4 Flashcards
Typical components of time series:
Trend is a long-term tendency (increasing, decreasing, or non-repetitively changing) of time series. Linear and exponent are popular types of trend.
Seasonal component — repeating relatively short-term component. Seasonality has a fixed period (or frequency). Seasonal component is usually caused by natural factors such as a year (earth’s revolution around the sun).
Random component — a component without trend or seasonality, a.k.a. remainder.
Forecasting strategies of time series components:
If trend follows linear, exponent, or other type of parametrically described function, the future values are estimated using this function.
Seasonal component is cyclically repeated for future times.
Remainder part is replaced by 0 (1 in multiplicative model) if behaviour of remainder is unpredictive. Otherwise it can be forecasted using some time
series models.
Properties of ACF
ACF(t, t) = 1: This says that the autocorrelation of a time point with itself (i.e., with zero lag) is always 1.
-1 ≤ ACF(t, t + τ) ≤ 1: This indicates that the autocorrelation at any lag τ will always be between -1 and 1. A value of -1 represents perfect negative correlation (as one value increases, the other one decreases), a value of 1 represents perfect positive correlation (both values increase or decrease together), and a value of 0 means no correlation at all.
ACF(t + τ, t) = ACF(t, t + τ): This property is showing the symmetry of the ACF. Autocorrelation at lag τ is the same forward and backward in time. This is because the relationship between the data points doesn’t change depending on the direction you look at the lag.
What is autocovariance function
is a tool used in time series analysis to measure how much two points in the same series, separated by a lag of τ time units, covary with each other.
What are properties of ACvF function?
ACvF(t, t + τ) = ACvF(τ): This states that the autocovariance at a lag of τ depends only on the lag itself and not on the specific time t. It’s a function of the lag τ alone, which makes the time series stationary.
ACvF(t, t) = ACvF(0) = var Yt: The autocovariance of a time series at lag 0 (τ = 0) is just the variance of the time series. It’s a special case where you’re measuring how much a point varies with itself, which is by definition the variance.
ACvF(t, t + τ) = ACvF(t + τ, t): This indicates the symmetry of autocovariance; the covariance between Yt and Yt+τ is the same as the covariance between Yt+τ and Yt.
ACvF(−τ) = ACvF(τ): The autocovariance function is even, meaning that it has symmetry around lag 0. The covariance is the same for τ and -τ, showing that the direction of the lag doesn’t matter.
How seasonality can be predicted?
Prior knowledge (ex., hourly observations of air temperature);
Observable pattern in a time series plot;
Repeated peaks in ACF (or ACvF) plot.
What is white noise?
is a random signal that has equal intensity at different frequencies, which makes it a constant power spectral density.
