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mtzl/fitting_dists.ipynb

Created November 1, 2016 21:50
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fitting_dists.ipynb
{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Fitting Distributions"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Author: Moritz Lotze <mlotze@km3net.de>"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Choose your dataset\n",
"\n",
"Change this value to the data you want to test:\n",
"\n",
"- noised_gauss\n",
"- bimodal_gauss"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"dataset = 'noised_gauss'"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"-----------------"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"# General imports\n",
"\n",
"import numpy as np\n",
"import matplotlib.pyplot as plt\n",
"import seaborn as sb\n",
"import statsmodels.api as sm\n",
"\n",
"from scipy.stats import norm\n",
"\n",
"dsets = {}"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Generate Pseudo Data"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Bimodal Gaussian + Noise"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stderr",
"output_type": "stream",
"text": [
"/usr/lib/python3.5/site-packages/ipykernel/__main__.py:3: VisibleDeprecationWarning: using a non-integer number instead of an integer will result in an error in the future\n",
" app.launch_new_instance()\n",
"/usr/lib/python3.5/site-packages/ipykernel/__main__.py:4: VisibleDeprecationWarning: using a non-integer number instead of an integer will result in an error in the future\n"
]
}
],
"source": [
"N = 100\n",
"\n",
"bmg = np.concatenate((np.random.normal(15, 1, 0.3 * N),\n",
" np.random.normal(20, 1, 0.7 * N)))\n",
"noise_bmg = 0.5\n",
"bimodal_gauss = np.random.normal(bmg, noise_bmg)[:, np.newaxis]\n",
"dsets['bimodal_gauss'] = bimodal_gauss"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Single Gaussian + Noise"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"mu_t = 21.01\n",
"sig_t = 3.01\n",
"gaussian = np.random.normal(mu_t, sig_t, N)\n",
"noise = 0.4\n",
"noised_gauss = np.random.normal(gaussian, noise)[:, np.newaxis]\n",
"dsets['noised_gauss'] = noised_gauss"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"# make X axis for plots\n",
"x = np.linspace(5, 35, N+1)"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"data = dsets[dataset]"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Histograms"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Fixed Number of Bins\n",
"15 bins, spaced from `data.min()` to `data.max()`."
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {
"collapsed": false
},
"outputs": [
{
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"### Knuth’s rule \n",
"A fixed-width, Bayesian approach to determining the optimal bin width of a histogram."
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"source": [
"from astropy.stats import knuth_bin_width\n",
"\n",
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"metadata": {},
"source": [
"## Fit a distribution (parametric)\n",
"\n",
"If we have a hypothesis what the distribution looks like (e.g. gaussian), and want to fit its parameters."
]
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{
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"metadata": {},
"source": [
"### Fit Distribution via Maximum Likelihood (recommended)\n",
"\n",
"The nice thing is, you can define your own PDFs in scipy and fit it."
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {
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"outputs": [],
"source": [
"from scipy.stats import norm\n",
"mu, sig = norm.fit(data)"
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"metadata": {
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"True: Mean 21.01, stdev 3.01\n",
"Fit: Mean 20.59, stdev 3.006\n"
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{
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"metadata": {},
"source": [
"### Fit Gaussian Mixture Model (GMM)\n",
"\n",
"Assuming the data is the sum of one or more gaussians. Easily handles multidimensinal case as well."
]
},
{
"cell_type": "code",
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"metadata": {
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"source": [
"from sklearn.mixture import GaussianMixture\n",
"\n",
"gmm = GaussianMixture(n_components=1)\n",
"gmm.fit(data)\n",
"mu = gmm.means_[0, 0]\n",
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"source": [
"plt.fill(x, norm(mu, sig).pdf(x), alpha=.3, label='Fitted')\n",
"plt.legend()\n",
"print('True: Mean {:.4}, stdev {:.4}'.format(mu_t, sig_t))\n",
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]
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"metadata": {},
"source": [
"## Fit Distribution (non-parametric)\n",
"\n",
"If we have no strong assumptions about the underlying pdf.\n",
" "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Kernel Density\n",
"\n",
"\"Smooth out\" each event with a kernel (e.g. gaussian) of a certain bandwidth, then add together all these mini-functions."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#### Implementation: Scikit-Learn"
]
},
{
"cell_type": "code",
"execution_count": 14,
"metadata": {
"collapsed": false
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{
"name": "stderr",
"output_type": "stream",
"text": [
"/usr/lib/python3.5/site-packages/sklearn/cross_validation.py:44: DeprecationWarning: This module was deprecated in version 0.18 in favor of the model_selection module into which all the refactored classes and functions are moved. Also note that the interface of the new CV iterators are different from that of this module. This module will be removed in 0.20.\n",
" \"This module will be removed in 0.20.\", DeprecationWarning)\n",
"/usr/lib/python3.5/site-packages/sklearn/grid_search.py:43: DeprecationWarning: This module was deprecated in version 0.18 in favor of the model_selection module into which all the refactored classes and functions are moved. This module will be removed in 0.20.\n",
" DeprecationWarning)\n"
]
},
{
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"best bandwidth: 1.5998587196060574\n"
]
}
],
"source": [
"# sklearn kernel density\n",
"from sklearn.grid_search import GridSearchCV\n",
"from sklearn.neighbors import KernelDensity\n",
"\n",
"# use grid search cross-validation to optimize the bandwidth\n",
"params = {'bandwidth': np.logspace(-2, 2, 50)}\n",
"grid = GridSearchCV(KernelDensity(), params)\n",
"grid.fit(data)\n",
"\n",
"print(\"best bandwidth: {0}\".format(grid.best_estimator_.bandwidth))\n",
"\n",
"# use the best estimator to compute the kernel density estimate\n",
"kde = grid.best_estimator_"
]
},
{
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"execution_count": 15,
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"source": [
"kde = dens.evaluate(x)\n",
"plt.fill(x, kde, alpha=.5, label='KDE')"
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"source": [
"## References\n",
"\n",
"### Density estimation:\n",
"\n",
"- B.W. Silverman, “Density Estimation for Statistics and Data Analysis”\n",
"- Hastie, Tibshirani and Friedman, “The Elements of Statistical Learning: Data Mining, Inference, and Prediction”, Springer, 2009.\n",
"- Liu, R., Yang, L. “Kernel estimation of multivariate cumulative distribution function.” Journal of Nonparametric Statistics (2008)"
]
}
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