{ "cells": [ { "cell_type": "markdown", "id": "daa444e1", "metadata": {}, "source": [ "| [**Lattice QCD I**](https://homepages.uni-regensburg.de/~lec17310/teaching/wise2122/lqft.html) | - | [**Christoph Lehner**](https://go.ur.de/lehner) |\n", "|:-|--|-:|\n", "\n", "# Chapter 3: statistics of continuous variables" ] }, { "cell_type": "markdown", "id": "8b33309c", "metadata": {}, "source": [ "## Univariate probability distribution and normal distribution\n", "\n", "Let us start with a random variable $x \\in \\mathbb{R}$ following the probability density function (PDF) $p:\\mathbb{R}\\to \\mathbb{R}$ with\n", "$$\n", "\\begin{align}\n", "\\int dx p(x) = 1 \\,.\n", "\\end{align}\n", "$$\n", "\n", "Since we only consider one variable, we call such distributions **univariate**.\n", "\n", "We define the expectation value of a function $f:\\mathbb{R}\\to\\mathbb{C}$ of this variable as\n", "$$\n", "\\begin{align}\n", "\\langle f(x) \\rangle = \\int dx p(x) f(x) \\,.\n", "\\end{align}\n", "$$\n", "Next, we call $\\langle x^m \\rangle$ the $m$-th **moment** of the distribution with first non-trivial case\n", "$$\n", "\\mu = \\langle x \\rangle\n", "$$\n", "the **mean** of the distribution. We refer to $\\langle (x - \\mu)^m \\rangle$ as the $m$-th **central moment** of the distribution and the first non-trivial case\n", "$$\n", "\\sigma^2 = \\langle (x - \\mu)^2 \\rangle\n", "$$\n", "as the **variance** of the distribution and call $\\sigma$ the **standard deviation**.\n", "\n", "We define the fraction of results that fall within $n$ $\\sigma$ around the mean as\n", "$$\n", "\\begin{align}\n", "f_n = \\int_{\\mu - n\\sigma}^{\\mu + n\\sigma} dx p(x)\n", "\\end{align}\n", "$$\n", "which depends on the PDF. A particularly important case is the **normal distribution**\n", "with\n", "$$\n", "\\begin{align}\n", "p(x) = \\frac1{\\sqrt{2\\pi \\sigma^2}} e^{-\\frac{(x-\\mu)^2}{2 \\sigma^2}}\n", "\\end{align}\n", "$$\n", "for which the PDF only depends on the mean and the variance. For the normal distribution,\n", "the $f_n$ are independent of $\\mu$ and $\\sigma$ (***Homework***: show this independence) and\n", "$$\n", "\\begin{align}\n", "f_1 &= 0.682\\ldots \\,, \\\\\n", "f_2 &= 0.954\\ldots \\,, \\\\\n", "f_3 &= 1 - 2.6\\ldots \\times 10^{-3} \\,, \\\\\n", "f_4 &= 1 - 6.3 \\ldots \\times 10^{-5} \\,, \\\\\n", "f_5 &= 1 - 5.7\\ldots \\times 10^{-7} \\,.\n", "\\end{align}\n", "$$\n", "\n", "So if we know the PDF, we can translate fluctuations of size $n$ $\\sigma$ to probabilities of such a fluctuation. And if the PDF is normal, the $f_n$ are fixed numbers.\n", "\n", "Interestingly, the normal distribution naturally appears in averaging processes. Let us first look at some data:" ] }, { "cell_type": "code", "execution_count": 1, "id": "bbb4ac2c", "metadata": {}, "outputs": [], "source": [ "import random\n", "import numpy as np\n", "from matplotlib import pyplot as plt\n", "\n", "random.seed(13)" ] }, { "cell_type": "code", "execution_count": 2, "id": "29ed61ed", "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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\n", 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\n", 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\n", 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "def histogram(X, l):\n", " v = [X() for i in range(1000)]\n", " plt.hist(v,label=l,density=True)\n", " mu = np.mean(v)\n", " sigma = np.std(v)\n", " x = np.linspace(0, 1.0, 100)\n", " y = [(2.*np.pi*sigma**2.)**-0.5*np.exp(-(xv-mu)**2./2/sigma**2.) for xv in x]\n", " plt.plot(x, y, label=\"Normal with same mu, sigma\")\n", " plt.legend()\n", " plt.show()\n", "\n", "# start with triangular distribution\n", "def p():\n", " return random.triangular(0.0,1.0,0.1)\n", "\n", "histogram(p, \"Triangular, no averaging\")\n", "histogram(lambda: sum([p() for i in range(2)]) / 2.0, \"Triangular, average 2\")\n", "histogram(lambda: sum([p() for i in range(4)]) / 4.0, \"Triangular, average 4\")\n", "histogram(lambda: sum([p() for i in range(8)]) / 8.0, \"Triangular, average 8\")\n", "histogram(lambda: sum([p() for i in range(16)]) / 16.0, \"Triangular, average 16\")" ] }, { "cell_type": "markdown", "id": "beb79640", "metadata": {}, "source": [ "***Homework***: Repeat this exercise with different starting distributions." ] }, { "cell_type": "markdown", "id": "82761013", "metadata": {}, "source": [ "## Central limit theorem" ] }, { "cell_type": "markdown", "id": "edfecfaf", "metadata": {}, "source": [ "The effect we just observed can be shown in a straightforward way.\n", "\n", "Consider $n$ random variables $x_1,\\ldots,x_n$ drawn from the identical PDF $p(x)$. They should be independent, i.e., their joint probability factors\n", "$$\n", "\\begin{align}\n", "p(x_1,\\ldots,x_n)=p(x_1)\\cdots p(x_n) \\,.\n", "\\end{align}\n", "$$\n", "We then define $\\tilde{x}$ as the sum of these $n$ random variables\n", "$$\n", "\\begin{align}\n", "\\tilde{x} = \\sum_{i=1}^n x_i \\,.\n", "\\end{align}\n", "$$\n", "\n", "We first note that\n", "$$\n", "\\begin{align}\n", "\\tilde{\\mu} &= \\langle \\tilde{x} \\rangle = \\langle \\sum_{i=1}^n x_i \\rangle = \n", "\\int dx_1 \\cdots dx_n p(x_1,\\ldots,x_n) \\sum_{i=1}^n x_i \\\\\n", "&= \\sum_{i=1}^n \\int dx_1 \\cdots dx_n p(x_1) \\cdots p(x_n) x_i \\\\\n", "&=\\sum_{i=1}^n \\int dx_i p(x_i) x_i \\\\\n", "&=\\sum_{i=1}^n \\mu = n\\mu\n", "\\end{align}\n", "$$\n", "and\n", "$$\n", "\\begin{align}\n", "\\tilde{\\sigma}^2 &= \\langle \\tilde{x}^2 \\rangle - \\tilde{\\mu}^2 \\\\\n", "&= \\sum_{i,j=1}^n \\langle x_i x_j \\rangle - n^2\\mu^2 \\\\\n", "&= \\sum_{i=1}^n \\langle x^2 \\rangle + \\sum_{i\\neq j} \\langle x \\rangle^2 - n^2\\mu^2 \\\\\n", "&= n (\\sigma^2 + \\mu^2) + n(n-1)\\mu^2 - n^2\\mu^2 \\\\\n", "&= n\\sigma^2 \\,.\n", "\\end{align}\n", "$$\n", "\n", "It is then natural to define the helper variable\n", "$$\n", "\\begin{align}\n", "z = \\frac{\\tilde{x} - n\\mu}{\\sqrt{n \\sigma^2}}\n", "\\end{align}\n", "$$\n", "which has mean 0 and variance 1.\n", "\n", "We then define the **characteristic function** of the PDF as its\n", "Fourier transform\n", "$$\n", "\\begin{align}\n", "\\varphi(k) &= \\int dx p(x) e^{ikx} \\,, \\\\\n", "p(x) &= \\frac1{2\\pi}\\int dk \\varphi(k) e^{-ikx} \\,.\n", "\\end{align}\n", "$$" ] }, { "cell_type": "markdown", "id": "66f488cc", "metadata": {}, "source": [ "The PDF for $z$ is then\n", "$$\n", "\\begin{align}\n", "p_z(z) &= \\int dx_1 \\cdots dx_n \\delta\\left(z - \\frac{\\sum_{i=1}^n x_i - n\\mu}{\\sqrt{n} \\sigma}\\right) p(x_1,\\ldots,x_n) \\,, \\\\\n", "&= \\int dx_1 \\cdots dx_n \\delta\\left(z - \\frac{\\sum_{i=1}^n x_i - n\\mu}{\\sqrt{n} \\sigma}\\right) p(x_1)\\cdots p(x_n) \\,, \\\\\n", "&= (2\\pi)^{-n}\\int dx_1 \\cdots dx_n dk_1 \\cdots dk_n \\delta\\left(z - \\frac{\\sum_{i=1}^n x_i - n\\mu}{\\sqrt{n} \\sigma}\\right) e^{-i \\sum_{i=1}^n x_i k_i} \\varphi(k_1)\\cdots \\varphi(k_n) \\,, \\\\\n", "\\end{align}\n", "$$\n", "and therefore the characteristic function for $z$ is\n", "$$\n", "\\begin{align}\n", "\\varphi_z(\\omega) &= (2\\pi)^{-n}\\int dz \\int dx_1 \\cdots dx_n dk_1 \\cdots dk_n \\delta\\left(z - \\frac{\\sum_{i=1}^n x_i - n\\mu}{\\sqrt{n} \\sigma}\\right) e^{-i \\sum_{i=1}^n x_i k_i+i\\omega z} \\varphi(k_1)\\cdots \\varphi(k_n) \\,, \\\\\n", "&= (2\\pi)^{-n}\\int dx_1 \\cdots dx_n dk_1 \\cdots dk_n e^{-i \\sum_{i=1}^n x_i \\left(k_i - \\frac{\\omega}{\\sqrt{n} \\sigma}\\right)-i\\omega \\sqrt{n}\\frac{\\mu}{\\sigma}} \\varphi(k_1)\\cdots \\varphi(k_n) \\,, \\\\\n", "&= \\int dk_1 \\cdots dk_n \\left[\\prod_{i=1}^n\\delta\\left(k_i - \\frac{\\omega}{\\sqrt{n} \\sigma}\\right)\\right] e^{-i\\omega \\sqrt{n}\\frac{\\mu}{\\sigma}} \\varphi(k_1)\\cdots \\varphi(k_n) \\,, \\\\\n", "&= e^{-i\\omega \\sqrt{n}\\frac{\\mu}{\\sigma}} \\varphi\\left(\\frac{\\omega}{\\sqrt{n} \\sigma}\\right)^n \\,, \\\\\n", "&= e^{-i\\omega \\sqrt{n}\\frac{\\mu}{\\sigma}} \\left(\\varphi(0) + \\varphi'(0) \\frac{\\omega}{\\sqrt{n} \\sigma}+ \\frac12 \\varphi''(0) \\frac{\\omega^2}{n \\sigma^2} + O(n^{-3/2})\\right)^n \\,.\n", "\\end{align}\n", "$$\n", "Since \n", "$$\n", "\\begin{align}\n", "\\varphi(0) &=\\int dx p(x)=1 \\,, \\\\\n", "\\varphi'(0) &=i\\int dx x p(x)=i\\mu \\,, \\\\\n", "\\varphi''(0) &= -\\int dx x^2 p(x)=-(\\sigma^2 + \\mu^2)\\,,\n", "\\end{align}\n", "$$\n", "and\n", "$$\n", "\\begin{align}\n", "\\log\\left(1 + \\varepsilon a_1 + \\varepsilon^2 a_2 + O(\\varepsilon^3)\\right)\n", "&=a_1 \\varepsilon + \\left(a_2 - \\frac12 a_1^2\\right) \\varepsilon^2 + O(\\varepsilon^3)\n", "\\end{align}\n", "$$\n", "we have\n", "$$\n", "\\begin{align}\n", "\\varphi_z(\\omega) &= e^{-i\\omega \\sqrt{n}\\frac{\\mu}{\\sigma}} \\left(1 + \\varphi'(0) \\frac{\\omega}{\\sqrt{n}\\sigma}+\\frac12 \\varphi''(0) \\frac{\\omega^2}{n \\sigma^2}+ O\\left(\\frac1{n^{3/2}}\\right)\\right)^n \\\\\n", "&= e^{-i\\omega \\sqrt{n}\\frac{\\mu}{\\sigma}} e^{ i\\mu \\sqrt{n} \\frac{\\omega}{\\sigma}- \\frac12 \\sigma^2 \\frac{\\omega^2}{\\sigma^2}} + O(n^{-1/2}) \\\\\n", "&= e^{- \\frac12 \\omega^2} + O\\left(\\frac1{\\sqrt{n}}\\right) \\,.\n", "\\end{align}\n", "$$\n", "A simple Gaussian integral then yields\n", "$$\n", "\\begin{align}\n", "p_z(z) = \\frac1{\\sqrt{2\\pi}} e^{-\\frac12 z^2} + O\\left(\\frac1{\\sqrt{n}}\\right) \\,.\n", "\\end{align}\n", "$$\n", "\n", "We have just proved the **central limit theorem** that states that the sum or average of independent identically distributed random variables tends towards a normal distribution in the limit of many terms in the sum. In the case of an average, the mean of the average is the mean of the original random variables, while the variance is reduced by $1/n$." ] }, { "cell_type": "markdown", "id": "e66c481c", "metadata": {}, "source": [ "## Estimators and bias\n", "\n", "Consider sampling $n$ variables $x_1,\\ldots,x_n \\in \\mathbb{R}$ from a PDF $p(x)$ and the samples to be independent of each other. Then we call\n", "$$\n", "\\begin{align}\n", "\\hat{f}(x_1,\\ldots,x_n)\n", "\\end{align}\n", "$$\n", "an **estimator**\n", "of a functional\n", "$$\n", "\\begin{align}\n", "f[p]\n", "\\end{align}\n", "$$\n", "if\n", "$$\n", "\\begin{align}\n", "\\lim_{n\\to\\infty} \\hat{f}(x_1,\\ldots,x_n) = f[p] \\,.\n", "\\end{align}\n", "$$\n", "\n", "We call such an estimator **unbiased** if also\n", "$$\n", "\\begin{align}\n", "\\langle \\hat{f}(x_1,\\ldots,x_n) \\rangle = \\int dx_1 \\cdots dx_n p(x_1,\\ldots,x_n) \\hat{f}(x_1,\\ldots,x_n) = f[p]\n", "\\end{align}\n", "$$ \n", "and **biased** if this property does not hold. An estimator is therefore unbiased if repeatedly sampling a small number of variables $n$ and averaging over the result of each gives the correct result.\n", "\n", "Let us first consider a simple unbiased estimator for the mean of a distribution,\n", "$$\n", "\\begin{align}\n", "\\hat{\\mu}(x_1,\\ldots,x_n) = \\frac1n \\sum_{i=1}^n x_i\n", "\\end{align}\n", "$$\n", "for which we already showed above that\n", "$$\n", "\\begin{align}\n", "\\langle \\hat{\\mu}(x_1,\\ldots,x_n) \\rangle &= \\int dx_1 \\cdots dx_n p(x_1,\\ldots,x_n)\\hat{\\mu}(x_1,\\ldots,x_n) \\\\\n", "&=\\frac1n\\sum_{i=1}^n \\int dx_1 \\cdots dx_n p(x_1) \\cdots p(x_n) x_i \\\\\n", "&=\\frac1n\\sum_{i=1}^n \\int dx_i p(x_i) x_i \\\\\n", "&=\\frac1n\\sum_{i=1}^n \\langle x \\rangle = \\langle x \\rangle \\,.\n", "\\end{align}\n", "$$\n", "\n", "Interestingly, the estimator of the variance\n", "$$\n", "\\begin{align}\n", "\\hat{\\sigma}^2_b(x_1,\\ldots,x_n) = \\frac1n \\sum_{i=1}^n \\left( x_i - \\hat{\\mu}(x_1,\\ldots,x_n) \\right)^2\n", "\\end{align}\n", "$$\n", "is biased, since\n", "$$\n", "\\begin{align}\n", "\\langle \\hat{\\sigma}^2_b(x_1,\\ldots,x_n)\\rangle &= \n", "\\frac1n \\sum_{i=1}^n \\langle x^2_i \\rangle - \\frac1{n^2} \\sum_{i,j=1}^n\\langle x_i x_j \\rangle \\\\\n", "&= \n", "\\left(\\frac1n - \\frac1{n^2}\\right) \\sum_{i=1}^n \\langle x^2_i \\rangle - \\frac1{n^2} \\sum_{i \\neq j}\\langle x_i \\rangle \\langle x_j \\rangle \\\\\n", "&=\n", "\\frac{n-1}{n} \\langle x^2 \\rangle - \\frac{n(n-1)}{n^2} \\langle x \\rangle^2 \\\\\n", "&=\n", "\\frac{n-1}{n} (\\langle x^2 \\rangle - \\langle x \\rangle^2 )\n", "\\end{align}\n", "$$\n", "which differs from the variance\n", "$$\n", "\\begin{align}\n", "\\sigma^2 = \\langle x^2 \\rangle - \\langle x \\rangle^2 \\,.\n", "\\end{align}\n", "$$\n", "\n", "The result, however, already tells us that an unbiased estimator can be defined by\n", "$$\n", "\\begin{align}\n", "\\hat{\\sigma}^2(x_1,\\ldots,x_n) &= \\frac{n}{n-1}\\hat{\\sigma}^2_b(x_1,\\ldots,x_n) \\\\\n", "&=\\frac1{n-1} \\sum_{i=1}^n \\left( x_i - \\hat{\\mu}(x_1,\\ldots,x_n) \\right)^2 \\,.\n", "\\end{align}\n", "$$\n", "\n", "Let us demonstrate this numerically:" ] }, { "cell_type": "code", "execution_count": 3, "id": "5704073b", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Biased estimator test for n=4: 1 == 0.7495601529974043 +- 0.006091813806822378\n", "Biased estimator test for n=8: 1 == 0.867362638559428 +- 0.004629796124124494\n", "Biased estimator test for n=16: 1 == 0.9411682174016318 +- 0.0034581022231148107\n", "Biased estimator test for n=32: 1 == 0.9674490769406684 +- 0.002452884914118586\n", "Biased estimator test for n=64: 1 == 0.9804292666298523 +- 0.0017597170299214334\n", "\n", "Unbiased estimator test for n=4: 1 == 1.0102138289446236 +- 0.008225561100470576\n", "Unbiased estimator test for n=8: 1 == 0.9942457249248356 +- 0.005346418402603227\n", "Unbiased estimator test for n=16: 1 == 0.9967384398055572 +- 0.003638493154759803\n", "Unbiased estimator test for n=32: 1 == 0.998204808485841 +- 0.002571835179560784\n", "Unbiased estimator test for n=64: 1 == 0.9971881428525637 +- 0.0017733381431177173\n" ] } ], "source": [ "random.seed(13)\n", "\n", "def estimate_variance_biased(x):\n", " mean = sum(x) / len(x)\n", " return sum([ (xi - mean) ** 2.0 for xi in x]) / len(x)\n", "\n", "def estimate_variance_unbiased(x):\n", " mean = sum(x) / len(x)\n", " return sum([ (xi - mean) ** 2.0 for xi in x]) / (len(x) - 1)\n", "\n", "def test_variance_estimator(est, n, N = 10000):\n", " sigma_expected = 1.0\n", " E = [est([random.gauss(mu=1.0, sigma=1.0)/sigma_expected**2. for i in range(n)]) for j in range(N)]\n", " val = np.mean(E)\n", " err = np.std(E) / N**0.5\n", " return f\"{val} +- {err}\"\n", "\n", "for n in [4,8,16,32,64]:\n", " print(f\"Biased estimator test for n={n}: 1 ==\", test_variance_estimator(estimate_variance_biased, n))\n", " \n", "print()\n", "\n", "for n in [4,8,16,32,64]:\n", " print(f\"Unbiased estimator test for n={n}: 1 ==\", test_variance_estimator(estimate_variance_unbiased, n))" ] }, { "cell_type": "markdown", "id": "81a7b859", "metadata": {}, "source": [ "## Functions of averages, jackknife and bootstrap variance estimators" ] }, { "cell_type": "markdown", "id": "2ac1c40a", "metadata": {}, "source": [ "Consider sampling $n$ variables $x_1,\\ldots,x_n \\in \\mathbb{R}^m$ from a PDF $p(x)$ and the samples to be independent of each other. We define the unbiased estimator\n", "$$\n", "\\begin{align}\n", "\\hat{x} = \\frac1n \\sum_{i=1}^n x_i\n", "\\end{align}\n", "$$\n", "of\n", "$$\n", "\\begin{align}\n", "\\overline{x} = \\langle x \\rangle \\,.\n", "\\end{align}\n", "$$\n", "\n", "Then a function $f:\\mathbb{R}^m \\to \\mathbb{R}$\n", "of the mean\n", "$$\n", "\\begin{align}\n", "f(\\overline{x})\n", "\\end{align}\n", "$$\n", "can be estimated by\n", "$$\n", "\\begin{align}\n", "f(\\hat{x}) \\,.\n", "\\end{align}\n", "$$\n", "If the function is linear, the estimator is also unbiased.\n", "\n", "Next, we can relate the variance of the estimator\n", "$$\n", "\\begin{align}\n", "\\textrm{Var}(f,x) = \\langle f(\\hat{x})^2 \\rangle - \\langle f(\\hat{x}) \\rangle^2\n", "\\end{align}\n", "$$\n", "to the variance of $\\hat{x}$ through\n", "$$\n", "\\begin{align}\n", "\\textrm{Var}(f,x) &= \\langle f(\\hat{x})^2 \\rangle - \\langle f(\\hat{x}) \\rangle^2 \\\\\n", "&=\\langle f(\\overline{x} + \\hat{x}-\\overline{x})^2 \\rangle - \\langle f(\\overline{x} + \\hat{x}-\\overline{x}) \\rangle^2 \\\\\n", "&=\n", "\\langle \\left(f(\\overline{x}) + \\frac{\\partial f}{\\partial x_i}(\\overline{x})(\\hat{x}-\\overline{x})_i + \\frac12 \\frac{\\partial^2 f}{\\partial x_i \\partial x_j}(\\overline{x})(\\hat{x}-\\overline{x})_i (\\hat{x}-\\overline{x})_j\\right)^2 \\rangle \\notag\\\\\n", "&\\quad - \\langle f(\\overline{x}) + \\frac{\\partial f}{\\partial x_i}(\\overline{x})(\\hat{x}-\\overline{x})_i + \\frac12 \\frac{\\partial^2 f}{\\partial x_i \\partial x_j}(\\overline{x})(\\hat{x}-\\overline{x})_i(\\hat{x}-\\overline{x})_j\\rangle^2 + \\langle O(\\hat{x}-\\overline{x})^3 \\rangle \\\\\n", "&=\n", "(f(\\overline{x}) \\frac{\\partial^2 f}{\\partial x_i \\partial x_j}(\\overline{x}) + \\frac{\\partial f}{\\partial x_i}\\frac{\\partial f}{\\partial x_j}(\\overline{x}))\\langle (\\hat{x}-\\overline{x})_i(\\hat{x}-\\overline{x})_j \\rangle \\notag\\\\\n", "&\\quad - f(\\overline{x}) \\frac{\\partial^2 f}{\\partial x_i \\partial x_j}(\\overline{x})\\langle (\\hat{x}-\\overline{x})_i(\\hat{x}-\\overline{x})_j \\rangle+ \\langle O(\\hat{x}-\\overline{x})^3 \\rangle \\\\\n", "&=\n", " \\frac{\\partial f}{\\partial x_i}(\\overline{x})\\frac{\\partial f}{\\partial x_j}(\\overline{x})\\langle (\\hat{x}-\\overline{x})_i(\\hat{x}-\\overline{x})_j \\rangle + \\langle O(\\hat{x}-\\overline{x})^3 \\rangle\n", "\\end{align}\n", "$$\n", "\n", "We have established above that due to the averaging, $\\hat{x}-\\overline{x} \\propto 1/\\sqrt{n}$. " ] }, { "cell_type": "markdown", "id": "a6e480dd", "metadata": {}, "source": [ "We now first define the **single-elimination Jackknife** resamples\n", "$$\n", "\\begin{align}\n", "\\hat{x}_j = \\frac1{n-1} \\sum_{i=1,i\\neq j}^n x_i=\\frac1{n-1}(n \\hat{x} - x_j)=\\frac1{n-1}((n-1) \\hat{x} + \\hat{x} - x_j) = \\hat{x} + \\frac{1}{n-1} (\\hat{x} - x_j)\n", "\\end{align}\n", "$$\n", "with which we can define the **single-elimination Jacknife variance estimator**\n", "$$\n", "\\begin{align}\n", "\\textrm{Var}^J(f,x) &= \\frac{n-1}{n} \\sum_{j=1}^n (f(\\hat{x}_j) - f(\\hat{x}))^2 \\\\\n", "&=\\frac{n-1}{n} \\sum_{j=1}^n \\left(\\frac{\\partial f}{\\partial x_i}(\\hat{x}) (\\hat{x}_j - \\hat{x})_i\\right)^2 (1 + O(1/\\sqrt{n})) \\\\\n", "&=\\frac{\\partial f}{\\partial x_i}(\\hat{x}) \\frac{\\partial f}{\\partial x_l}(\\hat{x}) \\frac{1}{n(n-1)} \\sum_{j=1}^n (x_j-\\hat{x})_i (x_j-\\hat{x})_l (1 + O(1/\\sqrt{n})) \\,.\n", "\\end{align}\n", "$$\n", "We can then show that\n", "$$\n", "\\begin{align}\n", "\\langle \\textrm{Var}^J(f,x) \\rangle &=\\frac{\\partial f}{\\partial x_i}(\\overline{x}) \\frac{\\partial f}{\\partial x_l}(\\overline{x}) \\frac{1}{n} \\langle \\frac1{n-1}\\sum_{j=1}^n (x_j-\\hat{x})_i (x_j-\\hat{x})_l \\rangle (1 + O(1/\\sqrt{n})) \\\\\n", "&=\\frac{\\partial f}{\\partial x_i}(\\overline{x}) \\frac{\\partial f}{\\partial x_l}(\\overline{x}) \\langle \\frac1{n-1} (x-\\hat{x})_i (x-\\hat{x})_l \\rangle (1 + O(1/\\sqrt{n}))\n", "\\end{align}\n", "$$\n", "since all $x_j$ are distributed following $p(x_j)$ and therefore $ \\langle (x_j-\\hat{x})_i (x_j-\\hat{x})_l \\rangle = \\langle (x-\\hat{x})_i (x-\\hat{x})_l \\rangle$.\n", "Analog to the proof of the biased estimator of $\\sigma^2$ above, we can show that $\\langle (x-\\hat{x})_i (x-\\hat{x})_l \\rangle = \\frac{n-1}{n} \\langle (x-\\overline{x})_i (x-\\overline{x})_l \\rangle$ and therefore\n", "$$\n", "\\begin{align}\n", "\\langle \\textrm{Var}^J(f,x) \\rangle &=\\frac{\\partial f}{\\partial x_i}(\\overline{x}) \\frac{\\partial f}{\\partial x_l}(\\overline{x}) \\frac{1}{n} \\langle (x - \\overline{x})_i(x-\\overline{x})_l \\rangle (1 + O(1/\\sqrt{n})) \\\\\n", "&=\\frac{\\partial f}{\\partial x_i}(\\overline{x}) \\frac{\\partial f}{\\partial x_l}(\\overline{x}) \\langle (\\hat{x} - \\overline{x})_i(\\hat{x}-\\overline{x})_l \\rangle (1 + O(1/\\sqrt{n})) \\\\\n", "&=\\textrm{Var}(f,x) (1 + O(1/\\sqrt{n}))\n", "\\end{align}\n", "$$\n", "since\n", "$$\n", "\\begin{align}\n", "\\frac{\\partial f}{\\partial x_i}(\\hat{x}) = \\frac{\\partial f}{\\partial x_i}(\\overline{x})(1+O(1/\\sqrt{n}) \\,.\n", "\\end{align}\n", "$$\n", "\n", "***Homework***: Show that this estimator is unbiased for linear $f$.\n", "\n", "An alternative very popular resampling technique is the **bootstrap** method. In this method, we\n", "create $S \\in \\mathbb{N}^+$ pseudosamples by randomly selecting $n$ elements from the set of $\\{x_1,\\ldots,x_n\\}$\n", "with replacement. The variance over these pseudosamples then estimates the distribution variance. We illustrate both the jackknife and bootstrap methods below numerically." ] }, { "cell_type": "code", "execution_count": 4, "id": "05e8f6ad", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "0.9997201133255009\n", "0.9943462780649956 0.06887494518918237\n" ] } ], "source": [ "def jackknife_variance_estimator(f, x):\n", " N = len(x)\n", " X = sum(x) / N\n", " mean = f(X)\n", " return sum([(f((N*X - x[j])/(N-1)) - mean)**2 for j in range(N)])*(N-1)/N\n", "\n", "res = [np.random.normal(2.0,0.5,2) for i in range(100)]\n", "\n", "sigma_expected = 0.5 / 2.0 * 2.0**0.5 / 100**0.5\n", "\n", "print(jackknife_variance_estimator(lambda x: x[0]/x[1], res)**0.5 / sigma_expected)\n", "\n", "\n", "tests_jk = [jackknife_variance_estimator(lambda x: x[0]/x[1], \n", " [np.random.normal(2.0,0.5,2) for i in range(100)])**0.5 / sigma_expected \n", " for l in range(100)]\n", "\n", "print(np.mean(tests_jk), np.std(tests_jk))" ] }, { "cell_type": "code", "execution_count": 5, "id": "7e30c8b7", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "1.103734908360207\n", "0.9880356393606512 0.11291216607476749\n" ] } ], "source": [ "def bootstrap_variance_estimator(f, x, S):\n", " N = len(x)\n", " X = sum(x) / N\n", " mean = f(X)\n", " return sum([ (f(sum([ random.choice(x) for l in range(N) ])/N) - mean)**2 for j in range(S) ])/S\n", "\n", "print(bootstrap_variance_estimator(lambda x: x[0]/x[1], res, 100)**0.5 / sigma_expected)\n", "\n", "tests_bs = [bootstrap_variance_estimator(lambda x: x[0]/x[1], \n", " [np.random.normal(2.0,0.5,2) for i in range(100)], 100)**0.5 / sigma_expected \n", " for l in range(100)]\n", "\n", "print(np.mean(tests_bs), np.std(tests_bs))" ] }, { "cell_type": "markdown", "id": "62f8b0da", "metadata": {}, "source": [ "***Homework***: repeat the calculation of the $E_1-E_0$ energy splitting of the Harmonic oscillator using the jackknife estimator to\n", " assess the statistical uncertainty of the extracted value." ] }, { "cell_type": "markdown", "id": "0efbc104", "metadata": {}, "source": [ "## Autocorrelated data and binning\n", "\n", "So far, we have assumed all data to be statistically independent. We have seen, however, in the last chapter that Markov chains typically yield autocorrelated data. A simple procedure to address this issue is the following.\n", "\n", "Consider a markov chain with elements $x_1,\\ldots,x_n \\in \\mathbb{R}^m$. Assuming a number $b$ that divides $n$, we could then average $b$ sequential elements to a block\n", "$$\n", "\\begin{align}\n", "\\tilde{x}_i = \\frac1b \\sum_{j=1}^b x_{ib + j} \\,.\n", "\\end{align}\n", "$$\n", "\n", "If we then define $\\tilde{n} = n / b$ and \n", "$$\n", "\\begin{align}\n", "\\hat{\\tilde{x}} = \\frac1{\\tilde{n}} \\sum_{j=1}^{\\tilde{n}} \\tilde{x}_j\n", "\\end{align}\n", "$$\n", "then\n", "$$\n", "\\begin{align}\n", "\\langle \\hat{\\tilde{x}} \\rangle = \\langle \\hat{x} \\rangle\n", "\\end{align}\n", "$$\n", "and\n", "$$\n", "\\begin{align}\n", "\\langle \\hat{\\tilde{x}}^2 \\rangle = \\langle \\hat{x}^2 \\rangle \\,.\n", "\\end{align}\n", "$$\n", "\n", "So binning does not affect the mean and variance but the binned results are generally less autocorrelated. Therefore after binning sufficiently, the data can be treated as if it were independent. If after binning too little data remains, the errors on the variance estimators, however, can grow significantly (see $1/\\sqrt{n}$ terms above, $n$ reduces by the amount of binning).\n", "\n", "Let us demonstrate this:" ] }, { "cell_type": "code", "execution_count": 6, "id": "0ab43a2d", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Acceptance rate: 0.62902\n", "1.0007439855609455\n", "1.000743985560925\n" ] }, { "data": { "image/png": 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h3/3QDfTSrNONXGOa5e1lnb5J+w8uJPXW5CmVutZhmA9ioJdmlndQrT1DsSgGemncQaV1y0CXpEIY6JJUCANdkgphoEtSIQx0SSqEgS5JhWjsl6IRsQx8fsynb6Zzz/U2srbxWNt4rG08s1zbszJzrteIxgK9johYWu2nr02ztvFY23isbTyl1uYpF0kqhIEuSYWY1UC/pekC+rC28VjbeKxtPEXWNpPn0CVJF5rVI3RJ0goGuiQVYuYCPSJ2R8SDEXEyIg40XU+3iHgkIj4TEfdERKP/jiki3h0RZyPieNewZ0TERyPis9Xf72tRbYsRcaZqu3siYk9DtW2LiCMRcSIi7o+IX6+GN952fWprvO0i4mkR8emIuLeq7WA1/PKI+FS1v34gIja1qLZbI+JzXe32vLWuravGDRFxd0R8uOofr91W++/RbXwAG4CHgSuATcC9wM6m6+qq7xFgc9N1VLW8FLgSON417C3Agar7APDmFtW2CPxmC9rtUuDKqvti4CFgZxvark9tjbcdEMDTq+6LgE8BLwRuB/ZVw98B/EqLarsVeGXT21xV1+uB9wEfrvrHardZO0K/CjiZmacy83HgNuDahmtqpcz8JPCVFYOvBd5Tdb8H+Om1rOmcVWprhcx8LDP/rer+b+ABYAstaLs+tTUuO/6n6r2oeiTwMuBD1fCm2m212lohIrYCPwm8q+oPxmy3WQv0LcCjXf2nackGXUngnyLiWETsb7qYHp6ZmY9V3f8JPLPJYnq4PiLuq07JNHI6qFtEbAeeT+eIrlVtt6I2aEHbVacN7gHOAh+l82n6a5n5RDVJY/vrytoy81y7valqtz+MiKc2URvwNuC3gW9X/d/PmO02a4Hedi/JzCuBa4DXRcRLmy5oNdn5LNeaoxTg7cAPAs8DHgP+oMliIuLpwN8Av5GZ/9U9rum261FbK9ouM/8vM58HbKXzafqHmqijl5W1RcRzgBvo1PhjwDOA31nruiLip4CzmXlsEvObtUA/A2zr6t9aDWuFzDxT/T0L/B2djbpNvhgRlwJUf882XM93ZOYXq53u28A7abDtIuIiOoH515n5t9XgVrRdr9ra1HZVPV8DjgAvAi6JiI3VqMb3167adlensDIzvwn8Bc2024uBvRHxCJ1TyC8D/ogx223WAv0osKO6ArwJ2AccargmACLiuyPi4nPdwE8Ax/s/a80dAl5ddb8a+IcGa3mSc2FZ+Rkaarvq/OWfAw9k5lu7RjXedqvV1oa2i4i5iLik6v4u4BV0zvEfAV5ZTdZUu/Wq7d+73qCDzjnqNW+3zLwhM7dm5nY6efaJzPx5xm23pq/ujnE1eA+dq/sPA29sup6uuq6g862be4H7m64NeD+dj9/fonMO7jV0zs19HPgs8DHgGS2q7b3AZ4D76ITnpQ3V9hI6p1PuA+6pHnva0HZ9amu87YDnAndXNRwHbqyGXwF8GjgJfBB4aotq+0TVbseBv6L6JkxTD2AX57/lMla7+dN/SSrErJ1ykSStwkCXpEIY6JJUCANdkgphoEtSIQx0SSqEgS5Jhfh/bnrBQH5n5PkAAAAASUVORK5CYII=\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "1.0007439855609455 +- 0.009350224119331632\n" ] } ], "source": [ "def metropolis_sample_xsqr(p, step, samples):\n", " x0 = 0.0\n", " res = []\n", " accept = 0\n", " for i in range(samples):\n", " x = x0 + step * random.uniform(-1.0, 1.0)\n", " r = p(x) / p(x0)\n", " l = random.uniform(0.0, 1.0)\n", " if l < r:\n", " accept += 1\n", " x0 = x\n", " res.append(x0**2)\n", " print(f\"Acceptance rate: {accept/samples}\")\n", " return res\n", "\n", "def bin_data(res, bin_size):\n", " assert len(res) % bin_size == 0\n", " n = len(res) // bin_size\n", " return [ sum(res[bin_size*i:bin_size*(i+1)])/bin_size for i in range(n) ]\n", "\n", "def avg(X):\n", " return sum(X) / len(X)\n", "\n", "res = metropolis_sample_xsqr(lambda x: np.exp(-x**2./2), 2.0, 100000)[100:]\n", "print(avg(res))\n", "\n", "res_binned = bin_data(res, 25)\n", "\n", "print(avg(res_binned))\n", "\n", "def autocovariance(Y, max_dt):\n", " mean = sum(Y)/len(Y)\n", " return [sum([ Y[t]*Y[t+dt] for t in range(len(Y) - dt) ])/(len(Y) - dt) - mean**2. for dt in range(max_dt) ]\n", "\n", "def autocorrelator(Y, max_dt):\n", " Cov = autocovariance(Y, max_dt)\n", " return [ Cov[t] / Cov[0] for t in range(len(Cov))]\n", "\n", "\n", "fig, ax = plt.subplots()\n", "\n", "plt.ylim([-0.1,1.1])\n", "ax.plot(range(40), autocorrelator(res, 40), marker='+', ls='', c='blue', label=\"autocorrelation original data\")\n", "ax.plot(range(40), autocorrelator(res_binned, 40), marker='+', ls='', c='red', label=\"autocorrelation bin-25 data\")\n", "plt.show()\n", "\n", "bin_size = [1, 5, 10, 20, 25, 50]\n", "error_versus_bin_size = [jackknife_variance_estimator(lambda x: x, bin_data(res, bs))**0.5 for bs in bin_size]\n", "fig, ax = plt.subplots()\n", "plt.ylim(0,0.02)\n", "ax.plot(bin_size, error_versus_bin_size, marker='+', ls='', c='red', label=\"autocorrelation bin-25 data\")\n", "plt.show()\n", "\n", "print(sum(res) / len(res),\"+-\",error_versus_bin_size[-2])" ] }, { "cell_type": "markdown", "id": "d584ac0f", "metadata": {}, "source": [ "## Combining statistically independent data\n", "\n", "Let $x_1,\\ldots,x_n$ be independent random variables, i.e., $p(x_1,\\ldots,x_n)=p(x_1)\\cdots p(x_n)$, then\n", "it can be shown in a similar way to the derivations above that the total variance of a function $f(x_1,\\ldots,x_n)$\n", "is the sum of the individual variances with respect to $x_i$ assuming that the fluctuations are sufficiently small that a linearization of $f$ is appropriate." ] }, { "cell_type": "markdown", "id": "e56da5db", "metadata": {}, "source": [ "## Bayes' theorem and finding a model that best describes a given data set\n", "\n", "We can write the joint probability of selecting a model $M$ and sampling a data set $D$ as\n", "$$\n", "\\begin{align}\n", "P(D \\cap M) = P(D\\vert M)P(M) = P(M\\vert D)P(D)\n", "\\end{align}\n", "$$\n", "from which Bayes' theorem\n", "$$\n", "\\begin{align}\n", "P(M\\vert D) = \\frac{P(D\\vert M)P(M)}{P(D)}\n", "\\end{align}\n", "$$\n", "follows. If we have no prior knowledge about which model is best, one typically uses a $P(M)$, which is independent of $M$. If prior knowledge was available, it can be included in a fitting procedure by appropriate choice of $P(M)$. Since also $P(D)$ is independent of $M$, without prior knowledge, the model which maximises $P(M\\vert D)$ also maximises $P(D\\vert M)$. Based on our earlier discussion of the central limit theorem, it is further reasonable to assume that $P(D\\vert M)$ follows a normal distribution, which for a model with parameter vector $p \\in \\mathbb{R}^{N_\\textrm{parameter}}$\n", "can be written as\n", "$$\n", "\\begin{align}\n", "P(D \\vert M) = (2\\pi)^{-N_\\textrm{data}/2} \\det(C)^{-1/2} e^{-\\frac12 (D_i - M_i(p)) C^{-1}_{ij} (D_j - M_j(p))} \\,,\n", "\\end{align}\n", "$$\n", "where $D \\in \\mathbb{R}^{N_\\textrm{data}}$ and $M:\\mathbb{R}^{N_\\textrm{parameter}} \\to \\mathbb{R}^{N_\\textrm{data}}$\n", "are the model predictions for the data. This is well defined for positive definite $C$. Then\n", "$$\n", "\\begin{align}\n", "\\langle D_i \\rangle = M_i(p)\n", "\\end{align}\n", "$$\n", "and $C \\in \\mathbb{R}^{N_\\textrm{data} \\times N_\\textrm{data}}$ with\n", "$$\n", "\\begin{align}\n", "\\langle (D_i - \\langle D_i \\rangle)(D_j - \\langle D_j \\rangle) \\rangle = C_{ij} \\,.\n", "\\end{align}\n", "$$\n", "\n", "If the individual data points are statistically independent, $C$ must be a diagonal matrix. In our example below, this will be the case. Since $C$ is symmetric, we can diagonalize it which yields new $N_\\textrm{data}$ statistically independent normal variables.\n", "\n", "The parameters that maximize $P(D \\vert M)$ are the same as the ones that minimize the variable\n", "$$\n", "\\begin{align}\n", "\\chi^2 = (D_i - M_i(p)) C^{-1}_{ij} (D_j - M_j(p)) \\,.\n", "\\end{align}\n", "$$\n", "We call the parameters that minimize $\\chi^2$ the fit result $p_0$ and\n", "the value of $\\chi^2$ at the minimum $\\chi^2_0$.\n", "\n", "It can be shown that $\\chi^2_0$ follows the PDF of a so-called $\\chi^2$-distribution with degrees of freedom $k = N_\\textrm{data} - N_\\textrm{parameter}$,\n", "$$\n", "\\begin{align}\n", "p(\\chi_0^2) = \\frac1{2^{k/2} \\Gamma(k/2)} (\\chi_0^2)^{k/2-1} e^{-\\chi_0^2/2}\n", "\\end{align}\n", "$$\n", "with mean $k$ and variance $2k$.\n", "\n", "The expectation value of $\\chi_0^2$ is therefore $k$ with fluctuations of size $\\sqrt{2k}$. It is common\n", "to quantify the likelyhood of the fit by considering the $P$-value defined as the likelyhood that the observed $\\chi_0^2$ or a larger value would be observed, i.e.,\n", "$$\n", "\\begin{align}\n", "P = \\int_{\\chi_0^2}^\\infty dx p(x) \\,.\n", "\\end{align}\n", "$$\n", "\n", "We now turn to a numerical demonstration based on data sampled from normal distribution with variance $f(a)$\n", "for different parameters $a$. We may think of the parameter $a$ as the lattice spacing of the path integral and $f(a)$ as the expected functional form of the continuum ($a\\to0$) limit." ] }, { "cell_type": "code", "execution_count": 7, "id": "89cfeee1", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Acceptance rate: 0.6392\n", "Acceptance rate: 0.6528\n", "Acceptance rate: 0.6646\n", "Acceptance rate: 0.6766\n" ] }, { "data": { "image/png": "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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "class xsqr_data:\n", " def __init__(self, lattice_spacings, f_true, nbin):\n", " self.lattice_spacings = lattice_spacings\n", " self.res = [bin_data(metropolis_sample_xsqr(\n", " lambda x: np.exp(-x**2./2.0/f_true(a)), 2.0, 10000)[100:],nbin) \n", " for a in lattice_spacings]\n", " self.xsqr_mean = [avg(x) for x in self.res]\n", " self.xsqr_err = [jackknife_variance_estimator(lambda x: x, X)**0.5 for X in self.res]\n", " \n", " def plot(self):\n", " fig, ax = plt.subplots()\n", " plt.xlim(0,self.lattice_spacings[-1]**2.*1.1)\n", " ax.errorbar([ a**2 for a in self.lattice_spacings], self.xsqr_mean, self.xsqr_err, \n", " marker='+', ls='', c='red')\n", " plt.xlabel(\"a^2\")\n", " plt.ylabel(\"\")\n", " plt.show()\n", " \n", "random.seed(13)\n", "d = xsqr_data([ 0.3, 0.5, 0.7, 0.9 ], lambda a: 1.0 + a**2*0.5, 25)\n", "d.plot()" ] }, { "cell_type": "code", "execution_count": 8, "id": "1c9e0ccd", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Covariance from a = 0.3 is [[ 0.00063662 -0.00106751]\n", " [-0.00106751 0.00179003]]\n", "Covariance from a = 0.5 is [[ 0.00021218 -0.00021919]\n", " [-0.00021919 0.00022642]]\n", "Covariance from a = 0.7 is [[5.32419334e-06 4.55058362e-05]\n", " [4.55058362e-05 3.88938018e-04]]\n", "Covariance from a = 0.9 is [[ 0.00012678 -0.00059282]\n", " [-0.00059282 0.00277204]]\n", "parameter[0] = 0.9725340439828718 +- 0.0313194600597356\n", "parameter[1] = 0.5325144648541262 +- 0.07195438673559529\n", "chi2 / dof = 0.645814241976392 / 2, i.e., p = 0.7240410994668618\n" ] } ], "source": [ "import scipy.optimize as opt\n", "from scipy.integrate import quad\n", "from scipy.special import gamma\n", "\n", "def jackknife_covariance_estimator(f, x):\n", " N = len(x)\n", " X = sum(x) / N\n", " mean = f(X)\n", " def outer_sqr(a):\n", " return np.outer(a,a)\n", " return sum([outer_sqr(f((N*X - x[j])/(N-1)) - mean) for j in range(N)])*(N-1)/N\n", "\n", "def get_p_value(chi2, dof):\n", " return quad(lambda x: 2**(-dof/2)/gamma(dof/2)*x**(dof/2-1)*np.exp(-x/2), chi2, np.inf)[0]\n", "\n", "class model_two_parameter:\n", " def __init__(self, power):\n", " self.power = power\n", " \n", " def __call__(self, a, p):\n", " return p[0] + p[1] * a**self.power\n", " \n", " def parameter_gradient(self, a, p):\n", " return np.array([1.0,a**self.power])\n", " \n", " def reasonable_parameter_values(self):\n", " return [0.8, 0.7]\n", " \n", "class fit:\n", " def __init__(self, data, lattice_spacings_to_fit, model):\n", " self.data = data\n", " self.lattice_spacings_to_fit = lattice_spacings_to_fit\n", " self.model = model\n", " \n", " def chi_square(self, data, parameter):\n", " return sum([ ((self.model(x, parameter) - y) / err)**2. for x, y, err in data ]) \n", "\n", " def most_likely_parameter(self, f, data):\n", " opt_res = opt.minimize(lambda p: self.chi_square(data, p), self.model.reasonable_parameter_values(),\n", " method = \"Nelder-Mead\", tol = 1e-7)\n", " assert opt_res.success == True\n", " return opt_res.x, opt_res.fun\n", " \n", " def select_lattices(self, data):\n", " return [data[i] for i in self.lattice_spacings_to_fit]\n", " \n", " def run(self):\n", " mean, chi2 = self.most_likely_parameter(\n", " self.model, self.select_lattices(\n", " list(zip(self.data.lattice_spacings, self.data.xsqr_mean, self.data.xsqr_err))\n", " )\n", " )\n", " \n", " covariances = []\n", " for i in range(len(self.data.res)):\n", " covariances.append(jackknife_covariance_estimator(\n", " lambda resi: self.most_likely_parameter(\n", " self.model, self.select_lattices(\n", " list(zip(self.data.lattice_spacings, \n", " self.data.xsqr_mean[:i] + [resi] + self.data.xsqr_mean[i+1:],\n", " self.data.xsqr_err))\n", " )\n", " )[0],\n", " self.data.res[i]\n", " ))\n", " print(\"Covariance from a =\",self.data.lattice_spacings[i],\"is\",covariances[i])\n", " \n", " self.best_parameter = mean\n", " self.best_parameter_cov = sum(covariances)\n", " self.chi2 = chi2\n", " self.dof = len(self.lattice_spacings_to_fit) - len(mean)\n", " self.p = get_p_value(self.chi2, self.dof)\n", " \n", " for i in range(len(mean)):\n", " print(f\"parameter[{i}] = {self.best_parameter[i]} +- {self.best_parameter_cov[i][i]**0.5}\")\n", " print(f\"chi2 / dof = {self.chi2} / {self.dof}, i.e., p = {self.p}\")\n", "\n", "f = fit(d, [0,1,2,3], model_two_parameter(2))\n", "f.run()" ] }, { "cell_type": "markdown", "id": "f2effb3a", "metadata": {}, "source": [ "The statistical estimates of both parameters agree within errors with the function that was used to generate the data.\n", "\n", "## Plotting fit results\n", "\n", "Once we have found the best fit parameters and have an estimate of their covariance, we can ask the question what the variance of the model prediction is for a given $a$ value. For sufficiently small fluctuations it is straightforward to show that\n", "$$\n", "\\begin{align}\n", "\\langle f(a,p)^2 \\rangle_p - \\langle f(a,p) \\rangle_p^2 &=\n", "\\frac{\\partial f}{\\partial p_i}(a,\\langle p \\rangle) \\frac{\\partial f}{\\partial p_j}(a,\\langle p \\rangle)\n", "\\langle (p - \\langle p \\rangle_p)_i (p - \\langle p \\rangle_p)_j \\rangle \\,.\n", "\\end{align}\n", "$$" ] }, { "cell_type": "code", "execution_count": 9, "id": "411be2c6", "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "def make_plot(data, fit):\n", " \n", " df_dp = fit.model.parameter_gradient\n", " \n", " def err_f(a,p,cov_p):\n", " return np.dot(df_dp(a,p), np.dot(cov_p, df_dp(a,p)))**0.5\n", "\n", " fxrange = np.arange(0.0, data.lattice_spacings[-1]*1.1, 0.01)\n", " fy = np.array([fit.model(asqr**0.5,fit.best_parameter) for asqr in fxrange])\n", " fyerr = np.array([err_f(asqr**0.5,fit.best_parameter,fit.best_parameter_cov) for asqr in fxrange])\n", " fig, ax = plt.subplots()\n", " plt.xlim(0,data.lattice_spacings[-1]**2.*1.1)\n", " ax.fill_between(fxrange,fy-fyerr,fy+fyerr,alpha=0.1,color=\"blue\")\n", " ax.plot(fxrange,fy,c=\"blue\")\n", " plt.xlabel(\"a^2\")\n", " plt.ylabel(\"\")\n", " ax.errorbar([ a**2 for a in data.lattice_spacings], data.xsqr_mean, data.xsqr_err, marker='+', ls='', c='red')\n", " plt.show()\n", " \n", "make_plot(d,f)" ] }, { "cell_type": "markdown", "id": "bcc04971", "metadata": {}, "source": [ "## Systematic uncertainties in fits\n", "\n", "The $P$-value can be used to detect models that are inconsistent with the data. One typically would reject fits with a $P$-value below 0.05 or 0.01. Note, however, that there may be models that are not rejected based on such a criterion which would, however, yield incorrect extrapolation results. It is therefore crucial to understand from first principles, what model is expected to describe the data. Let us, e.g., vary the power parameter of the model:" ] }, { "cell_type": "code", "execution_count": 10, "id": "8276993f", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Covariance from a = 0.3 is [[ 0.0002933 -0.00062207]\n", " [-0.00062207 0.00131937]]\n", "Covariance from a = 0.5 is [[ 0.00018441 -0.00031133]\n", " [-0.00031133 0.00052559]]\n", "Covariance from a = 0.7 is [[5.70044895e-05 6.94709773e-05]\n", " [6.94709773e-05 8.46638004e-05]]\n", "Covariance from a = 0.9 is [[ 2.89871654e-05 -3.63028438e-04]\n", " [-3.63028438e-04 4.54648273e-03]]\n", "parameter[0] = 1.0488704541569347 +- 0.023742388886163212\n", "parameter[1] = 0.5927260642234137 +- 0.08047425253679717\n", "chi2 / dof = 1.1672008310949793 / 2, i.e., p = 0.5578861244091011\n" ] }, { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "Covariance from a = 0.3 is [[ 0.00021279 -0.00051691]\n", " [-0.00051691 0.00125567]]\n", "Covariance from a = 0.5 is [[ 0.00016061 -0.0003554 ]\n", " [-0.0003554 0.00078639]]\n", "Covariance from a = 0.7 is [[ 8.93488411e-05 -1.22643890e-05]\n", " [-1.22643890e-05 1.68346079e-06]]\n", "Covariance from a = 0.9 is [[ 9.76342071e-06 -2.68073675e-04]\n", " [-2.68073675e-04 7.36048333e-03]]\n", "parameter[0] = 1.0768072629936558 +- 0.02173755115720664\n", "parameter[1] = 0.6882784657540444 +- 0.09697536912676169\n", "chi2 / dof = 5.04266567743004 / 2, i.e., p = 0.08035243850549526\n" ] }, { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "f = fit(d, [0,1,2,3], model_two_parameter(4))\n", "f.run()\n", "make_plot(d,f)\n", "\n", "f = fit(d, [0,1,2,3], model_two_parameter(6))\n", "f.run()\n", "make_plot(d,f)" ] }, { "cell_type": "markdown", "id": "f7fd5b55", "metadata": {}, "source": [ "Both models still yielded an acceptable $P$-value, however, the $a\\to 0$ result (paramter[0]) was inconsistent with the expected result of $1$.\n", "\n", "A more realistic case is that we know the asymptotic, say $a\\to0$, behavior but that at finite $a$ there are small deviations from the fit model. Let us investigate this case. In this case, we may devise a systematic error by investigating the effect of the data points that are least expected to follow the model (here the largest values of $a$):" ] }, { "cell_type": "code", "execution_count": 11, "id": "163701db", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Acceptance rate: 0.6363\n", "Acceptance rate: 0.6477\n", "Acceptance rate: 0.6539\n", "Acceptance rate: 0.635\n", "Covariance from a = 0.3 is [[ 0.00056386 -0.00085898]\n", " [-0.00085898 0.00130855]]\n", "Covariance from a = 0.5 is [[ 0.00021469 -0.00024639]\n", " [-0.00024639 0.00028278]]\n", "Covariance from a = 0.7 is [[2.79959999e-05 3.76234555e-05]\n", " [3.76234555e-05 5.05616674e-05]]\n", "Covariance from a = 0.9 is [[ 9.17326352e-05 -4.12311415e-04]\n", " [-4.12311415e-04 1.85321944e-03]]\n", "parameter[0] = 1.0523907312375735 +- 0.029971198544555266\n", "parameter[1] = 0.060524600513689505 +- 0.059119506270746136\n", "chi2 / dof = 5.119393026486666 / 2, i.e., p = 0.07732820496863554\n" ] }, { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "Covariance from a = 0.3 is [[ 0.00105505 -0.00245038]\n", " [-0.00245038 0.00569104]]\n", "Covariance from a = 0.5 is [[ 1.69798888e-04 -1.24159148e-04]\n", " [-1.24159148e-04 9.07867789e-05]]\n", "Covariance from a = 0.7 is [[ 0.00014609 -0.00103002]\n", " [-0.00103002 0.00726229]]\n", "Covariance from a = 0.9 is [[0. 0.]\n", " [0. 0.]]\n", "parameter[0] = 1.0053279073827692 +- 0.0370261851900245\n", "parameter[1] = 0.27205832729781265 +- 0.11421085408373621\n", "chi2 / dof = 0.43340017721134816 / 1, i.e., p = 0.5103252322066846\n" ] }, { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "Systematic error estimate: 0.047062823854804314\n" ] } ], "source": [ "random.seed(13)\n", "\n", "d2 = xsqr_data([ 0.3, 0.5, 0.7, 0.9 ], lambda a: 1.0 + a**2*0.5 - a**4*0.5, 25)\n", "f2_4a = fit(d2, [0,1,2,3], model_two_parameter(2))\n", "f2_4a.run()\n", "make_plot(d2,f2_4a)\n", "\n", "f2_3a = fit(d2, [0,1,2], model_two_parameter(2))\n", "f2_3a.run()\n", "make_plot(d2,f2_3a)\n", "\n", "print(f\"Systematic error estimate: {f2_4a.best_parameter[0] - f2_3a.best_parameter[0]}\")" ] }, { "cell_type": "markdown", "id": "53b03e24", "metadata": {}, "source": [ "After excluding the largest $a$ value, the fit results agree with the expected value and the difference of a fit to three and four lattice spacings can be used as a systematic error estimate.\n", "\n", "## Understanding the exponential noise growth in correlation functions\n", "\n", "Finally, we add a comment to the statistical noise of correlation functions. Consider the two-time correlator of the previous chapter,\n", "$$\n", "\\begin{align}\n", "\\langle x_f x_i \\rangle \\,,\n", "\\end{align}\n", "$$\n", "then its statistical variance is\n", "$$\n", "\\begin{align}\n", "\\langle x_f^2 x_i^2 \\rangle - \\langle x_f x_i \\rangle^2 \\,.\n", "\\end{align}\n", "$$\n", "Since both expectation values can be related to the spectrum and matrix elements of the Hamiltonian $H$, one can estimate the signal to noise of correlation functions in a Euclidean path integral. This is argument is typically credited to Peter Lepage.\n", "\n", "Consider the dominant long-time energy of $\\langle x_f^2 x_i^2 \\rangle$ to be $E_{x^2}$ and of $\\langle x_f x_i \\rangle$ to be $E_{x}$, then the noise to signal behaves as\n", "$$\n", "\\begin{align}\n", "\\frac{\\sigma^2}{\\mu^2} &= \\frac{\\langle x_f^2 x_i^2 \\rangle - \\langle x_f x_i \\rangle^2}{\\langle x_f x_i \\rangle^2} \\\\\n", "&=e^{-(t_f - t_i) (E_{x^2} - 2E_x)} + \\textrm{const} \\,.\n", "\\end{align}\n", "$$\n", "\n", "In the case of the Harmonic oscillator with parameters as in the last chapter, $E_{x^2}=0$ and $E_x=1$, so\n", "the relative noise grows exponentially!" ] }, { "cell_type": "markdown", "id": "520473ee", "metadata": {}, "source": [ "***Homework***: Perform the $E_1 - E_0$ study of the Harmonic oscillator for different lattice spacings $a$ and extrapolate to the continuum limit $a\\to 0$. Provide an estimate of both statistical and systematic uncertainties." ] } ], "metadata": { "kernelspec": { "display_name": "Python 3 (ipykernel)", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.12.0" } }, "nbformat": 4, "nbformat_minor": 5 }