{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Singular spectrum analysis for time series\n", "\n", "This notebook illustrates how to perform a singular spectrum analysis (SSA) with the pyActigraphy package.\n", "\n", "Briefly, the SSA is related to the Principal Component Analysis (PCA) for time series. The input signal is decomposed into additive components and their relative importance (i.e. variance) is quantified.\n", "\n", "\n", "More informations about the SSA methodology:\n", "\n", "- Vautard, R., Yiou, P., & Ghil, M. (1992). Singular-spectrum analysis: A toolkit for short, noisy chaotic signals. Physica D: Nonlinear Phenomena, 58(1–4), 95–126. https://doi.org/10.1016/0167-2789(92)90103-T\n", "- Golyandina, N., & Zhigljavsky, A. (2013). Singular Spectrum Analysis for Time Series. Berlin, Heidelberg: Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-642-34913-3\n", "\n", "\n", "In the context of Chronobiology, a nice overview of various technics (including SSA) is given in:\n", "\n", "- Fossion, R., Rivera, A. L., Toledo-Roy, J. C., & Angelova, M. (2018). Quantification of Irregular Rhythms in Chronobiology: A Time- Series Perspective. In Circadian Rhythm - Cellular and Molecular Mechanisms. InTech. https://doi.org/10.5772/intechopen.74742" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Imports and input data" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "ExecuteTime": { "end_time": "2023-01-30T15:47:30.220445Z", "start_time": "2023-01-30T15:47:28.539158Z" } }, "outputs": [], "source": [ "import pyActigraphy" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "ExecuteTime": { "end_time": "2023-01-30T15:47:30.223617Z", "start_time": "2023-01-30T15:47:30.221815Z" } }, "outputs": [], "source": [ "from pyActigraphy.analysis import SSA" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "ExecuteTime": { "end_time": "2023-01-30T15:47:30.227015Z", "start_time": "2023-01-30T15:47:30.225198Z" } }, "outputs": [], "source": [ "import numpy as np" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "ExecuteTime": { "end_time": "2023-01-30T15:47:30.230164Z", "start_time": "2023-01-30T15:47:30.228333Z" } }, "outputs": [], "source": [ "import plotly.graph_objs as go" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "ExecuteTime": { "end_time": "2023-01-30T15:47:30.233876Z", "start_time": "2023-01-30T15:47:30.231568Z" } }, "outputs": [], "source": [ "import os" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "ExecuteTime": { "end_time": "2023-01-30T15:47:30.238591Z", "start_time": "2023-01-30T15:47:30.235200Z" } }, "outputs": [], "source": [ "# Define the path to your input data\n", "fpath = os.path.join(os.path.dirname(pyActigraphy.__file__),'tests/data/')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Read test file:" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "ExecuteTime": { "end_time": "2023-01-30T15:47:30.256566Z", "start_time": "2023-01-30T15:47:30.239860Z" } }, "outputs": [], "source": [ "raw = pyActigraphy.io.read_raw_awd(fpath+'example_01.AWD', start_time='1918-01-24 08:00:00', period='9 days')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## SSA methodology" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Embedding" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Time-series $x=(x_0,x_1,\\dots,x_n,\\dots,x_{N−1})^\\intercal$, with length N, represents the signal under analysis. The mapping of this signal into a matrix A, of dimension L × K , assuming $L \\leq K$ , is called embedding, and can be defined as:\n", "\n", "\n", "$$A = \\begin{bmatrix}\n", " x_{0} & x_{1} & x_{2} & \\dots & x_{K-1} \\\\\n", " x_{1} & x_{2} & x_{3} & \\dots & x_{K} \\\\\n", " \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\n", " x_{L-1} & x_{L} & x_{L+1} & \\dots & x_{N-1}\n", "\\end{bmatrix}$$\n", "\n", "where L is the window length, or embedding dimension, and $K = N − L + 1$.\n", "A is a Hankel matrix, called the trajectory matrix." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "During the embedding step, the window length $L$ is a crucial parameter; only the components with a (quasi)periodicity $T \\leq L$ will be resolved." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In the context of human locomotor activities, a natural choice for $L$ is $24$ h." ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "ExecuteTime": { "end_time": "2023-01-30T15:47:30.261206Z", "start_time": "2023-01-30T15:47:30.257800Z" } }, "outputs": [], "source": [ "mySSA = SSA(raw.data,window_length='24h')" ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "ExecuteTime": { "end_time": "2023-01-30T15:47:30.313525Z", "start_time": "2023-01-30T15:47:30.264397Z" } }, "outputs": [ { "data": { "text/plain": [ "(1440, 11522)" ] }, "execution_count": 9, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Access the trajectory matrix\n", "mySSA.trajectory_matrix().shape" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In our example file, the acquisition frequency is $1$ min. Therefore, the trajectory matrix dimension is 1440, ie. $24$ h." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Singular value decomposition" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Factorization of the trajectory matrix A, using Singular Value Decomposition (SVD), yields to:\n", "$$ A = U\\Sigma V^\\intercal =\\sum_{r=1}^{R} \\sigma_r u_r v_{r}^\\intercal$$\n", "where $R = rank(A) \\leq L$, ${u_1,\\ldots, u_d }$ is the corresponding\n", "orthonormal system of the eigenvectors of the matrix $S = AA^\\intercal$ such as $ui \\cdot uj = 0$ for $i \\neq j$ and $\\lVert u_r \\rVert = 1$, $v_r = A^{\\intercal} u_r / \\sigma_r$, and $\\Sigma$ is a diagonal matrix $\\in \\mathbb{R}^{L×K}$ , whose diagonal elements ${\\sigma_r}$ are the singular values of A. The eigenvalues of $AA^\\intercal$ are given by $\\lambda_r = \\sigma_r^2$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Within the pyActigraphy package, the decomposition of the trajectory matrix is performed via the `fit` function:" ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "ExecuteTime": { "end_time": "2023-01-30T15:47:40.268748Z", "start_time": "2023-01-30T15:47:30.315401Z" } }, "outputs": [], "source": [ "mySSA.fit()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Fractional partial variances and Scree diagram\n", "\n", "One of the main results of SSA analysis is the so-called scree diagram that visually represents\n", "the partial variances $\\lambda_k = \\sigma^2_k$, ordered according to magnitude from the most to the least dominant, where $\\lambda_k$ can be interpreted as the variance of the “sub phase-space” of time-series component $g_k(n)$ and where $\\lambda_{tot} = \\sum^r_{k=1} \\lambda_k$ is the total variance of the phase space of the original time series x(n)." ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "ExecuteTime": { "end_time": "2023-01-30T15:47:40.273378Z", "start_time": "2023-01-30T15:47:40.269998Z" } }, "outputs": [ { "data": { "text/plain": [ "1.0" ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# By definition, the sum of the partial variances should be equal to 1:\n", "mySSA.lambda_s.sum()" ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "ExecuteTime": { "end_time": "2023-01-30T15:47:40.303133Z", "start_time": "2023-01-30T15:47:40.274540Z" }, "code_folding": [ 0 ] }, "outputs": [], "source": [ "layout = go.Layout(\n", " height=600,\n", " width=800,\n", " title=\"Scree diagram\",\n", " xaxis=dict(title=\"Singular value index\", type='log', showgrid=True, gridwidth=1, gridcolor='LightPink', title_font = {\"size\": 20}),\n", " yaxis=dict(title=r'$\\lambda_{k} / \\lambda_{tot}$', type='log', showgrid=True, gridwidth=1, gridcolor='LightPink', ),\n", " showlegend=False\n", ")" ] }, { "cell_type": "code", "execution_count": 13, "metadata": { "ExecuteTime": { "end_time": "2023-01-30T15:47:40.446048Z", "start_time": "2023-01-30T15:47:40.304112Z" } }, "outputs": [ { "data": { "text/html": [ " \n", " " ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "application/vnd.plotly.v1+json": { "config": { "plotlyServerURL": "https://plot.ly" }, "data": [ { "type": "scatter", "x": [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 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