{ "cells": [ { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": false }, "outputs": [ { "name": "stderr", "output_type": "stream", "text": [ ":0: FutureWarning: IPython widgets are experimental and may change in the future.\n" ] } ], "source": [ "%matplotlib inline\n", "from numpy import array, arange, exp, log, linspace\n", "from numpy.random import normal, random\n", "import lmfit\n", "from sklearn import cross_validation\n", "import seaborn as sns\n", "sns.set_style('ticks')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Underlying function\n", "\n", "$$\n", "f(t) = b e^{a t} \\\\\n", "g(t) = b^{at}\n", "$$" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": true }, "outputs": [], "source": [ "def f(t,a,b):\n", " return b*exp(-a*t)" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": true }, "outputs": [], "source": [ "def g(t,a,b):\n", " return b**(a*t)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Generate data" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAf8AAAFzCAYAAAAwgwAdAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAAIABJREFUeJzsvXlwHFl+mPlV4aj7wlEFEABJ8MruIXumu+fUzFgjWe7x\n", "2rLksOSI0VqWj9CuxrK8Ya+tsOQJ2+pdr7WjtSR7ZclaWbIk6wjLlizJocOW2yOpe0Yzfcyw2U2y\n", "ySTBBkgQQFWhCnWgjiwcVftH1nvILGZmAWiQIIj3RXQ0mFVZ+fJ6v/v3fJ1OB4VCoVAoFMcH/2EP\n", "QKFQKBQKxaNFCX+FQqFQKI4ZSvgrFAqFQnHMUMJfoVAoFIpjhhL+CoVCoVAcM5TwVygUCoXimKGE\n", "v0KhUCgUx4zBwx6AQqF4EE3Tfhr4s8Cv6rr+Tx7C738NCAIb3U2/ouv6jzl873PAd2HOFb+i6/r/\n", "0d0+DvwScBJoA9+j6/pX+hzzRSCj6/r37nGszwBfAW5bNn9G1/VbPd9zHZOmad8M/DAQAN4GvlvX\n", "9fW9jEOheJJQwl+heDz5HmBG1/Xlg/5hTdMiwBlgTNf1bY/v/XngLwPPYwrTP9A07R1d138d+Cng\n", "ZV3XP69p2geA39M07byu602PQ++3o9jHMZWgz/b5ntOYzgEx4OeBj+u6fkfTtM8Dnwe+b5/jUSiO\n", "PEr4KxSPGZqmfRHwAf9N07S/rev6lzy++zngL2Ja8RHg+3Vd/21N034A+I6er3eAPwN8AKgBv69p\n", 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100\n", "t = linspace(0,1)\n", "sig = 2\n", "a,b = 5,5\n", "data = array([f(t, a, b) + normal(0, sig**2, len(t)) for _ in range(N)])\n", "avg = data.mean(axis=0)\n", "sd = data.std(axis=0, ddof=1)\n", "for y in data:\n", " scatter(t, y, color='k', alpha=0.1, marker='o')\n", "fill_between(t, avg-sd, avg+sd, alpha=0.25)\n", "plot(t, avg, color='k', lw=2)\n", "title(\"f, a=%.2f, b=%.2f\" % (a,b))\n", "xlabel('t')\n", "ylabel('y')\n", "sns.despine()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Model fitting" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[[Model]]\n", " Model(f)\n", "[[Fit Statistics]]\n", " # function evals = 16\n", " # data points = 50\n", " # variables = 2\n", " chi-square = 8.101\n", " reduced chi-square = 0.169\n", "[[Variables]]\n", " a: 5.22585174 +/- 0.374737 (7.17%) (init= 1)\n", " b: 5.03812494 +/- 0.242209 (4.81%) (init= 1)\n", "[[Correlations]] (unreported correlations are < 0.100)\n", " C(a, b) = 0.669 \n" ] }, { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAfgAAAFqCAYAAAD2hkg3AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAAIABJREFUeJzt3Xl8VNX9//HXnS37xr5FwnrDDm6IggpSXGlRaa11qWvV\n", "2sXqF0UUwbpWi7/W2tZWLa27rSJabYWKKy6IC6tw2WVPAmTfZ3J/f0wSE5KQSZjMlvfz8ZhHMnfu\n", "zP3MEOZ9z73nnmPYto2IiIjEFke4CxAREZHgU8CLiIjEIAW8iIhIDFLAi4iIxCAFvIiISAxSwIuI\n", "iMQgV6g3aJrm7cB0wA08ZlnWP0Jdg4iISKwLaQveNM3TgQmWZZ0MnA4MDOX2RUREOotQt+CnAWtN\n", "01wMpAKzQrx9ERGRTiHUAd8dyATOw996fx3IDvTJpmm6gH7AbsuyvB1SoYiISAwIdcAfADbUhvMm\n", "0zQrTNPsZlnWgcNXNE1zPjCvuRdZtmxZx1YpIiISWYy2PiHUAb8c+CXwiGmafYAk4GBzK1qWNR+Y\n", "33CZaZpZwPYOrVBERCQGhLSTnWVZbwJfmab5Gf7D8z+1LEuz3YiIiARZyC+TsyzrtlBvU0REpLPR\n", "QDciIiIxSAEvIiISgxTwIiIiMUgBLyIiEoMU8CIiIjFIAS8iIhKDFPAiIhI0H374If/85z9bfPzV\n", "V1/lnXfeAeDZZ58N6Dl/+MMfePHFFxstKyws5Pzzz+fqq69uc42XXXYZ55xzTqNlS5cuJTs7m717\n", 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"execution_count": 6, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[[Model]]\n", " Model(g)\n", "[[Fit Statistics]]\n", " # function evals = 28\n", " # data points = 50\n", " # variables = 2\n", " chi-square = 77.206\n", " reduced chi-square = 1.608\n", "[[Variables]]\n", " a: 0.45969564 +/- 5.70e+06 (1239765152.84%) (init= 1)\n", " b: 0.08018864 +/- 2.48e+06 (3095810320.69%) (init= 1)\n", "[[Correlations]] (unreported correlations are < 0.100)\n", " C(a, b) = 1.000 \n" ] }, { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAfgAAAFqCAYAAAD2hkg3AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAAIABJREFUeJzt3XmYXFWB9/Hfra7qqt4XSNgSwxJykmEJDy8aRUVZxAXy\n", "GjTOqAhDHmBcZmGJ0RCyGpYohvcZQdQRiBoQEEUmig/wmnHQ+Mxk8opZDSdgwLCYhCy9d3V1dd/3\n", "j1rSnfRS1V11q+rW9/M8/XQtt+qeuunU75xz7znHcV1XAADAXwKFLgAAAMg9Ah4AAB8i4AEA8CEC\n", "HgAAHyLgAQDwIQIeAAAfCnq9Q2PMbZJmSwpJut9a+0OvywAAgN952oI3xnxQ0nustRdK+qCk073c\n", 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"1trvFKakpWukYyzp/yV/fjvgJf9qrX3a00KWuNH+jgds9/eSjLV2kfelLH0ZfF+kKlGOpN9ba28p\n", "TElLVwbH+BZJn1Xi+p1XJN2Y7DFBlowxpypR4b8wOZppTLlXtAEPAADGrpi76AEAwBgR8AAA+BAB\n", "DwCADxHwAAD4EAEPAIAPEfAAAPgQAQ9gVMaYBmPMzwtdDgCZI+ABZKJJ0nmFLgSAzBHwADLxLUkn\n", "G2N+VuiCAMgMAQ8gE/8s6S1rLYsOASWCgAeQCZazBUoMAQ8AgA8R8AAyEZcULHQhAGSOgAeQib2S\n", "9hhj1he6IAAyw3KxAAD4EC14AAB8iIAHAMCHCHgAAHyIgAcAwIcIeAAAfIiABwDAhwh4AAB86P8D\n", "OrnqfJhN5mAAAAAASUVORK5CYII=\n" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "model = lmfit.Model(g)\n", "params = model.make_params(a=1,b=1)\n", "\n", "gfit = model.fit(avg, t=t, params=params)\n", "print gfit.fit_report()\n", "gfit.plot_fit()\n", "sns.despine()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## [Cross-valiation](http://scikit-learn.org/stable/modules/cross_validation.html)" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "16.1754690489\n" ] } ], "source": [ "model = lmfit.Model(f)\n", "params = model.make_params(a=0,b=1)\n", "loocv = cross_validation.LeaveOneOut(N)\n", "fscore = 0\n", "for train_index, test_index in loocv:\n", " train, test = data[train_index,:], data[test_index,:]\n", " fit = model.fit(train.mean(axis=0), t=t, params=params)\n", " _a,_b = fit.best_values['a'],fit.best_values['b']\n", " fscore += ((test[0,:] - f(t, _a, _b))**2).sum()\n", "fscore /= float(len(t) * (N-1))\n", "print fscore" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "17.8122836437\n" ] } ], "source": [ "model = lmfit.Model(g)\n", "params = model.make_params(a=0,b=1)\n", "loocv = cross_validation.LeaveOneOut(N)\n", "gscore = 0\n", "for train_index, test_index in loocv:\n", " train, test = data[train_index,:], data[test_index,:]\n", " fit = model.fit(train.mean(axis=0), t=t, params=params)\n", " _a,_b = fit.best_values['a'],fit.best_values['b']\n", " gscore += ((test[0,:] - g(t, _a, _b))**2).sum()\n", "gscore /= float(len(t) * (N-1))\n", "print gscore" ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "0.90810753816950618" ] }, "execution_count": 9, "metadata": {}, "output_type": "execute_result" } ], "source": [ "fscore / gscore" ] } ], "metadata": { "kernelspec": { "display_name": "Python 2", "language": "python", "name": "python2" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 2 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython2", "version": "2.7.8" } }, "nbformat": 4, "nbformat_minor": 0 }