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<ul>
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<li><a href="#what-is-pyerrors">What is pyerrors?</a>
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<ul>
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<li><a href="#getting-started">Getting started</a></li>
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<li><a href="#basic-example">Basic example</a></li>
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</ul></li>
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<li><a href="#the-obs-class">The <code>Obs</code> class</a>
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<ul>
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@ -118,21 +118,16 @@ It is based on the <strong>gamma method</strong> <a href="https://arxiv.org/abs/
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<li><strong>real and complex matrix operations</strong> and their error propagation based on automatic differentiation (Cholesky decomposition, calculation of eigenvalues and eigenvectors, singular value decomposition...)</li>
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</ul>
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<h2 id="getting-started">Getting started</h2>
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<h2 id="basic-example">Basic example</h2>
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<div class="codehilite"><pre><span></span><code><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
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<span class="kn">import</span> <span class="nn">pyerrors</span> <span class="k">as</span> <span class="nn">pe</span>
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<span class="n">my_obs</span> <span class="o">=</span> <span class="n">pe</span><span class="o">.</span><span class="n">Obs</span><span class="p">([</span><span class="n">samples</span><span class="p">],</span> <span class="p">[</span><span class="s1">'ensemble_name'</span><span class="p">])</span>
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<span class="n">my_new_obs</span> <span class="o">=</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">my_obs</span><span class="p">)</span> <span class="o">/</span> <span class="n">my_obs</span> <span class="o">**</span> <span class="mi">2</span>
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<span class="n">my_new_obs</span><span class="o">.</span><span class="n">gamma_method</span><span class="p">()</span>
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<span class="nb">print</span><span class="p">(</span><span class="n">my_new_obs</span><span class="p">)</span>
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<span class="n">my_obs</span> <span class="o">=</span> <span class="n">pe</span><span class="o">.</span><span class="n">Obs</span><span class="p">([</span><span class="n">samples</span><span class="p">],</span> <span class="p">[</span><span class="s1">'ensemble_name'</span><span class="p">])</span> <span class="c1"># Initialize an Obs object with Monte Carlo samples</span>
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<span class="n">my_new_obs</span> <span class="o">=</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">my_obs</span><span class="p">)</span> <span class="o">/</span> <span class="n">my_obs</span> <span class="o">**</span> <span class="mi">2</span> <span class="c1"># Construct derived Obs object</span>
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<span class="n">my_new_obs</span><span class="o">.</span><span class="n">gamma_method</span><span class="p">()</span> <span class="c1"># Estimate the error with the gamma_method</span>
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<span class="nb">print</span><span class="p">(</span><span class="n">my_new_obs</span><span class="p">)</span> <span class="c1"># Print the result to stdout</span>
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<span class="o">></span> <span class="mf">0.31498</span><span class="p">(</span><span class="mi">72</span><span class="p">)</span>
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<span class="n">iamzero</span> <span class="o">=</span> <span class="n">my_new_obs</span> <span class="o">-</span> <span class="n">my_new_obs</span>
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<span class="n">iamzero</span><span class="o">.</span><span class="n">gamma_method</span><span class="p">()</span>
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<span class="nb">print</span><span class="p">(</span><span class="n">iamzero</span> <span class="o">==</span> <span class="mf">0.0</span><span class="p">)</span>
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<span class="o">></span> <span class="kc">True</span>
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</code></pre></div>
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<h1 id="the-obs-class">The <code>Obs</code> class</h1>
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@ -152,7 +147,7 @@ The second argument is a list containing the names of the respective Monte Carlo
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<h2 id="error-propagation">Error propagation</h2>
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<p>When performing mathematical operations on <code>Obs</code> objects the correct error propagation is intrinsically taken care using a first order Taylor expansion
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$$\delta_f^i=\sum_\alpha \bar{f}_\alpha \delta_\alpha^i\,,\quad \delta_\alpha^i=a_\alpha^i-\bar{a}_\alpha$$
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$$\delta_f^i=\sum_\alpha \bar{f}_\alpha \delta_\alpha^i\,,\quad \delta_\alpha^i=a_\alpha^i-\bar{a}_\alpha\,,$$
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as introduced in <a href="https://arxiv.org/abs/hep-lat/0306017">arXiv:hep-lat/0306017</a>.</p>
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<p>The required derivatives $\bar{f}_\alpha$ are evaluated up to machine precision via automatic differentiation as suggested in <a href="https://arxiv.org/abs/1809.01289">arXiv:1809.01289</a>.</p>
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@ -170,6 +165,11 @@ as introduced in <a href="https://arxiv.org/abs/hep-lat/0306017">arXiv:hep-lat/0
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<span class="n">my_sum</span> <span class="o">=</span> <span class="n">my_obs1</span> <span class="o">+</span> <span class="n">my_obs2</span>
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<span class="n">my_m_eff</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">my_obs1</span> <span class="o">/</span> <span class="n">my_obs2</span><span class="p">)</span>
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<span class="n">iamzero</span> <span class="o">=</span> <span class="n">my_m_eff</span> <span class="o">-</span> <span class="n">my_m_eff</span>
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<span class="c1"># Check that value and fluctuations are zero within machine precision</span>
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<span class="nb">print</span><span class="p">(</span><span class="n">iamzero</span> <span class="o">==</span> <span class="mf">0.0</span><span class="p">)</span>
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<span class="o">></span> <span class="kc">True</span>
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</code></pre></div>
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<h2 id="error-estimation">Error estimation</h2>
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@ -190,7 +190,7 @@ After having arrived at the derived quantity of interest the <code>gamma_method<
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</code></pre></div>
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<p>We use the following definition of the integrated autocorrelation time established in <a href="https://link.springer.com/article/10.1007/BF01022990">Madras & Sokal 1988</a>
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$$\tau_\mathrm{int}=\frac{1}{2}+\sum_{t=1}^{W}\rho(t)\geq \frac{1}{2}$$
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$$\tau_\mathrm{int}=\frac{1}{2}+\sum_{t=1}^{W}\rho(t)\geq \frac{1}{2}\,.$$
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The window $W$ is determined via the automatic windowing procedure described in <a href="https://arxiv.org/abs/hep-lat/0306017">arXiv:hep-lat/0306017</a>
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The standard value for the parameter $S$ of this automatic windowing procedure is $S=2$. Other values for $S$ can be passed to the <code>gamma_method</code> as parameter.</p>
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@ -206,12 +206,6 @@ The standard value for the parameter $S$ of this automatic windowing procedure i
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<p>The integrated autocorrelation time $\tau_\mathrm{int}$ and the autocorrelation function $\rho(W)$ can be monitored via the methods <code><a href="pyerrors/obs.html#Obs.plot_tauint">pyerrors.obs.Obs.plot_tauint</a></code> and <code><a href="pyerrors/obs.html#Obs.plot_tauint">pyerrors.obs.Obs.plot_tauint</a></code>.</p>
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<p>Example:</p>
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<div class="codehilite"><pre><span></span><code><span class="n">my_sum</span><span class="o">.</span><span class="n">plot_tauint</span><span class="p">()</span>
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<span class="n">my_sum</span><span class="o">.</span><span class="n">plot_rho</span><span class="p">()</span>
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</code></pre></div>
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<h3 id="exponential-tails">Exponential tails</h3>
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<p>Slow modes in the Monte Carlo history can be accounted for by attaching an exponential tail to the autocorrelation function $\rho$ as suggested in <a href="https://arxiv.org/abs/1009.5228">arXiv:1009.5228</a>. The longest autocorrelation time in the history, $\tau_\mathrm{exp}$, can be passed to the <code>gamma_method</code> as parameter. In this case the automatic windowing procedure is vacated and the parameter $S$ does not affect the error estimate.</p>
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@ -359,22 +353,17 @@ See <code><a href="pyerrors/obs.html#Obs.export_jackknife">pyerrors.obs.Obs.expo
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<span class="sd">- **non-linear fits with x- and y-errors** and exact linear error propagation based on automatic differentiation as introduced in [arXiv:1809.01289](https://arxiv.org/abs/1809.01289)</span>
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<span class="sd">- **real and complex matrix operations** and their error propagation based on automatic differentiation (Cholesky decomposition, calculation of eigenvalues and eigenvectors, singular value decomposition...)</span>
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<span class="sd">## Getting started</span>
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<span class="sd">## Basic example</span>
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<span class="sd">```python</span>
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<span class="sd">import numpy as np</span>
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<span class="sd">import pyerrors as pe</span>
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<span class="sd">my_obs = pe.Obs([samples], ['ensemble_name'])</span>
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<span class="sd">my_new_obs = 2 * np.log(my_obs) / my_obs ** 2</span>
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<span class="sd">my_new_obs.gamma_method()</span>
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<span class="sd">print(my_new_obs)</span>
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<span class="sd">my_obs = pe.Obs([samples], ['ensemble_name']) # Initialize an Obs object with Monte Carlo samples</span>
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<span class="sd">my_new_obs = 2 * np.log(my_obs) / my_obs ** 2 # Construct derived Obs object</span>
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<span class="sd">my_new_obs.gamma_method() # Estimate the error with the gamma_method</span>
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<span class="sd">print(my_new_obs) # Print the result to stdout</span>
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<span class="sd">> 0.31498(72)</span>
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<span class="sd">iamzero = my_new_obs - my_new_obs</span>
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<span class="sd">iamzero.gamma_method()</span>
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<span class="sd">print(iamzero == 0.0)</span>
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<span class="sd">> True</span>
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<span class="sd">```</span>
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<span class="sd"># The `Obs` class</span>
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<span class="sd">## Error propagation</span>
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<span class="sd">When performing mathematical operations on `Obs` objects the correct error propagation is intrinsically taken care using a first order Taylor expansion</span>
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<span class="sd">$$\delta_f^i=\sum_\alpha \bar{f}_\alpha \delta_\alpha^i\,,\quad \delta_\alpha^i=a_\alpha^i-\bar{a}_\alpha$$</span>
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<span class="sd">$$\delta_f^i=\sum_\alpha \bar{f}_\alpha \delta_\alpha^i\,,\quad \delta_\alpha^i=a_\alpha^i-\bar{a}_\alpha\,,$$</span>
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<span class="sd">as introduced in [arXiv:hep-lat/0306017](https://arxiv.org/abs/hep-lat/0306017).</span>
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<span class="sd">The required derivatives $\bar{f}_\alpha$ are evaluated up to machine precision via automatic differentiation as suggested in [arXiv:1809.01289](https://arxiv.org/abs/1809.01289).</span>
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@ -412,6 +401,11 @@ See <code><a href="pyerrors/obs.html#Obs.export_jackknife">pyerrors.obs.Obs.expo
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<span class="sd">my_sum = my_obs1 + my_obs2</span>
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<span class="sd">my_m_eff = np.log(my_obs1 / my_obs2)</span>
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<span class="sd">iamzero = my_m_eff - my_m_eff</span>
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<span class="sd"># Check that value and fluctuations are zero within machine precision</span>
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<span class="sd">print(iamzero == 0.0)</span>
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<span class="sd">> True</span>
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<span class="sd">```</span>
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<span class="sd">## Error estimation</span>
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@ -433,7 +427,7 @@ See <code><a href="pyerrors/obs.html#Obs.export_jackknife">pyerrors.obs.Obs.expo
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<span class="sd">```</span>
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<span class="sd">We use the following definition of the integrated autocorrelation time established in [Madras & Sokal 1988](https://link.springer.com/article/10.1007/BF01022990)</span>
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<span class="sd">$$\tau_\mathrm{int}=\frac{1}{2}+\sum_{t=1}^{W}\rho(t)\geq \frac{1}{2}$$</span>
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<span class="sd">$$\tau_\mathrm{int}=\frac{1}{2}+\sum_{t=1}^{W}\rho(t)\geq \frac{1}{2}\,.$$</span>
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<span class="sd">The window $W$ is determined via the automatic windowing procedure described in [arXiv:hep-lat/0306017](https://arxiv.org/abs/hep-lat/0306017)</span>
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<span class="sd">The standard value for the parameter $S$ of this automatic windowing procedure is $S=2$. Other values for $S$ can be passed to the `gamma_method` as parameter.</span>
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@ -450,12 +444,6 @@ See <code><a href="pyerrors/obs.html#Obs.export_jackknife">pyerrors.obs.Obs.expo
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<span class="sd">The integrated autocorrelation time $\tau_\mathrm{int}$ and the autocorrelation function $\rho(W)$ can be monitored via the methods `pyerrors.obs.Obs.plot_tauint` and `pyerrors.obs.Obs.plot_tauint`.</span>
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<span class="sd">Example:</span>
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<span class="sd">```python</span>
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<span class="sd">my_sum.plot_tauint()</span>
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<span class="sd">my_sum.plot_rho()</span>
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<span class="sd">```</span>
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<span class="sd">### Exponential tails</span>
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<span class="sd">Slow modes in the Monte Carlo history can be accounted for by attaching an exponential tail to the autocorrelation function $\rho$ as suggested in [arXiv:1009.5228](https://arxiv.org/abs/1009.5228). The longest autocorrelation time in the history, $\tau_\mathrm{exp}$, can be passed to the `gamma_method` as parameter. In this case the automatic windowing procedure is vacated and the parameter $S$ does not affect the error estimate.</span>
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