Classifications

• • G—PHYSICS
  • G06—COMPUTING; CALCULATING OR COUNTING
  • G06F—ELECTRIC DIGITAL DATA PROCESSING
  • G06F16/00—Information retrieval; Database structures therefor; File system structures therefor
  • G06F16/30—Information retrieval; Database structures therefor; File system structures therefor of unstructured textual data
  • G06F16/35—Clustering; Classification
• • G—PHYSICS
  • G06—COMPUTING; CALCULATING OR COUNTING
  • G06F—ELECTRIC DIGITAL DATA PROCESSING
  • G06F18/00—Pattern recognition
  • G06F18/20—Analysing
  • G06F18/23—Clustering techniques
  • G06F18/232—Non-hierarchical techniques
  • G06F18/2321—Non-hierarchical techniques using statistics or function optimisation, e.g. modelling of probability density functions
• • Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
  • Y10—TECHNICAL SUBJECTS COVERED BY FORMER USPC
  • Y10S—TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
  • Y10S707/00—Data processing: database and file management or data structures
  • Y10S707/99931—Database or file accessing
  • Y10S707/99937—Sorting
• • Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
  • Y10—TECHNICAL SUBJECTS COVERED BY FORMER USPC
  • Y10S—TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
  • Y10S707/00—Data processing: database and file management or data structures
  • Y10S707/99941—Database schema or data structure
  • Y10S707/99942—Manipulating data structure, e.g. compression, compaction, compilation

Definitions

• This invention relates generally to data modeling, and more particularly to efficient data modeling utilizing a sparse representation of the initial data set.
• a web site such as www.msnbc.com has many stories available on any given day or month.
• the operators of such a web site may desire to know whether there are any commonalties associated with the viewership of a given set of programs. That is, if a hypothetical user reads one given story, can with any probability it be said that the user is likely to read another given story. Yielding the answer to this type of inquiry allows the operators of the web site to better organize their site, for example, which may in turn yield increased readership .
• EM algorithm Expectation Maximization algorithm
• the EM algorithm permits a data analyst to assign each data record (e.g., the stories a given user has read) to a cluster of like data records, where each cluster has a probabilistic dependence for each story.
• assignment of a user into a cluster is useful to predict what other stories or types of stories a user may have also read, or may read in the future.
• a cluster is a collection of records, where each record has a similar set of attribute values.
• a disadvantage to utilization of the EM algorithm is that generally, as the size of the data set increases, the running time of the algorithm increases even moreso. This is because the EM algorithm typically cycles over the records in a data set many times, in an iterative manner, until a given convergence criterion is met. This is problematic for problems such as the web site example just described, because typically the data set can run into the millions of records, impeding timely analysis thereof. Thus, a data analyst may not utilize the EM algorithm as much as he or she would like to.
• a data set is first input.
• the data set has a plurality of records.
• Each record has at least one attribute, where each attribute has a default value.
• the attributes of one record can in one embodiment largely overlap with the attributes of another record, although the specific values of the attributes for one record may not so overlap.
• the method stores a sparse representation of each record, such that the value of each attribute of the record is stored only if it varies from the default value (that is, if the value equals the default value, it is not stored).
• a data model is then generated, utilizing the sparse representation, and the model is output.
• Embodiments of the invention by utilizing the sparse representation summarized in the preceding paragraph, provide for more efficient and less time-intensive data modeling of the data set. For example, in the context of the EM algorithm, the determination of a joint probability distribution over the cluster membership and the record values is generally required to be made for each record. This typically requires determination of a posterior probability for each attribute of each record, which in the prior art can be a time-consuming process. As a further example, embodiments of the invention provide for speeded-up clustering, as can be appreciated by those of ordinary skill within the art.
• determination of a posterior probability can involve only initially generating a joint probability based on the default value of an attribute, and determining the posterior probability based on this joint probability — such that this posterior probability is used for any record having the default value for this attribute.
• the posterior probability does not have to be generated anew for every record with this attribute, but rather once for the default value of the attribute, and then subsequently only for those records having a value for this attribute varying from the default value. This greatly decreases the amount of time it takes for the EM algorithm to be run.
• embodiments of the invention allow for use of the EM algorithm to cluster a sparse data set, without examining the value of every attribute of every record. In instances where each record contains, on average, only a small set of attributes that are not in their default state, embodiments of the invention can yield great decreases in run time.
• the invention includes computer-implemented methods, machine-readable media, computerized systems, and computers of varying scopes. Other aspects, embodiments and advantages of the invention, beyond those described here, will become apparent by reading the detailed description and with reference to the drawings. BRIEF DESCRIPTION OF THE DRAWINGS
• FIG. 1 is a diagram of an operating environment in conjunction with which embodiments of the invention can be practiced;
• FIG. 2 is a diagram of an illustrative Bayesian network showing a single node corresponding to the cluster membership, according to an embodiment of the invention
• FIG. 3 is a diagram showing a sparse representation of a data set according to an embodiment of the invention
• FIG. 4 is a flowchart of a method implementing a modified EM algorithm according to an embodiment of the invention.
• FIG. 5 is a diagram of a system according to an embodiment of the invention.
• FIG. 1 a diagram of the hardware and operating environment in conjunction with which embodiments of the invention may be practiced is shown.
• the description of FIG. 1 is intended to provide a brief, general description of suitable computer hardware and a suitable computing environment in conjunction with which the invention may be implemented.
• the invention is described in the general context of computer-executable instructions, such as program modules, being executed by a computer, such as a personal computer.
• program modules include routines, programs, objects, components, data structures, etc., that perform particular tasks or implement particular abstract data types.
• FIG. 1 for implementing the invention includes a general purpose computing device in the form of a computer 20, including a processing unit 21, a system memory 22, and a system bus 23 that operatively couples various system components include the system memory to the processing unit 21.
• the processor of computer 20 comprises a single central-processing unit (CPU), or a plurality of processing units, commonly referred to as a parallel processing environment.
• the computer 20 may be a conventional computer, a distributed computer, or any other type of computer; the invention is not so limited.
• the system bus 23 may be any of several types of bus structures including a memory bus or memory controller, a peripheral bus, and a local bus using any of a variety of bus architectures.
• the system memory may also be referred to as simply the memory, and includes read only memory (ROM) 24 and random access memory (RAM) 25.
• the computer 20 further includes a hard disk drive 27 for reading from and writing to a hard disk, not shown, a magnetic disk drive 28 for reading from or writing to a removable magnetic disk 29, and an optical disk drive 30 for reading from or writing to a removable optical disk 31 such as a CD ROM or other optical media.
• the hard disk drive 27, magnetic disk drive 28, and optical disk drive 30 are connected to the system bus 23 by a hard disk drive interface 32, a magnetic disk drive interface 33, and an optical disk drive interface 34, respectively.
• the drives and their associated computer-readable media provide nonvolatile storage of computer-readable instructions, data structures, program modules and other data for the computer 20. It should be appreciated by those skilled in the art that any type of computer-readable media which can store data that is accessible by a computer, such as magnetic cassettes, flash memory cards, digital video disks, Bernoulli cartridges, random access memories (RAMs), read only memories (ROMs), and the like, may be used in the exemplary operating environment.
• a number of program modules may be stored on the hard disk, magnetic disk 29, optical disk 31, ROM 24, or RAM 25, including an operating system 35, one or more application programs 36, other program modules 37, and program data 38.
• a user may enter commands and information into the personal computer 20 through input devices such as a keyboard 40 and pointing device 42.
• Other input devices may include a microphone, joystick, game pad, satellite dish, scanner, or the like.
• serial port interface 46 that is coupled to the system bus, but may be connected by other interfaces, such as a parallel port, game port, or a universal se ⁇ al bus (USB)
• a monitor 47 or other type of display device is also connected to the system bus 23 via an interface, such as a video adapter 48
• computers typically include other pe ⁇ pheral output devices (not shown), such as speakers and p ⁇ nters
• the computer 20 may operate in a networked environment using logical connections to one or more remote computers, such as remote computer 49
• remote computer 49 may be another computer, a server, a router, a network PC, a client, a peer device or other common network node, and typically includes many or all of the elements desc ⁇ bed above relative to the computer 20, although only a memory storage device 50 has been illustrated in FIG 1
• naive-Bayes model as known within the art, that can be utilized in conjunction with embodiments of the invention
• the model is used to represent a joint probability dist ⁇ bution over the set of all att ⁇ butes in a domain, and the cluster membership
• the model is desc ⁇ bed in conjunction with reference to FIG 2
• the naive-Bayes model desc ⁇ bed herein is not necessa ⁇ ly a limitation of all embodiments of the invention, that is, sparse representation as desc ⁇ bed in subsequent sections of the detailed desc ⁇ ption may be applicable for use with other models
• a na ⁇ ve-Bayes model that can be used in conjunction with at least some embodiments of the invention is used to model a particular data set, or domain
• the domain can be the domain of television shows that a person may watch, such that a data analyst is interested in clustenng people into three groups, based on their viewing behavior
• the attributes of the domain correspond to television
• a cluster of a na ⁇ ve-Bayes model is shown, as a Bayesian network
• the model includes a cluster node 200, and att ⁇ bute (or, va ⁇ able) nodes 202a, 202b, . ., 202n
• a Bayesian network includes one node for each vanable (or, att ⁇ bute), as well as a node that has a value for each cluster (the number of clusters is not explicitly shown in FIG 2)
• Directed edges 204a, 204b, . . ., 204n co ⁇ espond to probabilistic dependence - for example, for each attribute there is a probability distnbution for each value of its parent (the cluster node)
• Each node in the Bayesian network is annotated with a set of parameters.
• the parameters stored in each node define the probability dist ⁇ bution over the vanable, given the values of the parent vanable (it is noted that a cluster node, such as the cluster node 200, does not have a parent vanable or node).
• the symbol ⁇ is used to denote the set of all parameters stored in the model.
• r A i denotes the number of states for att ⁇ bute t
• the denominator is the same regardless of the value of c, and is simply the sum of the joint probability for each value of c. In practice, it is desirable to first compute and store away the joint probability for each value c, accumulating the sum as it is done. Then, each of the stored joint probabilities is divided by the sum to obtain the posteriors.
• a sparse representation is defined as: for each attribute in a domain, a default value is defined.
• the default value for an attribute is that value that occurs most frequently in the data set. For example, in the television-watching example of the preceding section, "did not watch” is likely the most frequent value for most television shows for a user (since a given user can only watch one television show at a time).
• the records of a sparse representation explicitly store only those values for attributes that do not have their default value. That is, in storing a sparse representation of each record, the value of those attributes of the record are stored only upon determining that such values vary from the default values for that attribute.
• FIG. 3 is a diagram of six records in an example data set, along with its co ⁇ esponding sparse representation.
• the data set 300 of FIG. 3 thus has three binary attributes A], A 2 , and A 3 , such that the default value for each attribute is 1.
• the data set 300 has six records. Therefore, in a sparse representation of the data set, as represented by 302 of FIG. 3, each record is stored as a list of attribute/ value pairs, one such pair for each attribute in the record that does not have its default value. Thus, for attributes of a record that have the default value, no such pair is stored.
• the representation of the data set as stored by embodiments of the invention is sparse, in that only attribute values of a record that vary from the default values for those attributes are stored.
• the list of pairs is traversed. If the attribute in question is encountered in the list, then the co ⁇ esponding value is returned. Otherwise, the default value of the attribute is returned.
• the data is stored in a sparse representation on disk. It is noted that using this representation can dramatically reduce the memory overhead for storing a large dataset. That is, where most attribute values for a given record in a given data set will be equal to the default values for those attributes, such attribute values are not stored in the sparse representation of the invention — rather only those attribute values that vary from the default values for those attributes are stored. As will be described in the next section of the detailed description, such a sparse representation provides for greatly reduced run times in generating a data model from the data set (for example, by the EM algorithm).
• the sparse representation 302 is generated from the data set 300 by a generating mechanism 304.
• the generating mechanism 304 can in one embodiment be a software program running on a computer such as that described in the first section of the detailed description, although the invention is not so limited.
• the data set 300 and the sparse representation 302 can be stored as data representing the data set 300 and the sparse representation 302, respectively, as residing on a computer-readable medium, such as a memory or a hard disk drive. Again, the invention is not so limited.
• the method is a computer-implemented method that generates a data model according to a (modified) EM algorithm, utilizing a sparse representation of a data set.
• the data model generated is a na ⁇ ve-Bayes model, as has been described, such that the EM algorithm is used to derive parameter values of the model, as those of ordinary skill within the art can appreciate.
• the computer-implemented method is desirably realized at least in part as one or more programs running on a computer — that is, as a program executed from a computer-readable medium such as a memory by a processor of a computer.
• the programs are desirably storable on a machine- readable medium such as a floppy disk or a CD-ROM, for distribution and installation and execution on another computer.
• Nc be the (hypothetical) known number of records that belong to cluster c. If there are N total records in the data, then the estimate for c can be
• This assignment is known and refe ⁇ ed to as the maximum likelihood estimation for the cluster parameter.
• the maximum likelihood estimation for the attribute parameters can also be derived from (hypothetical) known counts.
• the cluster assignment for each record is unknown, and consequently the sufficient statistics used to derive the parameter values are not available. Given a set of parameter values, however, the expected sufficient statistics can be derived using those counts instead. In particular,
• a data set having a plurality of records is input.
• the data set desirably represents a real-world event or events — for example, the stories read by users of a web site, the television shows watched by viewers, etc.
• Each record has at least one attribute.
• the set of all attributes is refe ⁇ ed to as the attribute set for the problem domain.
• Each attribute has a default value.
• the data set is desirably such that most attribute values for a given record are equal to the default value, although the invention is not so limited.
• a sparse representation of the data set is generated, such that a sparse representation of each record of the data set is stored. This is accomplished as has been described in the previous section of the detailed description. That is, only the values of the attributes of a given record are stored that vary from the default values for the attributes - in other words, the value of an attribute of a record is stored only upon determining that the attribute of the record has a value varying from the default value for this attribute.
• a data model of the data set is generated, utilizing the sparse representation generated and stored in 402.
• this is a na ⁇ ve-Bayes model, generated by utilizing the EM algorithm to determine sufficient statistics so that the parameters of the na ⁇ ve- Bayes model can be determined.
• the data model generated specifically refers to the na ⁇ ve-Bayes model in general and utilizing the EM algorithm in particular, although the invention is not necessarily so limited.
• the EM algorithm described herein is modified to take advantage of the sparse representation generated in 402.
• the parameters ⁇ of the model are initialized in accordance with a predetermined manner, as those of ordinary skill within the art can appreciate.
• the invention is not limited to the manner by which parameter initialization is accomplished.
• parameter initialization can be accomplished by randomly assigning values, or by marginal matching, both of which are known within the art.
• the EM algorithm is performed.
• the EM algorithm has two parts, the E part in 410 to update expected sufficient statistics for each record in the data set, and the M part in 412 to update parameters regarding the data model.
• the E part and the M part are repeated in an iterative process until a predetermined convergence criterion is met.
• the invention is not particularly limited to any given convergence criterion, however.
• the joint probability of each cluster and the default record is first calculated (determined) — that is, a joint probability based on the default value of each attribute is initially generated.
• PD(c) is used to denote these initially generated values.
• the default record for which this determination is made is a record that may or may not actually exist in the data set, for which every attribute has a value equal to the default value.
• PD(c) is multiplied, for each value c, by a set of co ⁇ ection terms, where there is a single co ⁇ ection term for each attribute that does not appear in its default state.
• the posterior probability is determined based on the joint probability, and the posterior probability is co ⁇ ected upon determining that the value of an attribute of the record varies from the default value.
• ⁇ dj, ..., d n ⁇ denote the default values for the attributes ⁇ A ⁇ , .... A grasp), respectively.
• the joint probability of each cluster c and the default case is (see above)
• attribute A takes on value a, ⁇ d,.
• PD(c) is co ⁇ ected for each such non-default attribute by multiplying by the following correction term L(c, A,, aj):
• ⁇ (n) denote the set of indices of the attributes in record n that do not have their default state in case n.
• ⁇ (l) ⁇ 3 ⁇
• ⁇ (2) ⁇ 1 ⁇
• ⁇ (3) ⁇ ⁇
• ⁇ (4) ⁇ 2 ⁇
• ⁇ (5) ⁇ 2,3 ⁇
• ⁇ v ⁇ ⁇ (6) ⁇ ⁇ .
• a is the (non-default) value for attribute A t in record n.
• the posterior probability of the cluster node is obtained by dividing each joint probability by the accumulated sum of joint probabilities, where the sum is taken over all values of c.
• first sufficient statistics are incremented by the posterior probability determined in 414.
• second sufficient statistics are incremented by the postenor probability determined in 414.
• the incrementation performed in 418 is also speeded up due to the sparse representation of the data set.
• the sufficient statistics co ⁇ esponding to the default values are derived from the others after the data has been scanned.
• the remainder of the sufficient statistics can be derived as follows:
• the second sufficient statistics are incremented only upon determining that the values of the attributes of the record vary from the default values (any of the values of any of the attributes).
• the method proceeds to 412, in which the M part of the algorithm is performed.
• the parameters regarding the data model are updated. More specifically, for each value c, the cluster parameters are reset as follows:
• the E part in 410 is again performed until a predetermined convergence criterion has been reached, as has been described. Thereafter, the method proceeds to 420.
• the resulting set of parameters obtained by the EM algorithm are used to assign cluster membership to each record in the data set.
• 420 is the last part of 404, in which a data model is generated.
• the data model is then output in 422.
• the invention is not limited to the manner by which the data model is output.
• it can be output to a further analysis program or software component, that allows for conclusions to be drawn regarding the data set, as the data model was generated therefrom (specifically, the individual clusters in which each record was assigned).
• a diagram of the data model an example of which was shown in FIG. 2, can be displayed on a displayed device, or printed to a printer, etc.
• the invention provides for in one embodiment speeded-up clustering as a part of the outputting in 422.
• the model generator 500 receives as an input a sparse representation of a data set 502, and generates as an output a data model 504 co ⁇ esponding to the data set.
• the model generator 500 is a computer program, although the invention is not so limited.
• the sparse representation 502 includes a plurality of records of a data set, where each record has at least one attribute, and each attribute has a default value, as has been described. That is, the sparse representation is such that it includes the value of an attribute of a record only if the value varies from the default value for that attribute.
• the data model 504 is based on the sparse representation 502, and in one embodiment is a na ⁇ ve-Bayes model as has been described.
• the generator 500 utilizes the modified EM algorithm as described in the previous section of the detailed description to generate the data model 504.
• the generator 500 thus utilizes the data model 504 generated to speed up clustering of the sparse representation of each record.
• embodiments of the invention can be used with both continuous variables and ordered variables, if they are used with domains for which the presence or absence of these values in the data is informative.
• a continuous variable can be defined as taking on values from one or more interval of the real line; temperature is an example.
• An ordered variable can be defined as taking on values from one or more (potentially infinite) intervals from the set of integers; years of education is an example of an ordered variable.
• a density function p assigns a value to each x such that the probability that x lies in some interval (x a , Xb) is:
• p could be a Gaussian (normal) distribution.
• Equation 1 is replaced with the following for continuous variables:
• p is replaced by the distribution over the integers, such as a poisson distribution.
• missing will co ⁇ espond to a known value. For instance, consider the television example, and assume modeling the duration of time that people watches each show is the point of interest. If for a given person, no length of time is given for a particular show, it is known that the person watched the show for a duration of zero minutes. For those people that watched the show, the point of interest may be in the average and standard deviation of the duration.

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• 1999
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