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)
“Art, it seems to me, should simplify. That, indeed, is very nearly the whole of the higher artistic process; finding what conventions of form and what detail one can do without and yet preserve the spirit of the whole—so that all that one has suppressed and cut away is there to the reader’s consciousness as much as if it were in type on the page.”
—Willa Cather (1873–1947)
“Lest darkness fall and time fall
In a long night when learned arteries
Mounting the ice and sum of barbarous time
Shall yield, without essence, perfect accident.
We are the eyelids of defeated caves.”
—Allen Tate (1899–1979)
“I thought I never wanted to be a father. A child seemed to be a series of limitations and responsibilities that offered no reward. But when I experienced the perfection of fatherhood, the rest of the world remade itself before my eyes.”
—Kent Nerburn (20th century)