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)
“Everything ponderous, viscous, and solemnly clumsy, all long- winded and boring types of style are developed in profuse variety among Germans—forgive me the fact that even Goethe’s prose, in its mixture of stiffness and elegance, is no exception, being a reflection of the “good old time” to which it belongs, and a reflection of German taste at a time when there still was a “German taste”Ma rococo taste in moribus et artibus.”
—Friedrich Nietzsche (1844–1900)
“If the technology cannot shoulder the entire burden of strategic change, it nevertheless can set into motion a series of dynamics that present an important challenge to imperative control and the industrial division of labor. The more blurred the distinction between what workers know and what managers know, the more fragile and pointless any traditional relationships of domination and subordination between them will become.”
—Shoshana Zuboff (b. 1951)