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{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Basic pyerrors example"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
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"Import pyerrors, as well as autograd wrapped numpy and matplotlib."
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]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"import numpy as np\n",
"import matplotlib.pyplot as plt\n",
"import pyerrors as pe"
]
},
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{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [],
"source": [
"plt.style.use('./base_style.mplstyle')\n",
"plt.rc('text', usetex=True)"
]
},
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{
"cell_type": "markdown",
"metadata": {},
"source": [
"We use numpy to generate some fake data"
]
},
{
"cell_type": "code",
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"execution_count": 3,
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"metadata": {},
"outputs": [],
"source": [
"test_sample1 = np.random.normal(2.0, 0.5, 1000)\n",
"test_sample2 = np.random.normal(1.0, 0.1, 1000)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"From this we can construct `Obs`, which are the basic object of `pyerrors`. For each sample we give to the obs, we also have to specify an ensemble/replica name. In this example we assume that both datasets originate from the same gauge field ensemble labeled 'ens1'."
]
},
{
"cell_type": "code",
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"execution_count": 4,
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"metadata": {},
"outputs": [],
"source": [
"obs1 = pe.Obs([test_sample1], ['ens1'])\n",
"obs2 = pe.Obs([test_sample2], ['ens1'])"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We can now combine these two observables into a third one:"
]
},
{
"cell_type": "code",
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"execution_count": 5,
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"metadata": {},
"outputs": [],
"source": [
"obs3 = np.log(obs1 ** 2 / obs2 ** 4)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"`pyerrors` overloads all basic math operations, the user can work with these `Obs` as if they were real numbers. The proper resampling is performed in the background via automatic differentiation."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"If we are now interested in the error of obs3, we can use the `gamma_method` to compute it and then print the object to the notebook"
]
},
{
"cell_type": "code",
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"execution_count": 6,
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"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
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"Obs[1.387(19)]\n"
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]
}
],
"source": [
"obs3.gamma_method()\n",
"print(obs3)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"With print level 1 we can take a look at the integrated autocorrelation time estimated by the automatic windowing procedure."
]
},
{
"cell_type": "code",
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"execution_count": 7,
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"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
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"Result\t 1.38669742e+00 +/- 1.94840399e-02 +/- 9.74201997e-04 (1.405%)\n",
" t_int\t 5.01998002e-01 +/- 4.47213596e-02 S = 2.00\n"
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]
}
],
"source": [
"obs3.print(1)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"As expected the random data from numpy exhibits no autocorrelation ($\\tau_\\text{int}\\,\\approx0.5$). It can still be interesting to have a look at the window size dependence of the integrated autocorrelation time"
]
},
{
"cell_type": "code",
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"execution_count": 8,
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"metadata": {},
"outputs": [
{
"data": {
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"image/png": "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"text/plain": [
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"<Figure size 640x395.55 with 1 Axes>"
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]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"obs3.plot_tauint()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"This figure shows the windowsize dependence of the integrated autocorrelation time. The red vertical line signalizes the window chosen by the automatic windowing procedure with $S=2.0$.\n",
"Choosing a larger windowsize would not significantly alter $\\tau_\\text{int}$, so everything seems to be correct here."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Correlated data"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We can now generate fake data with given covariance matrix and integrated autocorrelation times:"
]
},
{
"cell_type": "code",
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"execution_count": 9,
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"metadata": {},
"outputs": [],
"source": [
"cov = np.array([[0.5, -0.2], [-0.2, 0.3]]) # Covariance matrix\n",
"tau = [4, 8] # Autocorrelation times\n",
"c_obs1, c_obs2 = pe.misc.gen_correlated_data([2.8, 2.1], cov, 'ens1', tau)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"and once again combine the two `Obs` to a new one with arbitrary math operations"
]
},
{
"cell_type": "code",
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"execution_count": 10,
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"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
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"Result\t 3.27194697e-01 +/- 1.79228480e+00 +/- 3.07835024e-01 (547.773%)\n",
" t_int\t 5.31748262e+00 +/- 1.57262234e+00 S = 2.00\n"
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]
}
],
"source": [
"c_obs3 = np.sin(c_obs1 / c_obs2 - 1)\n",
"c_obs3.gamma_method()\n",
"c_obs3.print()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"This time we see a significant autocorrelation so it is worth to have a closer look at the normalized autocorrelation function (rho) and the integrated autocorrelation time"
]
},
{
"cell_type": "code",
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"execution_count": 11,
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"metadata": {},
"outputs": [
{
"data": {
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"image/png": "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"text/plain": [
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"<Figure size 640x395.55 with 1 Axes>"
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]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
},
{
"data": {
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"image/png": "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"text/plain": [
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"<Figure size 640x395.55 with 1 Axes>"
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]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"c_obs3.plot_rho()\n",
"c_obs3.plot_tauint()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We can now redo the error analysis and alter the value of S or attach a tail to the autocorrelation function to take into account long range autocorrelations"
]
},
{
"cell_type": "code",
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"execution_count": 12,
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"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
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"Result\t 3.27194697e-01 +/- 1.88231459e+00 +/- 2.01855751e-01 (575.289%)\n",
" t_int\t 5.86511391e+00 +/- 2.16269625e+00 tau_exp = 20.00, N_sigma = 1\n"
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]
},
{
"data": {
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"image/png": "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"text/plain": [
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"<Figure size 640x395.55 with 1 Axes>"
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]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"c_obs3.gamma_method(tau_exp=20)\n",
"c_obs3.print()\n",
"c_obs3.plot_tauint()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Jackknife"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"For comparison and as a crosscheck, we can do a jackknife binning analysis. We compare the result for different binsizes with the result from the gamma method. Besides the more robust approach of the gamma method, it can also be shown that the systematic error of the error decreases faster with $N$ in comparison to the binning approach (see hep-lat/0306017)"
]
},
{
"cell_type": "code",
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"execution_count": 13,
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"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Binning analysis:\n",
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"Result:\t 3.27194697e-01 +/- 1.30323584e+00 +/- 1.74847436e-01 (398.306%)\n",
"Result:\t 3.27194697e-01 +/- 1.42921199e+00 +/- 3.13124657e-01 (436.808%)\n",
"Result:\t 3.27194697e-01 +/- 1.36761713e+00 +/- 4.28131883e-01 (417.983%)\n"
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]
},
{
"data": {
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"image/png": "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"text/plain": [
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"<Figure size 640x395.55 with 1 Axes>"
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]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"Result from the automatic windowing procedure for comparison:\n",
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"Result\t 3.27194697e-01 +/- 1.78414777e+00 +/- 2.73504675e-01 (545.286%)\n",
" t_int\t 5.26930916e+00 +/- 1.36902941e+00 S = 1.50\n",
"Result\t 3.27194697e-01 +/- 1.79228480e+00 +/- 3.07835024e-01 (547.773%)\n",
" t_int\t 5.31748262e+00 +/- 1.57262234e+00 S = 2.00\n",
"Result\t 3.27194697e-01 +/- 1.67905409e+00 +/- 3.16358031e-01 (513.167%)\n",
" t_int\t 4.66682386e+00 +/- 1.53936903e+00 S = 3.00\n"
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]
},
{
"data": {
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"text/plain": [
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"<Figure size 640x395.55 with 1 Axes>"
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]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"import pyerrors.jackknifing as jn\n",
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"jack1 = jn.generate_jack(c_obs1, max_binsize=50)\n",
"jack2 = jn.generate_jack(c_obs2, max_binsize=50)\n",
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"jack3 = jn.derived_jack(lambda x: np.sin(x[0] / x[1] - 1), [jack1, jack2])\n",
"\n",
"print('Binning analysis:')\n",
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"jack3.print(binsize=10)\n",
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"jack3.print(binsize=25)\n",
"jack3.print(binsize=50)\n",
"\n",
"jack3.plot_tauint()\n",
"\n",
"print('Result from the automatic windowing procedure for comparison:')\n",
"c_obs3.gamma_method(S=1.5)\n",
"c_obs3.print()\n",
"c_obs3.gamma_method(S=2)\n",
"c_obs3.print()\n",
"c_obs3.gamma_method(S=3)\n",
"c_obs3.print()\n",
"\n",
"c_obs3.gamma_method(S=2)\n",
"c_obs3.plot_tauint()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"For this specific example the binned Jackknife procedure seems to underestimate the final error, the deduced intergrated autocorrelation time depends strongly on the chosen binsize. The automatic windowing procedure displayed for comparison gives more robust results for this example."
]
}
],
"metadata": {
"kernelspec": {
"display_name": "Python 3",
"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",
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"version": "3.6.9"
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}
},
"nbformat": 4,
"nbformat_minor": 4
}
