From 379b65d9090b7e2308061d21a2134f0f413dba9c Mon Sep 17 00:00:00 2001 From: Steve Mattis Date: Fri, 3 Jul 2015 16:13:56 -0500 Subject: [PATCH 1/7] adds contaminant transport example --- .../contaminantTransport.rst | 162 ++++++++++++++++++ doc/examples/overview.rst | 6 +- examples/contaminantTransport/contaminant.py | 107 ++++++++++++ .../contaminantTransport/files/Q_ref.txt.gz | Bin 0 -> 142 bytes .../contaminantTransport/files/data.txt.gz | Bin 0 -> 724090 bytes .../files/lam_domain.txt.gz | Bin 0 -> 109 bytes .../contaminantTransport/files/lam_ref.txt.gz | Bin 0 -> 79 bytes .../contaminantTransport/files/samples.txt.gz | Bin 0 -> 532215 bytes 8 files changed, 274 insertions(+), 1 deletion(-) create mode 100644 doc/examples/example_rst_files/contaminantTransport.rst create mode 100644 examples/contaminantTransport/contaminant.py create mode 100644 examples/contaminantTransport/files/Q_ref.txt.gz create mode 100644 examples/contaminantTransport/files/data.txt.gz create mode 100644 examples/contaminantTransport/files/lam_domain.txt.gz create mode 100644 examples/contaminantTransport/files/lam_ref.txt.gz create mode 100644 examples/contaminantTransport/files/samples.txt.gz diff --git a/doc/examples/example_rst_files/contaminantTransport.rst b/doc/examples/example_rst_files/contaminantTransport.rst new file mode 100644 index 00000000..5520a392 --- /dev/null +++ b/doc/examples/example_rst_files/contaminantTransport.rst @@ -0,0 +1,162 @@ +.. _contaminantTransport: + + +============================================================================ +Example: Concentration of Contaminant in Wells Based on Transport Parameters +=========================================================================== + +We will walk through the following `example +`. +This example takes uniformly distributed samples of parameters and +output data from a simple groundwater contaminant transport model, +and calculates solutions to the stochastic inverse problem. +The parameter domain is 5D, where the uncertain parameters are the x and y +locations of the source, the horizontal dispersivity, the flow angle, +and the contaminant flux. There are 11 measured QoIs in the data space +available. By changing the choice of QoIs that we use to solve the stochastic +inverse problem, we see the effect of geometric distinctness. +Probabilities in the parameter space are +calculated using the MC assumption. 1D and 2D marginals are calculated, +smoothed, and plotted. + +Import the necessary modules:: + + import numpy as np + import bet.calculateP as calculateP + import bet.postProcess as postProcess + import bet.calculateP.simpleFunP as simpleFunP + import bet.calculateP.calculateP as calculateP + import bet.postProcess.plotP as plotP + import bet.postProcess.plotDomains as plotD + + +Import the samples, data, reference data, and reference parameters:: + + lam_domain = np.loadtxt("files/lam_domain.txt.gz") #parameter domain + ref_lam = np.loadtxt("files/lam_ref.txt.gz") #reference parameter set + Q_ref = np.loadtxt("files/Q_ref.txt.gz") #reference QoI set + samples = np.loadtxt("files/samples.txt.gz") # uniform samples in parameter domain + data = np.loadtxt("files/data.txt.gz") # data from model + +Choose which subset of the 11 QoIs to use for inversion:: + + QoI_indices=[7,8,9,10] # Indices for output data with which you want to invert + +Choose the bin ratio for the uniform output probability:: + + bin_ratio = 0.25 #ratio of length of data region to invert + +Slice data and form approximate PDF for output for plotting:: + data = data[:,QoI_indices] + Q_ref=Q_ref[QoI_indices] + + dmax = data.max(axis=0) + dmin = data.min(axis=0) + dscale = bin_ratio*(dmax-dmin) + Qmax = Q_ref + 0.5*dscale + Qmin = Q_ref -0.5*dscale + def rho_D(x): + return np.all(np.logical_and(np.greater(x,Qmin), np.less(x,Qmax)),axis=1) + +Plot 2D projections of the data domain:: + + plotD.show_data(data, Q_ref = Q_ref, rho_D=rho_D, showdim=2) + +Suggested changes for user (1) +------------------------------ + +Change the ``QoI_indices`` and note how it changes the plots of the data +domain. The white points are ones with zero probability and the dark points +are those with nonzero probability. + + +Suggested changes for user (2) +------------------------------ + +Try different ways of discretizing the probability measure on +:math:`\mathcal{D}` defined as a uniform probability measure on a rectangle +(since :math:`\mathcal{D}` is 2-dimensional). + +* unif_unif creates a uniform measure on a hyperbox with dimensions relative to the size of the circumscribed hyperbox of the set :math:`\mathcal{D}` using the bin_ratio. A total of M samples are drawn within a slightly larger scaled hyperbox to discretize this measure defining M total generalized contour events in Lambda. The reason a slightly larger scaled hyperbox is used to draw the samples to discretize :math:`\mathcal{D}` is because otherwise every generalized contour event will have non-zero probability which obviously defeats the purpose of "localizing" the probability within a subset of :math:`\mathcal{D}`. + +* uniform_hyperrectangle uses the same measure defined in the same way as unif_unif, but the difference is in the discretization which is on a regular grid defined by ``center_pts_per_edge``. If ``center_pts_per_edge = 1``, then the contour event corresponding to the entire support of :math:`\rho_\mathcal{D}` is approximated as a single event. This is done by carefully placing a regular 3x3 grid (since :math:`dim(\mathcal{D})=2` in this case) of points in :math:`\mathcal{D}` with the center point of the grid in the center of the support of the measure and the other points placed outside of the rectangle defining the support to define a total of 9 contour events with 8 of them having exactly zero probability. + +Create a simple function approximation of the probablity measure on +:math:`\mathcal{D}`:: + + deterministic_discretize_D = True + + if deterministic_discretize_D == True: + (d_distr_prob, d_distr_samples, d_Tree) = simpleFunP.uniform_hyperrectangle(data=data, + Q_ref=Q_ref, bin_ratio=0.2, center_pts_per_edge = 1) + else: + (d_distr_prob, d_distr_samples, d_Tree) = simpleFunP.unif_unif(data=data, + Q_ref=Q_ref, M=50, bin_ratio=0.2, num_d_emulate=1E5) + + +Calculate probablities using the MC assumption:: + + (P, lam_vol, io_ptr) = calculateP.prob(samples=samples, + data=data, + rho_D_M=d_distr_prob, + d_distr_samples=d_distr_samples) + + +Calculate 2D marginal probs - Suggested changes for user (3) +------------------------------------------------------------- + +At this point, the only thing that should change in the plotP.* inputs +should be either the nbins values or sigma (which influences the kernel +density estimation with smaller values implying a density estimate that +looks more like a histogram and larger values smoothing out the values +more). + +There are ways to determine "optimal" smoothing parameters (e.g., see CV, GCV, +and other similar methods), but we have not incorporated these into the code +as lower-dimensional marginal plots have limited value in understanding the +structure of a high dimensional non-parametric probability measure. + +Plot the marginal probabilities:: + + (bins, marginals2D) = plotP.calculate_2D_marginal_probs(P_samples = P, samples = samples, lam_domain = lam_domain, nbins = [10, 10, 10]) + +Smooth 2d marginals probs (optional):: + + marginals2D = plotP.smooth_marginals_2D(marginals2D,bins, sigma=1.0) + +Plot 2d marginals probs:: + + plotP.plot_2D_marginal_probs(marginals2D, bins, lam_domain, filename = "contaminant_map", interactive=False, lam_ref=ref_lam, lambda_labels=labels) + +Calculate 1d marginal probs:: + + (bins, marginals1D) = plotP.calculate_1D_marginal_probs(P_samples = P, samples = samples, lam_domain = lam_domain, nbins = [10, 10, 10]) + +Smooth 1d marginal probs (optional):: + + marginals1D = plotP.smooth_marginals_1D(marginals1D, bins, sigma=1.0) + +Plot 1d marginal probs:: + + plotP.plot_1D_marginal_probs(marginals1D, bins, lam_domain, filename = "contaminant_map", interactive=False, lam_ref=ref_lam, lambda_labels=labels) + +Sort samples by highest probability density and take highest x percent:: + + (num_samples, P_high, samples_high, lam_vol_high, data_high)= postTools.sample_highest_prob(top_percentile=percentile, P_samples=P, samples=samples, lam_vol=lam_vol,data = data,sort=True) + +Print the number of these samples and the ratio of the volume they take up:: + + print (numsamples, np.sum(lam_vol_high) + + +Suggested changes for user (4): +------------------------------- +Notice how the marginal probabilites change with different choices of ``QoI_indices``. +Try choosing only 2 or 3, instead of 4, indices and notice the higher-dimensionality of the structure in the 2d marginals. Notice how some QoI concentrate the probability into smaller regions. These QoI are more geometrically distinct. + +Notice that the volume that the high-probability samples take up is smaller with more geometrically distinct QoIs. + +Suggested changes for user (5): +------------------------------- +Change ``percentile`` to values between 1.0 and 0.0. Notice that while the region of nonzero probabibilty may have a significant volume, much of this volume contains relatively low probability. Change the value to 0.95, 0.9, 0.75, and 0.5 and notice the volume decrease significantly. + diff --git a/doc/examples/overview.rst b/doc/examples/overview.rst index 093e9196..04afbb89 100644 --- a/doc/examples/overview.rst +++ b/doc/examples/overview.rst @@ -22,7 +22,7 @@ directory. Validation example ======================================= -See :ref:`validation` for an example . +See :ref:`validation` for an example. Linear Map Example @@ -73,3 +73,7 @@ Examples Estimating :math:`P_\Lambda` * :ref:`q1D` * :ref:`q2D` * :ref:`q3D` + +Contaminant Transport Example: +============================== +See :ref:`contaminantTransport` for an example. diff --git a/examples/contaminantTransport/contaminant.py b/examples/contaminantTransport/contaminant.py new file mode 100644 index 00000000..4f9c05e4 --- /dev/null +++ b/examples/contaminantTransport/contaminant.py @@ -0,0 +1,107 @@ +#! /usr/bin/env python + +# Copyright (C) 2014-2015 The BET Development Team + +""" +This example takes uniformly distributed samples of parameters and +output data from a simple groundwater contaminant transport model, +and calculates solutions to the stochastic inverse problem. +The parameter domain is 5D, where the uncertain parameters are the x and y +locations of the source, the horizontal dispersivity, the flow angle, +and the contaminant flux. There are 11 measured QoIs in the data space +available. By changing the choice of QoIs that we use to solve the stochastic +inverse problem, we see the effect of geometric distinctness. +Probabilities in the parameter space are +calculated using the MC assumption. 1D and 2D marginals are calculated, +smoothed, and plotted. The samples are then ranked by probability density +and the volume of high-probability regions are calculated. + +""" + +import numpy as np +import bet.calculateP as calculateP +import bet.postProcess as postProcess +import bet.calculateP.simpleFunP as simpleFunP +import bet.calculateP.calculateP as calculateP +import bet.postProcess.plotP as plotP +import bet.postProcess.plotDomains as plotD +import bet.postProcess.postTools as postTools + + +# Labels and descriptions of the uncertain parameters +labels = ['Source $y$ coordinate [L]', 'Source $x$ coordinate [L]', 'Dispersivity x [L]', 'Flow Angle [degrees]', 'Contaminant flux [M/T]'] + +# Load data from files +lam_domain = np.loadtxt("files/lam_domain.txt.gz") #parameter domain +ref_lam = np.loadtxt("files/lam_ref.txt.gz") #reference parameter set +Q_ref = np.loadtxt("files/Q_ref.txt.gz") #reference QoI set +samples = np.loadtxt("files/samples.txt.gz") # uniform samples in parameter domain +data = np.loadtxt("files/data.txt.gz") # data from model + +QoI_indices=[0,1,2,3] # Indices for output data with which you want to invert +bin_ratio = 0.25 #ratio of length of data region to invert + +data = data[:,QoI_indices] +Q_ref=Q_ref[QoI_indices] + +dmax = data.max(axis=0) +dmin = data.min(axis=0) +dscale = bin_ratio*(dmax-dmin) +Qmax = Q_ref + 0.5*dscale +Qmin = Q_ref -0.5*dscale +def rho_D(x): + return np.all(np.logical_and(np.greater(x,Qmin), np.less(x,Qmax)),axis=1) + +# Plot the data domain +plotD.show_data(data, Q_ref = Q_ref, rho_D=rho_D, showdim=2) + +# Whether or not to use deterministic description of simple function approximation of +# ouput probability +deterministic_discretize_D = True +if deterministic_discretize_D == True: + (d_distr_prob, d_distr_samples, d_Tree) = simpleFunP.uniform_hyperrectangle(data=data, + Q_ref=Q_ref, + bin_ratio=bin_ratio, + center_pts_per_edge = 1) +else: + (d_distr_prob, d_distr_samples, d_Tree) = simpleFunP.unif_unif(data=data, + Q_ref=Q_ref, + M=50, + bin_ratio=bin_ratio, + num_d_emulate=1E5) + +# calculate probablities making Monte Carlo assumption +(P, lam_vol, io_ptr) = calculateP.prob(samples=samples, + data=data, + rho_D_M=d_distr_prob, + d_distr_samples=d_distr_samples) + +# calculate 2D marginal probabilities +(bins, marginals2D) = plotP.calculate_2D_marginal_probs(P_samples = P, samples = samples, lam_domain = lam_domain, nbins = 10) + +# smooth 2D marginal probabilites for plotting (optional) +marginals2D = plotP.smooth_marginals_2D(marginals2D,bins, sigma=1.0) + +# plot 2D marginal probabilities +plotP.plot_2D_marginal_probs(marginals2D, bins, lam_domain, filename = "contaminant_map", + plot_surface=False, + lam_ref = ref_lam, + lambda_label=labels, + interactive=False) + +# calculate 1d marginal probs +(bins, marginals1D) = plotP.calculate_1D_marginal_probs(P_samples = P, samples = samples, lam_domain = lam_domain, nbins = 20) + +# smooth 1d marginal probs (optional) +marginals1D = plotP.smooth_marginals_1D(marginals1D, bins, sigma=1.0) + +# 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