{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Functional linear modelling" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "This notebook illustrates how to perform a functional linear modelling of actigraphy data with pyActigraphy" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Briefly speaking, the aim of such analysis is to find a \"smooth\" representation (functions) of the data for subsequent analysis." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Data can be converted to a functional form using either\n", "\n", "* basis function expansions (ex: Fourier, B-spline)\n", "* kernel functions\n", "\n", "\n", "Both types are implemented in `pyActigraphy`." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "NB: at the moment, the `pyActigraphy` package allows users to model the daily activity profile only. However, more profiles might be added in the future if needed." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The reference book for 'functional data analysis', including information about functional modelling:\n", "\n", "Ramsay, J. O., & Silverman, B. W. (2005). Functional Data Analysis. New York, NY: Springer New York. https://doi.org/10.1007/b98888" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In the context of actigraphy, a nice example can be found in:\n", "\n", "* Wang, J., Xian, H., Licis, A., Deych, E., Ding, J., McLeland, J., … Shannon, W. (2011). Measuring the impact of apnea and obesity on circadian activity patterns using functional linear modeling of actigraphy data. Journal of Circadian Rhythms, 9(1), 11. https://doi.org/10.1186/1740-3391-9-11" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Imports and input data" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "ExecuteTime": { "end_time": "2023-01-30T15:42:38.225384Z", "start_time": "2023-01-30T15:42:36.640761Z" } }, "outputs": [], "source": [ "import pyActigraphy\n", "from pyActigraphy.analysis import FLM" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "ExecuteTime": { "end_time": "2023-01-30T15:42:38.228792Z", "start_time": "2023-01-30T15:42:38.226933Z" } }, "outputs": [], "source": [ "import numpy as np" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "ExecuteTime": { "end_time": "2023-01-30T15:42:38.231916Z", "start_time": "2023-01-30T15:42:38.230288Z" } }, "outputs": [], "source": [ "import plotly.graph_objs as go" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "ExecuteTime": { "end_time": "2023-01-30T15:42:38.245943Z", "start_time": "2023-01-30T15:42:38.233206Z" } }, "outputs": [], "source": [ "# create objects for layout and traces\n", "layout = go.Layout(autosize=False, width=850, height=600, title=\"\",xaxis=dict(title=\"\"), shapes=[], showlegend=True)" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "ExecuteTime": { "end_time": "2023-01-30T15:42:38.249370Z", "start_time": "2023-01-30T15:42:38.247081Z" } }, "outputs": [], "source": [ "import os" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "ExecuteTime": { "end_time": "2023-01-30T15:42:38.252795Z", "start_time": "2023-01-30T15:42:38.250468Z" } }, "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 an example file:" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "ExecuteTime": { "end_time": "2023-01-30T15:42:38.271531Z", "start_time": "2023-01-30T15:42:38.253951Z" } }, "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": "code", "execution_count": 8, "metadata": { "ExecuteTime": { "end_time": "2023-01-30T15:42:38.276103Z", "start_time": "2023-01-30T15:42:38.273420Z" } }, "outputs": [], "source": [ "from pyActigraphy.analysis import FLM" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Basis function expansion" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The idea behind a basis function expansion is to represent the data $\\{x_i\\}_{i\\in[0,N]}$ as:\n", "\n", "$$ x_i = \\beta_{1} \\phi_1(t_i) + \\beta_{2} \\phi_2(t_i) +··· + \\beta_{n} \\phi_n(t_i)$$ \n", "where $\\{\\beta_{i}\\}_{i=1}^n$ are scalar coefficients and $\\{\\phi_i(·)\\}_{i=1}^n$ are basis functions.\n", "Possible basis functions include Fourier basis and splines." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Such expansion can be performed with a least-square fit" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Using a Fourier basis expansion" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In this case, the basis functions are simple (co)sine functions:\n", "$\\phi_{0}(t) = 1$, $\\phi_{2n}(t) = cos(n\\omega t)$ and $\\phi_{2n-1}(t) = sin(n\\omega t)$, $\\omega = \\frac{2\\pi}{T}$ where T is the period.\n", "\n", "For the daily activity profile, T = 24h." ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "ExecuteTime": { "end_time": "2023-01-30T15:42:38.281768Z", "start_time": "2023-01-30T15:42:38.279078Z" } }, "outputs": [], "source": [ "# Resampling frequency for the daily activity profile\n", "freq='1min'" ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "ExecuteTime": { "end_time": "2023-01-30T15:42:38.286124Z", "start_time": "2023-01-30T15:42:38.284156Z" } }, "outputs": [], "source": [ "# The number of basis functions is max_order*2+1 (1 constant + n cosine functions + n sine functions)\n", "max_order = 9" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "First, let's define a FLM object with \"Fourier\" functions as a basis:" ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "ExecuteTime": { "end_time": "2023-01-30T15:42:38.290036Z", "start_time": "2023-01-30T15:42:38.287733Z" } }, "outputs": [], "source": [ "flm = FLM(basis='fourier',sampling_freq=freq,max_order=max_order)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "To estimate the scalar coefficients of the basis expansion (\"beta\" coefficients), use the `fit` function:" ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "ExecuteTime": { "end_time": "2023-01-30T15:42:38.319178Z", "start_time": "2023-01-30T15:42:38.291549Z" }, "scrolled": true }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Create first the basis functions: fourier\n", " OLS Regression Results \n", "==============================================================================\n", "Dep. Variable: y R-squared: 0.731\n", "Model: OLS Adj. R-squared: 0.728\n", "Method: Least Squares F-statistic: 214.7\n", "Date: Mon, 30 Jan 2023 Prob (F-statistic): 0.00\n", "Time: 16:42:38 Log-Likelihood: -8390.5\n", "No. Observations: 1440 AIC: 1.682e+04\n", "Df Residuals: 1421 BIC: 1.692e+04\n", "Df Model: 18 \n", "Covariance Type: nonrobust \n", "==============================================================================\n", " coef std err t P>|t| [0.025 0.975]\n", "------------------------------------------------------------------------------\n", "const 170.9896 2.178 78.523 0.000 166.718 175.261\n", "x1 -139.4514 3.080 -45.283 0.000 -145.492 -133.410\n", "x2 -73.6459 3.080 -23.915 0.000 -79.687 -67.605\n", "x3 -10.5884 3.080 -3.438 0.001 -16.629 -4.547\n", "x4 -29.8752 3.080 -9.701 0.000 -35.916 -23.834\n", "x5 45.4418 3.080 14.756 0.000 39.401 51.483\n", "x6 58.3750 3.080 18.956 0.000 52.334 64.416\n", "x7 6.4342 3.080 2.089 0.037 0.393 12.475\n", "x8 -28.0195 3.080 -9.099 0.000 -34.060 -21.979\n", "x9 -13.4018 3.080 -4.352 0.000 -19.443 -7.361\n", "x10 -23.0809 3.080 -7.495 0.000 -29.122 -17.040\n", "x11 31.6536 3.080 10.279 0.000 25.613 37.695\n", "x12 -13.2986 3.080 -4.318 0.000 -19.340 -7.258\n", "x13 -27.9879 3.080 -9.088 0.000 -34.029 -21.947\n", "x14 16.3752 3.080 5.317 0.000 10.334 22.416\n", "x15 11.3963 3.080 3.701 0.000 5.355 17.437\n", "x16 -33.7290 3.080 -10.953 0.000 -39.770 -27.688\n", "x17 -15.9310 3.080 -5.173 0.000 -21.972 -9.890\n", "x18 3.9516 3.080 1.283 0.200 -2.089 9.993\n", "==============================================================================\n", "Omnibus: 379.235 Durbin-Watson: 0.633\n", "Prob(Omnibus): 0.000 Jarque-Bera (JB): 1490.128\n", "Skew: 1.222 Prob(JB): 0.00\n", "Kurtosis: 7.343 Cond. No. 1.41\n", "==============================================================================\n", "\n", "Notes:\n", "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n" ] } ], "source": [ "# By setting the \"verbose\" parameter to True, the result of least-square fit is displayed:\n", "flm.fit(raw,verbose=True)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now, to reconstruct the signal using its expansion up to the 9th order:" ] }, { "cell_type": "code", "execution_count": 13, "metadata": { "ExecuteTime": { "end_time": "2023-01-30T15:42:38.323982Z", "start_time": "2023-01-30T15:42:38.320950Z" } }, "outputs": [], "source": [ "flm_est = flm.evaluate(raw)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "And compare it to the original daily profile:" ] }, { "cell_type": "code", "execution_count": 14, "metadata": { "ExecuteTime": { "end_time": "2023-01-30T15:42:38.331607Z", "start_time": "2023-01-30T15:42:38.325093Z" } }, "outputs": [], "source": [ "daily_avg = raw.average_daily_activity(binarize=False,freq=freq)" ] }, { "cell_type": "code", "execution_count": 15, "metadata": { "ExecuteTime": { "end_time": "2023-01-30T15:42:38.341265Z", "start_time": "2023-01-30T15:42:38.332803Z" }, "code_folding": [ 0 ] }, "outputs": [], "source": [ "# set x-axis labels and their corresponding data values\n", "labels = ['00:00', '06:00', '12:00', '18:00']\n", "tickvals = ['00:00:00', '06:00:00', '12:00:00', '18:00:00']\n", "\n", "layout = go.Layout(\n", " autosize=False, width=900, height=600, \n", " title=\"Daily profile\",\n", " xaxis=dict(\n", " title=\"Time of day (HH:MM)\",\n", " ticktext=labels,\n", " tickvals=tickvals), \n", " yaxis=dict(title=\"Counts (a.u)\"),\n", " shapes=[], showlegend=True)" ] }, { "cell_type": "code", "execution_count": 16, "metadata": { "ExecuteTime": { "end_time": "2023-01-30T15:42:38.513351Z", "start_time": "2023-01-30T15:42:38.343191Z" } }, "outputs": [ { "data": { "text/html": [ " 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302.77777777777777, 241.11111111111111, 162, 197.33333333333334, 127.55555555555556, 137.55555555555554, 166.55555555555554, 267.1111111111111, 173.88888888888889, 279.55555555555554, 281.1111111111111, 267.8888888888889, 230.77777777777777, 247.33333333333334, 238.11111111111111, 140.77777777777777, 182.44444444444446, 180.33333333333334, 115.77777777777777, 185.33333333333334, 146.66666666666666, 151.44444444444446, 177, 148.66666666666666, 165.88888888888889, 142, 67.22222222222223, 112.77777777777777, 88.11111111111111, 150.55555555555554, 190.55555555555554, 223.88888888888889, 214, 328.77777777777777, 290.44444444444446, 270.77777777777777, 138.77777777777777, 190.66666666666666, 188.22222222222223, 130.66666666666666, 251, 203, 201.22222222222223, 229.11111111111111, 336.77777777777777, 289.8888888888889, 131.77777777777777, 177.55555555555554, 196.11111111111111, 209.11111111111111, 193.22222222222223, 122.44444444444444, 196, 228.11111111111111, 98.77777777777777, 183.44444444444446, 96.77777777777777, 33.77777777777778, 90.33333333333333, 123.77777777777777, 187.66666666666666, 122.77777777777777, 107.22222222222223, 105.33333333333333, 71.88888888888889, 37.55555555555556, 39.666666666666664, 62.22222222222222, 43, 47.55555555555556, 21, 75.88888888888889, 98.22222222222223, 109.11111111111111, 69.55555555555556, 44.44444444444444, 58.666666666666664, 75.55555555555556, 71.66666666666667, 112.55555555555556, 88.33333333333333, 99.33333333333333, 105.33333333333333, 106.77777777777777, 95.44444444444444, 157.88888888888889, 139.66666666666666, 71.44444444444444, 115.88888888888889, 73, 78.77777777777777, 90.55555555555556, 297.8888888888889, 152.44444444444446, 90.55555555555556, 147.77777777777777, 207.88888888888889, 249.22222222222223, 161.33333333333334, 122.88888888888889, 171.77777777777777, 238.88888888888889, 173.44444444444446, 376.55555555555554, 274.44444444444446, 307.55555555555554, 166.11111111111111, 196.88888888888889, 260.6666666666667, 249.66666666666666, 374.1111111111111, 349.3333333333333, 284, 215.11111111111111, 295.44444444444446, 359.44444444444446, 237.88888888888889, 153.44444444444446, 158.33333333333334, 135, 325.3333333333333, 345, 572.1111111111111, 262.6666666666667, 178.22222222222223, 147.11111111111111, 189.55555555555554, 168.11111111111111, 194.77777777777777, 385.8888888888889, 210.22222222222223, 186.77777777777777, 311.77777777777777, 323.77777777777777, 284.77777777777777, 269.77777777777777, 305.22222222222223, 411.1111111111111, 364.77777777777777, 236.44444444444446, 334.55555555555554, 303.6666666666667, 289.3333333333333, 383.1111111111111, 293.8888888888889, 401.3333333333333, 311.3333333333333, 204.44444444444446, 310.44444444444446, 376.22222222222223, 283.44444444444446, 216, 293.44444444444446, 359.1111111111111, 303.55555555555554, 380.6666666666667, 349.8888888888889, 322.6666666666667, 279.3333333333333, 380.8888888888889, 431.77777777777777, 419.8888888888889, 424.6666666666667, 419.8888888888889, 338, 244.55555555555554, 298, 374.22222222222223, 542.2222222222222, 349.77777777777777, 355.1111111111111, 420.77777777777777, 470.1111111111111, 340.1111111111111, 325.8888888888889, 240.11111111111111, 199, 207.55555555555554, 273.22222222222223, 155.22222222222223, 167.22222222222223, 219.22222222222223, 233.22222222222223, 329.3333333333333, 421.8888888888889, 315.1111111111111, 422, 262.1111111111111, 267.3333333333333, 296.22222222222223, 168.33333333333334, 168.66666666666666, 207.55555555555554, 269.1111111111111, 230.44444444444446, 217.55555555555554, 240.22222222222223, 280.6666666666667, 218.44444444444446, 267.44444444444446, 256.55555555555554, 386, 416.3333333333333, 292.22222222222223, 327.1111111111111, 302.1111111111111, 249.33333333333334, 222.88888888888889, 244.55555555555554, 324.1111111111111, 195.66666666666666, 220.55555555555554, 304.22222222222223, 215.44444444444446, 226.33333333333334, 277.6666666666667, 270.55555555555554, 326.3333333333333, 342.77777777777777, 363, 308.77777777777777, 225.22222222222223, 233.55555555555554, 255.44444444444446, 229.66666666666666, 293.22222222222223, 323, 437.6666666666667, 254.44444444444446, 194.33333333333334, 234.33333333333334, 278.44444444444446, 379.55555555555554, 235.44444444444446, 320.22222222222223, 272.22222222222223, 152.88888888888889, 295.77777777777777, 357.6666666666667, 363.3333333333333, 405.8888888888889, 432.55555555555554, 480.1111111111111, 505.6666666666667, 546.8888888888889, 375.1111111111111, 570.1111111111111, 592.3333333333334, 376.22222222222223, 481.77777777777777, 364.6666666666667, 360.6666666666667, 332.6666666666667, 324, 454.55555555555554, 350.6666666666667, 340.44444444444446, 372.1111111111111, 299.1111111111111, 343.8888888888889, 378.22222222222223, 488.6666666666667, 474.6666666666667, 420, 472.77777777777777, 444.77777777777777, 342.44444444444446, 413.22222222222223, 425.77777777777777, 471.77777777777777, 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236.44444444444446, 206.66666666666666, 149.33333333333334, 135, 110.33333333333333, 143.77777777777777, 148.66666666666666, 234.77777777777777, 202, 131.44444444444446, 121.55555555555556, 187.22222222222223, 177.77777777777777, 173, 235, 132.22222222222223, 149.33333333333334, 323.8888888888889, 254, 162.66666666666666, 225.44444444444446, 222.77777777777777, 265.44444444444446, 207, 244.22222222222223, 170.77777777777777, 167.11111111111111, 156.22222222222223, 187.66666666666666, 127.55555555555556, 211, 247.55555555555554, 282.55555555555554, 215, 315.44444444444446, 315, 463.55555555555554, 313.6666666666667, 344.6666666666667, 282.77777777777777, 269.77777777777777, 245, 365.3333333333333, 293.8888888888889, 293.6666666666667, 347.22222222222223, 280.6666666666667, 283.77777777777777, 210.77777777777777, 262.8888888888889, 217.77777777777777, 239.11111111111111, 281.6666666666667, 181.33333333333334, 110.88888888888889, 128.33333333333334, 118.11111111111111, 179, 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