Patent Publication Number: US-7716011-B2

Title: Strategies for identifying anomalies in time-series data

Description:
BACKGROUND 
   Many analysis tools monitor the behavior of an environment by analyzing time-series representations of the environment&#39;s key metrics. Consider a manufacturing process that employs one or more sensors. An analysis tool can assess the behavior of the manufacturing process by analyzing data generated by its sensors over a span of time (thus defining time-series data). Generally, an analysis tool can flag a dramatic change in the time-series data (e.g., a spike or a dip) as a potential malfunction within the environment. 
   In detecting spikes and dips, it is common to establish a model that defines the expected operation of the environment. For instance, an analysis tool may use various regression techniques to define a model that follows the general course of the measured time-series data. The analysis tool can then flag suspected anomalies by comparing each data point of the time-series data with the model. Large deviations are indicative of anomalies. 
   However, there are various challenges in properly detecting anomalies using the above-described approach. For instance, an environment may produce time-series data that is naturally noisy. This may make it difficult to define a model which accurately tracks the general course of the time-series data. Such difficulties may result in failing to identify actual anomalies, or in inaccurately labeling normal behavior as anomalous. 
   In addition to accuracy, in many monitoring environments, it is desirable to quickly identify anomalies soon after they occur. This may require an analysis tool to process a very large amount of data in a small amount of time, which, in turn, places certain constraints on the complexity of the algorithms used by the analysis tool. For instance, an algorithm may produce desirable accuracy with few false positives, yet may be too computationally complex to operate in a real-time manner. It is also desirable that the analysis tool scale well to evolving conditions within the environment being monitored. 
   In view of these illustrative factors, there is a need for effective strategies for detecting anomalies in time-series data. 
   SUMMARY 
   A strategy is described for identifying anomalies in time-series data. The time-series data may reflect any type of information collected from any type of environment. In one illustrative case, the time-series data may reflect traffic in a wide area network environment. In another illustrative case, the time-series data may reflect a flow of financial transactions. In another illustrative case, the time-series data may reflect data received from a physical sensor, and so on. In any case, the strategy can identify dramatic spikes and dips and/or or other deviant phenomena in the time-series data as potential anomalies. 
   The strategy operates by dividing the collected time-series data into a plurality of collected data segments. Then the strategy uses a modeling technique to fit a plurality of local models to the respective plurality of collected data segments. The term “local model” refers to a model of the local behavior of the time-series for a limited segment of time. In one case, the strategy can fit each local model subject to various constraints imposed by its neighboring local models. Considered in aggregate, the plurality of local models form a global model. The strategy identifies large differences between the collected time-series data and the global model as potential anomalies. 
   Given a space of models (such as a space of linear or n-polynomial models) the modeling technique can generate a local model to fit a data segment by choosing the model from the given space that minimizes a sum of error values. The error values correspond to differences between the model data and the collected time-series data. In a first exemplary approach, the modeling technique uses an absolute value (L 1 ) measure of the error values to generate all of the local models. The L 1  measure helps reduce the impact of Gibbs phenomenon. Gibbs phenomenon occurs when one or more outlying data points exert undue influence on the shape of the global model, potentially causing false positives and false negatives in the detection of anomalies in the time-series data. 
   In a second exemplary approach, the modeling technique uses the L 1  measure only for portions of the time-series data that are projected to be anomalous. The modeling technique can use a squared-term (L 2 ) measure of error values for portions of the time-series data that are projected to be normal. The strategy benefits from the use of the L 2  measure for normal portions because it is more computationally efficient than the L 1  measure. Moreover, a linear approximation can be used when performing computations based on the L 1  measure (for those portions of the time-series data that are considered anomalous). This further expedites the processing. 
   In a third exemplary approach, the modeling technique uses an iterative procedure to generate the local models. For example, the modeling technique can use an expectation-maximization procedure. In an expectation phase of the procedure, the modeling technique labels each portion of the time-series data as anomalous or normal. An anomalous portion is associated with a Laplacian probability distribution, while a normal portion is associated with a Gaussian (Normal) probability distribution. In a maximization phase, the modeling technique generates parameters that define the local models. In doing so, the maximization phase performs a cost-minimizing operation using the L 1  measure for anomalous portions and the L 2  measure for normal portions (where such portions have been labeled in the preceding expectation phase). The next iteration of the expectation phase uses the newly generated local models as a reference in re-labeling portions of the time-series data as either anomalous or normal The modeling technique repeats the above-described procedure for a predefined number of times or until it produces stable labeling results. 
   The strategy confers a number of benefits. According to one exemplary benefit, the strategy can provide reliable results in a computationally-efficient manner. This aspect makes the strategy suitable for the real-time analysis of large quantities of data. The strategy can be effectively used to analyze problems in a wide area network environment. 
   Additional exemplary implementations and attendant benefits are described in the following. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
       FIG. 1  shows an exemplary system for detecting anomalies in any type of environment being monitored. 
       FIG. 2  shows an exemplary application of the system of  FIG. 1  to the detection of traffic anomalies in a network environment. 
       FIG. 3  shows exemplary processing functionality that can be used to implement any aspect of the system of  FIG. 1 . 
       FIG. 4  shows an overview of an exemplary procedure for detecting anomalies. 
       FIG. 5  shows one exemplary procedure for analyzing time-series data collected using the procedure of  FIG. 4 . 
       FIG. 6  shows one exemplary procedure for generating a global model in the procedure of  FIG. 5  using a combination of L 1  and L 2  measures. 
       FIG. 7  shows an exemplary variation of the procedure of  FIG. 6 , in which an expectation-maximization approach is used to iteratively determine how the L 1  and L 2  measures are to be applied in generating the global model. 
       FIGS. 8-11  illustrate the fitting of local models to time-series data, and the use of the L 1  measure to reduce the effects of Gibbs phenomenon. 
       FIG. 12  shows an output of the system of  FIG. 1 , in which anomalies have been identified in the time-series data. 
       FIGS. 13-15  collectively show computer instructions that can be used to implement the system of  FIG. 1  according to one exemplary implementation. 
   

   The same numbers are used throughout the disclosure and figures to reference like components and features. Series  100  numbers refer to features originally found in  FIG. 1 , series  200  numbers refer to features originally found in  FIG. 2 , series  300  numbers refer to features originally found in  FIG. 3 , and so on. 
   DETAILED DESCRIPTION 
   This disclosure sets forth a strategy for detecting anomalies in time-series data. The strategy can be manifested in various systems, apparatuses, modules, procedures, storage mediums, data structures, and other forms. 
   This disclosure includes the following sections. Section A describes an exemplary system for detecting anomalies in time-series data. Section B describes exemplary procedures that explain the operation of the system of Section A. 
   A. Exemplary System 
   As a preliminary note, any of the functions described with reference to the figures can be implemented using software, firmware, hardware (e.g., fixed logic circuitry), manual processing, or a combination of these implementations. The term “logic, “module,” “component,” “system” or “functionality” as used herein generally represents software, firmware, hardware, or a combination of the elements. For instance, in the case of a software implementation, the term “logic,” “module,” “component,” “system,” or “functionality” represents program code that performs specified tasks when executed on a processing device or devices (e.g., CPU or CPUs). The program code can be stored in one or more computer readable memory devices. 
   More generally, the illustrated separation of logic, modules, components, systems, and functionality into distinct units may reflect an actual physical grouping and allocation of software, firmware, and/or hardware, or can correspond to a conceptual allocation of different tasks performed by a single software program, firmware program, and/or hardware unit. The illustrated logic, modules, components, systems, and functionality can be located at a single site (e.g., as implemented by a processing device), or can be distributed over plural locations. 
   The terms “machine-readable media” or the like refers to any kind of medium for retaining information in any form, including machine-readable storage media encompassing various kinds of storage devices (magnetic, optical, static, etc.), and machine-readable communication media encompassing transitory forms for representing information, including various hardwired and/or wireless links for transmitting the information from one point to another. 
   A.1. System Overview 
     FIG. 1  shows one exemplary framework for detecting anomalies in time-series data in any kind of environment  102  being monitored. The term “time-series data” refers to data that describes the behavior of the environment  102  over the course of time. The time-series data may sample the behavior at fixed or irregular intervals of time, or some combination thereof 
   The term “anomaly” is intended to have broad connotation. An anomaly represents an event that departs in some noted way from a general or expected trend in the time-series data. In a classic case, an anomaly may be associated with a dramatic change in the course of the time-series data, such as an upward spike or downward dip. In most of the examples presented here, an anomaly is associated with the improper functioning of the environment  102 . But in other cases, an anomaly may be associated with a desirable event. 
   The environment  102  can represent any target of monitoring activity. In one case, the environment  102  may correspond to any setting that includes one or more sensors. The sensors can generate time-series data that reflects changes in the physical characteristics of the environment  102 . For example, the environment  102  may correspond to a room, machine, process, etc., having temperature sensors, motion sensors, pressure sensors, and so on. In another case, the environment  102  may correspond to a financial system or business process. This environment  102  can generate time-series data that reflects a stream of financial transactions. In another case, the environment  102  may correspond to an electrical data processing device. Such an environment  102  may generate time-series data that reflects the memory consumption of the device, the processing speed of the device, and so on. In another case, the environment  102  may correspond to a network setting. Such an environment  102  may exhibit behaviors that can be monitored to provide time-series data reflecting processing loads experienced by different components in the network environment, the flow of messages exchanged among components, and so forth.  FIG. 2 , to be described in the subsection A.2 (below), provides additional details regarding a monitoring operation that can be conducted in one network-related environment. The environment  102  can be or represent yet other types of environmental settings. 
   In any event, the environment  102  may provide the time-series data using one or more collection modules  104 . The collection modules  104  monitor or measure the behavior(s) of the environment  102 . The collection modules  104  can correspond to one or more physical sensors, one or more data collection logs, a manual data collection operation, and so on. The collection modules  104  produce one or more instances of time-series data  106 . Each instance of the time-series data identifies some measured variable relative to the passage of time. More specifically, the time-series data  106  may include one or more fields that describe or indicate the value of the variable being monitored. For example, time-series data that reflects temperature within the environment  102  can include a field that identifies temperature measurements. Measured values of a variable can be expressed in numerical form. Alternatively, or in addition, the measured values can be expressed as a discrete set of values which can be converted into numerical form for analysis. The time-series data can also include one or more timestamp fields that identify that time at which the data has been collected (unless this timing information can be inferred, e.g., as in the case in which the data is collected at fixed intervals of time). 
   An analysis system  108  detects anomalies in the time-series data  106 . To perform this function, the analysis system  108  includes several modules, enumerated below. 
   The analysis system  108  includes a data receiving module  110  to receive the time-series data  106  from the environment  102  being monitored. The data receiving module  110  can operate using a push model, a pull model, or some combination of a push model and pull model. That is, the data receiving module  110  can passively receive data that is transmitted to it by the environment  102 , actively receive the data by polling the environment  102 , or by using some combination of passive receipt and active polling. The data receiving module  110  can optionally translate received values into a form usable by the analysis system  108 . Such translation can include, but is not limited to, any one of or combination of: translation of format; translation of discrete values into numerical values; conversions of units, and so on. The data receiving module  110  can store the collected data  106  in one or more data stores prior to analysis. 
   An anomaly analysis module (AAM)  112  receives the time-series data  106  from the data receiving module  110  and detects anomalies within this data  106 . Section B (below) describes exemplary techniques that the AAM  112  can employ to detect anomalies. 
   A result output module  114  conveys an output of the AAM  112 . For example, the result output module  114  can display a graph which plots the time-series data  106 . The output module  114  can indicate, such as by graphically marking, potential anomalies within the data  106 . In addition, or alternatively, the result output module  114  can provide a tabular output, indicating potential anomalies within a textual representation of the data  106  in any suitable manner, e.g., by highlighting information, changing the size or color of the information, and so on. The result output module  114  can produce yet other types of output presentations, including non-visual (e.g., audio) types of presentations. The output can also be stored for later retrieval in any manner. 
   An optional remediation module  116  can perform any type of action based on identified anomalies. For example, the remediation module  116  can take corrective action within the environment  102  based on the output of the AAM  112 . Consider the case in which the environment  102  is a process within a manufacturing plant. The remediation module  116  can change the behavior of the process to reduce the occurrence of the type of anomalies detected by the AAM  112 . 
   A.2. Application to Detection of Anomalies in a Network Environment 
     FIG. 2  shows an application of the analysis system  108  of  FIG. 1  to monitoring the behavior of a network environment  202 . 
   The network environment  202  can allow a plurality of client devices to interact with one or more content providers. That is, a client device may interact with a content provider to receive content from the provider, or more generally, to receive any service(s) being offered by the content provider. In one case, the content provider may receive searches from the client devices, and, in return, supply search results and other content to the requesting client devices. 
   The network environment  202  shows a representative content provider data center  204 . The content provider data center  204  may administer the services provided by the content provider in conjunction with various distribution nodes (not shown). The content provider data center  204  can include one or more server computers  206 , data storage devices, routers, and/or other data processing equipment. Although not shown, the content provider data center  204  may organize its resources in conventional fashion as a collection of front-end resources (with which the users directly interact) and a collection of back-end resources. 
   The network environment  202  shows a collection of client devices, including representative client device  208  and representative client device  210 . The client devices can generally correspond to any type of data processing device through which users may gain access to the network environment  202 .  FIG. 2  illustrates the client devices as personal computers, but the client devices can also correspond to personal digital assistant (PDA) devices, stylus-type input devices, mobile telephone devices, set-top box devices, game console devices, and so on. The client devices need not be the same; they can differ. As shown, representative client device  208  is attempting to interact with the content provider data center  204  (via connection  212 ). For example, the representative client device  208  may generate an HTTP request to retrieve information from the content provider data center  204 . 
   The network environment  202  may correspond to the Internet or some other wide area network. As such, the network environment  202  can integrate a large number of component networks. The term “component network” generally refers to network-related functionality that is a part of the more encompassing network environment  202 . For example, as broadly used herein, a network component may correspond to any kind of communication system, an Internet Service Provider (ISP) or any other network provider, an Autonomous System, an administrative domain, and so forth. The network components can be administered by different respective entities. Further, many of the network components can be considered independent in the sense that they are not generally under the direct control of the content provider. Each network component may include any type of data processing equipment, such as one or more server computers, data stores, data routers and so forth.  FIG. 2  specifically illustrates four representative network components ( 214 ,  216 ,  218 , and  220 ), although it will be appreciated that an actual network environment may include many more such network components. Representative network component  214  includes one or more server computers  222 . 
   The network environment  202  can include various data collection modules (denoted by the letter “C”) dispersed throughout the network environment  202 . For example, the content provider data center  204  can include zero, one or more collection modules  224 . Any client device can include one or more data collection modules; for instance, representative client device  210  includes zero, one or more data collection modules  226 . Any network component can also include one or more data collection modules; for instance, representative network component  214  includes zero, one or more data collection modules  228 . Still other components or locations within the network environment  202  may include data collection modules. 
   In the network environment  202 , a data collection module may comprise any functionality for monitoring the behavior of some aspect of the environment  202 . Each data collection module may have a unique vantage point as to what is happening in the network environment  202 . Accordingly, each data collection module may be able to monitor different types of events. For example, a data collection module can monitor any combination or permutation of: messages being sent, received, forwarded, processed, stored, and so on. A data collection module can log salient information regarding an identified message, such as information which identifies the type of message that has been detected, information which identifies the time at which the message was sent and/or received (or information which identifies any other action that was performed on or by the message), information which identifies any explicit status information associated with the message, and so forth. 
   In one case, the data collection modules may provide information which reveals a flow of message traffic through the network environment  212 . For example, the data collection modules can provide information regarding the number of messages that have been transmitted by any component, forwarded by any component, received by any component, and so on. 
   In addition to information that is directly communicated by the data collection modules, the network environment  202  or the analysis system  108  can infer other behavior of the network environment  202 . For example, the analysis system  108  can collect historical data that reflects the usual manner of operation of certain parts of the network environment  202 . If the network environment  202  becomes inactive relative to past performance, the analysis system  108  can record one or more events that identify this discrepancy. These are negative events in the sense that they register occurrences which should have happened (but have not). 
   The analysis system  108  can receive the time-series data generated by the various data collection modules. The analysis system  108  can identify anomalies in this time-series data. In one case, the analysis system  108  can combine the results of different data collection modules to form aggregated time-series data. This aggregating operation may involve removing redundant data that has been logged by multiple data collection modules. Alternatively, or in addition, the analysis system  108  can perform independent analysis on different streams of time-series data collected at different points in the network environment  202 . Due to its efficiency (to be described below), the analysis system  108  can perform analysis on multiple instances of the time-series data at the same time (e.g., in parallel). 
   According to one approach, the analysis system  108  may attempt to determine anomalies (e.g., spikes and/or dips) in message traffic within the network environment  202 . Spikes and dips may be indicative of anomalous behavior because a network environment characterized by heavy traffic volume is typically expected to vary in a relatively slow manner. A dip in traffic may be attributed to various causes. For example, one or more network components may not be functioning in a proper manner. For instance,  FIG. 2  shows that network component  220  may be malfunctioning, thus preventing (as per “blockage”  230 ) client requests from reaching the content provider data center  204 . Alternatively, or in addition, browser functionality of other local functionality employed by client devices may be functioning in an improper manner. Alternatively, or in addition, functionality at the content provider data center  204  (and/or at its associated distribution nodes) may be functioning in an improper manner. A spike in traffic may also be attributed to various causes. For example, an upward spike may indicate that a malicious entity is attacking part of the network environment  202  by generating a flood of messages. For example, an automated program may be generating these messages in a denial-of-service type of attack. 
   Upon detecting one or more anomalies within the network environment  202 , the analysis system  108  can take various types of remedial action. For example, the analysis system  108  can directly make changes in one or more network components to correct any problems in those components. In addition, or alternatively, the analysis system  108  can simply notify network components of anomalies, allowing these components to independently take corrective action. This might be appropriate where the network components that are responsible for the anomalies are not under the direct control of the entity that administers the analysis system  108 . 
   Consider the case in which the content provider administers the analysis system  108  for its own use. The content provider may be particularly concerned with detecting anomalies in the services it renders, even if those problems are the result of network components and other functionality that it has relatively no control over. These anomalies may lead to the loss of revenue, as customers are prevented from accessing the content provider&#39;s services. Moreover, if services are blocked, advertisers cannot disseminate ads to customers through these services. 
   A.3. Exemplary Processing Functionality 
     FIG. 3  sets forth exemplary processing functionality  302  that can be used to implement any aspect of the infrastructure shown in  FIGS. 1 and 2 . In one non-limiting case, for instance, the processing functionality  302  may represent any computer machine used to implement any aspect of the analysis system  108 . The processing functionality  302  can also represent any computer machine used in the general environment  102  of  FIG. 1  or the network environment  202  of  FIG. 2 . For instance, in the case of the network environment  202  of  FIG. 2 , the processing functionality  302  can be used to implement a user&#39;s client computer, a server computer in the content provider data center  204 , a server computer in a network component, and so on. 
   The processing functionality  302  can include various volatile and non-volatile memory, such as RAM  304  and ROM  306 , as well as one or more central processing units (CPUs)  308 . The processing functionality  302  can perform various operations identified above when the CPU  308  executes instructions that are maintained by memory (e.g.,  304 ,  306 , or elsewhere). The processing functionality  302  also optionally includes various media devices  310 , such as a hard disk module, an optical disk module, and so forth. 
   The processing functionality  302  also includes an input/output module  312  for receiving various inputs from the user (via input devices  314 ), and for providing various outputs to the user (via output devices  316 ). One particular output device may include a display apparatus and an associated graphical user interface (GUI)  318 . The processing functionality  302  can also include one or more network interfaces  320  for exchanging data with other devices via one or more communication conduits  322 . One or more communication buses  324  communicatively couple the above-described components together. 
   The communication conduits  322  can be implemented in different ways to suit different technical and commercial environments. For instance, the communication conduits  322  can include any kind of network (or combination of networks), such as a wide area network (e.g., the Internet), an intranet, Digital Subscriber Line (DSL) network infrastructure, point-to-point coupling infrastructure, and so on. In the case where one or more digital networks are used to exchange information, the communication conduits  322  can include various hardwired and/or wireless links, routers, gateways, name servers, and so on. The communication conduits  322  can be governed by any protocol or combination of protocols. (In the context of  FIG. 2 , the communication conduits  322  may loosely represent a wide area network, such as the Internet.) 
   B. Exemplary Procedures 
     FIGS. 4-7  show various procedures which explain the operation of the analysis system  108  of  FIG. 1  in flow chart form. To facilitate discussion, certain operations are described as constituting distinct blocks performed in a certain order. Such implementations are exemplary and non-limiting. Certain blocks described herein can be grouped together and performed in a single operation, and certain blocks can be performed in an order that differs from the order employed in the examples set forth in this disclosure. The blocks shown in the flowcharts can be implemented by software, firmware, hardware, manual processing, any combination of these implementations, and so on. 
   B.1. Overview of Entire Process 
     FIG. 4  is a procedure  400  which provides an overview of the operation of the analysis system  108  of  FIG. 1 . 
   In operation  402 , the data receiving module  110  receives time-series data from the environment  102 , either in a passive manner, an active manner, or in some hybrid passive-active manner. 
   In operation  404 , the anomaly analysis module (AAM)  112  or some other module can optionally perform pre-processing on the collected time-series data. For example, consider the case in which the collected time-series data reflects traffic within the network environment  202 . The pre-processing operation  404  may involve taking the square root of the collected data. This may be useful in some circumstances to help normalize variance across the peaks and valleys of the time-series data. 
   In operation  406 , the AAM  112  determines the presence of any anomalies in the time-series data. Later figures describe this process in greater detail. 
   In operation  408 , the AAM can optionally perform post-processing on the output of operation  406 . The post-processing may complement the pre-processing performed in operation  404 . For example, the post-processing may involve squaring the results of operation  406 . 
   In operation  410 , the result output module  114  outputs the results of the procedure  400 . The results may comprise a graphical depiction of the time-series data, with anomalous events appropriately marked. Alternatively, or in addition, the results may comprise a tabular presentation of alphanumeric data or some other information. 
   The procedure  400  can be performed for each instance of time-series data. Each instance of time-series data may reflect the output of a single data collection module, or perhaps the aggregate of outputs provided by a group of data collection modules. The AAM  112  can optionally perform analysis on multiple instance of time-series data at the same time, e.g., in parallel. This facilitates the rapid detection of anomalies within a data-rich environment, such as the network environment  202  of  FIG. 2 . 
   B.2. Overview of Analysis 
     FIG. 5  is a procedure that provides an overview of the anomaly analysis operation  406  of  FIG. 4 . To facilitate explanation, it will be assumed that the operations outlined in  FIG. 4  are performed on a single instance of time-series data. But as stated above, the AAM  112  can perform this analysis on multiple instances at the same time. 
   In operation  502 , the AAM  112  divides the collected time-series data into multiple segments, referred to herein as “collected data segments.” Operation  502  may allow the user to specify the duration of the data segments. It is preferable to set the length of the data segments to be short enough to facilitate accurate modeling, yet not so short that normal behavior cannot be distinguished from abnormal behavior. For frame of reference, without limitation, one exemplary implementation can set the length of the data segments to 60 minutes (although any shorter or longer interval can be used). 
   In operation  504 , the AAM  112  fits “local models” to the plurality of respective collected data segments. The term “local model” refers to a model of the local behavior of the time-series for a limited segment of time. The purpose here is to select local models which track the general path of the collected data segments, subject to certain boundary condition constraints to be discussed below. In one case, the AAM  112  can use regression to generate the local models. More specifically, the AAM  112  can use a spline curve fitting technique to generate the local models, where each local model corresponds to a polynomial. Operation  504  may allow the user to define the degree of the local models to be fit to the collected data segments. For instance, a user may select the degree as one of linear, quadratic, cubic, etc. Considered in aggregate, the plurality of local models form a “global model.” In other words, a “global model” refers to a plurality of local models, where the global model combines the predictions of one or more local models (e.g., via concatenating the local models in the time dimension). 
   In operation  506 , the AAM defines anomalies in the collected time-series data by detecting dramatic deviations between the collected time-series and the global model. A dramatic spike in message traffic may be indicative of a message-generating attack by an automated program within the network environment  202 . A dramatic dip in message traffic may be indicative of one or more parts (e.g., network components) of the network environment  202  that are no longer working properly. 
   Jumping ahead momentarily in the sequence of figures,  FIGS. 8-11  show how procedure  406  acts on representative time-series data. To facilitate discussion, these figures show how linear local models are fitted to the collected time-series data. However, the same principles conveyed by these figures can be applied to the case in which higher order local models are fitted to the time-series data. 
   To begin with,  FIG. 8  shows collected time-series data  802 . This data  802  may be supplied by one or more data collection modules within any kind of environment, such as, but not limited to, the network environment  202 . In this context, the vertical axis may represent message traffic within the network environment  202 , or some part thereof In accordance with operation  502 , the time-series data  802  is divided into a plurality of contiguous collected data segments, as shown in  FIG. 9 . 
     FIG. 9  shows, for tutorial purposes, how the AAM  112  would hypothetically fit local models to the collected data segments without any constraints imposed by boundary conditions. Segments  902  are three representative consecutive local models. Without boundary conditions, the endpoints of the local models do not necessarily connect together. In actual practice, a user can constrain the local models to ensure that the endpoints connect together. Further, the user can constrain the local models to ensure that the derivatives of the endpoints match up, which produces smoother transition points from one local model to the next. More specifically, the user can specify the order of derivative that is required to match at the endpoints, e.g., first derivative, second derivative, etc.  FIG. 10  shows fitting performed by the AAM  112  when boundary conditions are applied. As a result, the local models now form a continuous global model  1002 , e.g., without abrupt transitions from one local model to the next. 
     FIG. 10  also shows a marked deviation  1004  in the global model  1002 . The AAM  112  produces this deviation  1004  because it is attempting to fit one or more local models to a collection of time-series data that contains one or more outlying data points  1006 . These data points  1006  are outlying in the sense that they markedly depart from the general trend of the time-series data. While it is appropriate that the global model  1002  should track the actual time-series data, in this case, the outlying data points  1006  have exerted undue influence on the global model  1002 , drawing it too far downward. Because of this, the global model  1002  no longer accurately models the collected time-series data that occurs just before and after the outlying data points  1006 . This type of behavior is referred to as the Gibbs phenomenon. As an ultimate result, the Gibbs phenomenon may result in normal parts of the time-series data being labeled an anomalous (and/or, potentially, vice versa). 
   The Gibbs phenomenon may ensue, in part, from the modeling paradigm that is applied by the AAM  112 . By way of introduction, an error value defines a difference between a point in the collected time-series data and a corresponding point in the global model. For example, the difference between illustrative collected data point  1008  and model point  1010  defines an error value  1012 . For each local model, the AAM  112  generally operates by minimizing a sum of the error values to choose a final path of the model. In performing this operation, the AAM  112  can rely on various ways of representing the error values. In a first technique, the AAM  112  can use a square of the error values. This representation is referred to herein as a squared-term (L 2 ) measure. In a second technique, the AAM  112  can use an absolute value of an error values. This representation is referred to herein as an absolute value (L 1 ) measure. Although the use of L 1  and L 2  measures are described in detail herein, other ways of representing or estimating error values can be used. 
   There are certain advantages and disadvantage to both the L 2  measure and the L 1  measure. One advantage of the L 2  measure is that it allows modeling to be performed in a relatively efficient manner, e.g., by solving a system of linear equations. One disadvantage of the L 2  measure is that it is not robust to outliner values in the time-series data, and is thus prone to the type of Gibbs phenomenon illustrated by the deviation  1004  of  FIG. 10 . One advantage of the L 1  measure is that it is robust to outliner values in the time-series data and thus reduces the effects of the Gibbs phenomenon. One disadvantage of the L 1  measure is that it is more computationally complex to use in modeling (compared to modeling using the L 2  measure, which can use a linear technique). This is because the L 1  introduces nonlinear computations into the modeling operation. 
     FIG. 11  shows what happens when the AAM  112  uses the L 1  measure to generate all of the local models that compose the global model  1002 . As noted in region  1102  of  FIG. 11 , the choice of L 1  helps reduce the deviation  1004  of  FIG. 10  associated with the Gibbs phenomenon. 
   Returning to  FIG. 5 , the procedure  406  indicates that the AAM  112  can choose from among several different modeling techniques in performing operations  504  and  506 . 
   In a first technique, the AAM  112  can use the L 1  measure in generating all portions of all of the fitted local models, instead of the L 2  measure. As explained above, one advantage of this approach is that it reduces the Gibbs phenomenon. One disadvantage of this technique is that it is more computationally complex than the use of the L 2  measure, and therefore may potentially slow the processing performed by the AAM  112 . 
   In a second technique, the AAM  112  can use the L 1  measure, but only for those portions of the time-series data that are likely to represent anomalous events (as described more fully in the context of  FIG. 6  below). In the remaining portions, which likely represent normal behavior, the AAM  112  can use the L 2  measure. This has the dual advantage of reducing the Gibbs phenomenon in those areas where it is likely to manifest itself (e.g., in anomalous areas), while not otherwise slowing the AAM  112  by using the L 1  measure in normal areas where the Gibbs phenomenon is not typically a problem. To further expedite processing, the AAM  112  can use a linear approximation of the absolute value function in the regions in which L 1  is applied.  FIG. 6 , to be described below in turn, provides additional detail regarding the second technique. 
   In a third technique, the AAM  112  can use an expectation-maximization iterative procedure to apply a mixture of the L 1  measure and the L 2  measure in generating the global model (as described more fully in the context of  FIG. 7  below). In an expectation phase of the procedure, the AAM  112  labels each portion of the time-series data as anomalous or normal. An anomalous portion is associated with a Laplacian probability distribution, while a normal portion is associated with a Gaussian (Normal) probability distribution. In other words, abnormal points are assumed to fall around a local model with a Laplacian probability distribution and normal points are assumed to fall around the local model with a Gaussian probability distribution. Hence, the AAM  112  can classify whether each portion is anomalous (or not) based on the probability distribution under which the portion is more likely. In a maximization phase, the AAM  112  generates parameters that define the plurality of local models. In doing so, the maximization phase performs a cost-minimizing operation using the L 1  measure for anomalous portions and the L 2  measure for normal portions (where such portions have been labeled in the preceding expectation phase). The next iteration of the expectation phase uses the newly generated local models as a reference in re-labeling portions of the time-series data as either anomalous or normal. The modeling technique repeats the above-described procedure for a predefined number of times or until it produces stable labeling results.  FIG. 7 , to be described below in turn, provides additional detail regarding the third technique. 
   The second and third techniques are instances of one general mode of operation of the AAM  112 . According to this mode, the AAM  112  can choose among multiple curve-fitting paradigms in modeling different portions of the time-series data (not limited to two paradigms). The AAM  112  examines the characteristics of each portion of the time-series data and applies, in dynamic fashion, the curve-fitting paradigm deemed most suitable for that portion. For instance, in the specific examples identified above, the AAM  112  uses an error value metric to determine what paradigm to apply in modeling different portions of the time-series data. The AAM  112  determines what paradigms are most suitable for different error values based on accuracy-related considerations (to reduce the impact of Gibbs phenomenon) and complexity-related considerations (to reduce the impact of complex nonlinear computations on processing speed). These principles can be extended to other cases in which the AAM  112  uses other salient characteristics of the time-series data (besides an error value metric) to determine what curve-fitting paradigms to apply to different portions of the time-series data. The principles can likewise be extended to the use of other types of curve-fitting paradigms that achieve their distinguishing advantage through some other factor besides the way they define error value in a cost function. 
   As used herein, the term “portions” is intended to have broad connotation. For instance, in one case, the AAM  112  can apply the L 1  or L 2  measure when modeling all of the data points in an entire segment of the time-series data. In another case, the AAM  112  can apply the L 1  or L 2  measure with respect to each individual data point within the time-series data. In the latter case, the AAM  112  can apply two or more different paradigms for a single collected data segment, if warranted by the characteristics of the data. 
   B.3. Mixed Use of L 1  and L 2  Measures 
   Advancing to  FIG. 6 , this procedure provides further information regarding the above-identified second technique for fitting a global model to the time-series data, where the global model is composed of multiple local models in aggregate. Recall that in the second technique, the AAM  112  uses a combination of the L 1  measure and the L 2  measure to generate the global model. The second technique is a non-iterative solution (compared to the third technique, described in the next subsection). 
   In operation  602 , the AAM  112  identifies each portion the time-series data as likely representing normal data or anomalous data. The AAM  112  can apply one or more rules in making this decision. In one case, the AAM  112  can determine how far a portion diverges from a general trend established by the time-series data (as represented by the error value); if the portion diverges from the trend by more than a predetermined threshold, it can be labeled as anomalous. 
   In operation  604 , the AAM  112  can apply an L 2  measure when generating the global model for normal portions of the time-series data. The AAM  112  can apply the L 1  measure when generating the global model for anomalous portions of the time-series data. 
   B.4. Exemplary EM Procedure 
   Advancing to  FIG. 7 , this procedure provides further information regarding the above-described third technique for fitting a global model to the time-series data, where, again, the global model is formed by combined multiple contiguous local models. Recall that in the third technique, the AAM  112  uses an expectation-maximization approach to apply a mixture of the L 1  and L 2  measures. 
   In operation  702 , the AAM  112  starts the iterative procedure by computing initial local model values under the assumption that all points in the time-series data are normal. (To facilitate discussion, the local models are loosely referred to below as “splines.”) In this initial operation  702 , the AAM  112  can apply the L 2  measure to all collected data segments. 
   In operation  704 , the AAM  112  performs the first expectation phase computations. In this operation  704 , the AAM  112  labels each point of the time-series data as anomalous or normal based on its associated error value (e.g., the distance from the current estimated spline). Further information regarding this operation  704  is provided below. 
   In operation  708 , the AAM  112  updates the spline parameters using the labels generated in operation  704 . That is, the L 1  measure is applied for portions labeled as anomalous, while the L 2  measure is applied for portions labeled as normal. Further information regarding this operation  708  is provided below. 
   The loop shown in  FIG. 7  indicates that the operations  704  and  708  are repeated a plurality of times. Operation  706  determines when the iteration should terminate. In one case, the procedure terminates after a predetermined number of iterations (e.g., without limitation, 10 iterations). In another case, the operation terminates when it is observed that the labeling results produced in operation  704  have stabilized (meaning that these results do not substantially change with successive iterations). 
   Further information is now provided regarding operation  704 , in which the AAM  112  labels time-series points as anomalous or normal. According to one exemplary implementation, the AAM computes the variance of each time segment with respect to the estimated spline using the assumption that the errors follow a normal distribution. The probability of being anomalous is then computed for each point using both a Gaussian distribution and a Laplacian distribution. If the prior estimate of the probability that a point is anomalous is p, then the area of the Laplacian distribution is set to be p and the area of the Gaussian distribution is set to be (1-p). For example, if m anomalous points are expected in n samples, then p would be m/n. If the probability of a point is greater under the Laplacian distribution than under the Gaussian distribution, the point is labeled as anomalous; otherwise it is labeled as normal. 
   Further information is now provided regarding operation  708 , in which the AAM  112  updates the spline parameters. The AAM  112  updates the parameters of the spline by minimizing a cost function computed by summing the error squared for all points labeled as normal and the absolute value of the error for all points labeled as anomalous. 
   The error for anomalous points is far from zero, since points close to zero are labeled as normal. Based on this observation, the AAM  112  can approximate the absolute value cost function with two linear equations: e.g., a line with slope equal to +1 for positive error values and a line with slope equal to −1 for negative error values. This allows the AAM  112  to efficiently compute the spline parameters by solving a single system of linear equations in an efficient manner. 
   Note that the EM procedure works so long as the spline does not move so far in any single EM step that it advances to the other side of a point labeled as anomalous. If need be, this can be enforced by repeatedly halving the amount by which the parameters of the spline are changed, until this condition is satisfied. As an aside, note that the continuity and smoothness constraints on the boundaries of the local models are also linear constraints, and can be included in the same series of linear equations as the L 1  and L 2  cost functions. 
     FIGS. 13-15  show, without limitation, an exemplary computer program for implementing aspects of the EM procedure shown in  FIG. 7 . For instance, the series of instructions  1502  in  FIG. 15  implement the labeling of data points as anomalous or normal in operation  704 . 
   B.5. Exemplary Output 
   Finally,  FIG. 12  shows an exemplary type of output that can be produced by various implementations of the AAM  112 . The output shows the original time-series data  1202 , together with a global model  1204  that the AAM  112  has fit to the time-series data  1202 . The AAM  112  has marked two potentially anomalous events ( 1206 ,  1208 ) in the time-series data  1202 . These events ( 1206 ,  1208 ) are associated with large and rapid changes in the time-series data  1202 . 
   In closing, a number of features were described herein by first identifying exemplary problems that these features can address. This manner of explication does not constitute an admission that others have appreciated and/or articulated the problems in the manner specified herein. Appreciation and articulation of the problems present in the relevant art(s) is to be understood as part of the present invention. 
   More generally, although the invention has been described in language specific to structural features and/or methodological acts, it is to be understood that the invention defined in the appended claims is not necessarily limited to the specific features or acts described. Rather, the specific features and acts are disclosed as exemplary forms of implementing the claimed invention.