{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "## Estimators\n", "\n", "An **estimator** is a procedure applied to the data sample which gives a numerical value for a property of the parent population, or, as appropriate, a property or parameter of the parent distribution function. \n", "\n", "This is a general definition: The sample you have to study may have been drawn from a large *parent population*, for which you want to know some property. Alternatively, it may have been generated from a distribution function, arising from a basic law being investigated. This distribution function also has parameters and properties, and you may be trying to measure one of these. In many cases a parameter is also a property (e.g. $\\mu$ and $\\sigma$ for a gaussian).\n", "\n", "As an example, let's consider the case of attempting to find the average height of all students in a university on the basis of an (honestly selected) sample of $N$. Some estimators you can use are shown below. All of these _are_ estimators, but some are clearly better than others. We typically classify estimators as \"good\" or \"bad\", rather than \"right\" or \"wrong\"." ] }, { "cell_type": "code", "execution_count": 107, "metadata": {}, "outputs": [], "source": [ "import numpy as np\n", "%matplotlib inline\n", "\n", "#This will be the parent distribution, containing the heights of all 40k students at the university\n", "parentdist = np.random.normal(loc=168, scale=20, size=40000)\n", "\n", "#Here's our sample of N heights.\n", "N=100\n", "sampledist = np.random.choice(parentdist, size=N, replace=False)" ] }, { "cell_type": "code", "execution_count": 108, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "result of estimator #1: 169.69085043049358\n" ] } ], "source": [ "#Estimator 1: Add all the heights and divide by N\n", "def e1(sampledist):\n", " return np.sum(sampledist)/len(sampledist)\n", "\n", "print(\"result of estimator #1:\", e1(sampledist))\n" ] }, { "cell_type": "code", "execution_count": 109, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "result of estimator #2: 188.5453893672151\n" ] } ], "source": [ "#Estimator 2: Add up all the values and divide by N-10\n", "def e2(sampledist):\n", " return np.sum(sampledist)/(len(sampledist)-10)\n", "\n", "print(\"result of estimator #2:\", e2(sampledist))" ] }, { "cell_type": "code", "execution_count": 110, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "result of estimator #3: 180.0\n" ] } ], "source": [ "#Estimator 3: Throw away all the data and give the answer as 180cm\n", "def e3(sampledist):\n", " return 180.\n", "\n", "print(\"result of estimator #3:\", e3(sampledist))" ] }, { "cell_type": "code", "execution_count": 111, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "result of estimator #4: 164.53228420391866\n" ] } ], "source": [ "#Estimator 4: Add the tallest and shortest heights and divide by 2\n", "def e4(sampledist):\n", " return (max(sampledist)+min(sampledist))/2.\n", "\n", "print(\"result of estimator #4:\", e4(sampledist))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Picking a good estimator\n", "A \"good\" estimator is *consistent*, *unbiased*, and *efficient*. \n", "\n", "- An estimator is *consistent* if it tends to the true value as the number of data values tends to infinity.\n", "\n", "- An estimator is *unbiased* if its expectation value is equal to the true value.\n", "\n", "- An estimator is *efficient* if its variance is small.\n", "\n", "Let's take a look at some of our estimators from above and see how they do:" ] }, { "cell_type": "code", "execution_count": 112, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "import matplotlib.pyplot as plt\n", "#True average\n", "trueavg = np.average(parentdist)\n", "\n", "e1s = []\n", "e2s = []\n", "e3s = []\n", "e4s = []\n", "Ns = range(1000,10000,1000)\n", "for N in Ns:\n", " bigsample = np.random.choice(parentdist, size=N)\n", " e1s.append(e1(bigsample))\n", " e2s.append(e2(bigsample))\n", " e3s.append(e3(bigsample))\n", " e4s.append(e4(bigsample))\n", "\n", "plt.plot(Ns, e1s, label='estimator 1')\n", "plt.plot(Ns, e2s, label='estimator 2')\n", "plt.plot(Ns, e3s, label='estimator 3')\n", "plt.plot(Ns, e4s, label='estimator 4')\n", "plt.plot(Ns, trueavg*np.ones(len(Ns)), 'k--', label='true average')\n", "plt.xlabel('N (size of sample)')\n", "plt.ylabel('Estimator value')\n", "plt.legend(loc=0)\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Notice a few things:\n", "\n", "- Estimator 1 is consistent, unbiased, and efficient. This is a good estimator\n", "\n", "- Estimator 2 is consistent and efficient, but *not* unbiased, since if we calculate the expectation value for this estimator, we get:\n", "\n", "$\\left \\langle \\mu \\right \\rangle = \\frac{N}{N-10} \\mu \\neq \\mu$\n", "\n", "- Estimator 3 is efficient, but clearly not consistent or unbiased.\n", "\n", "- Estimator 4 is consistent and unbiased, but is not very efficient. This makes sense, because it really only uses 2 values in our sample. " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## The Likelihood Function\n", "\n", "One class of estimators that gets used a lot in physics is the *likelihood function*. For a set of data, {$x_1, x_2, x_3 ...x_N$} produced from some underlying distribution with parameter $a$, the probability of obtaining that particular data set is the product of the individual probabilities of obtaining each of the $x_i$. This product is called the *likelihood*:\n", "\n", "$L(x_1, x_2, x_3..x_N, a) = P(x_1;a)P(x_2;a)...P(x_N;a)$\n", "$L(x_1, x_2, x_3..x_N, a) = \\prod_{i=1}^N P(x_i;a)$\n", "\n", "\n", "We typically use this to attempt to fit the underlying distribution parameter $a$, given the data that we observe. \n", "\n", "To get some intuition for what's going on here, let's consider the following example: I have a random sample {$x_1, x_2, ...x_N$} where:\n", "\n", "- $x_i$ = 0 if a person is NOT wearing glasses\n", "- $x_i$ = 1 if a person is wearing glasses\n", "\n", "I want to create an estimator for $a$: the proportion of people who wear glasses. For the purposes of this example, let's treat the $x_i$ as independent bernoulli random variables. In this case, the probability mass function for each $x_i$ is just given by:\n", "\n", "$P(x_i;a) = a^{x_i}(1-a)^{1-x_i}$\n", "\n", "where again, $x_i$=0 or 1 and $0 < a < 1$. " ] }, { "cell_type": "code", "execution_count": 113, "metadata": {}, "outputs": [], "source": [ "#Fill in data here\n", "obsdata = [0,1,1,0,0,0,0,0,1]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The likelihood function as we've defined it above is then:\n", "\n", "$L(a) = \\prod_{i=1}^N P(x_i;a) = a^{x_1}(1-a)^{1-x_1} \\times a^{x_2}(1-a)^{1-x_2} \\times a^{x_3}(1-a)^{1-x_3} \\times ... a^{x_N}(1-a)^{1-x_N}$\n", "\n", "Since $L(a)$ is a function of $a$, we can put on our calculus hats to figure out the value of $a$ that maximizes our likelihood. This will be an estimator of the true value of $a$. " ] }, { "cell_type": "code", "execution_count": 114, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "#Calculate the likelihood for this data\n", "\n", "def llh(obsdata, a):\n", " prods = []\n", " for xi in obsdata:\n", " p_xi = (a**xi)*(1-a)**(1-xi)\n", " prods.append(p_xi)\n", " return np.prod(prods)\n", "\n", "#we're eventually going to want to maximize this with respect to a\n", "#so let's see what this looks like as a function of a\n", "\n", "avals = np.linspace(0,1,1000)\n", "llhs = [llh(obsdata, a) for a in avals]\n", "plt.plot(avals, llhs)\n", "plt.xlabel('a')\n", "plt.ylabel('llh value')\n", "plt.show()\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Oftentimes you might hear people talk about the \"log likelihood\". This is simply the log of the likelihood function: $log(L(a))$. We usually prefer to work with the log likelihood because in order to extremize our likelihood in terms of the parameter $a$, we're going to need to take the derivative of the likelihood. The log likelihood turns all the products in the likelihood into a sum, which is much easier to take the derivative of:\n", "\n", "$log(L(a)) = log(\\prod_i P(x_i;a)) = \\sum_i log(P(x_i;a))$\n", "\n", "Where here we have just used the property that $log(xy) = log(x)+log(y)$\n", "\n", "Additionally, you oftentimes might see people use the \"negative log likelihood\". This is just:\n", "\n", "negative log likelihood = $-1 \\times log(L(a))$\n", "\n", "And instead of maximizing the log likelihood, we will minimize the negative log likelihood. This is purely convention. " ] }, { "cell_type": "code", "execution_count": 115, "metadata": {}, "outputs": [ { "name": "stderr", "output_type": "stream", "text": [ "/usr/local/lib/python3.5/dist-packages/ipykernel_launcher.py:3: RuntimeWarning: divide by zero encountered in log10\n", " This is separate from the ipykernel package so we can avoid doing imports until\n" ] }, { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "#Let's plot the log likelihood\n", "\n", "log_llhs = np.log10(llhs)\n", "\n", "plt.plot(avals, log_llhs)\n", "plt.xlabel('a')\n", "plt.ylabel('log(llh)')\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The next step for obtaining our estimiator for $a$ is to extremize the log llh. This can be done numerically (pick your favorite minimizer/maximizer!), but in this case we can also do it analytically:\n", "\n", "$log(L(a)) = \\sum_i log(P(x_i;a)) = \\sum_i log[a^{x_i}(1-a)^{(1-x_i)}]$\n", "\n", "As per our calculus skills, if we want to extremize something we take the derivative and set it equal to zero:\n", "\n", "$\\frac{\\partial}{\\partial a} log(L(a)) = \\sum_i \\frac{(x-a)}{a-a^2}$\n", "\n", "$\\sum_i \\frac{(x-a)}{a-a^2}$ = 0\n", "\n", "$\\sum_i (x_i-a) = 0$\n", "\n", "$\\sum_i (x_i) - Na = 0$\n", "\n", "$\\hat{a} = \\frac{\\sum_i (x_i)} {N}$\n", "\n", "Where here $\\hat{a}$ is used to denote that this is an estimator for the true value of $a$. Let's check to make sure this makes sense:" ] }, { "cell_type": "code", "execution_count": 116, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Estimator for fraction of people who wear glasses: 0.3333333333333333\n" ] }, { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "def a_estimator(obsdata):\n", " ahat = float(sum(obsdata))/len(obsdata)\n", " return ahat\n", "\n", "ahat = a_estimator(obsdata)\n", "print(\"Estimator for fraction of people who wear glasses:\", ahat)\n", "\n", "#plot our llh function and the estimator value of a for our observed data\n", "plt.plot(avals, llhs)\n", "plt.axvline(ahat, color='steelblue')\n", "plt.xlabel('a')\n", "plt.ylabel('llh value')\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Looks like our estimator is in the right place! Or at the very least, it's properly identifying where the maximum of the likelihood is. \n", "\n", "So we now have a process for constructing an estimator for $a$, the proportion of people in the world who wear glasses, based off a single sample of data. Let's test this for a case where the true value of $a$ is $0.8$ (so 80% of people wear glasses)" ] }, { "cell_type": "code", "execution_count": 117, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Estimator for fraction of people who wear glasses: 0.83\n", "True value for fraction of people who wear glasses: 0.8\n" ] }, { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "def parentpop(truea):\n", " #truea is the true value for a that we can set for the parent population\n", " poparr = []\n", " for i in range(0,10000):\n", " rand = np.random.uniform(0,1)\n", " if rand < truea:\n", " poparr.append(1)\n", " else:\n", " poparr.append(0)\n", " return poparr\n", "\n", "def draw_subpop(truepop, N):\n", " #draw N observations from the true population\n", " whichvals = np.random.choice(truepop, N)\n", " return whichvals\n", "\n", "#create a parent population with a true value of a = 0.8\n", "atrue = 0.8\n", "parentarr = parentpop(atrue)\n", "#draw a test sample of 100 people from our parent population\n", "testpop = draw_subpop(parentarr, 100)\n", "#calculate our llh estimator value\n", "ahat = a_estimator(testpop)\n", "print(\"Estimator for fraction of people who wear glasses:\", ahat)\n", "print(\"True value for fraction of people who wear glasses:\", atrue)\n", "\n", "#calculate our llh for this test sample and plot our estimator\n", "avals = np.linspace(0,1,1000)\n", "llhs = [llh(testpop, a) for a in avals]\n", "plt.plot(avals, llhs)\n", "plt.axvline(ahat, color='steelblue', label='llh estimator value')\n", "plt.axvline(atrue, color='orange', label='True value')\n", "plt.xlabel('a')\n", "plt.ylabel('llh value')\n", "plt.show()\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The llh estimator can take on a number of different forms, depending on what our $P(x_i;a)$ actually look like. You'll see an example used for point source searches later this week. Likelihood maximization is most appropriate when you have a guess to what the _form_ of the underlying distribution is, and are looking to estimate the specific distribution parameters associated with that form.\n", "\n", "As a matter of fact, we might even have a distribution with *many* parameters! In this case, our individual observation probabilities would be functions of multiple parameters: $P(x_i;a,b,c...)$, and we would have to do a multi-dimensional maximization of the likelihood with respect to all the parameters $a, b, c...$. As an example of this, let's consider the problem of estimating the distribution of heights of a population based off a sample. Suppose we think that the true population of heights follows a gaussian distribution:\n", "\n", "$P(H|\\mu, \\sigma) = \\frac{1}{\\sqrt{2 \\pi \\sigma^2}}e^{\\frac{{-(H-\\mu)}^2}{2 \\sigma^2}}$\n", "\n", "where here $H$ refers to a height, and $P(H|\\mu, \\sigma)$ refers to the probability of a person having height $H$. We then have some sample data, consisting of the heights of each person in our sample:" ] }, { "cell_type": "code", "execution_count": 118, "metadata": {}, "outputs": [], "source": [ "sampleheights = [170., 165., 172., 175., 168., 167.]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "And our likelihood will simply be the product of the probabilities for each of our observations:\n", "\n", "$L(H_1, H_2, H_3..., \\mu, \\sigma) = P(H_1|\\mu, \\sigma) \\times P(H_2|\\mu, \\sigma) \\times P(H_3|\\mu, \\sigma)...$\n", "\n", "$L(H_1, H_2, H_3..., \\mu, \\sigma) = \\frac{1}{\\sqrt{2 \\pi \\sigma^2}}e^{\\frac{{-(H_1-\\mu)}^2}{2 \\sigma^2}} \\times \\frac{1}{\\sqrt{2 \\pi \\sigma^2}}e^{\\frac{{-(H_2-\\mu)}^2}{2 \\sigma^2}} \\times \\frac{1}{\\sqrt{2 \\pi \\sigma^2}}e^{\\frac{{-(H_3-\\mu)}^2}{2 \\sigma^2}}...$" ] }, { "cell_type": "code", "execution_count": 119, "metadata": {}, "outputs": [], "source": [ "#return the negative log likelihood for a set of sample heights given by Hs\n", "def log_llh(params,Hs):\n", " mu = params[0]\n", " sigma = params[1]\n", " ps = []\n", " for h in Hs:\n", " p = np.log(1./np.sqrt(2*np.pi*sigma**2.))+(-(h-mu)**2./(2.*sigma**2))\n", " ps.append(p)\n", " return -1.*np.sum(ps) " ] }, { "cell_type": "code", "execution_count": 120, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "#Let's plot some slices of our likelihood space to make sure everything is working okay\n", "#First let's fix sigma to some random value and look at how our llh varies with mu\n", "fixedsigma = []\n", "for havg in np.linspace(160,180,100):\n", " fixedsigma.append(log_llh([havg, 3.], sampleheights))\n", "\n", "#plot our results\n", "plt.plot(np.linspace(160,180,100), fixedsigma)\n", "plt.xlabel(\"H\")\n", "plt.ylabel(\"Log Likelihood\")\n", " \n", "plt.show()" ] }, { "cell_type": "code", "execution_count": 121, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "#now let's fix mu and vary sigma\n", "fixedmu = []\n", "for sigtest in np.linspace(0.5, 10., 100):\n", " fixedmu.append(log_llh([169.5, sigtest], sampleheights))\n", "\n", "\n", "plt.plot(np.linspace(0.5, 10, 100), fixedmu)\n", "plt.xlabel('$\\sigma$')\n", "plt.ylabel('Log Likelihood')\n", "plt.show()" ] }, { "cell_type": "code", "execution_count": 122, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ " fun: 15.684503219472408\n", " hess_inv: array([[1.74151787, 0.04242564],\n", " [0.04242564, 0.88728792]])\n", " jac: array([-1.66893005e-06, 1.90734863e-06])\n", " message: 'Optimization terminated successfully.'\n", " nfev: 40\n", " nit: 9\n", " njev: 10\n", " status: 0\n", " success: True\n", " x: array([169.49999668, 3.30403979])\n" ] } ], "source": [ "#Now let's go through and do the full 2d minimization \n", "#scipy has some build in minimization functions that are really useful for doing these types of things\n", "#all we have to do is give scipy our function to minimize and an initial guess\n", "#see documentation here: https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.minimize.html\n", "from scipy import optimize\n", "\n", "result = optimize.minimize(log_llh, [169., 2.0], args=sampleheights)\n", "print(result)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Confidence Intervals\n", "Suppose we want to know the value of a parameter $X$ and have estimated it from the data, giving result $x$. We know the resolution of our measurements, and thus the variance, $V(x)$ and the standard deviation $\\sigma$. Our goal is then to create a *confidence interval*: An interval within which there is a $C$ % chance that a single observation will lie within the interval, where $C$ is our *confidence level*. Formally, the confidence interval ($x_-, x_+$) for a given confidence level $C$ obeys the requirement that:\n", "\n", "$Prob(x_- \\leq x \\leq x_+) = \\int_{x_-}^{x_+}P(x)dx = C$\n", "\n", "There are a few possible ways to construct this interval:\n", "\n", "1. The *symmetric interval*: $x_-$ and $x_+$ are equidistant from the mean, $\\mu$, i.e $x_+ - \\mu = \\mu - x_-$\n", "\n", "2. The *central interval*: The probability above and below the interval are equal: $\\int_{-\\infty}^{x_-} P(x)dx = \\int_{x^+}^{\\infty} P(x)dx = (1-C)/2$\n", "\n", "Let's practice with a simple example. Suppose we measure the mass of an oreo to be 18 grams, with a standard deviation of 4 grams. Assuming that the distribution of oreo masses is a gaussian, what is the 90% confidence interval for the mass of a single oreo?" ] }, { "cell_type": "code", "execution_count": 123, "metadata": {}, "outputs": [ { "name": "stderr", "output_type": "stream", "text": [ "No handles with labels found to put in legend.\n" ] }, { "data": { "text/plain": [ "[]" ] }, "execution_count": 123, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "from scipy import stats\n", "# plot our distribution corresponding to our observation\n", "mu = 18\n", "sigma = 4\n", "x = np.linspace(mu-3*sigma, mu+3*sigma, 1000)\n", "\n", "plt.xlabel('Oreo Mass')\n", "plt.ylabel('PDF')\n", "plt.legend(loc=0)\n", "plt.plot(x, stats.norm.pdf(x, mu, sigma))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "For a 90% central confidence interval, we know that:\n", "\n", "$\\int_{-\\infty}^{x_-} \\frac{1}{\\sqrt{2 \\pi \\sigma^2}} e ^{\\frac{{x-\\mu}^2}{2 \\sigma^2}}dx = \\int_{x^+}^{\\infty} \\frac{1}{\\sqrt{2 \\pi \\sigma^2}} e ^{\\frac{{x-\\mu}^2}{2 \\sigma^2}}dx = (1-.9)/2$" ] }, { "cell_type": "code", "execution_count": 124, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "lower bound on central confidence interval: 11.420585492194109\n", "upper bound on central confidence interval: 24.57941450780589\n" ] } ], "source": [ "#stats.norm.ppf is the inverse of the stats.norm.cdf,\n", "#returning an x value at which integral(-inf, x) (normal dist)=input, in units of \"sigma\"\n", "#From the above, we want stats.norm.cdf(x-) = 0.05, so:\n", "howmanysigma = stats.norm.ppf(.05)\n", "\n", "#convert this to an oreo mass:\n", "oreomass = mu+howmanysigma*sigma\n", "print(\"lower bound on central confidence interval:\", oreomass)\n", "\n", "#Now let's do the same thing but to calculate the upper bound on the central condifdence interval\n", "howmanysigma2 = stats.norm.ppf(.95)\n", "\n", "#convert to an oreo mass:\n", "oreomass2 = mu+howmanysigma2*sigma\n", "print(\"upper bound on central confidence interval:\", oreomass2)" ] }, { "cell_type": "code", "execution_count": 125, "metadata": {}, "outputs": [ { "name": "stderr", "output_type": "stream", "text": [ "No handles with labels found to put in legend.\n" ] }, { "data": { "text/plain": [ "[]" ] }, "execution_count": 125, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "#plot our confidence interval\n", "mu = 18\n", "sigma = 4\n", "x = np.linspace(mu-3*sigma, mu+3*sigma, 1000)\n", "\n", "plt.xlabel('Oreo Mass')\n", "plt.ylabel('PDF')\n", "plt.legend(loc=0)\n", "plt.axvline(oreomass, color='steelblue')\n", "plt.axvline(oreomass2, color='steelblue')\n", "plt.plot(x, stats.norm.pdf(x, mu, sigma))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The interpretation here is that for a single oreo, if we say that it's mass is between 11.4 and 24.5, we will be correct 90% of the time. \n", "\n", "Notice that in this case, because we assumed a gaussian distribution, our central interval is also a symmetric interval. This isn't always true, since our underlying PDF may be asymmetric. \n", "\n", "We didn't necessarily have to choose a 90% confidence interval. The level of your confidence interval is up to you! Common values are 68% (1 $\\sigma$), 90% (1.64 $\\sigma$), and 99% (2.58 $\\sigma$). There is a trade off between a narrow interval and low confidence. You can say with great confidence that the mass will lie within very wide limits, but if you want to tie it down more precisely the confidence lessens. " ] }, { "cell_type": "code", "execution_count": 126, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "lower bound on 99% central confidence interval: 7.696682785804397\n", "upper bound on 99% central confidence interval: 28.3033172141956\n" ] }, { "data": { "text/plain": [ "[]" ] }, "execution_count": 126, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "#Let's calculate a 99% confidence interval\n", "howmanysigma = stats.norm.ppf(.005)\n", "\n", "#convert this to an oreo mass:\n", "oreo99lower = mu+howmanysigma*sigma\n", "print(\"lower bound on 99% central confidence interval:\", oreo99lower)\n", "\n", "#Now let's do the same thing but to calculate the upper bound on the central condifdence interval\n", "howmanysigma2 = stats.norm.ppf(.995)\n", "\n", "#convert to an oreo mass:\n", "oreo99upper = mu+howmanysigma2*sigma\n", "print(\"upper bound on 99% central confidence interval:\", oreo99upper)\n", "\n", "plt.axvline(oreo99lower, color='green', label='99% confidence interval')\n", "plt.axvline(oreo99upper, color='green')\n", "plt.axvline(oreomass, color='steelblue', label='90% confidence interval')\n", "plt.axvline(oreomass2, color='steelblue')\n", "plt.legend(loc=0)\n", "plt.plot(x, stats.norm.pdf(x, mu, sigma))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Other forms of confidence intervals that are useful to talk about are one-tailed limits. These are useful if we want to make statements like \"the mass of an oreo is at least $x$ grams\" or \"the mass of an oreo is less than $x$ grams\". These one-tailed limits satisfy the condition:\n", "\n", "$Prob(x < x_+) = \\int_{-\\infty}^{x_+}P(x)dx$ (for an upper limit $x_+$)\n", "\n", "$Prob(x > x_-) = \\int_{x_-}^{\\infty}P(x)dx$ (for a lower limit $x_-$)" ] }, { "cell_type": "code", "execution_count": 127, "metadata": { "scrolled": true }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "90% lower limit on oreo mass: 12.873793737821599\n" ] }, { "data": { "text/plain": [ "[]" ] }, "execution_count": 127, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "\n", "#for computing a 90% lower limit, we want to find the mass where 90% of the distribution is above this mass\n", "howmanysig = stats.norm.ppf(.1)\n", "lowerlim =mu+howmanysig*sigma\n", "\n", "#we can say, with 90% confidence, that the mass of an oreo is greater than this value\n", "print(\"90% lower limit on oreo mass:\", lowerlim)\n", "plt.axvline(lowerlim, label='90% lower limit')\n", "\n", "plt.xlabel('Oreo mass')\n", "plt.ylabel('PDF')\n", "plt.legend(loc=0)\n", "plt.plot(x, stats.norm.pdf(x, mu, sigma))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Notice that the 90% lower limit and the lower bound on the 90% central limit are NOT the same. This is because the 90% lower limit requires that all 10% of the \"non-included\" area be on one side of the distribution. " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let's test our interpretation of our lower limit. If we're correct, then if we look at an individual oreo, then 90% of the time it will have a mass greater than 12.87. " ] }, { "cell_type": "code", "execution_count": 128, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "number of oreos lighter than our lower limit: 102\n", "fraction: 0.102\n" ] } ], "source": [ "#simulate a package of oreos\n", "def oreo_package(N_oreos):\n", " return np.random.normal(loc=mu, scale=sigma, size=N_oreos)\n", "\n", "#simulate opening 1000 packages, and picking an oreo to test from each package\n", "lightoreos = 0\n", "Ntrials = 1000\n", "for k in range(0,Ntrials):\n", " onepackage = oreo_package(28) \n", " #a megastuff oreo package has 28 oreos\n", " one_oreo = np.random.choice(onepackage)\n", " if one_oreo < 12.87:\n", " lightoreos+=1\n", "\n", "print(\"number of oreos lighter than our lower limit:\", lightoreos)\n", "print(\"fraction:\", lightoreos/Ntrials)" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Python 2", "language": "python", "name": "python2" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.5.2" } }, "nbformat": 4, "nbformat_minor": 2 }