Generated gh-pages for commit 58cbb5a

Merge: 252fc58 a024fbe
Author: John Jakeman <[email protected]>

    Merge remote-tracking branch 'origin/devel'

_downloads/059572d8fc6c0ec9b2cfc587871e72dc/plot_multioutput_acv.ipynb

@@ -0,0 +1,68 @@
+{
+  "cells": [
+    {
+      "cell_type": "code",
+      "execution_count": null,
+      "metadata": {
+        "collapsed": false
+      },
+      "outputs": [],
+      "source": [
+        "Add latex macros$$\\newcommand{\\V}[1]{{\\boldsymbol{#1}}}\\newcommand{mean}[1]{{\\mathbb{E}\\left[#1\\right]}}\\newcommand{var}[1]{{\\mathbb{V}\\left[#1\\right]}}\\newcommand{covar}[2]{\\mathbb{C}\\text{ov}\\left[#1,#2\\right]}\\newcommand{corr}[2]{\\mathbb{C}\\text{or}\\left[#1,#2\\right]}\\newcommand{argmin}{\\mathrm{argmin}}\\def\\rv{z}\\def\\reals{\\mathbb{R}}\\def\\rvset{{\\mathcal{Z}}}\\def\\pdf{\\rho}\\def\\rvdom{\\Gamma}\\def\\coloneqq{\\colon=}\\newcommand{norm}{\\lVert #1 \\rVert}\\def\\argmax{\\operatorname{argmax}}\\def\\ai{\\alpha}\\def\\bi{\\beta}\\newcommand{\\dx}[1]{\\;\\text{d}#1}\\newcommand{\\mat}[1]{{\\boldsymbol{\\mathrm{#1}}}}$$"
+      ]
+    },
+    {
+      "cell_type": "markdown",
+      "metadata": {},
+      "source": [
+        "\n# Multioutput Approximate Control Variates\n\nThis tutorial demonstrates how computing statistics for multiple outputs simultaneoulsy can improve the accuracy of ACV estimates of individual statistics when compared to ACV applied to each output separately.\n\nThe optimal control variate weights are obtained by minimizing the estimator covariance [RM1985]_.\n"
+      ]
+    },
+    {
+      "cell_type": "code",
+      "execution_count": null,
+      "metadata": {
+        "collapsed": false
+      },
+      "outputs": [],
+      "source": [
+        "import numpy as np\nimport matplotlib.pyplot as plt\n\nfrom pyapprox import multifidelity as mf\nfrom pyapprox.benchmarks import setup_benchmark\nfrom pyapprox.util.visualization import mathrm_labels, mathrm_label\n\nnp.random.seed(1)\nbenchmark = setup_benchmark(\"multioutput_model_ensemble\")\ncosts = np.array([1, 0.01, 0.001])\nnmodels = 3\n\ncov = benchmark.covariance\n\nlabels = ([r\"$(f_{0})_{%d}$\" % (ii+1) for ii in range(benchmark.nqoi)] +\n          [r\"$(f_{2})_{%d}$\" % (ii+1) for ii in range(benchmark.nqoi)] +\n          [r\"$(f_{2})_{%d}$\" % (ii+1) for ii in range(benchmark.nqoi)])\nax = plt.subplots(1, 1, figsize=(8, 6))[1]\n_ = mf.plot_correlation_matrix(\n    mf.get_correlation_from_covariance(cov), ax=ax, model_names=labels,\n    label_fontsize=20)\n\ntarget_cost = 10\nstat = mf.multioutput_stats[\"mean\"](benchmark.nqoi)\nstat.set_pilot_quantities(cov)\nest = mf.get_estimator(\"gmf\", stat, costs)\nest.allocate_samples(target_cost)\n\n# get covariance of just first qoi\nqoi_idx = [0]\ncov_0 = stat.get_pilot_quantities_subset(\n    nmodels, benchmark.nqoi, [0, 1, 2], qoi_idx)[0]\nstat_0 = mf.multioutput_stats[\"mean\"](benchmark.nqoi)\nstat_0.set_pilot_quantities(cov_0)\nest_0 = mf.get_estimator(\"gmf\", stat_0, costs)\nest_0.allocate_samples(target_cost)\n\nest_labels = mathrm_labels([\"MOACV\", \"SOACV\"])\n\n# only works if qoi_idx = [0]\nfrom pyapprox.multifidelity.factory import ComparisonCriteria\nclass CustomComparisionCriteria(ComparisonCriteria):\n    def __call__(self, est_covariance, est):\n        return est_covariance[0, 0]\n\n\nax = plt.subplots(1, 1, figsize=(8, 6))[1]\n_ = mf.plot_estimator_variance_reductions(\n    [est, est_0], est_labels, ax, criteria=CustomComparisionCriteria())"
+      ]
+    },
+    {
+      "cell_type": "markdown",
+      "metadata": {},
+      "source": [
+        "## Video\nClick on the image below to view a video tutorial on multi-output approximate control variate Monte Carlo quadrature\n\n<img src=\"file://../../figures/multi-output-acv-thumbnail.png\" target=\"https://youtu.be/astvKKFh2yA?si=8vgmKRbjdhJYeUfq\">\n\n"
+      ]
+    },
+    {
+      "cell_type": "markdown",
+      "metadata": {},
+      "source": [
+        "## References\n.. [RM1985] [Reuven Y. Rubinstein and Ruth Marcus. Efficiency of multivariate control variates in monte carlo simulation. Operations Research, 33(3):661\u2013677, 1985.](https://doi.org/10.48550/arXiv.2310.00125)\n\n"
+      ]
+    }
+  ],
+  "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",
+      "version": "3.10.13"
+    }
+  },
+  "nbformat": 4,
+  "nbformat_minor": 0
+}

_downloads/0879eb12822e0af4ff688a990073fe18/plot_low_discrepancy_quadrature.ipynb

@@ -8,7 +8,7 @@
       },
       "outputs": [],
       "source": [
-        "Add latex macros$$\\newcommand{\\V}[1]{{\\boldsymbol{#1}}}\\newcommand{mean}[1]{{\\mathbb{E}\\left[#1\\right]}}\\newcommand{var}[1]{{\\mathbb{V}\\left[#1\\right]}}\\newcommand{covar}[2]{\\mathbb{C}\\text{ov}\\left[#1,#2\\right]}\\newcommand{corr}[2]{\\mathbb{C}\\text{or}\\left[#1,#2\\right]}\\newcommand{argmin}{\\mathrm{argmin}}\\def\\rv{z}\\def\\reals{\\mathbb{R}}\\def\\pdf{\\rho}\\def\\rvdom{\\Gamma}\\def\\coloneqq{\\colon=}\\newcommand{norm}{\\lVert #1 \\rVert}\\def\\argmax{\\operatorname{argmax}}\\def\\ai{\\alpha}\\def\\bi{\\beta}\\newcommand{\\dx}[1]{\\;\\mathrm{d}#1}$$"
+        "Add latex macros$$\\newcommand{\\V}[1]{{\\boldsymbol{#1}}}\\newcommand{mean}[1]{{\\mathbb{E}\\left[#1\\right]}}\\newcommand{var}[1]{{\\mathbb{V}\\left[#1\\right]}}\\newcommand{covar}[2]{\\mathbb{C}\\text{ov}\\left[#1,#2\\right]}\\newcommand{corr}[2]{\\mathbb{C}\\text{or}\\left[#1,#2\\right]}\\newcommand{argmin}{\\mathrm{argmin}}\\def\\rv{z}\\def\\reals{\\mathbb{R}}\\def\\rvset{{\\mathcal{Z}}}\\def\\pdf{\\rho}\\def\\rvdom{\\Gamma}\\def\\coloneqq{\\colon=}\\newcommand{norm}{\\lVert #1 \\rVert}\\def\\argmax{\\operatorname{argmax}}\\def\\ai{\\alpha}\\def\\bi{\\beta}\\newcommand{\\dx}[1]{\\;\\text{d}#1}\\newcommand{\\mat}[1]{{\\boldsymbol{\\mathrm{#1}}}}$$"
       ]
     },
     {
@@ -62,7 +62,7 @@
       },
       "outputs": [],
       "source": [
-        "from pyapprox.util.configure_plots import plt\nplt.plot(mc_samples[0, :], mc_samples[1, :], 'ko', label=\"MC\")\nplt.plot(sobol_samples[0, :], sobol_samples[1, :], 'rs', label=\"Sobol\")\nplt.legend()\nplt.show()\n\n#\n#Halton Sequences\n#================\n#Pyapprox also supports Halton Sequences\nhalton_samples = expdesign.halton_sequence(\n    benchmark.variable.num_vars(), nsamples, variable=benchmark.variable)\nvalues = benchmark.fun(halton_samples)\nprint_statistics(halton_samples, values)"
+        "import matplotlib.pyplot as plt\nplt.plot(mc_samples[0, :], mc_samples[1, :], 'ko', label=\"MC\")\nplt.plot(sobol_samples[0, :], sobol_samples[1, :], 'rs', label=\"Sobol\")\nplt.legend()\nplt.show()\n\n#\n#Halton Sequences\n#================\n#Pyapprox also supports Halton Sequences\nhalton_samples = expdesign.halton_sequence(\n    benchmark.variable.num_vars(), nsamples, variable=benchmark.variable)\nvalues = benchmark.fun(halton_samples)\nprint_statistics(halton_samples, values)"
       ]
     },
     {
@@ -89,7 +89,7 @@
       "name": "python",
       "nbconvert_exporter": "python",
       "pygments_lexer": "ipython3",
-      "version": "3.9.17"
+      "version": "3.10.13"
     }
   },
   "nbformat": 4,

_downloads/0ab8cf693453cc1c9b15d4fe2f93dccc/plot_low_discrepancy_quadrature.py

@@ -45,7 +45,7 @@
 #
 #Low-discrepancy sequences are typically more evenly space over the parameter
 #space. This can be seen by comparing the Monte Carlo and Sobol sequence samples
-from pyapprox.util.configure_plots import plt
+import matplotlib.pyplot as plt
 plt.plot(mc_samples[0, :], mc_samples[1, :], 'ko', label="MC")
 plt.plot(sobol_samples[0, :], sobol_samples[1, :], 'rs', label="Sobol")
 plt.legend()

_downloads/0f514da924c449ee666cf7265e7e7121/plot_gaussian_mfnets.ipynb

@@ -8,7 +8,7 @@
       },
       "outputs": [],
       "source": [
-        "Add latex macros$$\\newcommand{\\V}[1]{{\\boldsymbol{#1}}}\\newcommand{mean}[1]{{\\mathbb{E}\\left[#1\\right]}}\\newcommand{var}[1]{{\\mathbb{V}\\left[#1\\right]}}\\newcommand{covar}[2]{\\mathbb{C}\\text{ov}\\left[#1,#2\\right]}\\newcommand{corr}[2]{\\mathbb{C}\\text{or}\\left[#1,#2\\right]}\\newcommand{argmin}{\\mathrm{argmin}}\\def\\rv{z}\\def\\reals{\\mathbb{R}}\\def\\pdf{\\rho}\\def\\rvdom{\\Gamma}\\def\\coloneqq{\\colon=}\\newcommand{norm}{\\lVert #1 \\rVert}\\def\\argmax{\\operatorname{argmax}}\\def\\ai{\\alpha}\\def\\bi{\\beta}\\newcommand{\\dx}[1]{\\;\\mathrm{d}#1}$$"
+        "Add latex macros$$\\newcommand{\\V}[1]{{\\boldsymbol{#1}}}\\newcommand{mean}[1]{{\\mathbb{E}\\left[#1\\right]}}\\newcommand{var}[1]{{\\mathbb{V}\\left[#1\\right]}}\\newcommand{covar}[2]{\\mathbb{C}\\text{ov}\\left[#1,#2\\right]}\\newcommand{corr}[2]{\\mathbb{C}\\text{or}\\left[#1,#2\\right]}\\newcommand{argmin}{\\mathrm{argmin}}\\def\\rv{z}\\def\\reals{\\mathbb{R}}\\def\\rvset{{\\mathcal{Z}}}\\def\\pdf{\\rho}\\def\\rvdom{\\Gamma}\\def\\coloneqq{\\colon=}\\newcommand{norm}{\\lVert #1 \\rVert}\\def\\argmax{\\operatorname{argmax}}\\def\\ai{\\alpha}\\def\\bi{\\beta}\\newcommand{\\dx}[1]{\\;\\text{d}#1}\\newcommand{\\mat}[1]{{\\boldsymbol{\\mathrm{#1}}}}$$"
       ]
     },
     {
@@ -26,7 +26,7 @@
       },
       "outputs": [],
       "source": [
-        "import networkx as nx\nimport numpy as np\nfrom scipy import stats\nimport scipy\n\nfrom pyapprox.util.configure_plots import plt\nfrom pyapprox.bayes.laplace import (\n    laplace_posterior_approximation_for_linear_models\n)\nfrom pyapprox.bayes.gaussian_network import (\n    get_total_degree_polynomials, plot_1d_lvn_approx, GaussianNetwork,\n    cond_prob_variable_elimination,\n    convert_gaussian_from_canonical_form,\n    plot_peer_network_with_data\n)\n\nnp.random.seed(2)"
+        "import networkx as nx\nimport numpy as np\nimport scipy\nfrom scipy import stats\nimport matplotlib.pyplot as plt\n\nfrom pyapprox.bayes.laplace import (\n    laplace_posterior_approximation_for_linear_models)\nfrom pyapprox.bayes.gaussian_network import (\n    get_total_degree_polynomials, plot_1d_lvn_approx, GaussianNetwork,\n    cond_prob_variable_elimination,\n    convert_gaussian_from_canonical_form,\n    plot_peer_network_with_data)\n\nnp.random.seed(2)"
       ]
     },
     {
@@ -332,7 +332,7 @@
       "cell_type": "markdown",
       "metadata": {},
       "source": [
-        "### References\n.. [GJGEIJUQ2020] [A. Gorodetsky et al. MFNets: Multi-fidelity data-driven networks for bayesian learning and prediction, International Journal for Uncertainty Quantification, 2020.](https://www.alexgorodetsky.com/static/papers/gorodetsky_jakeman_geraci_eldred_mfnets_2020.pdf)\n\n.. [GJGJCP2020] [A. Gorodetsky et al. MFNets: Learning network representations for multifidelity surrogate modeling, 2020.](https://res.arxiv.org/abs/2008.02672)\n\n### Appendix\nThere is a strong connection between the mean of the Bayes posterior distribution of linear-Gaussian models with least squares regression. Specifically the mean of the posterior is equivalent to linear least-squares regression with a regulrization that penalizes deviations from the prior estimate of the parameters. Let the least squares objective function be\n\n\\begin{align}f(\\theta)=\\frac{1}{2}(y-A\\theta)^\\top\\Sigma_\\epsilon^{-1}(y-A\\theta)+\\frac{1}{2}(\\mu_\\theta-\\theta)^\\top\\Sigma_\\theta^{-1}(\\mu_\\theta-\\theta),\\end{align}\n\nwhere the first term on the right hand side is the usual least squares objective and the second is the regularization term. This regularized objective is minimized by setting its gradient to zero, i.e.\n\n\\begin{align}\\nabla_\\theta f(\\theta)=A^\\top\\Sigma_\\epsilon^{-1}(y-A\\theta)+\\Sigma_\\theta^{-1}(\\mu_\\theta-\\theta)=0,\\end{align}\n\nthus\n\n\\begin{align}A^\\top\\Sigma_\\epsilon^{-1}A\\theta+\\Sigma_\\theta^{-1}\\theta=A^\\top\\Sigma_\\epsilon^{-1}y+\\Sigma_\\theta^{-1}\\mu_\\theta\\end{align}\n\nand so\n\n\\begin{align}\\theta=\\left(A^\\top\\Sigma_\\epsilon^{-1}A\\theta+\\Sigma_\\theta^{-1}\\right)^{-1}\\left(A^\\top\\Sigma_\\epsilon^{-1}y+\\Sigma_\\theta^{-1}\\mu_\\theta\\right).\\end{align}\n\nNoting that $\\left(A^\\top\\Sigma_\\epsilon^{-1}A\\theta+\\Sigma_\\theta^{-1}\\right)^{-1}$ is the posterior covariance we obtain the usual expression for the posterior mean\n\n\\begin{align}\\mu^\\mathrm{post}=\\Sigma^\\mathrm{post}\\left(A^\\top\\Sigma_\\epsilon^{-1}y+\\Sigma_\\theta^{-1}\\mu_\\theta\\right)\\end{align}\n\n"
+        "### References\n.. [GJGEIJUQ2020] [A. Gorodetsky et al. MFNets: Multi-fidelity data-driven networks for bayesian learning and prediction, International Journal for Uncertainty Quantification, 2020.](https://www.alexgorodetsky.com/static/papers/gorodetsky_jakeman_geraci_eldred_mfnets_2020.pdf)\n\n..\n .. [GJGJCP2020] [A. Gorodetsky et al. MFNets: Learning network representations for multifidelity surrogate modeling, 2020.](https://res.arxiv.org/abs/2008.02672)\n\n### Appendix\nThere is a strong connection between the mean of the Bayes posterior distribution of linear-Gaussian models with least squares regression. Specifically the mean of the posterior is equivalent to linear least-squares regression with a regulrization that penalizes deviations from the prior estimate of the parameters. Let the least squares objective function be\n\n\\begin{align}f(\\theta)=\\frac{1}{2}(y-A\\theta)^\\top\\Sigma_\\epsilon^{-1}(y-A\\theta)+\\frac{1}{2}(\\mu_\\theta-\\theta)^\\top\\Sigma_\\theta^{-1}(\\mu_\\theta-\\theta),\\end{align}\n\nwhere the first term on the right hand side is the usual least squares objective and the second is the regularization term. This regularized objective is minimized by setting its gradient to zero, i.e.\n\n\\begin{align}\\nabla_\\theta f(\\theta)=A^\\top\\Sigma_\\epsilon^{-1}(y-A\\theta)+\\Sigma_\\theta^{-1}(\\mu_\\theta-\\theta)=0,\\end{align}\n\nthus\n\n\\begin{align}A^\\top\\Sigma_\\epsilon^{-1}A\\theta+\\Sigma_\\theta^{-1}\\theta=A^\\top\\Sigma_\\epsilon^{-1}y+\\Sigma_\\theta^{-1}\\mu_\\theta\\end{align}\n\nand so\n\n\\begin{align}\\theta=\\left(A^\\top\\Sigma_\\epsilon^{-1}A\\theta+\\Sigma_\\theta^{-1}\\right)^{-1}\\left(A^\\top\\Sigma_\\epsilon^{-1}y+\\Sigma_\\theta^{-1}\\mu_\\theta\\right).\\end{align}\n\nNoting that $\\left(A^\\top\\Sigma_\\epsilon^{-1}A\\theta+\\Sigma_\\theta^{-1}\\right)^{-1}$ is the posterior covariance we obtain the usual expression for the posterior mean\n\n\\begin{align}\\mu^\\mathrm{post}=\\Sigma^\\mathrm{post}\\left(A^\\top\\Sigma_\\epsilon^{-1}y+\\Sigma_\\theta^{-1}\\mu_\\theta\\right)\\end{align}\n\n"
       ]
     }
   ],
@@ -352,7 +352,7 @@
       "name": "python",
       "nbconvert_exporter": "python",
       "pygments_lexer": "ipython3",
-      "version": "3.9.17"
+      "version": "3.10.13"
     }
   },
   "nbformat": 4,

_downloads/1126141c04021d5845d61b810ec2c767/plot_bayesian_inference.ipynb

@@ -8,7 +8,7 @@
       },
       "outputs": [],
       "source": [
-        "Add latex macros$$\\newcommand{\\V}[1]{{\\boldsymbol{#1}}}\\newcommand{mean}[1]{{\\mathbb{E}\\left[#1\\right]}}\\newcommand{var}[1]{{\\mathbb{V}\\left[#1\\right]}}\\newcommand{covar}[2]{\\mathbb{C}\\text{ov}\\left[#1,#2\\right]}\\newcommand{corr}[2]{\\mathbb{C}\\text{or}\\left[#1,#2\\right]}\\newcommand{argmin}{\\mathrm{argmin}}\\def\\rv{z}\\def\\reals{\\mathbb{R}}\\def\\pdf{\\rho}\\def\\rvdom{\\Gamma}\\def\\coloneqq{\\colon=}\\newcommand{norm}{\\lVert #1 \\rVert}\\def\\argmax{\\operatorname{argmax}}\\def\\ai{\\alpha}\\def\\bi{\\beta}\\newcommand{\\dx}[1]{\\;\\mathrm{d}#1}$$"
+        "Add latex macros$$\\newcommand{\\V}[1]{{\\boldsymbol{#1}}}\\newcommand{mean}[1]{{\\mathbb{E}\\left[#1\\right]}}\\newcommand{var}[1]{{\\mathbb{V}\\left[#1\\right]}}\\newcommand{covar}[2]{\\mathbb{C}\\text{ov}\\left[#1,#2\\right]}\\newcommand{corr}[2]{\\mathbb{C}\\text{or}\\left[#1,#2\\right]}\\newcommand{argmin}{\\mathrm{argmin}}\\def\\rv{z}\\def\\reals{\\mathbb{R}}\\def\\rvset{{\\mathcal{Z}}}\\def\\pdf{\\rho}\\def\\rvdom{\\Gamma}\\def\\coloneqq{\\colon=}\\newcommand{norm}{\\lVert #1 \\rVert}\\def\\argmax{\\operatorname{argmax}}\\def\\ai{\\alpha}\\def\\bi{\\beta}\\newcommand{\\dx}[1]{\\;\\text{d}#1}\\newcommand{\\mat}[1]{{\\boldsymbol{\\mathrm{#1}}}}$$"
       ]
     },
     {
@@ -222,7 +222,7 @@
       "name": "python",
       "nbconvert_exporter": "python",
       "pygments_lexer": "ipython3",
-      "version": "3.9.17"
+      "version": "3.10.13"
     }
   },
   "nbformat": 4,