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)
“I find nothing healthful or exalting in the smooth conventions of society. I do not like the close air of saloons. I begin to suspect myself to be a prisoner, though treated with all this courtesy and luxury. I pay a destructive tax in my conformity.”
—Ralph Waldo Emerson (1803–1882)
“My friend devotes himself to his life, whenever he can find the spare time. His motto is: ‘Don’t just sit there: live!’ So he’s too busy to stand, to walk, to do anything, except to live. He even refused to kiss a girl, when invited, on the grounds that it was time again to be living. Schedules are sacred to him.”
—Marvin Cohen, U.S. author and humorist. The Self-Devoted Friend, New Directions (1967)
“Through a series of gradual power losses, the modern parent is in danger of losing sight of her own child, as well as her own vision and style. It’s a very big price to pay emotionally. Too bad it’s often accompanied by an equally huge price financially.”
—Sonia Taitz (20th century)