Spectral Density - Preliminary Conventions On Notations For Time Series

Preliminary Conventions On Notations For Time Series

The phrase time series has been defined as "... a collection of observations made sequentially in time." But it is also used to refer to a stochastic process that would be the underlying theoretical model for the process that generated the data (and thus include consideration of all the other possible sequences of data that might have been observed, but weren't). Furthermore, time can be either continuous or discrete. There are, therefore, four different but closely related definitions and formulas for the power spectrum of a time series.

If (discrete time) or (continuous time) is a stochastic process, we will refer to a possible time series of data coming from it as a sample or path or signal of the stochastic process. To avoid confusion, we will reserve the word process for a stochastic process, and use one of the words signal, or sample, to refer to a time series of data.

For X any random variable, standard notations of angle brackets or E will be used for ensemble average, also known as statistical expectation, and Var for the theoretical variance.

Read more about this topic:  Spectral Density

Famous quotes containing the words preliminary, conventions, time and/or series:

    Religion is the state of being grasped by an ultimate concern, a concern which qualifies all other concerns as preliminary and which itself contains the answer to the question of a meaning of our life.
    Paul Tillich (1886–1965)

    Languages exist by arbitrary institutions and conventions among peoples; words, as the dialecticians tell us, do not signify naturally, but at our pleasure.
    François Rabelais (1494–1553)

    I was surprised by Joe’s asking me how far it was to the Moosehorn. He was pretty well acquainted with this stream, but he had noticed that I was curious about distances, and had several maps. He and Indians generally, with whom I have talked, are not able to describe dimensions or distances in our measures with any accuracy. He could tell, perhaps, at what time we should arrive, but not how far it was.
    Henry David Thoreau (1817–1862)

    Life ... is not simply a series of exciting new ventures. The future is not always a whole new ball game. There tends to be unfinished business. One trails all sorts of things around with one, things that simply won’t be got rid of.
    Anita Brookner (b. 1928)