Parallel K-Means Clustering Based on MapReduce
k-Means-Clustering
合肥工业大学—数学建模组k-Means ClusteringOn this page…Introduction to k-Means Clustering Create Clusters and Determine Separation Determine the Correct Number of Clusters Avoid Local MinimaIntroduction to k-Means Clusteringk-means clustering is a partitioning method. The function kmeans partitions data into k mutuallyexclusive clusters, and returns the index of the cluster to which it has assigned each observation. Unlike hierarchical clustering, k-means clustering operates on actual observations (rather than the larger set of dissimilarity measures), and creates a single level of clusters. The distinctions mean that k-means clustering is often more suitable than hierarchical clustering for large amounts of data.kmeans treats each observation in your data as an object having a location in space. It finds apartition in which objects within each cluster are as close to each other as possible, and as far from objects in other clusters as possible. You can choose from five different distance measures, depending on the kind of data you are clustering.Each cluster in the partition is defined by its member objects and by its centroid, or center. The centroid for each cluster is the point to which the sum of distances from all objects in that clusteris minimized. kmeanscomputes cluster centroids differently for each distance measure, tominimize the sum with respect to the measure that you specify.kmeans uses an iterative algorithm that minimizes the sum of distances from each object to itscluster centroid, over all clusters. This algorithm moves objects between clusters until the sum cannot be decreased further. The result is a set of clusters that are as compact and well-separated as possible. You can control the details of the minimization using several optional inputparameters to kmeans, including ones for the initial values of the cluster centroids, and for themaximum number of iterations.Create Clusters and Determine SeparationThe following example explores possible clustering in four-dimensional data by analyzing the results of partitioning the points into three, four, and five clusters.Note Because each part of this example generates random numbers sequentially, i.e., without setting a new state, you must perform all steps in sequence to duplicate the results shown. If you perform the steps out of sequence, the answers will be essentially the same, but the intermediate results, number of iterations, or ordering of the silhouette plots may differ.王刚合肥工业大学—数学建模组 First, load some data:rng('default'); % For reproducibility load kmeansdata; size(X) ans =560 4 Even though these data are four-dimensional, and cannot be easily visualized, kmeans enables you to investigate whether a group structure exists in them. Call kmeans with k, the desired number of clusters, equal to 3. For this example, specify the city block distance measure, and usethe default starting method of initializing centroids from randomly selected data points.idx3 = kmeans(X,3,'distance','city');To get an idea of how well-separated the resulting clusters are, you can make a silhouette plotusing the cluster indices output from kmeans. The silhouette plot displays a measure of howclose each point in one cluster is to points in the neighboring clusters. This measure ranges from +1, indicating points that are very distant from neighboring clusters, through 0, indicating points that are not distinctly in one cluster or another, to -1, indicating points that are probably assignedto the wrong cluster. silhouette returns these values in its first output. [silh3,h] = silhouette(X,idx3,'city'); set(get(gca,'Children'),'FaceColor',[.8 .8 1]) xlabel('Silhouette Value') ylabel('Cluster')王刚合肥工业大学—数学建模组From the silhouette plot, you can see that most points in the second cluster have a large silhouette value, greater than 0.6, indicating that the cluster is somewhat separated from neighboring clusters. However, the third cluster contains many points with low silhouette values, and the first contains a few points with negative values, indicating that those two clusters are not well separated.Determine the Correct Number of ClustersIncrease the number of clusters to see if kmeans can find a better grouping of the data. This time, use the optional 'display' parameter to print information about each iteration.idx4 = kmeans(X,4, 'dist','city', 'display','iter');iter phasenumsum115602077.4321511778.643131771.14201771.1Best total sum of distances = 1771.1Notice that the total sum of distances decreases at each iteration as kmeans reassigns pointsbetween clusters and recomputes cluster centroids. In this case, the second phase of the algorithm did not make any reassignments, indicating that the first phase reached a minimum after five iterations. In some problems, the first phase might not reach a minimum, but the second phase always will.A silhouette plot for this solution indicates that these four clusters are better separated than the three in the previous solution.[silh4,h] = silhouette(X,idx4,'city'); set(get(gca,'Children'),'FaceColor',[.8 .8 1]) xlabel('Silhouette Value') ylabel('Cluster')王刚合肥工业大学—数学建模组A more quantitative way to compare the two solutions is to look at the average silhouette values for the two cases.cluster3 = mean(silh3) cluster4 = mean(silh4) cluster3 =0.5352 cluster4 =0.6400Finally, try clustering the data using five clusters.idx5 = kmeans(X,5,'dist','city','replicates',5); [silh5,h] = silhouette(X,idx5,'city'); set(get(gca,'Children'),'FaceColor',[.8 .8 1]) xlabel('Silhouette Value') ylabel('Cluster') mean(silh5) ans =0.5266王刚合肥工业大学—数学建模组This silhouette plot indicates that this is probably not the right number of clusters, since two of the clusters contain points with mostly low silhouette values. Without some knowledge of howmany clusters are really in the data, it is a good idea to experiment with a range of values for k.Avoid Local MinimaLike many other types of numerical minimizations, the solution that kmeans reaches often depends on the starting points. It is possible for kmeans to reach a local minimum, wherereassigning any one point to a new cluster would increase the total sum of point-to-centroid distances, but where a better solution does exist. However, you can use theoptional 'replicates' parameter to overcome that problem. For four clusters, specify five replicates, and use the 'display' parameter to print out the finalsum of distances for each of the solutions.[idx4,cent4,sumdist] = kmeans(X,4,'dist','city',... 'display','final','replicates',5);Replicate 1, 4 iterations, total sum of distances = 1771.1. Replicate 2, 7 iterations, total sum of distances = 1771.1. Replicate 3, 8 iterations, total sum of distances = 1771.1. Replicate 4, 5 iterations, total sum of distances = 1771.1. Replicate 5, 6 iterations, total sum of distances = 1771.1. Best total sum of distances = 1771.1王刚合肥工业大学—数学建模组In this example, kmeans found the same minimum in all five replications. However, even forrelatively simple problems, nonglobal minima do exist. Each of these five replicates began from adifferent randomly selected set of initial centroids, so sometimes kmeans finds more than one local minimum. However, the final solution that kmeans returns is the one with the lowest totalsum of distances, over all replicates.sum(sumdist) ans =1.7711e+03王刚。
kmeans 算法
kmeans 算法K-Means算法，也称为K均值聚类算法，是一种无监督机器学习方法，用于将数据集分成K个簇群。
该算法的核心思想是将数据点划分为不同的簇群，使得同一簇群内的点相似度尽可能高，而不同簇群之间的相似度尽可能低。
该算法可用于许多领域，如计算机视觉、医学图像处理、自然语言处理等。
1.工作原理K-Means算法的工作原理如下：1. 首先，从数据集中随机选择K个点作为初始簇群的中心点。
2. 接下来，计算每个数据点与K个中心点之间的距离，并将它们归入距离最近的簇群中。
这个过程称为“分配”。
3. 在所有数据点都被分配到簇群后，重新计算每个簇群的中心点，即将簇群中所有数据点的坐标取平均值得出新的中心点。
这个过程称为“更新”。
4. 重复执行2-3步骤，直到簇群不再发生变化或达到最大迭代次数为止。
2.优缺点1. 简单易懂，实现方便。
2. 可用于处理大量数据集。
1. 随机初始化可能导致算法无法找到全局最优解。
2. 结果受到初始中心点的影响。
3. 对离群值敏感，可能导致簇群数量不足或簇群数量偏多。
4. 对于非球形簇群，K-Means算法的效果可能较差。
3.应用场景K-Means算法可以广泛应用于许多领域，如：1. 机器学习和数据挖掘：用于聚类分析和领域分类。
2. 计算机视觉：用于图像分割和物体识别。
3. 自然语言处理：用于文本聚类和词向量空间的子空间聚类。
4. 财务分析：用于分析财务数据，比如信用评分和市场分析。
5. 医学图像处理：用于医学影像分析和分类。
总之，K-Means算法是一种简单有效的聚类算法，可用于处理大量数据集、连续型数据、图像和文本等多种形式数据。
但在实际应用中，需要根据具体情况选择合适的簇群数量和初始中心点，在保证算法正确性和有效性的同时，减少误差和提高效率。
k-means聚类的基本步骤
k-means聚类的基本步骤
嘿，朋友们！今天咱来聊聊 k-means 聚类的那些事儿哈。
你想啊，这k-means 聚类就好比是给一堆乱七八糟的东西分类整理。
首先呢，咱得确定要分成几类，这就好比你要决定把你的玩具分成几
堆一样。
这可不是随便定的哦，得根据实际情况好好琢磨琢磨。
然后呢，就像给每个东西找个家一样，随机选几个点作为初始的聚
类中心。
这就好像你先随便找几个地方放那几堆玩具。
接下来，就是把每个数据点都归到离它最近的那个聚类中心所属的
类里。
这就好像你把每个玩具都放到离它最近的那堆里去。
哎呀，是
不是挺形象的呀！
这还不算完呢，等都分好类了，还得重新计算每个类的中心。
这就
好比你重新调整一下那几堆玩具的位置，让它们更整齐。
然后再重复上面的过程，一直到这些聚类中心不再变化啦。
这就像
你反复调整玩具堆，直到你觉得满意为止。
你说这 k-means 聚类是不是挺有趣的呀？就像是在玩一个整理的游戏。
而且它用处可大了去了呢！比如说在数据分析里，能帮我们发现
一些隐藏的模式和规律。
你想想看，如果没有k-means 聚类，那面对一大堆杂乱无章的数据，我们得多头疼呀！但有了它，就好像有了一双神奇的手，能把这些乱
麻一样的数据整理得井井有条。
所以说呀，学会 k-means 聚类的基本步骤可太重要啦！咱可不能小
瞧了它，得好好研究研究，把它用在该用的地方，让它发挥出最大的
作用呀！这难道不是很有意义的事情吗？。
K-Means解析
Clustering中文翻译作“聚类”，简单地说就是把相似的东西分到一组，同Classification(分类)不同，对于一个classifier，通常需要你告诉它“这个东西被分为某某类”这样一些例子，理想情况下，一个classifier会从它得到的训练集中进行“学习”，从而具备对未知数据进行分类的能力，这种提供训练数据的过程通常叫做supervised learning(监督学习)，而在聚类的时候，我们并不关心某一类是什么，我们需要实现的目标只是把相似的东西聚到一起，因此，一个聚类算法通常只需要知道如何计算相似度就可以开始工作了，因此clustering通常并不需要使用训练数据进行学习，这在Machine Learning中被称作unsupervised learning(无监督学习)。
举一个简单的例子：现在有一群小学生，你要把他们分成几组，让组内的成员之间尽量相似一些，而组之间则差别大一些。
最后分出怎样的结果，就取决于你对于“相似”的定义了，比如，你决定男生和男生是相似的，女生和女生也是相似的，而男生和女生之间则差别很大”，这样，你实际上是用一个可能取两个值“男”和“女”的离散变量来代表了原来的一个小学生，我们通常把这样的变量叫做“特征”。
实际上，在这种情况下，所有的小学生都被映射到了两个点的其中一个上，已经很自然地形成了两个组，不需要专门再做聚类了。
另一种可能是使用“身高”这个特征。
我在读小学候，每周五在操场开会训话的时候会按照大家住的地方的地域和距离远近来列队，这样结束之后就可以结队回家了。
除了让事物映射到一个单独的特征之外，一种常见的做法是同时提取N种特征，将它们放在一起组成一个N维向量，从而得到一个从原始数据集合到N维向量空间的映射——你总是需要显式地或者隐式地完成这样一个过程，因为许多机器学习的算法都需要工作在一个向量空间中。
那么让我们再回到clustering的问题上，暂且抛开原始数据是什么形式，假设我们已经将其映射到了一个欧几里德空间上，为了方便展示，就使用二维空间吧，如下图所示：从数据点的大致形状可以看出它们大致聚为三个cluster，其中两个紧凑一些，剩下那个松散一些。
面向海量数据的K_means聚类优化算法
面向海量数据的K-means 聚类优化算法冀素琴，石洪波JI Suqin, SHI Hongbo山西财经大学信息管理学院，太原030031School of Information Management, Shanxi University of Finance & Economics, Taiyuan 030031, ChinaJI Suqin, SHI Hongbo. O ptimized K-means clustering algorithm for massive data. C omputer Engineering and Applications, 2014, 50（14）：143-147.Abstract：In order to solve the problem of the clustering on massive data under the framework of a centralized system, an optimized algorithm to K-means clustering based on MapReduce is proposed. By using MapReduce parallel programming framework and importing Canopy clustering, this algorithm optimizes initial clustering center, improves communication mode and calculation mode in iteration. The experimental results show that thi s algorithm can effectively improve the quality of clustering, and can have higher implementation efficiency, its good scalability, thus it fits to clustering analysis on massive data.Key words：massive data; clustering; MapReduce; K-means algorithm; Canopy algorithm摘要：针对集中式系统框架难以进行海量数据聚类分析的问题，提出基于MapReduce 的K-means 聚类优化算法。
k-means算法的并行化
聚类算法的目的就是获得能够反映N维空间中这些样本点的最本质的“类”的性质。这一步没有领域专家的参与,它除了集合知识外不考虑任何的领域知识,不考虑特征变量在其领域中的特定含义,仅仅认为它是特征空间中的一维而己。
Key Words:K-means；Parallel;Clustering;ClusteMining）,又称为数据库中的知识发现（简称KDD）,是从大量的、不完全的、有噪声的、模糊的、随机的数据中,提取隐含在其中未知的、有潜在应用价值的信息或模式的过程。计算机技术的迅猛发展以及网络的普及,使人们有更多机会使用便捷的方法与外界进行信息交流。可是,数据大量的涌入,增加了我们获取有用信息的难度。如何从大量的数据中获取有价值的信息,给数据挖掘系统的实现带来了难题,由于处理这些数据的复杂度很高,系统的计算能力很难达到要求,此时传统的单机服务器所能提供的有限计算资源往往不能满足要求,需要借助分布式计算技术来实现大规模并行计算。聚类是数据挖掘中的一项重要技术,是分析数据并从中发现有用信息的一种有效手段。基于“物以类聚”的思想,它将数据对象分组成为若干各类或簇,使得在同一个簇中的对象之间具有较高的相似度,而不同簇中的对象差别很大,通过聚类,人们能够识别密集和稀疏区域,发现全局的分布模式以及数据属性之间有趣的相互关系。K-means属于聚类分析中一种基本的划分方法,常采用误差平方和准则函数作为聚类准则。所以我们采用基于HADOOP的分布式聚类方法,以提高聚类的执行效率。
kmeans的聚类依据
kmeans的聚类依据摘要：一、kmeans 聚类的基本原理二、kmeans 聚类的应用场景三、kmeans 聚类的优缺点正文：一、kmeans 聚类的基本原理kmeans（K-means）聚类是一种基于距离的聚类方法，其主要思想是将数据集中的点分为K 个簇（cluster），使得每个簇的内部点之间的距离尽可能小，而不同簇之间的点之间的距离尽可能大。
kmeans 聚类的具体步骤如下：1.随机选择K 个数据点作为初始聚类中心。
2.将数据集中的每个点分配到距离它最近的聚类中心所在的簇。
3.根据上一步的结果，更新每个簇的聚类中心。
新的聚类中心是其所在簇内所有点的均值。
4.重复步骤2 和3，直到聚类中心的变化小于某个阈值或达到迭代次数限制。
二、kmeans 聚类的应用场景kmeans 聚类算法广泛应用于各种数据挖掘和分析任务中，例如：1.光伏和风力发电的典型场景分析。
通过kmeans 聚类可以找出光伏和风力发电的典型场景，从而优化能源配置和提高发电效率。
2.客户细分。
在市场营销中，通过分析客户的消费行为、偏好等数据，可以使用kmeans 聚类方法对客户进行细分，以便精准定位目标客户并制定有效的营销策略。
3.文本聚类。
在自然语言处理领域，kmeans 聚类可以用于对文本进行分类，例如新闻分类、情感分析等。
三、kmeans 聚类的优缺点kmeans 聚类算法具有以下优点：1.算法简单易懂，实现起来较为容易。
2.可以处理大规模数据集。
3.对于一些特定的数据分布，kmeans 聚类可以得到较好的结果。
然而，kmeans 聚类也存在一些缺点：1.算法的收敛性无法保证。
虽然迭代过程中聚类中心会不断更新，但无法确保最终结果是最优的。
2.对于某些数据分布，例如圆形分布，kmeans 聚类可能无法得到合理的结果。
3.需要事先指定聚类数量K，这在实际应用中可能较为困难。
kmeans 文本聚类 原理
kmeans 文本聚类原理
K均值（K-means）是一种常用的文本聚类算法，它的原理是基
于样本之间的相似度来将它们分成不同的簇。
在文本聚类中，K均
值算法首先需要将文本表示为特征向量，常用的方法包括词袋模型、TF-IDF权重等。
然后，算法随机初始化K个簇中心，接着将每个样
本分配到最近的簇中心，然后更新每个簇的中心为该簇所有样本的
平均值。
重复这个过程直到簇中心不再发生变化或者达到预定的迭
代次数。
K均值算法的核心思想是最小化簇内样本的方差，最大化簇间
样本的方差，从而实现簇内的相似度高、簇间的相似度低。
这样做
的目的是将相似的文本聚集到一起形成一个簇，并且使得不同簇之
间的文本尽可能地不相似。
需要注意的是，K均值算法对初始簇中心的选择比较敏感，可
能会收敛到局部最优解。
因此，通常会多次运行算法并选择最优的
聚类结果。
此外，K均值算法还需要事先确定簇的个数K，这通常需
要领域知识或者通过一些启发式方法来确定最佳的K值。
总的来说，K均值算法通过不断迭代更新簇中心来实现文本聚
类，其原理简单直观，易于实现。
然而，对初始簇中心的选择和簇个数的确定需要一定的经验和技巧。
k-means算法的并行化解析
题目：一种基于“云”计算平台的并行聚类——K-means算法设计与实现摘要云计算(Cloud Computing)是分布式计算(Distributed Computing)、并行计算(Parallel Computing)和网格计算(Grid Computing)的发展,云计算是一种新兴的分布式并行计算环境或模式,云计算的出现使得数据挖掘技术的网络化和服务化将成为新的趋势。
本文是对并行聚类算法K-means的研究。
首先介绍了K-means算法在单个计算机上的聚类算法的设计思想,其次重点对K-means算法在集群环境下聚类算法的设计思想进行具体阐述。
K-means聚类算法在面对海量数据时,时间和空间的复杂性已成为 K-means聚类算法的瓶颈。
本文在充分研究传统 K-Means聚类算法的基础上, 提出了基于的并行 K-Means聚类算法的设计思想, 给出了其加速比估算公式。
并通过实验证明了该算法的正确性和有效性。
关键字：K-means；并行；聚类；集群环境AbstractCloud Computing, which is a nascent distributed parallel computing environment or pattern, is the development of Distributed Computing, Parallel Computing and Grid Computing. The appearance of Cloud Computing makes the network and the service of the data mining technology become a new trend.The paper is a study of K-means which is among the parallel clustering algorithms. Firstly, it illustrates the design ideology of clustering algorithm of K-means algorithm on the every single computer. Secondly, it mainly elaborates the design ideology of K-means algorithm of clustering algorithms working in the clustering environment. Being confronted with a large quantity of data, the complexity of time and space has been the bottleneck of K-means. Based on the sufficient studies of traditional K-means, the paper puts forward the design ideology on the basis of the parallel K-means clustering algorithms and provides its estimation formula of speed-up ratio. The paper also proves the accuracy and the effectiveness of this algorithm by the means of the experiments.Key Words:K-means；Parallel; Clustering; Cluster environment目录摘要 (I)Abstract (II)目录 (III)1引言 (1)1.1研究意义 (1)1.2并行聚类算法国内外研究现状 (1) (2)2.1 聚类算法简介 (2)2.2 Hadoop平台简介 (3)3．K-means聚类算法分析 (4) (7)4.1 K-means聚类算法并行原理分析 (7)4.2 算法描述 (8) (9) (10)13 参考文献 (I)致谢 (III)1引言1.1研究意义数据挖掘（Data Mining）,又称为数据库中的知识发现（简称KDD）,是从大量的、不完全的、有噪声的、模糊的、随机的数据中,提取隐含在其中未知的、有潜在应用价值的信息或模式的过程。
各种聚类算法介绍及对比
各种聚类算法介绍及对比聚类算法是一种无监督学习的方法，目标是将数据集中的样本分成不同的组或簇，使得同一个簇内的样本相似度高，而不同簇之间的相似度低。
聚类算法主要有层次聚类、K-means、DBSCAN、谱聚类和密度聚类等。
下面将介绍这些聚类算法，并进行一些对比分析。
1. 层次聚类（Hierarchical Clustering）层次聚类算法可分为自上而下的凝聚聚类和自下而上的分裂聚类。
凝聚聚类从所有样本开始，逐步合并相似的样本，形成一个层次树状结构。
分裂聚类从一个单独的样本开始，逐步分裂为更小的簇，形成一个层次树状结构。
层次聚类的优点是可以根据需要选择得到任意数量的簇，但计算复杂度较高。
2. K-meansK-means是一种划分聚类算法，其步骤为：首先随机选择K个簇中心点，然后根据样本与簇中心的距离将样本划分至最近的簇，接着根据划分结果重新计算簇中心，重复上述过程直到算法收敛。
K-means算法简单高效，但对于非球形簇的数据集表现一般。
3. DBSCAN（Density-Based Spatial Clustering of Applications with Noise）DBSCAN是一种基于密度的聚类算法，不需要预先指定簇的数量。
DBSCAN将样本分为核心对象、边界对象和噪声对象，根据样本之间的密度和可达性关系进行聚类。
核心对象周围一定距离内的样本将被划分为同一个簇。
DBSCAN适用于有噪声数据和不规则形状簇的聚类，但对密度差异较大的数据集效果可能较差。
4. 谱聚类（Spectral Clustering）谱聚类算法先通过样本之间的相似度构建相似度矩阵，然后选取相似度矩阵的前k个最大特征值对应的特征向量作为样本的新表示。
接着将新表示的样本集采用K-means等方法进行聚类。
谱聚类算法在处理复杂几何结构、高维数据和大规模数据时表现出色，但需要选择合适的相似度计算方法和簇的数量。
5. 密度聚类（Density-Based Clustering）密度聚类算法通过估计样本的局部密度来发现簇。
k-means介绍
k-means介绍
K-means是一种基于聚类的分析方式，其主要用于在给定数据集中寻找相似的数据。
K-means算法旨在将数据分为不同的簇，以便深入研究每个簇的特点。
举个例子，假设我们有一个文本数据集，其中包括1000条新闻报道。
我们无法人工阅读每一条新闻，但我们可以使用K-means算法将
相似性高的新闻组合在一起，并向用户显示每个簇中的关键字，以便
用户可以快速了解这个主题的描述和内容。
K-means的思想是，以其中一个数据点为中心，选取固定数量的K
个点，将数据点分配到最近的中心点。
将每个中心点与其所分配的数
据点的平均值作为新的中心点。
重复这个步骤，直到达到最终收敛值。
这个算法会将当前的数据分为多个簇，每个簇包含了最近的中心点及
其分配的所有数据点。
K-means算法的优缺点
K-means算法有以下优点：
1.快速：K-means算法速度非常快，易于实现。
2.有效性：效果非常好，可以很好地识别相似的数据点并将它们分配到定义的簇中。
3.适用范围广：K-means适用于几乎所有类型的数据，因此被认为是一种广泛使用的算法。
但是，K-means算法也有一些缺点，如下所示：
1.初始点的选择非常重要：如果选取的K个点不是很好，那么整个算法的结果也不会很准确。
2.对数据密度敏感：K-means对于不同密度的数据不能进行很好地处理。
3.不适用于大型数据集：当数据集很大时，K-means的算法执行效率会非常低，而且可能会产生不可接受的结果。
kMeansClustering
k-Means ClusteringOn this page…Introduction to k-Means Clustering Create Clusters and Determine Separation Determine the Correct Number of Clusters Avoid Local MinimaIntroduction to k-Means Clusteringk-means clustering is a partitioning method. The function kmeans partitions data into k mutuallyexclusive clusters, and returns the index of the cluster to which it has assigned each observation. Unlike hierarchical clustering, k-means clustering operates on actual observations (rather than the larger set of dissimilarity measures), and creates a single level of clusters. The distinctions mean that k-means clustering is often more suitable than hierarchical clustering for large amounts of data.kmeans treats each observation in your data as an object having a location in space. It finds apartition in which objects within each cluster are as close to each other as possible, and as far from objects in other clusters as possible. You can choose from five different distance measures, depending on the kind of data you are clustering.Each cluster in the partition is defined by its member objects and by its centroid, or center. The centroid for each cluster is the point to which the sum of distances from all objects in that clusteris minimized. kmeanscomputes cluster centroids differently for each distance measure, tominimize the sum with respect to the measure that you specify.kmeans uses an iterative algorithm that minimizes the sum of distances from each object to itscluster centroid, over all clusters. This algorithm moves objects between clusters until the sum cannot be decreased further. The result is a set of clusters that are as compact and well-separated as possible. You can control the details of the minimization using several optional inputparameters to kmeans, including ones for the initial values of the cluster centroids, and for themaximum number of iterations.Create Clusters and Determine SeparationThe following example explores possible clustering in four-dimensional data by analyzing the results of partitioning the points into three, four, and five clusters.Note Because each part of this example generates random numbers sequentially, i.e., without setting a new state, you must perform all steps in sequence to duplicate the results shown. If you perform the steps out of sequence, the answers will be essentially the same, but the intermediate results, number of iterations, or ordering of the silhouette plots may differ.First, load some data:rng('default'); % For reproducibility load kmeansdata; size(X) ans =560 4 Even though these data are four-dimensional, and cannot be easily visualized, kmeans enables you to investigate whether a group structure exists in them. Call kmeans with k, the desired number of clusters, equal to 3. For this example, specify the city block distance measure, and usethe default starting method of initializing centroids from randomly selected data points.idx3 = kmeans(X,3,'distance','city');To get an idea of how well-separated the resulting clusters are, you can make a silhouette plotusing the cluster indices output from kmeans. The silhouette plot displays a measure of howclose each point in one cluster is to points in the neighboring clusters. This measure ranges from +1, indicating points that are very distant from neighboring clusters, through 0, indicating points that are not distinctly in one cluster or another, to -1, indicating points that are probably assignedto the wrong cluster. silhouette returns these values in its first output.[silh3,h] = silhouette(X,idx3,'city'); set(get(gca,'Children'),'FaceColor',[.8 .8 1]) xlabel('Silhouette Value') ylabel('Cluster')From the silhouette plot, you can see that most points in the second cluster have a large silhouette value, greater than 0.6, indicating that the cluster is somewhat separated from neighboring clusters. However, the third cluster contains many points with low silhouette values, and the first contains a few points with negative values, indicating that those two clusters are not well separated.Determine the Correct Number of ClustersIncrease the number of clusters to see if kmeans can find a better grouping of the data. This time, use the optional 'display' parameter to print information about each iteration.idx4 = kmeans(X,4, 'dist','city', 'display','iter');iter phasenumsum115602077.4321511778.643131771.14201771.1Best total sum of distances = 1771.1Notice that the total sum of distances decreases at each iteration as kmeans reassigns pointsbetween clusters and recomputes cluster centroids. In this case, the second phase of the algorithm did not make any reassignments, indicating that the first phase reached a minimum after five iterations. In some problems, the first phase might not reach a minimum, but the second phase always will.A silhouette plot for this solution indicates that these four clusters are better separated than the three in the previous solution.[silh4,h] = silhouette(X,idx4,'city'); set(get(gca,'Children'),'FaceColor',[.8 .8 1]) xlabel('Silhouette Value') ylabel('Cluster')A more quantitative way to compare the two solutions is to look at the average silhouette values for the two cases.cluster3 = mean(silh3) cluster4 = mean(silh4)cluster3 = 0.5352cluster4 = 0.6400Finally, try clustering the data using five clusters.idx5 = kmeans(X,5,'dist','city','replicates',5); [silh5,h] = silhouette(X,idx5,'city'); set(get(gca,'Children'),'FaceColor',[.8 .8 1]) xlabel('Silhouette Value') ylabel('Cluster') mean(silh5) ans =0.5266This silhouette plot indicates that this is probably not the right number of clusters, since two of the clusters contain points with mostly low silhouette values. Without some knowledge of howmany clusters are really in the data, it is a good idea to experiment with a range of values for k.Avoid Local MinimaLike many other types of numerical minimizations, the solution that kmeans reaches often depends on the starting points. It is possible for kmeans to reach a local minimum, wherereassigning any one point to a new cluster would increase the total sum of point-to-centroid distances, but where a better solution does exist. However, you can use theoptional 'replicates' parameter to overcome that problem. For four clusters, specify five replicates, and use the 'display' parameter to print out the finalsum of distances for each of the solutions.[idx4,cent4,sumdist] = kmeans(X,4,'dist','city',... 'display','final','replicates',5);Replicate 1, 4 iterations, total sum of distances = 1771.1.Replicate 2, 7 iterations, total sum of distances = 1771.1.Replicate 3, 8 iterations, total sum of distances = 1771.1.Replicate 4, 5 iterations, total sum of distances = 1771.1.Replicate 5, 6 iterations, total sum of distances = 1771.1.Best total sum of distances = 1771.1In this example, kmeans found the same minimum in all five replications. However, even forrelatively simple problems, nonglobal minima do exist. Each of these five replicates began from adifferent randomly selected set of initial centroids, so sometimes kmeans finds more than one local minimum. However, the final solution that kmeans returns is the one with the lowest totalsum of distances, over all replicates.sum(sumdist)ans = 1.7711e+03。
基于MapReduce的并行K-Means聚类
2.MapReduce框架下的并行K-Means算法
K-Meaቤተ መጻሕፍቲ ባይዱs算法：
首先从n个数据对象任意选择 k 个对象作为初始聚类中心；而对于所 剩下其它对象，则根据它们与这些聚类中心的相似度（距离），分别将它 们分配给与其最相似的（聚类中心所代表的）聚类；然 后再计算每个所获 新聚类的聚类中心（该聚类中所有对象的均值）；不断重复这一过程直到 标准测度函数开始收敛为止。一般都采用均方差作为标准测度函数。k个聚 类具有以下特点：各聚类本身尽可能的紧凑，而各聚类之间尽可能的分开。
1.假设所有的objects都可以同时存在于主存中； 2.并行系统中提供了受限制的程序模型并且使用这种限制去进行并行的自动 计算。 两个假设都禁止包含数以万计objects的大型数据集。因此，面向大型数据 集的并行聚类算法应该被设计出来。
1.简介
在许多应用领域，数据聚类得到广泛的重视。例如：数据挖掘，文档检 索，图像分割和模式分类。随着科技的进步，信息用量越来越大，使得对大 规模数据的聚类成为了一种严峻的挑战，为了解决这个问题，许多研究者尝 试着设计更高效的平行聚类算法。 在本文中，我们基于提出了一种并行K-Means聚类算法。 MapReduce是一种简单的但是很强大的并行编程模型。用户只要详细定 义map函数和reduce函数，关于并行计算、处理机器故障、跨机器交流的日 志安排等都是潜在的在大规模集群计算机上执行。
可伸缩性
4.总结
本文对基于云计算平台Hadoop的并行K-Means算法，设计进行了深人的研 究。首先，简要介绍了Hadoop 平台的基本组成，包括HDFS框架 MapReduce各个阶段的工作流程以及结构关系。然后，给出基于Hadoop的 并行k-means算法设计时需要思考的主要问题、算法设计的主要流程以及 方法和策略等。最后，通过在多组不同大小数据集上的实验表明，我们 设计的并行聚类算法PKMeans适合运行于大规模云计算平台，可以有效地 应用于实际中海量数据的分析和挖掘。
k均值聚类（k-meansclustering）
k均值聚类（k-meansclustering）k均值聚类（k-means clustering）算法思想起源于1957年Hugo Steinhaus[1]，1967年由J.MacQueen在[2]第⼀次使⽤的，标准算法是由Stuart Lloyd在1957年第⼀次实现的，并在1982年发布[3]。
简单讲，k-means clustering是⼀个根据数据的特征将数据分类为k组的算法。
k是⼀个正整数。
分组是根据原始数据与聚类中⼼（cluster centroid）的距离的平⽅最⼩来分配到对应的组中。
例⼦：假设我们有4个对象作为训练集，每个对象都有两个属性见下。
可根据x,y坐标将数据表⽰在⼆维坐标系中。
object Atrribute 1 (x):weight indexAttribute 2 (Y):pHMedicine A11Medicine B21Medicine C43Medicine D54表⼀原始数据并且我们知道这些对象可依属性被分为两组（cluster 1和cluster 2）。
问题在于如何确定哪些药属于cluster 1，哪些药属于cluster 2。
k-means clustering实现步骤很简单。
刚开始我们需要为各个聚类中⼼设置初始位置。
我们可以从原始数据中随机取出⼏个对象作为聚类中⼼。
然后k means算法执⾏以下三步直⾄收敛（即每个对象所属的组都不改变）。
1.确定中⼼的坐标2.确定每个对象与每个中⼼的位置3.根据与中⼼位置的距离，每个对象选择距离最近的中⼼归为此组。
图1 k means流程图对于表1中的数据，我们可以得到坐标系中的四个点。
1.初始化中⼼值：我们假设medicine A和medicine B作为聚类中⼼的初值。
⽤c1和c2表⽰中⼼的坐标，c1=(1,1)，c2=(2,1)。
2对象-中⼼距离：利⽤欧式距离（d = sqrt((x1-x2)^2+(y1-y2)^2)）计算每个对象到每个中⼼的距离。
一种基于CUDA的K-Means多级并行优化方法
收稿日期:2020-06-17收修改稿日期=2020-08-10作者简介:方玉玲，女，1990年生，博士，讲师,CCF会员，研究方向为模式识别、并行计 算等;那丽春，女,1967年生，硕士，副教授，研究方向为并行数据库.
小型微型计算系统
Journal of Chinese Computer Systems
2021年7月第7期
Vol. 42 No. 7 2021
一种基于CUDA的K-Means多级并行优化方法
方玉玲，那丽春
（上海立信会计金融学院信息管理学院，上海201209） E-mail: fangyl@ lixin. edu. cn
结果表明，在保证实验结果准确性的情况下，与其它优化并行算法相比，本文方法最高加速比达到了 39.7% ,平均加速比达到
T 22.3%,同时降低了 GPU资源占用率.
关键词:K-Means；并行计算；CUDA；多级并行优化
中图分类号:TP391
文献标识码:A
文章编号:1000-1220（2021）07-1547-07
1引言
作为机器学习中主要算法之一，聚类分析在多个领域得 到充分应用，如数据挖掘，大数据分析，推荐系统等m”.通过 聚类算法不仅能够对用户关注的类别进行区分，还能用于进 行个性化推荐，如淘宝、微博的智能兴趣推荐中最常使用的就 是聚类技术.从使用效果上看，聚类分析也是数据分类的一 种，但与分类技术还存在一定的差别，最大不同之处在于聚类 处理数据的所属类是未知的，它是一个无监督过程⑶.是在 没有相关经验的基础上，对数据进行处理，分析出数据间内在
kmeans算法用法
K-means算法用法1. 介绍K-means算法是一种基于距离度量的聚类算法，它将数据集划分为K个不重叠的簇。
每个簇都由离其质心最近的数据点组成，质心是簇内所有点的均值。
K-means算法是一种迭代算法，通过不断更新簇的质心和重新分配数据点来优化聚类结果。
K-means算法的核心思想是最小化簇内数据点与质心之间的距离平方和，也称为误差平方和（SSE）。
K-means算法的优点是简单、高效，适用于大规模数据集。
但它也有一些限制，例如对初始质心的选择敏感，容易陷入局部最优解。
2. 算法步骤K-means算法的步骤如下：步骤1：选择初始质心随机选择K个数据点作为初始质心，或者使用一些启发式方法来选择初始质心，例如K-means++算法。
步骤2：分配数据点到最近的质心对于每个数据点，计算它与每个质心之间的距离，并将其分配到距离最近的质心所属的簇中。
步骤3：更新质心对于每个簇，计算簇内所有数据点的均值，将均值作为新的质心。
步骤4：重复步骤2和步骤3，直到质心不再改变或达到最大迭代次数。
3. 簇数K的选择在使用K-means算法时，需要事先指定簇的数量K。
选择合适的K值对聚类结果的准确性和可解释性都有重要影响。
3.1 肘部法则肘部法则是一种常用的方法，用于选择K的值。
它基于SSE与K的关系，SSE在K增大时逐渐减小。
当K增大到一定程度时，SSE的下降速度会显著变缓，形成一个类似手肘的曲线。
选择肘部对应的K值作为最佳的簇数。
3.2 轮廓系数轮廓系数是一种衡量聚类结果质量的指标，它同时考虑了簇内紧密度和簇间分离度。
轮廓系数的取值范围在-1到1之间，值越接近1表示聚类结果越好。
4. K-means算法的改进K-means算法有一些改进的版本，用于克服其局限性。
以下是一些常见的改进方法：4.1 K-means++K-means++是一种改进的初始质心选择方法，它通过引入概率来选择初始质心，使得初始质心之间的距离更加均匀，从而提高了算法的收敛速度和聚类质量。
简述k-means聚类算法
简述k-means聚类算法k-means 是一种基于距离的聚类算法，它通常被用于将数据集中的对象分成若干个不同的组（簇），每个簇中的对象彼此相似，而不同簇中的对象则彼此差别较大。
该算法最早由美国数学家 J. MacQueen 在 1967 年提出，被称为“是一种对大规模数据集进行聚类分析的算法”。
K-means 算法的步骤如下：1. 随机选取 k 个中心点（centroid）作为起点。
这些中心点可以是来自于数据集的 k 个随机点，或者是由领域知识人员事先指定的。
2. 对于数据集中的每一个点，计算它和 k 个中心点之间的距离，然后将该点分配给距离最短的中心点（即所属簇）。
3. 对于每个簇，重新计算中心点的位置。
中心点位置是该簇中所有点的平均位置。
4. 重复步骤 2 和 3，直到中心点的位置不再发生变化或者达到了预设的最大迭代次数。
最终，k-means 算法会生成 k 个簇，每个簇中包含了若干个相似的对象。
使用 k-means 算法时需要注意的几点：1. 确定 k 值。
K 值的选择至关重要，因为它直接影响到聚类的效果。
如果 k 值过大，可能会导致某些簇内只有极少数的数据点，甚至完全没有数据点。
如果 k 值过小，则簇之间的差别可能会被忽略，影响聚类的精度。
因此，需要通过试错法和业务需求等多方面考虑，选择一个合适的 k 值。
2. 初始中心点的选取。
在 k-means 算法中，初始中心点的位置对聚类结果有很大的影响。
如果它们被随机选取，可能会导致算法陷入局部最优解。
因此，有些研究者提出了一些改进方法，如 K-means++ 算法等，来优化初始中心点的选取。
3. 处理异常值。
由于 k-means 算法是基于距离的，因此对于离群点（outliers）可能会产生较大的影响。
一种处理方法是将它们剔除或者加权处理。
总的来说，k-means 算法是一种简单而有效的聚类算法，可以应用于许多领域，如图像处理、自然语言处理、数据挖掘等。
Parallel K-Means Clustering Based on MapReduce
Parallel K-Means Clustering Based onMapReduceWeizhong Zhao1,2,Huifang Ma1,2,and Qing He11The Key Laboratory of Intelligent Information Processing,Institute of ComputingTechnology,Chinese Academy of Sciences2Graduate University of Chinese Academy of Sciences{zhaowz,mahf,heq}@Abstract.Data clustering has been received considerable attention inmany applications,such as data mining,document retrieval,image seg-mentation and pattern classification.The enlarging volumes of informa-tion emerging by the progress of technology,makes clustering of verylarge scale of data a challenging task.In order to deal with the problem,many researchers try to design efficient parallel clustering algorithms.In this paper,we propose a parallel k-means clustering algorithm basedon MapReduce,which is a simple yet powerful parallel programmingtechnique.The experimental results demonstrate that the proposed algo-rithm can scale well and efficiently process large datasets on commodityhardware.Keywords:Data mining;Parallel clustering;K-means;Hadoop;MapRe-duce.1IntroductionWith the development of information technology,data volumes processed by many applications will routinely cross the peta-scale threshold,which would in turn increase the computational requirements.Efficient parallel clustering algorithms and implementation techniques are the key to meeting the scalability and performance requirements entailed in such scientific data analyses.So far, several researchers have proposed some parallel clustering algorithms[1,2,3]. All these parallel clustering algorithms have the following drawbacks:a)They assume that all objects can reside in main memory at the same time;b)Their parallel systems have provided restricted programming models and used the restrictions to parallelize the computation automatically.Both assumptions are prohibitive for very large datasets with millions of objects.Therefore,dataset oriented parallel clustering algorithms should be developed.MapReduce[4,5,6,7]is a programming model and an associated implementa-tion for processing and generating large datasets that is amenable to a broad variety of real-world ers specify the computation in terms of a map and a reduce function,and the underlying runtime system automatically parallelizes the computation across large-scale clusters of machines,handles machine fail-ures,and schedules inter-machine communication to make efficient use of theM.G.Jaatun,G.Zhao,and C.Rong(Eds.):CloudCom2009,LNCS5931,pp.674–679,2009.c Springer-Verlag Berlin Heidelberg2009Parallel K-Means Clustering Based on MapReduce675 network and disks.Google and Hadoop both provide MapReduce runtimes with fault tolerance and dynamicflexibility support[8,9].In this paper,we adapt k-means algorithm[10]in MapReduce framework which is implemented by Hadoop to make the clustering method applicable to large scale data.By applying proper<key,value>pairs,the proposed algorithm can be parallel executed effectively.We conduct comprehensive experiments to evaluate the proposed algorithm.The results demonstrate that our algorithm can effectively deal with large scale datasets.The rest of the paper is organized as follows.In Section2,we present our parallel k-means algorithm based on MapReduce framework.Section3shows ex-perimental results and evaluates our parallel algorithm with respect to speedup, scaleup,and sizeup.Finally,we offer our conclusions in Section4.2Parallel K-Means Algorithm Based on MapReduceIn this section we present the main design for Parallel K-Means(PKMeans) based on MapReduce.Firstly,we give a brief overview of the k-means algorithm and analyze the parallel parts and serial parts in the algorithms.Then we explain how the necessary computations can be formalized as map and reduce operations in detail.2.1K-Means AlgorithmK-means algorithm is the most well-known and commonly used clustering method.It takes the input parameter,k,and partitions a set of n objects into k clusters so that the resulting intra-cluster similarity is high whereas the inter-cluster similarity is low.Cluster similarity is measured according to the mean value of the objects in the cluster,which can be regarded as the cluster’s”center of gravity”.The algorithm proceeds as follows:Firstly,it randomly selects k objects from the whole objects which represent initial cluster centers.Each remaining object is assigned to the cluster to which it is the most similar,based on the distance between the object and the cluster center.The new mean for each cluster is then calculated.This process iterates until the criterion function converges.In k-means algorithm,the most intensive calculation to occur is the calcu-lation of distances.In each iteration,it would require a total of(nk)distance computations where n is the number of objects and k is the number of clusters being created.It is obviously that the distance computations between one ob-ject with the centers is irrelevant to the distance computations between other objects with the corresponding centers.Therefore,distance computations be-tween different objects with centers can be parallel executed.In each iteration, the new centers,which are used in the next iteration,should be updated.Hence the iterative procedures must be executed serially.676W.Zhao,H.Ma,and Q.He2.2PKMeans Based on MapReduceAs the analysis above,PKMeans algorithm needs one kind of MapReduce job. The map function performs the procedure of assigning each sample to the closest center while the reduce function performs the procedure of updating the new centers.In order to decrease the cost of network communication,a combiner function is developed to deal with partial combination of the intermediate values with the same key within the same map task.Map-function The input dataset is stored on HDFS[11]as a sequencefile of<key,value>pairs,each of which represents a record in the dataset.The key is the offset in bytes of this record to the start point of the datafile,and the value is a string of the content of this record.The dataset is split and globally broadcast to all mappers.Consequently,the distance computations are parallel executed.For each map task,PKMeans construct a global variant centers which is an array containing the information about centers of the clusters.Given the information,a mapper can compute the closest center point for each sample. The intermediate values are then composed of two parts:the index of the closest center point and the sample information.The pseudocode of map function is shown in Algorithm1.Algorithm1.map(key,value)Input:Global variable centers,the offset key,the sample valueOutput:<key’,value’>pair,where the key’is the index of the closest center point and value’is a string comprise of sample information1.Construct the sample instance from value;2.minDis=Double.MAX V ALUE;3.index=-1;4.For i=0to centers.length dodis=ComputeDist(instance,centers[i]);If dis<minDis{minDis=dis;index=i;}5.End For6.Take index as key’;7.Construct value’as a string comprise of the values of different dimensions;8.output<key ,value >pair;9.EndNote that Step2and Step3initialize the auxiliary variable minDis and index; Step4computes the closest center point from the sample,in which the function ComputeDist(instance,centers[i])returns the distance between instance and the center point centers[i];Step8outputs the intermediate data which is used in the subsequent procedures.Combine-function.After each map task,we apply a combiner to combine the intermediate data of the same map task.Since the intermediate data is stored in local disk of the host,the procedure can not consume the communication cost. In the combine function,we partially sum the values of the points assigned to the same cluster.In order to calculate the mean value of the objects for eachParallel K-Means Clustering Based on MapReduce677 cluster,we should record the number of samples in the same cluster in the same map task.The pseudocode for combine function is shown in Algorithm2.bine(key,V)Input:key is the index of the cluster,V is the list of the samples assigned to the same cluster Output:<key ,value >pair,where the key’is the index of the cluster,value’is a string comprised of sum of the samples in the same cluster and the sample number1.Initialize one array to record the sum of value of each dimensions of the samples contained inthe same cluster,i.e.the samples in the list V;2.Initialize a counter num as0to record the sum of sample number in the same cluster;3.while(V.hasNext()){Construct the sample instance from V.next();Add the values of different dimensions of instance to the arraynum++;4.}5.Take key as key’;6.Construct value’as a string comprised of the sum values of different dimensions and num;7.output<key ,value >pair;8.EndReduce-function.The input of the reduce function is the data obtained from the combine function of each host.As described in the combine function,the data includes partial sum of the samples in the same cluster and the sample number.In reduce function,we can sum all the samples and compute the total number of samples assigned to the same cluster.Therefore,we can get the new centers which are used for next iteration.The pseudocode for reduce function is shown in Algorithm3.Algorithm3.reduce(key,V)Input:key is the index of the cluster,V is the list of the partial sums from different host Output:<key ,value >pair,where the key’is the index of the cluster,value’is a string repre-senting the new center1.Initialize one array record the sum of value of each dimensions of the samples contained in thesame cluster,e.g.the samples in the list V;2.Initialize a counter NUM as0to record the sum of sample number in the same cluster;3.while(V.hasNext()){Construct the sample instance from V.next();Add the values of different dimensions of instance to the arrayNUM+=num;4.}5.Divide the entries of the array by NUM to get the new center’s coordinates;6.Take key as key’;7.Construct value’as a string comprise of the center’s coordinates;8.output<key ,value >pair;9.End3Experimental ResultsIn this section,we evaluate the performance of our proposed algorithm with respect to speedup,scaleup and sizeup[12].Performance experiments were run678W.Zhao,H.Ma,and Q.He(a)Speedup(b)Scaleup(c)SizeupFig.1.Evaluations resultson a cluster of computers,each of which has two2.8GHz cores and4GB of memory.Hadoop version0.17.0and Java1.5.014are used as the MapReduce system for all experiments.To measure the speedup,we kept the dataset constant and increase the num-ber of computers in the system.The perfect parallel algorithm demonstrates lin-ear speedup:a system with m times the number of computers yields a speedup of m.However,linear speedup is difficult to achieve because the communication cost increases with the number of clusters becomes large.We have performed the speedup evaluation on datasets with different sizes and systems.The number of computers varied from1to4.The size of the dataset increases from1GB to8GB.Fig.1.(a)shows the speedup for different datasets.As the result shows,PKMeans has a very good speedup performance. Specifically,as the size of the dataset increases,the speedup performs better. Therefore,PKMeans algorithm can treat large datasets efficiently.Scaleup evaluates the ability of the algorithm to grow both the system and the dataset size.Scaleup is defined as the ability of an m-times larger system to perform an m-times larger job in the same run-time as the original system.To demonstrate how well the PKMeans algorithm handles larger datasets when more computers are available,we have performed scaleup experiments where we have increased the size of the datasets in direct proportion to the number of com-puters in the system.The datasets size of1GB,2GB,3GB and4GB are executed on1,2,3and4computers respectively.Fig.1.(b)shows the performance results of the datasets.Clearly,the PKMeans algorithm scales very well.Sizeup analysis holds the number of computers in the system constant,and grows the size of the datasets by the factor m.Sizeup measures how much longer it takes on a given system,when the dataset size is m-times larger than the original dataset.To measure the performance of sizeup,we havefixed the number of computers to1,2,3,and4respectively.Fig.1.(c)shows the sizeup results on different computers.The graph shows that PKMeans has a very good sizeup performance. 4ConclusionsAs data clustering has attracted a significant amount of research attention, many clustering algorithms have been proposed in the past decades.However,theParallel K-Means Clustering Based on MapReduce679 enlarging data in applications makes clustering of very large scale of data a chal-lenging task.In this paper,we propose a fast parallel k-means clustering algo-rithm based on MapReduce,which has been widely embraced by both academia and industry.We use speedup,scaleup and sizeup to evaluate the performances of our proposed algorithm.The results show that the proposed algorithm can process large datasets on commodity hardware effectively.Acknowledgments.This work is supported by the National Science Foun-dation of China(No.60675010,60933004,60975039),863National High-Tech Program(No.2007AA01Z132),National Basic Research Priorities Programme (No.2007CB311004)and National Science and Technology Support Plan (No.200-6BAC08B06).References1.Rasmussen,E.M.,Willett,P.:Efficiency of Hierarchical Agglomerative ClusteringUsing the ICL Distributed Array Processor.Journal of Documentation45(1),1–24 (1989)2.Li,X.,Fang,Z.:Parallel Clustering Algorithms.Parallel Computing11,275–290(1989)3.Olson, C.F.:Parallel Algorithms for Hierarchical Clustering.Parallel Comput-ing21(8),1313–1325(1995)4.Dean,J.,Ghemawat,S.:MapReduce:Simplified Data Processing on Large Clus-ters.In:Proc.of Operating Systems Design and Implementation,San Francisco, CA,pp.137–150(2004)5.Dean,J.,Ghemawat,S.:MapReduce:Simplified Data Processing on Large Clus-munications of The ACM51(1),107–113(2008)6.Ranger,C.,Raghuraman,R.,Penmetsa,A.,Bradski,G.,Kozyrakis,C.:Evaluat-ing MapReduce for Multi-core and Multiprocessor Systems.In:Proc.of13th Int.Symposium on High-Performance Computer Architecture(HPCA),Phoenix,AZ (2007)mmel,R.:Google’s MapReduce Programming Model-Revisited.Science ofComputer Programming70,1–30(2008)8.Hadoop:Open source implementation of MapReduce,/hadoop/9.Ghemawat,S.,Gobioff,H.,Leung,S.:The Google File System.In:Symposium onOperating Systems Principles,pp.29–43(2003)10.MacQueen,J.:Some Methods for Classification and Analysis of Multivariate Ob-servations.In:Proc.5th Berkeley Symp.Math.Statist,Prob.,vol.1,pp.281–297 (1967)11.Borthakur,D.:The Hadoop Distributed File System:Architecture and Design(2007)12.Xu,X.,Jager,J.,Kriegel,H.P.:A Fast Parallel Clustering Algorithm for LargeSpatial Databases.Data Mining and Knowledge Discovery3,263–290(1999)。
多核CPU下的K-means遥感影像分类并行方法
多核CPU下的K-means遥感影像分类并行方法吴洁璇;陈振杰;张云倩;骈宇哲;周琛【摘要】针对海量遥感影像快速分类的应用需求,提出一种基于K-means算法的遥感影像并行分类方法.该方法结合CPU下进程级与线程级模式的并行特征,设计融合进程级与线程级并行的两阶段数据粒度划分方法和任务调度方法,在保证精度的基础上实现并行加速.利用大数据量的多尺度遥感影像进行实验,结果表明:所提并行方法可大大减少遥感影像的分类时间,取得了良好的加速比(13.83),并可达到负载均衡,从而解决了大区域遥感影像快速分类的问题.【期刊名称】《计算机应用》【年(卷),期】2015(035)005【总页数】6页(P1296-1301)【关键词】K-means算法;并行计算;负载均衡;数据粒度划分;消息传递接口;OpenMP【作者】吴洁璇;陈振杰;张云倩;骈宇哲;周琛【作者单位】江苏省地理信息技术重点实验室(南京大学),南京210023;江苏省地理信息技术重点实验室(南京大学),南京210023;江苏省地理信息技术重点实验室(南京大学),南京210023;江苏省地理信息技术重点实验室(南京大学),南京210023;江苏省地理信息技术重点实验室(南京大学),南京210023【正文语种】中文【中图分类】TP751遥感影像分类是遥感影像处理的重要研究内容[1]。
K-means算法作为典型的非监督分类方法之一，具有操作简单、快速、自动化程度高的特点，在遥感影像分类中应用广泛[2]。
多年来，众多学者针对K-means算法在遥感影像分类中的应用展开了研究：有的学者侧重于考虑遥感影像的地物特征并对K-means算法进行改进[3-5]；有的学者将K-means算法与多种分类算法进行结合，研究多分类器集成技术[6-7]。
然而，随着对地观测技术的不断进步，复杂地理计算和大区域空间分析所涉及的遥感影像数据量日益增加，现有算法的串行模式和单机的硬件平台，已无法满足大区域、高精度遥感影像快速处理的需求[8]。
k-means算法的的基本原理 -回复
k-means算法的的基本原理-回复K-means算法是一种常用的聚类算法，其基本原理是将n个样本数根据特征相似性划分为k个簇（cluster），使得簇内的样本相似度较高，而簇间的相似度较低。
在这篇文章中，我们将详细介绍K-means算法的基本原理，并逐步回答相关问题。
一、什么是K-means算法？K-means是一种无监督学习算法，主要用于对数据进行聚类分析。
它通过将数据分为若干个簇，使得同一簇内的数据之间的相似度最大化，而不同簇之间的相似度最小化。
二、K-means算法的步骤是什么？1. 初始化：随机选择k个样本作为初始质心（centroid）。
2. 分配：将剩下的样本逐个与质心进行比较，将其分配给与之最近的质心所属的簇。
3. 更新质心：对每个簇，计算其中样本的平均值，将该平均值作为新的质心。
4. 重复2和3步骤，直到簇的分配不再发生变化或达到最大迭代次数。
三、K-means算法的优缺点是什么？K-means算法有以下优点：1. 实现简单：K-means算法的步骤清晰明了，容易理解和实现。
2. 对大型数据集有良好的可伸缩性：K-means算法适用于大型数据集和高维数据。
3. 可解释性强：K-means的结果对于解释聚类效果是非常直观的。
但是K-means算法也有一些缺点：1. 对初始质心敏感：初始质心的选择对最终的聚类结果有影响。
因此，多次随机初始化并选择最优结果是常见的策略。
2. 对于非球形簇效果不佳：K-means算法将簇视为球形，对于非球形簇效果相对较差。
3. 容易陷入局部最优解：K-means算法可能陷入局部最优解，得到不太理想的聚类结果。
四、如何选择合适的K值？选择合适的K值是至关重要的，不同的K值可能会导致完全不同的聚类结果。
常用的选择方法有下面两种：1. 经验判断：根据经验知识或者对数据集的初步了解，选择一个可能合适的K值作为初始值。
2. Elbow方法：计算不同K值下的聚类结果的平均损失函数（如误差平方和），并绘制成K值和损失函数值之间的关系图。
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Parallel K-Means Clustering Based onMapReduceWeizhong Zhao1,2,Huifang Ma1,2,and Qing He11The Key Laboratory of Intelligent Information Processing,Institute of ComputingTechnology,Chinese Academy of Sciences2Graduate University of Chinese Academy of Sciences{zhaowz,mahf,heq}@Abstract.Data clustering has been received considerable attention inmany applications,such as data mining,document retrieval,image seg-mentation and pattern classification.The enlarging volumes of informa-tion emerging by the progress of technology,makes clustering of verylarge scale of data a challenging task.In order to deal with the problem,many researchers try to design efficient parallel clustering algorithms.In this paper,we propose a parallel k-means clustering algorithm basedon MapReduce,which is a simple yet powerful parallel programmingtechnique.The experimental results demonstrate that the proposed algo-rithm can scale well and efficiently process large datasets on commodityhardware.Keywords:Data mining;Parallel clustering;K-means;Hadoop;MapRe-duce.1IntroductionWith the development of information technology,data volumes processed by many applications will routinely cross the peta-scale threshold,which would in turn increase the computational requirements.Efficient parallel clustering algorithms and implementation techniques are the key to meeting the scalability and performance requirements entailed in such scientific data analyses.So far, several researchers have proposed some parallel clustering algorithms[1,2,3]. All these parallel clustering algorithms have the following drawbacks:a)They assume that all objects can reside in main memory at the same time;b)Their parallel systems have provided restricted programming models and used the restrictions to parallelize the computation automatically.Both assumptions are prohibitive for very large datasets with millions of objects.Therefore,dataset oriented parallel clustering algorithms should be developed.MapReduce[4,5,6,7]is a programming model and an associated implementa-tion for processing and generating large datasets that is amenable to a broad variety of real-world ers specify the computation in terms of a map and a reduce function,and the underlying runtime system automatically parallelizes the computation across large-scale clusters of machines,handles machine fail-ures,and schedules inter-machine communication to make efficient use of theM.G.Jaatun,G.Zhao,and C.Rong(Eds.):CloudCom2009,LNCS5931,pp.674–679,2009.c Springer-Verlag Berlin Heidelberg2009Parallel K-Means Clustering Based on MapReduce675 network and disks.Google and Hadoop both provide MapReduce runtimes with fault tolerance and dynamicflexibility support[8,9].In this paper,we adapt k-means algorithm[10]in MapReduce framework which is implemented by Hadoop to make the clustering method applicable to large scale data.By applying proper<key,value>pairs,the proposed algorithm can be parallel executed effectively.We conduct comprehensive experiments to evaluate the proposed algorithm.The results demonstrate that our algorithm can effectively deal with large scale datasets.The rest of the paper is organized as follows.In Section2,we present our parallel k-means algorithm based on MapReduce framework.Section3shows ex-perimental results and evaluates our parallel algorithm with respect to speedup, scaleup,and sizeup.Finally,we offer our conclusions in Section4.2Parallel K-Means Algorithm Based on MapReduceIn this section we present the main design for Parallel K-Means(PKMeans) based on MapReduce.Firstly,we give a brief overview of the k-means algorithm and analyze the parallel parts and serial parts in the algorithms.Then we explain how the necessary computations can be formalized as map and reduce operations in detail.2.1K-Means AlgorithmK-means algorithm is the most well-known and commonly used clustering method.It takes the input parameter,k,and partitions a set of n objects into k clusters so that the resulting intra-cluster similarity is high whereas the inter-cluster similarity is low.Cluster similarity is measured according to the mean value of the objects in the cluster,which can be regarded as the cluster’s”center of gravity”.The algorithm proceeds as follows:Firstly,it randomly selects k objects from the whole objects which represent initial cluster centers.Each remaining object is assigned to the cluster to which it is the most similar,based on the distance between the object and the cluster center.The new mean for each cluster is then calculated.This process iterates until the criterion function converges.In k-means algorithm,the most intensive calculation to occur is the calcu-lation of distances.In each iteration,it would require a total of(nk)distance computations where n is the number of objects and k is the number of clusters being created.It is obviously that the distance computations between one ob-ject with the centers is irrelevant to the distance computations between other objects with the corresponding centers.Therefore,distance computations be-tween different objects with centers can be parallel executed.In each iteration, the new centers,which are used in the next iteration,should be updated.Hence the iterative procedures must be executed serially.676W.Zhao,H.Ma,and Q.He2.2PKMeans Based on MapReduceAs the analysis above,PKMeans algorithm needs one kind of MapReduce job. The map function performs the procedure of assigning each sample to the closest center while the reduce function performs the procedure of updating the new centers.In order to decrease the cost of network communication,a combiner function is developed to deal with partial combination of the intermediate values with the same key within the same map task.Map-function The input dataset is stored on HDFS[11]as a sequencefile of<key,value>pairs,each of which represents a record in the dataset.The key is the offset in bytes of this record to the start point of the datafile,and the value is a string of the content of this record.The dataset is split and globally broadcast to all mappers.Consequently,the distance computations are parallel executed.For each map task,PKMeans construct a global variant centers which is an array containing the information about centers of the clusters.Given the information,a mapper can compute the closest center point for each sample. The intermediate values are then composed of two parts:the index of the closest center point and the sample information.The pseudocode of map function is shown in Algorithm1.Algorithm1.map(key,value)Input:Global variable centers,the offset key,the sample valueOutput:<key’,value’>pair,where the key’is the index of the closest center point and value’is a string comprise of sample information1.Construct the sample instance from value;2.minDis=Double.MAX V ALUE;3.index=-1;4.For i=0to centers.length dodis=ComputeDist(instance,centers[i]);If dis<minDis{minDis=dis;index=i;}5.End For6.Take index as key’;7.Construct value’as a string comprise of the values of different dimensions;8.output<key ,value >pair;9.EndNote that Step2and Step3initialize the auxiliary variable minDis and index; Step4computes the closest center point from the sample,in which the function ComputeDist(instance,centers[i])returns the distance between instance and the center point centers[i];Step8outputs the intermediate data which is used in the subsequent procedures.Combine-function.After each map task,we apply a combiner to combine the intermediate data of the same map task.Since the intermediate data is stored in local disk of the host,the procedure can not consume the communication cost. In the combine function,we partially sum the values of the points assigned to the same cluster.In order to calculate the mean value of the objects for eachParallel K-Means Clustering Based on MapReduce677 cluster,we should record the number of samples in the same cluster in the same map task.The pseudocode for combine function is shown in Algorithm2.bine(key,V)Input:key is the index of the cluster,V is the list of the samples assigned to the same cluster Output:<key ,value >pair,where the key’is the index of the cluster,value’is a string comprised of sum of the samples in the same cluster and the sample number1.Initialize one array to record the sum of value of each dimensions of the samples contained inthe same cluster,i.e.the samples in the list V;2.Initialize a counter num as0to record the sum of sample number in the same cluster;3.while(V.hasNext()){Construct the sample instance from V.next();Add the values of different dimensions of instance to the arraynum++;4.}5.Take key as key’;6.Construct value’as a string comprised of the sum values of different dimensions and num;7.output<key ,value >pair;8.EndReduce-function.The input of the reduce function is the data obtained from the combine function of each host.As described in the combine function,the data includes partial sum of the samples in the same cluster and the sample number.In reduce function,we can sum all the samples and compute the total number of samples assigned to the same cluster.Therefore,we can get the new centers which are used for next iteration.The pseudocode for reduce function is shown in Algorithm3.Algorithm3.reduce(key,V)Input:key is the index of the cluster,V is the list of the partial sums from different host Output:<key ,value >pair,where the key’is the index of the cluster,value’is a string repre-senting the new center1.Initialize one array record the sum of value of each dimensions of the samples contained in thesame cluster,e.g.the samples in the list V;2.Initialize a counter NUM as0to record the sum of sample number in the same cluster;3.while(V.hasNext()){Construct the sample instance from V.next();Add the values of different dimensions of instance to the arrayNUM+=num;4.}5.Divide the entries of the array by NUM to get the new center’s coordinates;6.Take key as key’;7.Construct value’as a string comprise of the center’s coordinates;8.output<key ,value >pair;9.End3Experimental ResultsIn this section,we evaluate the performance of our proposed algorithm with respect to speedup,scaleup and sizeup[12].Performance experiments were run678W.Zhao,H.Ma,and Q.He(a)Speedup(b)Scaleup(c)SizeupFig.1.Evaluations resultson a cluster of computers,each of which has two2.8GHz cores and4GB of memory.Hadoop version0.17.0and Java1.5.014are used as the MapReduce system for all experiments.To measure the speedup,we kept the dataset constant and increase the num-ber of computers in the system.The perfect parallel algorithm demonstrates lin-ear speedup:a system with m times the number of computers yields a speedup of m.However,linear speedup is difficult to achieve because the communication cost increases with the number of clusters becomes large.We have performed the speedup evaluation on datasets with different sizes and systems.The number of computers varied from1to4.The size of the dataset increases from1GB to8GB.Fig.1.(a)shows the speedup for different datasets.As the result shows,PKMeans has a very good speedup performance. Specifically,as the size of the dataset increases,the speedup performs better. Therefore,PKMeans algorithm can treat large datasets efficiently.Scaleup evaluates the ability of the algorithm to grow both the system and the dataset size.Scaleup is defined as the ability of an m-times larger system to perform an m-times larger job in the same run-time as the original system.To demonstrate how well the PKMeans algorithm handles larger datasets when more computers are available,we have performed scaleup experiments where we have increased the size of the datasets in direct proportion to the number of com-puters in the system.The datasets size of1GB,2GB,3GB and4GB are executed on1,2,3and4computers respectively.Fig.1.(b)shows the performance results of the datasets.Clearly,the PKMeans algorithm scales very well.Sizeup analysis holds the number of computers in the system constant,and grows the size of the datasets by the factor m.Sizeup measures how much longer it takes on a given system,when the dataset size is m-times larger than the original dataset.To measure the performance of sizeup,we havefixed the number of computers to1,2,3,and4respectively.Fig.1.(c)shows the sizeup results on different computers.The graph shows that PKMeans has a very good sizeup performance. 4ConclusionsAs data clustering has attracted a significant amount of research attention, many clustering algorithms have been proposed in the past decades.However,theParallel K-Means Clustering Based on MapReduce679 enlarging data in applications makes clustering of very large scale of data a chal-lenging task.In this paper,we propose a fast parallel k-means clustering algo-rithm based on MapReduce,which has been widely embraced by both academia and industry.We use speedup,scaleup and sizeup to evaluate the performances of our proposed algorithm.The results show that the proposed algorithm can process large datasets on commodity hardware effectively.Acknowledgments.This work is supported by the National Science Foun-dation of China(No.60675010,60933004,60975039),863National High-Tech Program(No.2007AA01Z132),National Basic Research Priorities Programme (No.2007CB311004)and National Science and Technology Support Plan (No.200-6BAC08B06).References1.Rasmussen,E.M.,Willett,P.:Efficiency of Hierarchical Agglomerative ClusteringUsing the ICL Distributed Array Processor.Journal of Documentation45(1),1–24 (1989)2.Li,X.,Fang,Z.:Parallel Clustering Algorithms.Parallel Computing11,275–290(1989)3.Olson, C.F.:Parallel Algorithms for Hierarchical Clustering.Parallel Comput-ing21(8),1313–1325(1995)4.Dean,J.,Ghemawat,S.:MapReduce:Simplified Data Processing on Large Clus-ters.In:Proc.of Operating Systems Design and Implementation,San Francisco, CA,pp.137–150(2004)5.Dean,J.,Ghemawat,S.:MapReduce:Simplified Data Processing on Large Clus-munications of The ACM51(1),107–113(2008)6.Ranger,C.,Raghuraman,R.,Penmetsa,A.,Bradski,G.,Kozyrakis,C.:Evaluat-ing MapReduce for Multi-core and Multiprocessor Systems.In:Proc.of13th Int.Symposium on High-Performance Computer Architecture(HPCA),Phoenix,AZ (2007)mmel,R.:Google’s MapReduce Programming Model-Revisited.Science ofComputer Programming70,1–30(2008)8.Hadoop:Open source implementation of MapReduce,/hadoop/9.Ghemawat,S.,Gobioff,H.,Leung,S.:The Google File System.In:Symposium onOperating Systems Principles,pp.29–43(2003)10.MacQueen,J.:Some Methods for Classification and Analysis of Multivariate Ob-servations.In:Proc.5th Berkeley Symp.Math.Statist,Prob.,vol.1,pp.281–297 (1967)11.Borthakur,D.:The Hadoop Distributed File System:Architecture and Design(2007)12.Xu,X.,Jager,J.,Kriegel,H.P.:A Fast Parallel Clustering Algorithm for LargeSpatial Databases.Data Mining and Knowledge Discovery3,263–290(1999)。