Time-series Analysis

Intelligent data analysis often requires one to extract meaningful conclusions about a complicated system using data from a single sensor. If the system is linear, a wealth of well-established, powerful time-series analysis techniques is available to the analyst. If it is not, the problem is much harder and one must resort to nonlinear dynamics theory in order to infer useful information from the data. IDA problems are often complicated by a simultaneous overabundance and lack of data: reams of information about the voltage output of a power substation, for instance, but no data about other important quantities, such as the temperatures inside the transformers. Data-mining techniquess14] provide some useful ways to deal successfully with the sheer volume of information that constitutes one part of this problem. The second part of the problem is much harder. If the target system is highly complex | say, an electromechanical device whose dynamics is governed by three metal blocks, two springs, a pulley, several magnets, and a battery | but only one of its important properties (e.g., the position of one of the masses) is sensor-accessible, the data analysis procedure would appear to be fundamentally limited. Delay-coordinate embedding, a technique developed by the dynamics community , is one way to get around this problem; it lets one reconstruct the internal dynamics of a complicated nonlinear system from a single time series. That is, one can often use delay-coordinate embedding to infer useful information about internal (and unmeasurable) transformer temperatures using only their output voltages. The reconstruction produced by delay-coordinate embedding is not, of course, completely equivalent to the internal dynamics in all situations, or embedding would amount to a general solution to control theory's observer problem: how to identify all of the internal state variables of a system and infer their values from the signals that can be observed. However, a single-sensor reconstruction, if done right, can still be extremely useful because its results are guaranteed to be topologically (i.e., qualitatively) identical to the internal dynamics. This means that conclusions drawn about the reconstructed dynamics are also true of the internal dynamics of the system inside the black box. If the system under examination is linear, the analysis is comparatively straightforward and the tools | the topic of section 1 of this chapter | are well developed. One can characterize the data using statistics (mean, standard deviation, etc.), t curves to them (functional approximation), …