Note
Go to the end to download the full example code.
Uncertainty Quantification in one-dimensional deconvolution#
This tutorial walks through the process of solving a simple 1D deconvolution problem in a Bayesian setting. It also shows how to define such a convolution model in CUQIpy.
Setup#
We start by importing the necessary modules
import cuqi
import numpy as np
import matplotlib.pyplot as plt
Setting up the forward model#
We start by defining the forward model. In this case, we will use a simple convolution model. The forward model is defined by the following equation:
\[\mathbf{y} = \mathbf{A} \mathbf{x}\]
where \(\mathbf{y}\) is the data, \(\mathbf{A}\) is the convolution (forward model) operator, and \(\mathbf{x}\) is the solution.
The easiest way to define the forward model is to use the testproblem module.
This module contains a number of pre-defined test problems that contain the
forward model and synthetic data. In this case, we will use the
cuqi.testproblem.Deconvolution1D test problem. We extract the forward model
and synthetic data from the test problem by calling the get_components()
method.
# Forward model and data
A, y_data, info = cuqi.testproblem.Deconvolution1D().get_components()
There are many parameters that can be set when creating the test problem. For more details
see the cuqi.testproblem.Deconvolution1D documentation. In this case, we will use
the default parameters. The get_components() method returns the forward model,
synthetic data, and a ProblemInfo object that contains information about the
test problem.
Let’s take a look at the forward model
print(A)
CUQI LinearModel: Continuous1D[128] -> Continuous1D[128].
Forward parameters: ['x'].
We see that the forward model is a a LinearModel object. This
object contains the forward model and the adjoint model. We also see that the domain and
range of the forward model are both continuous 1D spaces. Finally, we see that the default
forward parameters are set to \(\mathbf{x}\).
Let’s take a look at the synthetic data and compare with the exact solution
that we can find in the ProblemInfo object.
y_data.plot(label="Synthetic data")
info.exactSolution.plot(label="Exact solution")
plt.title("Deconvolution 1D problem")
plt.legend()
<matplotlib.legend.Legend object at 0x7fad7829caa0>
Setting up the prior#
We now need to define the prior distribution for the solution. In this case, we will use
a Gaussian Markov Random Field (GMRF) prior. For more details on the GMRF prior, see the
cuqi.distribution.GMRF documentation.
x = cuqi.distribution.GMRF(np.zeros(A.domain_dim), 200)
Setting up the likelihood#
We now need to define the likelihood. First let us take a look at the information provided by the test problem.
print(info.infoString)
Noise type: Additive Gaussian with std: 0.01
We see that the noise level is known and that the noise is Gaussian. We can use this
information to define the likelihood. In this case, we will use a Gaussian
distribution.
y = cuqi.distribution.Gaussian(A @ x, 0.01**2)
Bayesian problem (Joint distribution)#
After defining the prior and likelihood, we can now define the Bayesian problem. The Bayesian problem is defined by the joint distribution of the solution and the data. This can be seen when we print the Bayesian problem.
BP = cuqi.problem.BayesianProblem(y, x)
print(BP)
BayesianProblem with target:
JointDistribution(
Equation:
p(y,x) = p(y|x)p(x)
Densities:
y ~ CUQI Gaussian. Conditioning variables ['x'].
x ~ CUQI GMRF.
)
Setting the data (posterior)#
Now to set the data, we need to call the set_data()
BP.set_data(y=y_data)
print(BP)
BayesianProblem with target:
Posterior(
Equation:
p(x|y) ∝ L(x|y)p(x)
Densities:
y ~ CUQI Gaussian Likelihood function. Parameters ['x'].
x ~ CUQI GMRF.
)
Sampling from the posterior#
We can then use the automatic sampling method to sample from the posterior distribution.
samples = BP.sample_posterior(1000)
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!! Automatic sampler selection is a work-in-progress. !!!
!!! Always validate the computed results. !!!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Using cuqi.sampler LinearRTO sampler.
burn-in: 20%
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Elapsed time: 2.9734134674072266
Plotting the results#
samples.plot_ci(exact=info.exactSolution)
[<matplotlib.lines.Line2D object at 0x7fad73fd1b50>, <matplotlib.lines.Line2D object at 0x7fad73fd0dd0>, <matplotlib.collections.FillBetweenPolyCollection object at 0x7fad73d45550>]
Unknown noise level#
In the previous example, we assumed that we knew the noise level of the data. In
many cases, this is not the case. If we do not know the noise level, we can
use a Gamma distribution to model the noise level.
s = cuqi.distribution.Gamma(1, 1e-4)
Update likelihood with unknown noise level#
y = cuqi.distribution.Gaussian(A @ x, prec=lambda s: s)
Bayesian problem (Joint distribution)#
BP = cuqi.problem.BayesianProblem(y, x, s)
print(BP)
BayesianProblem with target:
JointDistribution(
Equation:
p(y,x,s) = p(y|x,s)p(x)p(s)
Densities:
y ~ CUQI Gaussian. Conditioning variables ['x', 's'].
x ~ CUQI GMRF.
s ~ CUQI Gamma.
)
Setting the data (posterior)#
BP.set_data(y=y_data)
print(BP)
BayesianProblem with target:
JointDistribution(
Equation:
p(x,s|y) ∝ L(x,s|y)p(x)p(s)
Densities:
y ~ CUQI Gaussian Likelihood function. Parameters ['x', 's'].
x ~ CUQI GMRF.
s ~ CUQI Gamma.
)
Sampling from the posterior#
samples = BP.sample_posterior(1000)
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!! Automatic sampler selection is a work-in-progress. !!!
!!! Always validate the computed results. !!!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Using Gibbs sampler
burn-in: 20%
Automatically determined sampling strategy:
x: LinearRTO
s: Conjugate
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Elapsed time: 4.619251728057861
Plotting the results#
Let is first look at the estimated noise level and compare it with the true noise level
samples["s"].plot_trace(exact=1/0.01**2)
array([[<Axes: title={'center': 's'}>, <Axes: title={'center': 's'}>]],
dtype=object)
We see that the estimated noise level is close to the true noise level. Let’s now look at the estimated solution
samples["x"].plot_ci(exact=info.exactSolution)
[<matplotlib.lines.Line2D object at 0x7fad78127f50>, <matplotlib.lines.Line2D object at 0x7fad78126510>, <matplotlib.collections.FillBetweenPolyCollection object at 0x7fad785b8bc0>]
We can even plot traces of “x” for a few cases and compare
samples["x"].plot_trace(exact=info.exactSolution)
Selecting 5 randomly chosen variables
array([[<Axes: title={'center': 'x13'}>, <Axes: title={'center': 'x13'}>],
[<Axes: title={'center': 'x17'}>, <Axes: title={'center': 'x17'}>],
[<Axes: title={'center': 'x39'}>, <Axes: title={'center': 'x39'}>],
[<Axes: title={'center': 'x56'}>, <Axes: title={'center': 'x56'}>],
[<Axes: title={'center': 'x119'}>,
<Axes: title={'center': 'x119'}>]], dtype=object)
And finally we note that the UQ method does this analysis automatically and shows a selected number of plots
BP.UQ(exact={"x": info.exactSolution, "s": 1/0.01**2})
Computing 1000 samples
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!! Automatic sampler selection is a work-in-progress. !!!
!!! Always validate the computed results. !!!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Using Gibbs sampler
burn-in: 20%
Automatically determined sampling strategy:
x: LinearRTO
s: Conjugate
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Elapsed time: 4.604166030883789
Plotting results
{'x': CUQIpy Samples:
---------------
Ns (number of samples):
1000
Geometry:
_DefaultGeometry1D[128]
Shape:
(128, 1000)
Samples:
[[ 0.0618453 0.07626861 -0.06397197 ... 0.03660061 -0.018834
0.02208633]
[ 0.04090545 0.10575296 -0.07316161 ... 0.07386206 -0.01817452
0.0461615 ]
[ 0.12200632 0.06148444 0.02467105 ... 0.04250902 0.00584573
0.01069121]
...
[ 0.05785521 -0.01794829 0.03422982 ... 0.09473283 0.05314016
0.08696787]
[ 0.0555219 -0.02355431 0.09301284 ... -0.01661903 0.04886413
0.05206315]
[ 0.00247182 -0.01837786 -0.0454012 ... -0.00911954 0.02045178
-0.03140952]]
, 's': CUQIpy Samples:
---------------
Ns (number of samples):
1000
Geometry:
_DefaultGeometry1D[1]
Shape:
(1, 1000)
Samples:
[[10594.12379585 12121.81508892 11384.40144924 13206.41902302
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12857.39922942 10595.53164607 10994.2787046 10972.7584469
9933.35655342 9972.88349757 13340.88668782 9511.72000169
11421.33066053 11203.56883978 11173.06595463 10971.93398827
10879.54381593 9646.0988863 12519.25709992 14589.94679495
10524.57845522 10049.6470148 11620.7068093 11223.75361607
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10921.93011504 17138.94615697 10849.47540006 10910.14964701
10450.56686927 10993.54900787 12287.43011962 11925.44963422
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Total running time of the script: (0 minutes 13.075 seconds)