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Jupyter notebook with sympy for finding an improved moment-based estimator of Signal Noise ratio. No pasaran.
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| { | |
| "cells": [ | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "## Improved estimation of Signal Noise Ratio via moments\n", | |
| "\n", | |
| "Let $\\mu$, and $\\sigma$ be the mean and standard deviation of the returns of an asset. Then $\\zeta = \\mu / \\sigma$ is the \"Signal to Noise Ratio\" (SNR). Typically the SNR is estimated with the _Sharpe Ratio_, defined as $\\hat{\\zeta} = \\hat{\\mu} / \\hat{\\sigma}$, where $\\hat{\\mu}$ and $\\hat{\\sigma}$ are the vanilla sample estimates. Can we gain efficiency in the case where the returns have significant skew and kurtosis?\n", | |
| "\n", | |
| "Here we consider an estimator of the form\n", | |
| "$$\n", | |
| "v = a_0 + \\frac{a_1 + \\left(1+a_2\\right)\\hat{\\mu} + a_3 \\hat{\\mu}^2}{\\hat{\\sigma}} + a_4 \\left(\\frac{\\hat{\\mu}}{\\hat{\\sigma}}\\right)^2.\n", | |
| "$$\n", | |
| "The Sharpe Ratio corresponds to $a_0 = a_1 = a_2 = a_3 = a_4 = 0$. Note that we were inspired by \n", | |
| "Norman Johnson's [work on t-tests under skewed distributions](http://www.jstor.org/stable/2286597). Johnson considered a similar setup, but with only $a_1, a_2,$ and $a_3$ free, and was concerned with the problem of hypothesis testing on $\\mu$. \n", | |
| "\n", | |
| "Below, following Johnson, I will use the Cornish Fisher expansions of $\\hat{\\mu}$ and $\\hat{\\sigma}$ to approximate $v$ as a function of the first few cumulants of the distribution, and some normal variates. I will then compute the mean square error, $E\\left[\\left(v - \\zeta\\right)^2\\right],$ and take its derivative with respect to $a_i$. Unfortunately, we will find that the first order conditions are solved by $a_i=0$, which is to say that the vanilla Sharpe has the lowest MSE of estimators of this kind. Our adventure will take us far, but we will return home empty handed.\n", | |
| "\n", | |
| "We proceed." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 1, | |
| "metadata": { | |
| "collapsed": true | |
| }, | |
| "outputs": [], | |
| "source": [ | |
| "# load what we need from sympy\n", | |
| "from __future__ import division\n", | |
| "from sympy import *\n", | |
| "from sympy import Order\n", | |
| "from sympy.assumptions.assume import global_assumptions\n", | |
| "from sympy.stats import P, E, variance, Normal\n", | |
| "init_printing()\n", | |
| "nfactor = 4\n", | |
| "\n", | |
| "# define some symbols.\n", | |
| "a0, a1, a2, a3, a4 = symbols('a_0 a_1 a_2 a_3 a_4',real=True)\n", | |
| "n, sigma = symbols('n \\sigma',real=True,positive=True)\n", | |
| "zeta, mu3, mu4 = symbols('\\zeta \\mu_3 \\mu_4',real=True)\n", | |
| "mu = zeta * sigma" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "We now express $\\hat{\\mu}$ and $\\hat{\\sigma}^2$ by the [Cornish Fisher expansion](https://en.wikipedia.org/wiki/Cornish%E2%80%93Fisher_expansion). This is an expression of the distribution of a random variable in terms of its cumulants and a normal variate. The expansion is ordered in a way such that when applied to the _mean_ of independent draws of a distribution, the terms are clustered by the order of $n$. The Cornish Fisher expansion also involves the [Hermite polynomials](https://en.wikipedia.org/wiki/Hermite_polynomials). The expansions of $\\hat{\\mu}$ and $\\hat{\\sigma}^2$ are not independent. We follow Johnson in expression the correlation of normals and truncating: " | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 2, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
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| |
| "text/latex": [ | |
| "$$\\sigma \\zeta + \\frac{\\sigma}{\\sqrt{n}} \\left(\\frac{\\mu_3^{2}}{\\sigma^{3.0} n} \\left(- 0.0392837100659193 \\sqrt{2} z_{1}^{3} + 0.0982092751647983 \\sqrt{2} z_{1}\\right) + \\frac{\\mu_3}{\\sigma^{1.5} \\sqrt{n}} \\left(0.166666666666667 z_{1}^{2} - 0.166666666666667\\right) + \\frac{\\mu_4}{\\sigma^{4} n} \\left(0.0294627825494395 \\sqrt{2} z_{1}^{3} - 0.0883883476483184 \\sqrt{2} z_{1}\\right) + z_{1}\\right)$$" | |
| ], | |
| "text/plain": [ | |
| " ⎛ 2 -3.0 ⎛ 3 \n", | |
| " ⎜\\mu₃ ⋅\\sigma ⋅⎝- 0.0392837100659193⋅√2⋅z₁ + 0.09820\n", | |
| " \\sigma⋅⎜───────────────────────────────────────────────────────\n", | |
| " ⎜ n \n", | |
| " ⎝ \n", | |
| "\\sigma⋅\\zeta + ───────────────────────────────────────────────────────────────\n", | |
| " \n", | |
| "\n", | |
| " ⎞ -1.5 ⎛ 2 \n", | |
| "92751647983⋅√2⋅z₁⎠ \\mu₃⋅\\sigma ⋅⎝0.166666666666667⋅z₁ - 0.16666666666666\n", | |
| "────────────────── + ─────────────────────────────────────────────────────────\n", | |
| " √n \n", | |
| " \n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " √n \n", | |
| "\n", | |
| " ⎞ ⎛ 3 ⎞ ⎞\n", | |
| "7⎠ \\mu₄⋅⎝0.0294627825494395⋅√2⋅z₁ - 0.0883883476483184⋅√2⋅z₁⎠ ⎟\n", | |
| "── + ─────────────────────────────────────────────────────────── + z₁⎟\n", | |
| " 4 ⎟\n", | |
| " \\sigma ⋅n ⎠\n", | |
| "──────────────────────────────────────────────────────────────────────\n", | |
| " " | |
| ] | |
| }, | |
| "execution_count": 2, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "# probabilist's hermite polynomials\n", | |
| "def Hen(x,n):\n", | |
| " return (2**(-n/2) * hermite(n,x/sqrt(2)))\n", | |
| "\n", | |
| "# this comes out of the wikipedia page:\n", | |
| "h1 = lambda x : Hen(x,2) / 6\n", | |
| "h2 = lambda x : Hen(x,3) / 24\n", | |
| "h11 = lambda x : - (2 * Hen(x,3) + Hen(x,1)) / 36\n", | |
| "\n", | |
| "# mu3 is the 3rd centered moment of x\n", | |
| "gamma1 = (mu3 / (sigma**(3/2))) / sqrt(n)\n", | |
| "gamma2 = (mu4 / (sigma**4)) / n\n", | |
| "\n", | |
| "# grab two normal variates with correlation rho\n", | |
| "# which happens to take value:\n", | |
| "# rho = mu3 / sqrt(sigma**2 * (mu4 - sigma**4))\n", | |
| "z1 = Normal('z_1',0,1)\n", | |
| "z3 = Normal('z_3',0,1)\n", | |
| "rho = symbols('\\\\rho',real=True)\n", | |
| "z2 = rho * z1 + sqrt(1-rho**2)*z3\n", | |
| "\n", | |
| "# this is out of Johnson, but we call it mu hat instead of x bar:\n", | |
| "muhat = mu + (sigma/sqrt(n)) * (z1 + gamma1 * h1(z1) + gamma2 * h2(z1) + gamma1**2 * h11(z1))\n", | |
| "muhat" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 3, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXUAAAA/BAMAAADtZ0VFAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n", | |
| "text/latex": [ | |
| "$$\\frac{1}{\\sigma} \\left(1 - \\frac{\\sqrt{\\mu_4 - \\sigma^{4}}}{2 \\sigma^{3} \\sqrt{n}} \\left(\\rho z_{1} + \\sqrt{- \\rho^{2} + 1} z_{3}\\right)\\right)$$" | |
| ], | |
| "text/plain": [ | |
| " ________________ ⎛ _____________ ⎞\n", | |
| " ╱ 4 ⎜ ╱ 2 ⎟\n", | |
| " ╲╱ \\mu₄ - \\sigma ⋅⎝\\rho⋅z₁ + ╲╱ - \\rho + 1 ⋅z₃⎠\n", | |
| "1 - ───────────────────────────────────────────────────\n", | |
| " 3 \n", | |
| " 2⋅\\sigma ⋅√n \n", | |
| "───────────────────────────────────────────────────────\n", | |
| " \\sigma " | |
| ] | |
| }, | |
| "execution_count": 3, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "addo = sqrt((mu4 - sigma**4) / (n * sigma**4)) * z2\n", | |
| "# this is s^2 in Johnson:\n", | |
| "sighat2 = (sigma**2) * (1 + addo)\n", | |
| "# use Taylor's theorem to express sighat^-1:\n", | |
| "invs = (sigma**(-1)) * (1 - (1/(2*sigma)) * addo)\n", | |
| "invs" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 4, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
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T22MfKDpThW5L+zrGz6Xdmu+l3yoMs/aqhRqh06Hp3q7ZZ8t6JCydb4TxL/4xbaBe4Efu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\n", | |
| "text/latex": [ | |
| "$$a_{0} + \\frac{1}{\\sigma} \\left(1 - \\frac{\\sqrt{\\mu_4 - \\sigma^{4}}}{2 \\sigma^{3} \\sqrt{n}} \\left(\\rho z_{1} + \\sqrt{- \\rho^{2} + 1} z_{3}\\right)\\right) \\left(a_{1} + a_{3} \\left(\\sigma \\zeta + \\frac{\\sigma}{\\sqrt{n}} \\left(\\frac{\\mu_3^{2}}{\\sigma^{3.0} n} \\left(- 0.0392837100659193 \\sqrt{2} z_{1}^{3} + 0.0982092751647983 \\sqrt{2} z_{1}\\right) + \\frac{\\mu_3}{\\sigma^{1.5} \\sqrt{n}} \\left(0.166666666666667 z_{1}^{2} - 0.166666666666667\\right) + \\frac{\\mu_4}{\\sigma^{4} n} \\left(0.0294627825494395 \\sqrt{2} z_{1}^{3} - 0.0883883476483184 \\sqrt{2} z_{1}\\right) + z_{1}\\right)\\right)^{2} + \\left(a_{2} + 1\\right) \\left(\\sigma \\zeta + \\frac{\\sigma}{\\sqrt{n}} \\left(\\frac{\\mu_3^{2}}{\\sigma^{3.0} n} \\left(- 0.0392837100659193 \\sqrt{2} z_{1}^{3} + 0.0982092751647983 \\sqrt{2} z_{1}\\right) + \\frac{\\mu_3}{\\sigma^{1.5} \\sqrt{n}} \\left(0.166666666666667 z_{1}^{2} - 0.166666666666667\\right) + \\frac{\\mu_4}{\\sigma^{4} n} \\left(0.0294627825494395 \\sqrt{2} z_{1}^{3} - 0.0883883476483184 \\sqrt{2} z_{1}\\right) + z_{1}\\right)\\right)\\right) + \\frac{a_{4}}{\\sigma^{2}} \\left(1 - \\frac{\\sqrt{\\mu_4 - \\sigma^{4}}}{2 \\sigma^{3} \\sqrt{n}} \\left(\\rho z_{1} + \\sqrt{- \\rho^{2} + 1} z_{3}\\right)\\right)^{2} \\left(\\sigma \\zeta + \\frac{\\sigma}{\\sqrt{n}} \\left(\\frac{\\mu_3^{2}}{\\sigma^{3.0} n} \\left(- 0.0392837100659193 \\sqrt{2} z_{1}^{3} + 0.0982092751647983 \\sqrt{2} z_{1}\\right) + \\frac{\\mu_3}{\\sigma^{1.5} \\sqrt{n}} \\left(0.166666666666667 z_{1}^{2} - 0.166666666666667\\right) + \\frac{\\mu_4}{\\sigma^{4} n} \\left(0.0294627825494395 \\sqrt{2} z_{1}^{3} - 0.0883883476483184 \\sqrt{2} z_{1}\\right) + z_{1}\\right)\\right)^{2}$$" | |
| ], | |
| "text/plain": [ | |
| " ⎛ \n", | |
| " ⎜ ⎛ \n", | |
| " ⎜ ⎜ \n", | |
| " ⎛ ________________ ⎛ _____________ ⎞⎞ ⎜ ⎜ \n", | |
| " ⎜ ╱ 4 ⎜ ╱ 2 ⎟⎟ ⎜ ⎜ \n", | |
| " ⎜ ╲╱ \\mu₄ - \\sigma ⋅⎝\\rho⋅z₁ + ╲╱ - \\rho + 1 ⋅z₃⎠⎟ ⎜ ⎜ \n", | |
| " ⎜1 - ───────────────────────────────────────────────────⎟⋅⎜a₁ + a₃⋅⎜\\sigm\n", | |
| " ⎜ 3 ⎟ ⎝ ⎝ \n", | |
| " ⎝ 2⋅\\sigma ⋅√n ⎠ \n", | |
| "a₀ + ─────────────────────────────────────────────────────────────────────────\n", | |
| " \n", | |
| " \n", | |
| "\n", | |
| " \n", | |
| " ⎛ 2 -3.0 ⎛ 3 \n", | |
| " ⎜\\mu₃ ⋅\\sigma ⋅⎝- 0.0392837100659193⋅√2⋅z₁ + 0.0982092751\n", | |
| " \\sigma⋅⎜────────────────────────────────────────────────────────────\n", | |
| " ⎜ n \n", | |
| " ⎝ \n", | |
| "a⋅\\zeta + ────────────────────────────────────────────────────────────────────\n", | |
| " \n", | |
| " \n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " \n", | |
| " \n", | |
| "\n", | |
| " \n", | |
| " ⎞ -1.5 ⎛ 2 ⎞ \n", | |
| "647983⋅√2⋅z₁⎠ \\mu₃⋅\\sigma ⋅⎝0.166666666666667⋅z₁ - 0.166666666666667⎠ \n", | |
| "───────────── + ─────────────────────────────────────────────────────────── + \n", | |
| " √n \n", | |
| " \n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " √n \n", | |
| " \n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " \n", | |
| " \n", | |
| "\n", | |
| " 2 \n", | |
| " ⎛ 3 ⎞ ⎞⎞ \n", | |
| "\\mu₄⋅⎝0.0294627825494395⋅√2⋅z₁ - 0.0883883476483184⋅√2⋅z₁⎠ ⎟⎟ \n", | |
| "─────────────────────────────────────────────────────────── + z₁⎟⎟ \n", | |
| " 4 ⎟⎟ \n", | |
| " \\sigma ⋅n ⎠⎟ \n", | |
| "─────────────────────────────────────────────────────────────────⎟ + (a₂ + 1)\n", | |
| " ⎠ \n", | |
| " \n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " \\sigma \n", | |
| " \n", | |
| "\n", | |
| " \n", | |
| " ⎛ ⎛ 2 -3.0 ⎛ 3 \n", | |
| " ⎜ ⎜\\mu₃ ⋅\\sigma ⋅⎝- 0.0392837100659193⋅√2⋅z₁ + 0.098\n", | |
| " ⎜ \\sigma⋅⎜─────────────────────────────────────────────────────\n", | |
| " ⎜ ⎜ n \n", | |
| " ⎜ ⎝ \n", | |
| "⋅⎜\\sigma⋅\\zeta + ─────────────────────────────────────────────────────────────\n", | |
| " ⎝ \n", | |
| " \n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " \n", | |
| " \n", | |
| "\n", | |
| " \n", | |
| " ⎞ -1.5 ⎛ 2 \n", | |
| "2092751647983⋅√2⋅z₁⎠ \\mu₃⋅\\sigma ⋅⎝0.166666666666667⋅z₁ - 0.166666666666\n", | |
| "──────────────────── + ───────────────────────────────────────────────────────\n", | |
| " √n \n", | |
| " \n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " √n \n", | |
| " \n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " \n", | |
| " \n", | |
| "\n", | |
| " ⎞ \n", | |
| " ⎞ ⎛ 3 ⎞ ⎞⎞⎟ \n", | |
| "667⎠ \\mu₄⋅⎝0.0294627825494395⋅√2⋅z₁ - 0.0883883476483184⋅√2⋅z₁⎠ ⎟⎟⎟ \n", | |
| "──── + ─────────────────────────────────────────────────────────── + z₁⎟⎟⎟ \n", | |
| " 4 ⎟⎟⎟ \n", | |
| " \\sigma ⋅n ⎠⎟⎟ \n", | |
| "────────────────────────────────────────────────────────────────────────⎟⎟ a\n", | |
| " ⎠⎠ \n", | |
| " \n", | |
| "────────────────────────────────────────────────────────────────────────── + ─\n", | |
| " \n", | |
| " \n", | |
| "\n", | |
| " \n", | |
| " ⎛ \n", | |
| " 2 ⎜ \n", | |
| " ⎛ ________________ ⎛ _____________ ⎞⎞ ⎜ \\\n", | |
| " ⎜ ╱ 4 ⎜ ╱ 2 ⎟⎟ ⎜ \n", | |
| " ⎜ ╲╱ \\mu₄ - \\sigma ⋅⎝\\rho⋅z₁ + ╲╱ - \\rho + 1 ⋅z₃⎠⎟ ⎜ \n", | |
| "₄⋅⎜1 - ───────────────────────────────────────────────────⎟ ⋅⎜\\sigma⋅\\zeta + ─\n", | |
| " ⎜ 3 ⎟ ⎝ \n", | |
| " ⎝ 2⋅\\sigma ⋅√n ⎠ \n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " \n", | |
| " \n", | |
| "\n", | |
| " \n", | |
| " ⎛ 2 -3.0 ⎛ 3 \n", | |
| " ⎜\\mu₃ ⋅\\sigma ⋅⎝- 0.0392837100659193⋅√2⋅z₁ + 0.0982092751647983⋅√2⋅z\n", | |
| "sigma⋅⎜───────────────────────────────────────────────────────────────────────\n", | |
| " ⎜ n \n", | |
| " ⎝ \n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " \n", | |
| " \n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " 2 \n", | |
| " \\sigma \n", | |
| "\n", | |
| " \n", | |
| " ⎞ -1.5 ⎛ 2 ⎞ ⎛ \n", | |
| "₁⎠ \\mu₃⋅\\sigma ⋅⎝0.166666666666667⋅z₁ - 0.166666666666667⎠ \\mu₄⋅⎝0.029\n", | |
| "── + ─────────────────────────────────────────────────────────── + ───────────\n", | |
| " √n \n", | |
| " \n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " √n \n", | |
| " \n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " \n", | |
| " \n", | |
| "\n", | |
| " 2\n", | |
| " 3 ⎞ ⎞⎞ \n", | |
| "4627825494395⋅√2⋅z₁ - 0.0883883476483184⋅√2⋅z₁⎠ ⎟⎟ \n", | |
| "──────────────────────────────────────────────── + z₁⎟⎟ \n", | |
| " 4 ⎟⎟ \n", | |
| " \\sigma ⋅n ⎠⎟ \n", | |
| "──────────────────────────────────────────────────────⎟ \n", | |
| " ⎠ \n", | |
| " \n", | |
| "────────────────────────────────────────────────────────\n", | |
| " \n", | |
| " " | |
| ] | |
| }, | |
| "execution_count": 4, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "# the new statistic; it is v = part1 + part2 + part3\n", | |
| "part1 = a0\n", | |
| "part2 = (a1 + (1+a2)*muhat + a3 * muhat**2) * invs\n", | |
| "part3 = a4 * (muhat*invs)**2\n", | |
| "\n", | |
| "v = part1 + part2 + part3\n", | |
| "v" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "That's a bit hairy. Here I truncate that statistic in $n$. This was hard for me to figure out in sympy, so I took a limit. (I like how 'oo' is infinity in sympy.)" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 5, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
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sdDE6XLhhAdPv4Ic3pHRQSkcGr5aMWTJZNA0JlMzJT8dhvYIRNRTxitckbxBja21i4y3Z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\nxJPZjIxARiAjkBHYUwj8P+DbE5JurVIyAAAAAElFTkSuQmCC\n", | |
| "text/latex": [ | |
| "$$\\frac{1}{\\sigma^{17.0} \\sqrt{n}} \\left(- 0.5 \\rho \\sigma^{13.0} a_{1} \\sqrt{\\mu_4 - \\sigma^{4}} z_{1} - 1.0 \\rho \\sigma^{14.0} \\zeta^{2} a_{4} \\sqrt{\\mu_4 - \\sigma^{4}} z_{1} - 0.5 \\rho \\sigma^{14.0} \\zeta a_{2} \\sqrt{\\mu_4 - \\sigma^{4}} z_{1} - 0.5 \\rho \\sigma^{14.0} \\zeta \\sqrt{\\mu_4 - \\sigma^{4}} z_{1} - 0.5 \\rho \\sigma^{15.0} \\zeta^{2} a_{3} \\sqrt{\\mu_4 - \\sigma^{4}} z_{1} - 0.5 \\sigma^{13.0} a_{1} \\sqrt{\\mu_4 - \\sigma^{4}} \\sqrt{- \\rho^{2} + 1} z_{3} - 1.0 \\sigma^{14.0} \\zeta^{2} a_{4} \\sqrt{\\mu_4 - \\sigma^{4}} \\sqrt{- \\rho^{2} + 1} z_{3} - 0.5 \\sigma^{14.0} \\zeta a_{2} \\sqrt{\\mu_4 - \\sigma^{4}} \\sqrt{- \\rho^{2} + 1} z_{3} - 0.5 \\sigma^{14.0} \\zeta \\sqrt{\\mu_4 - \\sigma^{4}} \\sqrt{- \\rho^{2} + 1} z_{3} - 0.5 \\sigma^{15.0} \\zeta^{2} a_{3} \\sqrt{\\mu_4 - \\sigma^{4}} \\sqrt{- \\rho^{2} + 1} z_{3} + 2.0 \\sigma^{17.0} \\zeta a_{4} z_{1} + 1.0 \\sigma^{17.0} a_{2} z_{1} + 1.0 \\sigma^{17.0} z_{1} + 2.0 \\sigma^{18.0} \\zeta a_{3} z_{1}\\right) + \\frac{1}{\\sigma} \\left(\\sigma^{2} \\zeta^{2} a_{3} + \\sigma \\zeta^{2} a_{4} + \\sigma \\zeta a_{2} + \\sigma \\zeta + \\sigma a_{0} + a_{1}\\right)$$" | |
| ], | |
| "text/plain": [ | |
| " ⎛ ________________ \n", | |
| " -17.0 ⎜ 13.0 ╱ 4 \n", | |
| "0.25⋅\\sigma ⋅⎝- 2.0⋅\\rho⋅\\sigma ⋅a₁⋅╲╱ \\mu₄ - \\sigma ⋅z₁ - 4.0⋅\\rho⋅\\\n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " \n", | |
| "\n", | |
| " ________________ _\n", | |
| " 14.0 2 ╱ 4 14.0 ╱ \n", | |
| "sigma ⋅\\zeta ⋅a₄⋅╲╱ \\mu₄ - \\sigma ⋅z₁ - 2.0⋅\\rho⋅\\sigma ⋅\\zeta⋅a₂⋅╲╱ \n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " \n", | |
| "\n", | |
| "_______________ ________________ \n", | |
| " 4 14.0 ╱ 4 \n", | |
| "\\mu₄ - \\sigma ⋅z₁ - 2.0⋅\\rho⋅\\sigma ⋅\\zeta⋅╲╱ \\mu₄ - \\sigma ⋅z₁ - 2.0⋅\\r\n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " \n", | |
| "\n", | |
| " ________________ ________\n", | |
| " 15.0 2 ╱ 4 13.0 ╱ \n", | |
| "ho⋅\\sigma ⋅\\zeta ⋅a₃⋅╲╱ \\mu₄ - \\sigma ⋅z₁ - 2.0⋅\\sigma ⋅a₁⋅╲╱ \\mu₄ - \n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " \n", | |
| "\n", | |
| "________ _____________ ________________ \n", | |
| " 4 ╱ 2 14.0 2 ╱ 4 \n", | |
| "\\sigma ⋅╲╱ - \\rho + 1 ⋅z₃ - 4.0⋅\\sigma ⋅\\zeta ⋅a₄⋅╲╱ \\mu₄ - \\sigma ⋅╲╱\n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " √n \n", | |
| "\n", | |
| " _____________ ________________ ___________\n", | |
| "╱ 2 14.0 ╱ 4 ╱ 2 \n", | |
| " - \\rho + 1 ⋅z₃ - 2.0⋅\\sigma ⋅\\zeta⋅a₂⋅╲╱ \\mu₄ - \\sigma ⋅╲╱ - \\rho + \n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " \n", | |
| "\n", | |
| "__ ________________ _____________ \n", | |
| " 14.0 ╱ 4 ╱ 2 \n", | |
| "1 ⋅z₃ - 2.0⋅\\sigma ⋅\\zeta⋅╲╱ \\mu₄ - \\sigma ⋅╲╱ - \\rho + 1 ⋅z₃ - 2.0⋅\\si\n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " \n", | |
| "\n", | |
| " ________________ _____________ \n", | |
| " 15.0 2 ╱ 4 ╱ 2 17.0 \n", | |
| "gma ⋅\\zeta ⋅a₃⋅╲╱ \\mu₄ - \\sigma ⋅╲╱ - \\rho + 1 ⋅z₃ + 8.0⋅\\sigma ⋅\\ze\n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " \n", | |
| "\n", | |
| " \n", | |
| " 17.0 17.0 18.0 \n", | |
| "ta⋅a₄⋅z₁ + 4.0⋅\\sigma ⋅a₂⋅z₁ + 4.0⋅\\sigma ⋅z₁ + 8.0⋅\\sigma ⋅\\zeta⋅a₃⋅\n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " \n", | |
| "\n", | |
| " ⎞ \n", | |
| " ⎟ 2 2 2 \n", | |
| "z₁⎠ \\sigma ⋅\\zeta ⋅a₃ + \\sigma⋅\\zeta ⋅a₄ + \\sigma⋅\\zeta⋅a₂ + \\sigma⋅\\zeta + \n", | |
| "─── + ────────────────────────────────────────────────────────────────────────\n", | |
| " \\sigma \n", | |
| "\n", | |
| " \n", | |
| " \n", | |
| "\\sigma⋅a₀ + a₁\n", | |
| "──────────────\n", | |
| " " | |
| ] | |
| }, | |
| "execution_count": 5, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "#show nothing\n", | |
| "v_0 = limit(v,n,oo)\n", | |
| "v_05 = v_0 + (limit(sqrt(n) * (v - v_0),n,oo) / sqrt(n))\n", | |
| "v_05" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "Now we define the error as $v - \\zeta$ and compute the approximate bias and variance of the error. We sum the variance and squared bias to get mean square error." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 6, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "image/png": "iVBORw0KGgoAAAANSUhEUgAAAQ0AAAAlBAMAAACqtKHUAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n257XTkjT3qbbjmnatz2vnZCmPfAXIjgWuEqELtAAAAAASUVORK5CYII=\n", | |
| "text/latex": [ | |
| "$$\\sigma \\zeta^{2} a_{3} + \\zeta^{2} a_{4} + \\zeta a_{2} + a_{0} + \\frac{a_{1}}{\\sigma}$$" | |
| ], | |
| "text/plain": [ | |
| " 2 2 a₁ \n", | |
| "\\sigma⋅\\zeta ⋅a₃ + \\zeta ⋅a₄ + \\zeta⋅a₂ + a₀ + ──────\n", | |
| " \\sigma" | |
| ] | |
| }, | |
| "execution_count": 6, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "staterr = v_05 - zeta\n", | |
| "# mean squared error of the statistic v, is\n", | |
| "# MSE = E((newstat - zeta)**2)\n", | |
| "# this is too slow, though, so evaluate them separately instead:\n", | |
| "bias = E(staterr)\n", | |
| "simplify(bias)" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 7, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
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3rBVzoNt88Jpyd/WzLvAYsrt2KLyKYhJ3DCBlJcl8DKAN8wZgjIJ7nUtJFAGF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\n", | |
| "text/latex": [ | |
| "$$\\left(- \\zeta + \\frac{1}{\\sigma} \\left(\\sigma^{2} \\zeta^{2} a_{3} + \\sigma \\zeta^{2} a_{4} + \\sigma \\zeta a_{2} + \\sigma \\zeta + \\sigma a_{0} + a_{1}\\right)\\right)^{2} + \\frac{1}{n} \\left(\\frac{0.25 \\mu_4}{\\sigma^{8.0}} a_{1}^{2} + \\frac{1.0 \\mu_4}{\\sigma^{7.0}} \\zeta^{2} a_{1} a_{4} + \\frac{0.5 \\mu_4}{\\sigma^{7.0}} \\zeta a_{1} a_{2} + \\frac{0.5 \\mu_4}{\\sigma^{7.0}} \\zeta a_{1} + \\frac{1.0 \\mu_4}{\\sigma^{6.0}} \\zeta^{4} a_{4}^{2} + \\frac{1.0 \\mu_4}{\\sigma^{6.0}} \\zeta^{3} a_{2} a_{4} + \\frac{1.0 \\mu_4}{\\sigma^{6.0}} \\zeta^{3} a_{4} + \\frac{0.5 \\mu_4}{\\sigma^{6.0}} \\zeta^{2} a_{1} a_{3} + \\frac{0.25 \\mu_4}{\\sigma^{6.0}} \\zeta^{2} a_{2}^{2} + \\frac{0.5 \\mu_4}{\\sigma^{6.0}} \\zeta^{2} a_{2} + \\frac{0.25 \\mu_4}{\\sigma^{6.0}} \\zeta^{2} + \\frac{1.0 \\mu_4}{\\sigma^{5.0}} \\zeta^{4} a_{3} a_{4} + \\frac{0.5 \\mu_4}{\\sigma^{5.0}} \\zeta^{3} a_{2} a_{3} + \\frac{0.5 \\mu_4}{\\sigma^{5.0}} \\zeta^{3} a_{3} + \\frac{0.25 \\mu_4}{\\sigma^{4.0}} \\zeta^{4} a_{3}^{2} - \\frac{2.0 \\rho}{\\sigma^{4.0}} \\zeta a_{1} a_{4} \\sqrt{\\mu_4 - \\sigma^{4}} - \\frac{1.0 \\rho}{\\sigma^{4.0}} a_{1} a_{2} \\sqrt{\\mu_4 - \\sigma^{4}} - \\frac{1.0 \\rho}{\\sigma^{4.0}} a_{1} \\sqrt{\\mu_4 - \\sigma^{4}} - \\frac{4.0 \\rho}{\\sigma^{3.0}} \\zeta^{3} a_{4}^{2} \\sqrt{\\mu_4 - \\sigma^{4}} - \\frac{4.0 \\rho}{\\sigma^{3.0}} \\zeta^{2} a_{2} a_{4} \\sqrt{\\mu_4 - \\sigma^{4}} - \\frac{4.0 \\rho}{\\sigma^{3.0}} \\zeta^{2} a_{4} \\sqrt{\\mu_4 - \\sigma^{4}} - \\frac{2.0 \\rho}{\\sigma^{3.0}} \\zeta a_{1} a_{3} \\sqrt{\\mu_4 - \\sigma^{4}} - \\frac{1.0 \\rho}{\\sigma^{3.0}} \\zeta a_{2}^{2} \\sqrt{\\mu_4 - \\sigma^{4}} - \\frac{2.0 \\rho}{\\sigma^{3.0}} \\zeta a_{2} \\sqrt{\\mu_4 - \\sigma^{4}} - \\frac{1.0 \\rho}{\\sigma^{3.0}} \\zeta \\sqrt{\\mu_4 - \\sigma^{4}} - \\frac{6.0 \\rho}{\\sigma^{2.0}} \\zeta^{3} a_{3} a_{4} \\sqrt{\\mu_4 - \\sigma^{4}} - \\frac{3.0 \\rho}{\\sigma^{2.0}} \\zeta^{2} a_{2} a_{3} \\sqrt{\\mu_4 - \\sigma^{4}} - \\frac{3.0 \\rho}{\\sigma^{2.0}} \\zeta^{2} a_{3} \\sqrt{\\mu_4 - \\sigma^{4}} - \\frac{2.0 \\rho}{\\sigma^{1.0}} \\zeta^{3} a_{3}^{2} \\sqrt{\\mu_4 - \\sigma^{4}} - \\frac{0.25 a_{1}^{2}}{\\sigma^{4.0}} - \\frac{1.0 a_{1}}{\\sigma^{3.0}} \\zeta^{2} a_{4} - \\frac{0.5 \\zeta}{\\sigma^{3.0}} a_{1} a_{2} - \\frac{0.5 \\zeta}{\\sigma^{3.0}} a_{1} - \\frac{1.0 \\zeta^{4}}{\\sigma^{2.0}} a_{4}^{2} - \\frac{1.0 a_{2}}{\\sigma^{2.0}} \\zeta^{3} a_{4} - \\frac{1.0 a_{4}}{\\sigma^{2.0}} \\zeta^{3} - \\frac{0.5 a_{1}}{\\sigma^{2.0}} \\zeta^{2} a_{3} - \\frac{0.25 \\zeta^{2}}{\\sigma^{2.0}} a_{2}^{2} - \\frac{0.5 a_{2}}{\\sigma^{2.0}} \\zeta^{2} - \\frac{0.25 \\zeta^{2}}{\\sigma^{2.0}} - \\frac{1.0 a_{3}}{\\sigma^{1.0}} \\zeta^{4} a_{4} - \\frac{0.5 a_{2}}{\\sigma^{1.0}} \\zeta^{3} a_{3} - \\frac{0.5 a_{3}}{\\sigma^{1.0}} \\zeta^{3} + 8.0 \\sigma^{1.0} \\zeta^{2} a_{3} a_{4} + 4.0 \\sigma^{1.0} \\zeta a_{2} a_{3} + 4.0 \\sigma^{1.0} \\zeta a_{3} + 4.0 \\sigma^{2.0} \\zeta^{2} a_{3}^{2} - 0.25 \\zeta^{4} a_{3}^{2} + 4.0 \\zeta^{2} a_{4}^{2} + 4.0 \\zeta a_{2} a_{4} + 4.0 \\zeta a_{4} + 1.0 a_{2}^{2} + 2.0 a_{2} + 1.0\\right)$$" | |
| ], | |
| "text/plain": [ | |
| " \n", | |
| "⎛ 2 2 2 \n", | |
| "⎜ \\sigma ⋅\\zeta ⋅a₃ + \\sigma⋅\\zeta ⋅a₄ + \\sigma⋅\\zeta⋅a₂ + \\sigma⋅\\zet\n", | |
| "⎜-\\zeta + ────────────────────────────────────────────────────────────────────\n", | |
| "⎝ \\sigma \n", | |
| "\n", | |
| " 2 \n", | |
| " ⎞ -8.0 2 -7.0 2 \n", | |
| "a + \\sigma⋅a₀ + a₁⎟ 0.25⋅\\mu₄⋅\\sigma ⋅a₁ + 1.0⋅\\mu₄⋅\\sigma ⋅\\zeta ⋅a\n", | |
| "──────────────────⎟ + ───────────────────────────────────────────────────────\n", | |
| " ⎠ \n", | |
| "\n", | |
| " \n", | |
| " -7.0 -7.0 \n", | |
| "₁⋅a₄ + 0.5⋅\\mu₄⋅\\sigma ⋅\\zeta⋅a₁⋅a₂ + 0.5⋅\\mu₄⋅\\sigma ⋅\\zeta⋅a₁ + 1.0⋅\\m\n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " \n", | |
| "\n", | |
| " \n", | |
| " -6.0 4 2 -6.0 3 -\n", | |
| "u₄⋅\\sigma ⋅\\zeta ⋅a₄ + 1.0⋅\\mu₄⋅\\sigma ⋅\\zeta ⋅a₂⋅a₄ + 1.0⋅\\mu₄⋅\\sigma \n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " \n", | |
| "\n", | |
| " \n", | |
| "6.0 3 -6.0 2 -6.0 2\n", | |
| " ⋅\\zeta ⋅a₄ + 0.5⋅\\mu₄⋅\\sigma ⋅\\zeta ⋅a₁⋅a₃ + 0.25⋅\\mu₄⋅\\sigma ⋅\\zeta \n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " \n", | |
| "\n", | |
| " \n", | |
| " 2 -6.0 2 -6.0 2 \n", | |
| "⋅a₂ + 0.5⋅\\mu₄⋅\\sigma ⋅\\zeta ⋅a₂ + 0.25⋅\\mu₄⋅\\sigma ⋅\\zeta + 1.0⋅\\mu₄⋅\n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " \n", | |
| "\n", | |
| " \n", | |
| " -5.0 4 -5.0 3 -5\n", | |
| "\\sigma ⋅\\zeta ⋅a₃⋅a₄ + 0.5⋅\\mu₄⋅\\sigma ⋅\\zeta ⋅a₂⋅a₃ + 0.5⋅\\mu₄⋅\\sigma \n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " \n", | |
| "\n", | |
| " \n", | |
| ".0 3 -4.0 4 2 -4.0 \n", | |
| " ⋅\\zeta ⋅a₃ + 0.25⋅\\mu₄⋅\\sigma ⋅\\zeta ⋅a₃ - 2.0⋅\\rho⋅\\sigma ⋅\\zeta⋅a₁⋅\n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " \n", | |
| "\n", | |
| " ________________ ________________ \n", | |
| " ╱ 4 -4.0 ╱ 4 \n", | |
| "a₄⋅╲╱ \\mu₄ - \\sigma - 1.0⋅\\rho⋅\\sigma ⋅a₁⋅a₂⋅╲╱ \\mu₄ - \\sigma - 1.0⋅\\\n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " \n", | |
| "\n", | |
| " ________________ ____\n", | |
| " -4.0 ╱ 4 -3.0 3 2 ╱ \n", | |
| "rho⋅\\sigma ⋅a₁⋅╲╱ \\mu₄ - \\sigma - 4.0⋅\\rho⋅\\sigma ⋅\\zeta ⋅a₄ ⋅╲╱ \\mu\n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " \n", | |
| "\n", | |
| "____________ ________________ \n", | |
| " 4 -3.0 2 ╱ 4 \n", | |
| "₄ - \\sigma - 4.0⋅\\rho⋅\\sigma ⋅\\zeta ⋅a₂⋅a₄⋅╲╱ \\mu₄ - \\sigma - 4.0⋅\\rho\n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " \n", | |
| "\n", | |
| " ________________ \n", | |
| " -3.0 2 ╱ 4 -3.0 \n", | |
| "⋅\\sigma ⋅\\zeta ⋅a₄⋅╲╱ \\mu₄ - \\sigma - 2.0⋅\\rho⋅\\sigma ⋅\\zeta⋅a₁⋅a₃⋅╲╱\n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " \n", | |
| "\n", | |
| " ________________ ________________ \n", | |
| "╱ 4 -3.0 2 ╱ 4 \n", | |
| " \\mu₄ - \\sigma - 1.0⋅\\rho⋅\\sigma ⋅\\zeta⋅a₂ ⋅╲╱ \\mu₄ - \\sigma - 2.0⋅\\r\n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " n \n", | |
| "\n", | |
| " ________________ ____\n", | |
| " -3.0 ╱ 4 -3.0 ╱ \n", | |
| "ho⋅\\sigma ⋅\\zeta⋅a₂⋅╲╱ \\mu₄ - \\sigma - 1.0⋅\\rho⋅\\sigma ⋅\\zeta⋅╲╱ \\mu\n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " \n", | |
| "\n", | |
| "____________ ________________ \n", | |
| " 4 -2.0 3 ╱ 4 \n", | |
| "₄ - \\sigma - 6.0⋅\\rho⋅\\sigma ⋅\\zeta ⋅a₃⋅a₄⋅╲╱ \\mu₄ - \\sigma - 3.0⋅\\rho\n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " \n", | |
| "\n", | |
| " ________________ \n", | |
| " -2.0 2 ╱ 4 -2.0 2 \n", | |
| "⋅\\sigma ⋅\\zeta ⋅a₂⋅a₃⋅╲╱ \\mu₄ - \\sigma - 3.0⋅\\rho⋅\\sigma ⋅\\zeta ⋅a₃⋅╲\n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " \n", | |
| "\n", | |
| " ________________ ________________ \n", | |
| " ╱ 4 -1.0 3 2 ╱ 4 \n", | |
| "╱ \\mu₄ - \\sigma - 2.0⋅\\rho⋅\\sigma ⋅\\zeta ⋅a₃ ⋅╲╱ \\mu₄ - \\sigma - 0.25\n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " \n", | |
| "\n", | |
| " \n", | |
| " -4.0 2 -3.0 2 -3.0 \n", | |
| "⋅\\sigma ⋅a₁ - 1.0⋅\\sigma ⋅\\zeta ⋅a₁⋅a₄ - 0.5⋅\\sigma ⋅\\zeta⋅a₁⋅a₂ - 0\n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " \n", | |
| "\n", | |
| " \n", | |
| " -3.0 -2.0 4 2 -2.0 3 \n", | |
| ".5⋅\\sigma ⋅\\zeta⋅a₁ - 1.0⋅\\sigma ⋅\\zeta ⋅a₄ - 1.0⋅\\sigma ⋅\\zeta ⋅a₂⋅\n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " \n", | |
| "\n", | |
| " \n", | |
| " -2.0 3 -2.0 2 -2.0 \n", | |
| "a₄ - 1.0⋅\\sigma ⋅\\zeta ⋅a₄ - 0.5⋅\\sigma ⋅\\zeta ⋅a₁⋅a₃ - 0.25⋅\\sigma ⋅\n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " \n", | |
| "\n", | |
| " \n", | |
| " 2 2 -2.0 2 -2.0 2 -1.\n", | |
| "\\zeta ⋅a₂ - 0.5⋅\\sigma ⋅\\zeta ⋅a₂ - 0.25⋅\\sigma ⋅\\zeta - 1.0⋅\\sigma \n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " \n", | |
| "\n", | |
| " \n", | |
| "0 4 -1.0 3 -1.0 3 \n", | |
| " ⋅\\zeta ⋅a₃⋅a₄ - 0.5⋅\\sigma ⋅\\zeta ⋅a₂⋅a₃ - 0.5⋅\\sigma ⋅\\zeta ⋅a₃ + 8.0⋅\n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " \n", | |
| "\n", | |
| " \n", | |
| " 1.0 2 1.0 1.0 \n", | |
| "\\sigma ⋅\\zeta ⋅a₃⋅a₄ + 4.0⋅\\sigma ⋅\\zeta⋅a₂⋅a₃ + 4.0⋅\\sigma ⋅\\zeta⋅a₃ + \n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " \n", | |
| "\n", | |
| " \n", | |
| " 2.0 2 2 4 2 2 2 \n", | |
| "4.0⋅\\sigma ⋅\\zeta ⋅a₃ - 0.25⋅\\zeta ⋅a₃ + 4.0⋅\\zeta ⋅a₄ + 4.0⋅\\zeta⋅a₂⋅a₄ \n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " \n", | |
| "\n", | |
| " \n", | |
| " 2 \n", | |
| "+ 4.0⋅\\zeta⋅a₄ + 1.0⋅a₂ + 2.0⋅a₂ + 1.0\n", | |
| "───────────────────────────────────────\n", | |
| " " | |
| ] | |
| }, | |
| "execution_count": 7, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "# variance of the error:\n", | |
| "varerr = variance(staterr)\n", | |
| "MSE = (bias**2) + varerr \n", | |
| "collect(MSE,n)" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "That's really involved, and finding the derivative will be ugly. Instead we truncate at $n^{-1}$, which leaves us terms constant in $n$. Looking above, you will see that removing terms in $n^{-1}$ leaves some quantity squared. That is what we will minimize. The way forward is fairly clear from here." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 8, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
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JRU5ErkJggg==\n", | |
| "text/latex": [ | |
| "$$\\frac{1}{\\sigma^{2}} \\left(\\sigma^{4} \\zeta^{4} a_{3}^{2} + 2 \\sigma^{3} \\zeta^{4} a_{3} a_{4} + 2 \\sigma^{3} \\zeta^{3} a_{2} a_{3} + 2 \\sigma^{3} \\zeta^{2} a_{0} a_{3} + \\sigma^{2} \\zeta^{4} a_{4}^{2} + 2 \\sigma^{2} \\zeta^{3} a_{2} a_{4} + 2 \\sigma^{2} \\zeta^{2} a_{0} a_{4} + 2 \\sigma^{2} \\zeta^{2} a_{1} a_{3} + \\sigma^{2} \\zeta^{2} a_{2}^{2} + 2 \\sigma^{2} \\zeta a_{0} a_{2} + \\sigma^{2} a_{0}^{2} + 2 \\sigma \\zeta^{2} a_{1} a_{4} + 2 \\sigma \\zeta a_{1} a_{2} + 2 \\sigma a_{0} a_{1} + a_{1}^{2}\\right)$$" | |
| ], | |
| "text/plain": [ | |
| " 4 4 2 3 4 3 3 \n", | |
| "\\sigma ⋅\\zeta ⋅a₃ + 2⋅\\sigma ⋅\\zeta ⋅a₃⋅a₄ + 2⋅\\sigma ⋅\\zeta ⋅a₂⋅a₃ + 2⋅\\sigm\n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " \n", | |
| " \n", | |
| "\n", | |
| " 3 2 2 4 2 2 3 2 \n", | |
| "a ⋅\\zeta ⋅a₀⋅a₃ + \\sigma ⋅\\zeta ⋅a₄ + 2⋅\\sigma ⋅\\zeta ⋅a₂⋅a₄ + 2⋅\\sigma ⋅\\zet\n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " \n", | |
| " \\\n", | |
| "\n", | |
| " 2 2 2 2 2 2 2 \n", | |
| "a ⋅a₀⋅a₄ + 2⋅\\sigma ⋅\\zeta ⋅a₁⋅a₃ + \\sigma ⋅\\zeta ⋅a₂ + 2⋅\\sigma ⋅\\zeta⋅a₀⋅a₂\n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " 2 \n", | |
| "sigma \n", | |
| "\n", | |
| " 2 2 2 \n", | |
| " + \\sigma ⋅a₀ + 2⋅\\sigma⋅\\zeta ⋅a₁⋅a₄ + 2⋅\\sigma⋅\\zeta⋅a₁⋅a₂ + 2⋅\\sigma⋅a₀⋅a₁\n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " \n", | |
| " \n", | |
| "\n", | |
| " 2\n", | |
| " + a₁ \n", | |
| "──────\n", | |
| " \n", | |
| " " | |
| ] | |
| }, | |
| "execution_count": 8, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "# truncate!\n", | |
| "MSE_0 = limit(collect(MSE,n),n,oo)\n", | |
| "MSE_1 = MSE_0 + (limit(n * (MSE - MSE_0),n,oo)/n)\n", | |
| "MSE_0" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "Now we take the derivative of the Mean Square Error with respect to the $a_i$. In each case we will get an equation linear in all the $a_i$. The first order condition, which corresponds to minimizing the MSE, occurs for $a_i=0$." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 9, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "image/png": "iVBORw0KGgoAAAANSUhEUgAAAUAAAAAqBAMAAAAwpp8RAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n", | |
| "text/latex": [ | |
| "$$2 \\sigma \\zeta^{2} a_{3} + 2 \\zeta^{2} a_{4} + 2 \\zeta a_{2} + 2 a_{0} + \\frac{2 a_{1}}{\\sigma}$$" | |
| ], | |
| "text/plain": [ | |
| " 2 2 2⋅a₁ \n", | |
| "2⋅\\sigma⋅\\zeta ⋅a₃ + 2⋅\\zeta ⋅a₄ + 2⋅\\zeta⋅a₂ + 2⋅a₀ + ──────\n", | |
| " \\sigma" | |
| ] | |
| }, | |
| "execution_count": 9, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "# a_0\n", | |
| "simplify(diff(MSE_0,a0))" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 10, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "image/png": "iVBORw0KGgoAAAANSUhEUgAAAUIAAAAsBAMAAADiCqwxAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n+B/d1NS/DOg6FwAAAABJRU5ErkJggg==\n", | |
| "text/latex": [ | |
| "$$2 \\zeta^{2} a_{3} + \\frac{2 a_{4}}{\\sigma} \\zeta^{2} + \\frac{2 \\zeta}{\\sigma} a_{2} + \\frac{2 a_{0}}{\\sigma} + \\frac{2 a_{1}}{\\sigma^{2}}$$" | |
| ], | |
| "text/plain": [ | |
| " 2 \n", | |
| " 2 2⋅\\zeta ⋅a₄ 2⋅\\zeta⋅a₂ 2⋅a₀ 2⋅a₁ \n", | |
| "2⋅\\zeta ⋅a₃ + ─────────── + ────────── + ────── + ───────\n", | |
| " \\sigma \\sigma \\sigma 2\n", | |
| " \\sigma " | |
| ] | |
| }, | |
| "execution_count": 10, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "# a_1\n", | |
| "simplify(diff(MSE_0,a1))" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 11, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "image/png": "iVBORw0KGgoAAAANSUhEUgAAAWUAAAAsBAMAAABPi7SYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nVwAAAABJRU5ErkJggg==\n", | |
| "text/latex": [ | |
| "$$\\frac{2 \\zeta}{\\sigma} \\left(\\sigma^{2} \\zeta^{2} a_{3} + \\sigma \\zeta^{2} a_{4} + \\sigma \\zeta a_{2} + \\sigma a_{0} + a_{1}\\right)$$" | |
| ], | |
| "text/plain": [ | |
| " ⎛ 2 2 2 \n", | |
| "2⋅\\zeta⋅⎝\\sigma ⋅\\zeta ⋅a₃ + \\sigma⋅\\zeta ⋅a₄ + \\sigma⋅\\zeta⋅a₂ + \\sigma⋅a₀ + \n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " \\sigma \n", | |
| "\n", | |
| " ⎞\n", | |
| "a₁⎠\n", | |
| "───\n", | |
| " " | |
| ] | |
| }, | |
| "execution_count": 11, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "# a_2\n", | |
| "simplify(diff(MSE_0,a2))" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 12, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "image/png": "iVBORw0KGgoAAAANSUhEUgAAAWoAAAAcBAMAAAC6+ui4AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nI7uKfwG28pR9BuU/vwAAAABJRU5ErkJggg==\n", | |
| "text/latex": [ | |
| "$$2 \\zeta^{2} \\left(\\sigma^{2} \\zeta^{2} a_{3} + \\sigma \\zeta^{2} a_{4} + \\sigma \\zeta a_{2} + \\sigma a_{0} + a_{1}\\right)$$" | |
| ], | |
| "text/plain": [ | |
| " 2 ⎛ 2 2 2 \n", | |
| "2⋅\\zeta ⋅⎝\\sigma ⋅\\zeta ⋅a₃ + \\sigma⋅\\zeta ⋅a₄ + \\sigma⋅\\zeta⋅a₂ + \\sigma⋅a₀ +\n", | |
| "\n", | |
| " ⎞\n", | |
| " a₁⎠" | |
| ] | |
| }, | |
| "execution_count": 12, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "# a_3\n", | |
| "simplify(diff(MSE_0,a3))" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 13, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "image/png": "iVBORw0KGgoAAAANSUhEUgAAAW4AAAAvBAMAAAAx/z3BAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n", | |
| "text/latex": [ | |
| "$$\\frac{2 \\zeta^{2}}{\\sigma} \\left(\\sigma^{2} \\zeta^{2} a_{3} + \\sigma \\zeta^{2} a_{4} + \\sigma \\zeta a_{2} + \\sigma a_{0} + a_{1}\\right)$$" | |
| ], | |
| "text/plain": [ | |
| " 2 ⎛ 2 2 2 \n", | |
| "2⋅\\zeta ⋅⎝\\sigma ⋅\\zeta ⋅a₃ + \\sigma⋅\\zeta ⋅a₄ + \\sigma⋅\\zeta⋅a₂ + \\sigma⋅a₀ +\n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " \\sigma \n", | |
| "\n", | |
| " ⎞\n", | |
| " a₁⎠\n", | |
| "────\n", | |
| " " | |
| ] | |
| }, | |
| "execution_count": 13, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "# a_4\n", | |
| "simplify(diff(MSE_0,a4))" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "To recap, the minimal MSE occurs for $a_0 = a_1 = a_2 = a_3 = a_4 = 0$. We must try another approach." | |
| ] | |
| } | |
| ], | |
| "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": "2.7.12" | |
| } | |
| }, | |
| "nbformat": 4, | |
| "nbformat_minor": 0 | |
| } |
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