Abstract:
A method, system, and computer program product provides automated determination of the size of the sample that is to be used in training a neural network data mining model that is large enough to properly train the neural network data mining model, yet is no larger than is necessary. A method of performing training of a neural network data mining model comprises the steps of: a) providing a training dataset for training an untrained neural network data mining model, the first training dataset comprising a plurality of rows of data, b) selecting a row of data from the training dataset for performing training processing on the neural network data mining model, c) computing an estimate of a gradient or cost function of the neural network data mining model, d) determining whether the gradient or cost function of the neural network data mining model has converged, based on the computed estimate of the gradient or cost function of the neural network data mining model, e) repeating steps b)-d), if the gradient or cost function of the neural network data mining model has not converged, and f) updating weights of the neural network data mining model, if the gradient or cost function of the neural network data mining model has converged.

Description:
FIELD OF THE INVENTION 
     The present invention relates to automated determination of the size of the sample that is to be used in training a neural network data mining model. 
     BACKGROUND OF THE INVENTION 
     Data mining is a technique by which hidden patterns may be found in a group of data. True data mining doesn&#39;t just change the presentation of data, but actually discovers previously unknown relationships among the data. Data mining is typically implemented as software in or in association with database systems. Data mining includes several major steps. First, data mining models are generated by based on one or more data analysis algorithms. Initially, the models are “untrained”, but are “trained” by processing training data and generating information that defines the model. The generated information is then deployed for use in data mining, for example, by providing predictions of future behavior based on specific past behavior. 
     One type of modeling that is useful in building data mining models is neural network modeling. Generally, a neural network is a set of connected input/output units where each connection has a weight associated with it. During the learning phase, the network learns by adjusting the weights so as to be able to more accurately generate an output based on the input samples. 
     Traditionally, neural network models are trained using batch methods, in which large amounts of data are used to train the models. However, problems arise with these batch-training methods because the size of the data sample to be used for training must be specified. For large datasets, if all the rows of data in the dataset are used, the computation of necessary information, such as gradient and cost function information, becomes too computationally expensive. One solution to this problem is to sample the data in the dataset and only use the sample to train the model. However, this present a problem because the proper sample size must be chosen for best results. If the sample chosen is too large, the computation is still too expensive, while if the sample chosen is too small, the trained model is not adequately predictive of the dataset. Thus, the sample size must be chosen intelligently. Because each model or type of model requires different sample sizes, there is no fixed sample size that will work properly for all cases. 
     A need arises for an automated technique that determines the size of the sample that is to be used in training a neural network data mining model. 
     SUMMARY OF THE INVENTION 
     The present invention is a method, system, and computer program product that provides automated determination of the size of the sample that is to be used in training a neural network data mining model. The present invention provides a subsample of the training dataset that is large enough to properly train the neural network data mining model, yet is no larger than is necessary 
     In one embodiment, a method of performing training of a neural network data mining model comprises the steps of: a) providing a training dataset for training an untrained neural network data mining model, the first training dataset comprising a plurality of rows of data, b) selecting a row of data from the training dataset for performing training processing on the neural network data mining model, c) computing an estimate of a gradient or cost function of the neural network data mining model, d) determining whether the gradient or cost function of the neural network data mining model has converged, based on the computed estimate of the gradient or cost function of the neural network data mining model, e) repeating steps b)-d), if the gradient or cost function of the neural network data mining model has not converged, and f) updating weights of the neural network data mining model, if the gradient or cost function of the neural network data mining model has converged. 
     In one aspect of the present invention, the selecting step comprises the step of randomly selecting the row of data from the training dataset for performing training processing on the neural network data mining model. 
     In one aspect of the present invention, the method further comprises the step of storing a number of rows of data that were selected. 
     In one aspect of the present invention, the method further comprises the step of performing additional training processing to the neural network data mining model using the training dataset, wherein the additional training processing is performed using a number of rows of data equal to the stored number of rows that were selected. 
     In one aspect of the present invention, the method further comprises the step of performing additional training processing to the neural network data mining model by repeating steps b)-f) until a decrease in an error of the neural network data mining model is less than a predefined threshold. 
     In one embodiment, a method of performing training of a neural network data mining model comprises the steps of: a) providing a training dataset for training an untrained neural network data mining model, the first training dataset comprising a plurality of rows of data, b) selecting, in each of a plurality of threads, a row of data from the training dataset for performing training processing on the neural network data mining model, c) computing, in each of the plurality of threads, an estimate of a gradient or cost function of the neural network data mining model, d) consolidating the computed estimate of the gradient or cost function of the neural network data mining model from each thread to form a consolidated gradient or cost function, e) determining whether the consolidated gradient or cost function of the neural network data mining model has converged, based on the consolidated computed estimate of the gradient or cost function of the neural network data mining model, e) repeating steps b)-d), if the consolidated gradient or cost function of the neural network data mining model has not converged, and f) updating weights of the neural network data mining model, if the consolidated gradient or cost function of the neural network data mining model has converged. 
     In one aspect of the present invention, the selecting step comprises the step of randomly selecting, in each of the plurality of threads, the row of data from the training dataset for performing training processing on the neural network data mining model. 
     In one aspect of the present invention, the method further comprises the steps of consolidating a number of rows of data that were selected by each thread to from a consolidated gradient or cost function, and storing the consolidated number of rows. 
     In one aspect of the present invention, the method further comprises the steps of performing additional training processing to the neural network data mining model using the training dataset, wherein the additional training processing is performed using a number of rows of data equal to the stored consolidated number of rows. 
     In one aspect of the present invention, each thread processes an equal portion of the number of rows of data processed. 
     In one aspect of the present invention, the method further comprises the steps of performing additional training processing to the neural network data mining model by repeating steps b)-f) until a decrease in an error of the neural network data mining model is less than a predefined threshold. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The details of the present invention, both as to its structure and operation, can best be understood by referring to the accompanying drawings, in which like reference numbers and designations refer to like elements. 
         FIG. 1  is an exemplary block diagram of a data mining system, in which the present invention may be implemented. 
         FIG. 2  is an exemplary block diagram of a database/data mining system shown in  FIG. 1 . 
         FIG. 3  is an exemplary block diagram of a database/data mining system shown in  FIG. 1 . 
         FIG. 4  is an exemplary data flow diagram of a data mining process, which may be implemented in the system shown in  FIG. 1 . 
         FIG. 5  is an exemplary block diagram of an artificial neural network of the type that may be used in a neural network data mining model. 
         FIG. 6  is an exemplary flow diagram of backpropagation learning process in the artificial neural network shown in  FIG. 5 . 
         FIG. 7  is an exemplary flow diagram of a subsampling process for use in a single threaded embodiment of the present invention. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     An exemplary data mining system  100 , in which the present invention may be implemented, is shown in  FIG. 1 . System  100  includes a database/data mining system  102  that is connected to a variety of sources of data. For example, system  102  may be connected to a plurality of internal or proprietary data sources, such as systems  104 A- 104 N. Systems  104 A- 104 N may be any type of data source, warehouse, or repository, including those that are not publicly accessible. Examples of such systems include inventory control systems, accounting systems, scheduling systems, etc. System  102  may also be connected to a plurality of proprietary data sources that are accessible in some way over the Internet  108 . Such systems include systems  106 A- 106 N, shown in  FIG. 1 . Systems  106 A- 106 N may be publicly accessible over the Internet  108 , they may be privately accessible using a secure connection technology, or they may be both publicly and privately accessible. System  102  may also be connected to other systems over the Internet  108 . For example, system  110  may be privately accessible to system  102  over the Internet  108  using a secure connection, while system  112  may be publicly accessible over the Internet  108 . 
     The common theme to the systems connected to system  102  is that the connected systems all are potential sources of data for system  102 . The data involved may be of any type, from any original source, and in any format. System  102  has the capability to utilize and all such data that is available to it. 
     An exemplary embodiment of database/data mining system  102  is shown in  FIG. 2 . System  102  is a database management system that includes data mining functionality. Database management system  202  is connected to data sources  204 , such as the proprietary and public data sources shown in  FIG. 1 . Database management system includes two main components, data  206 , and database management system (DBMS) engine  208 . Data  206  includes data, typically arranged as a plurality of data tables, such as relational data tables, as well as indexes and other structures that facilitate access to the data. DBMS engine  208  typically includes software that receives and processes queries of the database, obtains data satisfying the queries, and generates and transmits responses to the queries. DBMS engine  208  also includes data mining block  210 , which provides DBMS engine  208  with the capability to obtain data and perform data mining processing on that data, so as to respond to requests for data mining processed data from one or more users, such as user  212 . 
     An exemplary block diagram of a database/data mining system  102 , shown in  FIG. 1 , is shown in  FIG. 3 . Database/data mining system  102  is typically a programmed general-purpose computer system, such as a personal computer, workstation, server system, and minicomputer or mainframe computer. Database/data mining system  102  includes processor (CPU)  302 , input/output circuitry  304 , network adapter  306 , and memory  308 . CPU  302  executes program instructions in order to carry out the functions of the present invention. Typically, CPU  302  is a microprocessor, such as an INTEL PENTIUM® processor, but may also be a minicomputer or mainframe computer processor. Input/output circuitry  304  provides the capability to input data to, or output data from, database/data mining system  102 . For example, input/output circuitry may include input devices, such as keyboards, mice, touchpads, trackballs, scanners, etc., output devices, such as video adapters, monitors, printers, etc., and input/output devices, such as, modems, etc. Network adapter  306  interfaces database/data mining system  102  with network  310 . Network  310  may be any standard local area network (LAN) or wide area network (WAN), such as Ethernet, Token Ring, the Internet, or a private or proprietary LAN/WAN. 
     Memory  308  stores program instructions that are executed by, and data that are used and processed by, CPU  302  to perform the functions of the database/data mining system  102 . Memory  308  may include electronic memory devices, such as random-access memory (RAM), read-only memory (ROM), programmable read-only memory (PROM), electrically erasable programmable read-only memory (EEPROM), flash memory, etc., and electro-mechanical memory, such as magnetic disk drives, tape drives, optical disk drives, etc., which may use an integrated drive electronics (IDE) interface, or a variation or enhancement thereof, such as enhanced IDE (EIDE) or ultra direct memory access (UDMA), or a small computer system interface (SCSI) based interface, or a variation or enhancement thereof, such as fast-SCSI, wide-SCSI, fast and wide-SCSI, etc, or a fiber channel-arbitrated loop (FC-AL) interface. 
     Memory  308  includes data  206 , database management processing routines  312 , data mining processing routines  314 , and operating system  316 . Data  206  includes data, typically arranged as a plurality of data tables, such as relational database tables, as well as indexes and other structures that facilitate access to the data. Database management processing routines  312  are software routines that provide database management functionality, such as database query processing. Data mining processing routines  314  are software routines that implement the data mining processing performed by the present invention. In particular, data mining processing routines  314  include software routines that perform the center based cluster analysis of the present invention. Preferably, this data mining processing is integrated with database management processing. For example, data mining processing may be initiated by receipt of a database query, either in standard SQL or in the form of extended SQL statements, or data mining processing may be initiated from a programming language, such as JAVA. Operating system  320  provides overall system functionality. 
     Although the example shown in  FIG. 3  illustrates a single processor system, the present invention contemplates implementation on a system or systems that provide multi-processor, multi-tasking, multi-process, and/or multi-thread computing. Multi-processor computing involves performing computing using more than one processor. Multi-tasking computing involves performing computing using more than one operating system task. A task is an operating system concept that refers to the combination of a program being executed and bookkeeping information used by the operating system. Whenever a program is executed, the operating system creates a new task for it. The task is like an envelope for the program in that it identifies the program with a task number and attaches other bookkeeping information to it. Many operating systems, including UNIX®, OS/2®, and WINDOWS®, are capable of running many tasks at the same time and are called multitasking operating systems. Multi-tasking is the ability of an operating system to execute more than one executable at the same time. Each executable is running in its own address space, meaning that the executables have no way to share any of their memory. This has advantages, because it is impossible for any program to damage the execution of any of the other programs running on the system. However, the programs have no way to exchange any information except through the operating system (or by reading files stored on the file system). Multi-process computing is similar to multi-tasking computing, as the terms task and process are often used interchangeably, although some operating systems make a distinction between the two. 
     In most operating systems, there is a one-to-one relationship between the task and the program, but some operating systems allow a program to be divided into multiple tasks. Such systems are called multithreading operating systems. A thread—sometimes called an execution context or a lightweight process—is a single sequential flow of control within a program. A thread itself is not a program; it cannot run on its own. Rather, it runs within a program. A thread is similar to a real process in that a thread and a running program are both a single sequential flow of control. However, a thread is considered lightweight because it runs within the context of a full-blown program and takes advantage of the resources allocated for that program and the program&#39;s environment. As a sequential flow of control, a thread must carve out some of its own resources within a running program. (It must have its own execution stack and program counter for example.) The code running within the thread works only within that context. A thread may also be referred as a lightweight process or as an execution context. 
     A thread is a semi-process, which has its own stack, and executes a given piece of code. Unlike a real process, the thread normally shares its memory with other threads (where as for processes we usually have a different memory area for each one of them). A Thread Group is a set of threads all executing inside the same process. They all share the same memory, and thus can access the same global variables, same heap memory, same set of file descriptors, etc. All these threads execute in parallel (i.e. using time slices, or if the system has several processors, then really in parallel). 
     The advantage of using a thread group instead of a normal serial program is that several operations may be carried out in parallel, and thus events can be handled immediately as they arrive (for example, if we have one thread handling a user interface, and another thread handling database queries, we can execute a heavy query requested by the user, and still respond to user input while the query is executed). 
     The advantage of using a thread group over using a process group is that context switching between threads is much faster then context switching between processes (context switching means that the system switches from running one thread or process, to running another thread or process). Also, communications between two threads is usually faster and easier to implement then communications between two processes. On the other hand, because threads in a group all use the same memory space, if one of them corrupts the contents of its memory, other threads might suffer as well. With processes, the operating system normally protects processes from one another, and thus if one corrupts its own memory space, other processes won&#39;t suffer. Another advantage of using processes is that they can run on different machines, while all the threads have to run on the same machine (at least normally). 
     An exemplary data flow diagram of a data mining process, including building and scoring of models and generation of predictions/recommendations, is shown in  FIG. 4 . The training/model building step  402  involves generating the models that are used to perform data mining recommendation and prediction. The inputs to training/model building step  402  include training parameters  404 , training data  406 , and untrained models  408 . Untrained models  408  include algorithms that process the training data  406  in order to actually build the models. In particular, untrained models  408  includes algorithms that are used to build data mining models that are based on neural networks. Training parameters  404  are parameters that are input to the data-mining model building algorithms to control how the algorithms build the models. Training data  406  is data that is input to the algorithms and which is used to actually build the models. 
     Training/model building step  402  invokes the data mining model building algorithms included in untrained models  408 , initializes the algorithms using the training parameters  404 , processes training data  406  using the algorithms to build the model, and generates trained model  410 . Trained model  410  includes information, such as functions, that implements the conditions and decisions that make up an operational model. In particular, neural network models implement a mapping between the input space and the output space. This mapping may be implemented, for example, by a combination of basis functions, which define the neural network topology, and transfer functions, which define the transfer of information between nodes in the network. Trained model  410  may also be evaluated and adjusted in order to improve the quality, i.e. prediction accuracy, of the model. Trained model  410  is then encoded in an appropriate format and deployed for use in making predictions or recommendations. 
     Scoring step  412  involves using the deployed trained model  410  to make predictions or recommendations based on new data that is received. Trained model  410 , prediction parameters  414 , and prediction data  416  are input to scoring step  412 . Trained models  410  include information defining the model that was generated by model building step  402 . Prediction parameters  414  are parameters that are input to the scoring step  418  to control the scoring of scoring data  416  against trained model  410  and are input to the selection and prediction/recommendation step  420  to control the selection of the scored data and the generation of predictions and recommendations 
     Scoring data  416  is processed according trained model  410 , as controlled by prediction parameters  414 , to generate one or more scores for each row of data in scoring data  416 . The scores for each row of data indicate how closely the row of data matches attributes of the model, how much confidence may be placed in the prediction, how likely each output prediction/recommendation to be true, and other statistical indicators. Scored data  418  is output from scoring step  412  and includes predictions or recommendations, along with corresponding probabilities for the scored data. 
     Scored model  418  is input to selection and prediction/recommendation generation step, which evaluates the probabilities associated with the predictions/recommendations and selects at least a portion of the predictions/recommendations. The selected predictions/recommendations are those having probabilities meeting the selection criteria. The selection criteria may be defined by desired results data and/or by predefined or default criteria included in selection/generation step  420 . In addition, the selection criteria may include a limit on the number of predictions/recommendations that are to be selected, or may indicate that the predictions/recommendations are to be sorted based on their associated probabilities. The selected predictions/recommendations are output  422  from step  420  for use in data mining. 
     An example of an artificial neural network of the type that may be used in a neural network data mining model is shown in  FIG. 5 . Neural networks, such as network  500 , are typically organized in layers. Layers are made up of a number of interconnected nodes, such as nodes  502 A and  502 B, each of which contains an activation function. Patterns are presented to the network via the input layer  504 , which communicates to one or more hidden layers  506  where the actual processing is done via a system of weighted connections  508 . The hidden layers then link to an output layer  510  where the answer is output. 
     Most artificial neural networks contain some form of learning rule, which modifies the weights of the connections according to the input patterns that are presented. In a sense, artificial neural networks learn by example as do their biological counterparts. 
     There are many different kinds of learning rules used by neural networks. A typical well-known learning rule is the delta rule. The delta rule is often utilized by the most common class of artificial neural networks, which are called backpropagational neural networks (BPNNs). Backpropagation refers to the backwards propagation of error in the neural network. 
     With the delta rule, as with other types of backpropagation, learning is a supervised process that occurs with each cycle or epoch. This backpropagation learning process is shown in  FIG. 6 . Each time the network is presented with a new input pattern  602  (a new cycle or epoch), the input is filtered through a weight function  604 , such as the function: I=f(ΣW i ·Input), where W i  are the weights. The output  606  is generated by a combination of various nodes in the hidden layers  608 . Within each hidden layer node is an activation function that polarizes network activity and helps stabilize it. The weights are adjusted through a forward activation flow of outputs, and the backwards error propagation of weight adjustments using a backpropagation function, such as 
               W   new     =       W   old     -       β   ⁢           ⁢       ∇   E     .       W             
More simply, when a neural network is initially presented with a pattern, it makes a random guess as to what it might be. It then sees how far its answer was from the actual one and makes an appropriate adjustment to its connection weights.
 
     Backpropagation performs a gradient descent within the solution&#39;s vector space towards a global minimum along the steepest vector of the error surface. The global minimum is that theoretical solution with the lowest possible error. The error surface itself is a hyperparaboloid but is seldom smooth. Indeed, in most problems, the solution space is quite irregular with numerous ‘pits’ and ‘hills’, which may cause the network to settle down in a local minimum, which is not the best overall solution. When the error has reached the global minimum, or at least a local minimum of acceptable quality, the error function may be said to have converged. Alternatively to using just the error function, a cost function that takes into account the cost of the various nodal transitions, as well as the error, may be used. 
     An exemplary flow diagram of a subsampling process  700 , for use in a single threaded embodiment of the present invention, is shown in  FIG. 7 . Process  700  begins with step  702 , in which a training epoch is started. In step  704 , a randomly selected row of data in the training dataset is processed for training of a neural network model. In step  706 , an estimate of the gradient of an error or cost function of the neural network model being trained is computed. For example, error or cost functions, such as those described above, may be used. In step  708 , it is determined whether the gradient descent of the error or cost functions has converged. The determination of whether the gradient descent of the error or cost functions has converged is performed at regular intervals of N rows, where N≧1. Where N=1, the determination is performed for each row. Where N&gt;1, the determination is performed less frequently, which improves processing performance of the system. If the gradient descent of the error or cost functions has not converged, then the process loops back to step  704 , and an additional row of data in the training dataset is processed. If the gradient descent of the error or cost functions has converged, then the process continues with step  710 , in which the weights of the neural network model are updated. In addition, the number of rows that were processed in order to achieve convergence is made available. The number of rows that were processed are a subset of the total number of rows in the training dataset, yet convergence of the neural network data mining model has been achieved. Thus, process  700  provides a subsample of the training dataset that is large enough to properly train the neural network data mining model, yet is no larger than is necessary. 
     In step  712 , it is determined, for each training epoch, whether the error of the neural network data mining model has decreased significantly from the previous training epoch. A significant decrease is typically defined as being greater than a predefined threshold; thus, a decrease that is not significant is a decrease that is less than the predefined threshold. If, in step  714 , it is determined that the decrease in the error of the neural network data mining model is significant, then the process continues with step  714 , in which at least one additional training epoch is performed. If, in step  712 , it is determined that the decrease in the error of the neural network data mining model is not significant, then the process continues with step  716 , in which the training of the neural network data mining model is complete. 
     In step  714 , additional training epochs are performed. Each additional training epoch uses the same dataset, but, since in step  704 , the rows of data actually processed are randomly selected, each epoch uses a different selection of rows of data. In one embodiment, the number of rows of data used in each successive training epoch is the number of rows that were processed in order to achieve convergence in the first training epoch. Processing of this number of rows will provide to each later epoch a subsample of the training dataset that is large enough to properly train the neural network data mining model, yet is no larger than is necessary. Alternatively, steps  702 - 712  may be repeated by later training epochs, in order to more precisely tailor the number of rows of data processed to the particular rows of data that are selected in the epoch. Repeating steps  702 - 712  will ensure that the neural network data mining model will properly converge, even if the data selected in later training epochs differs significantly from the data selected in the initial training epoch. 
     Process  700  may be implemented in a single-threaded computing environment, or may also be advantageously implemented in a multi-threaded environment. In a multi-threaded environment, process  700  begins with step  702 , in which a training epoch is started. In step  704 , each thread processes a randomly selected row of data in the training dataset for training of a neural network model. In step  706 , each thread computes an estimate of the gradient of an error or cost function of the neural network model being trained. For example, error or cost functions, such as those described above, may be used. In step  708 , the computed gradients from each thread are consolidated to form a single gradient, and it is determined whether the consolidated gradient descent of the error or cost functions has converged. The determination of whether the gradient descent of the error or cost functions has converged is performed at regular intervals of N rows, where N≧1. Where N=1, the determination is performed for each row. Where N&gt;1, the determination is performed less frequently, which improves processing performance of the system. If the gradient descent of the error or cost functions has not converged, then the process loops back to step  704 , and each thread processes an additional row of data in the training dataset. If the gradient descent of the error or cost functions has converged, then the process continues with step  710 , in which the weights of the neural network model are updated. In addition, the number of rows that were processed by each thread in order to achieve convergence is consolidated and made available. The consolidated number of rows that were processed are a subset of the total number of rows in the training dataset, yet convergence of the neural network data mining model has been achieved. Thus, process  700  provides a subsample of the training dataset that is large enough to properly train the neural network data mining model, yet is no larger than is necessary. 
     In step  712 , it is determined, for each training epoch, whether the error of the neural network data mining model has decreased significantly from the previous training epoch. A significant decrease is typically defined as being greater than a predefined threshold; thus, a decrease that is not significant is a decrease that is less than the predefined threshold. If, in step  714 , it is determined that the decrease in the error of the neural network data mining model is significant, then the process continues with step  714 , in which at least one additional training epoch is performed. If, in step  712 , it is determined that the decrease in the error of the neural network data mining model is not significant, then the process continues with step  716 , in which the training of the neural network data mining model is complete. 
     In step  714 , additional training epochs are performed. Each additional training epoch uses the same dataset, but, since in step  704 , the rows of data actually processed are randomly selected, each epoch uses a different selection of rows of data. In one embodiment, the number of rows of data used in each successive training epoch is the consolidated number of rows that were processed by all threads in order to achieve convergence in the first training epoch. Processing of this number of rows will provide to each later epoch a subsample of the training dataset that is large enough to properly train the neural network data mining model, yet is no larger than is necessary. The consolidated number of rows is used as the total number of rows, and an equal portion of that total number of rows may be processed by each thread. Alternatively, steps  702 - 712  may be repeated by later training epochs, in order to more precisely tailor the number of rows of data processed to the particular rows of data that are selected in the epoch. Repeating steps  702 - 712  will ensure that the neural network data mining model will properly converge, even if the data selected in later training epochs differs significantly from the data selected in the initial training epoch. 
     It is important to note that while the present invention has been described in the context of a fully functioning data processing system, those of ordinary skill in the art will appreciate that the processes of the present invention are capable of being distributed in the form of a computer readable medium of instructions and a variety of forms and that the present invention applies equally regardless of the particular type of signal bearing media actually used to carry out the distribution. Examples of computer readable media include recordable-type media such as floppy disc, a hard disk drive, RAM, and CD-ROM&#39;s, as well as transmission-type media, such as digital and analog communications links. 
     Although specific embodiments of the present invention have been described, it will be understood by those of skill in the art that there are other embodiments that are equivalent to the described embodiments. Accordingly, it is to be understood that the invention is not to be limited by the specific illustrated embodiments, but only by the scope of the appended claims.