from uqpylab import sessions
import scipy.io

# Start the session
mySession = sessions.cloud()
# (Optional) Get a convenient handle to the command line interface
uq = mySession.cli
# Reset the session
mySession.reset()

Processing .

.

 done!

 uqpylab.sessions :: INFO     :: This is UQ[py]Lab, version 1.00, running on https://uqcloud.ethz.ch. 
                                 UQ[py]Lab is free software, published under the open source BSD 3-clause license.
                                 To request special permissions, please contact:
                                  - Stefano Marelli (marelli@ibk.baug.ethz.ch).
                                 A new session (96e6d355df7b488c83b390c4c1df33e0) started.

 uqpylab.sessions :: INFO     :: Reset successful.

uq.rng(100, 'twister');

file_location = './Bayesian_Data'
mat = scipy.io.loadmat('{}/{}'.format(file_location,'uq_Example_BayesianLinearRegression.mat'))
Data = mat['Data'].tolist()
A = mat['Model']['A'][0,0]
M = A.shape[0]

ModelOpts = {'Type': 'Model', 
             'ModelFun':'LinearModel.model',
             'Parameters': {
                 'A': A.tolist()
                 }
             }
myForwardModel = uq.createModel(ModelOpts)

PriorOpts = {}
PriorOpts['Marginals'] = []
for i in range(M):
    PriorOpts['Marginals'].append({
        'Type': 'Gaussian',
        'Parameters': [0, 1]
    })
    
myPriorDist = uq.createInput(PriorOpts)

myData = {
    'y': Data
}

Solver = {
    'Type': 'MCMC',
    'MCMC': {
        'Sampler': 'AM',
        'Steps': 1e2,
        'NChains': 1e2,
        'TO': 1e2,
        'Proposal': {
            'PriorScale': 0.1
        }
    }
}

DiscrepancyOptsKnown = {
    'Type': 'Gaussian',
    'Parameters': 1
}

BayesOpts = {
    'Type': 'Inversion',
    'Data': myData,
    'Discrepancy': DiscrepancyOptsKnown,
    'Solver': Solver,
    'Prior': myPriorDist['Name']
}

myBayesianAnalysisKnownDisc = uq.createAnalysis(BayesOpts)

 uqpylab.sessions :: INFO     :: Received intermediate compute request, function: LinearModel.model.

 uqpylab.sessions :: INFO     :: Carrying out local computation...

 uqpylab.sessions :: INFO     :: Local computation complete.

 uqpylab.sessions :: INFO     :: Starting transmission of intermediate compute results ((100, 24))...

 uqpylab.sessions :: INFO     :: Intermediate compute results sent.

 uqpylab.sessions :: INFO     :: Received intermediate compute request, function: LinearModel.model.

 uqpylab.sessions :: INFO     :: Carrying out local computation...

 uqpylab.sessions :: INFO     :: Local computation complete.

 uqpylab.sessions :: INFO     :: Starting transmission of intermediate compute results ((100, 24))...

 uqpylab.sessions :: INFO     :: Intermediate compute results sent.

uq.print(myBayesianAnalysisKnownDisc)

%----------------------- Inversion output -----------------------%
   Number of calibrated model parameters:         8
   Number of non-calibrated model parameters:     0

   Number of calibrated discrepancy parameters:   0

%------------------- Data and Discrepancy
%  Data-/Discrepancy group 1:
   Number of independent observations:            1

   Discrepancy:
      Type:                                       Gaussian
      Discrepancy family:                         Scalar
      Discrepancy parameters known:               Yes

   Associated outputs:
      Model 1: 
         Output dimensions:                       1
                                                  to
                                                  24


%------------------- Solver
   Solution method:                               MCMC

   Algorithm:                                     AM
   Duration (HH:MM:SS):                           00:01:53
   Number of sample points:                       1.00e+04

%------------------- Posterior Marginals
----------------------------------------------------------
| Parameter | Mean  | Std  | (0.025-0.97) Quant. | Type  |
----------------------------------------------------------
| X1        | 0.73  | 0.4  | (-0.17 - 1.5)       | Model |
| X2        | -0.91 | 0.39 | (-1.6 - -0.063)     | Model |
| X3        | -0.48 | 0.28 | (-0.99 - 0.09)      | Model |
| X4        | 0.91  | 0.46 | (-0.19 - 1.8)       | Model |
| X5        | -0.16 | 0.28 | (-0.72 - 0.4)       | Model |
| X6        | 0.42  | 0.35 | (-0.36 - 1.1)       | Model |
| X7        | 1.3   | 0.37 | (0.44 - 1.8)        | Model |
| X8        | -0.28 | 0.38 | (-0.95 - 0.56)      | Model |
----------------------------------------------------------

%------------------- Point estimate
--------------------------------------
| Parameter | Mean  | Parameter Type |
--------------------------------------
| X1        | 0.73  | Model          |
| X2        | -0.91 | Model          |
| X3        | -0.48 | Model          |
| X4        | 0.91  | Model          |
| X5        | -0.16 | Model          |
| X6        | 0.42  | Model          |
| X7        | 1.3   | Model          |
| X8        | -0.28 | Model          |
--------------------------------------

%------------------- Correlation matrix, 6 most important (model parameters)
----------------------------------------------------------------
|    |  X1        X2       X5        X6       X7       X8      |
----------------------------------------------------------------
| X1 |  1         -0.58    0.48      0.63     0.066    -0.039  |
| X2 |  -0.58     1        -0.59     -0.34    0.049    0.027   |
| X5 |  0.48      -0.59    1         0.25     0.053    -0.029  |
| X6 |  0.63      -0.34    0.25      1        0.2      -0.13   |
| X7 |  0.066     0.049    0.053     0.2      1        -0.36   |
| X8 |  -0.039    0.027    -0.029    -0.13    -0.36    1       |
----------------------------------------------------------------

uq.display(myBayesianAnalysisKnownDisc);

SigmaOpts = {
  'Marginals': [
    {
      'Name': 'Sigma2', 
      'Type': 'Uniform',
      'Parameters': [0, 20]
    }
  ]
}

mySigmaDist = uq.createInput(SigmaOpts)

DiscrepancyOptsUnknownDisc = {
    'Type': 'Gaussian',
    'Prior': mySigmaDist['Name']
}

BayesOpts['Discrepancy'] = DiscrepancyOptsUnknownDisc

BayesOpts['Prior'] = myPriorDist['Name']

myBayesianAnalysisUnknownDisc = uq.createAnalysis(BayesOpts)

 uqpylab.sessions :: INFO     :: Received intermediate compute request, function: LinearModel.model.

 uqpylab.sessions :: INFO     :: Carrying out local computation...

 uqpylab.sessions :: INFO     :: Local computation complete.

 uqpylab.sessions :: INFO     :: Starting transmission of intermediate compute results ((100, 24))...

 uqpylab.sessions :: INFO     :: Intermediate compute results sent.

 uqpylab.sessions :: INFO     :: Received intermediate compute request, function: LinearModel.model.

 uqpylab.sessions :: INFO     :: Carrying out local computation...

 uqpylab.sessions :: INFO     :: Local computation complete.

 uqpylab.sessions :: INFO     :: Starting transmission of intermediate compute results ((93, 24))...

 uqpylab.sessions :: INFO     :: Intermediate compute results sent.

uq.print(myBayesianAnalysisUnknownDisc)

%----------------------- Inversion output -----------------------%
   Number of calibrated model parameters:         8
   Number of non-calibrated model parameters:     0

   Number of calibrated discrepancy parameters:   1

%------------------- Data and Discrepancy
%  Data-/Discrepancy group 1:
   Number of independent observations:            1

   Discrepancy:
      Type:                                       Gaussian
      Discrepancy family:                         Scalar
      Discrepancy parameters known:               No

   Associated outputs:
      Model 1: 
         Output dimensions:                       1
                                                  to
                                                  24


%------------------- Solver
   Solution method:                               MCMC

   Algorithm:                                     AM
   Duration (HH:MM:SS):                           00:01:53
   Number of sample points:                       1.00e+04

%------------------- Posterior Marginals
----------------------------------------------------------------
| Parameter | Mean  | Std  | (0.025-0.97) Quant. | Type        |
----------------------------------------------------------------
| X1        | 0.55  | 0.58 | (-0.65 - 1.6)       | Model       |
| X2        | -0.68 | 0.63 | (-1.8 - 0.65)       | Model       |
| X3        | -0.37 | 0.44 | (-1.2 - 0.6)        | Model       |
| X4        | 0.7   | 0.66 | (-0.71 - 1.9)       | Model       |
| X5        | -0.17 | 0.46 | (-1.1 - 0.7)        | Model       |
| X6        | 0.27  | 0.56 | (-0.98 - 1.3)       | Model       |
| X7        | 1     | 0.59 | (-0.4 -  2)         | Model       |
| X8        | -0.28 | 0.5  | (-1.2 - 0.71)       | Model       |
| Sigma2    | 5.9   | 4.1  | (1.1 - 17)          | Discrepancy |
----------------------------------------------------------------

%------------------- Point estimate
--------------------------------------
| Parameter | Mean  | Parameter Type |
--------------------------------------
| X1        | 0.55  | Model          |
| X2        | -0.68 | Model          |
| X3        | -0.37 | Model          |
| X4        | 0.7   | Model          |
| X5        | -0.17 | Model          |
| X6        | 0.27  | Model          |
| X7        | 1     | Model          |
| X8        | -0.28 | Model          |
| Sigma2    | 5.9   | Discrepancy    |
--------------------------------------

%------------------- Correlation matrix, 6 most important (model parameters)
---------------------------------------------------------------
|    |  X1       X2       X4       X5        X6       X7      |
---------------------------------------------------------------
| X1 |  1        -0.52    0.35     0.38      0.48     0.16    |
| X2 |  -0.52    1        -0.17    -0.49     -0.23    -0.11   |
| X4 |  0.35     -0.17    1        0.019     0.22     0.14    |
| X5 |  0.38     -0.49    0.019    1         0.13     0.0018  |
| X6 |  0.48     -0.23    0.22     0.13      1        0.19    |
| X7 |  0.16     -0.11    0.14     0.0018    0.19     1       |
---------------------------------------------------------------

uq.display(myBayesianAnalysisUnknownDisc);

mySession.quit()

 uqpylab.sessions :: INFO     :: Session 96e6d355df7b488c83b390c4c1df33e0 terminated.

True

INVERSION: BAYESIAN LINEAR REGRESSION¶

Package imports¶

Start a remote UQCloud session¶

Set the random seed for reproducibility¶

Forward model¶

Prior distribution of the model parameters¶

Measurement data¶

Solver options¶

Bayesian analysis with known measurement error¶

Discrepancy model¶

Bayesian analysis¶

Bayesian analysis with unknown measurement error¶

Discrepancy model¶

Bayesian analysis¶

Terminate the remote UQCloud session¶