{ "cells": [ { "cell_type": "markdown", "id": "329b70ac-fb96-46f9-93a5-a935043d835d", "metadata": {}, "source": [ "
\n", " Climate of the Earth system\n", "

Prof. Dr. Markus Meier
\n", " Leibniz Institute for Baltic Sea Research Warnemünde (IOW)
\n", " E-Mail: markus.meier@io-warnemuende.de

\n", "
" ] }, { "cell_type": "markdown", "id": "519b554f-e050-4dc0-9cf9-c8dd075a8033", "metadata": {}, "source": [ "# Auto-covariance function\n", "\n", "- auto-covariance function $\\gamma$ 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\n", "- non-parametric estimator of the auto-correlation function:\n", "```{math}\n", "\\rho(\\tau) = \\frac{\\gamma(\\tau)}{\\gamma(0)},\n", "```\n", "- where $\\gamma(\\tau)$ is the sample auto-covariance function, (covariance of X with itself at a time lag of $\\tau$)\n", "```{math}\n", "\\gamma(\\tau) = \\frac{1}{T} \\sum_{t=|\\tau|+1}^T \\mathbf{X}'_{t-|\\tau|} \\mathbf{X}'_t\n", "```\n", "- characteristics of $\\gamma$:\n", " - $\\gamma$ is symmetric\n", " - $\\gamma(0) = Var(\\mathbf{X})$\n", " - $\\gamma(\\tau) \\leq \\gamma(0)$\n", " - $|\\rho(\\tau)| \\leq 1$\n", " - $\\rho(\\tau) < 0~\\Rightarrow$ oscillation\n", "\n", "- $\\rho(\\tau)$ illustrates how the dynamical system responds to a disturbance from the equilibrium, persistence of a forecast skill\n", "\n", "```{figure} figures/L15/L15_1_temperature_series.PNG\n", "---\n", "width: 30%\n", "---\n", "

Figure 1: 50 years long time series of monthly mean temperatures of the atmosphere in 2mter height at location in northern Asia (upper), of the sea surface in the North Atlantic (middle) and in 180meters depth in the North Atlantic Ocean (lower).

\n", "```\n", "\n", "```{figure} figures/L15/L15_3_elnino.PNG\n", "---\n", "width: 60%\n", "---\n", "

Figure 2: El Nino SST time series (left) and an estimate of the spectrum together with a theoretical spectrum (right). The theoretical spectrum is that of an auto-regressive process of the first order, where the parameters of the process have been fitted to those of the El Nino SST time series.

\n", "```\n", "\n", "- auto-correlation functions of the above time series in Figure 3 below\n", "```{figure} figures/L15/L15_2_acf.PNG\n", "---\n", "width: 60%\n", "---\n", "

Figure 3: Left: The auto-correlation function of the monthly mean temperature time series as shown in Fig. 1. The atmosphere in 2meter height at location in central northern Asia (upper), of the sea surface in the North Atlantic (middle) and in 180meters depth in the North Atlantic Ocean (lower). Right: The auto-correlation function of the observed time series of El Nino monthly mean SST, as shown in Fig. 2.

\n", "```\n", "\n", "#### Autoregressive (AR) processes\n", "- 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:\n", "```{math}\n", "X_t = \\alpha_1X_{t-1} + \\alpha_2X_{t-2} + ... \\alpha_pX_{t-p} + \\varepsilon_t\n", "```\n", "$\\alpha_1,\\alpha_2,...,\\alpha_p$ - autoregressive coefficients determining the dependance on the past values, \n", "$\\varepsilon_t$ - random error term or white noise at time $t$\n", "- order p of the autoregressive process determines how many past values have to be considered in the predicition of the current value\n", "- AR(1) processes rely only on the previous value\n", "```{math}\n", "X_t = \\alpha X_{t-1} + \\varepsilon_t,\n", "```\n", "- while AR(2) processes rely on the previous two values\n", "```{math}\n", "X_t = \\alpha_1X_{t-1} + \\alpha_2X_{t-2} + \\varepsilon_t,\n", "```\n", "\n", "```{figure} figures/L15/L15_4_AR1.png\n", "---\n", "width: 60%\n", "---\n", "

Figure 4: Left: Different realization of AR(1)-processes with different $\\alpha_1=0.5$, $0.9$, $0.99$ but with identical unit variance normal white noise processes $\\mathbf{Z_t}$. Note the different y-axis scaling. Right: The auto-correlation function corresponding to the AR(1)-time series to the left.

\n", "```\n", "```{figure} figures/L15/L15_5_AR2.png\n", "---\n", "width: 60%\n", "---\n", "

Figure 5: Left: Different realization of AR(2)-processes with different $\\alpha_1$ and $\\alpha_2$, but with identical unit variance normal white noise processes $\\mathbf{Z_t}$. Note the different y-axis scaling. Right: The auto-correlation function corresponding to the AR(2)-time series to the left.

\n", "```\n", "\n", "#### Decorrelation time\n", "- 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\n", "- 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:\n", "```{math}\n", "Var(\\mathbf{\\bar{X}}) = \\frac{\\sigma^2_X}{n_X} \\Rightarrow n_X\n", "```\n", "- relation between the true number of time steps used and $n_X$ is the decorrelation time\n", "```{math}\n", "\\begin{align*}\n", "\\tau_D &= \\frac{n}{n_X}\\\\\n", " &= 1+2\\sum_{k=1}^{\\infty}\\rho(k),~~~~~\\mathrm{for~ the~ mean}\\\\\n", "&=1+2\\sum_{k=1}^{\\infty}\\rho(k)^2,~~~\\mathrm{for~ the~ variance}\n", "\\end{align*}\n", "```" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3 (ipykernel)", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.11.6" } }, "nbformat": 4, "nbformat_minor": 5 }