Many types of variable objects exist in the universe, including stars with predictable behavior (e.g., Cepheids), objects with behavior that is inherently unpredictable (e.g, AGN), and objects with both predictable and irregular variability patterns (e.g., X-ray binaries). Constant monitoring of variable objects has been a continuing interest in astronomy, beginning with 16th century astronomer David Fabricius, and extending through history to Herschel, Leavitt and others. Today, monitoring is done by a wide variety of techniques, observers and instruments, from dedicated amateurs, to professional astronomers interested in intensive monitoring of individual objects, to all-sky monitors such as the RXTE ASM and BATSE aboard CGRO.
Despite being a new tool, all-sky monitors have already made important, if not decisive contributions to solving some of astronomy's most persistent mysteries, such as the cosmological origin of gamma-ray bursts and linking emission regions in AGN. With major initiatives such as the Large-Area Synoptic Survey Telescope (LSST) and Supernova Acceleration Probe (SNAP), all-sky monitors are poised to become a major discovery tool in astronomy. To maximize the utility of large monitoring programs, it is important to devise ways of handling large amounts of data in real time and find not only variability but also predictive patterns among these large data streams. It was with these goals in mind that we undertook this project.
The need to understand time series data is not unique to astronomy: time series data exists in a wide variety of fields including geology and atmospheric science as well as many business applications. The common problem is how to efficiently find patterns in the data. Fourier transform and power spectrum methods, commonly used in astronomy research, are well suited for finding patterns with well-determined periodicities (e.g., see Scargle 1997). However, they may be less helpful for objects with irregular behavior, and are not optimized to serve as predictive tools. Our approach was to employ clustering and rule discovery, which are optimized for finding patterns that do not rely on a regular periodicity.
We first attempt to represent the dataset as a collection of patterns, by sliding a window through the time series to get subsequences. We then cluster these points using the greedy clustering algorithm explained in Das, Gunopulos & Mannila (1997). Once a good fit is achieved, these clusters can then be considered as the basic shapes of the time series, with the entire series composed of superpositions and/or combinations of these basic shapes. The next and main step of the process is to find interesting probabilistic rules between the clusters in the two time series. These rules are of the form: ``If a cluster A occurs in time series 1 then we can say with confidence c that B will occur in time series 2 within time T". Figure 1 displays a flow chart of the basic algorithm used in predictive time series analysis.
Two goals are possible with this approach, each requiring a slightly different analysis tree: (1) to schedule a telescope more intelligently to observe objects when they are at their brightest or most interesting stage, or (2) as an analysis tool to find patterns within a (possibly multivariate) time series (see Figure 2).
Within the time-series analysis (TSA) module, the user can control three main parameters: the window size, the minimum cluster distance and the time period for prediction. The TSA module analyzes the time series data and performs trend prediction and rule discovery on both real time and historical data. This output can either be used directly for data analysis for a paper or to forecast both the activity level of an object and the ability of the telescope or satellite to perform the desired observations. All of these factors could be considered in the light of the constraints imposed by individual investigators as well as the review committee. The goal of such a process would be to maximize the scientific usefulness of all the observations. If, for example, a given observation required that a source be above a certain flux level, such a program might have a better chance of ensuring that this was so during the observations.
To test and validate the toolbox, we used RXTE ASM datasets for various objects, spanning several different object classes. In Figure 3, we show one particularly interesting dataset, from SMC X-1. SMC X-1 shows interesting periodic behavior on several different timescales: a long time-period behavior as well as short time-scale variations superposed upon the long-period behavior, but occurring only when the object is bright (see e.g., Kahabka & Li 1999). The toolkit was found to be particularly effective for predicting short-period rules. Longer-period rules can be found by varying the window size, although possibly at the expense of losing short time-scale rules as the window size becomes larger than the rules in question. We can also find longer-period rules with smoothing techniques.
The current implementation of the toolkit consist of a rule inference engine, programmed in C and a visualizer, programmed in Java. The current toolkit has several limitations. First, the current toolkit can analyze only a single time series at any time. This obviously gives it limited functionality, as many objects are studied with multiple instruments. Second, it has limited ability to deal with missing data. In addition, one could envision multiple processing options, including the incorporation of dynamic time warping techniques, which could be particularly useful to spot quasi-periodic oscillations.
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