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Created February 21, 2014 18:26
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Numerical Cramer-Rao bound in Python
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{
"metadata": {
"name": "Cramer-Rao"
},
"nbformat": 3,
"nbformat_minor": 0,
"worksheets": [
{
"cells": [
{
"cell_type": "code",
"collapsed": false,
"input": "%pylab inline\nfrom scipy.misc import derivative\nfrom scipy.optimize import curve_fit\nimport inspect",
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": "Populating the interactive namespace from numpy and matplotlib\n"
}
],
"prompt_number": 1
},
{
"cell_type": "heading",
"level": 1,
"metadata": {},
"source": "Numerical Cramer-Rao bound in Python"
},
{
"cell_type": "markdown",
"metadata": {},
"source": "[Cram\u00e9r-Rao bound](http://en.wikipedia.org/wiki/Cram%C3%A9r%E2%80%93Rao_bound) is a lower limit on the variance of estimators of a deterministic parameter."
},
{
"cell_type": "markdown",
"metadata": {},
"source": "Let's take an oscillating signal as an example, with three parameters: amplitude, frequency and initial phase. We know it will be sampled at 100Hz with noise level of 0.1."
},
{
"cell_type": "code",
"collapsed": false,
"input": "def signal(t, a, f, ph):\n return a * sin(2 * pi * f * t + ph)\n\np0 = [2, 8, 0]\nnoise = 0.1\n\nT = arange(0, 1, 0.01)\nplot(T, signal(T, *p0), '.-k')\nxlabel('time (s)')",
"language": "python",
"metadata": {},
"outputs": [
{
"metadata": {},
"output_type": "pyout",
"prompt_number": 2,
"text": "<matplotlib.text.Text at 0x7f14fde6d490>"
},
{
"metadata": {},
"output_type": "display_data",
"png": 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9PcRPRqq2OQT1qZ5nn30WpaWlOHfuHG688UY8++yzHq/TaDSoqKhA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\nVAIM1fGrqKioBBj/D7H7J/gMMbyEAAAAAElFTkSuQmCC\n",
"text": "<matplotlib.figure.Figure at 0x7f14fde59a50>"
}
],
"prompt_number": 2
},
{
"cell_type": "markdown",
"metadata": {},
"source": "For convenience let's produce an array of parameters' names and a dictionary mapping those names to their values."
},
{
"cell_type": "code",
"collapsed": false,
"input": "labels = inspect.getargspec(signal)[0][1:]\np0dict = dict(zip(labels, p0))\nprint('labels:', labels)\nprint('p0dict:', p0dict)",
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": "labels: ['a', 'f', 'ph']\np0dict: {'a': 2, 'ph': 0, 'f': 8}\n"
}
],
"prompt_number": 3
},
{
"cell_type": "markdown",
"metadata": {},
"source": "Cramer-Rao bounds come form inverse of [Fisher information matrix](http://en.wikipedia.org/wiki/Fisher_information). Calculating it may be difficult in general case, but for a sample from [multivariate normal distribution](http://en.wikipedia.org/wiki/Multivariate_normal_distribution) is [really simple](http://en.wikipedia.org/wiki/Fisher_information#Multivariate_normal_distribution). Most measurements are in fact a sample from multivariate normal distribution, a vector $\\mu$, with number of dimensions equal to number of samples. If measurements are not correlated and have the same variance $\\sigma^2$ then an element of Fisher information matrix is:\n$$\\mathcal{I}_{mn} = \\frac{1}{\\sigma^2} \\frac{\\partial \\mu^\\mathrm{T}}{\\partial \\theta_m} \\frac{\\partial \\mu}{\\partial \\theta_n} = \\frac{1}{\\sigma^2} \\sum_k \\frac{\\partial \\mu_k}{\\partial \\theta_m} \\frac{\\partial \\mu_k}{\\partial \\theta_n}$$\nwhere $\\theta$ is a vector of parameters, in our case $[a, f, ph]^T$.\n\nWe can define: $D_{ik} := \\frac{\\partial \\mu_k}{\\partial \\theta_i}$ to get:\n$$\\mathcal{I}_{mn} = \\frac{1}{\\sigma^2} \\sum_k \\frac{\\partial \\mu_k}{\\partial \\theta_m} \\frac{\\partial \\mu_k}{\\partial \\theta_n} = \\frac{1}{\\sigma^2} \\sum_k D_{mk} D_{nk}$$\n\n"
},
{
"cell_type": "markdown",
"metadata": {},
"source": "Let's first calculate matrix $D_{ik}$:"
},
{
"cell_type": "code",
"collapsed": false,
"input": "D = zeros((len(p0), len(T)))\n\n# for every parameter\nfor i, argname in enumerate(labels):\n # for every sample\n for k, t in enumerate(T):\n # Define a function to be derivated.\n # It's our signal at a given time t with all parameters fixed, except the one named 'argname'.\n func = lambda x: signal(t, **dict(p0dict, **{argname: x}))\n \n # Calulate derivative of func at point given by p0\n # Be careful with dx, since it's 1 by default!\n D[i,k] = derivative(func, p0dict[argname], dx=0.0001)",
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 4
},
{
"cell_type": "markdown",
"metadata": {},
"source": "We can take a closer look at the $D_{ik}$ matrix to better understand what it is."
},
{
"cell_type": "code",
"collapsed": false,
"input": "plot(T, signal(T, *p0), '--k', lw=2, label='signal')\n\nfor Di, argname in zip(D, labels):\n plot(T, Di, '.-', label=argname)\n \nlegend(loc='best')\nxlabel('time (s)')",
"language": "python",
"metadata": {},
"outputs": [
{
"metadata": {},
"output_type": "pyout",
"prompt_number": 5,
"text": "<matplotlib.text.Text at 0x7f14fd6169d0>"
},
{
"metadata": {},
"output_type": "display_data",
"png": 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AV39+pXH+77//juPHj5tMvgNpB9AzpKfR53UJ6oITmSdQXlmOtLQ07N69G99+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\nnhBCbMD/B/0O5oKkGCn4AAAAAElFTkSuQmCC\n",
"text": "<matplotlib.figure.Figure at 0x7f14fd74b790>"
}
],
"prompt_number": 5
},
{
"cell_type": "markdown",
"metadata": {},
"source": "$D_{ik}$ matrix expresses how much $k\\text{th}$ sample affects $i\\text{th}$ parameter. So we see that amplitude is affected by samples that are in sine's apices. On the other hand, these points don't affect phase much. We also see that as signal lasts samples get more and more sensitive to frequency. It's intuitive, since even a very small change in frequency, given enough time, can cause an arbitrarily big change in signal's phase."
},
{
"cell_type": "markdown",
"metadata": {},
"source": "Recall how to get the Fisher information matrix from $D$:\n$$\\mathcal{I}_{m,n}= \\frac{1}{\\sigma^2} \\sum_k D_{mk} D_{nk}$$\n\nNumpy provides an excellent function to sum arrays with [Einstein notation](http://en.wikipedia.org/wiki/Einstein_notation): [```einsum```](http://docs.scipy.org/doc/numpy/reference/generated/numpy.einsum.html)"
},
{
"cell_type": "code",
"collapsed": false,
"input": "I = 1/noise**2 * einsum('mk,nk', D, D) # How cool is that?\nprint(I)",
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": "[[ 5.00000000e+03 -5.71453545e+02 -1.55465640e-09]\n [ -5.71453545e+02 2.55477009e+05 6.15752139e+04]\n [ -1.55465640e-09 6.15752139e+04 1.99999999e+04]]\n"
}
],
"prompt_number": 6
},
{
"cell_type": "markdown",
"metadata": {},
"source": "Cramer-Rao bounds lie on a diagonal of the inverse of $\\mathcal{I}$:"
},
{
"cell_type": "code",
"collapsed": false,
"input": "iI = inv(I)\n\nfor argname, variance in zip(labels, iI.diagonal()):\n print('{}: {:.2g}'.format(argname, sqrt(variance)))",
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": "a: 0.014\nf: 0.0039\nph: 0.014\n"
}
],
"prompt_number": 7
},
{
"cell_type": "markdown",
"metadata": {},
"source": "If we now try to fit the signal with the exact model we should be exactly at Cramer-Rao bound with accuracy of the fitted parameters. In reality we will not, since variances yielded by fitting procedure are estimators themselves and thus have certain spread."
},
{
"cell_type": "code",
"collapsed": false,
"input": "S = signal(T, *p0) + randn(T.size) * noise\nplot(T, S, '.-k')\n\npopt, pcov = curve_fit(signal, T, S, p0=p0)\n\nfor argname, variance in zip(labels, pcov.diagonal()):\n print('{}: {:.2g}'.format(argname, sqrt(variance)))\n\nTl = linspace(T[0], T[-1], 10000)\nplot(Tl, signal(Tl, *popt))",
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": "a: 0.013\nf: 0.0037\nph: 0.013\n"
},
{
"metadata": {},
"output_type": "pyout",
"prompt_number": 8,
"text": "[<matplotlib.lines.Line2D at 0x7f14fd585d10>]"
},
{
"metadata": {},
"output_type": "display_data",
"png": 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PRhEEPw5H8eGXnTvriccdpDVsBTI/D0ajvyiNeoHu7grASySi3fkcGYmWFMIE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\ne/bsEdkiBmNmCF7SyGA4IgrFyObiN0M1BIMxG7BIneFU3CyrXxkMW8x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"text": "<matplotlib.figure.Figure at 0x7f14fd648450>"
}
],
"prompt_number": 8
},
{
"cell_type": "markdown",
"metadata": {},
"source": "A function to calculate inverse of Fisher information matrix:"
},
{
"cell_type": "code",
"collapsed": false,
"input": "def cramer_rao(model, p0, X, noise, show_plot=False):\n \"\"\"Calulate inverse of the Fisher information matrix for model\n sampled on grid X with parameters p0. Assumes samples are not\n correlated and have equal variance noise^2.\n \n Parameters\n ----------\n model : callable\n The model function, f(x, ...). It must take the independent\n variable as the first argument and the parameters as separate\n remaining arguments.\n X : array\n Grid where model is sampled.\n p0 : M-length sequence\n Point in parameter space where Fisher information matrix is\n evaluated.\n noise: scalar\n Squared variance of the noise in data.\n show_plot : boolean\n If True shows plot.\n \n Returns\n -------\n iI : 2d array\n Inverse of Fisher information matrix.\n \"\"\"\n labels = inspect.getargspec(model)[0][1:]\n p0dict = dict(zip(inspect.getargspec(model)[0][1:], p0))\n \n D = zeros((len(p0), X.size))\n for i, argname in enumerate(labels):\n D[i,:] = [ derivative(lambda p: model(x, **dict(p0dict, **{argname: p})),\n p0dict[argname],\n dx=0.0001)\n for x in X ]\n \n if show_plot:\n plot(X, model(X, *p0), '--k', lw=2, label='signal')\n for d, label in zip(D, labels):\n plot(X, d, '.-', label=label)\n legend(loc='best')\n title('Parameter dependence on particular data point')\n \n I = 1/noise**2 * einsum('mk,nk', D, D)\n iI = inv(I)\n \n return iI",
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 9
},
{
"cell_type": "markdown",
"metadata": {},
"source": "We can now use this function to show a funny little fact.\n\nNon-diagonal elements of inverse of the Fisher information matrix show correlations between parameters. Let's see how correlation between frequency and phase looks depending on where we put zero in the time series. Zero is special, because that's where the phase is fixed to ```ph```."
},
{
"cell_type": "code",
"collapsed": false,
"input": "N = []\nT0 = arange(0, 11)\nfor t0 in T0:\n T = arange(-t0, 10 - t0, 0.01)\n N.append( cramer_rao(signal, p0, T, noise)[2,1] )\n \nplot(T0, N, 'o-')\naxhline(0, color='black')\nxlabel('zero location - point with fixed phase')\nylabel('frequency - phase correlation')",
"language": "python",
"metadata": {},
"outputs": [
{
"metadata": {},
"output_type": "pyout",
"prompt_number": 10,
"text": "<matplotlib.text.Text at 0x7f14fd5192d0>"
},
{
"metadata": {},
"output_type": "display_data",
"png": 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xs5PF586dY/Dgwezfv58yZcpQo0YNVqxYQfXq1a0fnEwWZ8mBA9otoNHR2h1B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\nnCQCIYRwcJIIhBDCwf0/0s7J1r+hQmUAAAAASUVORK5CYII=\n",
"text": "<matplotlib.figure.Figure at 0x7f14fd641f90>"
}
],
"prompt_number": 10
},
{
"cell_type": "markdown",
"metadata": {},
"source": "The correlation drops out only when we look at the phase at the middle of the signal. Then the $D_{ik}$ matrix looks like this:"
},
{
"cell_type": "code",
"collapsed": false,
"input": "T = arange(-0.5, 0.51, 0.01)\ncramer_rao(signal, p0, T, noise, show_plot=True);\nxlabel('time (s)')",
"language": "python",
"metadata": {},
"outputs": [
{
"metadata": {},
"output_type": "pyout",
"prompt_number": 11,
"text": "<matplotlib.text.Text at 0x7f14fd524ad0>"
},
{
"metadata": {},
"output_type": "display_data",
"png": 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GKbux+NcqdT6Pu3VdV/Z5PF6DbkBKLkiGm4Wb1nxe1l6IFcY2iAzquJV6C0ce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vLy8rB27Vrp57S0NIwfP75JjnXkyBHMnTuXdXtsbCzeeeedJjk2B4c6cH7qHK2OpKQkvPLKK7h7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\nlgCFAAAAAElFTkSuQmCC\n",
"text": "<matplotlib.figure.Figure at 0x7f14fd515050>"
}
],
"prompt_number": 11
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": "References"
},
{
"cell_type": "markdown",
"metadata": {},
"source": "D. Rife, R. Boorstyn _\"[Single-Tone Parameter Estimation from Discrete-Time Observations](http://ieeexplore.ieee.org/xpl/articleDetails.jsp?tp=&arnumber=1055282&url=http%3A%2F%2Fieeexplore.ieee.org%2Fxpls%2Fabs_all.jsp%3Farnumber%3D1055282)\"_, IEEE Transactions on Information Theory **20**, 5, 1974"
}
],
"metadata": {}
}
]
}
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