{ "metadata": { "name": "", "signature": "sha256:6053b3c73be13fa5d5b8eba5479d8b17db084214bf0843fdd393a9a6d7c0022a" }, "nbformat": 3, "nbformat_minor": 0, "worksheets": [ { "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Solution: Frequentist Model Fitting Breakout\n", "\n", "Following is an example solution to the [Model Fitting Breakout](01.2-Model-Fitting-Breakout.ipynb)." ] }, { "cell_type": "code", "collapsed": false, "input": [ "%matplotlib inline\n", "import numpy as np\n", "import matplotlib.pyplot as plt\n", "\n", "# use seaborn plotting defaults\n", "# If this causes an error, you can comment it out.\n", "import seaborn as sns\n", "sns.set()" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 1 }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Part I: Fourier Fit to RR Lyrae" ] }, { "cell_type": "code", "collapsed": false, "input": [ "from fig_code import sample_light_curve\n", "t, y, dy = sample_light_curve()" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "[10003298 10004892 10013411 ..., 9984569 9987252 999528]\n" ] } ], "prompt_number": 2 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Visualize the data:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "plt.errorbar(t, y, dy, fmt='o')\n", "plt.gca().invert_yaxis();" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "display_data", "png": 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"text": [ "" ] } ], "prompt_number": 3 }, { "cell_type": "markdown", "metadata": {}, "source": [ "This data has already been phased, so if we just take the fractional part of *t* then we can see the folded light curve" ] }, { "cell_type": "code", "collapsed": false, "input": [ "plt.errorbar(t % 1, y, dy, fmt='o')\n", "plt.gca().invert_yaxis();" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "display_data", "png": 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"text": [ "" ] } ], "prompt_number": 4 }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 1. Iterative Solution\n", "\n", "We'll start with an iterative solution to the problem.\n", "\n", "1. Create a function which evaluates the model given an array of times *t*, a base frequency $\\omega$, and an array of coefficients $\\theta$. **For this section, we will not treat $\\omega$ as a model parameter, but as a constant.** For your model, you can use $\\omega = 2\\pi$ (we'll relax this assumption in part II).\n", "\n", "2. Create a function which evaluates the log-likelihood as a function of $\\theta$, using the above function. **Keep in mind that ``theta`` must be a one-dimensional array (this is what ``optimize.fmin()`` requires)**\n", "\n", "3. Use ``scipy.optimize.fmin`` to maximize your log-likelihood (i.e. minimize the negative log-likelihood) to find the optimal model.\n", "\n", "4. Plot this model over the data to see how it looks.\n", "\n", "5. Use new data, from ``fig_code.sample_light_curve_2()``, and apply your code again (hopefully you've written the functions so that they can easily be reused, right?" ] }, { "cell_type": "code", "collapsed": false, "input": [ "# Part 1:\n", "def model(theta, t, omega=2*np.pi):\n", " return (theta[0] +\n", " theta[1] * np.sin(omega * t) + theta[2] * np.cos(omega * t) +\n", " theta[3] * np.sin(2 * omega * t) + theta[4] * np.cos(2 * omega * t))" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 5 }, { "cell_type": "code", "collapsed": false, "input": [ "# Part 2:\n", "def log_likelihood(theta, t, y, dy, omega=2*np.pi):\n", " return -0.5 * np.sum(np.log(2 * np.pi * dy ** 2) +\n", " ((y - model(theta, t, omega)) / dy) ** 2)" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 6 }, { "cell_type": "code", "collapsed": false, "input": [ "# Part 3:\n", "from scipy import optimize\n", "\n", "# fmin minimizes, not maximizes, a function: use the negative log-likelihood\n", "def neg_log_likelihood(theta, t, y, dy):\n", " return - log_likelihood(theta, t, y, dy)\n", "\n", "theta_guess = [0, 1, 1, 1, 1]\n", "theta_fit = optimize.fmin(neg_log_likelihood, theta_guess, args=(t, y, dy))" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "Optimization terminated successfully.\n", " Current function value: 2478.505453\n", " Iterations: 482\n", " Function evaluations: 781\n" ] } ], "prompt_number": 7 }, { "cell_type": "code", "collapsed": false, "input": [ "# Part 4:\n", "\n", "# We'll plot the phased data for clarity\n", "phase_fit = np.linspace(0, 1)\n", "plt.errorbar(t % 1, y, dy, fmt='o')\n", "plt.plot(phase_fit, model(theta_fit, phase_fit));\n", "plt.gca().invert_yaxis()" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "display_data", "png": 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u6KO2kcas6hF418t4RzC7MulZpyFnDIHae5c4rmQ0p5WMYVfNHrY3bPR9fcve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\n8PtomMu2YOpxGE99Ac79I3Tb3RUSsNK5YlqqRO1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"text": [ "" ] } ], "prompt_number": 8 }, { "cell_type": "code", "collapsed": false, "input": [ "# Part 5:\n", "\n", "from fig_code import sample_light_curve_2\n", "t, y, dy = sample_light_curve_2()\n", "\n", "theta_fit = optimize.fmin(neg_log_likelihood, theta_guess, args=(t, y, dy))\n", "\n", "plt.errorbar(t % 1, y, dy, fmt='o')\n", "plt.plot(phase_fit, model(theta_fit, phase_fit));\n", "plt.gca().invert_yaxis();" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "[10003298 10004892 10013411 ..., 9984569 9987252 999528]\n", "Optimization terminated successfully." ] }, { "output_type": "stream", "stream": "stdout", "text": [ "\n", " Current function value: 8062.414006\n", " Iterations: 392\n", " Function evaluations: 634\n" ] }, { "metadata": {}, "output_type": "display_data", "png": 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KuySHaeCbIvfUF6GvVdFMrbxZ8xJuj5tQ5EZPiMwkwTpNfXDwY5qdLZw3/Gyy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\ni4FV/oH65HPO9f75ZFLaDyIEaoCUFuZ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"text": [ "" ] } ], "prompt_number": 9 }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 2. Direct Solution\n", "\n", "Now that you've done this iteratively, let's compute the direct solution using linear algebra.\n", "\n", "1. Create a function which will construct the design matrix ($X$) given *t*, $\\omega$, and a specified number of terms\n", "\n", "2. Create a function which, given this design matrix, will compute the maximum likelihood parameters using ``np.linalg.dot`` and ``np.linalg.solve``.\n", "\n", "3. Plot this answer over your previous answer. Hint: they should agree!" ] }, { "cell_type": "code", "collapsed": false, "input": [ "# Part 1:\n", "\n", "def construct_X(t, omega, nterms):\n", " # nterms should be odd\n", " assert nterms % 2 == 1\n", " \n", " X = np.zeros((len(t), nterms))\n", " \n", " N = int(nterms / 2)\n", " X[:, 0] = 1\n", " X[:, 1::2] = np.sin(omega * t[:, None] * np.arange(1, N + 1))\n", " X[:, 2::2] = np.cos(omega * t[:, None] * np.arange(1, N + 1))\n", " \n", " return X\n", "\n", "def model2(theta, t, omega=2 * np.pi):\n", " t = np.atleast_1d(t)\n", " return np.dot(construct_X(t, omega, len(theta)),\n", " theta)\n", "\n", "# Just double-check that this is computing the same thing:\n", "print(model(theta_fit, t[:5]))\n", "print(np.dot(construct_X(t[:5], 2*np.pi, 5),\n", " theta_fit))" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "[ 13.97156005 13.90883202 13.86268342 13.83231379 13.82840569]\n", "[ 13.97156005 13.90883202 13.86268342 13.83231379 13.82840569]\n" ] } ], "prompt_number": 10 }, { "cell_type": "code", "collapsed": false, "input": [ "# Part 2:\n", "\n", "def fourier_solve(t, y, dy, nterms, omega=2*np.pi):\n", " X = construct_X(t, omega, nterms)\n", " return np.linalg.solve(np.dot(X.T / dy ** 2, X),\n", " np.dot(X.T / dy ** 2, y))" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 11 }, { "cell_type": "code", "collapsed": false, "input": [ "# Part 3:\n", "theta_fit2 = fourier_solve(t, y, dy, 5)\n", "\n", "plt.errorbar(t % 1, y, dy, fmt='o')\n", "plt.plot(phase_fit, model(theta_fit, phase_fit), label='iterative');\n", "plt.plot(phase_fit, model(theta_fit2, phase_fit), label='direct');\n", "plt.gca().invert_yaxis()\n", "plt.legend();" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "display_data", "png": 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d/1tO6wOF5HVIoefgc+v8zHnqfMbEoSFtr6jRvWNqnRkQ15smd/kGrvI4haWZ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\n/+XCPb9jjtXwRa9z6DlojM/rJyfE+vwl8nW6jfP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"text": [ "" ] } ], "prompt_number": 12 }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Part II: Bonus \u2013 finding the optimal phase\n", "\n", "It's possible (though much more difficult) to use maximum-likelihood estimation to find the best phase. Loading the data this way gives the raw MJD values of the observations." ] }, { "cell_type": "code", "collapsed": false, "input": [ "t, y, dy = sample_light_curve(phased=False)\n", "plt.errorbar(t, y, dy, fmt='o');" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "[10003298 10004892 10013411 ..., 9984569 9987252 999528]\n" ] }, { "metadata": {}, "output_type": "display_data", "png": 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"text": [ "" ] } ], "prompt_number": 13 }, { "cell_type": "markdown", "metadata": {}, "source": [ "The model we want to fit is:\n", "\n", "$$\n", "y = a_0 + a_1\\sin(\\omega t) + b_1\\cos(\\omega t)\n", "$$\n", "\n", "except this time our parameter vector is $\\theta = [a_0, a_1, a_2, \\omega]$. Since we're fitting for $\\omega$ itself, this is **not a linear model**, so the closed-form solution will not work. Furthermore, this is a **non-convex** problem, so standard optimization will not work either!\n", "\n", "Instead, we'll do a bit of a hack: for each value of $\\omega$, we find the best $(a_0, a_1, b_1)$, and then report the value of the log-likelihood as a function of $\\omega$.\n", "\n", "Your task is to plot $\\omega$ vs. $\\log L_{max}(\\omega)$ and find the $\\omega$ which best-fits the data. Once you've done this, divide *t* by the phase, and re-produce the plots you did above." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We'll see later in the week that there is a more efficient way of doing this procedure, called the *Lomb-Scargle Periodogram*. At its core, however, it's essentially doing exactly what we do here, just much more efficiently." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Solution:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "t, y, dy = sample_light_curve(phased=False)\n", "\n", "period = np.linspace(0.1, 1.1, 10000)\n", "omega = 2 * np.pi / period\n", "log_like = np.zeros_like(omega)\n", "\n", "for i in range(len(omega)):\n", " theta_fit = fourier_solve(t, y, dy, nterms=5, omega=omega[i])\n", " log_like[i] = log_likelihood(theta_fit, t, y, dy, omega=omega[i])\n", " \n", "plt.plot(period, log_like, linewidth=0.5);" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "[10003298 10004892 10013411 ..., 9984569 9987252 999528]\n" ] }, { "metadata": {}, "output_type": "display_data", "png": 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J79rbC3NXpyUoTL9yphu++adMa9bO8s10Q33vyPawgek9/WmDQZPsZ4R4IYgi\nzpu7MbwuKgQEa4b3+/vHpfbPQF+g3reW3kP4+1u9zZnZuAG1dXbjjicWZdgrmitq0/YWfBlhc+KW\nna3JgO1GVYFfPLIA9zzzaTKwb9WXic1WrSDMvPAVNgOVvPLb/G69WXrJDy992AB8z0rwepbZL5Oq\nDQxzWHMFACDX7cTqzU14N+qFo1wSbr8ufrgryutysl5t29mKp95cgZ4eNTnY0WgVTO+Ssvsw+80p\nhfWwaY5L9d0GA30BW7ulOdASjdab2KsfBnhmuc9rojvRE3l/npNMN7ieHtV55LZl86b6Fs+Dxppb\nu1Kb0PXPeG3BemwzrQ3fuLsTb7s+5EX7Do9lYTyCdfUvAKEigpFd1taAoCPj3/xIyxfr6O+Uo5uO\nPedT+9abJ95Ybrt2QlScmuAXr2zAsnU7sWqTt3M9ipYkK2MxnHyqjba0d2P5+l1Yv203fvvUJ5Ec\n09wyZFswc3if53yJUeBnoM+ybC7C8NL8NZDrvTd1AcCiFdtDf27C56TZtvZudHb34I//dlvNLpoF\nc7wEmCffWJHaxeChL87Lz6j1F/buuMQ0Utv8oJxEj+ppZTJfbBL4rmXRHgDY1FdLkpqovaWBjIwl\nlMNeNm8s3GC7dkIkDy9RIlhlUv+CdiPTo0iiOfta27uzNjCvqVVr3cmU5ORCNaanx3n5ie+0TGPd\ntL2ld8yLipTMcrv2n5vzpWk/7f/Pm7uvbNQ3tuOP/w7eRZpPGOiz7KY/ZPGhL1ng5QZ7/X1zba9s\npylFfgoXT7yxPNT60ioy9zW3dnRpK87pN4moKj6KkQDdig29zePp4+U9DJTIRAVenLcGmxtabQfv\n/e315cn74HK9QJj8ffOkstLU0onn3uu94Zpryq8v7G1p8pLczQ0tGceC+H2YjVnj7t7WM9eBoXZr\nMTj8pgmbloGEl6ZsHxav3I5/ZRjcZsvHhWFXqMzIw9dcunYnYCpA3/PMYuxo6l1gSrvmtNdSxoFY\njv383DXa/qbvZKzx75Scnc0daORDbciLhiZ/C1D0BVUFHnllme1rdiPEra0Sza3Rr2hmfMKqjU1o\n0efvOy3ekbHWnvEG5TQaN/gNVkurEl389HiTNUYnL1tr37JjHMZY19+1qVix/hENRQE++GILVuvN\n2ebgZr6RNrd2YnODkU4VT77hPhXOatGK7Wkjz81BAQCa9fEFQRYwevhl7UEwros7hRkEoPvLrGVp\nu/j+RUzQuV9xAAAgAElEQVTnsqr3BVkHqc76wGZmQcCf/skM0xYtietNlwcpy9WmDOrUGM8E8HL9\nernEo3x0br5goC9gYYLKu59swsb63Z6a4Y1H0VoXbAlNv37XbW3GjmSBqPdbbbZO49Kv0st+8zaA\n3vnf2ksqvtzU5Ds4RKmhyflhLWmv2f14Pqc8bNze2wzvdNO0qzGmfBZSa6rZ0tDUkZyWZ36WvTmu\nmJM6z/L4UrsZEnbPAbB+W8fpbBG3aPg+XNAHGXl829adbSndFAqAmS8tTdnn6be9j2F46+MNvrvs\nrBQlqq6T1BLQ2i3NuOupTxB4Yq8lT3Px6NxsY6DPor5YJznMZfPbv3+CphaH2rkp7UbN+urfmxbQ\nsXy1dVubMzZzpdWE9GNY54B7nea1cXvvtKn123YHXjvd7cExfu9LTj95pgWBPlnpvXvDz7z8p99J\nvZkbebRqY+9gsdcWaE3kgc4lhy/sduqnFEocPtSt0GR49BVpd/BIuHU3uZ0Tu1u7nOd1m/RFz0my\n3Bjow1TMX7IFv3p0IQDg8deW47p7U9fZ8HN/M//myfwJmwmm9yd6VMwztVioUB1nTsDyWqDlqQsM\nA30eCFMgsHvnQy987mthDjt2U16cBhbKdTvx8vtr05ZPXbhsW0qfqNMF9YSpFh40K1ZsaEzenD23\nnur/f3b2l9jd1uW46Iw/wb7AivW7zEPYXd1oN+4j08fqr6/bqhWO7J4J7+T2xz5yfK3DeAKgj1LC\nC3p/qdvbvJ0HqTvNej/4Ijdpi80E9J7DGg7/sNSe+2RAvALT7A7/n/jWxxuxalNTcoCkeVrlM++s\nsl1L3s4mU8tT0O+dHMCnpgYt8/GeN51XlIqBPg888NwSbKz39rhKL4UCP3PsG1vs+8Hve9Z+6U87\nvzHmyFuSNnvxpmS/pi39Ko1i2t2aLU1Ytm5nynEdP1axDySLVmxPy1//S5R63C/Di26tGo6tMCGo\nquo4ONHthu7UnGtXqOsdL2AaOOi4epl9Dilwzru2jvSWmbs8TuX6U4gFqAwdnc4tQ2HO8e2Nbe7N\nCA4vrd+6OzlFM0yL+YKlqevfq6qKTdtbPK9UmVbpMAVt43iZvL5wPa7+/Wwk557CvSb+b8tAu+RH\n62+xPskwj2YiZgUDfR5oaGxHp0tt0mhu3bKjNW3tbccakYeGQ0VR8IbDc9I7Mj2r3cOV0d6ZwIoN\njY5zre36htdsaU4OxssWLxe10ZztVxSNgEGOYbzHa0uONQ8u+83byTEYfj7/i5APpnE+fzW3PPyh\n43s32HSHWM97p6WEo26sVQDc9Kf5yb8zsX7+2i3NaYNP2/WCw41/mB+oqesB4+mAIb+sNRD/9MH3\nA7ZCKuhK9HjuejJ/xqqNTb0rTKq9/3Pq808pTNp4e5HWStHW0e1twZ4Cx0CfRX5OH7frxni4TGdX\nwjRozZkC72s0By7JWo6vKM4jzv1+RpAgay5EONWGP5L1yZtMpn454+En1uVro5IWkJXeG7vT4jDJ\nTXZp1zf98pGF9h+oH9ToXrH79g2NmfvFrYzm3LQct/mAIFOV1m/bjdcdzof121JbwRQl4OJPDpye\nZ277qFMAieS5lZm19v/S/DVYujb1CXler+FPMkxfVaGm/D5zPvUyFc75qrUO9PPj2XdXmYKw8xdM\nabmxfpT+77c/DrL6oJrSRffkmysw//P4Db6zYqDPMi+Xg5drxrYE7TSgOuD0Mt+fn75Tyj+NJl+/\n5eVAtxCld5S+0/f/cnOjttCJ/noUA4DNI/8BBBoxb7xvtukG7Jo2l8+we5a6n3T1NafvaT6V5qYM\nsuo7L8xbY7s9TA3QuI7ufTb16X8LZb17YHc5IdpdugwA4C8vL0u5qP7ysv3U2lS9ibFbf97tftCd\n6HHsxjCXbx2fA69Amztv2uGzLxvsd/ZJBTD/863pG/X0dHVH92jdfMJAXyB6l/r0d5MJO/LfW+nf\nve80WXv1GFjNg/7ec1ji1O7zM+lfVZ61J+AZwozgLfGYQdkOdlH0V7aHfBa5OR+bWjvxTpAFWULK\ndOnENSh44ZY17322OX1evXELMJ1cy11W9TS3Srqdj2FnxZhntP7j7ZWRLc+bbxjoC0RHVyJQc9lP\nH/rAZmvm6U0Go/TvZVC306GSa457jFDmfl/XwXw+jR1e0/uPbEXLMMcNEWEzFTC8jJA2Tq8ossZr\n94t5nEZql0Xvn35nQwSdZumX07gao3DttYxtTO/0M97u539ZYLuf8/ujGW62bO3OwMVRRUn9HotX\nOdfSzXkRZR+69dkgcz7dnHLtGI/Z9vvgpnzHQJ9NUQYT07H8rJ+fcclPh0OltQS4/9OV46ItPkWV\nnVHd9OyESmOIN2d6q9PDX7LG43e540n7pxq6tkRlOLan0810/E9X2fdxZzpvFcvfQRvPSvSo9qlL\n4LNaG+SxzxEwfq+Ey4A6BXpwtrvMUiewOx/D1HTvZ90IAFilF2rtKkZ2v2mnzcDj2x51GOtSoBjo\nsy2imGI+Pe+0PPLVvGSj+4NjnG3O8LCTO574ODU9Nk1gGW+wEcfXR0wLppiv6dWbm2wLOF4ex5vL\nruw33Z5o15cCRKwoakCqw99eGefAYg+LD5mfSHj30/ZTSd0WUgJSH1rkNb1tNn3XRrrXbnWeYmuk\n5ed/XWD7IJxMolzVWAVcn+jY2NKJeUu22I4dsFuoJq1OoQJwGFDsZUnxX7ms++CVW0GmEDHQ54mM\nfbums9464Mq8ZKOxaIzfe/WHlrmyVsv1QWfGaHRrjSubT+nza8WGRvzkwffTtienYylaE6SdT+we\n4+pD0GwIm3s7mzrw+sL1js8H8CJMw4t12mdYwfJRiyJufb9RCvLMh83WQZge9Tg8H2CJx0FqfgL9\nuq3NWL3Zf4vBvCVaq5GR1odeSF8z3q41zXovefD5z1GieLsmom6bM5KSj88oCYOBPou8D8wKf7qa\nR3H7Kr37/Ginvt5MBYVscxq/8PnqHbbb/Swq5E+waLnbFDSCHOGTldsjW+c/V0U28y/o1nTvfF35\n6xuPlJqeKmPa1vJ1vYXKWR/YT9kzjuH4kunvm/443/QZ0Z/H2zLMQXey5Ev7ay2T//1bamtha0c3\noKQv1JNkzgyXm13cF8Hxg4E+y6LqD96ll+Kdmjdvf0zrU+rq7tEHr7jcNFLG4tmvhpbpvTkJBm7d\ntg6vvdXHzeFB88VagMrlTSqK39Yc9up3+Z+fH4yS9tlWUbY8/dv0eF03u1Nq/umfb17e1ZHDa07j\nC9J5O6MU9C60kw12XfQtNt0+iqI4DsT1+gvubuvC7MV9P1sjHzHQ56EeVU0bDPLKB+u0S9XjWR72\nfpZxGprN8fviIT5O/KzdntUgmj89GIHtjKLZ0pQPYW+2vqdEuvwGUT6pz+u4hJQHurikrcNlup5T\n4cVr3hi/wWOv2TwIKOVzQnBaE8F0XK/XntdlJNz2W7J6h+NiS8WGgb4Pvez1oRuq89rYG7e39C4M\n4+OzMwXCFRu89mvqzaN5FtH8dFd8rPfDG89yj1J+5UowUU5pDMotIJqfvGcW5fRAv1T0XmPWFfu8\n2u3S5x9VGTrTGIEwhWCnNN7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"text": [ "" ] } ], "prompt_number": 14 }, { "cell_type": "markdown", "metadata": {}, "source": [ "See why this can't be iteratively optimized? There are **way** too many local extrema! Let's find the best period:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "best_period = period[np.argmax(log_like)]\n", "print(best_period)" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "0.580348034803\n" ] } ], "prompt_number": 15 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Just to make sure, let's zoom-in and get a more precise value for the maximum:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "# Zoom-in on this value and find a more precise result\n", "\n", "period = np.linspace(0.575, 0.585, 10000)\n", "omega = 2 * np.pi / period\n", "log_like = np.zeros_like(omega)\n", "\n", "for i in range(len(omega)):\n", " theta_fit = fourier_solve(t, y, dy, nterms=5, omega=omega[i])\n", " log_like[i] = log_likelihood(theta_fit, t, y, dy, omega=omega[i])\n", " \n", "plt.plot(period, log_like, linewidth=0.5)\n", " \n", "best_period = period[np.argmax(log_like)]\n", "print(best_period)" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "0.580317531753\n" ] }, { "metadata": {}, "output_type": "display_data", "png": 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9nJTys50eHxEBEVWBMWA9AFpEga6biEUjPX/tJ+Q8njuxgg/+2JU9f22iQdNRD4AQ4tMA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\npOIazizmNl+v2gOQzQdfCridsitpVYu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"text": [ "" ] } ], "prompt_number": 16 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now given this period, let's compute the phase and plot the folded light curve:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "phase = t % best_period\n", "plt.errorbar(phase, y, dy, fmt='o')\n", "plt.gca().invert_yaxis();" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "display_data", "png": 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"text": [ "" ] } ], "prompt_number": 17 }, { "cell_type": "markdown", "metadata": {}, "source": [ "We've managed to find the period associated with the maximum likelihood!" ] } ], "metadata": {} } ] }