{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Least squares fitting of models to data" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "This is a quick introduction to `statsmodels` for physical scientists (e.g. physicists, astronomers) or engineers.\n", "\n", "Why is this needed?\n", "\n", "Because most of `statsmodels` was written by statisticians and they use a different terminology and sometimes methods, making it hard to know which classes and functions are relevant and what their inputs and outputs mean." ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": false, "execution": { "iopub.execute_input": "2021-02-02T06:54:46.244699Z", "iopub.status.busy": "2021-02-02T06:54:46.236513Z", "iopub.status.idle": "2021-02-02T06:54:47.563348Z", "shell.execute_reply": "2021-02-02T06:54:47.564786Z" } }, "outputs": [], "source": [ "import numpy as np\n", "import pandas as pd\n", "import statsmodels.api as sm" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Linear models" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Assume you have data points with measurements `y` at positions `x` as well as measurement errors `y_err`.\n", "\n", "How can you use `statsmodels` to fit a straight line model to this data?\n", "\n", "For an extensive discussion see [Hogg et al. (2010), \"Data analysis recipes: Fitting a model to data\"](https://arxiv.org/abs/1008.4686) ... we'll use the example data given by them in Table 1.\n", "\n", "So the model is `f(x) = a * x + b` and on Figure 1 they print the result we want to reproduce ... the best-fit parameter and the parameter errors for a \"standard weighted least-squares fit\" for this data are:\n", "* `a = 2.24 +- 0.11`\n", "* `b = 34 +- 18`" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": false, "execution": { "iopub.execute_input": "2021-02-02T06:54:47.571311Z", "iopub.status.busy": "2021-02-02T06:54:47.569610Z", "iopub.status.idle": "2021-02-02T06:54:47.610557Z", "shell.execute_reply": "2021-02-02T06:54:47.611793Z" } }, "outputs": [ { "data": { "text/html": [ "
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" ], "text/plain": [ " x y y_err\n", "0 201.0 592.0 61.0\n", "1 244.0 401.0 25.0\n", "2 47.0 583.0 38.0\n", "3 287.0 402.0 15.0\n", "4 203.0 495.0 21.0" ] }, "execution_count": 1, "metadata": {}, "output_type": "execute_result" } ], "source": [ "data = \"\"\"\n", " x y y_err\n", "201 592 61\n", "244 401 25\n", " 47 583 38\n", "287 402 15\n", "203 495 21\n", " 58 173 15\n", "210 479 27\n", "202 504 14\n", "198 510 30\n", "158 416 16\n", "165 393 14\n", "201 442 25\n", "157 317 52\n", "131 311 16\n", "166 400 34\n", "160 337 31\n", "186 423 42\n", "125 334 26\n", "218 533 16\n", "146 344 22\n", "\"\"\"\n", "try:\n", " from StringIO import StringIO\n", "except ImportError:\n", " from io import StringIO\n", "data = pd.read_csv(StringIO(data), delim_whitespace=True).astype(float)\n", "\n", "# Note: for the results we compare with the paper here, they drop the first four points\n", "data.head()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "To fit a straight line use the weighted least squares class [WLS](https://www.statsmodels.org/devel/generated/statsmodels.regression.linear_model.WLS.html) ... the parameters are called:\n", "* `exog` = `sm.add_constant(x)`\n", "* `endog` = `y`\n", "* `weights` = `1 / sqrt(y_err)`\n", "\n", "Note that `exog` must be a 2-dimensional array with `x` as a column and an extra column of ones. Adding this column of ones means you want to fit the model `y = a * x + b`, leaving it off means you want to fit the model `y = a * x`.\n", "\n", "And you have to use the option `cov_type='fixed scale'` to tell `statsmodels` that you really have measurement errors with an absolute scale. If you do not, `statsmodels` will treat the weights as relative weights between the data points and internally re-scale them so that the best-fit model will have `chi**2 / ndf = 1`." ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": false, "execution": { "iopub.execute_input": "2021-02-02T06:54:47.617329Z", "iopub.status.busy": "2021-02-02T06:54:47.615550Z", "iopub.status.idle": "2021-02-02T06:54:47.660633Z", "shell.execute_reply": "2021-02-02T06:54:47.661822Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ " WLS Regression Results \n", "==============================================================================\n", "Dep. Variable: y R-squared: 0.400\n", "Model: WLS Adj. R-squared: 0.367\n", "Method: Least Squares F-statistic: 193.5\n", "Date: Tue, 02 Feb 2021 Prob (F-statistic): 4.52e-11\n", "Time: 06:54:47 Log-Likelihood: -119.06\n", "No. Observations: 20 AIC: 242.1\n", "Df Residuals: 18 BIC: 244.1\n", "Df Model: 1 \n", "Covariance Type: fixed scale \n", "==============================================================================\n", " coef std err z P>|z| [0.025 0.975]\n", "------------------------------------------------------------------------------\n", "const 213.2735 14.394 14.817 0.000 185.062 241.485\n", "x 1.0767 0.077 13.910 0.000 0.925 1.228\n", "==============================================================================\n", "Omnibus: 0.943 Durbin-Watson: 2.901\n", "Prob(Omnibus): 0.624 Jarque-Bera (JB): 0.181\n", "Skew: -0.205 Prob(JB): 0.914\n", "Kurtosis: 3.220 Cond. No. 575.\n", "==============================================================================\n", "\n", "Notes:\n", "[1] Standard Errors are based on fixed scale\n" ] } ], "source": [ "exog = sm.add_constant(data['x'])\n", "endog = data['y']\n", "weights = 1. / (data['y_err'] ** 2)\n", "wls = sm.WLS(endog, exog, weights)\n", "results = wls.fit(cov_type='fixed scale')\n", "print(results.summary())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Check against scipy.optimize.curve_fit" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false, "execution": { "iopub.execute_input": "2021-02-02T06:54:47.667500Z", "iopub.status.busy": "2021-02-02T06:54:47.665762Z", "iopub.status.idle": "2021-02-02T06:54:47.691285Z", "shell.execute_reply": "2021-02-02T06:54:47.692553Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "a = 1.077 +- 0.077\n", "b = 213.273 +- 14.394\n" ] } ], "source": [ "# You can use `scipy.optimize.curve_fit` to get the best-fit parameters and parameter errors.\n", "from scipy.optimize import curve_fit\n", "\n", "def f(x, a, b):\n", " return a * x + b\n", "\n", "xdata = data['x']\n", "ydata = data['y']\n", "p0 = [0, 0] # initial parameter estimate\n", "sigma = data['y_err']\n", "popt, pcov = curve_fit(f, xdata, ydata, p0, sigma, absolute_sigma=True)\n", "perr = np.sqrt(np.diag(pcov))\n", "print('a = {0:10.3f} +- {1:10.3f}'.format(popt[0], perr[0]))\n", "print('b = {0:10.3f} +- {1:10.3f}'.format(popt[1], perr[1]))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Check against self-written cost function" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": false, "execution": { "iopub.execute_input": "2021-02-02T06:54:47.697971Z", "iopub.status.busy": "2021-02-02T06:54:47.696277Z", "iopub.status.idle": "2021-02-02T06:54:47.818357Z", "shell.execute_reply": "2021-02-02T06:54:47.819621Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "a = 1.077\n", "b = 213.274\n" ] } ], "source": [ "# You can also use `scipy.optimize.minimize` and write your own cost function.\n", "# This does not give you the parameter errors though ... you'd have\n", "# to estimate the HESSE matrix separately ...\n", "from scipy.optimize import minimize\n", "\n", "def chi2(pars):\n", " \"\"\"Cost function.\n", " \"\"\"\n", " y_model = pars[0] * data['x'] + pars[1]\n", " chi = (data['y'] - y_model) / data['y_err']\n", " return np.sum(chi ** 2)\n", "\n", "result = minimize(fun=chi2, x0=[0, 0])\n", "popt = result.x\n", "print('a = {0:10.3f}'.format(popt[0]))\n", "print('b = {0:10.3f}'.format(popt[1]))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Non-linear models" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": false, "execution": { "iopub.execute_input": "2021-02-02T06:54:47.825392Z", "iopub.status.busy": "2021-02-02T06:54:47.823692Z", "iopub.status.idle": "2021-02-02T06:54:47.829670Z", "shell.execute_reply": "2021-02-02T06:54:47.830848Z" } }, "outputs": [], "source": [ "# TODO: we could use the examples from here:\n", "# http://probfit.readthedocs.org/en/latest/api.html#probfit.costfunc.Chi2Regression" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "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": "3.7.9" } }, "nbformat": 4, "nbformat_minor": 0 }