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Kyle Cranmer added file figures/james-stein/JamesStein.v3.ipynb
almost 9 years ago
Commit id: 6211b1f6f2af4fc99c221604937063724d690238
deletions | additions
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"source": [
"# A Quick Demo of the James-Stein Estimator\n",
"\n",
"Kyle Cranmer, June 25, 2015\n",
"\n",
"[]( https://creativecommons.org/licenses/by/4.0/)"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
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"source": [
"## The Problem\n",
"\n",
"\n",
"Consider a standard multivariate Gaussian distribution for $\\vec x$ in $n$ dimensions centered around $\\vec\\mu$\n",
"\n",
"\\begin{equation}\n",
"f(\\vec{x}|\\vec{\\mu}) = \\prod_{i=1}^{n} \\frac{1}{\\sqrt{2\\pi}} \\exp\\left(-\\frac{(x_i- \\mu_i)^2}{2}\\right)\\;.\n",
"\\end{equation}\n",
"\n",
"\n",
"We want to compare the mean squared error\n",
"\\begin{equation}\n",
"E[||\\bar{\\vec x}-\\vec\\mu||^2])\n",
"\\end{equation}\n",
"\n",
"of the sample mean (an unbiased estimator)\n",
"\\begin{equation}\n",
"\\overline x_i = \\frac{1}{m} \\sum_{j=1}^m x_{ij}\n",
"\\end{equation}\n",
"\n",
"to the James-Stein estimator\n",
"\\begin{equation}\n",
"x_{JS} = \\left( 1 - \\frac{n-2}{||\\bar{x}||^2} \\right) \\bar{x} \n",
"\\end{equation}\n",
"\n",
"\n",
"(where $i$ is the component of the vector and $j$ is an index over the samples).\n"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"%pylab inline"
],
"language": "python",
"metadata": {
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},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"Populating the interactive namespace from numpy and matplotlib\n"
]
}
],
"prompt_number": 1
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{
"cell_type": "code",
"collapsed": false,
"input": [
"def jamesSteinEstimator(x):\n",
" return x*(1. - (x.size-2.)/sum(x*x))"
],
"language": "python",
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"outputs": [],
"prompt_number": 2
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{
"cell_type": "code",
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"input": [
"def RMS(x,mean):\n",
" return sqrt(sum((x-mean)*(x-mean)))"
],
"language": "python",
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"outputs": [],
"prompt_number": 3
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{
"cell_type": "code",
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"input": [
"def doRun(nDim,nSamp=1000):\n",
" mean=zeros(nDim)\n",
" data = zeros(nDim*nSamp).reshape(nDim,nSamp)\n",
" for i in range(nDim):\n",
" mean[i] = 0.5 #set value of the mean's components here \n",
" data[i,:] = random.normal(mean[i],1,nSamp).T\n",
" data=data.T\n",
"\n",
" avRMS_js = 0\n",
" avRMS_mle = 0\n",
" for x in data:\n",
" avRMS_js += RMS(jamesSteinEstimator(x),mean)\n",
" avRMS_mle += RMS(x,mean)\n",
" avRMS_js /= nSamp\n",
" avRMS_mle /= nSamp\n",
" return (avRMS_js, avRMS_mle)"
],
"language": "python",
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"outputs": [],
"prompt_number": 4
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"nDimToTest = linspace(2,20)\n",
"av_js = zeros(nDimToTest.size)\n",
"av_mle = zeros(nDimToTest.size)\n",
"for i,nDim in enumerate(nDimToTest):\n",
" [av_js[i], av_mle[i]] = doRun(int(nDim))"
],
"language": "python",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
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"prompt_number": 5
},
{
"cell_type": "code",
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"input": [
"plt.plot(nDimToTest,av_js,c='b') #James Stein in blue\n",
"plt.plot(nDimToTest,av_mle,c='r') #MLE in red\n",
"plt.ylabel('Mean Squared Error')\n",
"plt.xlabel('Dimensionality')\n",
"plt.legend(('James-Stein', 'MLE'), loc='upper left')\n",
"plt.savefig('james-stein.pdf')"
],
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"text": [
""
]
}
],
"prompt_number": 6
},
{
"cell_type": "code",
"collapsed": false,
"input": [],
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": null
},
{
"cell_type": "code",
"collapsed": true,
"input": [],
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": null
}
],
"metadata": {}
}
]
}
