05_Trajectory Pattern Mining

Chapter5

Trajectory Pattern Mining

Hoyoung Jeung,Man Lung Yiu,and Christian S.Jensen

Abstract:

In step with the rapidly growing volumes of available moving-object trajectory data,there is also an increasing need for techniques that enable the analysis of tra-jectories.Such functionality may bene?t a range of application area and services, including transportation,the sciences,sports,and prediction-based and social ser-vices,to name but a few.The chapter?rst provides an overview trajectory patterns and a categorization of trajectory patterns from the literature.Next,it examines relative motion patterns,which serve as fundamental background for the chapter’s subsequent discussions.Relative patterns enable the speci?cation of patterns to be identi?ed in the data that refer to the relationships of motion attributes among mov-ing objects.The chapter then studies disc-based and density-based patterns,which address some of the limitations of relative motion patterns.The chapter also reviews indexing structures and algorithms for trajectory pattern mining.

5.1Introduction

We are witnessing a rapid and continued diffusion of mobile devices such as smart-phones,personal navigation devices and tablet computers.Further,these devices are increasingly being geo-positioned using satellite navigation systems,e.g.,GPS,sys-tems that exploit the wireless communication infrastructure,and proximity-based systems,e.g.,RFID-based systems.

Hoyoung Jeung

′Ecole Polytechnique F′e d′e rale de Lausanne,Switzerland,e-mail:hoyoung.jeung@epfl.ch Man Lung Yiu

Department of Computing,Hong Kong Polytechnic University,Hong Kong,e-mail:csmlyiu@ https://www.wendangku.net/doc/2010106986.html,.hk

Christian S.Jensen

Department of Computer Science,Aarhus University,Denmark,e-mail:csj@cs.au.dk

143

144Hoyoung Jeung,Man Lung Yiu,and Christian S.Jensen The resulting location-aware devices?nd widespread use in various business and personal settings in society.As a consequence of this development,increasing vol-umes of position data are being accumulated,and the capability of analyzing large volumes of trajectory data is in increasing demand in a wide spectrum of applica-tions.

Signi?cant applications in various domains need to identify and utilize groups of trajectories that exhibit similar patterns from a collection of trajectories.Example applications include transportation optimization,prediction-enabled services,scien-ti?c and social analysis applications,sports analyses,as well as crowd and outlier analyses.

?Transportation optimization applications[25]need to?nd groups of similar tra-jectories that indicates that the corresponding objects traveled together.For in-stance,a car pooling application may connect drivers in the same trajectory group in order to reduce their travel expenses.A logistics application may ex-amine the delivery trucks in the same group in order to achieve better planning.?Prediction methods[50]may exploit knowledge of trajectory groups for the understanding of object behavior.Such knowledge can be used for offering ef-fective noti?cations,for the delivery of advertisements to targeted audience,and for providing customized location-based services.

?Scienti?c studies may call for the identi?cation of groups of animals that moved together.They are useful in discovering animal movement patterns(e.g.,bumble bees,a variety of birds,sea turtles,whales,and?sh)[2],in?nding herds of animals,and in studying animal behavior patterns in habitats.

Similarly,social analysis studies may aim to identify socio-economic pattern-s[23]from typical movement patterns of individuals.

?Team sports events[30,1](soccer,baseball,hockey,rugby,digital battle?elds) also provide valuable trajectory data that capture the players’movements.By studying a game as groups of trajectories,it may be possible to better understand the game[30],to analyze the tactics used in the game,and to even extract the location and time of using a certain strategy.

?Traf?c analysis applications may utilize trajectory groups for the study of crowds and outliers.In this scenario,a moving object can be either a vehicles on the road or a pedestrian on the street.A large trajectory group is likely to indicate a crowd behavior.By identifying crowds from the trajectories,a better understanding of crowds is possible,e.g.,the times and places when and where crowds form and dissolve.Such information may be exploited for managing transportation infrastructures effectively.

It is also of interest to mine outliers,which do not belong to any trajectory group.This may be used for detecting and removing errors in the trajectory data(e.g.,?nding a device with a malfunctioning GPS receiver).It may also be applied for identifying dangerous driving behaviors.

Trajectory pattern mining is an emerging and rapidly developing topic in the areas of data mining and query processing that aims at discovering groups of tra-

5Trajectory Pattern Mining145 jectories based on their proximity in either a spatial or a spatiotemporal sense.The literature contains a variety of recent proposals in this area.

Existing proposals represent different trade-offs in the expressiveness of the tra-jectory patterns studied.Considering only restricted patterns may result in not being able to identify interesting phenomena from the data,whereas considering quite re-laxed patterns may lead to the reporting of insigni?cant patterns.

Existing proposals also come with their own index structures and mining algo-rithms that aim to enable ef?cient and scalable discovery of patterns in large trajec-tory datasets.This chapter presents an overview of the key concepts and discovery techniques in state-of-the-art studies in the mining of trajectory patterns.

The rest of the chapter is organized as follows.Section5.2introduces the concept of trajectory pattern and provides a categorization of patterns.We then study rela-tive motion patterns in Section5.3,presenting disc-based patterns in Section5.4, and examining density-based patterns in Section5.5.Section5.6covers distance measures and methods for mining trajectory patterns.We summarize the chapter in Section5.7.

5.2Overview of Trajectory Patterns

5.2.1Pattern Discovery Process

A trajectory of a moving object is a continuous function from the time domain to the domain in which the movement occurs.In animal tracking,the objects typically move freely in the water,on the surface of the Earth,or in the air.In such cases,the movement domain is often modeled as two-or three-dimensional Euclidean space. In settings where the objects are vehicle,the movement domain is often modeled as a spatial graph that models a road network.Further,the spatial extent of a moving object is typically ignored so that the position of an object at a given time is modeled as a point.

Figure5.1illustrates the architectural context for managing trajectories.Multiple objects,each with a mobile device with a unique identi?er id,contribute data.Each device samples the location of its object according to some policy.For example, devices may take a position sample at a?xed frequency such as every second.A location record has the format(t,x t,y t,id),where the sampled location at time t is (x t,y t).

The true trajectory of an object is unknown.The location records available for an object are used to create an approximation of the object’s true trajectory.Specif-ically,existing techniques for trajectory pattern mining typically assume that the trajectory of an object is given by a polyline,i.e.,a sequence of connected line seg-ments.

A server is employed for storing and managing the collected trajectories from all devices.Important steps in the handling of trajectories are described next.

146Hoyoung Jeung,Man Lung Yiu,and Christian S.Jensen Device 1

Device 2

online

Collected trajectories Refined trajectories 1. Data collection

Server

2. Pre-processing Device n

……Device 3offline 3. Trajectory mining

Patterns Fig.5.1Architecture.

1.Data collection.

In this step,devices submit their location records to the server.Online devices may submit their data in real time or delayed,in batches.For example,a device may submit records as soon they become available on the device,or it may collect a small batch of records before submitting in order to save bandwidth.Of?ine devices accumulate their trajectory data,which is transferred to the serv-er in batches via other means.For example,an of?ine navigation device in a vehicle may occasionally be connected to an online personal computer in order to transfer location records to the server.

2.Pre-processing.

Due in part to imperfections in the positioning technologies employed and the use of different sampling policies,collected trajectories may be inhomogeneous and may not be of the desired quality.For instance,some of trajectories may lack location records for times where other trajectories posses such records.In pre-processing,the server “cleans”the data and converts it into a standard format in preparation for the data mining.As part of the pre-processing,the granularity and the representation of the re?ned trajectories need to be decided by the server.For example,a ?nest sampling rate is often chosen,and it is de-cided whether a trajectory be stored as a sequence of location records or should be approximated by a piecewise linear function from time to space.

3.Mining trajectories.

The re?ned trajectories are made available for the mining algorithms to discover patterns (e.g.,clusters of trajectories).

4.Post-processing.

This step enables applications to analyze or tune the discovery process.For ex-ample,mined patterns can be displayed interactively using visualization tools,which can suggest more appropriate parameter values for the mining algorithms or ?nding further research issues.

5Trajectory Pattern Mining147 5.2.2Classi?cation of Trajectory Pattern Concepts and Techniques The term trajectory pattern covers many different types of patterns that can be mined from trajectory data[29].As a result,the concepts and techniques underlying trajectory pattern discovery are classi?ed by according to a variety of aspects[19]. In the following,we describe key classi?cations of trajectory pattern concepts and techniques while emphasizing the intuitions behind the classi?cations.

5.2.2.1Mining Tasks on Trajectories

Two typical mining tasks on trajectories are clustering and join.Clustering aims at discovering groups of similar objects from a single trajectory collection.On the other hand,a join is a speci?c operation that computes pairs of similar objects from two trajectory collections.

Clustering of Trajectories

Clustering is the process of organizing objects into groups so that the members of a group are similar and so that members of distinct groups are dissimilar,according to some speci?c de?nition of similarity.Clustering is a fundamental concept in data mining,due to its wide spectrum of applications.

A clustering technique is classi?ed according to the type of data that it is applica-ble to.Clustering techniques for different types of data are likely to vary signi?cant-ly with respect to the underlying concepts and the speci?c techniques employed.

In the context of trajectory data,clustering attempts to group trajectories based on their geometric proximity in either spatial or spatiotemporal space.Clustering of trajectories is perhaps one of the most fundamental operations used in various types of trajectory pattern mining,since the discovery of trajectory patterns typically involves the process of grouping similar positions,trajectories,and objects.

Assuming the polyline representation of a trajectory,one fundamental approach to trajectory clustering treats a segment of a trajectory as a minimum unit when computing the distance between two trajectories.To this end,it is essential to design an effective distance function for measuring the distance between two trajectories (segment).We offer details on trajectory distance measures in Section5.6.1.An-other fundamental approach views a cluster of trajectories as a sequence of spatial clusters.Assuming that all objects have position samples at the same points in time, such spatial clusters can be obtained by applying some spatial clustering technique to the positions for each point in time.

This chapter describes in detail the core ideas and concepts of state-of-the-art algorithms for clustering of trajectories:relative motion patterns,?ock,TRACLAUS, moving cluster,convoy,and swarm.

Trajectory Join

Some trajectory patterns are de?ned and computed by means of join queries.Given two data sets P1and P2,spatio-temporal joins?nd pairs of elements from the two sets that satisfy a given predicate with both spatial and temporal attributes[31,17].The

148Hoyoung Jeung,Man Lung Yiu,and Christian S.Jensen study of airplane or vessel trajectories with the objective of?nding incidents may be accomplished using joins.Since joins may involve the comparison of all trajectories in data set P1with all trajectories in data set P2,which is computationally expensive, a common approach for join processing involves the use if indexing techniques to avoid unnecessary distance computations.For example,Tao et al.[53]show how join queries are processed by using the time-parameterized methods[56,52,51]. The close-pair join[13]reports all object pairs(o1,o2)from P1×P2with distance Dτ(o1,o2)≤e within a time intervalτ,where e is a user-speci?ed distance.Plane-

sweep techniques[6,61]have been proposed for evaluating spatio-temporal joins. Zhou et al.[61]use join predicates that de?ne a rectangular region in time and space.An index structure(MTSB-tree)is introduced to enable ef?cient retrieval of the pairs of trajectories that satisfy the join predicates.Instead of using an index, Arumugam and Jermaine[6]utilize MBR approximations of trajectory segments to reduce the computation of query processing.

As does the close-pair join,the trajectory join[10]aims at retrieving all pairs of similar trajectories in two data sets.Bakalov et al.[9,10]represent trajectories as se-quences of symbols,and apply sliding window techniques to measure the symbolic distance between possible pairs.

5.2.2.2Spatial and Spatiotemporal Patterns

Groups of similar trajectories carry different semantics depending on whether they are grouped according to spatial or spatiotemporal geometric proximity.Figure5.2 demonstrates the difference between spatial and spatiotemporal trajectory patterns. At time t=1,two objects o1and o2start moving closely together in the direction of the upper-right corner.At time t=2,a new object o3appears and starts to move from a close location to where o1and o2were at time t=1.Meanwhile,o1and o2 have reached the center of the movement space and continue to move.At time t=3, o3keeps following a path similar to those that o1and o2have been following,while o1and o2complete their journeys.Finally,o3?nishes its journey at time t=4close to where o1and o2stopped.

W

Fig.5.2Construction of three objects’trajectories during t=[1,4].

Given the trajectories of o1,o2,and o3collected until t=4,the embeddings of the trajectories into the movement space look similar.When we cluster the trajecto-

5Trajectory Pattern Mining149 ries according to the similarity of the embeddings,the three trajectories may form one cluster,although o3has never traveled together with o1and o2,but has merely followed the same route as these.

The above is an example of a spatial trajectory pattern(Figure5.3(a)).Despite the fact that the concept of trajectory encompasses the time domain,a variety of applications do not need to consider the temporal aspects of trajectories.

For example,analyzing a set of hurricane trajectories collected over several years for forecasting the trajectories of future hurricanes may be done best by considering simply the embeddings into the movement space of the past hurricanes so that is becomes the similarity of the routes of the hurricanes that is studied.Put differently, the routes of the past hurricanes may be the most important aspect when aiming to offer pre-warnings to regions of potential damage.

In contrast to spatial trajectory patterns,spatiotemporal trajectory patterns take the time information in a trajectory into account.For example,o3in Figure5.2may not belong to the same cluster as the one o1and o2belong to(see Figure5.3(b)), since o3always trailed behind o1and o2.Having a time constraint,a spatiotemporal

D VSDWLDO SDWWHUQ

E VSDWLRWHPSRUDO SDWWHUQ

Fig.5.3Spatial versus spatiotemporal trajectory patterns.

trajectory pattern is therefore stricter than a spatial trajectory pattern.This time con-straint can help discover more speci?c patterns in many cases of mining trajectory data.For instance,identifying objects that have traveled together for the purpose of given car pooling recommendations should consider the temporal information of trajectories.Discovering places in a road network where congestion may occur based on vehicle trajectories should take into account the time information of the trajectories.

5.2.2.3Granularity of Trajectory Patterns

The granularity of a trajectory pattern can be characterized by the time interval during which the pattern holds and the number of objects that are involved in the pattern.

Global Vs.Partial Patterns

Trajectory patterns can be classi?ed based on the temporal extent of the trajectories

150Hoyoung Jeung,Man Lung Yiu,and Christian S.Jensen that participate in the patterns.In global trajectory patterns,trajectories are viewed as non-decomposable,i.e.,the basic unit of pattern discovery is a whole trajectory. For example,Gaffney et al.[24]propose a model-based clustering algorithm that groups trajectories with overall similar routes using a regression mixture model and the EM algorithm.This algorithm generate clusters of trajectories with respect to the overall distances among whole trajectories.

In contrast,partial trajectory patterns consider partial trajectories in the pattern discovery process.The idea behind this approach is that the clustering of whole trajectories may not detect trajectories with similar sub-trajectories.In general,a trajectory may be long and complex.Hence,even though some parts of trajectories are similar,the full trajectories might not be similar.Based in this view,Lee et al.[42]propose to partition trajectories into line segments and to build groups of close trajectory segments.Recent studies on trajectory clustering algorithms,such as moving clusters[36],?ocks[11],convoys[35],and swarms[44],also consider patterns that appear in sub-trajectories.

Individual Vs.Group Patterns

Another distinction relates to whether a set of individual trajectories are retrieved that satisfy a pattern speci?ed in query,called individual trajectory pattern re-trieval[28,32,48],or whether sets of trajectories are retrieved so that the trajec-tories in a set exhibit a similar pattern according to some speci?c notion of pattern, called group trajectory pattern discovery[27,33]

So-called spatiotemporal pattern queries[28,48]illustrate well the retrieval of trajectories that satisfy a particular pattern as speci?ed in a query.Examples of such queries include the?nding of?ights that descended into an airport,but did not land; identifying?ights that had to make several approaches before entering the airport; and discovering vessels that changed course to avoid a collision.

The discovery of groups of trajectories that share patterns[57]is fundamental-ly different in that the objects returned are to be similar to each other according to a given notion of similarity rather than similar to a query pattern.The discov-ery of group patterns may enable different forms of sharing among the objects who have similar trajectories,e.g.,in car pooling or delivery truck logistics.This class of patterns include concurrence[39],trend-setter[39],?ock[11,27,26],lead-ership[5,4],convergence[11],encounter[11],convoy[33],and swarm[44]. We describe each of these pattern concepts in detail in Sections5.4and5.5.

5.2.2.4Constrained Trajectory Patterns

Some trajectory patterns can be associated with constraints along the spatial and temporal dimensions.

Spatial Constraints:Movement on Spatial Networks

Many types of objects move in a spatial network,such as a road network,a rail net-work,or the kind of network made up by the corridors used by commercial aircraft. Since those objects are always located somewhere in the networks,raw trajectory

5Trajectory Pattern Mining151 data is typically modeled as or transformed to network-based trajectories,e.g.,edge sequences in a road network graph[34,60,18].As a result,trajectory patterns of network-constrained objects also have different forms and sometimes carry differ-ent semantics from the pattern types generally considered in unconstrained object trajectories.

As an example,Figure5.4(a)illustrates a road network,with its graph model shown in Figure5.4(b).All junctions form vertices,and each edge contains the corresponding road segment’s information such as a weight w(distance).In this

Fig.5.4An example of road network-constrained trajectory model.

network model,an object’s location is represented as a tuple l=(e,d,t),meaning that the object is located on the edge e=(v i,v j),at distance d from v i at time t.Table5.1shows an example of network-constrained objects’trajectories over2 days.

object trajectory ID sequence of network positions

o11(e1,1.2,t1),(e1,2.9,t2),(e2,0.2,t3)

o12(e1,1.3,t2),(e1,3.4,t3)

o23(e3,7.6,t1),(e5,1.5,t2),(e5,5.3,t3)

o24(e1,1.2,t1),(e1,2.9,t2),(e2,0.2,t3)

Table5.1An example of network-constrained trajectories.

Given such network constrained trajectories,trajectory patterns reveal hidden, but useful,information.Finding popular routes,for example,can indicate reliable paths when drivers travel to unfamiliar destinations,as well as suggest higher-quality roads for truck delivery services[18].They also enable location prediction that in turn can enable services that report relevant traf?c conditions and upcoming points of interest(POIs)such as gas stations to users[12,34].In addition,Internet map services(e.g.,Google Maps,Yahoo Maps)can be enriched based on trajectory patterns.

Temporal Constraints:Periodicity

In the real world,various types of objects exhibit periodic movement patterns.For example,many individuals go to work every weekday following the same or similar routes each day,public transportation is governed by time schedules,and animals

152Hoyoung Jeung,Man Lung Yiu,and Christian S.Jensen annually migrate to reproduce or seek warmer climates.Findings in the literature suggest that car drivers tend to follow regular trajectories more than70%of the time[37].

Periodic pattern mining of trajectory data concerns the discovery of periodic ob-ject behavior[47,45],i.e.,objects that follow the same routes(approximately)over regular time intervals.These periodic patterns provide an insight into,and concise explanation of,periodic behaviors(e.g.,daily,weekly,monthly,and yearly)across long movement histories.

Periodic patterns are also useful for compressing movement data[14],since they summarize movement trajectories into a compact format.In addition,periodic pat-terns can serve as a basis for future-movement prediction[32].Moreover,if an ob-ject fails to follow an established,regular periodic behavior,this could be a signal of an abnormal environmental change or an accident.

When considering object movement,it is typically unreasonable to expect an ob-ject to repeat its behavior exactly during each time period considered.This implies that patterns to identify should not be rigid,but that object behavior should be al-lowed to differ slightly from one period to the next while still resulting in a pattern. Next,behaviors that make up patterns may also be shifted in time(e.g.,due to traf?c delays).

The approximate nature of patterns in the spatiotemporal domain increases the complexity of mining tasks.In addition,the periods that yield patterns may be un-known.Further,multiple periods(e.g.,day and week)may exist that yield different patterns in the same data.As a result,periodic pattern mining of trajectory data takes into account a wide variety of modeling approaches as well as ef?cient discovery algorithms.

5.3Relative Motion Patterns

Given a collection of trajectories,it is challenging to capture and compare motion events of individuals and groups of individuals.To facilitate trajectory data analysis, the analysis concept RElative MOtion(REMO)by Laube et al.[39,40,38,41] enables the identi?cation of similar movements in a collection of moving object trajectories.The phrase“relative motion”refers to the relationships among motion attributes of different moving objects over space and over time.

REMO aims to enable the formulation and discovery of meaningful trajecto-ry patterns based on motion attributes(i.e.,speed,change of speed or motion az-imuth),as extracted from raw trajectory data.Figure5.5demonstrates the pro-cess of building motion azimuth-based trajectories from raw trajectories.The left part of the?gure shows trajectories of three objects moving in two-dimensional space.We then consider eight possible movement directions(motion azimuth):↑, ,→, ,↓, ,←, .The right part shows the motion azimuth-based represen-tation of the trajectories,where a trajectory is represented by a direction for each

5Trajectory Pattern Mining153 of four time points.This latter space of trajectories is called the analysis space in REMO.

t1t2t3t4

o1 →

o2 →→→

o3 →

(a)raw trajectories(b)motion-azimuth trajectories

Fig.5.5Motion azimuth-based trajectory representation.

REMO de?nes three sets of relative motion patterns:basic motion patterns,spa-tial motion patterns,and aggregate/segregate motion patterns.These pattern con-cepts serve as fundamental background for subsequent studies of moving objects patterns.This section summarizes the key ideas in REMO.

5.3.1Basic Motion Patterns

The?rst set of REMO patterns describe motion events and patterns that explicitly disregard the absolute position of the moving objects.This means that a set of geo-graphically distant objects can be identi?ed as belonging to the same group,as long as they exhibit the same prede?ned motion properties.The basic motion patterns in-clude(i)patterns over time,(ii)patterns across objects,and(iii)patterns over time and across objects.Such patters are exempli?ed next.

Constance

This pattern is de?ned as a sequence of equal motion attribute values for some con-secutive time points.As an example in Figure5.5,object o2moves with a constant motion azimuth→during the interval of[t2,t4].

Concurrence

A concurrence motion pattern captures the incidence of having multiple objects with similar motion attributes.For example,in Figure5.5,objects o1,o2and o3 move with the same motion azimuth at time t1.

Trendsetter

This motion pattern attempts to?nd objects that anticipate a certain motion pattern that is shared by a set of other objects in future.Thus,this complex pattern combines

154Hoyoung Jeung,Man Lung Yiu,and Christian S.Jensen the simple patterns concurrence and constance.For example,o2anticipates at t2the motion azimuth→that is shared by all other objects at time t4.

5.3.2Spatial Motion Patterns

Based on the motion patterns in the set of patterns just covered,REMO identi?es three important trajectory patterns that are all classi?ed as spatial motion patterns. This second set of motion patterns includes spatial constraints regarding the abso-lute positions of the moving objects[40,5],meaning that certain objects’motions appear within a spatial range(e.g.,rectangle,circle,or ellipse)for some duration of time.Therefore,these patterns capture objects that are geographically close to one another,which is different from the basic motion patterns.

Figure5.6illustrates examples of the following three spatial motion patterns. Each spatial constraint is a circle.

(a)track(b)?ock(c)leadership

Fig.5.6Examples of REMO spatial motion patterns.

Track

Given a spatial range,e.g.,a disc,a track motion pattern?nds objects who trav-el within the range while keeping the same motion.For instance,an airplane that ?ies at constant speed and direction can be captured by the track motion pattern. Essentially,track combines constance pattern and a spatial constraint.

Flock

The concept of?ock pattern in REMO covers not only a group of animals that live, travel,or feed together(e.g.,a pride of lions,a school of?sh,a gam of whales,a gaggle of geese,a murder of crows,or a swarm of insects),but also a group or crowd of people that move together.

In order to capture this concept using the basic motion patterns,the de?nition of?ock consists of the concurrence pattern combined with a spatial constraint.

5Trajectory Pattern Mining155 Intuitively,this patter may help identify groups of objects that travel together,i.e., within spatial proximity,for some duration of time.

Leadership

Leadership is a widespread phenomenon in social settings.In particular,the animal behavior research community studies the general topic of group decision-making in animals,searching for evidence for groups led in their activities by some domi-nant individuals,e.g.,dominant breeding wolves frequently show signi?cant frontal leadership,leading the pack during travel.

The leadership pattern corresponds to the trendsetter pattern,but includes a spatial constraint.For example,followers must lie within a given geographical re-gion when they join the motion of the trendsetter.As a result,this pattern captures well the case where a group of people are under the leadership of one person.

5.3.3Aggregate/Segregate Motion Patterns

The third set of relative motion patterns describe aggregation and segregation of objects’movements.

Convergence

This pattern describes a set of m objects during a time interval k that share motion azimuth vectors intersecting within a given spatial range,e.g.,a disc with radius r. This pattern captures the behavior of a group of objects that converge in a certain region.An example of this pattern is wild animals that are heading in a synchronized fashion for a mating place.

Encounter

Suppose that some antelopes distributed over a?eld are heading for a location.At some later time,they will thus meet at the location,according to their current mo-tions.To capture such an extrapolated(future)meeting within a spatial range,the encounter pattern is de?ned as a set of m objects that will arrive in a given spa-tial range r concurrently k time points later(i.e.,the extrapolations of the objects’current motions intersect with r).

Divergence

This pattern is an opposite concept of convergence,integrating a spatial diver-gence pattern with the temporal constraint of a preceding meeting in a region.The graphical representation of the divergence pattern is that of“heading backwards”instead of forwards,relative to the direction of motion.

Breakup

Like the convergence pattern is an opposite concept of convergence,the con-cept of breakup is an opposite of the concept of encounter.This pattern captures objects’behaviors,such as departing from a meeting point.

156Hoyoung Jeung,Man Lung Yiu,and Christian S.Jensen 5.3.4Discussion

Although the REMO patterns constitute important mining concepts for trajectory data,they pose several open problems that should be addressed.We cover such open problems brie?y.

?In the REMO framework,it is dif?cult to determine an absolute distance be-tween two objects because the pattern discovery process is performed over the motion attributes(i.e.speed,change of speed,or motion azimuth)that are de-rived from raw trajectories.As a result,some pattern analysis tasks,such as ?nding the k nearest neighbor objects of a given object(trajectory),are dif?cult to support in REMO.

?Although the motion attributes in REMO encompass the objects’speeds and changes of speeds,the objects’motions are mainly analyzed considering the motion azimuths that capture the direction of a trajectory at a point in time.Full motion analysis considering all the motion attributes simultaneously is indeed not a simple task,but requires complex mechanisms and heavy computation.?The default motion azimuths in REMO consists of eight different https://www.wendangku.net/doc/2010106986.html,ing a?ner or coarser angle granularity would substantially impact the effectiveness and ef?ciency of trajectory pattern discovery.For example,the classi?cation of motion azimuths into only the two classes East and West would reveal a lot of presumably meaningless constancy patterns.In contrast,every constancy pattern found with360azimuth classes may be unnecessary for typical applica-tions.It is non-trivial to determine an appropriate angle granularity for a given trajectory data set.

?During the detection of relative motion patterns,time intervals with uncertain or missing data points may reduce the accuracy and effectiveness of pattern dis-covery signi?cantly.In addition,the REMO framework assumes all trajectories have the same sampling rate as the granularity used in the analysis task,which may not hold for real-world trajectory data.Before employing the concept of relative motion patterns,pre-processing steps are required(e.g.,interpolating missing samples and smoothing trajectories)in order to ensure a uniform time granularity.

5.4Disc-Based Trajectory Patterns

Subsequent to the introduction of relative motion patterns by Laube et al.,a sub-stantial body of studies have continued the study of trajectory data pattern min-ing[11,35,27,54,44,8,5,4].These subsequent studies have rede?ned,extended, and further developed the concepts of trajectory patterns substantially.The advances over the original relative motion pattern concepts relate mainly to three aspects. 1.Distance-based trajectory analysis:the core idea of REMO—i.e.,extracting

and utilizing motion features of trajectories for pattern analysis—is no longer

5Trajectory Pattern Mining157 considered in further studies.Instead of using motion attributes,the subsequent studies consider Euclidean distance for measuring the proximity of positions of trajectories at time points.As a result,the importance of the basic relative motion patterns constance,concurrence,trend-setter decreases in trajectory analysis.

2.Disc-based spatial range:while REMO allows regions with arbitrary geomet-

ric shapes(e.g.,ellipse,rectangle,disc)to be used as spatial ranges in motion patterns(Section5.3.2),the subsequent studies of REMO take into account on-ly circular ranges,i.e.,discs.This approach is easy to apply,and discs capture intuitively the ranges of moving-object groups.

3.The REMO framework lacks time constraints,meaning that the relative motion

patterns do not include an explicit parameter for specifying the time lengths of motion patterns.A relative motion pattern that,e.g.,lasts for a few seconds may be too short to identify a common behavior among objects.Yet such a pattern can be identi?ed with REMO.Subsequent studies include an explicit time duration constraint in order to alleviate this problem.

Built on the above principles,a rich body of studies have re?ned or rede?ned the original concepts and de?nitions of relative motion patterns.They have also introduced a set of new pattern types,mainly based on the REMO concept.

We categorize these newly introduced patterns into two classes:?Prospective patterns that are likely to occur some time in the near future.This

class covers encounter and convergence.

?Flock-based patterns that correspond to the original?ock pattern of REMO.

This class includes rede?nitions of?ock,meet,and leadership.

This section discusses in detail concepts that relate to these patterns.

5.4.1Prospective Patterns

Gudmundsson et al.[27]revisit the encounter and convergence patterns in RE-MO,providing generic de?nitions based on the geometric arrangements of the mov-ing objects.The resulting new patterns thus do not require the motion attributes in the REMO framework.The new pattern de?nitions consider the future trajectories of moving objects,meaning that the new encounter and convergence patterns capture future events that are likely to occur,based on the current motions of mov-ing objects.Speci?cally,they are de?ned as follows.

Encounter(m,r)

Satis?ed by a group of at least m objects that will arrive simultaneously in a disc with radius r,assuming that they keep their current speeds and directions. Convergence(m,r)

Satis?ed by a group of at least m objects that will pass through a disc with radius r

158Hoyoung Jeung,Man Lung Yiu,and Christian S.Jensen (not necessarily at the same time),assuming that they keep their current movement directions.

Figure 5.7demonstrates the current and future trajectories obtained from ?ve ob-jects o 1,o 2,o 3,o 4,and o 5.The arrows show the current positions and their current motions,i.e.,speed and direction;the dotted lines indicate the objects’future trajec-tories based on the current motions.According to the future trajectories,o 2,o 3,o 4,and o 5will pass the disc r ,forming a convergence pattern if given an m that does not exceed 45.If these objects arrive at r at the same time,they also satisfy the encounter pattern with the same parameters.

R

R U R

R

R

Fig.5.7Examples of the encounter and convergence patterns.5.4.2Flock-Driven Patterns

Gudmundsson et al.[11,27,26,3,54]have studied extensions of the ?ock pat-tern introduced by Laube et al.[39,40]intensively.The studies have resulted in a wide range of interesting trajectory pattern concepts and discovery techniques.This section summarizes key ideas in relation to the extensions and variants of ?ock .Flock (m ,r ,k )

In one study [11],two types of ?ock patterns are de?ned:one concerns continuous object movement,while the other concerns discrete object movement.The latter is widely used in database and data mining research.A discrete ?ock occurs if at least m objects move together for at least k consecutive time points while staying within a disc with radius r .

Based on this ?ock concept,two interesting variants,meet [11]and leader-ship [5],are de?ned:

Meet (m ,r ,k )

A meet pattern occurs if at least m objects stay together in a stationary disc with

5Trajectory Pattern Mining 159radius r for at least k consecutive time points.Unlike the de?nition of ?ock ,the disc speci?ed in meet has a ?xed location.Thus,the concept of meet resembles a past variant of encounter .

Figure 5.8illustrates the concepts of ?ock and meet .In Figure 5.8(a ),a ?ock of objects o 1,o 2,and o 3is found during the time points [t 7,t 8,and t 9,while the meet pattern of objects o 1and o 2is identi?ed during the time points t 3,...,t 8(Fig -ure 5.8(b )).

D IOR ck

E PHHW

R R R

R R

R

Fig.5.8Examples of the ?ock and meet patterns.Leadership (m ,r ,k )

Andersson et al.[4]present a clear concept and de?nition of a leadership pattern.Their leadership pattern is essentially an extension of ?ock .Yet it captures the spatial constraint of the leader being ahead of the followers.Formally,the pattern occurs when a set of at least m objects move together for at least k consecutive time points while staying within a disc with radius r ,and when at least one of the objects is/was heading in the leader’s direction.

Andersson et al.[5]also introduce a variant of leadership where two new pa-rameters are added to the de?nition of leadership .In Leadership (m ,r ,k ,α,β),parameter αin?uences the spatial extent of a pattern,and parameter βdetermines the spatial characteristics of a pattern.

5.4.3Discussion

The disc-based trajectory patterns are satis?ed by groups of objects that move to-gether within a disc with some user-speci?ed extent.As a result,the chosen disc extent has a substantial effect on the result of the discovery process.Although the disc concept is intuitive,it raises potential problems when specifying trajectory pat-terns,as discussed next.

160Hoyoung Jeung,Man Lung Yiu,and Christian S.Jensen ?The selection of a proper disc size turns out to be dif?cult,as situations can occur where objects that intuitively belong together or do not belong together are not quite within any disk of the given size or are within such a disk.In Figure5.9,for example,all objects travel together in a natural group.However, object o4does not enter the disc and is not discovered as a member of the?ock.

This problem occurs because what constitutes a?ock is very sensitive to the user-speci?ed disc size,which does not take the data distribution into account.

Indeed,if a?ock must contain at least4objects,no?ock would be found in this example.

?For some data sets,no single appropriate disc size may exist that works well for all parts of the(space,time)domain.A herd of animals,for instance,consists of individual that move together,but the herd may expand or contract over time.

This can not be captured by a?xed-size disc.

?The use of a circular shape may not always be appropriate.For example,sup-pose that two different groups of cars move across a river and each group has a long linear form along roads.A suf?cient disc size for capturing one group may also capture the other group as one?ock.Ideally,no particular shape should be ?xed apriori.

R

R

R

Fig.5.9Lossy-?ock problem.

5.5Density-Based Trajectory Patterns

In order to avoid the size and shape restrictions inherent to disc-based trajectory pat-terns,density-based trajectory patterns have been introduced.These employ density concepts[22]that allow the capture of generic trajectory pattern of any shape and any extent.In this section,we?rst review the concept of density-based clustering and then describe in detail the key concepts and de?nitions relating to density-based trajectory patterns.

5Trajectory Pattern Mining 161

5.5.1Density Notions

As a precursor to de?ning density-based trajectory patterns,we need to understand the notion of density connection [22].The main advantage of this concept is that it is able to capture clusters of any size and shape,as long as the cluster members meet certain distance-related conditions.

?Given a distance threshold e and a set of points S ,the e -neighborhood of a point p is given as NH e (p )={q ∈S |D (p ,q )≤e }.?Given a distance threshold e and an integer m ,a point p is directly density-reachable from a point q with respect to e and m if p ∈NH e (q )and |NH e (q )|≥m .

?A point p is said to be density-reachable from a point q with respect to e and m if there exists a chain of points p 1,p 2,...,p n in set S such that p 1=q ,p n =p ,and p i +1is directly density-reachable from p i with respect to e and m ,i ∈{1,...,n ?1}.

?Given a set of points S ,a point p ∈S is density-connected to a point q ∈S with respect to e and m if there exists a point x ∈S such that both p and q are density-reachable from x with respect to e and m .

Figure 5.10(a )exempli?es a density-connected cluster when assuming that m =

4.Each dashed circle indicates an e -neighborhood of some object.For instance,o 1is directly density reachable from o 3with respect to threshold e and m =4because it belongs to the e -neighborhood of o 3that contains a total of 4objects.Also,o 1and o 9are density-connected through the following chain of points: o 3,o 4,o 6,o 7 .The collective extent of the cluster can exceed e ,and the cluster can have arbitrary shape.Since the parameter m is used to determine whether a point is directly den-sity reachable from another,the size of a density-connected cluster is at least m .In addition,the cluster is preserved at time t =2in Figure

5.10(b ),although the size of the cluster and the topology of the cluster members are changed.

(a ) ti me t 1(b ) ti me

t 2o o 9 D WLPH W

E WLPH W Fig.5.10An example of a density-connected cluster.

162Hoyoung Jeung,Man Lung Yiu,and Christian S.Jensen As shown in the above example,the de?nition of density-connection permits us to capture a group of“connected”points with arbitrary shape and extent.It thus allows us to overcome the main drawbacks of the concept of disc-based trajecto-ry patterns.We proceed to introduce trajectory patterns built upon the concepts of density-connected objects.

5.5.2Moving Objects Clustering

Recently,a wide variety of clustering concepts and algorithms for trajectory data have appeared in the literature.The different proposals generally assume different data and application requirements,and they have different advantages and disadvan-tages.In this section,we present a clear,systematic,and comprehensive overview of recent works on the clustering of trajectories in order to offer a good foundation for subsequent studies of trajectory pattern mining.

5.5.2.1TRACLUS

Lee et al.introduce trajectory clustering[42]for the purpose of grouping trajectory segments.Speci?cally,they proposed a partition-and-group framework TRACLUS that clusters trajectories in the following three steps:

?Partitioning:each trajectory is partitioned into a set of line segments.This pro-cess?nds the points where the behavior of a trajectory changes,called its char-acteristic points.The process adopts the minimum description length(MDL) principle,which is used widely in information theory.

?Grouping:trajectory segments that are close to each other according to a cer-tain distance measure are grouped into a cluster.For this process,Lee et al.

de?ne a set of distance measures for trajectory segments(details are presented in Section5.6.1)that are used for density-based clustering using the DBSCAN algorithm[22].This allows the clusters of trajectories obtained by TRACLUS to form any shape and size.

?Representing:given a cluster,this process derives a representative trajectory for the cluster that describes the overall movement of the trajectory partitions that belong to the cluster.Basically,this process averages the lengths and angles of the entire trajectory segments belonging to the cluster,thus constructing a new trajectory segment,i.e.,a representative trajectory.

Figure5.11illustrates the result of the above three processing steps of the partition-and-group framework for trajectory clustering.The trajectory segments shown in grey in Figure5.11(b)denote those that are not included in the result clus-ters c1and c2.The dotted line segments“in”each cluster in Figure5.11(b)show the representative trajectory segments of c1and c2.

力 扣 数 据 结 构 与 算 法

前端如何搞定数据结构与算法(先导篇) 「观感度：?」 「口味：锅包肉」 「烹饪时间：20min」 本文已收录在Github? 为什么要学习数据结构与算法？ 在0202年的今天，由于每天被无数的信息轰炸，大多数人已经变得越来越浮躁了，并且丧失了独立思考的能力。 你可能会经常听到这样的感慨： 技术人究竟能走多远？我遇到了天花板 35岁的程序员要如何面对中年危机？ 技术更新太快，好累，学不动了 然后，你也变得焦虑起来。那你有没有静下心来想过，如何才能抵御年龄增长并且使自己增值呢？ 无非是终身学习，持续修炼自己的内功。内功也就是基础知识和核心概念，这些轰轰烈烈发展的技术本质，其实都是基础知识，也就是我们在大学里学过的基础课-程。 操作系统 计算机组成原理 计算机网络 编译原理

设计模式 数据结构与算法 这也就是为什么越靠谱的面试官越注重你基础知识的掌握程度，为什么越牛的的企业越重视你的算法能力。因为当你拥有了这些，你已经比大多数人优秀了。你的天花板由你自己来决定，大家口中的中年危机可能并不会成为你的危机。新技术来临时，你对它的本质会看得更加透彻，学起来会一通百通。这样的人才，公司培养你也会花费更少的成本。 (不过，一辈子做个开开心心的 CRUD Boy 也是一种选择。) 数据结构与算法之间的关系 Rob Pikes 5 Rules of Programming中的第五条是这样说的： Data dominates. If youve chosen the right data structures and organized things well, the algorithms will almost always be self-evident. Data structures, not algorithms, are central to programming. 数据占主导。如果您选择了正确的数据结构并组织得当，那么这些算法几乎总是不言而喻的。数据结构而非算法是编程的核心。 瑞士计算机科学家，Algol W，Modula，Oberon 和 Pascal 语言的设计师 Niklaus Emil Wirth 写过一本非常经典的书《Algorithms + Data Structures = Programs》，即算法 + 数据结构 = 程序。 我们可以得出结论，数据结构与算法之间是相辅相成的关系。数据结构服务于算法，算法作用于特定的数据结构之上。 数据结构与算法好难，怎么学？

数据挖掘中的聚类分析方法

计算机工程应用技术本栏目责任编辑：贾薇薇 数据挖掘中的聚类分析方法 黄利文 （泉州师范学院理工学院，福建泉州３６２０００） 摘要：聚类分析是多元统计分析的重要方法之一，该方法在许多领域都有广泛的应用。本文首先对聚类的分类做简要的介绍，然后给出了常用的聚类分析方法的基本思想和优缺点，并对常用的聚类方法作比较分析，以便人们根据实际的问题选择合适的聚类方法。 关键词：聚类分析；数据挖掘 中图分类号：ＴＰ３１１文献标识码：Ａ文章编号：１００９－３０４４（２００８）１２－２０５６４－０２ ＣｌｕｓｔｅｒＡｎｌａｙｓｉｓＭｅｔｈｏｄｓｏｆＤａｔａＭｉｎｉｎｇ ＨＵＡＮＧＬｉ－ｗｅｎ （ＳｃｈｏｏｌｏｆＳｃｉｅｎｃｅ，ＱｕａｎｚｈｏｕＮｏｒｍａｌＵｎｉｖｅｒｓｉｔｙ，Ｑｕａｎｚｈｏｕ３６２０００，Ｃｈｉｎａ） Ａｂｓｔｒａｃｔ：Ｃｌｕｓｔｅｒａｎａｌｙｓｉｓｉｓｏｎｅｏｆｔｈｅｉｍｐｏｒｔａｎｔｍｅｔｈｏｄｓｏｆｍｕｌｔｉｖａｒｉａｔｅｓｔａｔｉｓｔｉｃａｌａｎａｌｙｓｉｓ，ａｎｄｔｈｉｓｍｅｔｈｏｄｈａｓａｗｉｄｅｒａｎｇｅｏｆａｐｐｌｉｃａ－ｔｉｏｎｓｉｎｍａｎｙｆｉｅｌｄｓ．Ｉｎｔｈｉｓｐａｐｅｒ，ｔｈｅｃｌａｓｓｉｆｉｃａｔｉｏｎｏｆｔｈｅｃｌｕｓｔｅｒｉｓｉｎｔｒｏｄｕｃｅｄｂｒｉｅｆｌｙ，ａｎｄｔｈｅｎｇｉｖｅｓｓｏｍｅｃｏｍｍｏｎｍｅｔｈｏｄｓｏｆｃｌｕｓｔｅｒａｎａｌｙｓｉｓａｎｄｔｈｅａｄｖａｎｔａｇｅｓａｎｄｄｉｓａｄｖａｎｔａｇｅｓｏｆｔｈｅｓｅｍｅｔｈｏｄｓ，ａｎｄｔｈｅｓｅｃｌｕｓｔｅｒｉｎｇｍｅｔｈｏｄｗｅｒｅｃｏｍｐａｒｅｄａｎｄａｎｓｌｙｚｅｄｓｏｔｈａｔｐｅｏｐｌｅｃａｎｃｈｏｓｅｓｕｉｔａｂｌｅｃｌｕｓｔｅｒｉｎｇｍｅｔｈｏｄｓａｃｃｏｒｄｉｎｇｔｏｔｈｅａｃｔｕａｌｉｓｓｕｅｓ． Ｋｅｙｗｏｒｄｓ：ＣｌｕｓｔｅｒＡｎａｌｙｓｉｓ；ＤａｔａＭｉｎｉｎｇ １引言 聚类分析是数据挖掘中的重要方法之一，它把一个没有类别标记的样本集按某种准则划分成若干个子类，使相似的样品尽可能归为一类，而不相似的样品尽量划分到不同的类中。目前，该方法已经被广泛地应用于生物、气候学、经济学和遥感等许多领域，其目的在于区别不同事物并认识事物间的相似性。因此，聚类分析的研究具有重要的意义。 本文主要介绍常用的一些聚类方法，并从聚类的可伸缩性、类的形状识别、抗“噪声”能力、处理高维能力和算法效率五个方面对其进行比较分析，以便人们根据实际的问题选择合适的聚类方法。 ２聚类的分类 聚类分析给人们提供了丰富多彩的分类方法，这些方法大致可归纳为以下几种［１，２，３，４］：划分方法、层次方法、基于密度的聚类方法、基于网格的聚类方法和基于模型的聚类方法。 ２．１划分法（ｐａｒｔｉｔｉｏｎｇｉｎｇｍｅｔｈｏｄｓ） 给定一个含有ｎ个对象（或元组）的数据库，采用一个划分方法构建数据的ｋ个划分，每个划分表示一个聚簇，且ｋ≤ｎ。在聚类的过程中，需预先给定划分的数目ｋ，并初始化ｋ个划分，然后采用迭代的方法进行改进划分，使得在同一类中的对象之间尽可能地相似，而不同类的中的对象之间尽可能地相异。这种聚类方法适用于中小数据集，对大规模的数据集进行聚类时需要作进一步的改进。 ２．２层次法（ｈｉｅｔａｒｃｈｉｃａｌｍｅｔｈｏｄｓ） 层次法对给定数据对象集合按层次进行分解，分解的结果形成一颗以数据子集为节点的聚类树，它表明类与类之间的相互关系。根据层次分解是自低向上还是自顶向下，可分为凝聚聚类法和分解聚类法：凝聚聚类法的主要思想是将每个对象作为一个单独的一个类，然后相继地合并相近的对象和类，直到所有的类合并为一个，或者符合预先给定的终止条件；分裂聚类法的主要思想是将所有的对象置于一个簇中，在迭代的每一步中，一个簇被分裂为更小的簇，直到最终每个对象在单独的一个簇中，或者符合预先给定的终止条件。在层次聚类法中，当数据对象集很大，且划分的类别数较少时，其速度较快，但是，该方法常常有这样的缺点：一个步骤（合并或分裂）完成，它就不能被取消，也就是说，开始错分的对象，以后无法再改变，从而使错分的对象不断增加，影响聚类的精度，此外，其抗“噪声”的能力也较弱，但是若把层次聚类和其他的聚类技术集成，形成多阶段聚类，聚类的效果有很大的提高。２．３基于密度的方法（ｄｅｎｓｉｔｙ－ｂａｓｅｄｍｅｔｈｏｄｓ） 该方法的主要思想是只要临近区域的密度（对象或数据点的数目）超过某个阈值，就继续聚类。也就是说，对于给定的每个数据点，在一个给定范围的区域中必须至少包含某个数目的点。这样的方法就可以用来滤处＂噪声＂孤立点数据，发现任意形状的簇。２．４基于网格的方法（ｇｒｉｄ－ｂａｓｅｄｍｅｔｈｏｄｓ） 这种方法是把对象空间量化为有限数目的单元，形成一个网格结构。所有的聚类操作都在这个网格结构上进行。用这种方法进行聚类处理速度很快，其处理时间独立于数据对象的数目，只与量化空间中每一维的单元数目有关。 ２．５基于模型的方法（ｍｏｄｅｌ－ｂａｓｅｄｍｅｔｈｏｄ） 基于模型的方法为每个簇假定一个模型，寻找数据对给定模型的最佳拟合。该方法经常基于这样的假设：数据是根据潜在的概 收稿日期：２００８－０２－１７ 作者简介：黄利文（１９７９－），男，助教。

几种排序算法分析

《几种排序算法的分析》 摘要: 排序算法是在C++中经常要用到的一种重要的算法。如何进行排序，特别是高效率的排序是是计算机应用中的一个重要课题。同一个问题可以构造不同的算法，最终选择哪一个好呢？这涉及如何评价一个算法好坏的问题，算法分析就是评估算法所消耗资源的方法。可以对同一问题的不同算法的代价加以比较，也可以由算法设计者根据算法分析判断一种算法在实现时是否会遇到资源限制的问题。排序的目的之一就是方便数据的查找。在实际生活中，应根据具体情况悬着适当的算法。一般的，对于反复使用的程序，应选取时间短的算法；对于涉及数据量较大，存储空间较小的情况则应选取节约存储空间的算法。本论文重点讨论时间复杂度。时间复杂度就是一个算法所消耗的时间。算法的效率指的是最坏情况下的算法效率。 排序分为内部排序和外部排序。本课程结业论文就内部排序算法（插入排序，选择排序，交换排序，归并排序和基数排序）的基本思想，排序步骤和实现算法等进行介绍。 本论文以较为详细的文字说明，表格对比，例子阐述等方面加以比较和总结，通过在参加数据的规模，记录说带的信息量大小，对排序稳定的要求，关键字的分布情况以及算法的时间复杂度和空间复杂度等方面进行比较，得出它们的优缺点和不足，从而加深了对它们的认识和了解，进而使自己在以后的学习和应用中能够更好的运用。

1.五种排序算法的实例： 1.1.插入排序 1.1.1.直接插入排序 思路：将数组分为无序区和有序区两个区，然后不断将无序区的第一个元素按大小顺序插入到有序区中去，最终将所有无序区元素都移动到有序区完成排序。 要点：设立哨兵，作为临时存储和判断数组边界之用。 实现： Void InsertSort(Node L[],int length) { Int i,j;//分别为有序区和无序区指针 for(i=1;i=1)//直到增量缩小为1 { Shell(L,d); d=d/2;//缩小增量 } } Void Shell(Node L[],int d) {

数据结构经典例题

数据结构例题（及答案） 项目一习题(答案) 一选择题 1. 算法的计算量的大小称为计算的(B )。 A( 效率 B. 复杂性 C. 现实性 D. 难度 2.算法的时间复杂度取决于(C ) A(问题的规模 B. 待处理数据的初态 C. A和B 3(从逻辑上可以把数据结构分为(C )两大类。 A(动态结构、静态结构 B(顺序结构、链式结构 C(线性结构、非线性结构 D(初等结构、构造型结构 4(连续存储设计时，存储单元的地址(A )。 A(一定连续 B(一定不连续 C(不一定连续 D(部分连续，部分不连续 5. 以下属于逻辑结构的是(C )。 A(顺序表 B. 哈希表 C.有序表 D. 单链表 二、判断题 1. 数据元素是数据的最小单位。(×) 2. 记录是数据处理的最小单位。(×) 3. 数据的逻辑结构是指数据的各数据项之间的逻辑关系;(×) 4(程序一定是算法。(×) 5. 在顺序存储结构中，有时也存储数据结构中元素之间的关系。(×) 6. 顺序存储方式的优点是存储密度大，且插入、删除运算效率高。(×) 7. 数据结构的基本操作的设置的最重要的准则是，实现应用程序与存储结构的独立。(?)

8. 数据的逻辑结构说明数据元素之间的顺序关系,它依赖于计算机的储存结构. (×) 三、填空 1(数据的物理结构包括数据元素的表示和数据元素间关系的表示。 2. 对于给定的n个元素,可以构造出的逻辑结构有集合，线性结构，树形 结构，图状结构或网状结构四种。 3(数据的逻辑结构是指数据的组织形式，即数据元素之间逻辑关系的总体。而 逻辑关系是指数据元素之间的关联方式或称“邻接关系”。 4(一个数据结构在计算机中表示(又称映像) 称为存储结构。 5(抽象数据类型的定义仅取决于它的一组逻辑特性，而与在计算机内部如何表 示和实现无关，即不论其内部结构如何变化，只要它的数学特性不变，都不影响 其外部使用。 6(数据结构中评价算法的两个重要指标是算法的时间复杂度和空间复杂度。 7. 数据结构是研讨数据的逻辑结构和物理结构，以及它们之间的相互 关系，并对与这种结构定义相应的操作(运算)，设计出相应的算法。 ( 一个算法具有5个特性: 有穷性、确定性、可行性，有零个或多个输入、 有一个或多个输8 出。 四、应用题 1. 1. 数据结构是一门研究什么内容的学科, 答:数据结构是一门研究在非数值计算的程序设计问题中，计算机的操作对象 及对象间的关系和施加于对象的操作等的学科 2. 2. 数据元素之间的关系在计算机中有几种表示方法,各有什么特点, 答:四 种表示方法

聚类分析算法解析.doc

聚类分析算法解析 一、不相似矩阵计算 1.加载数据 data(iris) str(iris) 分类分析是无指导的分类，所以删除数据中的原分类变量。 iris$Species<-NULL 2. 不相似矩阵计算 不相似矩阵计算，也就是距离矩阵计算，在R中采用dist()函数，或者cluster包中的daisy()函数。dist()函数的基本形式是 dist(x, method = "euclidean", diag = FALSE, upper = FALSE, p = 2) 其中x是数据框（数据集），而方法可以指定为欧式距离"euclidean", 最大距离"maximum", 绝对值距离"manhattan", "canberra", 二进制距离非对称"binary" 和明氏距离"minkowski"。默认是计算欧式距离，所有的属性必须是相同的类型。比如都是连续类型，或者都是二值类型。 dd<-dist(iris) str(dd) 距离矩阵可以使用as.matrix()函数转化了矩阵的形式，方便显示。Iris数据共150例样本间距离矩阵为150行列的方阵。下面显示了1~5号样本间的欧式距离。 dd<-as.matrix(dd)

二、用hclust()进行谱系聚类法（层次聚类） 1.聚类函数 R中自带的聚类函数是hclust()，为谱系聚类法。基本的函数指令是 结果对象 <- hclust（距离对象, method=方法） hclust()可以使用的类间距离计算方法包含离差法"ward"，最短距离法"single"，最大距离法"complete"，平均距离法"average"，"mcquitty"，中位数法 "median" 和重心法"centroid"。下面采用平均距离法聚类。 hc <- hclust(dist(iris), method="ave") 2.聚类函数的结果 聚类结果对象包含很多聚类分析的结果，可以使用数据分量的方法列出相应的计算结果。 str(hc) 下面列出了聚类结果对象hc包含的merge和height结果值的前6个。其行编号表示聚类过程的步骤，X1，X2表示在该步合并的两类，该编号为负代表原始的样本序号，编号为正代表新合成的类；变量height表示合并时两类类间距离。比如第1步，合并的是样本102和143，其样本间距离是0.0，合并后的类则使用该步的步数编号代表，即样本-102和-143合并为1类。再如第6行表示样本11和49合并，该两个样本的类间距离是0.1，合并后的类称为6类。 head (hc$merge,hc$height)

几种常见内部排序算法比较

常见内部排序算法比较 排序算法是数据结构学科经典的内容，其中内部排序现有的算法有很多种，究竟各有什么特点呢？本文力图设计实现常用内部排序算法并进行比较。分别为起泡排序，直接插入排序，简单选择排序，快速排序，堆排序，针对关键字的比较次数和移动次数进行测试比较。 问题分析和总体设计 ADT OrderableList { 数据对象：D={ai| ai∈IntegerSet,i=1,2,…,n,n≥0} 数据关系：R1={〈ai-1，ai〉|ai-1, ai∈D, i=1,2,…,n} 基本操作： InitList(n) 操作结果：构造一个长度为n，元素值依次为1，2，…，n的有序表。Randomizel(d,isInverseOrser) 操作结果：随机打乱 BubbleSort( ) 操作结果：进行起泡排序 InserSort( ) 操作结果：进行插入排序 SelectSort( ) 操作结果：进行选择排序 QuickSort( ) 操作结果：进行快速排序 HeapSort( ) 操作结果：进行堆排序 ListTraverse(visit( )) 操作结果：依次对L种的每个元素调用函数visit( ) }ADT OrderableList 待排序表的元素的关键字为整数.用正序,逆序和不同乱序程度的不同数据做测试比较,对关键字的比较次数和移动次数(关键字交换计为3次移动)进行测试比较.要求显示提示信息,用户由键盘输入待排序表的表长(100-1000)和不同测试数据的组数(8-18).每次测试完毕,要求列表现是比较结果. 要求对结果进行分析.

详细设计 1、起泡排序 算法：核心思想是扫描数据清单，寻找出现乱序的两个相邻的项目。当找到这两个项目后，交换项目的位置然后继续扫描。重复上面的操作直到所有的项目都按顺序排好。 bubblesort(struct rec r[],int n) { int i,j; struct rec w; unsigned long int compare=0,move=0; for(i=1;i<=n-1;i++) for(j=n;j>=i+1;j--) { if(r[j].key

大数据时代的空间数据挖掘综述

第37卷第7期测绘与空间地理信息 GEOMATICS ＆SPATIAL INFOＲMATION TECHNOLOGY Vol．37，No．7收稿日期：2014－01－22 作者简介：马宏斌（1982－），男，甘肃天水人，作战环境学专业博士研究生，主要研究方向为地理空间信息服务。 大数据时代的空间数据挖掘综述 马宏斌1 ，王 柯1，马团学 2（1．信息工程大学地理空间信息学院，河南郑州450000；2．空降兵研究所，湖北孝感432000） 摘 要：随着大数据时代的到来，数据挖掘技术再度受到人们关注。本文回顾了传统空间数据挖掘面临的问题， 介绍了国内外研究中利用大数据处理工具和云计算技术，在空间数据的存储、管理和挖掘算法等方面的做法，并指出了该类研究存在的不足。最后，探讨了空间数据挖掘的发展趋势。关键词：大数据；空间数据挖掘；云计算中图分类号：P208 文献标识码：B 文章编号：1672－5867（2014）07－0019－04 Spatial Data Mining Big Data Era Ｒeview MA Hong －bin 1，WANG Ke 1，MA Tuan －xue 2 （1．Geospatial Information Institute ，Information Engineering University ，Zhengzhou 450000，China ； 2．Airborne Institute ，Xiaogan 432000，China ） Abstract ：In the era of Big Data ，more and more researchers begin to show interest in data mining techniques again．The paper review most unresolved problems left by traditional spatial data mining at first．And ，some progress made by researches using Big Data and Cloud Computing technology is introduced．Also ，their drawbacks are mentioned．Finally ，future trend of spatial data mining is dis-cussed． Key words ：big data ；spatial data mining ；cloud computing 0引言 随着地理空间信息技术的飞速发展，获取数据的手 段和途径都得到极大丰富，传感器的精度得到提高和时空覆盖范围得以扩大，数据量也随之激增。用于采集空间数据的可能是雷达、红外、光电、卫星、多光谱仪、数码相机、成像光谱仪、全站仪、天文望远镜、电视摄像、电子 显微镜、CT 成像等各种宏观与微观传感器或设备，也可能是常规的野外测量、人口普查、土地资源调查、地图扫描、 地图数字化、统计图表等空间数据获取手段，还可能是来自计算机、 网络、GPS ，ＲS 和GIS 等技术应用和分析空间数据。特别是近些年来，个人使用的、携带的各种传感器（重力感应器、电子罗盘、三轴陀螺仪、光线距离感应器、温度传感器、红外线传感器等），具备定位功能电子设备的普及，如智能手机、平板电脑、可穿戴设备（GOOGLE GLASS 和智能手表等），使人们在日常生活中产生了大量具有位置信息的数据。随着志愿者地理信息（Volunteer Geographic Information ）的出现，使这些普通民众也加入到了提供数据者的行列。 以上各种获取手段和途径的汇集，就使每天获取的 数据增长量达到GB 级、 TB 级乃至PB 级。如中国遥感卫星地面站现在保存的对地观测卫星数据资料达260TB ，并以每年15TB 的数据量增长。比如2011年退役的Landsat5卫星在其29年的在轨工作期间，平均每年获取8．6万景影像，每天获取67GB 的观测数据。而2012年发射的资源三号（ZY3）卫星，每天的观测数据获取量可以达到10TB 以上。类似的传感器现在已经大量部署在卫 星、 飞机等飞行平台上，未来10年，全球天空、地空间部署的百万计传感器每天获取的观测数据将超过10PB 。这预示着一个时代的到来，那就是大数据时代。大数据具有 “4V ”特性，即数据体量大（Volume ）、数据来源和类型繁多（Variety ）、数据的真实性难以保证（Veracity ）、数据增加和变化的速度快（Velocity ）。对地观测的系统如图1所示。 在这些数据中，与空间位置相关的数据占了绝大多数。传统的空间知识发现的科研模式在大数据情境下已经不再适用，原因是传统的科研模型不具有普适性且支持的数据量受限， 受到数据传输、存储及时效性需求的制约等。为了从存储在分布方式、虚拟化的数据中心获取信息或知识，这就需要利用强有力的数据分析工具来将

数据结构(C语言)【经典题库】含标准答案

《数据结构与算法》复习题 选择题 1．在数据结构中，从逻辑上可以把数据结构分为 C 。 A．动态结构和静态结构 B．紧凑结构和非紧凑结构 C．线性结构和非线性结构 D．内部结构和外部结构 2．数据结构在计算机内存中的表示是指 A 。 A．数据的存储结构 B．数据结构 C．数据的逻辑结构 D．数据元素之间的关系 3．在数据结构中，与所使用的计算机无关的是数据的 A 结构。 A．逻辑 B．存储 C．逻辑和存储 D．物理 4．在存储数据时，通常不仅要存储各数据元素的值，而且还要存储 C 。A．数据的处理方法 B．数据元素的类型 C．数据元素之间的关系 D．数据的存储方法 5．在决定选取何种存储结构时，一般不考虑 A 。 A．各结点的值如何 B．结点个数的多少 C．对数据有哪些运算 D．所用的编程语言实现这种结构是否方便。 6．以下说法正确的是 D 。 A．数据项是数据的基本单位 B．数据元素是数据的最小单位

C．数据结构是带结构的数据项的集合 D．一些表面上很不相同的数据可以有相同的逻辑结构 7．算法分析的目的是 C ，算法分析的两个主要方面是 A 。（1）A．找出数据结构的合理性 B．研究算法中的输入和输出的关系C．分析算法的效率以求改进 C．分析算法的易读性和文档性（2）A．空间复杂度和时间复杂度 B．正确性和简明性 C．可读性和文档性 D．数据复杂性和程序复杂性 8．下面程序段的时间复杂度是 O(n2) 。 s =0; for( I =0; i

数据挖掘考试题精编版

数据挖掘考试题 公司内部编号：（GOOD-TMMT-MMUT-UUPTY-UUYY-DTTI-

数据挖掘考试题 一．选择题 1. 当不知道数据所带标签时，可以使用哪种技术促使带同类标签的数据与带其他标签的数据相分离( ) A.分类 B.聚类 C.关联分析 D.主成分分析 2. ( )将两个簇的邻近度定义为不同簇的所有点对邻近度的平均值，它是一种凝聚层次聚类技术。 A.MIN(单链) B.MAX(全链) C.组平均 D.Ward方法 3.数据挖掘的经典案例“啤酒与尿布试验”最主要是应用了( )数据挖掘方法。 A 分类 B 预测 C关联规则分析 D聚类 4.关于K均值和DBSCAN的比较，以下说法不正确的是( ) A.K均值丢弃被它识别为噪声的对象，而DBSCAN一般聚类所有对象。 B.K均值使用簇的基于原型的概念，DBSCAN使用基于密度的概念。 C.K均值很难处理非球形的簇和不同大小的簇，DBSCAN可以处理不同大小和不同形状的簇 D.K均值可以发现不是明显分离的簇，即便簇有重叠也可以发现，但是DBSCAN 会合并有重叠的簇 5.下列关于Ward’s Method说法错误的是：( ) A.对噪声点和离群点敏感度比较小 B.擅长处理球状的簇

C.对于Ward方法，两个簇的邻近度定义为两个簇合并时导致的平方误差 D.当两个点之间的邻近度取它们之间距离的平方时，Ward方法与组平均非常相似 6.下列关于层次聚类存在的问题说法正确的是：( ) A.具有全局优化目标函数 B.Group Average擅长处理球状的簇 C.可以处理不同大小簇的能力 D.Max对噪声点和离群点很敏感 7.下列关于凝聚层次聚类的说法中，说法错误的事：( ) A.一旦两个簇合并，该操作就不能撤销 B.算法的终止条件是仅剩下一个簇 C.空间复杂度为()2m O D.具有全局优化目标函数 8.规则{牛奶，尿布}→{啤酒}的支持度和置信度分别为：( ) 9.下列( )是属于分裂层次聚类的方法。 A.Min B.Max C.Group Average D.MST 10.对下图数据进行凝聚聚类操作，簇间相似度使用MAX计算，第二步是哪两个簇合并：( ) A.在{3}和{l,2}合并 B.{3}和{4,5}合并 C.{2,3}和{4,5}合并

数据结构经典例题

数据结构经典例题 1.设计一个算法将L拆分成两个带头节点的单链表L1和L2。 void split(LinkList *&L,LinkList *&L1,LinkList *&L2) { LinkList *p=L->next,*q,*r1; //p指向第1个数据节点 L1=L; //L1利用原来L的头节点 r1=L1; //r1始终指向L1的尾节点 L2=(LinkList *)malloc(sizeof(LinkList));//创建L2的头节点 L2->next=NULL; //置L2的指针域为NULL while (p!=NULL) { r1->next=p; //采用尾插法将*p(data值为ai)插入L1中 r1=p; p=p->next; //p移向下一个节点(data值为bi) q=p->next; //由于头插法修改p的next域,故用q保存*p的后继节点 p->next=L2->next; //采用头插法将*p插入L2中 L2->next=p; p=q; //p重新指向ai+1的节点 } r1->next=NULL; //尾节点next置空 } 2.查找链表中倒数第k个位置上的节点（k为正整数）。若查找成功，算法输出该节点的data域的值，并返回1；否则，只返回0。 typedef struct LNode {int data; struct LNode *link; } *LinkList; int Searchk(LinkList list,int k) { LinkList p,q; int count=0; p=q=list->link; while (p!=NULL) { if (countlink; p=p->link; } if (count

聚类分析K-means算法综述

聚类分析K-means算法综述 摘要：介绍K-means聚类算法的概念，初步了解算法的基本步骤，通过对算法缺点的分析，对算法已有的优化方法进行简单分析，以及对算法的应用领域、算法未来的研究方向及应用发展趋势作恰当的介绍。 关键词：K-means聚类算法基本步骤优化方法应用领域研究方向应用发展趋势 算法概述 K-means聚类算法是一种基于质心的划分方法，输入聚类个数k，以及包含n个数据对象的数据库，输出满足方差最小标准的k个聚类。 评定标准：同一聚类中的对象相似度较高；而不同聚类中的对象相似度较小。聚类相似度是利用各聚类中对象的均值所获得一个“中心对象”（引力中心）来进行计算。 解释：基于质心的划分方法就是将簇中的所有对象的平均值看做簇的质心，然后根据一个数据对象与簇质心的距离，再将该对象赋予最近的簇。 k-means 算法基本步骤 （1）从n个数据对象任意选择k 个对象作为初始聚类中心 （2）根据每个聚类对象的均值（中心对象），计算每个对象与这些中心对象的距离；并根据最小距离重新对相应对象进行划分 （3）重新计算每个（有变化）聚类的均值（中心对象） （4）计算标准测度函数，当满足一定条件，如函数收敛时，则算法终止；如果条件不满足则回到步骤（2） 形式化描述 输入：数据集D，划分簇的个数k 输出：k个簇的集合 （1）从数据集D中任意选择k个对象作为初始簇的中心； （2）Repeat （3）For数据集D中每个对象P do （4）计算对象P到k个簇中心的距离 （5）将对象P指派到与其最近（距离最短）的簇；

（6）End For （7）计算每个簇中对象的均值，作为新的簇的中心； （8）Until k个簇的簇中心不再发生变化 对算法已有优化方法的分析 (1)K-means算法中聚类个数K需要预先给定 这个K值的选定是非常难以估计的，很多时候,我们事先并不知道给定的数据集应该分成多少个类别才最合适，这也是K一means算法的一个不足"有的算法是通过类的自动合并和分裂得到较为合理的类型数目k，例如Is0DAIA算法"关于K一means算法中聚类数目K 值的确定，在文献中，根据了方差分析理论，应用混合F统计量来确定最佳分类数，并应用了模糊划分嫡来验证最佳分类数的正确性。在文献中，使用了一种结合全协方差矩阵RPCL算法，并逐步删除那些只包含少量训练数据的类。文献中针对“聚类的有效性问题”提出武汉理工大学硕士学位论文了一种新的有效性指标：V(k km) = Intra(k) + Inter(k) / Inter(k max)，其中k max是可聚类的最大数目,目的是选择最佳聚类个数使得有效性指标达到最小。文献中使用的是一种称为次胜者受罚的竞争学习规则来自动决定类的适当数目"它的思想是：对每个输入而言不仅竞争获胜单元的权值被修正以适应输入值，而且对次胜单元采用惩罚的方法使之远离输入值。 (2)算法对初始值的选取依赖性极大以及算法常陷入局部极小解 不同的初始值，结果往往不同。K-means算法首先随机地选取k个点作为初始聚类种子，再利用迭代的重定位技术直到算法收敛。因此，初值的不同可能导致算法聚类效果的不稳定，并且，K-means算法常采用误差平方和准则函数作为聚类准则函数(目标函数)。目标函数往往存在很多个局部极小值，只有一个属于全局最小，由于算法每次开始选取的初始聚类中心落入非凸函数曲面的“位置”往往偏离全局最优解的搜索范围，因此通过迭代运算，目标函数常常达到局部最小，得不到全局最小。对于这个问题的解决，许多算法采用遗传算法(GA)，例如文献中采用遗传算法GA进行初始化，以内部聚类准则作为评价指标。 (3)从K-means算法框架可以看出，该算法需要不断地进行样本分类调整，不断地计算调整后的新的聚类中心，因此当数据量非常大时,算法的时间开销是非常大 所以需要对算法的时间复杂度进行分析，改进提高算法应用范围。在文献中从该算法的时间复杂度进行分析考虑，通过一定的相似性准则来去掉聚类中心的候选集，而在文献中，使用的K-meanS算法是对样本数据进行聚类。无论是初始点的选择还是一次迭代完成时对数据的调整，都是建立在随机选取的样本数据的基础之上，这样可以提高算法的收敛速度。

十 大 经 典 排 序 算 法 总 结 超 详 细

数据挖掘十大经典算法，你都知道哪些？ 当前时代大数据炙手可热，数据挖掘也是人人有所耳闻，但是关于数据挖掘更具体的算法，外行人了解的就少之甚少了。 数据挖掘主要分为分类算法，聚类算法和关联规则三大类，这三类基本上涵盖了目前商业市场对算法的所有需求。而这三类里又包含许多经典算法。而今天，小编就给大家介绍下数据挖掘中最经典的十大算法，希望它对你有所帮助。 一、分类决策树算法C4.5 C4.5，是机器学习算法中的一种分类决策树算法，它是决策树(决策树，就是做决策的节点间的组织方式像一棵倒栽树)核心算法ID3的改进算法，C4.5相比于ID3改进的地方有： 1、用信息增益率选择属性 ID3选择属性用的是子树的信息增益，这里可以用很多方法来定义信息，ID3使用的是熵（shang），一种不纯度度量准则，也就是熵的变化值，而 C4.5用的是信息增益率。区别就在于一个是信息增益，一个是信息增益率。 2、在树构造过程中进行剪枝，在构造决策树的时候，那些挂着几个元素的节点，不考虑最好，不然容易导致过拟。 3、能对非离散数据和不完整数据进行处理。 该算法适用于临床决策、生产制造、文档分析、生物信息学、空间数据建模等领域。 二、K平均算法

K平均算法（k-means algorithm）是一个聚类算法，把n个分类对象根据它们的属性分为k类（kn）。它与处理混合正态分布的最大期望算法相似，因为他们都试图找到数据中的自然聚类中心。它假设对象属性来自于空间向量，并且目标是使各个群组内部的均方误差总和最小。 从算法的表现上来说，它并不保证一定得到全局最优解，最终解的质量很大程度上取决于初始化的分组。由于该算法的速度很快，因此常用的一种方法是多次运行k平均算法，选择最优解。 k-Means 算法常用于图片分割、归类商品和分析客户。 三、支持向量机算法 支持向量机(Support Vector Machine)算法，简记为SVM，是一种监督式学习的方法，广泛用于统计分类以及回归分析中。 SVM的主要思想可以概括为两点： (1)它是针对线性可分情况进行分析，对于线性不可分的情况，通过使用非线性映射算法将低维输入空间线性不可分的样本转化为高维特征空间使其线性可分； (2)它基于结构风险最小化理论之上，在特征空间中建构最优分割超平面，使得学习器得到全局最优化，并且在整个样本空间的期望风险以某个概率满足一定上界。 四、The Apriori algorithm Apriori算法是一种最有影响的挖掘布尔关联规则频繁项集的算法，其核心是基于两阶段“频繁项集”思想的递推算法。其涉及到的关联规则在分类上属于单维、单层、布尔关联规则。在这里，所有支持度大于最小支

聚类算法分析报告汇总

嵌入式方向工程设计实验报告 学院班级：130712 学生学号：13071219 学生姓名：杨阳 同作者：无 实验日期：2010年12月

聚类算法分析研究 1 实验环境以及所用到的主要软件 Windows Vista NetBeans6.5.1 Weka3.6 MATLAB R2009a 2 实验内容描述 聚类是对数据对象进行划分的一种过程，与分类不同的是，它所划分的类是未知的，故此，这是一个“无指导的学习” 过程，它倾向于数据的自然划分。其中聚类算法常见的有基于层次方法、基于划分方法、基于密度以及网格等方法。本文中对近年来聚类算法的研究现状与新进展进行归纳总结。一方面对近年来提出的较有代表性的聚类算法，从算法思想。关键技术和优缺点等方面进行分析概括；另一方面选择一些典型的聚类算法和一些知名的数据集，主要从正确率和运行效率两个方面进行模拟实验，并分别就同一种聚类算法、不同的数据集以及同一个数据集、不同的聚类算法的聚类情况进行对比分析。最后通过综合上述两方面信息给出聚类分析的研究热点、难点、不足和有待解决的一些问题等。 实验中主要选择了K 均值聚类算法、FCM 模糊聚类算法并以UCI Machine Learning Repository 网站下载的IRIS 和WINE 数据集为基础通过MATLAB 实现对上述算法的实验测试。然后以WINE 数据集在学习了解Weka 软件接口方面的基础后作聚类分析，使用最常见的K 均值（即K-means ）聚类算法和FCM 模糊聚类算法。下面简单描述一下K 均值聚类的步骤。 K 均值算法首先随机的指定K 个类中心。然后： （1）将每个实例分配到距它最近的类中心，得到K 个类； （2）计分别计算各类中所有实例的均值，把它们作为各类新的类中心。 重复（1）和（2），直到K 个类中心的位置都固定，类的分配也固定。 在实验过程中通过利用Weka 软件中提供的simpleKmeans （也就是K 均值聚类算法对WINE 数据集进行聚类分析，更深刻的理解k 均值算法，并通过对实验结果进行观察分析，找出实验中所存在的问题。然后再在学习了解Weka 软件接口方面的基础上对Weka 软件进行一定的扩展以加入新的聚类算法来实现基于Weka 平台的聚类分析。 3 实验过程 3.1 K 均值聚类算法 3.1.1 K 均值聚类算法理论 K 均值算法是一种硬划分方法，简单流行但其也存在一些问题诸如其划分结果并不一定完全可信。K 均值算法的划分理论基础是 2 1 min i c k i k A i x v ∈=-∑∑ （1） 其中c 是划分的聚类数，i A 是已经属于第i 类的数据集i v 是相应的点到第i 类的平均距离，即

数据挖掘主要算法

朴素贝叶斯： 有以下几个地方需要注意： 1. 如果给出的特征向量长度可能不同，这是需要归一化为通长度的向量（这里以文本分类为例），比如说是句子单词的话，则长度为整个词汇量的长度，对应位置是该单词出现的次数。 2. 计算公式如下： 其中一项条件概率可以通过朴素贝叶斯条件独立展开。要注意一点就是的计算方法，而由朴素贝叶斯的前提假设可知， = ，因此一般有两种，一种是在类别为ci的那些样本集中，找到wj出现次数的总和，然后除以该样本的总和；第二种方法是类别为ci的那些样本集中，找到wj出现次数的总和，然后除以该样本中所有特征出现次数的总和。 3. 如果中的某一项为0，则其联合概率的乘积也可能为0，即2中公式的分子为0，为了避免这种现象出现，一般情况下会将这一项初始化为1，当然为了保证概率相等，分母应对应初始化为2（这里因为是2类，所以加2，如果是k类就需要加k，术语上叫做laplace 光滑, 分母加k的原因是使之满足全概率公式）。 朴素贝叶斯的优点： 对小规模的数据表现很好，适合多分类任务，适合增量式训练。 缺点： 对输入数据的表达形式很敏感。 决策树： 决策树中很重要的一点就是选择一个属性进行分枝，因此要注意一下信息增益的计算公式，并深入理解它。 信息熵的计算公式如下:

其中的n代表有n个分类类别（比如假设是2类问题，那么n=2）。分别计算这2类样本在总样本中出现的概率p1和p2，这样就可以计算出未选中属性分枝前的信息熵。 现在选中一个属性xi用来进行分枝，此时分枝规则是：如果xi=vx的话，将样本分到树的一个分支；如果不相等则进入另一个分支。很显然，分支中的样本很有可能包括2个类别，分别计算这2个分支的熵H1和H2,计算出分枝后的总信息熵H’=p1*H1+p2*H2.，则此时的信息增益ΔH=H-H’。以信息增益为原则，把所有的属性都测试一边，选择一个使增益最大的属性作为本次分枝属性。 决策树的优点： 计算量简单，可解释性强，比较适合处理有缺失属性值的样本，能够处理不相关的特征； 缺点： 容易过拟合（后续出现了随机森林，减小了过拟合现象）； Logistic回归： Logistic是用来分类的，是一种线性分类器，需要注意的地方有： 1. logistic函数表达式为： 其导数形式为： 2. logsitc回归方法主要是用最大似然估计来学习的，所以单个样本的后验概率为： 到整个样本的后验概率：

十大经典排序算法

.1 算法分类 十种常见排序算法可以分为两大类： ?比较类排序：通过比较来决定元素间的相对次序，由于其时间复杂度不能突破O(nlogn)，因此也称为非线性时间比较类排序。 ?非比较类排序：不通过比较来决定元素间的相对次序，它可以突破基于比较排序的时间下界，以线性时间运行，因此也称为线性时间非比较类排序。 0.2 算法复杂度

0.3 相关概念 ?稳定：如果a原本在b前面，而a=b，排序之后a仍然在b的前面。 ?不稳定：如果a原本在b的前面，而a=b，排序之后a 可能会出现在b 的后面。 ?时间复杂度：对排序数据的总的操作次数。反映当n变化时，操作次数呈现什么规律。 ?空间复杂度：是指算法在计算机 内执行时所需存储空间的度量，它也是数据规模n的函数。 1、冒泡排序（Bubble Sort） 冒泡排序是一种简单的排序算法。它重复地走访过要排序的数列，一次比较两个元素，如果它们的顺序错误就把它们交换过来。走访数列的工作是重复地进行直到没有再需要交换，也就是说该数列已经排序完成。这个算法的名字由来是因为越小的元素会经由交换慢慢“浮”到数列的顶端。

1.1 算法描述 ?比较相邻的元素。如果第一个比第二个大，就交换它们两个； ?对每一对相邻元素作同样的工作，从开始第一对到结尾的最后一对，这样在最后的元素应该会是最大的数； ?针对所有的元素重复以上的步骤，除了最后一个； ?重复步骤1~3，直到排序完成。 1.2 动图演示 1.3 代码实现 ?

2、选择排序（Selection Sort） 选择排序(Selection-sort)是一种简单直观的排序算法。它的工作原理：首先在未排序序列中找到最小（大）元素，存放到排序序列的起始位置，然后，再从剩余未排序元素中继续寻找最小（大）元素，然后放到已排序序列的末尾。以此类推，直到所有元素均排序完毕。 2.1 算法描述 n个记录的直接选择排序可经过n-1趟直接选择排序得到有序结果。具体算法描述如下： ?初始状态：无序区为R[1..n]，有序区为空； ?第i趟排序(i=1,2,3…n-1)开始时，当前有序区和无序区分别为R[1..i-1]和R(i..n）。该趟排序从当前无序区中-选出关键字最小的记录R[k]，将它与无序区的第1个记录R交换，使R[1..i]和R[i+1..n)分别变为记录个数增加1个的新有序区和记录个数减少1个的新无序区； ?n-1趟结束，数组有序化了。 2.2 动图演示 2.3 代码实现 ?