h-hops away from C J perform two tasks:they send their own mem-ory information to the nodes located h−1hops away;and they forward the information they have received from the nodes located between h+1and k hops away from C J.If we denote by M i the size of the memory information for one node,the total energy cost of collecting the information on available memory is:E meminfo4=kh=1((E t+E r)(N h n−N h−1n)M i+(E t+E r)(N k n−N h n)M i)=(E t+E r)(kN k n−k−1h=1N h n)M iNote that(N h n−N h−1n)represents the number of nodesh-hops away and(N k n−N h−1n)represents the number of nodes located more than h and up to k hops away from C J. The total energy cost of the fourth step of DIJ is:E4=E memreq4+E meminfo4.Note that the costs of Step4do not depend on the join re-lations directly,but through k which determines the size of the join region R J and it is determined by the size of the join relation R d.Let B s be the average size(in bits)of the buffer space that each node in R J can allocate for processing the query. The minimum number of nodes that must be used to store relation R d in region R J is||R d||B s,where||R||denotes the size(in bits)of relation R.Since nodes are added to R J in groups based on their hop distance,k is the lowest number of hops such that the nodes within k hops from C J have sufficient buffer space to buffer R d:k={min h|N h n B s≥||R d||}.。

单细胞转录组测序数据分析方法单细胞转录组测序（single-cell RNA sequencing，scRNA-seq）是一种能够测量每个细胞内大量基因表达的技术。

与传统的全组细胞转录组测序相比，scRNA-seq可以更细致地研究不同表型细胞的异质性，从而深入了解细胞发育、组织构建以及疾病的发病机制。

然而，由于单细胞转录组数据规模庞大，独特的数据结构和差异化的表达模式，分析这些数据也面临着挑战。

下面将介绍几种常见的单细胞转录组测序数据分析方法。

1. 数据预处理在进行单细胞转录组测序数据分析之前，首先需要对原始数据进行预处理。

常见的预处理步骤包括去除低质量的细胞、去除批次效应、进行基因表达量的归一化以及异常值的处理。

去除低质量的细胞通常可以根据细胞的表达量进行筛选。

在大多数情况下，保留表达量高于一定阈值的细胞可以有效去除噪音和低质量的数据。

批次效应是由不同实验批次或处理过程引入的技术差异。

为了消除批次效应对分析结果的影响，可以应用一些统计方法，例如ComBat算法，对数据进行批次校正。

基因表达量的归一化是将不同细胞之间、不同基因之间的表达量进行统一的过程。

常见的归一化方法有TPM (Transcripts Per Million)、FPKM (Fragments Per Kilobase of transcript per Million mapped reads)以及CPM (Counts per Million)等。

异常值的处理是要将表达量异常的基因或细胞进行处理，以保证数据的准确性。

一种常见的方法是将异常值置为缺失值或使用统计方法进行调整。

2. 细胞聚类细胞聚类是将单细胞数据根据其表达模式的相似性进行分组的方法。

通过聚类分析，我们可以将同一类型细胞的数据聚集在一起，便于后续的细胞识别和功能注释。

常见的细胞聚类算法包括K-means、层次聚类（hierarchical clustering）、DBSCAN（Density-Based Spatial Clustering of Applications with Noise）等。

EE693H Fall2007Game TheoryTR,12:00pm–1:15pm,Holmes389Course InformationGame theory provides the most natural framework to study the strategic interactions between self-interested decision makers.Due to the emergence of distributed complex systems made up of many autonomous agents (such as the Internet),there has been a resurgence of interest in game theory within the engineering and the computer science communities.This course will introduce the students to the fundamentals of noncoopera-tive game theory as well as the computational tools provided by noncooperative game theory.Emphasis will be on the engineering applications such as control,communications,transportation systems,and resource allocation problems.The course is intended for mathematically inclined students with some background on probability theory.Instructor:G¨u rdal Arslan,Holmes440,Phone:956–3432,E-mail:*****************Office Hours:OpenRecommended Texts:Dynamic Noncooperative Game Theory by Bas.ar and OlsderGame Theory by Fudenberg and Tirole,Webpage:/∼gurdal/EE693H.htmSite of announcements,handouts,homeworks,etc.Grading:Homework30%;Mid-term35%;Project35%.Policies:No credit will be given to late homeworks.Exams must be taken at the announced times.(Tentative)Topics•Introduction(1Lecture)–Examples and various solution concepts•Zero-Sum Finite Games in Normal Form(2Lecture)–Security strategies–Lower and upper values–Saddle-point equilibrium–Mixed strategies–Minmax theorem–Computation of saddle-point equilibria by graphical solution and LP approaches–Dominated strategies–Iterative elimination of dominated strategies•Normal Form Games(6Lecture)–Pure and mixed strategies–Dominated strategies and solution by iterated dominance–Nash equilibrium–Pure equilibrium,Strict equilibrium–Examples of pure equilibrium(Cournot’s model of oligopoly,CDMA uplink power control)–Existence of mixed equilibria infinite normal games(Best response correspondence,Kakutani’s fixed point theorem)–Existence of pure equilibrium in infinite games with continuous payoffs(Quasi-concavity of player payoffs in its own decisions)–Sufficient conditions for the uniqueness of pure equilibrium in infinite games with continuous payoffs(Diagonal strict concavity condition)–Existence of mixed equilibrium in infinite games with continuous payoffs–Discontinuous games–Computation of Nash equilibria infinite normal-form games(algebraic approach,optimization approach)–Correlated equilibrium,coarse correlated equilibrium,correlated equilibrium with information partitions•Well-known Classes of Non-Zero-Sum Games(7Lecture)–Generalized ordinal potential games and existence of pure equilibria–Finite improvement property–Characterization of potential games–Weighted potential games–Congestion games–Inefficiency of Nash equilibria in congestion games,Tolls minimizing the total congestion,Braess’paradox–Price of anarchy and price of stability in congestion games–Infinite potential games–Efficiency loss in resource allocation games–(Weakly)acyclic games–Consensus problem–Supermodular games•Learning in games(8Lecture)–Cournot’s adjustment process–Fictitious play,Asymptotic behavior,Convergence of beliefs in certain classes of games,Shapley’s example,Lack of payoffconsistency,–Stochasticfictitious play,Payoffconsistency,Perturbed equilibria,Convergence of intended be-havior via stochastic approximation theory–Computation,memory,and observation requirements offictitious play–Regret based dynamics,Utility basedfictitious play–Finite memory variants offictitious play,Adaptive play,Elements of Markov processes,Perturbed Markov processes,Stochastic stability•Repeated Games•Auctions;Mechanism design;Incentive design•Games with incomplete/imperfect information;•Extensive form games•Dynamic games;Markov games。

工业工程专业术语工业工程英语术语工业工程专业是为满足国家经济开展和参加WTO对人才的迫切需要而建立的。

该学科主要是以生产过程为研究对象，以提高劳动生产率、保证质量和降低本钱为目标，特别注重研究人的因素，充分发挥投入资源的作用。

以下是PINCAI收集的工业工程专业术语，仅供大家阅读参考!1.工业工程的定义：工业工程是对人员，物料，设备，能源和信息所组成的集成系统，进展设计，改善和设置的一门学科。

它综合运用数学，物理学和社会科学方面的专门知识和技术，以及工程分析和设计的原理与方法，对该系统所取得的成果进展确定，预测和评价2.生产率的定义：一般定义：产出和投入之比概念：是衡量生产要素(资源)使用效率的尺度3.生产率管理的定义：是对一个生产系统的生产率进展规划，测定，评价，控制和提高的系统管理过程4.生产过程的定义：是从产品投产前一系列生产技术组织开始，直到把它生产出来为止的过程(自然过程，劳动过程)劳动过程：生产准备过程，根本生产过程，辅助生产过程，生产效劳过程根本生产过程：工艺检验运输5.生产流程分析的定义：生产流程分析是以产品的整个制造过程为研究对象的一种系统分析6.作业测定的定义：作业测定(工作衡量)是运用各种技术来确定合格工人按规定作业标准完成某项工作所需的时间7.标准时间的定义：在适宜的操作条件下，用最适宜的操作方法，以普通熟练工人的正常速度完成标准作业所需的劳动时间8.时间研究的定义：时间研究是一种作业测定技术，旨在决定一位合格，适当，训练有素的操作者，在标准状态下，对一特定的工作以正常速度操作所需的时间9.工作日写实的定义：按照工作时间消耗的顺序对工人在整个工作时间或部分工作时间的一切情况，进展实地观察，记录和分析的一种方法10.标准资料的定义：将直接由作业测定所获得的大量测定值或经历值分析，编制而成的某种构造的作业要素(根本操作单元)正常时间值的数据库11.劳动定额的定义：指在一定生产技术，组织条件下，采用科学合理的方法，为生产一定量的合格产品或完成一定量的工作，所预先规定的劳动消耗量的限额12.劳动定额标准的定义：是对劳动定额制定，实施，统计分析和修订的各个环节中重复性事物所作的统一规定13.现场管理的定义：是对生产现场的一切活动，按照企业的经营目标，进展方案，组织，协调，控制和鼓励的总称14.定置管理的定义：定置管理是企业对生产现场中的人，物，场所三者之间的关系进展科学地分析研究，使之到达最正确结合状态的一门科学管理方法，它以物在场所的科学定置为前提，以完整的信息系统为媒介，以实现人和物的有效结合为目的，通过对生产现场的，整顿，把生产现场中不需要的物品去除掉，把需要的物品放在规定位置上，使其随手可得，促进生产现场管理文明化，科学化，到达高效生产，优质生产，平安生产。

测绘工程专业英语翻译各位读友大家好，此文档由网络收集而来，欢迎您下载，谢谢篇一：测绘工程专业英语课文翻译Unit 9 Basic Statistical Analysis of Random Errors （随机误差的统计学基本分析）Random errors are those variables that remain after mistakes are detected and eliminated and all systematic errors have been removed or corrected from the measured 后，并且所有系统误差被从测量值中移除或修正后，保留下的那些变量variable变量、变化n.）They are beyond the control of the the random errors are errors the occurrence of which does not follow a deterministic pattern.确定性的模式pattern而发生的误差）In mathematical statistics, they areconsidered as stochastic variables, and despite their irregular behavior, the study of random errors in any well-conducted measuring process or experiment has indicated that random errors follow the following empirical rules:mathematical statistics中，它们被当成随机变量stochastic variable，尽管它们的行为无规律，在任一正确的well-conducted原意为品行端正的，这里指测量实验和活动是无误的测量活动和实验中，对的随机误差的研究显示indicate随机误差遵循以下经验法则empirical⑴A random error will not exceed a certain amount.（随即误差不会超过一个确定的值）⑵Positive and negative random errors may occur at the same frequency.（正负误差出现的频率相同）⑶Errors that are small in magnitude are more likely to occur than those that are larger in magnitude.比数值大的误差出现可能性大be likely to 可能）⑷The mean of random errors tends to zero as the sample size tends to infinite.随机误差的平均值趋近于0）In mathematical statistics, random errors follow statistical behavioral laws such as the laws of 行为behavioral行为的规律，如概率法则）A characteristic theoretical pattern of error distribution occurs upon analysis of a large number of repeated measurements of a quantity, which conform to normal or Gaussian distribution.观测分析analysisn.中的误差分布的一个特征理论模式，遵照conform to遵照正态或高斯分布）在对一个量进行大量重复观测分析后，得到一个误差分布的理论特征——正态或高斯分布The plot of error sizes versus probabilities would approach a smooth curve of the characteristic bell-shape.与……相对概率的关系图，接近一条光滑的特有的characteristic特有的钟形曲线。

数学专业英语词汇(S)s admissible s容许的s matrix s矩阵saddle point 鞍点saddle point method 鞍点法salient angle 凸角salient point 折点saltusfunction 跳跃函数sample 样本sample correlation coefficient 样本相关系数sample correlation matrix 样本相关阵sample covariance 样本协方差sample covariance matrix 样本协方差阵sample dispersion 样本方差sample function 样本函数sample mean 样本均值sample median 样本中位数sample moment 样本矩sample point 样本点sample quartile 样本四分位数sample size 样本的大小sample space 样本空间sample survey 样本甸sample unit 抽样单位sample variable 样本变量sample variance 样本方差sampling 抽样sampling distribution 样本分布sampling error 抽样误差sampling fraction 抽样比sampling inspection 抽样检查sampling method 抽样法sampling moment 样本矩sampling ratio 抽样比sampling survey 样本甸sampling unit 抽样单位sarrus rule 萨律法satisfiable 可满足的satisfy 满足saturated set 浸润集saturated subset 浸润子集saturated vertex 饱和顶点saturation curve 饱和曲线scalar 纯量scalar curvature 纯量曲率scalar density 纯量密度scalar field 纯量场scalar flow of vector field 向量场的纯量流scalar matrix 纯量阵scalar multiplication 纯量乘法scalar product 纯量积scalar quantity 纯量scalar triple product 纯量三重积scalar valued function 纯量值函数scalar valued map 纯量值映射scale 图度scale factor 标度因子scale mark 分核度scale parameter 尺度参数scale transformation 标度变换法scalene triangle 不规则三角形scalenohedron 偏三角面体scalenous triangle 不规则三角形scatter 散布scatter diagram 散布图scattered set 无核集scattergram 点状图scattering theory 散射理论schauder theorem 肖德不动点定理scheduling and production planning 生产计划理论scheduling problem 日程计划问题schematic diagram 简图scheme 模式schlicht 单叶的schlicht domain 单叶域schlicht function 单叶函数schlicht mapping 单叶映射schlicht surface 单叶曲面schmidt orthogonalization 施密特正交化法schwartz space 施瓦尔兹空间schwarz formula 施瓦尔兹公式schwarz inequality 施瓦尔兹不等式scope 辖域screw 螺旋体search process 搜她程secant 正割secant curve 正割曲线second axiom of countability 第二可数公理second boundary condition 诺伊曼边界条件second boundary value problem 诺伊曼问题second comparison test 第二比较检验second component 第二分量second countability axiom 第二可数公理second derivative 二次导数second factor 第二因子second fundamental form 第二基本形式second limit theorem 第二极限定理second variation 第二变分secondary diagonal 次对角线secondary extremal 配连极值secondary obstruction 第二障碍section 截口section functor 截面函子section graph 部分等距算子sectional area 截面积sectional curvature 截面曲率sector 扇形sectorial harmonic 扇低函数secular equation 长期方程sedenion 十六元数segment 线段segment of a circle 弓形segment of a curve 弧段segmentation 分割selection 选择self adjoint boundary value problem 自伴边值问题self adjoint differential equation 自伴微分方程self adjoint eigenvalue problem 自伴特盏问题self adjoint element 自伴元self adjoint linear mapping 自伴线性映射self adjointness 自伴性self checking code 自校验代吗self complementary graph 自补图self conjugate 自共轭的self conjugate partition 自共轭分拆self dual 自对偶的self dual category 自对偶范畴self dual group 自对偶群self intersection number 自交数self loop 自身环self orthogonal submodule 自成正交子模self polar curve 自配极曲线self polar tetrahedron 自配极四面形self polar triangle 自配极三角形self tangency 自切self weighting sample 自加权样本selfadjoint linear subspace 自伴线性子空间selfadjoint operator 自伴算子selfconjugate latin square 自共轭拉丁方selfosculating point of curve 曲线的自密切点semantic equivalence 语义等价semantical decision problem 语义判定问题semantical paradox 语义悖论semantics 语义学semi exact 半正合的semi interquartile 半内四分位数间距semi invariant 半不变式semi invariant exterior derivative 半不变外微分semi logarithmic representation 半对数表示semiadditive category 半加性范畴semianalytic set 半解析集合semiangle 半角semiaxis 半轴semibounded 半有界的semibounded operator 半有界算子semicircle 半圆semicircular 半圆的semicircular domain 半圆域semicircular protractor 量角器分度规semicontinuity 半连续性semicontinuous 半连续的semicontinuous function 半连续函数semicontinuum 半连续统semiconvergent series 半收敛级数semicubical parabola 半三次抛物线semidefinite eigenvalue problem 半定特盏问题semidefinite kernel 半定核semidefinite operator 半定算子semidefinite quadratic form 半定二次形式semidefinite variational problem 半定变分问题semidiameter 半径semidirect product 半直积semidiscretization 半离散化semifinite 半有限的semifinite trace 半有限迹semigroup 半群semigroup algebra 半群代数semigroup of operators 算子半群semihereditary ring 半遗传环semilinear 半线性的semilinear mapping 半线性映射semilinear substitution 半线性代换semilinear transformation 半线性变换semilocal ring 半局部环semilogarithmic diagram 半对数图semilogarithmic paper 半对数坐标纸semimagic square 半幻方semimajor axis 半长轴semimajorant 半强函数semimartingale 半semimean axis 半中轴semimetric 半度量semiminor axis 半短轴semiminorant 半弱函数semimodular lattice 半模格semimodularity 半模性seminorm 半范数semiorder 半有序semiordered banach space 半有序巴拿赫空间semiordered set 半有序集semipath 半通路semiperiod 半周期semipolar set 半极集semiprimary ring 半准素环semiprime ideal 半素理想semiprimitive ring 半本原环semiquaternion 半四元数semireductive 半可简约的semireflexive 半自反的semireflexive space 半自反空间semireflexivity 半自反性semiregular point 半正则点semiregular space 半正则空间semiregular topology 半正则拓扑semiscalar product 半纯量积semisimple algebra 半单代数semisimple group 半单群semisimple module 半单模semisimple representation 半单表示semisimple ring 半单环semisimplicial complex 半单纯复形semisimplicial map 半单纯映射semisphere 半球semispherical 半球的semitangent 半切线semitransverse axis 半贯轴semiuniform space 半一致空间sense of rotation 旋转指向senseclass 指向类sensepreserving mapping 保向映射sensitivity analysis 灵敏度分析sentence 命题sentential calculus 命题演算sentential connective 命题联结词sentential function 谓词separability 可分离性separable closure 可分闭包separable degree 分离度separable element 可分元separable extension 可分扩张separable field 可分域separable game 可分对策separable graph 可分图separable polynomial 可分多项式separable sets 可分集separable space 可分空间separable stochastic process 可分随机过程separable topological space 可分拓扑空间separable transcendental extension 可分超越扩张separate 分离separated equation 分离变数方程separated sets 隔离集separated space 分离空间separating edge 分离棱separating plane 分离平面separating transcendence basis 可分超越基separation 分离separation axiom 分离公理separation of the zeros 零点分离separation of variables 分离变量separation principle 分离原理separation relation 分离关系separation theorem 分离定理separator 分隔符sequence 序列sequence convergent almost everywhere 几乎处处收敛列sequence of arcs 弧序列sequence of complex numbers 复数序列sequence of differences 差分序列sequence of distinct points 一一序列sequence of functions 函数序列sequence of iterations 迭代序列sequence of numbers 数列sequence of partial sums 部分和列sequence of points 点序列sequence of sets 集序列sequence of signs 符号序列sequence space 序列空间sequencing 排序sequencing problem 日程计划问题sequent 串联的sequential analysis 序贯分析sequential compactness 列紧性sequential control 顺序控制sequential estimation 序列估计sequential likelihood ratio test 序贯似然比值检验sequential programming 顺序规划sequential sampling 序贯抽样sequential sampling plan 序列抽样法sequential switching circuit 依次转接电路sequential test 序贯检定sequential word function 序贯字函数sequentially compact set 列紧集sequentially complete space 序列完备空间serial correlation 自相关serial correlation coefficient 序列相关系数series 级数series development 级数展开series expansion 级数展开series of functions 函数级数series solution 级数解serpentine 蛇形线service model 服务模型sesquilinear form 半双线性形式set 集set algebra 集代数set function 集函数set of condensation points 凝聚点集set of continuum power 连续统势的集set of limit 极限集合set of lower bounds 下界集set of measure zero 零测度集set of numbers 数集set of points 点集set of sets 集的集set of strategies 策略集set of ternary numbers 三进制数集set of upper bounds 上界集set theoretic addition 集论的加法set theoretic image 集合论的象set theoretic intersection 集论的交set theoretic operation 集论的运算set theoretic proof 集论的证set theoretic union 集论的并集set theoretical 集论的set theory 集论set topology 集论拓扑set valued functor 集值函子set valued mapping 集值映射sexadecimal digit 十六进制数字sexadecimal notation 十六进记法sexadecimal number system 十六进制数系sexagesimal arithmetic 六十进算术sexagesimal system 六十进制sextant 六分仪sextic 六次曲线sextic equation 六次方程shadow 影shadow price 影子价格shape 形状shape theory 形状理论sheaf 层sheaf homomorphism 层同态sheaf of germs of continuous functions 连续函数的芽层sheaf of planes 平面束sheaf theoretic 层理论的shear 剪切shearing modulus 刚性模量shearing strain 切应变shearing stress 切应力sheet 叶sheets of rieman surface 黎曼曲面的叶shell 壳层shift 移动shift operator 位移算子shift register 移位寄存器shilov boundary 希洛夫边界short division 短除法shorten 缩短shortening 缩短shortest 最短线shortest confidence interval 最短置信区间shortest distance 最短距离shortest path 最短道路shortest path problem 最短道路问题shortest route 最短道路shortest route problem 最短道路问题side 边side condition 边条件side elevation 侧视图sieve 筛sieve method 筛法sieve of eratosthenes 厄拉多塞筛sigma additivity 可列可加性sigma algebra 代数sigma compact space 紧空间sigma compactness 紧性sigma complete filter 完备滤子sigma complete lattice 完全铬sigma complete lower semilattice 完备下半格sigma complete semilattice 完备半格sigma discrete family of subsets 子集的离散族sigma finite measure 有限测度sigma function 函数sigma locally finite family of subsets 子集的局部有限族sigma monogenic function 单演函数sigma space 空间sigmacompleteness 完备性sigmafield of sets 集的域sigmalattice 格sigmoid s形曲线sign 符号sign digit 符号数字sign of equality 等号sign of inclusion 包含记号sign of inequality 不等号sign of intersection 相交记号sign of multiplication 乘号sign of permutation 置换的符号sign of subtraction 减法sign of summation 连加号sign of the membership relation 从属关系记号sign of union 并号signal 信号signature of permutation 置换的符号signed numbers 带符号数signed rank test 符号秩检验signed tree 指定符号的树significance level 显著性水平significance of a deviation 偏差的显著性significance test 显著性检定significant 有效的significant digit 有效数字significant figure 有效数字signum 正负号函数similar figures 相似形similar function 相似函数similar matrix 相似矩阵similar ordered set 相似有序集similar region 相似域similar test 相似检验similar triangles 相似三角形similarity 相似similarity principle 相似性原理similarity theorem 相似性定理similarity transformation 相似变换similitude 相似similitude transformation 相似变换simple 单的simple abelian variety 单阿贝耳簇simple algebra 单代数simple arc 简单弧simple branch point 单分枝点simple broken line 单折曲线simple chain 简单链simple character 简单特贞simple circuit 简单围道simple closed curve 简单闭曲线simple component 单分量simple compression 单压缩simple connectedness 单连通性simple connectivity 单连通性simple continued fraction 正则连分数simple continued fraction expansion 简单连分数展开simple convergence 点态收敛simple correlation coefficient 单相关系数simple cycle 简单循环simple domain 单叶域simple eigenvalue 简单特盏simple elongation 单伸长simple event 简单事件simple extension 单扩张simple extension field 单扩张域simple fixed point 单纯不动点simple fraction 普通分数simple function 单叶函数simple graph 简单图simple group 单群simple harmonic motion 简谐运动simple hypothesis 简单假设simple integral 单积分simple intersection point 单纯交点simple iterative method 单迭代法simple lattice 单格simple lie algebra 单李代数simple module 单模simple object 简单对象simple path 简单道路simple pendulum 单摆simple point 单点simple polygon 简单多边形simple polyhedron 简单多面体simple product 简单积simple proper value 简单特盏simple quadrilateral 简单四边形simple random sampling 简单随机样本simple regression 简单回归simple regression coefficient 单回归系数simple ring 单环simple root 单根simple sample 简单样本simple sampling 简单抽样simple series 简单级数simple set 单集simple spectrum 单谱simple surface 简单曲面simple tangent 单切线simple theory of types 简单类型论simple transcendental extension 单超越扩张simplex 单形simplex method 单形法simplex multiplier 单形乘数simplex tableau 单形表simplex theorem 单形定理simplicial approximation 单纯逼近simplicial cell 单纯胞腔simplicial chainmapping 单纯链映射simplicial cochain complex 单纯上链复形simplicial cohomology group 单纯上同岛simplicial complex 单纯复形simplicial homology 单纯同调simplicial map 单形映射simplicial mapping cylinder 单形映射柱simplicial pair 单形对simplicialapproximation theorem 单纯逼近定理simplification 简化simplified fraction 简化分数simplified newton method 简化牛顿法simply connected group 单连通群simply connected region 单连通区域simply connected spatial domain 单连通空间域simply convergent filter 单收敛滤子simply ordered group 全有序群simply periodic function 单周期函数simplyconnected domain 单连通域simpson rule 辛卜生法则simulation 模拟simultaneity 同时性simultaneous confidence intervals 联合置信区间simultaneous diagonalization 同时对角化simultaneous differential equation 联立微分方程simultaneous differential equations 联立微分方程simultaneous equations 方程组simultaneous estimation 联立估计simultaneous invariant 联立不变式simultaneous substitution 同时代入sine 正弦函数sine curve 正弦曲线sine function 正弦函数sine integral 正弦积分sine integral function 正弦积分函数sine law 正弦定律sine spiral 正弦螺线sine theorem 正弦定理sine wave 正弦波single 单的single address 单地址的single address code 一地址代码single address instruction 单地址指令single address system 单地址系统single factor method 单因子法single step method 单步法single step process 单步法single valued 单值的single valued analytic function 单值解析函数single valued correspondence 单值对应single valued function 单值函数single valued operation 单值运算single valued relation 单值关系single valuedness 单值性singlevalued mapping 单值映射singly periodic function 单周期函数singular 奇异的singular automorphism 奇异自同构singular bivariate normal distribution 奇异二元正态分布singular boundary 奇异边界singular boundary point 奇异边界点singular chain 奇异链singular chain complex 奇异链复形singular cohomology group 奇异上同岛singular complex 奇异复形singular conic 奇二次曲线singular correspondence 奇对应singular cycle 连续循环singular distribution 退化分布singular element 奇元素singular elliptic function 奇异椭圆函数singular function 奇异函数singular function of bounded variation 有界变差奇异函数singular graph 奇异图singular homology 奇异下同调singular homology class 奇异同掂singular homology group 连续同岛singular integral 奇解singular integral element 奇异积分元素singular integral equation 奇异积分方程singular kernel 奇核singular line element 奇异线素singular linear operator 奇异线性算子singular linear transformation 奇异线性变换singular locus 奇轨迹singular mapping 奇异映射singular matrix 退化阵singular operator 奇异算子singular ordinal 特异序数singular part 奇异部分singular plane 奇异平面singular point 奇点singular proposition 特称命题singular quadric 奇异二次曲面singular series 奇异级数singular solution 奇解singular space 奇异空间singular submodule 奇子模singular subspace 奇异子空间singular surface 奇曲面singular transformation 奇异变换singular value 奇异值singular variational problem 奇异变分问题singular vector 奇异向量singularity 奇点sink 收点sinusoid 正弦摆线sinusoidal 正弦的sinusoidal function 正弦函数sinusoidal law 正弦定律sinusoidal spiral 正弦螺线situation 情况size 样本的大小skeleton 骨架skew curve 空间曲线skew derivation 斜微分skew determinant 斜对称行列式skew distribution 偏斜分布skew field 非交换域skew hermitian form 斜埃尔米德型skew hermitian matrix 斜埃尔米德矩阵skew lines 偏斜线skew position 歪扭位置skew quadrilateral 挠四边形skew surface 非可展直纹曲面skew symmetric 反对称的skew symmetric determinant 斜对称行列式skew symmetric matrix 斜对称矩阵skew symmetric tensor 斜对称张量slack variable 松弛变量slide rule 计算尺sliding vector 滑动向量slitregion 裂纹区域slope 斜率slope function 斜率函数slope intercept form 斜截式slope line 倾斜线slope of a curve 曲线的斜率slowly increasing sequence 缓增序列small circle 小圆small inductive dimension 小归纳维数small sample 小样本small set 小集smallest element 最小元smash 收缩smooth curve 平滑曲线smooth map 光滑映射smooth morphism 光滑射smooth projective plane curve 光滑射影平面曲线smoothing 光滑化smoothness 光滑度sobolev embedding theorem 水列夫嵌入定理sobolev space 水列夫空间sojourn time 逗留时间solenoidal group 螺线群solenoidal vector field 螺线向量场solid 立体;固体solid angle 立体角solid geometry 立体几何solid n sphere n维球体solid of revolution 旋转体soliton 孤立子soluble 可解的solution 解solution curve 积分曲线solution domain 解域solution formula 解公式solution set of equation 方程的解集solution space 解空间solution surface 积分曲面solution tree 解树解答树solution vector 解向量solvability 可解性solvable 可解的solvable equation 可解方程solvable group 可解群solvable ideal 可解理想solve 解sorter 分类器source 发点source free vector field 无源向量场source function 格林函数source of field 场源source program 源程序space 空间space coordinates 空间坐标space curve 空间曲线space integral 体积积分space like manifold 类空廖space of left cosets 左傍系空间space of matrices 矩阵空间space of quaternions 四元数空间space of right cosets 右傍系空间space quadratic transformation 空间二次变换space region 空间区域span 生成spanning tree 最大树生成树sparse matrix 稀巯阵spatial 空间的spatial co ordinate 空间坐标spatial isomorphism 空间同构spatial point 空间点special divisor class 特殊除子类special functional equations 特殊函数方程special functions 特殊函数special group 特殊群special homology manifold 特殊同滴special jordan algebra 特殊约当代数special linear group 特殊线性群special linear homogeneous group 特殊线性齐次群special orthogonal group 特殊正交群special purpose computer 专用计算机special representation 特殊表示special unitarian group 特殊酉群special valuation 特殊赋值specialization 特定化specific 特殊的specific address 绝对地址specific heat 比热specificity 特性spectral analysis of operators 算子的谱分析spectral decomposition 谱表示spectral density 谱线密度spectral distribution 谱分布spectral distribution curve 光谱分布曲线spectral function 谱函数spectral functor 谱函子spectral geometry 谱几何spectral integral 谱积分spectral invariant 谱不变量spectral line 谱线spectral mapping theorem 谱映射定理spectral measure 谱测度spectral multiplicity 谱重度spectral norm 谱模spectral point 谱点spectral property 谱性质spectral radius 谱半径spectral representation 谱表示spectral sequence 谱序列spectral set 谱集spectral space 谱空间spectral subspace 谱子空间spectral synthesis 谱综合spectral theorem 谱定理spectrum 谱spectrum of a matrix 阵的谱speed 速度sphere 球sphere of contact 相切球面sphere of the inversion 反演球spherical 球形的spherical angle 球面角spherical astronomy 球面天文学spherical asymptote 球面渐近线spherical bessel function 球贝塞耳函数spherical cap 球冠spherical cohomology class 球上同掂spherical coordinates 球极坐标spherical curvature 球面曲率spherical curve 球面曲线spherical cyclic curve 球面循环曲线spherical derivative 球面导数spherical domain 球面域spherical epicycloid 球面外摆线spherical excess 球面角盈spherical fiber 球面纤维spherical function 球函数spherical geometry 球面几何学spherical harmonic function 球面低函数spherical harmonics 球面低函数spherical helix 球面螺旋线spherical homology class 球面同掂spherical image 球面象spherical indicatrix 球面指标spherical indicatrix of binormal 副法线球面指标spherical indicatrix of principal normal 吱线球面指标spherical indicatrix of tangents 切线球面指标spherical mean 球中值spherical neighborhood 球形邻域spherical parallelogram 球面平行四边形spherical polar coordinates 球极坐标spherical polygon 球面多边形spherical pyramid 球面棱锥spherical sector 球心角体spherical segment 球截形spherical shell 球壳spherical space 球面空间spherical surface 球面spherical triangle 球面三角形spherical triangular coordinates 球面三角坐标spherical trigonometry 球面三角学spherics 球面几何学sphero quartic 球面四次曲线spheroid 回转椭圆面spheroidal coordinates 球体坐标spheroidal function 球体低函数spheroidal harmonic 球体低函数spheroidal wave function 球体波函数spinode 第一类尖点spinor 旋子spinor field 旋量场spinor genus 旋量狂spinor group 旋子群spinor representation 旋量表示spiral 螺线spiral point 螺线极点spiral surface 螺面spline 样条spline function 样条函数spline interpolation 样条内插splitting 分裂splittingfield 分裂域spur 迹squarable 可平方的square 正方形square bracket 方括弧square contingency 平方列联square deviation 平方偏差square matrix 方阵square measure 平方测度square mesh 正方网格square root 平方根square root transformation 平方根变换square summable function 平方可积函数squareintegrable function 平方可积函数stability 稳定性stability conditions 稳定条件stability criterion 稳定性判据stability group 稳定群stability number 稳定数stability region 稳定区stabilization 稳定stabilization method 稳定法stabilizer 迷向群stable convergence 稳定收敛stable equilibrium 稳定平衡stable homotopy group 稳定同伦群stable manifold 稳定廖stable orbit 稳定轨道stable point 稳定点stable process 稳定过程stable set 稳定集合stable solution 稳定解stable state 稳定状态stalk 茎standard complex 标准复形standard deviation 标准差standard equation 标准方程standard form 标准型standard isobaric surfaces 标准等压面standard n simplex 标准n单形standard regression coefficient 标准回归系数standard simplex 标准单形standard topology 标准拓扑standardization 标准化standardize 使标准化standardized normal distribution 标准化正态分布standardized normal variate 标准化正态变量standardized variable 标准化变量standardized variate 标准化变量star body 星形体star covering 星形覆盖star neighborhood 星形邻域star of a simplex 单形的星形star region 拟星形域star shaped domain 拟星形域starlike domain 拟星形域starlike mapping 拟星形映射starlike set 拟星集starrefinement 星型加细start time 开始时间state 状态state coordinates 状态坐标state function 状态函数state of equilibrium 平衡状态state region 状态区域state space 状态空间state variable 状态变数state vector 状态向量statement 命题static model 静态模型statics 静力学stationary 平稳的stationary curve 平稳曲线stationary distribution 平稳分布stationary flow 平稳流定常流stationary function 平稳函数stationary osculating plane 平稳密切面stationary point 平稳点stationary point process 平稳点过程stationary process 平稳随机过程stationary state 定态stationary time series 平稳时间序列stationary value 平稳值statistic 统计量statistical 统计的statistical accuracy 统计精确度statistical analysis 统计分析statistical data analysis 统计数据分析statistical decision function 统计判决函数statistical decision problem 统计判决问题statistical decision procedure 统计判决程序statistical decision process 统计判决过程statistical distribution 统计分布statistical ergodic theorem 平均遍历定理statistical error 统计误差statistical estimate 估计statistical estimation 统计估计statistical hypothesis 统计假设statistical hypothesis testing 统计假设检验statistical inference 统计推断statistical mechanics 统计力学statistical method 平均法statistical model 随机性模型statistical optimization 统计最佳化statistical quality control 统计质量管理statistical thermodynamics 统计热力学statistics 统计学statistics of extreme values 极值统计statistics of extremes 极值统计steady 平稳的steady state 定态steenrod algebra 斯丁洛特代数steenrod operation 斯丁洛特运算steepest descent method 最速下降法steering program 痔序steering routine 痔序steinberg group 斯坦因伯格群step 步长step function 阶梯函数step length 步长stepping stone method 起脚石法steradian 球面度stereoangle 立体角stereogram 立体频数stereographic projection 球极平面射影stereography 立体平画法stereometry 立体几何stiefel whitney class 斯蒂费尔惠特尼类stieltjes integral 斯蒂尔吉斯积分stirling formula 斯特林公式stochastic 随机的stochastic approximation 随机逼近stochastic automaton 随机自动机stochastic connection 随机联络stochastic control 随机控制stochastic dependence 随机相依stochastic differential equation 随机微分方程stochastic differentiation 随机微分法stochastic dynamic model 随机动态模型stochastic dynamic programming 随机动态规划stochastic filtering 随机滤波stochastic game 随机对策stochastic independence 随机独立stochastic integral 随机积分stochastic integral equation 随机积分方程stochastic integration 随机积分stochastic matrix 随机阵stochastic maximum principle 随机极大原理stochastic model 随机性模型stochastic optimization 随机规划法stochastic process 随机过程stochastic programming 随机规划法stochastic variable 随机变数stochastically dependent event 随机相依事件stochastically independent event 随机独立事件stokes integral theorem 斯托克斯定理stokes theorem 斯托克斯定理stop time 停止时间stopped process 停止过程stopping rule 停止规则storage 存储store 存储器store capacity 存储容量store cell 存储单元straight 直的straight angle 平角straighten 弄平straightline 直线strain 应变strain tensor 应变张量strategic equivalence 策略等价性strategic model 策略模型strategy 策略strategy of a game 对策的策略strategy polygon 策略多角形stratification 层化stratified sample 分层样本stratified sampling 分层抽样stratified selection 分层抽样stratum 层stream function 怜数streamline 吝strength 强度stress 应力stress ellipsoid 应力椭球stress function 应力函数stress of a body 体应力stress tensor 应力张量stretching transformation 伸缩变换strict convexity 严格凸性strict decreasing 严格递减strict epimorphism 严格满射strict extremum 严格极值strict implication 严格蕴涵strict increasing 严格递增strict inductive limit 严格归纳极限strict inequality 严格不等式strict isotonicity 严格保序性strict isotony 严格保序性strict minimum 严格极小strict morphism 严格射strict solution 严格解strict upper bound 严格上界strictly concave function 严格凹函数strictly convex function 严格凸函数strictly convex space 严格凸空间strictly decreasing 严格减少的strictly dominant strategy 严格优策略strictly finer topology 严格较细拓扑strictly increasing 严格递增的strictly increasing mapping 严格递增映射strictly lower triangular matrix 严格下三角矩阵strictly monotone decreasing 严格单递减的strictly monotone increasing 严格单递增的strictly monotonic function 严格单弹数strictly monotonic mapping 严格单党射strictly monotonic sequence 严格单凋列strictly normed linear space 严格赋范线性空间strictly positive measure 严格正测度strictly upper triangular matrix 严格上三角矩阵strip 带strip of conditional convergence 条件收敛带strip region 带形区域strong component 强分支strong convergence 强收敛strong convergent operator 强收敛算子strong deformation retract 强形变收缩核strong discontinuity 强间断strong dual 强对偶strong ellipticity 强椭圆型strong epimorphism 强满射strong extremum 强相对极值strong inequality 严格不等式strong law of large numbers 强大数定律strong markov process 强马尔可夫过程strong monomorphism 强单射strong operator topology 强算子拓扑strong solution 强解strong summability 强可和性strong topology 强拓扑strongly coercive 强强制的strongly connected compactum 强连通紧统strongly connected graph 强连通图strongly continuous map 强连续映射strongly elliptic operator 强椭圆算子strongly inaccessible cardinal 强不可达基数strongly inaccessible ordinal 强不可达序数strongly mixing transformation 强混合变换strongly paracompact space 强仿紧空间strongly plurisubharmonic function 强多重次低函数strongly pseudoconvex domain 强伪凸域strophoid 环诉structural morphism 结构射structural stability 构造稳定性structure 结构structure constant 构造常数structure equations 结构方程structure formula 结构公式structure function 结构函数structure group 结构群structure morphism 结构射structure sheaf 结构层structure theorem 结构定理structured complex 结构复形student t distribution 学生t分布sturm chain 斯图谟链sturm liouville eigenvalue problem 斯图谟刘维尔特盏问题subadditive function 次加性函数subadditive functional 次加性泛函subadditive interval function 次加性区间函数subadditive set function 次加性集函数subadditivity 次可加性subalgebra 子代数subautomaton 子自动机subbase 子基subbundle 子丛subcategory 子范畴subchain 子链subclass 子类subcoalgebra 子上代数subcomplex 子复形subcontinuum 子连续统subcovering 子覆盖subdeterminant 子行列式subdifferential 次微分subdirect sum 次直和subdivide 细分。

vit中可视化方法
在VIT（视觉Transformer）中，常用的可视化方法包括：
1. 特征图的可视化：可将网络中间层产生的特征图进行可视化，以便观察模型对于不同输入的响应情况。

2. 注意力图的可视化：通过可视化注意力机制，可以观察到模型在处理输入时关注的区域或者图像的局部信息。

3. CAM（Class Activation Map）：通过CAM方法，可以将模
型预测的类别与图像特征进行关联，从而可视化出模型对于不同类别的关注区域。

4. Grad-CAM（Gradient-weighted Class Activation Mapping）：
通过Grad-CAM方法，可以将网络最后一层的梯度与卷积层
特征图相结合，从而可视化出模型对于不同类别的关注区域。

5. TSNE（t-Distributed Stochastic Neighbor Embedding）：将高
维特征映射为二维或三维空间，可用于可视化数据集中的样本分布情况。

6. UMAP（Uniform Manifold Approximation and Projection）：
类似于TSNE，UMAP也是一种降维方法，可用于可视化高维
特征在低维空间中的分布。

7. PCA（Principal Component Analysis）：通过主成分分析方法，可将高维特征映射为低维空间，用于可视化数据集中的样本分布情况。

8. 图片重建：通过输入图片与模型的特征逆向传播，可以实现对抗网络中的生成器网络进行图片重建，从而可以观察到模型对于输入的理解情况。

这些可视化方法可以帮助我们更好地理解模型的处理过程和结果，进一步改进模型的性能。

机器学习中的数据预处理：PCA、TSNE和UMAP的区别本文将从机器学习中的数据预处理的视角来介绍PCA、TSNE和UMAP这三个常见的降维方法以及它们之间的区别。

1. PCA（主成分分析）PCA是一种常用的线性降维方法，它通过线性变换将高维数据压缩到低维空间中。

PCA的核心思想是将原始数据映射到一个新的低维空间中，使得新的变量之间的协方差为0，即去除原始数据之间的冗余性。

这个新的低维空间的基向量就被称为主成分。

在PCA中，我们需要解决的是如何选择保留哪些主成分以达到最佳的降维效果。

优点：（1）PCA是一种无监督的方法，不需要指定任何标签信息；（2）PCA能够在降维的同时保留尽可能多的原始信息，需要减少数据集的维度但又不希望失去太多有用的信息时，PCA往往是一个很好的选择。

缺点：（1）PCA只能对线性可分数据进行降维，对于非线性数据，PCA 的效果很差；（2）PCA选择的主成分超出了必要的数量时，有可能会导致过度拟合的问题。

2. TSNE（t-Distributed Stochastic Neighbor Embedding）TSNE是一种基于概率的非线性降维方法，它能够将高维数据映射到二维或三维空间中，以帮助我们直观地观察数据的分布情况。

TSNE 将高维数据的相似性用高斯分布表示，然后在低维空间中，通过学习使得高维数据对应的低维点对应的概率分布尽可能地地接近。

TSNE的核心思想是保留高维数据的局部结构特征。

优点：（1）TSNE能够在低维空间中比较好地保留高维数据的相对距离关系，使数据间的相似性在低维空间中更加明显，进而有助于我们的聚类或分类；（2）TSNE能够对于非线性数据集进行有效降维，具有良好的可视化效果。

缺点：（1）TSNE的计算复杂度比较高，时间、空间成本大，当数据量较为庞大时，效率会降低；（2）TSNE没有捕获全局结构信息，因为它主要是保留了数据点的局部结构和相对距离关系，而没有考虑全局结构之间的关系，可能导致在处理全局关系较为复杂或加噪数据下的不准确性。