{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Generating a number range with normal spacing density\n", "\n", "We want to generate N numbers between x_min and x_max with both limits included. This is trivial to do with a uniform distribution, for example with np.linspace(x_min, x_max, N) in NumPy.\n", "\n", "But sometimes it can be helpful to have a range of numbers that are more concentrated around a specific area. For example if they are used as input for monte-carlo simulations. Ideally their density would be proportional to a [normal distribution](http://en.wikipedia.org/wiki/Normal_distribution). That shall be the goal of this notebook! Source is [on github](https://github.com/s9w/articles).\n", "\n", "We'll assume a normal definition with a standard deviation of $\\sigma$ and a center of $\\mu$:\n", "\n", "$$f(x) = \\frac{1}{\\sigma \\sqrt{2\\pi}}\\exp\\left(-\\frac{(x-\\mu)^2}{2\\sigma^2}\\right)$$\n", "\n", "Let's have a look:" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": false }, "outputs": [], "source": [ "import numpy as np\n", "import matplotlib as mpl\n", "import matplotlib.pyplot as plt\n", "from scipy.stats import norm\n", "%matplotlib inline\n", "mpl.rcParams['lines.linewidth'] = 2\n", "mpl.rcParams['axes.grid'] = True\n", "mpl.rcParams['legend.labelspacing'] = 0.0" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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Aqhnyqmh0jsiDxpArZc1mE/3796dBgwblbgMHDgQqTk27fv36E+rzpmwg9JXmiy++4I47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\n4YOugchI+PVX2L8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OYPAGM7PTBefDh2CnXJ3O0fjVVwdNkixXnUePwrBh+vXLL+t1Lgsd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\ndDz/OI3fbEx2QTZ7H99Li3ohtArg9u0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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "x = np.linspace(-3, 5, 100)\n", "parameters = [(1.0, 1.0), (1.0, 0.5), (0.0, 0.7)]\n", "for mu, sigma in parameters:\n", " plt.plot(x, norm.pdf(x, mu, sigma), label=\"$\\\\mu$={} $\\\\sigma$={}\".format(mu, sigma))\n", "plt.legend();" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The key idea for our range is that the points divide a normal distribution into segments of equal areas. So first we need to look at the integrals. Normal distributions are always normalized, so $\\displaystyle\\int_{-\\infty}^\\infty f(x) dx=1$. The integral up to a certain point is given by the cumulative distribution function $\\displaystyle\\text{cdf}(x)=\\int_{-\\infty}^x f(x) dx$. For the normal distrubution, that is\n", "\n", "$$\\text{cdf}(x) = \\frac 1 2 \\left(1+\\text{erf}\\left(\\frac{x-\\mu}{\\sigma\\sqrt 2}\\right)\\right)$$\n", "\n", "Let's have a look:" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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R+7EFC+Dee83myKtWwW23WVPOphObqPN5HXI5crGr1y7KhpdNey+ncuQZff21\n5t13FQULOtmyxUGJEjlWtBAukxy5uKboaDNKBWDYMOuCOMDrf7yORtOjbo/LgrhdnnhC0bChWfa2\na1fPHpIoRFZIILeBJ+T/evWCo0ehUaP0B51W+PPgn/y661fyBubljSZv/Of9nMyRp3I44P33zXos\nv/3m4JtvcrwKV+UJ94WnkLZwjQRyPzRzJkyZAnnymMWwAgKsKUdrTf+FZgeKlxu8TNG8Ra0p6AYU\nKwZvvGFu/xdf1Bw5YnOFhMgGyZH7mehoqFEDjh+HTz81a6pYZfbO2dw/7X4K5ynMnhf3kD84/3/O\nsSNHnkpr6N7dyZIlDlq3dvLrrw63LtMrhDtIjlz8R58+JojfdRf0zN4+DteV7Exm4B8DAXi98etX\nDeJ2UwreesukWObO9ZwUixCukkBuA7vyf3PmwDffmFUNJ0507/riV5qyeQpbTm6hTFgZnqv73DXP\nsyNHntGVKZZjx+yri+SF00lbuEYCuZ84exa6dzdfv/UWVK5sXVkXky7y5uI3ARgWMYzgXG5ezNzN\n2rWDJk2cnD2r6NXL8yYKCZEZyZH7iZ494bPPzIJYK1da94ATYORfI+kzvw81itRgY4+NBDiuXZid\nOfKMjh6FNm00Fy4ofvnFBHchPIFbcuRKqZZKqR1Kqd1KqX7XOe8OpVSSUqr9jVRWWGflShg71kz8\nmTDB2iAemxDL20vNFPz3W7x/3SDuSUqWhL59ze/Kc8/J2uXCu1w3kCulAoAxQEugOtBZKVXtGucN\nB+YB8tw/EzmZ/0tMhG7dzAiNV16BWrWsLe/DFR9yOv40jcs0pk2lNpmeb3eOPKNHH4WaNZ0cO+ag\nf/+cT7FIXjidtIVrMuuR1wP+0Vrv11pfAqYBV/vQ2Qv4EfjXzfUT2fThh7B1K1SoAIMGWVvWsbhj\nfLzyYwCGtxiO8rKxfAEB8O67Zq/Pzz93sHy53TUSImsyC+SlgEMZjg+nfC+NUqoUJriPS/mWJMIz\nERERkSPl7NplHmwCfPEFhIRYW97QJUPTlqltULpBln6mSZMm1lbKRZUrw7PPmq+7dXOSmJhzZefU\nfeENpC1ck1kgz0pQHgn0T3mSqZDUikfQGp5/Hi5ehKeeguYWb8az89ROJqybgEM5eLf5u9YWZrHn\nnlOULu1k+3YHI0ZIv0R4vlyZvH8EKJ3huDSmV57R7cC0lI/RhYFWSqlLWutZV14sMjKSsmXLAhAe\nHk6dOnUt23EIAAAeH0lEQVTS/vKm5sT84Thj/s+q8t54I4pFi6BQoQg++sj6f1/XUV1JPphMt4e6\nUbVw1Sz/fEJCAoUK1WTVKnNcv755387j4GDo3HkpH3wAb73VlE6d4MAB97bX1Y43bNjASy+9ZNn1\nvel45MiRfh0fJk+eDJAWLzNz3eGHSqlcwE6gOXAUWA101lpvv8b5XwGztdY/X+U9GX6YIioqKu1/\noBXOnIGqVeHkSZg0KX2VQ6ssO7CMJpObkDcwL7t77aZEvqyvCztv3jwKFapp+/DDq+nTx8z4bNnS\n/NfqlL/V94U3kbZIl+3hh1rrJKAnMB/YBnyvtd6ulOqulOruvqr6F6tv0IEDTRBv3NikVayktebV\n318F4JWGr7gUxMHzcuQZDRzoIDRUM2+egx9/tL48CVzppC1cIxOCfMxff0HDhmYExoYNZoEsK03f\nOp1HfnyEYnmL8c+L/xAaFOrSz3vKhKBrmToVhgyB4sWd7NzpIL/nLRkjfJwsmuWhMubI3SkpCZ57\nLn3MuNVBPDE5kQGLBgAwNGKoy0EcPGsc+dV07GjGlh8/7mDwYGs7IlbdF95I2sI1Esh9yNixphd+\n883w5pvWlzfu73HsObOHaoWr0fW2rtYXaAOHA4YNc+BwaEaPho0b7a6REP8lqRUfcewYVKkCcXEw\naxbcd5+15UXHR1NxVEXOJJxhVqdZ3Fflxgr09NRKqrff1vzf/ynuvNPJ8uUOS1eOFCIjSa34kZdf\nNkH8/vutD+IAby15izMJZ2hWrhltK7e1vkCb9e6tKFzYyV9/OfjqK7trI8TlJJDbwN35v0WLzEO5\nkBCz64/Vdp3exZi/x6BQjLhnRLam4nt6jjxVvnwwYID5dXn1VSenTrm/DMkLp5O2cI0Eci938SK8\n8IL5+s03IYvzB7Kl38J+JDmT6FKnC3WK17G+QA/Rpg3Ur+/kzBl7FtUS4lokR+7l3n0XXn/d5Mc3\nbYKgIGvLW7xvMc2+aXZDk3+uxlty5Kn27oX77tMkJSlWrIAGWVtSRogbJjlyH7dvX/qiWGPHWh/E\nk53JvLzgZQD639U/20HcG5UvD11TBuj06OEkKcne+ggBEsht4a78X+/ekJBg1tFu1swtl7yurzZ8\nxfrj67kp/030bdDXLdf0lhx5Rs89pyhRwsmmTQ7GjnXfdSUvnE7awjUSyL3UrFkwezbkzw8ffWR9\nebEJsQxcNBCAD1p8QJ7APNYX6qFCQuDNN82vzuuv27thsxAgOXKvdOECVK8OBw6YUSovvmh9mS/P\nf5mP//qYRqUbsazLMrdtGuFtOfKMund3EhXloFMnJ1OnSp9IWENy5D7qnXdMEK9Tx6w5brWdp3Yy\navUoFIpRrUZ53c4/VnnzTQfBwZpp0xwsWmR3bYQ/k0Bug+zk/7ZvN9u3QfqGylbrM78PSc4kut7a\nldtK3ObWa3tjjjzVTTeZtW0Ann/eycWL2bue5IXTSVu4RgK5F0nd9efSJbOhck4MfZuzaw6//fMb\n+YPz807zd6wv0Mt07aooV87Jrl0OPvxQUofCHpIj9yLffgtPPAGFC8OOHVCokLXlJSQlUGNsDfae\n2cuIe0a4baRKRt6cI0+1ciVERkLu3Jpt2xTlytldI+FLJEfuQ86cMeupAHzwgfVBHODD5R+y98xe\nahSpQa96vawv0Es1aABt2jhJSFD07OlE+isip0kgt8GN5P9ef93s+nPXXdbv+gOw78w+3v3TbKL8\nWevPCAwItKQcb86RZzRggIPQULMl3IwZN3YNyQunk7ZwjQRyL7BqFXz+udn1Z9w4cmQJ1T7z+5CQ\nlMCjNR+ladmm1hfo5YoUgb59zf+YXr2cxMXZXCHhVyRH7uGSkqBuXbOhwWuvwfDh1pc5d/dc2kxp\nQ76gfOzouYOS+UpaVpYv5MhTJSfDI4842bLFwYsvaj79VIZpiuyTHLkP+PRTE8TLloVBg6wvL/5S\nPL1+M/nwoRFDLQ3iviYgAN5+2+wmNGYMrFtnd42Ev5BAboOs5v8OHEgP3p99BnnzWlenVG8vfZu9\nZ/ZSs2hNetbraXl5vpIjT1WtmhlZ5HQqunVzkpyc9Z+VvHA6aQvXSCD3UFpDr15mOv7DD0Pr1taX\nufXkVj5Y8QEKxRdtv7DsAaev691bUayYk3XrHIwZY3dthD+QHLmH+ukn6NDBLIq1fTuUtDjD4dRO\nmnzVhOWHlvNc3ecY28aNy/pdhy/lyDNauNBs+JE3rxlbXqaM3TUS3kpy5F4qJgZ6pmQ13n3X+iAO\nMHHdRJYfWk7x0OK82/xd6wv0cS1awN13Ozl/XvHcczK2XFhLArkNMsv/vfYaHD8ODRumr+VhpRPn\nTvDawtcAGHnvSMJzh1tfaApfy5FnNGiQg9BQzdy5Dn74IfPzJS+cTtrCNRLIPUxUFIwfb3b7GT8+\nZ8aMvzD3BWISYmhZsSWP1HjE+gL9RNGi8Npr5hNxr15OoqNtrpDwWZIj9yDx8VC7NuzeDUOH5sxw\nw5+2/USH6R0IDQpl6/NbKROWs8lcX82Rp3I64bHHzIPPyEgnX30lfSfhGsmRe5m33jJBvEYN6N/f\n+vKi46N5Ye4LAAxvMTzHg7g/cDjM2PKgIM3kyQ4WLLC7RsIXSSC3wdXyf2vWmMWwlEpPrVitz/w+\nnDh/giY3N6FH3R7WF3gVvpwjT1WhQvrD62eeufb0fckLp5O2cI0Ecg9w8SJ06WKmeL/0Us6sMz53\n91y+2fgNuXPlZsJ9E3AouRWs1LWronp1J4cOOejXz2l3dYSPkRy5Bxg0yKRVKlY00/HzWLyvcXR8\nNDXH1eRo3FE+vPtDXmn4irUFXoev58gz2rED2rfXJCcroqKgqaxFJrJAcuReYP16eO89k1KZNMn6\nIA7Qc25PjsYdpWHphvS5s4/1BQoAqlaFHikZrKefdnL+vL31Eb5DArkNUvN/iYkmpZKUZHKojRtb\nX/YPW39g6pap5A3MyzcPfEOAI8D6Qq/DH3LkGfXooahUycnevQ5effXyT6iSF04nbeEaCeQ2GjLE\npFLKlze9cqsdizvGc3PMDKOP7vmICgUrWF+ouExQEHz4oYNcuTTjxil+/93uGglfIDlym6xYYXrg\nWsPSpWbnHytprWk7tS1zd8/l3gr38ttjv6GU/etl+1OOPKNx4zQjRypKljTrlxcoYHeNhKeSHLmH\nOncudalTMx3f6iAOMGb1GObunkuB3AWYeP9Ejwji/qxbN0WtWk6OHnXQs6eMYhHZk6VArpRqqZTa\noZTarZTqd5X3H1NKbVRKbVJKLVdK1XJ/VX1H585R7N0LtWqZGZxW23h8I6/8bkamTLh/AqXye07v\n199y5Kly5YIPPnAQHKyZMsXBjz9KXjgjaQvXZBrIlVIBwBigJVAd6KyUqnbFaXuBJlrrWsBbwJfu\nrqiv+PVX8woKgm+/heBga8s7n3ieTj91IjE5kR6396B9tfbWFiiyrFw584kM4NlnzebaQtyIrPTI\n6wH/aK33a60vAdOAdhlP0Fqv1FrHphyuAm5ybzV9w9GjZpQKRPDOO1CzpvVlvjTvJXac2kH1ItUZ\nce8I6wt0UZMmTeyugq0efRQ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KFRhodjOqXNm8KlaEdes2UKJEBGXKuG/ehy/LLLNcD/hH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\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "x = np.linspace(-2, 3, 100)\n", "x_min=0.0; x_max=1.0; mu0=0.0; sigma0=0.7\n", "plt.plot(x, norm.pdf(x, mu0, sigma0), label=\"Normal dist\")\n", "plt.fill_between(x, 0, norm.pdf(x, mu0, sigma0), where=(x_max>x)&(x>x_min), alpha=0.15)\n", "plt.plot(x, norm.cdf(x, mu0, sigma0), label=\"CDF\")\n", "plt.legend();" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We want to look at the case where $f(x)$ is only defined on some interval $[x_\\text{min}, x_\\text{max}]$, that's hinted by the shaded area. That area is $A=\\text{cdf}(x_\\text{max})-\\text{cdf}(x_\\text{min})$. So the total integral is divided into $A$ + the area left of it ($\\text{cdf}(x_\\text{min})$) + the area right of it (not important). Since we know all areas, we'll solve $\\text{cdf}(x)$ for $x$:\n", "\n", "$$x=\\sigma\\sqrt 2\\cdot\\text{erf}^{-1}(2\\cdot\\text{cdf}(x)-1)+\\mu$$\n", "\n", "To generate our desired range, we'll evaluate that equation for different values of cdf(x). We have to start at $\\text{cdf}(x_\\text{min})$ and step through $N-1$ parts of $A$. So we divide $A$ into $A_0=\\frac{A}{N-1}$ and can write down the result. The cases i=0 and i=N-1 are identical to x_min and x_max.\n", "\n", "Technical note: SciPy has the inverse error function and the pdf included. In C++ we need Boost." ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "from scipy.special import erfinv\n", "\n", "def normalSpace(x_min, x_max, N, mu, sigma):\n", " area_min = norm.cdf(x_min, mu, sigma)\n", " area_max = norm.cdf(x_max, mu, sigma)\n", " A = area_max - area_min\n", " A0 = A/(N-1)\n", " seq = [x_min] + [np.sqrt(2)*sigma*erfinv((area_min+i*A0)*2-1)+mu for i in range(1,N-1)] + [x_max]\n", " return np.asarray(seq)\n", "\n", "fig = plt.subplot(xlim=(-0.2,4))\n", "parameters = [(1.75, 0.5), (1.75, 0.7), (1.5, 1.0), (0.0, 0.6), (2.0, 0.7)]\n", "for i, (mu, sigma) in enumerate(parameters):\n", " x_min=0.5; x_max=3.0; N=25\n", " points = normalSpace(x_min, x_max, N, mu, sigma)\n", " plt.plot(points, np.full(N, -i-0.3), \".\", markersize=5, label=\"$\\sigma$={:.1f}\".format(sigma))\n", " plt.legend()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Works like a charm! Example C++ implementation:\n", "\n", "`cpp\n", "#include \n", "#include \n", "\n", "std::vector normalSpace(double x_min, double x_max, unsigned int N, double mu, double sigma){\n", " boost::math::normal normal_dist(mu, sigma);\n", " double area_min = boost::math::cdf(normal_dist, x_min);\n", " double area_max = boost::math::cdf(normal_dist, x_max);\n", " double A = area_max - area_min;\n", " double A0 = A/(N-1.0);\n", " double area;\n", " std::vector x;\n", " for(int i=0; i