{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Moderne Methoden der Datenanalyse SS2021\n",
"# Practical Exercise 6"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Exercise 6.1: Hypothesis Testing\n",
"\n",
"\"Is this a new discovery or just a statistical fluctuation?\" Statistics offers some methods to give a quantitative answer. But these methods should not be used blindly. In particular, one should know exactly what the obtained numbers mean and what they don't mean."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Exercise 6.1.1 (obligatory to solve either 6.1.1 or 6.1.2)\n",
"\n",
"The following table shows the number of winners in a horse race for different track numbers:\n",
"\n",
"| track ||| 1 || 2 || 3 || 4 || 5 || 6 || 7 || 8 |\n",
"|------------|||------||------||------||------||------||------||------||------|\n",
"| # winners ||| 29 || 19 || 18 || 25 || 17 || 10 || 15 || 11 |\n",
"\n",
"Use a $\\chi ^2$ test to check the hypothesis that the track number has *no* influence on the chance to win. Define a significance level, e.g., $\\alpha = 5 \\, \\%$ or $\\alpha = 1 \\, \\%$, *before* you do the test."
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"# pick your poison\n",
"from ROOT import TMath\n",
"from scipy.stats import chi2\n",
"\n",
"def exercise6_1_1(confidenceLevel):\n",
" \n",
" # numbers given in the exercise\n",
" nTracks = 8\n",
" nWin = [29, 19, 18, 25, 17, 10, 15, 11]\n",
" \n",
" return"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Exercise 6.1.2 (obligatory to solve either 6.1.1 or 6.1.2)\n",
"\n",
"In a counting experiment, 5 events are observed while $\\mu _\\mathrm{B} = 1.8$ background events are expected. Is this a significant ($= 3 \\sigma$) excess? Calculate the probability of observing 5 or more events when the expectation value is 1.8 using Poisson statistics."
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"import numpy as np\n",
"\n",
"# pick your poison\n",
"from ROOT import gRandom, TMath, Math\n",
"from scipy.stats import poisson, norm"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"def exercise6_1_2():\n",
" \n",
" # numbers given in the exercise\n",
" nBackground = 1.8\n",
" nObserved = 5\n",
"\n",
" # calculate the probability to observe 5 or more events\n",
" \n",
" # optional: make toy experiments\n",
" \n",
" return"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Exercise 6.2: Parameter Estimation"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Exercise 6.2.1 (voluntary)\n",
"\n",
"Consider the following set of values approximately following a Gaussian distribution (see also excercise_6_2_1.csv):\n",
"\n",
"\n",
"\n",
" \n",
"\n",
"\n",
"$$ y(x) = a _\\mathrm{1} \\cdot \\exp \\{-\\frac{1}{2} (\\frac{ x-a _\\mathrm{2} }{ a _\\mathrm{3} } )^2 \\} $$\n",
"\n",
"\n",
"with $\\sigma _i$ being the uncertainty on $y _i$.\n",
"\n",
" - Determine the values of the three parameters $a _\\mathrm{1}$, $a _\\mathrm{2}$, and $a _\\mathrm{3}$ as well as their uncertainties.\n",
" \n",
" - Afterwards, use the transformation $z = \\ln (y)$ to obtain the linear function $z(x) = b _\\mathrm{1} + b _\\mathrm{2} x + b _\\mathrm{3} x^2$. Determine the new parameters $b _\\mathrm{1}$, $b _\\mathrm{2}$, and $b _\\mathrm{3}$ and uncertainties in two ways and compare the results: first, by fitting this new function; second, by a calculation using the results for $a _\\mathrm{j}$ and the transformation $z = \\ln (y)$."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#### ROOT approach"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"from ROOT import TF1, TGraphErrors, TCanvas\n",
"import numpy as np\n",
"\n",
"def exercise6_2_1(verbose = 0):\n",
" # read exercise_6_2_1.csv or type are values explicitly\n",
" \n",
" # fit a Gaussian function\n",
" \n",
" #print(\" -------- Fit Parameters ---------- \")\n",
" #print(\" a1 = {0:.3f} +/- {1:.3f} \".format(a1, err_a1))\n",
" #print(\" a2 = {0:.3f} +/- {1:.3f} \".format(a2, err_a2))\n",
" #print(\" a3 = {0:.3f} +/- {1:.3f} \".format(a3, err_a3))\n",
" #print(\" chi2 = {0:.3f} \".format(chi))\n",
" \n",
"\n",
" # now with log\n",
" \n",
" # calculation\n",
" \n",
" \n",
" #print(\" -------- Fit Parameters ---------- \")\n",
" #print(\" b1 = {0:.3f} +/- {1:.3f} \".format(b1, err_b1))\n",
" #print(\" b2 = {0:.3f} +/- {1:.3f} \".format(b2, err_b2))\n",
" #print(\" b3 = {0:.3f} +/- {1:.3f} \".format(b3, err_b3))\n",
" #print(\" chi2 = {0:.3f} \".format(chi2))\n",
" \n",
" #print(\"\\n -------- Estimated Parameters ---------- \")\n",
" #print(\" b1 = {0:.3f} +/- {1:.3f} \".format(b1est, err_b1est))\n",
" #print(\" b2 = {0:.3f} +/- {1:.3f} \".format(b2est, err_b2est))\n",
" #print(\" b3 = {0:.3f} +/- {1:.3f} \".format(b3est, err_b3est))\n",
"\n",
" \n",
" # draw graphs\n",
"\n",
" return"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#### Python approach"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"import numpy as np\n",
"import matplotlib.pyplot as plt\n",
"import scipy.optimize"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"def exercise_6_2_1():\n",
" # read exercise_6_2_1.csv or type are values explicitly\n",
"\n",
" # fit a Gaussian function\n",
"\n",
" #print(\"Fit Result:\")\n",
" #print(\"a1 = {0:.3f} +- {3:.3f}\\na2 = {1:.3f} +- {4:.3f}\\na3 = {2:.3f} +- {5:.3f}\".\n",
" # format(a1, a2, a3, a1err, a2err, a3err))\n",
" \n",
" \n",
" # now with log\n",
" \n",
" # calculation\n",
" \n",
"\n",
" #print(\"Fit Result:\")\n",
" #print(\"b1 = {0:.3f} +- {3:.3f}\\nb2 = {1:.3f} +- {4:.3f}\\nb3 = {2:.3f} +- {5:.3f}\".\n",
" # format(*bopt, np.sqrt(bcov[0, 0]), np.sqrt(bcov[1, 1]), np.sqrt(bcov[2, 2])))\n",
" #print(\"Transformed Result:\")\n",
" #print(\"b1 = {0:.3f} +- {3:.3f}\\nb2 = {1:.3f} +- {4:.3f}\\nb3 = {2:.3f} +- {5:.3f}\".\n",
" # format(b1, b2, b3, b1err, b2err, b3err))\n",
" \n",
" # plot\n",
" \n",
" return"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Exercise 6.2.2 (obligatory)\n",
"\n",
"This exercise aims at constructing the error band around a function, $f(x)$, fitted to data points $(x,y)$ - i.e., the errors on the fitted parameters are transformed into errors on the value of the function at each value of $x$. (If you do not want to use `ROOT` to optimize the curve fit in this exercise, you can instead use the fitting algorithm provided by scipy, e.g., scipy.optimize.curve_fit - see Python approach)."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Let us attack the problem in steps:\n",
"\n",
" - First, define a function for our problem: a straight line as $f(x) = a + bx$ with parameters a and b.\n",
" \n",
" - Next, consider `npoints=11` data points in the interval `[xof, xof+npoints-1]` with `xof=10` and y(x) = x. To simulate measurement errors, shift the data points in y-direction by a random shift, drawn from a Gaussian distribution with $\\sigma = 0.5$ and $\\mu = 0.0$.\n",
"\n",
" - Now fit the straight line defined above to your data points.\n",
" \n",
" Draw the data points and the fit result. Print the correlation coefficient of the rrors on the parameters $a$ and $b$.\n",
" \n",
" Try to give an intuitive argument why the two parameters $a$ (axis intercept) and $b$ (slope) are strongly correlated.\n",
" \n",
" - Next, construct the error band around the fit function: write a function `make_band(f, cov, withCorr=0)`, which takes the function $f$ and the covariance matrix, as input arguments and calculates for each value of x the error on $y$, $\\Delta _y (x)$. As a first approach, use the simple formula for error propagation, which, in this case, results in $\\Delta _y (x) ^2 = \\Delta _a ^2 + (x \\cdot \\Delta _b )^2$ if correlations are neglected.\n",
" \n",
" Draw the curves $y(x) \\pm \\Delta_y (x)$, i.e., the \"error band\" on top of your data and fit result. Does this look correct?\n",
" \n",
" - Derive the appropriate formula for error propagation taking into account the correlation of the errors. Re-calculate $\\Delta _y (x)$ and plot the corresponding error band. Compare with the above result obtained without correlations.\n",
" \n",
"To get a better idea of what happens here and what the effect of the correlations is, you might want to repeat the whole exercise setting `xof=-5.`, meaning now that the mean of the $x$-values of the data points is approximately at $0$. Alternatively, you may choose the parametrisation of the function as `y(x) = a' + b(x-(xof+(npoints-1)/2))`."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#### ROOT Apprach"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"from ROOT import TF1, gRandom, TGraphErrors, TAxis, TVirtualFitter, TMatrixD, TH1, TCanvas, TFile, TLegend\n",
"from array import array\n",
"from math import fabs, sqrt"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"# Evaluates error band with/without taking into account the correlation\n",
"def make_band_r (f, cov, withCorr=0):\n",
" # As first approach, use the simple formula for error propagation\n",
" \n",
" # Derive the appropriate formula taking into account the correlation of the errors\n",
" \n",
" return"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"def drawUncertaintyBand(npoints=11, xof=10, sigma=0.5):\n",
" # First: Define a function (e.g. a straight line)\n",
" \n",
"\n",
" # Next: `npoints` data points in the interval [xof, xof+npoints-1] with y(x) = x.\n",
" # Shift the data points in y-direction\n",
"\n",
" \n",
" # Store the values in a TGraphErrors\n",
" \n",
"\n",
" # Now fit the straight line defined above to your data points\n",
"\n",
" \n",
" # Access the correlation matrix\n",
" # Hint: you can use the code in the appendix (see bottom of this notebook!)\n",
" \n",
" \n",
" # Define above the function `make_band` and draw the \"error band\" on top of your data and fit result\n",
" # w/o correlations \n",
"\n",
" # w/ correlations\n",
" \n",
" \n",
" return"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#### Python approach"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"import numpy as np\n",
"import matplotlib.pyplot as plt\n",
"import scipy.optimize"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"# First: Define a function (e.g. a straight line)\n",
"def linear(x, *p):"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"def make_band_p(function, popt, pcov, range, use_correlation=False):\n",
" # As first approach, use the simple formula for error propagation\n",
" \n",
" # Derive the appropriate formula taking into account the correlation of the errors\n",
" "
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"def exercise_6_2_2(xof=10):\n",
" # Next: `npoints` data points in the interval [xof, xof+npoints-1] with y(x) = x.\n",
" # Shift the data points in y-direction\n",
"\n",
" # Now fit the straight line defined above to your data points\n",
"\n",
" # Define above the function `make_band` and draw the \"error band\" on top of your data and fit result\n",
" # w/o correlation\n",
"\n",
" # w/ correlation\n",
" "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Appendix: Accessing the covariance matrix\n",
"\n",
"#### Example code for pyROOT / JupyROOT:"
]
},
{
"cell_type": "raw",
"metadata": {},
"source": [
"fitrp = TVirtualFitter.GetFitter()\n",
"nPar = fitrp.GetNumberTotalParameters()\n",
"\n",
"covmat = TMatrixD(nPar, nPar, fitrp.GetCovarianceMatrix())\n",
"\n",
"covmat.Print()\n",
"\n",
"cormat = TMatrixD(covmat)\n",
"for i in range(nPar):\n",
" for j in range(nPar):\n",
" cormat[i][j] = cormat[i][j] / (sqrt(covmat[i][i]) * sqrt(covmat[j][j]))\n",
"\n",
"cormat.Print()"
]
}
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