Auto-covariance function

Auto-covariance function#

auto-covariance function $γ$ describes the time scale behavior of a time series: how likely a variable depends on its own time series. an example of autocorrelation: if todays weather is warm there is a high likelyhood that tomorrows weather is also warm
non-parametric estimator of the auto-correlation function:

ρ (τ) = \frac{γ (τ)}{γ (0)},

where $γ (τ)$ is the sample auto-covariance function, (covariance of X with itself at a time lag of $τ$ )

γ (τ) = \frac{1}{T} \sum_{t = | τ | + 1}^{T} X_{t - | τ |}^{'} X_{t}^{'}

characteristics of $γ$ :
- $γ$ is symmetric
- $γ (0) = V a r (X)$
- $γ (τ) \leq γ (0)$
- $| ρ (τ) | \leq 1$
- $ρ (τ) < 0 \Rightarrow$ oscillation
$ρ (τ)$ illustrates how the dynamical system responds to a disturbance from the equilibrium, persistence of a forecast skill

auto-correlation functions of the above time series in Figure 3 below

Autoregressive (AR) processes#

in an AR process each value in a time series is modeled as a linear combination of its previous values, along with some random noise. a general AR(p) process can be represented as follows:

X_{t} = α_{1} X_{t - 1} + α_{2} X_{t - 2} + . . . α_{p} X_{t - p} + ε_{t}

$α_{1}, α_{2}, . . ., α_{p}$ - autoregressive coefficients determining the dependance on the past values,
$ε_{t}$ - random error term or white noise at time $t$

order p of the autoregressive process determines how many past values have to be considered in the predicition of the current value
AR(1) processes rely only on the previous value

X_{t} = α X_{t - 1} + ε_{t},

while AR(2) processes rely on the previous two values

X_{t} = α_{1} X_{t - 1} + α_{2} X_{t - 2} + ε_{t},

Decorrelation time#

describes characteristic timescales of stochastic processes: how fast the autocorrelation decreases with time lag as a measure of the memory and persistence of the processes
number degree of freedom $n_{X}$ of $X$ (initially we had the definition for $n_{X}$ that $X_{i}$ and $X_{j}$ have to be independent, meaning uncorrelated), central limit theorem:

V a r (\bar{X}) = \frac{σ_{X}^{2}}{n_{X}} \Rightarrow n_{X}

relation between the true number of time steps used and $n_{X}$ is the decorrelation time

\begin{array}{r} \begin{aligned} τ_{D} & = \frac{n}{n_{X}} \\ = 1 + 2 \sum_{k = 1}^{\infty} ρ (k), for the mean \\ = 1 + 2 \sum_{k = 1}^{\infty} ρ (k)^{2}, for the variance \end{aligned} \end{array}

Auto-covariance function

Contents

Auto-covariance function#

Autoregressive (AR) processes#

Decorrelation time#