"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
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],
"source": [
"# Make the exponential PDF\n",
"k = minkit.Parameter('k', -0.1, bounds=(-1, 0))\n",
"\n",
"e = minkit.Exponential('e', x, k)\n",
"\n",
"# Add the two PDFs\n",
"y = minkit.Parameter('y', 0.5, bounds=(0, 1))\n",
"\n",
"pdf = minkit.AddPDFs.two_components('pdf', g, e, y)\n",
"\n",
"data = pdf.generate(10000)\n",
"\n",
"# Keep the initial values\n",
"initials = pdf.get_values()\n",
"\n",
"# Minimize the PDF\n",
"with minkit.minimizer('uml', pdf, data) as minimizer:\n",
" minimizer.migrad()\n",
"\n",
"values, edges = minkit.data_plotting_arrays(data, bins=100)\n",
"\n",
"centers = 0.5 * (edges[1:] + edges[:-1])\n",
"\n",
"plt.hist(centers, bins=edges, weights=values, histtype='step', color='k', label='data');\n",
"\n",
"pdf_centers, pdf_values = minkit.pdf_plotting_arrays(pdf, values, edges)\n",
"_, g_values = minkit.pdf_plotting_arrays(pdf, values, edges, component='g')\n",
"_, e_values = minkit.pdf_plotting_arrays(pdf, values, edges, component='e')\n",
"\n",
"plt.plot(pdf_centers, e_values, color='orange', linestyle=':', label='Exponential')\n",
"plt.plot(pdf_centers, g_values, color='green', linestyle='--', label='Gaussian')\n",
"plt.plot(pdf_centers, pdf_values, color='blue', label='PDF')\n",
"\n",
"plt.legend();\n",
"\n",
"# Reset values\n",
"pdf.set_values(**initials)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Let's now assume we want to know the two yields of the PDFs. To do this we need to modify the model, and do and use an extended likelihood during the minimization."
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"\n",
" \n",
" FCN = -122354.9230188445 \n",
" TOTAL NCALL = 124 \n",
" NCALLS = 124 \n",
" \n",
" \n",
" EDM = 2.1664581983657007e-05 \n",
" GOAL EDM = 1e-05 \n",
" \n",
" UP = 1.0 \n",
" \n",
"
\n",
"\n",
" \n",
" Valid \n",
" Valid Param \n",
" Accurate Covar \n",
" PosDef \n",
" Made PosDef \n",
" \n",
" \n",
" True \n",
" True \n",
" True \n",
" True \n",
" False \n",
" \n",
" \n",
" Hesse Fail \n",
" HasCov \n",
" Above EDM \n",
" Reach calllim \n",
" \n",
" \n",
" False \n",
" True \n",
" False \n",
" False \n",
" \n",
"
"
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"metadata": {},
"output_type": "display_data"
},
{
"data": {
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" Limit+ \n",
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"data": {
"text/html": [
""
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"output_type": "display_data"
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"source": [
"# Build the extended PDF\n",
"ng = minkit.Parameter('ng', 5000, bounds=(0, 10000))\n",
"ne = minkit.Parameter('ne', 5000, bounds=(0, 10000))\n",
"\n",
"pdf = minkit.AddPDFs.two_components('pdf', g, e, ng, ne)\n",
"\n",
"initials = pdf.get_values()\n",
"\n",
"# Generate and fit\n",
"data = pdf.generate(int(ng.value + ne.value))\n",
"\n",
"ng.value = 6000 # vary the yields to flee from the minimum\n",
"ne.value = len(data) - ng.value\n",
"\n",
"with minkit.minimizer('ueml', pdf, data) as minimizer:\n",
" minimizer.migrad()\n",
"\n",
"values, edges = minkit.data_plotting_arrays(data, bins=100)\n",
"\n",
"centers = 0.5 * (edges[1:] + edges[:-1])\n",
"\n",
"plt.hist(centers, bins=edges, weights=values, histtype='step', color='k', label='data');\n",
"\n",
"pdf_centers, pdf_values = minkit.pdf_plotting_arrays(pdf, values, edges)\n",
"_, g_values = minkit.pdf_plotting_arrays(pdf, values, edges, component='g')\n",
"_, e_values = minkit.pdf_plotting_arrays(pdf, values, edges, component='e')\n",
"\n",
"plt.plot(pdf_centers, e_values, color='orange', linestyle=':', label='Exponential')\n",
"plt.plot(pdf_centers, g_values, color='green', linestyle='--', label='Gaussian')\n",
"plt.plot(pdf_centers, pdf_values, color='blue', label='PDF')\n",
"\n",
"plt.legend();\n",
"\n",
"# Reset the values\n",
"pdf.set_values(**initials)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Note that we have specified the *ueml* (unbinned extended maximum likelihood) FCN. If we set it to *uml*, we would get wrong values. Let's now see what happens with binned data sets. In this case, we will use the *bml* FCN."
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"\n",
" \n",
" FCN = 85.809988390516 \n",
" TOTAL NCALL = 100 \n",
" NCALLS = 100 \n",
" \n",
" \n",
" EDM = 2.6451594374739883e-07 \n",
" GOAL EDM = 1e-05 \n",
" \n",
" UP = 1.0 \n",
" \n",
"
\n",
"\n",
" \n",
" Valid \n",
" Valid Param \n",
" Accurate Covar \n",
" PosDef \n",
" Made PosDef \n",
" \n",
" \n",
" True \n",
" True \n",
" True \n",
" True \n",
" False \n",
" \n",
" \n",
" Hesse Fail \n",
" HasCov \n",
" Above EDM \n",
" Reach calllim \n",
" \n",
" \n",
" False \n",
" True \n",
" False \n",
" False \n",
" \n",
"
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"metadata": {},
"output_type": "display_data"
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{
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"data": {
"text/html": [
""
]
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"metadata": {
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],
"source": [
"import numpy as np\n",
"\n",
"values, edges = np.histogram(data['x'].as_ndarray(), bins=100, range=x.bounds)\n",
"\n",
"binned_data = minkit.BinnedDataSet.from_ndarray(edges, x, values)\n",
"\n",
"with minkit.minimizer('bml', pdf, binned_data) as minimizer:\n",
" minimizer.migrad()\n",
"\n",
"values, edges = minkit.data_plotting_arrays(data, bins=100)\n",
"\n",
"centers = 0.5 * (edges[1:] + edges[:-1])\n",
"\n",
"plt.hist(centers, bins=edges, weights=values, histtype='step', color='k', label='data');\n",
"\n",
"pdf_centers, pdf_values = minkit.pdf_plotting_arrays(pdf, values, edges)\n",
"_, g_values = minkit.pdf_plotting_arrays(pdf, values, edges, component='g')\n",
"_, e_values = minkit.pdf_plotting_arrays(pdf, values, edges, component='e')\n",
"\n",
"plt.plot(pdf_centers, e_values, color='orange', linestyle=':', label='Exponential')\n",
"plt.plot(pdf_centers, g_values, color='green', linestyle='--', label='Gaussian')\n",
"plt.plot(pdf_centers, pdf_values, color='blue', label='PDF')\n",
"\n",
"plt.legend();\n",
"\n",
"# Reset the values\n",
"pdf.set_values(**initials)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Similarly, with the FCN *chi2* we would be doing a fit to the chi-square, that is, we would be assuming that the input values, instead of following a Poisson distribution, they follow a Gaussian."
]
}
],
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