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清华大学软件学院课件_工业大数据及其软件_外部

URL服务器网页爬虫网页索引
一次写、多次读
锚索引服务网页库大量网页数据，价值密度很低，数据桶
词汇
URL解析
成本十分敏感
链接
文档索引
数据排序
网络用户
网页排序
Google的DIY硬件平台
10 来源：Mass Data Processing Technology on Large Scale Clusters
接收方
满足了大量设备监测对海量数据传输的可靠性要求
3 海量数据存储与处理
基于自由表的大规模状态监测数据存储框架，达到每秒数十万数据点的实时写入
泵送设备
温度压力振动功率
港口机械
电压速度位置
路面机械
里程转速电流
挖掘机械
油位倾角 …
结构化存储
工况
自由表存储
时标
时间
设备
。。。。。。
效率：2CPU节点可以管理超过9万个用户
Google的贡献
颠覆
One size fits all 的惯性思维
IBM、Oracle、Microsoft产业巨头的垄断
贡献
引爆了开源大数据系统的野蛮生长
Hadoop生态系统
You Say, “tomato…”
Google calls it: MapReduce GFS Bigtable Chubby Hadoop equivalent: Hadoop HDFS HBase Zookeeper
混凝土输送缸
油缸密封套腐蚀
内壁刮花
密封环损坏
阀块受损
着眼于求解眼前的问题，还需寻找深层次原因
应用实践3——故障分析新手段

华为 DCG5000C7100-A Data Center Network Manager 7.0.

white paper1CAPACITY PLANNING IS ONE OF THE MOST DIFFICULT ASPECTS OF BUILDING A DATA CENTER TODAY, given the complexity andnumber of variables to consider. Proactive capacity management ensures optimal availability of fourcritical data center resources: rack space, power, cooling and network connectivity. All four of thesemust be in balance for the data center to function most efficiently in terms of operations, resourcesand associated costs. Putting in place a holistic capacity plan prior to building a data center is a bestpractice that goes far to ensure optimal operations.Unfortunately, once the data center is in operation, it is all too common for it to fall out of balanceover time due to organic growth and ad hoc decisions on factors like power, cooling or networkmanagement, or equipment selection and placement. The result is inefficiency and in the worst-case scenario, data center downtime. For example, carrying out asset moves, adds and changes(MACs) without full insight into the impact of asset power consumption, heat dissipation and networkconnectivity changes can create an imbalance that can seriously compromise the data center’soverall resilience and, in turn, its stability and uptime.Once the data center becomes “fragmented” and diverges from the plan set out during the initialbuild, power and cooling are no longer used efficiently, and space and connectivity often becomelimited. The inability to utilize one or more of these resources due to the lack of other resourcesleads to stranded capacity, which ultimately equates with stranded capital, and the prematuredeath of the data center.IT and facility managers face a host of infrastructure challenges, with capacity issues at the topof the list:n Stranded/lost capacity/fragmentationn Running out of data center resourcesn Finding optimal space for critical business assetsnRequirement for CapEx spending to address capacity issuesCapacity Management via DCIM:Real-Time Data Center Intelligence Pays Offof the problem. Symptoms of stranded capacity include hot spots (where there is not enough cooling) or difficulty in deploying equipment due to insufficient rack space. Another possible outcome is that data center specifications indicate there should be more than enough power capacity, but somehow there isn’t. Stranded data center capacity is often not identified until resources are running low and issues have cropped up, but by then, the problem can be serious.Without data center infrastructure management (DCIM) solutions, many data center managers cannot access data center intelligence to understand and improve capacity utilization, to determine if there is stranded capacity (and then free it up), or to proactively provision capacity when adding floor space or building a new data center. DCIM Helps Optimize Capacity Utilization And Reduce Fragmentation Leveraging real-time infrastructure data and analytics provided by DCIM software helps maximize capacity utili-zation (whether for a greenfield or existing data center) and reduce fragmentation, saving the cost of retrofitting a data center or building a new one. Automating data collection via sensors and instrumentation throughout the data center generates positive return on investment (ROI) when combined with DCIM software to yield insightsfor better decision making.The power of DCIM is that it gives data center managers the visibility and intelligence they need to address challenges like fragmentation and stranded capacity. After all, you can’t solve problems you don’t know about.RACK SPACE,POWER, COOLINGAND NETWORKCONNECTIVITYMUST BE INBALANCE FORTHE DATACENTER TOFUNCTION ATPEAK EFFICIENCY CAPACITY MANAGEMENT: KEEPING FOUR AREAS IN BALANCE Data center capacity management demands optimizing four areas at the same time:n RACK SPACE. “When rack space runs short, it tends to be obvious,” says Alexandra Bannerman, Intel-ligent Management Systems solutions manager for Panduit Corp., a provider of data center infrastruc-ture management solutions. Bannerman continues, “It is a primary operational problem. Data center managers need to deploy assets but find that there just isn’t space for them, due to a lack of forward planning. It could be the case that the data center is completely full, but more often than not there is available rack space that can’t be used because of previous asset placements and their impact on the surroundings, and it’s difficult to move these assets once they’re running.” Being able to plan right from the start and use the space effectively is the optimal strategy.n NETWORK CONNECTIVITY. “This area is much trickier,” says Bannerman. “Many, if not most, datacenter managers do not track connectivity in detail, so it’s often unknown to them exactly which ports are taken up, and by what. Many ignore connectivity, when in fact it is just as important a consider-ation for overall efficiency as space, power and cooling,” she says. “It is frustrating to be in the process of deploying assets only to discover they can’t be placed in the available space because there isno network connectivity,” Bannerman adds. The answer is to use DCIM solutions that monitor port connections and connected devices.n POWER. Energy costs are continually rising. “Without granular and holistic visibility into data centerpower consumption, it is very difficult to ensure efficient energy utilization or to meet corporateenergy-reduction goals,” says Bannerman. A DCIM solution that provides detailed granular and holistic information regarding power usage is important.n COOLING. “If you can monitor exactly where things are hot and where they’re cold, you can workproductively to utilize the available cooling more efficiently, which means not having to overcool the data center,” says Bannerman. She adds, “Over time, the best way to manage cooling and avoid hot spots is to extract information regarding the data center infrastructure from DCIM software, andfeed it into a computational fluid dynamics [CFD] modeling software tool to get an understanding of thermal characteristics, and how to improve them.” CFD is a numerical technique that provides insight into thermal and airflow behavior by modeling what-if scenarios to optimize a data center layout and address hot spots.Assessing and monitoring each zone of the data center independently gives you the detailed insight to reveal and address infrastructure challenges, including optimizing and reclaiming capacity; managing your space and network; monitoring and managing cooling; power and energy monitoring and efficiency; accurate and detailed reporting; and interdepartmental visibility.Panduit SmartZone ™ DCIM Software Solutions: Enabling “Best Fits” Panduit SmartZone ™ DCIM Software Solutions help data center managers determine the ”Best Fits” for capacity in their facilities. As a data center manager, you need to understand where the best places are to deploy assets, answering questions such as “Where do I have sufficient power, space, cooling and connectivity to place these assets?” “Best Fit” automates the provisioning process by determining the optimal placement location based on available resources.SmartZone ™ DCIM Software leverages the automatically updated database of space, connectivity, power and environmental data provided by the DCIM software modules and hardware to aid in comprehensive capacity management. Users can easily identify capacity that can’t be used by IT loads due to a lack of one or more of the resources related to floor and rack space, power, power distribution, network, cooling and cooling distribu-tion, and then are able to create work orders to reclaim the stranded capacity. In addition, the system identifies where contiguous cabinet space exists for placement of assets that need to be grouped in the same physical location—eliminating the need for physical inspection of cabinets.“Determining ‘Best Fits’ speeds the asset deployment decision-making process and helps prevent stranded capacity, giving multiple placement options,” says Khaled Nassoura, director of Intelligent Management Systems, Panduit Corp. “The Best Fits Connectivity View also prevents patching errors and makes deployment clearer and even actionable by non-IT workers,” adds Nassoura.When building a new data center, understanding capacity (and provisioning resources) are fairly straightforward tasks. “You look at the total capacity that is supplied into the data center across the four key resources of rack space, power, cooling and network connectivity,” says Nassoura. “This is what you design on day one. For themost part, that total supply is static.”But—much more commonly—for companies that may not have started with proactive capacity planning, the mission is to identify and free up stranded resources. First, you determine the total capacity per thedesign documents and then subtract out the used capacity. This is very difficult without using a DCIM tool like SmartZone ™ DCIM Software. “Based on data collected within SmartZone™ Solutions, we can come up with the recommendations for how the data center can be reconfigured to maximize capacity utilization.”Nassoura cautions that reconfiguration is not necessarily an easy exercise as there is some manual effort required to move things around to remove the capacity blockages. “The important thing is to document resource utilization and track it in a DCIM application like SmartZone™ DCIM Software,” he says.With SmartZone ™ Solutions in place, “when you make a change and introduce new systems, our DCIM Software is intelligent enough through features like auto-discover to look for resources and find the optimum intersection between them to be able to make the best deployment decision,” says Nassoura.Proven Benefits of Reclaiming Stranded CapacityReclaiming stranded capacity allows you to reduce OpEx and/or prevent CapEx spending on new facilities by extending the life of your data center. Reclaiming capacity gives you the flexibility of further equipment deploy-ments, increased loading, reduced power consumption and increased cooling efficiency. Companies of different sizes across a variety of industries have realized the benefits of proactive capacity management.PROACTIVE CAPACITY PLANNING HELPS ZEN INTERNET REDUCE ENERGY CONSUMPTIONFor example, Zen Internet of Rochdale, United Kingdom, implemented Panduit SmartZone ™ DCIM Solutions to realize energy and operational efficiencies in its new, state-of-the-art data center as well as its existing data centers. Zen Internet needed proactive capacity management to control costs by ensuring the most efficient use of power and cooling to support the ever-increasing demands of its cloud and hosting services.“WHEN YOU MAKE A CHANGE AND INTRODUCE NEW SYSTEMS, OUR SOFTWARE IS INTELLIGENT ENOUGH THROUGH FEATURES LIKE AUTO-DISCOVER TO LOOK FOR RESOURCES AND FIND THE OPTIMUM INTER-SECTION BETWEEN THEM TO BE ABLE TO MAKE THE BEST DEPLOYMENTDECISION.”—Khaled Nassoura,Director of IntelligentManagement Systems, Panduit Corp.Visit www.panduit. com/dcimfor more information. gent hardware provided granular data regarding a broad range of metrics from power usage, humidity, tempera-ture and emissions, to leak detection, security and rack-level assets.The information gathered by the SmartZone™ Solutions in Zen Internet’s racks and cabinets enabled the company to measure and control power usage and infrastructure efficiency. Zen Internet was then able to see its capacity, power, environmental and connectivity status for each data center, allowing more informed deci-sion making regarding energy/power, cooling and capacity planning.Deploying SmartZone™ power and environmental monitoring hardware within its racks enabled Zen Internet to monitor capacity and determine whether it was under- or overprovisioning power and space for its customers. Leveraging SmartZone™ Sensors to monitor temperature and humidity levels allowed Zen Internet to take measurements and trending information to identify opportunities for cooling cost reductions, and accurately advise new co-location customers coming into the data center.Zen Internet achieved these hard benefits as a result of the deployment:n Increased energy efficiency across Zen Internet’s data centers, achieving a PUE rating of 1.6 (with 1 being the optimum)n Exceeded annual 5% energy-reduction target goal by achieving an 8% reduction in energy consumption and carbon emissionsn Deployed additional SmartZone™ Power Distribution Units (PDUs), Gateways and Sensors on a modular basis, resulting in improved capacity management, cost savings and improved energy efficiencyThe Panduit SmartZone™ DCIM Solutions helped address the fast-growing company’s power and energy usage challenges, capacity constraints and environmental issues (temperature, humidity and carbon footprint) to provide the tools and information needed to make intelligent decisions for its data operations.FINANCIAL COMPANY MEETS 20% ENERGY-REDUCTION GOALA financial giant was targeting double-digit reductions in energy usage and greenhouse gas emissions from its European-based real estate footprint and data centers. With such aggressive energy-reduction targets, it was important for the financial institution to upgrade its technology, including deployment of Panduit SmartZone™DCIM Solutions for data center visibility and intelligence. At the same time, the company also needed to improve data center operational efficiency as nearly 80% of its servers were operating at 5% of capacity.With integrated threshold monitoring and early warning alerting on power, humidity, temperature and other variables at both cabinet and room level, Panduit was able to help the organization understand the interdepen-dencies between power, rack space and cooling within its data center environment, including past trends, to improve capacity.Most notably, the company has achieved its goal of a nearly 20% reduction in energy consumption across its data center and facilities footprint within a six-year period, four years ahead of schedule. SmartZone™ DCIM Solutions played a key role in its development of industry-leading energy-efficiency programs. Energy-efficiency projects completed in the past year are projected to save more than 50,000 megawatt hours of electricity annually. In less than a decade, the organization has realized an estimated $200 million reduction in energy costs from energy-efficient projects.Capacity Management: Optimize ResourcesOver time, dynamic, virtualized workloads together with the need to provision new applications quickly and the lack of insight into available power and cooling resources have resulted in underutilized capacity, decreased energy efficiency and the need to build new data center space.Panduit SmartZone™ data center infrastructure management solutions provide granular data center information and comprehensive capacity management capabilities in order to help you avoid stranded capacity, through efficient utilization of the resources you build into your data center plans on day one. The end result is improved data center operations and efficiency, reduced operational expenditure and, ultimately, the ability to avoid unnecessary capital expenditure by extending the life of your data center. n。

olap_精品文档

olapOLAP: Understanding Online Analytical ProcessingIntroduction:In today's data-driven world, businesses are constantly seeking ways to efficiently analyze vast amounts of data to gain insights and make informed decisions. OLAP, which stands for Online Analytical Processing, is a powerful technology that enables businesses to perform complex analytical queries on their data. This document aims to provide a comprehensive overview of OLAP, including its definition, architecture, benefits, and real-world applications.Definition of OLAP:OLAP is a data processing technology that allows users to quickly and interactively analyze large volumes of data from multiple perspectives. It was first introduced in the 1960s and has since become a vital component of business intelligence and decision support systems. Unlike traditional database systems that focus on transaction processing (OLTP), OLAP focuses on analytical processing. It facilitates complex dataanalysis through its ability to store, organize, and retrieve data in a multidimensional database structure.Architecture of OLAP:OLAP systems are typically built on a multidimensional database structure, which organizes data into cubes or hypercubes. These cubes consist of dimensions and measures. Dimensions represent the characteristics by which the data is analyzed, such as time, product, location, or customer. Measures, on the other hand, represent the values that are being analyzed, such as sales revenue, profit, or customer count.OLAP systems provide users with a user-friendly interface known as an OLAP cube browser or pivot table. This interface allows users to select dimensions and measures and manipulate them to generate interactive reports. OLAP cubes can support advanced analytical functionalities such as drill-down, roll-up, slice-and-dice, and drill-across, enabling users to analyze data from various perspectives and levels of detail.Benefits of OLAP:OLAP offers numerous benefits to businesses seeking to gain insights from their data:1. Rapid analysis: OLAP enables users to perform complex analytical queries on large data sets in real-time. This allows businesses to quickly identify patterns, trends, and anomalies in their data, leading to faster and more accurate decision-making.2. Interactive reporting: OLAP cubes provide a user-friendly interface that allows users to interactively navigate through data, drill down into specific dimensions, and perform ad-hoc analysis. This empowers users to explore data intuitively and gain deeper insights into their business operations.3. Multidimensional analysis: OLAP's multidimensional database structure allows for sophisticated and flexible analysis. Users can analyze data at different levels of granularity and from multiple dimensions, enabling them to gain a comprehensive understanding of their data.4. Scalability: OLAP systems are designed to handle large volumes of data efficiently. They can process and analyze terabytes of data quickly and accurately, making them suitable for businesses with massive data sets.5. Integration with other systems: OLAP systems can seamlessly integrate with other business intelligence tools, data warehouses, and data sources, allowing users to extract data from multiple systems and perform comprehensive analysis.Applications of OLAP:OLAP has found applications in various industries and business functions:1. Sales and Marketing: OLAP helps businesses analyze sales performance, product profitability, customer segmentation, and market trends. It enables marketing teams to identify high-value customers, analyze campaign effectiveness, and optimize sales strategies.2. Finance and Accounting: OLAP assists finance departments in budget planning, financial reporting, and analysis of financial ratios. It allows for the analysis of revenue, expenses, cash flow, and profitability across different dimensions.3. Supply Chain Management: OLAP helps optimize supply chain operations by analyzing inventory levels, demand patterns, supplier performance, and transportation costs. It enables businesses to identify inefficiencies, mitigate risks, and improve overall supply chain performance.4. Human Resources: OLAP can be utilized to analyze employee performance, workforce planning, and talent management. It enables businesses to identify skill gaps, track training effectiveness, and optimize workforce allocation.Conclusion:OLAP is a powerful technology that empowers businesses to analyze large volumes of data quickly and interactively. Its ability to provide multidimensional analysis, scalable performance, and integration with other systems makes it a valuable tool for business intelligence and decision support. Whether in sales, finance, supply chain, or human resources, OLAP offers businesses the ability to gain insights, optimize operations, and make data-driven decisions.。

【计算机科学】_神经网络学习_期刊发文热词逐年推荐_20140724

科研热词神经网络风险评估集成学习遗传规划遗传算法进化泛函网络语音识别粒子群神经元函数特征项混合基函数权重敏感系数支持向量机小波神经网络(wnn) 小波分析函数逼近入侵检测信息 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
2011年序号 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
2011年科研热词神经网络特征抽取深网入口机器学习预测误差粗逻辑神经网络粗糙集粗糙神经元短期负荷预测模糊神经网络模糊推理数字识别改进bp网络安全态势天气预测 l-m优化法 bp算法推荐指数 4 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1
2012年序号 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
科研热词隐神经元贡献因子蛋白质二级预测结构信息离散hopfield 神经网络特征提取正交基函数正交化样本属性权值与结构确定法权值最优结构径向基网络学习算法奇异值分解多输入噪声数字识别函数逼近人脸识别 rbf神经网络 laguerre正交多项式
推荐指数 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
2013年序号 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
科研热词集成学习连续学习软件可靠性早期预测蚁群算法矩阵伪逆特征选择混沌神经网络泛函神经元概率神经网络时空总和忆阻器布尔函数学习算法奇偶校验问题基函数分类二进神经网络 lvq神经网络 lasso回归方法 lars算法 bp神经网络 bagging

【神经网络与深度学习】ZLIB介绍

【神经⽹络与深度学习】ZLIB介绍zlib类库提供了很多种压缩和解压缩的⽅式，由于时间的关系我只学习⼀下内容，以下是我在实现web 服务器压缩数据⽹页中使⽤到⼀些函数和常⽤数据结构、常量等。

zlib使⽤过程压缩过程：deflateInit() ->deflate() ->deflateEnd(); 对应的解压过程 inflateInit() -> inflate() -> inflateEnd();压缩过程：deflateInit2() ->deflate() ->deflateEnd(); 对应的解压过程 inflateInit2() -> inflate() -> inflateEnd();zlib使⽤的实例请看:注释内容详细（英⽂）（来⾃百度百科，本⼈未加正式，请谅解）常⽤的数据结构typedef struct z_stream_s {z_const Bytef *next_in; //要压缩数据的⾸地址uInt avail_in; //压缩数据的长度uLong total_in; //压缩数据缓冲区的长度Bytef *next_out; //压缩数据保存位置。

uInt avail_out; //存放压缩数据位置的⾸地址uLong total_out; //存放压缩数据位置的⼤⼩z_const char *msg; //存放最近的错误信息，NULL表⽰没有错误struct internal_state FAR *state; /* not visible by applications */alloc_func zalloc; /* used to allocate the internal state */free_func zfree; /* used to free the internal state */voidpf opaque; /* private data object passed to zalloc and zfree */int data_type; // 表⽰数据类型，⽂本或者⼆进制uLong adler; /* adler32 value of the uncompressed data */uLong reserved; /* reserved for future use */} z_stream;对于z_stream我们⼀般使⽤z_stream stream;在deflateInit()或者inflateInit()前设置的参数,初始化参数设置stream.zalloc = Z_NULL;stream.zfree = Z_NULL;stream.opaque = Z_NULL;stream.avail_in = 0;stream.next_in = Z_NULL;在deflate()或inflate前设置的参数，压缩前的参数设置strm.avail_in = in_len;strm.next_in = in;strm.avail_out = out_len;strm.next_out = out;常⽤的常量⽤来设置压缩和解压缩时，结果数据输出的⽅式，具体区别没有看懂的（为了避免误导⼤家，⼤家尽量看/manual.html英⽂帮助吧）#define Z_NO_FLUSH 0 //没有缓存，直接写⼊到结果中#define Z_PARTIAL_FLUSH 1#define Z_SYNC_FLUSH 2#define Z_FULL_FLUSH 3#define Z_FINISH 4 //采⽤此种⽅式，压缩将会变成单步执⾏。

5G赋能试题汇总

单选题(1/3) 本题分数:20 1、以下哪个不是容器技术平台对比维度（）。 A.功能集 B.调度 C.优缺点 D.空间大小标准答案：D 单选题(2/3) 本题分数:20 2、 Swarm与kubernetes、Mesos相比，功能较弱的是（）。 A.Swarm B.kubernetes C.Mesos D.以上都是标准答案：A 单选题(3/3) 本题分数:20 3、在Docker中，不同用户的进程是通过（）namespace隔离开的。 B.pid C.mnt D.uts 标准答案：B 多选题(1/2) 本题分数:20
A.iSCSI存储 B.NFS文件存储 C.资料存储 D.计算能力标准答案：CD
单选题(1/3) 本题分数:20 1、对ESXi的管理有2种方式,使用vsphere client直接管理esxi主机和使用vcenter server来管理，vsphere client和vcenter Server分别访问ESXi的什么服务 A.hostd,vpxa B.hostd,ipfx C.vpfa,hostd D.ipx,vps 标准答案：A 单选题(2/3) 本题分数:20 2、以下哪个不是VMkernel可以提供的核心功能 A.资源调度 B.I/O 堆栈 C.虚拟机发布 D.设备驱动程序标准答案：C 单选题(3/3) 本题分数:20 3、要使端口组到达其他VLAN上的端口组,必须将VLAN ID设置为() A.80 B.4095 C.8080 D.3306 标准答案：B 多选题(1/2) 本题分数:20 1、与其他Hypervisor相比,ESXi具有以下的优点: A.简化部署和配置 B.减少管理开销 C.简化程序的修补和更新 D.提高可靠性和安全性标准答案：ABCD 多选题(2/2) 本题分数:20 2、在整个vCenter Server的体系架构中，包括了哪几部分 A.vCenter Server核心模块 B.数据库服务 C.AD服务 D.管理客户端标准答案：ABCD

Atomic Decomposition by Basis pursuit

SIAM R EVIEWc2001Society for Industrial and Applied Mathematics Vol.43,No.1,pp.129–159Atomic Decomposition by BasisPursuit ∗Scott Shaobing Chen †David L.Donoho ‡Michael A.Saunders §Abstract.The time-frequency and time-scale communities have recently developed a large number ofovercomplete waveform dictionaries—stationary wavelets,wavelet packets,cosine packets,chirplets,and warplets,to name a few.Decomposition into overcomplete systems is not unique,and several methods for decomposition have been proposed,including the method of frames (MOF),matching pursuit (MP),and,for special dictionaries,the best orthogonal basis (BOB).Basis pursuit (BP)is a principle for decomposing a signal into an “optimal”superpo-sition of dictionary elements,where optimal means having the smallest l 1norm of coef-ﬁcients among all such decompositions.We give examples exhibiting several advantages over MOF,MP,and BOB,including better sparsity and superresolution.BP has interest-ing relations to ideas in areas as diverse as ill-posed problems,abstract harmonic analysis,total variation denoising,and multiscale edge denoising.BP in highly overcomplete dictionaries leads to large-scale optimization problems.With signals of length 8192and a wavelet packet dictionary,one gets an equivalent linear program of size 8192by 212,992.Such problems can be attacked successfully only because of recent advances in linear and quadratic programming by interior-point methods.We obtain reasonable success with a primal-dual logarithmic barrier method and conjugate-gradient solver.Key words.overcomplete signal representation,denoising,time-frequency analysis,time-scale anal-ysis, 1norm optimization,matching pursuit,wavelets,wavelet packets,cosine pack-ets,interior-point methods for linear programming,total variation denoising,multiscale edges,MATLAB code AMS subject classiﬁcations.94A12,65K05,65D15,41A45PII.S003614450037906X1.Introduction.Over the last several years,there has been an explosion of in-terest in alternatives to traditional signal representations.Instead of just represent-ing signals as superpositions of sinusoids (the traditional Fourier representation)we now have available alternate dictionaries—collections of parameterized waveforms—of which the wavelets dictionary is only the best known.Wavelets,steerable wavelets,segmented wavelets,Gabor dictionaries,multiscale Gabor dictionaries,wavelet pack-∗Publishedelectronically February 2,2001.This paper originally appeared in SIAM Journal onScientiﬁc Computing ,Volume 20,Number 1,1998,pages 33–61.This research was partially sup-ported by NSF grants DMS-92-09130,DMI-92-04208,and ECS-9707111,by the NASA Astrophysical Data Program,by ONR grant N00014-90-J1242,and by other sponsors./journals/sirev/43-1/37906.html†Renaissance Technologies,600Route 25A,East Setauket,NY 11733(schen@).‡Department of Statistics,Stanford University,Stanford,CA 94305(donoho@).§Department of Management Science and Engineering,Stanford University,Stanford,CA 94305(saunders@).129D o w n l o a d e d 08/09/14 t o 58.19.126.38. R e d i s t r i b u t i o n s u b j e c t t o S I A M l i c e n s e o r c o p y r i g h t ; s e e h t t p ://w w w .s i a m .o r g /j o u r n a l s /o j s a .p h p130S.S.CHEN,D.L.DONOHO,AND M.A.SAUNDERSets,cosine packets,chirplets,warplets,and a wide range of other dictionaries are now available.Each such dictionary D is a collection of waveforms (φγ)γ∈Γ,with γa parameter,and we envision a decomposition of a signal s ass =γ∈Γαγφγ,(1.1)or an approximate decomposition s =m i =1αγi φγi +R (m ),(1.2)where R (m )is a residual.Depending on the dictionary,such a representation de-composes the signal into pure tones (Fourier dictionary),bumps (wavelet dictionary),chirps (chirplet dictionary),etc.Most of the new dictionaries are overcomplete ,either because they start out that way or because we merge complete dictionaries,obtaining a new megadictionary con-sisting of several types of waveforms (e.g.,Fourier and wavelets dictionaries).The decomposition (1.1)is then nonunique,because some elements in the dictionary have representations in terms of other elements.1.1.Goals of Adaptive Representation.Nonuniqueness gives us the possibility of adaptation,i.e.,of choosing from among many representations one that is most suited to our purposes.We are motivated by the aim of achieving simultaneously the following goals .•Sparsity.We should obtain the sparsest possible representation of the object—the one with the fewest signiﬁcant coeﬃcients.•Superresolution.We should obtain a resolution of sparse objects that is much higher resolution than that possible with traditional nonadaptive approaches.An important constraint ,which is perhaps in conﬂict with both the goals,follows.•Speed.It should be possible to obtain a representation in order O (n )or O (n log(n ))time.1.2.Finding a Representation.Several methods have been proposed for obtain-ing signal representations in overcomplete dictionaries.These range from general approaches,like the method of frames (MOF)[9]and the method of matching pursuit (MP)[29],to clever schemes derived for specialized dictionaries,like the method of best orthogonal basis (BOB)[7].These methods are described brieﬂy in section 2.3.In our view,these methods have both advantages and shortcomings.The principal emphasis of the proposers of these methods is on achieving suﬃcient computational speed.While the resulting methods are practical to apply to real data,we show below by computational examples that the methods,either quite generally or in important special cases,lack qualities of sparsity preservation and of stable superresolution.1.3.Basis Pursuit.Basis pursuit (BP)ﬁnds signal representations in overcom-plete dictionaries by convex optimization:it obtains the decomposition that minimizes the 1normof the coeﬃcients occurring in the representation.Because of the nondif-ferentiability of the 1norm,this optimization principle leads to decompositions that can have very diﬀerent properties fromthe MOF—in particular,they can be m uch sparser.Because it is based on global optimization,it can stably superresolve in ways that MP cannot.D o w n l o a d e d 08/09/14 t o 58.19.126.38. R e d i s t r i b u t i o n s u b j e c t t o S I A M l i c e n s e o r c o p y r i g h t ; s e e h t t p ://w w w .s i a m .o r g /j o u r n a l s /o j s a .p h pATOMIC DECOMPOSITION BY BASIS PURSUIT131BP can be used with noisy data by solving an optimization problem trading oﬀa quadratic misﬁt measure with an 1normof coeﬃcients.Examples show that it can stably suppress noise while preserving structure that is well expressed in the dictionary under consideration.BP is closely connected with linear programming.Recent advances in large-scale linear programming—associated with interior-point methods—can be applied to BP and can make it possible,with certain dictionaries,to nearly solve the BP optimization problem in nearly linear time.We have implemented primal-dual log barrier interior-point methods as part of a MATLAB [31]computing environment called Atomizer,which accepts a wide range of dictionaries.Instructions for Internet access to Atomizer are given in section 7.3.Experiments with standard time-frequency dictionaries indicate some of the potential beneﬁts of BP.Experiments with some nonstandard dictionaries,like the stationary wavelet dictionary and the heaviside dictionary,indicate important connections between BP and methods like Mallat and Zhong’s [29]multiscale edge representation and Rudin,Osher,and Fatemi’s [35]total variation-based denoising methods.1.4.Contents.In section 2we establish vocabulary and notation for the rest of the article,describing a number of dictionaries and existing methods for overcomplete representation.In section 3we discuss the principle of BP and its relations to existing methods and to ideas in other ﬁelds.In section 4we discuss methodological issues associated with BP,in particular some of the interesting nonstandard ways it can be deployed.In section 5we describe BP denoising,a method for dealing with problem (1.2).In section 6we discuss recent advances in large-scale linear programming (LP)and resulting algorithms for BP.For reasons of space we refer the reader to [4]for a discussion of related work in statistics and analysis.2.Overcomplete Representations.Let s =(s t :0≤t <n )be a discrete-time signal of length n ;this may also be viewed as a vector in R n .We are interested in the reconstruction of this signal using superpositions of elementary waveforms.Traditional methods of analysis and reconstruction involve the use of orthogonal bases,such as the Fourier basis,various discrete cosine transformbases,and orthogonal wavelet bases.Such situations can be viewed as follows:given a list of n waveforms,one wishes to represent s as a linear combination of these waveforms.The waveforms in the list,viewed as vectors in R n ,are linearly independent,and so the representation is unique.2.1.Dictionaries and Atoms.A considerable focus of activity in the recent sig-nal processing literature has been the development of signal representations outside the basis setting.We use terminology introduced by Mallat and Zhang [29].A dic-tionary is a collection of parameterized waveforms D =(φγ:γ∈Γ).The waveforms φγare discrete-time signals of length n called atoms .Depending on the dictionary,the parameter γcan have the interpretation of indexing frequency,in which case the dictionary is a frequency or Fourier dictionary,of indexing time-scale jointly,in which case the dictionary is a time-scale dictionary,or of indexing time-frequency jointly,in which case the dictionary is a time-frequency ually dictionaries are complete or overcomplete,in which case they contain exactly n atoms or more than n atoms,but one could also have continuum dictionaries containing an inﬁnity of atoms and undercomplete dictionaries for special purposes,containing fewer than n atoms.Dozens of interesting dictionaries have been proposed over the last few years;we focusD o w n l o a d e d 08/09/14 t o 58.19.126.38. R e d i s t r i b u t i o n s u b j e c t t o S I A M l i c e n s e o r c o p y r i g h t ; s e e h t t p ://w w w .s i a m .o r g /j o u r n a l s /o j s a .p h p132S.S.CHEN,D.L.DONOHO,AND M.A.SAUNDERSin this paper on a half dozen or so;much of what we do applies in other cases as well.2.1.1.T rivial Dictionaries.We begin with some overly simple examples.The Dirac dictionary is simply the collection of waveforms that are zero except in one point:γ∈{0,1,...,n −1}and φγ(t )=1{t =γ}.This is of course also an orthogonal basis of R n —the standard basis.The heaviside dictionary is the collection of waveforms that jump at one particular point:γ∈{0,1,...,n −1};φγ(t )=1{t ≥γ}.Atoms in this dictionary are not orthogonal,but every signal has a representation s =s 0φ0+n −1 γ=1(s γ−s γ−1)φγ.(2.1)2.1.2.Frequency Dictionaries.A Fourier dictionary is a collection of sinusoidalwaveforms φγindexed by γ=(ω,ν),where ω∈[0,2π)is an angular frequency variable and ν∈{0,1}indicates phase type:sine or cosine.In detail,φ(ω,0)=cos(ωt ),φ(ω,1)=sin(ωt ).For the standard Fourier dictionary,we let γrun through the set of all cosines with Fourier frequencies ωk =2πk/n ,k =0,...,n/2,and all sines with Fourier frequencies ωk ,k =1,...,n/2−1.This dictionary consists of n waveforms;it is in fact a basis,and a very simple one:the atoms are all mutually orthogonal.An overcomplete Fourier dictionary is obtained by sampling the frequencies more ﬁnely.Let be a whole number >1and let Γ be the collection of all cosines with ωk =2πk/( n ),k =0,..., n/2,and all sines with frequencies ωk ,k =1,..., n/2−1.This is an -fold overcomplete system.We also use complete and overcomplete dictionaries based on discrete cosine transforms and sine transforms.2.1.3.Time-Scale Dictionaries.There are several types of wavelet dictionaries;to ﬁx ideas,we consider the Haar dictionary with “father wavelet”ϕ=1[0,1]and “mother wavelet”ψ=1(1/2,1]−1[0,1/2].The dictionary is a collection of transla-tions and dilations of the basic mother wavelet,together with translations of a father wavelet.It is indexed by γ=(a,b,ν),where a ∈(0,∞)is a scale variable,b ∈[0,n ]indicates location,and ν∈{0,1}indicates gender.In detail,φ(a,b,1)=ψ(a (t −b ))·√a,φ(a,b,0)=ϕ(a (t −b ))·√a.For the standard Haar dictionary,we let γrun through the discrete collection ofmother wavelets with dyadic scales a j =2j /n ,j =j 0,...,log 2(n )−1,and locations that are integer multiples of the scale b j,k =k ·a j ,k =0,...,2j −1,and the collection of father wavelets at the coarse scale j 0.This dictionary consists of n waveforms;it is an orthonormal basis.An overcomplete wavelet dictionary is obtained by sampling the locations more ﬁnely:one location per sample point.This gives the so-called sta-tionary Haar dictionary,consisting of O (n log 2(n ))waveforms.It is called stationary since the whole dictionary is invariant under circulant shift.A variety of other wavelet bases are possible.The most important variations are smooth wavelet bases,using splines or using wavelets deﬁned recursively fromtwo-scale ﬁltering relations [10].Although the rules of construction are more complicated (boundary conditions [33],orthogonality versus biorthogonality [10],etc.),these have the same indexing structure as the standard Haar dictionary.In this paper,we use symmlet -8smooth wavelets,i.e.,Daubechies nearly symmetric wavelets with eight vanishing moments;see [10]for examples.D o w n l o a d e d 08/09/14 t o 58.19.126.38. R e d i s t r i b u t i o n s u b j e c t t o S I A M l i c e n s e o r c o p y r i g h t ; s e e h t t p ://w w w .s i a m .o r g /j o u r n a l s /o j s a .p h pATOMIC DECOMPOSITION BY BASIS PURSUIT133Time 00.5100.20.40.60.81(c) Time DomainFig.2.1Time-frequency phase plot of a wavelet packet atom.2.1.4.Time-Frequency Dictionaries.Much recent activity in the wavelet com-munities has focused on the study of time-frequency phenomena.The standard ex-ample,the Gabor dictionary,is due to Gabor [19];in our notation,we take γ=(ω,τ,θ,δt ),where ω∈[0,π)is a frequency,τis a location,θis a phase,and δt is the duration,and we consider atoms φγ(t )=exp {−(t −τ)2/(δt )2}·cos(ω(t −τ)+θ).Such atoms indeed consist of frequencies near ωand essentially vanish far away from τ.For ﬁxed δt ,discrete dictionaries can be built fromtim e-frequency lattices,ωk =k ∆ωand τ = ∆τ,and θ∈{0,π/2};with ∆τand ∆ωchosen suﬃciently ﬁne these are complete.For further discussions see,e.g.,[9].Recently,Coifman and Meyer [6]developed the wavelet packet and cosine packet dictionaries especially to meet the computational demands of discrete-time signal pro-cessing.For one-dimensional discrete-time signals of length n ,these dictionaries each contain about n log 2(n )waveforms.A wavelet packet dictionary includes,as special cases,a standard orthogonal wavelets dictionary,the Dirac dictionary,and a collec-tion of oscillating waveforms spanning a range of frequencies and durations.A cosine packet dictionary contains,as special cases,the standard orthogonal Fourier dictio-nary and a variety of Gabor-like elements:sinusoids of various frequencies weighted by windows of various widths and locations.In this paper,we often use wavelet packet and cosine packet dictionaries as exam-ples of overcomplete systems,and we give a number of examples decomposing signals into these time-frequency dictionaries.A simple block diagram helps us visualize the atoms appearing in the decomposition.This diagram,adapted from Coifman and Wickerhauser [7],associates with each cosine packet or wavelet packet a rectangle in the time-frequency phase plane.The association is illustrated in Figure 2.1for a cer-tain wavelet packet.When a signal is a superposition of several such waveforms,we indicate which waveforms appear in the superposition by shading the corresponding rectangles in the time-frequency plane.D o w n l o a d e d 08/09/14 t o 58.19.126.38. R e d i s t r i b u t i o n s u b j e c t t o S I A M l i c e n s e o r c o p y r i g h t ; s e e h t t p ://w w w .s i a m .o r g /j o u r n a l s /o j s a .p h p134S.S.CHEN,D.L.DONOHO,AND M.A.SAUNDERS2.1.5.Further Dictionaries.We can always merge dictionaries to create mega-dictionaries;examples used below include mergers of wavelets with heavisides.2.2.Linear Algebra.Suppose we have a discrete dictionary of p waveforms and we collect all these waveforms as columns of an n -by-p matrix Φ,say.The decompo-sition problem(1.1)can be written Φα=s ,(2.2)where α=(αγ)is the vector of coeﬃcients in (1.1).When the dictionary furnishes a basis,then Φis an n -by-n nonsingular matrix and we have the unique representation α=Φ−1s .When the atoms are,in addition,mutually orthonormal,then Φ−1=ΦT and the decomposition formula is very simple.2.2.1.Analysis versus Synthesis.Given a dictionary of waveforms,one can dis-tinguish analysis from synthesis .Synthesis is the operation of building up a signal by superposing atoms;it involves a matrix that is n -by-p :s =Φα.Analysis involves the operation of associating with each signal a vector of coeﬃcients attached to atoms;it involves a matrix that is p -by-n :˜α=ΦT s .Synthesis and analysis are very diﬀer-ent linear operations,and we must take care to distinguish them.One should avoid assuming that the analysis operator ˜α=ΦT s gives us coeﬃcients that can be used as is to synthesize s .In the overcomplete case we are interested in,p n and Φis not invertible.There are then many solutions to (2.2),and a given approach selects a particular solution.One does not uniquely and automatically solve the synthesis problemby applying a sim ple,linear analysis operator.We now illustrate the diﬀerence between synthesis (s =Φα)and analysis (˜α=ΦTs ).Figure 2.2a shows the signal Carbon .Figure 2.2b shows the time-frequency structure of a sparse synthesis of Carbon ,a vector αyielding s =Φα,using a wavelet packet dictionary.To visualize the decomposition,we present a phase-plane display with shaded rectangles,as described above.Figure 2.2c gives an analysis of Carbon ,with the coeﬃcients ˜α=ΦT s ,again displayed in a phase plane.Once again,between analysis and synthesis there is a large diﬀerence in sparsity.In Figure 2.2d we compare the sorted coeﬃcients of the overcomplete representation (synthesis)with the analysis coeﬃcients.putational Complexity of Φand ΦT .Diﬀerent dictionaries can im-pose drastically diﬀerent computational burdens.In this paper we report compu-tational experiments on a variety of signals and dictionaries.We study primarily one-dimensional signals of length n ,where n is several thousand.Signals of this length occur naturally in the study of short segments of speech (a quarter-second to a half-second)and in the output of various scientiﬁc instruments (e.g.,FT-NMR spec-trometers).In our experiments,we study dictionaries overcomplete by substantial factors,say,10.Hence the typical matrix Φwe are interested in is of size “thousands”by “tens-of-thousands.”The nominal cost of storing and applying an arbitrary n -by-p matrix to a p -vector is a constant times np .Hence with an arbitrary dictionary of the sizes we are interested in,simply to verify whether (1.1)holds for given vectors αand s would require tens of millions of multiplications and tens of millions of words of memory.In contrast,most signal processing algorithms for signals of length 1000require only thousands of memory words and a few thousand multiplications.Fortunately,certain dictionaries have fast implicit algorithms .By this we mean that Φαand ΦT s can be computed,for arbitrary vectors αand s ,(a)without everD o w n l o a d e d 08/09/14 t o 58.19.126.38. R e d i s t r i b u t i o n s u b j e c t t o S I A M l i c e n s e o r c o p y r i g h t ; s e e h t t p ://w w w .s i a m .o r g /j o u r n a l s /o j s a .p h pATOMIC DECOMPOSITION BY BASIS PURSUIT135Time0.5100.20.40.60.81Time0.5100.20.40.60.81(d) Sorted CoefficientsSynthesis: SolidAnalysis: Dashed Fig.2.2Analysis versus synthesis of the signal Carbon .storing the matrices Φand ΦT ,and (b)using special properties of the matrices to accelerate computations.The most well-known example is the standard Fourier dictionary for which we have the fast Fourier transform algorithm.A typical implementation requires 2·n storage locations and 4·n ·J multiplications if n is dyadic:n =2J .Hence for very long signals we can apply Φand ΦT with much less storage and time than the matrices would nominally require.Simple adaptation of this idea leads to an algorithm for overcomplete Fourier dictionaries.Wavelets give a more recent example of a dictionary with a fast implicit algorithm;if the Haar or S8-symmlet is used,both Φand ΦT may be applied in O (n )time.For the stationary wavelet dictionary,O (n log(n ))time is required.Cosine packets and wavelet packets also have fast implicit algorithms.Here both Φand ΦT can be applied in order O (n log(n ))time and order O (n log(n ))space—much better than the nominal np =n 2log 2(n )one would expect fromnaive use of the m atrix deﬁnition.For the viewpoint of this paper,it only makes sense to consider dictionaries with fast implicit algorithms.Among dictionaries we have not discussed,such algorithms may or may not exist.2.3.Existing Decomposition Methods.There are several currently popular ap-proaches to obtaining solutions to (2.2).2.3.1.Frames.The MOF [9]picks out,among all solutions of (2.2),one whose coeﬃcients have minimum l 2norm:min α 2subject toΦα=s .(2.3)The solution of this problemis unique;label it α†.Geometrically,the collection of all solutions to (2.2)is an aﬃne subspace in R p ;MOF selects the element of this subspace closest to the origin.It is sometimes called a minimum-length solution.There is aD o w n l o a d e d 08/09/14 t o 58.19.126.38. R e d i s t r i b u t i o n s u b j e c t t o S I A M l i c e n s e o r c o p y r i g h t ; s e e h t t p ://w w w .s i a m .o r g /j o u r n a l s /o j s a .p h p136S.S.CHEN,D.L.DONOHO,AND M.A.SAUNDERSTime0.5100.20.40.60.81Time0.5100.20.40.60.81Fig.2.3MOF representation is not sparse.matrix Φ†,the generalized inverse of Φ,that calculates the minimum-length solution to a systemof linear equations:α†=Φ†s =ΦT (ΦΦT )−1s .(2.4)For so-called tight frame dictionaries MOF is available in closed form.A nice example is the standard wavelet packet dictionary.One can compute that for all vectors v ,ΦT v 2=L n · v 2,L n =log 2(n ).In short Φ†=L −1n ΦT .Notice that ΦTis simply the analysis operator.There are two key problems with the MOF.First,MOF is not sparsity preserving .If the underlying object has a very sparse representation in terms of the dictionary,then the coeﬃcients found by MOF are likely to be very much less sparse.Each atom in the dictionary that has nonzero inner product with the signal is,at least potentially and also usually,a member of the solution.Figure 2.3a shows the signal Hydrogen made of a single atom in a wavelet packet dictionary.The result of a frame decomposition in that dictionary is depicted in a phase-plane portrait;see Figure 2.3c.While the underlying signal can be synthesized from a single atom,the frame decomposition involves many atoms,and the phase-plane portrait exaggerates greatly the intrinsic complexity of the object.Second,MOF is intrinsically resolution limited .No object can be reconstructed with features sharper than those allowed by the underlying operator Φ†Φ.Suppose the underlying object is sharply localized:α=1{γ=γ0}.The reconstruction will not be α,but instead Φ†Φα,which,in the overcomplete case,will be spatially spread out.Figure 2.4presents a signal TwinSine consisting of the superposition of two sinusoids that are separated by less than the so-called Rayleigh distance 2π/n .We analyze these in a fourfold overcomplete discrete cosine dictionary.In this case,reconstruction by MOF (Figure 2.4b)is simply convolution with the Dirichlet kernel.The result is the synthesis fromcoeﬃcients with a broad oscillatory appearance,consisting not of twoD o w n l o a d e d 08/09/14 t o 58.19.126.38. R e d i s t r i b u t i o n s u b j e c t t o S I A M l i c e n s e o r c o p y r i g h t ; s e e h t t p ://w w w .s i a m .o r g /j o u r n a l s /o j s a .p h pATOMIC DECOMPOSITION BY BASIS PURSUIT137Fig.2.4Analyzing TwinSine with a fourfold overcomplete discrete cosine dictionary.but of many frequencies and giving no visual clue that the object may be synthesized fromtwo frequencies alone.2.3.2.Matching Pursuit.Mallat and Zhang [29]discussed a general method for approximate decomposition (1.2)that addresses the sparsity issue directly.Starting froman initial approxim ation s (0)=0and residual R (0)=s ,it builds up a sequence of sparse approximations stepwise.At stage k ,it identiﬁes the dictionary atomthat best correlates with the residual and then adds to the current approximation a scalar multiple of that atom,so that s (k )=s (k −1)+αk φγk ,where αk = R (k −1),φγk and R (k )=s −s (k ).After m steps,one has a representation of the form(1.2),with residual R =R (m ).Similar algorithms were proposed by Qian and Chen [39]for Gabor dictionaries and by Villemoes [48]for Walsh dictionaries.A similar algorithm was proposed for Gabor dictionaries by Qian and Chen [39].For an earlier instance of a related algorithm,see [5].An intrinsic feature of the algorithmis that when stopped after a few steps,it yields an approximation using only a few atoms.When the dictionary is orthogonal,the method works perfectly.If the object is made up of only m n atoms and the algorithmis run for m steps,it recovers the underlying sparse structure exactly.When the dictionary is not orthogonal,the situation is less clear.Because the algorithmis m yopic,one expects that,in certain cases,it m ight choose wrongly in the ﬁrst few iterations and end up spending most of its time correcting for any mistakes made in the ﬁrst few terms.In fact this does seem to happen.To see this,we consider an attempt at superresolution.Figure 2.4a portrays again the signal TwinSine consisting of sinusoids at two closely spaced frequencies.When MP is applied in this case (Figure 2.4c),using the fourfold overcomplete discrete cosine dictionary,the initial frequency selected is in between the two frequencies making up the signal.Because of this mistake,MP is forced to make a series of alternating corrections that suggest a highly complex and organized structure.MPD o w n l o a d e d 08/09/14 t o 58.19.126.38. R e d i s t r i b u t i o n s u b j e c t t o S I A M l i c e n s e o r c o p y r i g h t ; s e e h t t p ://w w w .s i a m .o r g /j o u r n a l s /o j s a .p h p138S.S.CHEN,D.L.DONOHO,AND M.A.SAUNDERSFig.2.5Counterexamples for MP.misses entirely the doublet structure.One can certainly say in this case that MP has failed to superresolve.Second,one can give examples of dictionaries and signals where MP is arbitrarily suboptimal in terms of sparsity.While these are somewhat artiﬁcial,they have a character not so diﬀerent fromthe superresolution exam ple.DeVore and Temlyakov’s Example.Vladimir Temlyakov,in a talk at the IEEE Confer-ence on Information Theory and Statistics in October 1994,described an example in which the straightforward greedy algorithmis not sparsity preserving.In our adapta-tion of this example,based on Temlyakov’s joint work with DeVore [12],one constructs a dictionary having n +1atoms.The ﬁrst n are the Dirac basis;the ﬁnal atomin-volves a linear combination of the ﬁrst n with decaying weights.The signal s has an exact decomposition in terms of A atoms,but the greedy algorithm goes on forever,with an error of size O (1/√m )after m steps.We illustrate this decay in Figure 2.5a.For this example we set A =10and choose the signal s t =10−1/2·1{1≤t ≤10}.The dictionary consists of Dirac elements φγ=δγfor 1≤γ≤n andφn +1(t )=c,1≤t ≤10,c/(t −10),10<t ≤n,with c chosen to normalize φn +1to unit norm.Shaobing Chen’s Example.The DeVore–Temlyakov example applies to the original MP algorithmas announced by Mallat and Zhang in 1992.A later reﬁnem ent of the algorithm(see Pati,Rezaiifar,and Krishnaprasad [38]and Davis,Mallat,and Zhang [11])involves an extra step of orthogonalization.One takes all m terms that have entered at stage m and solves the least-squares problemmin (αi )s −m i =1αi φγi2D o w n l o a d e d 08/09/14 t o 58.19.126.38. R e d i s t r i b u t i o n s u b j e c t t o S I A M l i c e n s e o r c o p y r i g h t ; s e e h t t p ://w w w .s i a m .o r g /j o u r n a l s /o j s a .p h p。

AxiChrom

Differences between pilot and process scale
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Reproducibility
8000 7000 6000 5000 Plates/m 4000 3000 2000 1000 0 1 2 3 Run num ber 4 5 Plates/m Asymmetry 0,80 1,60 1,50 1,40 1,30 1,20 1,10 1,00 0,90
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数学建模美国赛历年试题.

历年美国大学生数学建模赛题目录MCM85问题-A 动物群体的管理 (3)MCM85问题-B 战购物资储备的管理 (3)MCM86问题-A 水道测量数据 (4)MCM86问题-B 应急设施的位置 (4)MCM87问题-A 盐的存贮 (4)MCM87问题-B 停车场 (5)MCM88问题-A 确定毒品走私船的位置 (5)MCM88问题-B 两辆铁路平板车的装货问题 (5)MCM89问题-A 蠓的分类 (5)MCM89问题-B 飞机排队 (6)MCM90-A 药物在脑内的分布 (6)MCM90问题-B 扫雪问题 (6)MCM91问题-B 通讯网络的极小生成树 (6)MCM 91问题-A 估计水塔的水流量 (7)MCM92问题-A 空中交通控制雷达的功率问题 (7)MCM 92问题-B 应急电力修复系统的修复计划 (7)MCM93问题-A 加速餐厅剩菜堆肥的生成 (7)MCM93问题-B 倒煤台的操作方案 (8)MCM94问题-A 住宅的保温 (8)MCM 94问题-B 计算机网络的最短传输时间 (9)MCM-95问题-A 单一螺旋线 (9)MCM95题-B A1uacha Balaclava学院 (10)MCM96问题-A 噪音场中潜艇的探测 (10)MCM96问题-B 竞赛评判问题 (10)MCM97问题-A Velociraptor(疾走龙属)问题 (11)MCM97问题-B为取得富有成果的讨论怎样搭配与会成员 (11)MCM98问题-A 磁共振成像扫描仪 (12)MCM98问题-B 成绩给分的通胀 (13)MCM99问题-A 大碰撞 (13)MCM99问题-B “非法”聚会 (13)MCM2000问题-A空间交通管制 (13)MCM2000问题-B: 无线电信道分配 (14)MCM2001问题- A: 选择自行车车轮 (14)MCM2001问题-B 逃避飓风怒吼（一场恶风...） .. (15)MCM2001问题-C我们的水系-不确定的前景 (15)MCM2002问题-A风和喷水池 (15)MCM2002问题-B航空公司超员订票 (16)MCM2002问题-C (16)MCM2003问题-A: 特技演员 (17)MCM2003问题-B: Gamma刀治疗方案 (18)MCM2003问题-C航空行李的扫描对策 (18)MCM2004问题-A：指纹是独一无二的吗？ (18)MCM2004问题-B：更快的快通系统 (18)MCM2004问题-C安全与否？ (19)MCM2005问题A.水灾计划 (19)MCM2005B.Tollbooths (19)MCM2005问题C：不可再生的资源 (20)MCM2006问题A: 用于灌溉的自动洒水器的安置和移动调度 (20)MCM2006问题B: 通过机场的轮椅 (20)MCM2006问题C : 抗击艾滋病的协调 (21)MCM2008问题A:给大陆洗个澡 (23)MCM2008问题B：建立数独拼图游戏 (23)MCM85问题-A 动物群体的管理在一个资源有限，即有限的食物、空间、水等等的环境里发现天然存在的动物群体。

5速来!数据科学工具包-几百种工具-经典收藏版!

速来！数据科学工具包-几百种工具-经典收藏版！一、数据科学工具包数据科学融合了多门学科并且建立在这些学科的理论和技术之上，包括数学、概率模型、统计学、机器学习、数据仓库、可视化等。

在实际应用中，数据科学包括数据的收集、清洗、分析、可视化以及数据应用整个迭代过程，最终帮助组织制定正确的发展决策数据科学的从业者称为数据科学家。

数据科学家有其独特的基本思路与常用工具，秦陇纪全面梳理数据分析师和数据科学家使用的工具包，包括开源的技术平台相关工具、挖掘分析处理工具、其它常见工具等几百种，几十个大类，部分网址，欢迎大家积极传播！数据科学家是有着开阔视野的复合型人才，他们既有坚实的数据科学基础，如数学、统计学、计算机学等，又具备广泛的业务知识和经验数据科学家通过精深的技术和专业知识在某些科学学科领域解决复杂的数据问题，从而制定出适合不同决策人员的大数据计划和策略。

数据分析师和数据科学家使用的工具在网上的MOOC有提供，比如2016年2月1日约翰-霍普金斯大学Coursera数据科学专业化课程等网络课程。

数据科学家的常用工具与基本思路，并对数据、相关问题和数据分析师和数据科学家使用的工具做了综合概述。

数据科学家和大数据技术人员的工具包：A.大数据技术平台相关2015最佳工具，B.开源大数据处理工具汇总，C.常见的数据挖掘分析处理工具。

A.大数据技术平台相关2015最佳工具InfoWorld在分布式数据处理、流式数据分析、机器学习以及大规模数据分析领域精选出了2015年的开源工具获奖者，下面我们来简单介绍下这些获奖的技术工具。

1. Spark在Apache的大数据项目中，Spark是最火的一个，特别是像IBM这样的重量级贡献者的深入参与，使得Spark的发展和进步速度飞快。

与Spark产生最甜蜜的火花点仍然是在机器学习领域。

去年以来DataFrames API取代SchemaRDD API，类似于R和Pandas的发现，使数据访问比原始RDD接口更简单。

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NetCube:AScalableToolforFastDataMiningandCompression

DimitrisMargaritisComputerScienceDept.CarnegieMellonUniversityPittsburgh,PA15213,U.S.A.D.Margaritis@cs.cmu.eduChristosFaloutsosComputerScienceDept.CarnegieMellonUniversityPittsburgh,PA15213,U.S.A.C.Faloutsos@cs.cmu.eduSebastianThrunComputerScienceDept.CarnegieMellonUniversityPittsburgh,PA15213,U.S.A.S.Thrun@cs.cmu.edu

AbstractWeproposeannovelmethodofcomputingandstoringDataCubes.OurideaistouseBayesianNetworks,whichcangenerateapprox-imatecountsforanyquerycombinationofat-tributevaluesand“don’tcares.”ABayesiannet-workrepresentstheunderlyingjointprobabilitydistributionofthedatathatwereusedtogener-ateit.Bymeansofsuchanetworktheproposedmethod,NetCube,exploitscorrelationsamongat-tributes.Ourproposedpreprocessingalgorithmscaleslinearlyonthesizeofthedatabase,andisthusscalable;itisalsoparallelizablewithastraightforwardparallelimplementation.More-over,wegiveanalgorithmtoestimatecountsofarbitraryqueriesthatisfast(constantonthedatabasesize).ExperimentalresultsshowthatNetCubeshavefastgenerationanduse(afewthemselvesforansweringDataCubequeries.Havingsaidthat,therealchallengeliesinhowtoconstructamodelofthedatathatisgoodenoughforourpurposes.Forthis,therearetwoimportantconsiderationsthatarerelevanttotheproblemthatweareaddressing:One,themodelshouldbeanaccuratedescriptionofourdata,orattheveryleastofthesequantitiesderivedfromthemthatareofinterest.Inthisproblemthequantitiesarethecountsinthedatabaseofeveryinterestingcountquerythatcanbeappliedtothem(i.e.querieswithsomemini-mumsupportsuchas1%;otherqueryresultscanbeduetonoiseanderrorsinthedata).Second,themodelshouldbesimpleenoughsothatusingitinsteadoftheactualdatatoansweraqueryshouldnottakeanexorbitantamountoftimeorconsumeanenormousamountofspace,moresoperhapsthanusingtherawdatathemselves.Thesetwoissuesareconﬂicting,andtheproblemofbal-ancingthemisacentralissueintheAIﬁeldofmachinelearning(whichconcernsitselfwiththedevelopmentofmodelsofdata):itisalwayspossibletodescribethedata(orthederivedquantitiesweareinterestedin)better,oratleastaswell,withincreasinglycomplexmodels.How-ever,thecostofsuchmodelsincreaseswithcomplexity,intermsofbothsizetostorethemodelparametersandtimethatittakestouseitforcomputingtherelevantquantities(thequerycountsinourcase).InthispaperwechosetouseBayesiannetworks(BNs).Suchmodelsarenottheonlychoicepossible,butwepickedthembecausetheyareamature,broadlyacceptableandwellrespectedmethodofmodelingdatainthemachinelearningcommunity.Thisacceptanceandrespectcomesnotonlyfromtheirpracti-caleffectiveness,butalsofromtheirsoundmathematicalfoundationsinprobabilitytheory,asopposedtoamulti-tudeofotheradhocapproachesthatexistintheliterature.ThemethodofproducingtheBNsfromdatathatweuseisonethathasproventobescientiﬁcallyacceptableinthemachinelearningcommunityandgoodinpractice[16,23].Theremainderofthepaperisorganizedasfollows.Insection2webrieﬂyreviewthecurrentliteratureonDat-aCubesandtheprevalentcurrentimplementation,bitmaps,andalsoofBayesiannetworks.Insection3wepresentasimpleintroductiontoBayesiannetworksandmethodsofinducingtheirstructurefromdata.Insection4wedescribeourapproach,andweshowsomeexperimentalresultsinsection5.Weconcludewithadiscussionofrelevantissuesanddirectionsoffutureresearchinsection6.2RelatedWorkDataCubeswereintroducedin[8].Theymaybeused,intheory,toansweranyqueryquickly(e.g.constanttimeforatable-lookuprepresentation).Inpracticehowevertheyhaveprovenexceedinglydifﬁculttocomputeandstorebe-causeoftheirinherentlyexponentialnature.Tosolvethisproblem,severalapproacheshavebeenproposed.[9]sug-gestmaterializingonlyasubsetofviewsandproposeaprincipledwayofselectingwhichonestoprefer.Theirsystemcomputesthequeryfromthoseviewsatruntime.Cubescontainingonlycellsofsomeminimumsupportaresuggestedin[2]andacoarse-to-ﬁnetraversalisproposedthatimprovesspeedbycondensingcellsoflessthattheminimumsupport.Histogram-basedapproachesalsoex-ist[13],aswellasapproximationssuchashistogramcom-pressionusingtheDCTtransform[17]orwavelets[24].Perhapsclosesttoourapproachis[1],whichuseslinearregressiontomodelDataCubes.Bitmapsarerelativelyre-centmethodforefﬁcientlycomputingcountsfromhighlycompressedbitmappedinformationaboutthepropertiesofrecordsinthedatabase.Theyareexacttechniques.Un-liketheDataCubeandBayesiannetworks,bitmapsdonotmaintaincounts,butinsteadperformapassoverseveralbitmapsatruntimeinordertoansweranaggregatequery[14,3].Queryoptimizersforbitmapsalsoexist[25].TherehasnotbeenmuchworkonapplyingBayesiannetworkstodatabases.Anexceptionis[21],wherepossi-blecausalrelationsfromdataarecomputedforpurposesofdatamining.Also,[6]usedBayesiannetworksforloss-lessdatacompressionappliedtorelativelysmalldatasets.Dataminingresearchitselfhasbeenmostlyfocusedondis-coveringassociationrulesfromdata[18,15],whichcanbeviewedofasaspecialcaseofBayesiannetworkinduction.Bayesiannetworkresearchontheotherhandhasﬂour-ishedinthelastdecade,spurredmostlybyPearl’ssem-inalbook[19].[10]containsacomprehensiveoverviewofapproachestoinferenceandstructureinduction.Re-strictedclassesofBayesiannetworkssuchastreeshavebeensolvedoptimally[5]inthepast.However,thegen-eralproblemisNP-complete[4].Thereexisttwogeneralapproaches:thehill-climbingapproachbasedontheMDLscore[16,23],theprevalent,morepracticalonewhichisusedhere,andtheconstraint-basedapproach.Constraint-basedalgorithmsarecoveredin[22].