AMICI Python example “steadystate”

This is an example using the [model_steadystate_scaled.sbml] model to demonstrate and test SBML import and AMICI Python interface.

[1]:
# SBML model we want to import
sbml_file = 'model_steadystate_scaled_without_observables.xml'
# Name of the model that will also be the name of the python module
model_name = 'model_steadystate_scaled'
# Directory to which the generated model code is written
model_output_dir = model_name

import libsbml
import importlib
import amici
import os
import sys
import numpy as np
import matplotlib.pyplot as plt

The example model

Here we use libsbml to show the reactions and species described by the model (this is independent of AMICI).

[2]:
sbml_reader = libsbml.SBMLReader()
sbml_doc = sbml_reader.readSBML(sbml_file)
sbml_model = sbml_doc.getModel()
dir(sbml_doc)

print('Species: ', [s.getId() for s in sbml_model.getListOfSpecies()])

print('\nReactions:')
for reaction in sbml_model.getListOfReactions():
    reactants = ' + '.join(['%s %s'%(int(r.getStoichiometry()) if r.getStoichiometry() > 1 else '', r.getSpecies()) for r in reaction.getListOfReactants()])
    products  = ' + '.join(['%s %s'%(int(r.getStoichiometry()) if r.getStoichiometry() > 1 else '', r.getSpecies()) for r in reaction.getListOfProducts()])
    reversible = '<' if reaction.getReversible() else ''
    print('%3s: %10s %1s->%10s\t\t[%s]' % (reaction.getId(),
                        reactants,
                        reversible,
                         products,
                        libsbml.formulaToL3String(reaction.getKineticLaw().getMath())))

Species:  ['x1', 'x2', 'x3']

Reactions:
 r1:       2 x1  ->        x2                [p1 * x1^2]
 r2:   x1 +  x2  ->        x3                [p2 * x1 * x2]
 r3:         x2  ->      2 x1                [p3 * x2]
 r4:         x3  ->  x1 +  x2                [p4 * x3]
 r5:         x3  ->                          [k0 * x3]
 r6:             ->        x1                [p5]

Importing an SBML model, compiling and generating an AMICI module

Before we can use AMICI to simulate our model, the SBML model needs to be translated to C++ code. This is done by amici.SbmlImporter.

[3]:
# Create an SbmlImporter instance for our SBML model
sbml_importer = amici.SbmlImporter(sbml_file)

In this example, we want to specify fixed parameters, observables and a \(\sigma\) parameter. Unfortunately, the latter two are not part of the SBML standard. However, they can be provided to amici.SbmlImporter.sbml2amici as demonstrated in the following.

Constant parameters

Constant parameters, i.e. parameters with respect to which no sensitivities are to be computed (these are often parameters specifying a certain experimental condition) are provided as a list of parameter names.

[4]:
constantParameters = ['k0']

Observables

Specifying observables is beyond the scope of SBML. Here we define them manually.

If you are looking for a more scalable way for defining observables, then checkout PEtab. Another possibility is using SBML’s `AssignmentRules <https://sbml.org/software/libsbml/5.18.0/docs/formatted/python-api/classlibsbml_1_1_assignment_rule.html>`__ to specify model outputs within the SBML file.

[5]:
# Define observables
observables = {
    'observable_x1': {'name': '', 'formula': 'x1'},
    'observable_x2': {'name': '', 'formula': 'x2'},
    'observable_x3': {'name': '', 'formula': 'x3'},
    'observable_x1_scaled': {'name': '', 'formula': 'scaling_x1 * x1'},
    'observable_x2_offsetted': {'name': '', 'formula': 'offset_x2 + x2'},
    'observable_x1withsigma': {'name': '', 'formula': 'x1'}
}

\(\sigma\) parameters

To specify measurement noise as a parameter, we simply provide a dictionary with (preexisting) parameter names as keys and a list of observable names as values to indicate which sigma parameter is to be used for which observable.

[6]:
sigmas = {'observable_x1withsigma': 'observable_x1withsigma_sigma'}

Generating the module

Now we can generate the python module for our model. amici.SbmlImporter.sbml2amici will symbolically derive the sensitivity equations, generate C++ code for model simulation, and assemble the python module. Standard logging verbosity levels can be passed to this function to see timestamped progression during code generation.

[7]:
import logging
sbml_importer.sbml2amici(model_name,
                         model_output_dir,
                         verbose=logging.INFO,
                         observables=observables,
                         constant_parameters=constantParameters,
                         sigmas=sigmas)
2021-11-30 16:57:57.997 - amici.sbml_import - INFO - Finished gathering local SBML symbols        + (8.05E-03s)
2021-11-30 16:57:58.047 - amici.sbml_import - INFO - Finished processing SBML parameters          + (4.64E-02s)
2021-11-30 16:57:58.054 - amici.sbml_import - INFO - Finished processing SBML compartments        + (2.69E-04s)
2021-11-30 16:57:58.063 - amici.sbml_import - INFO - Finished processing SBML species initials   ++ (1.47E-03s)
2021-11-30 16:57:58.067 - amici.sbml_import - INFO - Finished processing SBML rate rules         ++ (7.65E-05s)
2021-11-30 16:57:58.068 - amici.sbml_import - INFO - Finished processing SBML species             + (9.09E-03s)
2021-11-30 16:57:58.076 - amici.sbml_import - INFO - Finished processing SBML reactions           + (4.32E-03s)
2021-11-30 16:57:58.086 - amici.sbml_import - INFO - Finished processing SBML rules               + (6.88E-03s)
2021-11-30 16:57:58.093 - amici.sbml_import - INFO - Finished processing SBML initial assignments + (5.51E-05s)
2021-11-30 16:57:58.097 - amici.sbml_import - INFO - Finished processing SBML species references  + (8.19E-04s)
2021-11-30 16:57:58.105 - amici.sbml_import - INFO - Finished processing SBML events              + (1.00E-04s)
2021-11-30 16:57:58.105 - amici.sbml_import - INFO - Finished importing SBML                        (1.19E-01s)
2021-11-30 16:57:58.180 - amici.sbml_import - INFO - Finished processing SBML observables           (6.99E-02s)
2021-11-30 16:57:58.202 - amici.ode_export - INFO - Finished running smart_multiply               + (3.40E-03s)
2021-11-30 16:57:58.210 - amici.ode_export - INFO - Finished importing SbmlImporter                 (1.42E-02s)
2021-11-30 16:57:58.286 - amici.ode_export - INFO - Finished simplifying Jy                     +++ (6.10E-02s)
2021-11-30 16:57:58.288 - amici.ode_export - INFO - Finished computing Jy                        ++ (6.69E-02s)
2021-11-30 16:57:58.299 - amici.ode_export - INFO - Finished simplifying y                      +++ (6.08E-04s)
2021-11-30 16:57:58.300 - amici.ode_export - INFO - Finished computing y                         ++ (5.38E-03s)
2021-11-30 16:57:58.308 - amici.ode_export - INFO - Finished simplifying sigmay                 +++ (1.19E-04s)
2021-11-30 16:57:58.310 - amici.ode_export - INFO - Finished computing sigmay                    ++ (5.58E-03s)
2021-11-30 16:57:58.351 - amici.ode_export - INFO - Finished writing Jy.cpp                       + (1.33E-01s)
2021-11-30 16:57:58.421 - amici.ode_export - INFO - Finished running smart_jacobian             +++ (5.81E-02s)
2021-11-30 16:57:58.452 - amici.ode_export - INFO - Finished simplifying dJydsigma              +++ (2.69E-02s)
2021-11-30 16:57:58.453 - amici.ode_export - INFO - Finished computing dJydsigma                 ++ (9.43E-02s)
2021-11-30 16:57:58.466 - amici.ode_export - INFO - Finished writing dJydsigma.cpp                + (1.09E-01s)
2021-11-30 16:57:58.505 - amici.ode_export - INFO - Finished running smart_jacobian             +++ (2.48E-02s)
2021-11-30 16:57:58.559 - amici.ode_export - INFO - Finished simplifying dJydy                  +++ (4.86E-02s)
2021-11-30 16:57:58.560 - amici.ode_export - INFO - Finished computing dJydy                     ++ (8.27E-02s)
2021-11-30 16:57:58.586 - amici.ode_export - INFO - Finished writing dJydy.cpp                    + (1.12E-01s)
2021-11-30 16:57:58.599 - amici.ode_export - INFO - Finished simplifying root                   +++ (5.50E-05s)
2021-11-30 16:57:58.600 - amici.ode_export - INFO - Finished computing root                      ++ (4.63E-03s)
2021-11-30 16:57:58.601 - amici.ode_export - INFO - Finished writing root.cpp                     + (8.96E-03s)
2021-11-30 16:57:58.624 - amici.ode_export - INFO - Finished simplifying w                     ++++ (7.65E-03s)
2021-11-30 16:57:58.626 - amici.ode_export - INFO - Finished computing w                        +++ (1.21E-02s)
2021-11-30 16:57:58.641 - amici.ode_export - INFO - Finished running smart_jacobian             +++ (1.04E-02s)
2021-11-30 16:57:58.647 - amici.ode_export - INFO - Finished simplifying dwdp                   +++ (1.45E-03s)
2021-11-30 16:57:58.648 - amici.ode_export - INFO - Finished computing dwdp                      ++ (3.77E-02s)
2021-11-30 16:57:58.658 - amici.ode_export - INFO - Finished writing dwdp.cpp                     + (5.28E-02s)
2021-11-30 16:57:58.680 - amici.ode_export - INFO - Finished running smart_jacobian             +++ (1.15E-02s)
2021-11-30 16:57:58.687 - amici.ode_export - INFO - Finished simplifying dwdx                   +++ (1.95E-03s)
2021-11-30 16:57:58.687 - amici.ode_export - INFO - Finished computing dwdx                      ++ (2.17E-02s)
2021-11-30 16:57:58.696 - amici.ode_export - INFO - Finished writing dwdx.cpp                     + (3.37E-02s)
2021-11-30 16:57:58.705 - amici.ode_export - INFO - Finished running smart_jacobian             +++ (1.05E-04s)
2021-11-30 16:57:58.710 - amici.ode_export - INFO - Finished simplifying dwdw                   +++ (6.07E-04s)
2021-11-30 16:57:58.711 - amici.ode_export - INFO - Finished computing dwdw                      ++ (8.67E-03s)
2021-11-30 16:57:58.713 - amici.ode_export - INFO - Finished writing dwdw.cpp                     + (1.25E-02s)
2021-11-30 16:57:58.729 - amici.ode_export - INFO - Finished simplifying xdot                  ++++ (5.40E-03s)
2021-11-30 16:57:58.729 - amici.ode_export - INFO - Finished computing xdot                     +++ (8.78E-03s)
2021-11-30 16:57:58.746 - amici.ode_export - INFO - Finished running smart_jacobian             +++ (1.30E-02s)
2021-11-30 16:57:58.752 - amici.ode_export - INFO - Finished simplifying dxdotdw                +++ (2.97E-04s)
2021-11-30 16:57:58.753 - amici.ode_export - INFO - Finished computing dxdotdw                   ++ (3.42E-02s)
2021-11-30 16:57:58.765 - amici.ode_export - INFO - Finished writing dxdotdw.cpp                  + (4.89E-02s)
2021-11-30 16:57:58.777 - amici.ode_export - INFO - Finished running smart_jacobian             +++ (9.65E-05s)
2021-11-30 16:57:58.780 - amici.ode_export - INFO - Finished simplifying dxdotdx_explicit       +++ (7.85E-05s)
2021-11-30 16:57:58.781 - amici.ode_export - INFO - Finished computing dxdotdx_explicit          ++ (8.68E-03s)
2021-11-30 16:57:58.782 - amici.ode_export - INFO - Finished writing dxdotdx_explicit.cpp         + (1.24E-02s)
2021-11-30 16:57:58.793 - amici.ode_export - INFO - Finished running smart_jacobian             +++ (9.18E-05s)
2021-11-30 16:57:58.797 - amici.ode_export - INFO - Finished simplifying dxdotdp_explicit       +++ (1.22E-04s)
2021-11-30 16:57:58.798 - amici.ode_export - INFO - Finished computing dxdotdp_explicit          ++ (8.08E-03s)
2021-11-30 16:57:58.799 - amici.ode_export - INFO - Finished writing dxdotdp_explicit.cpp         + (1.36E-02s)
2021-11-30 16:57:58.816 - amici.ode_export - INFO - Finished running smart_jacobian            ++++ (1.92E-03s)
2021-11-30 16:57:58.821 - amici.ode_export - INFO - Finished simplifying dydx                  ++++ (1.55E-04s)
2021-11-30 16:57:58.822 - amici.ode_export - INFO - Finished computing dydx                     +++ (1.20E-02s)
2021-11-30 16:57:58.832 - amici.ode_export - INFO - Finished running smart_jacobian            ++++ (1.11E-04s)
2021-11-30 16:57:58.837 - amici.ode_export - INFO - Finished simplifying dydw                  ++++ (4.09E-04s)
2021-11-30 16:57:58.837 - amici.ode_export - INFO - Finished computing dydw                     +++ (9.68E-03s)
2021-11-30 16:57:58.844 - amici.ode_export - INFO - Finished simplifying dydx                   +++ (1.97E-04s)
2021-11-30 16:57:58.845 - amici.ode_export - INFO - Finished computing dydx                      ++ (3.77E-02s)
2021-11-30 16:57:58.849 - amici.ode_export - INFO - Finished writing dydx.cpp                     + (4.51E-02s)
2021-11-30 16:57:58.862 - amici.ode_export - INFO - Finished running smart_jacobian            ++++ (1.74E-03s)
2021-11-30 16:57:58.866 - amici.ode_export - INFO - Finished simplifying dydp                  ++++ (3.88E-04s)
2021-11-30 16:57:58.867 - amici.ode_export - INFO - Finished computing dydp                     +++ (9.93E-03s)
2021-11-30 16:57:58.872 - amici.ode_export - INFO - Finished simplifying dydp                   +++ (1.78E-04s)
2021-11-30 16:57:58.873 - amici.ode_export - INFO - Finished computing dydp                      ++ (1.84E-02s)
2021-11-30 16:57:58.876 - amici.ode_export - INFO - Finished writing dydp.cpp                     + (2.33E-02s)
2021-11-30 16:57:58.887 - amici.ode_export - INFO - Finished running smart_jacobian             +++ (1.58E-03s)
2021-11-30 16:57:58.892 - amici.ode_export - INFO - Finished simplifying dsigmaydp              +++ (2.41E-04s)
2021-11-30 16:57:58.893 - amici.ode_export - INFO - Finished computing dsigmaydp                 ++ (1.08E-02s)
2021-11-30 16:57:58.894 - amici.ode_export - INFO - Finished writing dsigmaydp.cpp                + (1.50E-02s)
2021-11-30 16:57:58.901 - amici.ode_export - INFO - Finished writing sigmay.cpp                   + (2.68E-03s)
2021-11-30 16:57:58.909 - amici.ode_export - INFO - Finished computing stau                      ++ (1.35E-04s)
2021-11-30 16:57:58.910 - amici.ode_export - INFO - Finished writing stau.cpp                     + (4.10E-03s)
2021-11-30 16:57:58.916 - amici.ode_export - INFO - Finished computing deltax                    ++ (1.34E-04s)
2021-11-30 16:57:58.916 - amici.ode_export - INFO - Finished writing deltax.cpp                   + (3.34E-03s)
2021-11-30 16:57:58.924 - amici.ode_export - INFO - Finished computing deltasx                   ++ (1.99E-04s)
2021-11-30 16:57:58.925 - amici.ode_export - INFO - Finished writing deltasx.cpp                  + (3.88E-03s)
2021-11-30 16:57:58.934 - amici.ode_export - INFO - Finished writing w.cpp                        + (5.23E-03s)
2021-11-30 16:57:58.945 - amici.ode_export - INFO - Finished simplifying x0                     +++ (7.68E-05s)
2021-11-30 16:57:58.946 - amici.ode_export - INFO - Finished computing x0                        ++ (4.22E-03s)
2021-11-30 16:57:58.949 - amici.ode_export - INFO - Finished writing x0.cpp                       + (1.04E-02s)
2021-11-30 16:57:58.961 - amici.ode_export - INFO - Finished simplifying x0_fixedParameters     +++ (4.96E-05s)
2021-11-30 16:57:58.962 - amici.ode_export - INFO - Finished computing x0_fixedParameters        ++ (4.51E-03s)
2021-11-30 16:57:58.963 - amici.ode_export - INFO - Finished writing x0_fixedParameters.cpp       + (8.84E-03s)
2021-11-30 16:57:58.982 - amici.ode_export - INFO - Finished running smart_jacobian             +++ (2.15E-04s)
2021-11-30 16:57:58.987 - amici.ode_export - INFO - Finished simplifying sx0                    +++ (1.20E-04s)
2021-11-30 16:57:58.988 - amici.ode_export - INFO - Finished computing sx0                       ++ (9.27E-03s)
2021-11-30 16:57:58.989 - amici.ode_export - INFO - Finished writing sx0.cpp                      + (2.26E-02s)
2021-11-30 16:57:58.999 - amici.ode_export - INFO - Finished running smart_jacobian             +++ (3.06E-05s)
2021-11-30 16:57:59.003 - amici.ode_export - INFO - Finished running smart_jacobian             +++ (2.44E-05s)
2021-11-30 16:57:59.008 - amici.ode_export - INFO - Finished simplifying sx0_fixedParameters    +++ (6.08E-05s)
2021-11-30 16:57:59.009 - amici.ode_export - INFO - Finished computing sx0_fixedParameters       ++ (1.28E-02s)
2021-11-30 16:57:59.009 - amici.ode_export - INFO - Finished writing sx0_fixedParameters.cpp      + (1.59E-02s)
2021-11-30 16:57:59.021 - amici.ode_export - INFO - Finished writing xdot.cpp                     + (7.26E-03s)
2021-11-30 16:57:59.028 - amici.ode_export - INFO - Finished writing y.cpp                        + (2.13E-03s)
2021-11-30 16:57:59.036 - amici.ode_export - INFO - Finished simplifying x_rdata                +++ (5.62E-05s)
2021-11-30 16:57:59.038 - amici.ode_export - INFO - Finished computing x_rdata                   ++ (5.16E-03s)
2021-11-30 16:57:59.041 - amici.ode_export - INFO - Finished writing x_rdata.cpp                  + (9.60E-03s)
2021-11-30 16:57:59.050 - amici.ode_export - INFO - Finished simplifying total_cl               +++ (4.08E-05s)
2021-11-30 16:57:59.051 - amici.ode_export - INFO - Finished computing total_cl                  ++ (3.64E-03s)
2021-11-30 16:57:59.052 - amici.ode_export - INFO - Finished writing total_cl.cpp                 + (7.19E-03s)
2021-11-30 16:57:59.064 - amici.ode_export - INFO - Finished simplifying x_solver               +++ (7.69E-05s)
2021-11-30 16:57:59.065 - amici.ode_export - INFO - Finished computing x_solver                  ++ (4.62E-03s)
2021-11-30 16:57:59.068 - amici.ode_export - INFO - Finished writing x_solver.cpp                 + (1.13E-02s)
2021-11-30 16:57:59.079 - amici.ode_export - INFO - Finished generating cpp code                    (8.66E-01s)
2021-11-30 16:58:07.834 - amici.ode_export - INFO - Finished compiling cpp code                     (8.75E+00s)

Importing the module and loading the model

If everything went well, we need to add the previously selected model output directory to our PYTHON_PATH and are then ready to load newly generated model:

[8]:
sys.path.insert(0, os.path.abspath(model_output_dir))
model_module = importlib.import_module(model_name)

And get an instance of our model from which we can retrieve information such as parameter names:

[9]:
model = model_module.getModel()

print("Model name:", model.getName())
print("Model parameters:", model.getParameterIds())
print("Model outputs:   ", model.getObservableIds())
print("Model states:    ", model.getStateIds())
Model name: model_steadystate_scaled
Model parameters: ('p1', 'p2', 'p3', 'p4', 'p5', 'scaling_x1', 'offset_x2', 'observable_x1withsigma_sigma')
Model outputs:    ('observable_x1', 'observable_x2', 'observable_x3', 'observable_x1_scaled', 'observable_x2_offsetted', 'observable_x1withsigma')
Model states:     ('x1', 'x2', 'x3')

Running simulations and analyzing results

After importing the model, we can run simulations using amici.runAmiciSimulation. This requires a Model instance and a Solver instance. Optionally you can provide measurements inside an ExpData instance, as shown later in this notebook.

[10]:
# Create Model instance
model = model_module.getModel()

# set timepoints for which we want to simulate the model
model.setTimepoints(np.linspace(0, 60, 60))

# Create solver instance
solver = model.getSolver()

# Run simulation using default model parameters and solver options
rdata = amici.runAmiciSimulation(model, solver)
[11]:
print('Simulation was run using model default parameters as specified in the SBML model:')
print(model.getParameters())
Simulation was run using model default parameters as specified in the SBML model:
(1.0, 0.5, 0.4, 2.0, 0.1, 2.0, 3.0, 0.2)

Simulation results are provided as numpy.ndarrays in the returned dictionary:

[12]:
#np.set_printoptions(threshold=8, edgeitems=2)
for key, value in rdata.items():
    print('%12s: ' % key, value)
          ts:  [ 0.          1.01694915  2.03389831  3.05084746  4.06779661  5.08474576
  6.10169492  7.11864407  8.13559322  9.15254237 10.16949153 11.18644068
 12.20338983 13.22033898 14.23728814 15.25423729 16.27118644 17.28813559
 18.30508475 19.3220339  20.33898305 21.3559322  22.37288136 23.38983051
 24.40677966 25.42372881 26.44067797 27.45762712 28.47457627 29.49152542
 30.50847458 31.52542373 32.54237288 33.55932203 34.57627119 35.59322034
 36.61016949 37.62711864 38.6440678  39.66101695 40.6779661  41.69491525
 42.71186441 43.72881356 44.74576271 45.76271186 46.77966102 47.79661017
 48.81355932 49.83050847 50.84745763 51.86440678 52.88135593 53.89830508
 54.91525424 55.93220339 56.94915254 57.96610169 58.98305085 60.        ]
           x:  [[0.1        0.4        0.7       ]
 [0.57995052 0.73365809 0.0951589 ]
 [0.55996496 0.71470091 0.0694127 ]
 [0.5462855  0.68030366 0.06349394]
 [0.53561883 0.64937432 0.05923555]
 [0.52636487 0.62259567 0.05568686]
 [0.51822013 0.59943346 0.05268079]
 [0.51103767 0.57935661 0.05012037]
 [0.5047003  0.56191592 0.04793052]
 [0.49910666 0.54673518 0.0460508 ]
 [0.49416809 0.53349812 0.04443205]
 [0.48980687 0.52193767 0.04303399]
 [0.48595476 0.51182731 0.04182339]
 [0.48255176 0.50297412 0.04077267]
 [0.47954511 0.49521318 0.03985882]
 [0.47688833 0.48840304 0.03906254]
 [0.47454049 0.48242198 0.03836756]
 [0.47246548 0.47716502 0.0377601 ]
 [0.47063147 0.47254128 0.03722844]
 [0.46901037 0.46847202 0.03676259]
 [0.46757739 0.46488881 0.03635397]
 [0.46631065 0.46173207 0.03599523]
 [0.46519082 0.45894987 0.03568002]
 [0.46420083 0.45649684 0.03540285]
 [0.4633256  0.45433332 0.03515899]
 [0.4625518  0.45242457 0.03494429]
 [0.46186768 0.45074016 0.03475519]
 [0.46126282 0.44925337 0.03458856]
 [0.46072804 0.44794075 0.03444166]
 [0.46025521 0.44678168 0.03431212]
 [0.45983714 0.44575804 0.03419784]
 [0.45946749 0.44485388 0.03409701]
 [0.45914065 0.44405514 0.03400802]
 [0.45885167 0.44334947 0.03392946]
 [0.45859615 0.44272595 0.03386009]
 [0.45837021 0.44217497 0.03379883]
 [0.45817043 0.44168805 0.03374473]
 [0.45799379 0.44125772 0.03369693]
 [0.4578376  0.44087738 0.03365471]
 [0.45769949 0.44054121 0.0336174 ]
 [0.45757737 0.44024405 0.03358444]
 [0.45746939 0.43998137 0.03355531]
 [0.45737391 0.43974917 0.03352956]
 [0.45728948 0.43954389 0.03350681]
 [0.45721483 0.43936242 0.0334867 ]
 [0.45714882 0.43920198 0.03346892]
 [0.45709045 0.43906014 0.03345321]
 [0.45703884 0.43893474 0.03343932]
 [0.4569932  0.43882387 0.03342704]
 [0.45695285 0.43872584 0.03341618]
 [0.45691717 0.43863917 0.03340658]
 [0.45688561 0.43856254 0.0333981 ]
 [0.45685771 0.43849478 0.0333906 ]
 [0.45683304 0.43843488 0.03338397]
 [0.45681123 0.4383819  0.0333781 ]
 [0.45679194 0.43833507 0.03337292]
 [0.45677488 0.43829365 0.03336833]
 [0.4567598  0.43825703 0.03336428]
 [0.45674646 0.43822466 0.0333607 ]
 [0.45673467 0.43819603 0.03335753]]
          x0:  [0.1 0.4 0.7]
        x_ss:  [nan nan nan]
          sx:  None
         sx0:  None
       sx_ss:  None
           y:  [[0.1        0.4        0.7        0.2        3.4        0.1       ]
 [0.57995052 0.73365809 0.0951589  1.15990103 3.73365809 0.57995052]
 [0.55996496 0.71470091 0.0694127  1.11992992 3.71470091 0.55996496]
 [0.5462855  0.68030366 0.06349394 1.092571   3.68030366 0.5462855 ]
 [0.53561883 0.64937432 0.05923555 1.07123766 3.64937432 0.53561883]
 [0.52636487 0.62259567 0.05568686 1.05272975 3.62259567 0.52636487]
 [0.51822013 0.59943346 0.05268079 1.03644027 3.59943346 0.51822013]
 [0.51103767 0.57935661 0.05012037 1.02207533 3.57935661 0.51103767]
 [0.5047003  0.56191592 0.04793052 1.00940059 3.56191592 0.5047003 ]
 [0.49910666 0.54673518 0.0460508  0.99821331 3.54673518 0.49910666]
 [0.49416809 0.53349812 0.04443205 0.98833618 3.53349812 0.49416809]
 [0.48980687 0.52193767 0.04303399 0.97961374 3.52193767 0.48980687]
 [0.48595476 0.51182731 0.04182339 0.97190952 3.51182731 0.48595476]
 [0.48255176 0.50297412 0.04077267 0.96510352 3.50297412 0.48255176]
 [0.47954511 0.49521318 0.03985882 0.95909022 3.49521318 0.47954511]
 [0.47688833 0.48840304 0.03906254 0.95377667 3.48840304 0.47688833]
 [0.47454049 0.48242198 0.03836756 0.94908097 3.48242198 0.47454049]
 [0.47246548 0.47716502 0.0377601  0.94493095 3.47716502 0.47246548]
 [0.47063147 0.47254128 0.03722844 0.94126293 3.47254128 0.47063147]
 [0.46901037 0.46847202 0.03676259 0.93802074 3.46847202 0.46901037]
 [0.46757739 0.46488881 0.03635397 0.93515478 3.46488881 0.46757739]
 [0.46631065 0.46173207 0.03599523 0.9326213  3.46173207 0.46631065]
 [0.46519082 0.45894987 0.03568002 0.93038164 3.45894987 0.46519082]
 [0.46420083 0.45649684 0.03540285 0.92840166 3.45649684 0.46420083]
 [0.4633256  0.45433332 0.03515899 0.92665119 3.45433332 0.4633256 ]
 [0.4625518  0.45242457 0.03494429 0.9251036  3.45242457 0.4625518 ]
 [0.46186768 0.45074016 0.03475519 0.92373536 3.45074016 0.46186768]
 [0.46126282 0.44925337 0.03458856 0.92252564 3.44925337 0.46126282]
 [0.46072804 0.44794075 0.03444166 0.92145608 3.44794075 0.46072804]
 [0.46025521 0.44678168 0.03431212 0.92051041 3.44678168 0.46025521]
 [0.45983714 0.44575804 0.03419784 0.91967427 3.44575804 0.45983714]
 [0.45946749 0.44485388 0.03409701 0.91893498 3.44485388 0.45946749]
 [0.45914065 0.44405514 0.03400802 0.91828131 3.44405514 0.45914065]
 [0.45885167 0.44334947 0.03392946 0.91770333 3.44334947 0.45885167]
 [0.45859615 0.44272595 0.03386009 0.91719229 3.44272595 0.45859615]
 [0.45837021 0.44217497 0.03379883 0.91674042 3.44217497 0.45837021]
 [0.45817043 0.44168805 0.03374473 0.91634087 3.44168805 0.45817043]
 [0.45799379 0.44125772 0.03369693 0.91598758 3.44125772 0.45799379]
 [0.4578376  0.44087738 0.03365471 0.9156752  3.44087738 0.4578376 ]
 [0.45769949 0.44054121 0.0336174  0.91539898 3.44054121 0.45769949]
 [0.45757737 0.44024405 0.03358444 0.91515474 3.44024405 0.45757737]
 [0.45746939 0.43998137 0.03355531 0.91493878 3.43998137 0.45746939]
 [0.45737391 0.43974917 0.03352956 0.91474782 3.43974917 0.45737391]
 [0.45728948 0.43954389 0.03350681 0.91457897 3.43954389 0.45728948]
 [0.45721483 0.43936242 0.0334867  0.91442966 3.43936242 0.45721483]
 [0.45714882 0.43920198 0.03346892 0.91429764 3.43920198 0.45714882]
 [0.45709045 0.43906014 0.03345321 0.91418091 3.43906014 0.45709045]
 [0.45703884 0.43893474 0.03343932 0.91407768 3.43893474 0.45703884]
 [0.4569932  0.43882387 0.03342704 0.91398641 3.43882387 0.4569932 ]
 [0.45695285 0.43872584 0.03341618 0.9139057  3.43872584 0.45695285]
 [0.45691717 0.43863917 0.03340658 0.91383433 3.43863917 0.45691717]
 [0.45688561 0.43856254 0.0333981  0.91377123 3.43856254 0.45688561]
 [0.45685771 0.43849478 0.0333906  0.91371543 3.43849478 0.45685771]
 [0.45683304 0.43843488 0.03338397 0.91366609 3.43843488 0.45683304]
 [0.45681123 0.4383819  0.0333781  0.91362246 3.4383819  0.45681123]
 [0.45679194 0.43833507 0.03337292 0.91358388 3.43833507 0.45679194]
 [0.45677488 0.43829365 0.03336833 0.91354976 3.43829365 0.45677488]
 [0.4567598  0.43825703 0.03336428 0.9135196  3.43825703 0.4567598 ]
 [0.45674646 0.43822466 0.0333607  0.91349292 3.43822466 0.45674646]
 [0.45673467 0.43819603 0.03335753 0.91346934 3.43819603 0.45673467]]
      sigmay:  [[1.  1.  1.  1.  1.  0.2]
 [1.  1.  1.  1.  1.  0.2]
 [1.  1.  1.  1.  1.  0.2]
 [1.  1.  1.  1.  1.  0.2]
 [1.  1.  1.  1.  1.  0.2]
 [1.  1.  1.  1.  1.  0.2]
 [1.  1.  1.  1.  1.  0.2]
 [1.  1.  1.  1.  1.  0.2]
 [1.  1.  1.  1.  1.  0.2]
 [1.  1.  1.  1.  1.  0.2]
 [1.  1.  1.  1.  1.  0.2]
 [1.  1.  1.  1.  1.  0.2]
 [1.  1.  1.  1.  1.  0.2]
 [1.  1.  1.  1.  1.  0.2]
 [1.  1.  1.  1.  1.  0.2]
 [1.  1.  1.  1.  1.  0.2]
 [1.  1.  1.  1.  1.  0.2]
 [1.  1.  1.  1.  1.  0.2]
 [1.  1.  1.  1.  1.  0.2]
 [1.  1.  1.  1.  1.  0.2]
 [1.  1.  1.  1.  1.  0.2]
 [1.  1.  1.  1.  1.  0.2]
 [1.  1.  1.  1.  1.  0.2]
 [1.  1.  1.  1.  1.  0.2]
 [1.  1.  1.  1.  1.  0.2]
 [1.  1.  1.  1.  1.  0.2]
 [1.  1.  1.  1.  1.  0.2]
 [1.  1.  1.  1.  1.  0.2]
 [1.  1.  1.  1.  1.  0.2]
 [1.  1.  1.  1.  1.  0.2]
 [1.  1.  1.  1.  1.  0.2]
 [1.  1.  1.  1.  1.  0.2]
 [1.  1.  1.  1.  1.  0.2]
 [1.  1.  1.  1.  1.  0.2]
 [1.  1.  1.  1.  1.  0.2]
 [1.  1.  1.  1.  1.  0.2]
 [1.  1.  1.  1.  1.  0.2]
 [1.  1.  1.  1.  1.  0.2]
 [1.  1.  1.  1.  1.  0.2]
 [1.  1.  1.  1.  1.  0.2]
 [1.  1.  1.  1.  1.  0.2]
 [1.  1.  1.  1.  1.  0.2]
 [1.  1.  1.  1.  1.  0.2]
 [1.  1.  1.  1.  1.  0.2]
 [1.  1.  1.  1.  1.  0.2]
 [1.  1.  1.  1.  1.  0.2]
 [1.  1.  1.  1.  1.  0.2]
 [1.  1.  1.  1.  1.  0.2]
 [1.  1.  1.  1.  1.  0.2]
 [1.  1.  1.  1.  1.  0.2]
 [1.  1.  1.  1.  1.  0.2]
 [1.  1.  1.  1.  1.  0.2]
 [1.  1.  1.  1.  1.  0.2]
 [1.  1.  1.  1.  1.  0.2]
 [1.  1.  1.  1.  1.  0.2]
 [1.  1.  1.  1.  1.  0.2]
 [1.  1.  1.  1.  1.  0.2]
 [1.  1.  1.  1.  1.  0.2]
 [1.  1.  1.  1.  1.  0.2]
 [1.  1.  1.  1.  1.  0.2]]
          sy:  None
     ssigmay:  None
           z:  None
          rz:  None
      sigmaz:  None
          sz:  None
         srz:  None
     ssigmaz:  None
        sllh:  None
       s2llh:  None
           J:  [[-2.04603669  0.57163267  2.        ]
 [ 0.69437133 -0.62836733  2.        ]
 [ 0.21909801  0.22836733 -3.        ]]
        xdot:  [-1.08967281e-05 -2.64534209e-05 -2.92761862e-06]
      status:  0.0
         llh:  nan
        chi2:  nan
         res:  [0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.
 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.
 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.
 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.
 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.
 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.
 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.
 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.
 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.
 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.
 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.
 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.
 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.
 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.
 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.]
        sres:  None
         FIM:  None
           w:  [[3.4        0.1        0.2        0.4        0.1        0.7
  0.01       0.02       0.16       1.4        0.7        0.1       ]
 [3.73365809 0.57995052 1.15990103 0.73365809 0.57995052 0.0951589
  0.3363426  0.21274269 0.29346324 0.1903178  0.0951589  0.1       ]
 [3.71470091 0.55996496 1.11992992 0.71470091 0.55996496 0.0694127
  0.31356076 0.20010373 0.28588036 0.13882541 0.0694127  0.1       ]
 [3.68030366 0.5462855  1.092571   0.68030366 0.5462855  0.06349394
  0.29842785 0.18582001 0.27212146 0.12698788 0.06349394 0.1       ]
 [3.64937432 0.53561883 1.07123766 0.64937432 0.53561883 0.05923555
  0.28688753 0.17390856 0.25974973 0.1184711  0.05923555 0.1       ]
 [3.62259567 0.52636487 1.05272975 0.62259567 0.52636487 0.05568686
  0.27705998 0.16385625 0.24903827 0.11137372 0.05568686 0.1       ]
 [3.59943346 0.51822013 1.03644027 0.59943346 0.51822013 0.05268079
  0.26855211 0.15531924 0.23977338 0.10536158 0.05268079 0.1       ]
 [3.57935661 0.51103767 1.02207533 0.57935661 0.51103767 0.05012037
  0.2611595  0.14803652 0.23174264 0.10024074 0.05012037 0.1       ]
 [3.56191592 0.5047003  1.00940059 0.56191592 0.5047003  0.04793052
  0.25472239 0.14179957 0.22476637 0.09586103 0.04793052 0.1       ]
 [3.54673518 0.49910666 0.99821331 0.54673518 0.49910666 0.0460508
  0.24910746 0.13643958 0.21869407 0.0921016  0.0460508  0.1       ]
 [3.53349812 0.49416809 0.98833618 0.53349812 0.49416809 0.04443205
  0.2442021  0.13181887 0.21339925 0.08886411 0.04443205 0.1       ]
 [3.52193767 0.48980687 0.97961374 0.52193767 0.48980687 0.04303399
  0.23991077 0.12782433 0.20877507 0.08606799 0.04303399 0.1       ]
 [3.51182731 0.48595476 0.97190952 0.51182731 0.48595476 0.04182339
  0.23615203 0.12436246 0.20473093 0.08364678 0.04182339 0.1       ]
 [3.50297412 0.48255176 0.96510352 0.50297412 0.48255176 0.04077267
  0.2328562  0.12135552 0.20118965 0.08154533 0.04077267 0.1       ]
 [3.49521318 0.47954511 0.95909022 0.49521318 0.47954511 0.03985882
  0.22996351 0.11873853 0.19808527 0.07971763 0.03985882 0.1       ]
 [3.48840304 0.47688833 0.95377667 0.48840304 0.47688833 0.03906254
  0.22742248 0.11645686 0.19536122 0.07812507 0.03906254 0.1       ]
 [3.48242198 0.47454049 0.94908097 0.48242198 0.47454049 0.03836756
  0.22518867 0.11446438 0.19296879 0.07673511 0.03836756 0.1       ]
 [3.47716502 0.47246548 0.94493095 0.47716502 0.47246548 0.0377601
  0.22322363 0.112722   0.19086601 0.0755202  0.0377601  0.1       ]
 [3.47254128 0.47063147 0.94126293 0.47254128 0.47063147 0.03722844
  0.22149398 0.1111964  0.18901651 0.07445688 0.03722844 0.1       ]
 [3.46847202 0.46901037 0.93802074 0.46847202 0.46901037 0.03676259
  0.21997073 0.10985912 0.18738881 0.07352518 0.03676259 0.1       ]
 [3.46488881 0.46757739 0.93515478 0.46488881 0.46757739 0.03635397
  0.21862862 0.10868575 0.18595552 0.07270794 0.03635397 0.1       ]
 [3.46173207 0.46631065 0.9326213  0.46173207 0.46631065 0.03599523
  0.21744562 0.10765529 0.18469283 0.07199046 0.03599523 0.1       ]
 [3.45894987 0.46519082 0.93038164 0.45894987 0.46519082 0.03568002
  0.2164025  0.10674963 0.18357995 0.07136003 0.03568002 0.1       ]
 [3.45649684 0.46420083 0.92840166 0.45649684 0.46420083 0.03540285
  0.21548241 0.10595311 0.18259874 0.0708057  0.03540285 0.1       ]
 [3.45433332 0.4633256  0.92665119 0.45433332 0.4633256  0.03515899
  0.21467061 0.10525213 0.18173333 0.07031797 0.03515899 0.1       ]
 [3.45242457 0.4625518  0.9251036  0.45242457 0.4625518  0.03494429
  0.21395417 0.1046349  0.18096983 0.06988859 0.03494429 0.1       ]
 [3.45074016 0.46186768 0.92373536 0.45074016 0.46186768 0.03475519
  0.21332175 0.10409116 0.18029606 0.06951039 0.03475519 0.1       ]
 [3.44925337 0.46126282 0.92252564 0.44925337 0.46126282 0.03458856
  0.21276339 0.10361194 0.17970135 0.06917712 0.03458856 0.1       ]
 [3.44794075 0.46072804 0.92145608 0.44794075 0.46072804 0.03444166
  0.21227033 0.10318943 0.1791763  0.06888332 0.03444166 0.1       ]
 [3.44678168 0.46025521 0.92051041 0.44678168 0.46025521 0.03431212
  0.21183485 0.1028168  0.17871267 0.06862424 0.03431212 0.1       ]
 [3.44575804 0.45983714 0.91967427 0.44575804 0.45983714 0.03419784
  0.21145019 0.10248805 0.17830322 0.06839569 0.03419784 0.1       ]
 [3.44485388 0.45946749 0.91893498 0.44485388 0.45946749 0.03409701
  0.21111037 0.10219795 0.17794155 0.06819402 0.03409701 0.1       ]
 [3.44405514 0.45914065 0.91828131 0.44405514 0.45914065 0.03400802
  0.21081014 0.10194188 0.17762206 0.06801603 0.03400802 0.1       ]
 [3.44334947 0.45885167 0.91770333 0.44334947 0.45885167 0.03392946
  0.21054485 0.10171582 0.17733979 0.06785891 0.03392946 0.1       ]
 [3.44272595 0.45859615 0.91719229 0.44272595 0.45859615 0.03386009
  0.21031042 0.10151621 0.17709038 0.06772018 0.03386009 0.1       ]
 [3.44217497 0.45837021 0.91674042 0.44217497 0.45837021 0.03379883
  0.21010325 0.10133992 0.17686999 0.06759766 0.03379883 0.1       ]
 [3.44168805 0.45817043 0.91634087 0.44168805 0.45817043 0.03374473
  0.20992015 0.1011842  0.17667522 0.06748945 0.03374473 0.1       ]
 [3.44125772 0.45799379 0.91598758 0.44125772 0.45799379 0.03369693
  0.20975831 0.10104665 0.17650309 0.06739386 0.03369693 0.1       ]
 [3.44087738 0.4578376  0.9156752  0.44087738 0.4578376  0.03365471
  0.20961527 0.10092512 0.17635095 0.06730942 0.03365471 0.1       ]
 [3.44054121 0.45769949 0.91539898 0.44054121 0.45769949 0.0336174
  0.20948882 0.10081774 0.17621648 0.0672348  0.0336174  0.1       ]
 [3.44024405 0.45757737 0.91515474 0.44024405 0.45757737 0.03358444
  0.20937705 0.10072286 0.17609762 0.06716887 0.03358444 0.1       ]
 [3.43998137 0.45746939 0.91493878 0.43998137 0.45746939 0.03355531
  0.20927824 0.10063901 0.17599255 0.06711061 0.03355531 0.1       ]
 [3.43974917 0.45737391 0.91474782 0.43974917 0.45737391 0.03352956
  0.20919089 0.1005649  0.17589967 0.06705912 0.03352956 0.1       ]
 [3.43954389 0.45728948 0.91457897 0.43954389 0.45728948 0.03350681
  0.20911367 0.1004994  0.17581756 0.06701361 0.03350681 0.1       ]
 [3.43936242 0.45721483 0.91442966 0.43936242 0.45721483 0.0334867
  0.2090454  0.10044151 0.17574497 0.06697339 0.0334867  0.1       ]
 [3.43920198 0.45714882 0.91429764 0.43920198 0.45714882 0.03346892
  0.20898505 0.10039033 0.17568079 0.06693784 0.03346892 0.1       ]
 [3.43906014 0.45709045 0.91418091 0.43906014 0.45709045 0.03345321
  0.20893168 0.1003451  0.17562406 0.06690641 0.03345321 0.1       ]
 [3.43893474 0.45703884 0.91407768 0.43893474 0.45703884 0.03343932
  0.2088845  0.10030511 0.1755739  0.06687863 0.03343932 0.1       ]
 [3.43882387 0.4569932  0.91398641 0.43882387 0.4569932  0.03342704
  0.20884279 0.10026976 0.17552955 0.06685407 0.03342704 0.1       ]
 [3.43872584 0.45695285 0.9139057  0.43872584 0.45695285 0.03341618
  0.20880591 0.10023851 0.17549034 0.06683236 0.03341618 0.1       ]
 [3.43863917 0.45691717 0.91383433 0.43863917 0.45691717 0.03340658
  0.2087733  0.10021088 0.17545567 0.06681317 0.03340658 0.1       ]
 [3.43856254 0.45688561 0.91377123 0.43856254 0.45688561 0.0333981
  0.20874446 0.10018646 0.17542502 0.0667962  0.0333981  0.1       ]
 [3.43849478 0.45685771 0.91371543 0.43849478 0.45685771 0.0333906
  0.20871897 0.10016486 0.17539791 0.0667812  0.0333906  0.1       ]
 [3.43843488 0.45683304 0.91366609 0.43843488 0.45683304 0.03338397
  0.20869643 0.10014577 0.17537395 0.06676793 0.03338397 0.1       ]
 [3.4383819  0.45681123 0.91362246 0.4383819  0.45681123 0.0333781
  0.2086765  0.10012889 0.17535276 0.0667562  0.0333781  0.1       ]
 [3.43833507 0.45679194 0.91358388 0.43833507 0.45679194 0.03337292
  0.20865887 0.10011396 0.17533403 0.06674583 0.03337292 0.1       ]
 [3.43829365 0.45677488 0.91354976 0.43829365 0.45677488 0.03336833
  0.20864329 0.10010077 0.17531746 0.06673667 0.03336833 0.1       ]
 [3.43825703 0.4567598  0.9135196  0.43825703 0.4567598  0.03336428
  0.20862951 0.1000891  0.17530281 0.06672856 0.03336428 0.1       ]
 [3.43822466 0.45674646 0.91349292 0.43822466 0.45674646 0.0333607
  0.20861733 0.10007878 0.17528986 0.06672139 0.0333607  0.1       ]
 [3.43819603 0.45673467 0.91346934 0.43819603 0.45673467 0.03335753
  0.20860656 0.10006966 0.17527841 0.06671506 0.03335753 0.1       ]]
  preeq_wrms:  nan
     preeq_t:  nan
preeq_numlinsteps:  None
preeq_numsteps:  [[0 0 0]]
preeq_numstepsB:  0.0
preeq_status:  [[0 0 0]]
preeq_cpu_time:  0.0
preeq_cpu_timeB:  0.0
 posteq_wrms:  nan
    posteq_t:  nan
posteq_numlinsteps:  None
posteq_numsteps:  [[0 0 0]]
posteq_numstepsB:  0.0
posteq_status:  [[0 0 0]]
posteq_cpu_time:  0.0
posteq_cpu_timeB:  0.0
    numsteps:  [  0 100 144 165 181 191 200 207 213 218 223 228 233 237 241 245 249 252
 255 258 261 264 266 269 272 275 278 282 286 290 293 296 299 303 307 311
 314 317 321 325 328 333 337 340 342 344 346 348 350 352 354 356 358 359
 360 361 362 363 364 365]
 numrhsevals:  [  0 114 160 193 212 227 237 248 255 260 267 272 277 282 287 292 296 300
 303 306 309 312 315 318 322 325 329 333 337 342 345 348 352 358 365 369
 372 376 381 385 389 395 400 403 405 407 409 411 413 415 417 419 421 422
 424 426 427 428 429 430]
numerrtestfails:  [0 1 1 3 3 4 4 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 6 6 6
 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6]
numnonlinsolvconvfails:  [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
       order:  [0 5 5 5 5 5 4 5 4 5 4 4 4 4 4 4 5 5 5 5 5 5 5 5 4 4 5 5 5 4 4 4 5 5 5 4 4
 4 4 5 5 4 5 5 5 5 5 5 5 4 4 4 4 4 4 4 4 4 4 4]
    cpu_time:  3.176
   numstepsB:  None
numrhsevalsB:  None
numerrtestfailsB:  None
numnonlinsolvconvfailsB:  None
   cpu_timeB:  0.0
[13]:
print(model.getParameters())
(1.0, 0.5, 0.4, 2.0, 0.1, 2.0, 3.0, 0.2)

Plotting trajectories

The simulation results above did not look too appealing. Let’s plot the trajectories of the model states and outputs them using matplotlib.pyplot:

[14]:
import amici.plotting
amici.plotting.plotStateTrajectories(rdata, model = None)
amici.plotting.plotObservableTrajectories(rdata, model = None)
_images/ExampleSteadystate_26_0.png
_images/ExampleSteadystate_26_1.png

Computing likelihood

Often model parameters need to be inferred from experimental data. This is commonly done by maximizing the likelihood of observing the data given to current model parameters. AMICI will compute this likelihood if experimental data is provided to amici.runAmiciSimulation as optional third argument. Measurements along with their standard deviations are provided through an amici.ExpData instance.

[15]:
# Create model instance and set time points for simulation
model = model_module.getModel()
model.setTimepoints(np.linspace(0, 10, 11))

# Create solver instance, keep default options
solver = model.getSolver()

# Run simulation without experimental data
rdata = amici.runAmiciSimulation(model, solver)

# Create ExpData instance from simulation results
edata = amici.ExpData(rdata, 1.0, 0.0)

# Re-run simulation, this time passing "experimental data"
rdata = amici.runAmiciSimulation(model, solver, edata)

print('Log-likelihood %f' % rdata['llh'])
Log-likelihood -97.118555

Simulation tolerances

Numerical error tolerances are often critical to get accurate results. For the state variables, integration errors can be controlled using setRelativeTolerance and setAbsoluteTolerance. Similar functions exist for sensitivities, steadystates and quadratures. We initially compute a reference solution using extremely low tolerances and then assess the influence on integration error for different levels of absolute and relative tolerance.

[16]:
solver.setRelativeTolerance(1e-16)
solver.setAbsoluteTolerance(1e-16)
solver.setSensitivityOrder(amici.SensitivityOrder.none)
rdata_ref = amici.runAmiciSimulation(model, solver, edata)

def get_simulation_error(solver):
    rdata = amici.runAmiciSimulation(model, solver, edata)
    return np.mean(np.abs(rdata['x']-rdata_ref['x'])), np.mean(np.abs(rdata['llh']-rdata_ref['llh']))

def get_errors(tolfun, tols):
    solver.setRelativeTolerance(1e-16)
    solver.setAbsoluteTolerance(1e-16)
    x_errs = []
    llh_errs = []
    for tol in tols:
        getattr(solver, tolfun)(tol)
        x_err, llh_err = get_simulation_error(solver)
        x_errs.append(x_err)
        llh_errs.append(llh_err)
    return x_errs, llh_errs

atols = np.logspace(-5,-15, 100)
atol_x_errs, atol_llh_errs = get_errors('setAbsoluteTolerance', atols)

rtols = np.logspace(-5,-15, 100)
rtol_x_errs, rtol_llh_errs = get_errors('setRelativeTolerance', rtols)

fig, axes = plt.subplots(1, 2, figsize=(15, 5))

def plot_error(tols, x_errs, llh_errs, tolname, ax):
    ax.plot(tols, x_errs, 'r-', label='x')
    ax.plot(tols, llh_errs, 'b-', label='llh')
    ax.set_xscale('log')
    ax.set_yscale('log')
    ax.set_xlabel(f'{tolname} tolerance')
    ax.set_ylabel('average numerical error')
    ax.legend()

plot_error(atols, atol_x_errs, atol_llh_errs, 'absolute', axes[0])
plot_error(rtols, rtol_x_errs, rtol_llh_errs, 'relative', axes[1])

# reset relative tolerance to default value
solver.setRelativeTolerance(1e-8)
solver.setRelativeTolerance(1e-16)
_images/ExampleSteadystate_30_0.png

Sensitivity analysis

AMICI can provide first- and second-order sensitivities using the forward- or adjoint-method. The respective options are set on the Model and Solver objects.

Forward sensitivity analysis

[17]:
model = model_module.getModel()
model.setTimepoints(np.linspace(0, 10, 11))
model.requireSensitivitiesForAllParameters()              # sensitivities w.r.t. all parameters
# model.setParameterList([1, 2])                          # sensitivities
# w.r.t. the specified parameters
model.setParameterScale(amici.ParameterScaling.none)         # parameters are used as-is (not log-transformed)

solver = model.getSolver()
solver.setSensitivityMethod(amici.SensitivityMethod.forward)        # forward sensitivity analysis
solver.setSensitivityOrder(amici.SensitivityOrder.first) # first-order sensitivities

rdata = amici.runAmiciSimulation(model, solver)

# print sensitivity-related results
for key, value in rdata.items():
    if key.startswith('s'):
        print('%12s: ' % key, value)
          sx:  [[[ 0.00000000e+00  0.00000000e+00  0.00000000e+00]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00]]

 [[-2.00747250e-01  1.19873139e-01 -9.44167985e-03]
  [-1.02561396e-01 -1.88820454e-01  1.01855972e-01]
  [ 4.66193077e-01 -2.86365372e-01  2.39662449e-02]
  [ 4.52560294e-02  1.14631370e-01 -3.34067919e-02]
  [ 4.00672911e-01  1.92564093e-01  4.98877759e-02]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00]]

 [[-2.23007240e-01  1.53979022e-01 -1.26885280e-02]
  [-1.33426939e-01 -3.15955239e-01  9.49575030e-02]
  [ 5.03470377e-01 -3.52731535e-01  2.81567412e-02]
  [ 3.93630714e-02  1.10770683e-01 -1.05673869e-02]
  [ 5.09580304e-01  4.65255489e-01  9.24843702e-02]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00]]

 [[-2.14278104e-01  1.63465064e-01 -1.03268418e-02]
  [-1.60981967e-01 -4.00490452e-01  7.54810648e-02]
  [ 4.87746419e-01 -3.76014315e-01  2.30919334e-02]
  [ 4.28733680e-02  1.15473583e-01 -6.63571687e-03]
  [ 6.05168647e-01  7.07226039e-01  1.23870914e-01]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00]]

 [[-2.05888038e-01  1.69308689e-01 -7.93085660e-03]
  [-1.84663809e-01 -4.65451966e-01  5.95026117e-02]
  [ 4.66407064e-01 -3.87612079e-01  1.76410128e-02]
  [ 4.52451104e-02  1.19865712e-01 -4.73313094e-03]
  [ 6.90798449e-01  9.20396633e-01  1.49475827e-01]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00]]

 [[-1.98803165e-01  1.73327268e-01 -6.03008179e-03]
  [-2.04303740e-01 -5.16111388e-01  4.68785776e-02]
  [ 4.47070326e-01 -3.94304029e-01  1.32107437e-02]
  [ 4.69732048e-02  1.22961727e-01 -3.35899442e-03]
  [ 7.68998995e-01  1.10844286e+00  1.70889328e-01]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00]]

 [[-1.92789113e-01  1.75978657e-01 -4.54517629e-03]
  [-2.20500138e-01 -5.55540705e-01  3.68776526e-02]
  [ 4.30424855e-01 -3.97907706e-01  9.75257113e-03]
  [ 4.82793652e-02  1.24952071e-01 -2.30991637e-03]
  [ 8.40805131e-01  1.27504628e+00  1.89020151e-01]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00]]

 [[-1.87672774e-01  1.77588334e-01 -3.38318222e-03]
  [-2.33807210e-01 -5.86081383e-01  2.89236334e-02]
  [ 4.16201399e-01 -3.99295277e-01  7.06598588e-03]
  [ 4.92546648e-02  1.26089711e-01 -1.50412006e-03]
  [ 9.06806543e-01  1.42334018e+00  2.04522708e-01]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00]]

 [[-1.83320440e-01  1.78410042e-01 -2.47240692e-03]
  [-2.44690164e-01 -6.09568485e-01  2.25774266e-02]
  [ 4.04061655e-01 -3.99063012e-01  4.97908386e-03]
  [ 4.99612484e-02  1.26581014e-01 -8.85891342e-04]
  [ 9.67473970e-01  1.55589415e+00  2.17895305e-01]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00]]

 [[-1.79620591e-01  1.78640114e-01 -1.75822439e-03]
  [-2.53540123e-01 -6.27448857e-01  1.75019839e-02]
  [ 3.93704970e-01 -3.97656641e-01  3.35895484e-03]
  [ 5.04492282e-02  1.26586733e-01 -4.13401240e-04]
  [ 1.02322336e+00  1.67481439e+00  2.29524046e-01]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00]]

 [[-1.76478441e-01  1.78430281e-01 -1.19867662e-03]
  [-2.60686971e-01 -6.40868686e-01  1.34365068e-02]
  [ 3.84873835e-01 -3.95414931e-01  2.10369522e-03]
  [ 5.07601805e-02  1.26231631e-01 -5.46465317e-05]
  [ 1.07443160e+00  1.78183962e+00  2.39710937e-01]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00]]]
         sx0:  [[0. 0. 0.]
 [0. 0. 0.]
 [0. 0. 0.]
 [0. 0. 0.]
 [0. 0. 0.]
 [0. 0. 0.]
 [0. 0. 0.]
 [0. 0. 0.]]
       sx_ss:  [[nan nan nan]
 [nan nan nan]
 [nan nan nan]
 [nan nan nan]
 [nan nan nan]
 [nan nan nan]
 [nan nan nan]
 [nan nan nan]]
      sigmay:  [[1.  1.  1.  1.  1.  0.2]
 [1.  1.  1.  1.  1.  0.2]
 [1.  1.  1.  1.  1.  0.2]
 [1.  1.  1.  1.  1.  0.2]
 [1.  1.  1.  1.  1.  0.2]
 [1.  1.  1.  1.  1.  0.2]
 [1.  1.  1.  1.  1.  0.2]
 [1.  1.  1.  1.  1.  0.2]
 [1.  1.  1.  1.  1.  0.2]
 [1.  1.  1.  1.  1.  0.2]
 [1.  1.  1.  1.  1.  0.2]]
          sy:  [[[ 0.00000000e+00  0.00000000e+00  0.00000000e+00  0.00000000e+00
    0.00000000e+00  0.00000000e+00]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00  0.00000000e+00
    0.00000000e+00  0.00000000e+00]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00  0.00000000e+00
    0.00000000e+00  0.00000000e+00]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00  0.00000000e+00
    0.00000000e+00  0.00000000e+00]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00  0.00000000e+00
    0.00000000e+00  0.00000000e+00]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00  1.00000000e-01
    0.00000000e+00  0.00000000e+00]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00  0.00000000e+00
    1.00000000e+00  0.00000000e+00]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00  0.00000000e+00
    0.00000000e+00  0.00000000e+00]]

 [[-2.00747250e-01  1.19873139e-01 -9.44167985e-03 -4.01494500e-01
    1.19873139e-01 -2.00747250e-01]
  [-1.02561396e-01 -1.88820454e-01  1.01855972e-01 -2.05122791e-01
   -1.88820454e-01 -1.02561396e-01]
  [ 4.66193077e-01 -2.86365372e-01  2.39662449e-02  9.32386154e-01
   -2.86365372e-01  4.66193077e-01]
  [ 4.52560294e-02  1.14631370e-01 -3.34067919e-02  9.05120589e-02
    1.14631370e-01  4.52560294e-02]
  [ 4.00672911e-01  1.92564093e-01  4.98877759e-02  8.01345822e-01
    1.92564093e-01  4.00672911e-01]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00  5.80072436e-01
    0.00000000e+00  0.00000000e+00]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00  0.00000000e+00
    1.00000000e+00  0.00000000e+00]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00  0.00000000e+00
    0.00000000e+00  0.00000000e+00]]

 [[-2.23007240e-01  1.53979022e-01 -1.26885280e-02 -4.46014480e-01
    1.53979022e-01 -2.23007240e-01]
  [-1.33426939e-01 -3.15955239e-01  9.49575030e-02 -2.66853878e-01
   -3.15955239e-01 -1.33426939e-01]
  [ 5.03470377e-01 -3.52731535e-01  2.81567412e-02  1.00694075e+00
   -3.52731535e-01  5.03470377e-01]
  [ 3.93630714e-02  1.10770683e-01 -1.05673869e-02  7.87261427e-02
    1.10770683e-01  3.93630714e-02]
  [ 5.09580304e-01  4.65255489e-01  9.24843702e-02  1.01916061e+00
    4.65255489e-01  5.09580304e-01]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00  5.60534516e-01
    0.00000000e+00  0.00000000e+00]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00  0.00000000e+00
    1.00000000e+00  0.00000000e+00]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00  0.00000000e+00
    0.00000000e+00  0.00000000e+00]]

 [[-2.14278104e-01  1.63465064e-01 -1.03268418e-02 -4.28556209e-01
    1.63465064e-01 -2.14278104e-01]
  [-1.60981967e-01 -4.00490452e-01  7.54810648e-02 -3.21963935e-01
   -4.00490452e-01 -1.60981967e-01]
  [ 4.87746419e-01 -3.76014315e-01  2.30919334e-02  9.75492839e-01
   -3.76014315e-01  4.87746419e-01]
  [ 4.28733680e-02  1.15473583e-01 -6.63571687e-03  8.57467361e-02
    1.15473583e-01  4.28733680e-02]
  [ 6.05168647e-01  7.07226039e-01  1.23870914e-01  1.21033729e+00
    7.07226039e-01  6.05168647e-01]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00  5.46870655e-01
    0.00000000e+00  0.00000000e+00]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00  0.00000000e+00
    1.00000000e+00  0.00000000e+00]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00  0.00000000e+00
    0.00000000e+00  0.00000000e+00]]

 [[-2.05888038e-01  1.69308689e-01 -7.93085660e-03 -4.11776077e-01
    1.69308689e-01 -2.05888038e-01]
  [-1.84663809e-01 -4.65451966e-01  5.95026117e-02 -3.69327617e-01
   -4.65451966e-01 -1.84663809e-01]
  [ 4.66407064e-01 -3.87612079e-01  1.76410128e-02  9.32814128e-01
   -3.87612079e-01  4.66407064e-01]
  [ 4.52451104e-02  1.19865712e-01 -4.73313094e-03  9.04902208e-02
    1.19865712e-01  4.52451104e-02]
  [ 6.90798449e-01  9.20396633e-01  1.49475827e-01  1.38159690e+00
    9.20396633e-01  6.90798449e-01]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00  5.36280366e-01
    0.00000000e+00  0.00000000e+00]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00  0.00000000e+00
    1.00000000e+00  0.00000000e+00]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00  0.00000000e+00
    0.00000000e+00  0.00000000e+00]]

 [[-1.98803165e-01  1.73327268e-01 -6.03008179e-03 -3.97606330e-01
    1.73327268e-01 -1.98803165e-01]
  [-2.04303740e-01 -5.16111388e-01  4.68785776e-02 -4.08607480e-01
   -5.16111388e-01 -2.04303740e-01]
  [ 4.47070326e-01 -3.94304029e-01  1.32107437e-02  8.94140651e-01
   -3.94304029e-01  4.47070326e-01]
  [ 4.69732048e-02  1.22961727e-01 -3.35899442e-03  9.39464097e-02
    1.22961727e-01  4.69732048e-02]
  [ 7.68998995e-01  1.10844286e+00  1.70889328e-01  1.53799799e+00
    1.10844286e+00  7.68998995e-01]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00  5.27091252e-01
    0.00000000e+00  0.00000000e+00]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00  0.00000000e+00
    1.00000000e+00  0.00000000e+00]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00  0.00000000e+00
    0.00000000e+00  0.00000000e+00]]

 [[-1.92789113e-01  1.75978657e-01 -4.54517629e-03 -3.85578227e-01
    1.75978657e-01 -1.92789113e-01]
  [-2.20500138e-01 -5.55540705e-01  3.68776526e-02 -4.41000277e-01
   -5.55540705e-01 -2.20500138e-01]
  [ 4.30424855e-01 -3.97907706e-01  9.75257113e-03  8.60849709e-01
   -3.97907706e-01  4.30424855e-01]
  [ 4.82793652e-02  1.24952071e-01 -2.30991637e-03  9.65587304e-02
    1.24952071e-01  4.82793652e-02]
  [ 8.40805131e-01  1.27504628e+00  1.89020151e-01  1.68161026e+00
    1.27504628e+00  8.40805131e-01]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00  5.18989205e-01
    0.00000000e+00  0.00000000e+00]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00  0.00000000e+00
    1.00000000e+00  0.00000000e+00]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00  0.00000000e+00
    0.00000000e+00  0.00000000e+00]]

 [[-1.87672774e-01  1.77588334e-01 -3.38318222e-03 -3.75345548e-01
    1.77588334e-01 -1.87672774e-01]
  [-2.33807210e-01 -5.86081383e-01  2.89236334e-02 -4.67614420e-01
   -5.86081383e-01 -2.33807210e-01]
  [ 4.16201399e-01 -3.99295277e-01  7.06598588e-03  8.32402797e-01
   -3.99295277e-01  4.16201399e-01]
  [ 4.92546648e-02  1.26089711e-01 -1.50412006e-03  9.85093296e-02
    1.26089711e-01  4.92546648e-02]
  [ 9.06806543e-01  1.42334018e+00  2.04522708e-01  1.81361309e+00
    1.42334018e+00  9.06806543e-01]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00  5.11829985e-01
    0.00000000e+00  0.00000000e+00]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00  0.00000000e+00
    1.00000000e+00  0.00000000e+00]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00  0.00000000e+00
    0.00000000e+00  0.00000000e+00]]

 [[-1.83320440e-01  1.78410042e-01 -2.47240692e-03 -3.66640879e-01
    1.78410042e-01 -1.83320440e-01]
  [-2.44690164e-01 -6.09568485e-01  2.25774266e-02 -4.89380329e-01
   -6.09568485e-01 -2.44690164e-01]
  [ 4.04061655e-01 -3.99063012e-01  4.97908386e-03  8.08123310e-01
   -3.99063012e-01  4.04061655e-01]
  [ 4.99612484e-02  1.26581014e-01 -8.85891342e-04  9.99224969e-02
    1.26581014e-01  4.99612484e-02]
  [ 9.67473970e-01  1.55589415e+00  2.17895305e-01  1.93494794e+00
    1.55589415e+00  9.67473970e-01]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00  5.05500234e-01
    0.00000000e+00  0.00000000e+00]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00  0.00000000e+00
    1.00000000e+00  0.00000000e+00]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00  0.00000000e+00
    0.00000000e+00  0.00000000e+00]]

 [[-1.79620591e-01  1.78640114e-01 -1.75822439e-03 -3.59241183e-01
    1.78640114e-01 -1.79620591e-01]
  [-2.53540123e-01 -6.27448857e-01  1.75019839e-02 -5.07080247e-01
   -6.27448857e-01 -2.53540123e-01]
  [ 3.93704970e-01 -3.97656641e-01  3.35895484e-03  7.87409940e-01
   -3.97656641e-01  3.93704970e-01]
  [ 5.04492282e-02  1.26586733e-01 -4.13401240e-04  1.00898456e-01
    1.26586733e-01  5.04492282e-02]
  [ 1.02322336e+00  1.67481439e+00  2.29524046e-01  2.04644672e+00
    1.67481439e+00  1.02322336e+00]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00  4.99901907e-01
    0.00000000e+00  0.00000000e+00]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00  0.00000000e+00
    1.00000000e+00  0.00000000e+00]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00  0.00000000e+00
    0.00000000e+00  0.00000000e+00]]

 [[-1.76478441e-01  1.78430281e-01 -1.19867662e-03 -3.52956882e-01
    1.78430281e-01 -1.76478441e-01]
  [-2.60686971e-01 -6.40868686e-01  1.34365068e-02 -5.21373942e-01
   -6.40868686e-01 -2.60686971e-01]
  [ 3.84873835e-01 -3.95414931e-01  2.10369522e-03  7.69747670e-01
   -3.95414931e-01  3.84873835e-01]
  [ 5.07601805e-02  1.26231631e-01 -5.46465317e-05  1.01520361e-01
    1.26231631e-01  5.07601805e-02]
  [ 1.07443160e+00  1.78183962e+00  2.39710937e-01  2.14886320e+00
    1.78183962e+00  1.07443160e+00]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00  4.94949118e-01
    0.00000000e+00  0.00000000e+00]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00  0.00000000e+00
    1.00000000e+00  0.00000000e+00]
  [ 0.00000000e+00  0.00000000e+00  0.00000000e+00  0.00000000e+00
    0.00000000e+00  0.00000000e+00]]]
     ssigmay:  [[[0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 1.]]

 [[0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 1.]]

 [[0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 1.]]

 [[0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 1.]]

 [[0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 1.]]

 [[0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 1.]]

 [[0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 1.]]

 [[0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 1.]]

 [[0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 1.]]

 [[0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 1.]]

 [[0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 0.]
  [0. 0. 0. 0. 0. 1.]]]
      sigmaz:  None
          sz:  None
         srz:  None
     ssigmaz:  None
        sllh:  [nan nan nan nan nan nan nan nan]
       s2llh:  None
      status:  0.0
        sres:  [[0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0.]]

Adjoint sensitivity analysis

[18]:
# Set model options
model = model_module.getModel()
p_orig = np.array(model.getParameters())
p_orig[list(model.getParameterIds()).index('observable_x1withsigma_sigma')] = 0.1 # Change default parameter
model.setParameters(p_orig)
model.setParameterScale(amici.ParameterScaling.none)
model.setTimepoints(np.linspace(0, 10, 21))

solver = model.getSolver()
solver.setMaxSteps(10**4) # Set maximum number of steps for the solver

# simulate time-course to get artificial data
rdata = amici.runAmiciSimulation(model, solver)
edata = amici.ExpData(rdata, 1.0, 0)
edata.fixedParameters = model.getFixedParameters()
# set sigma to 1.0 except for observable 5, so that p[7] is used instead
# (if we have sigma parameterized, the corresponding ExpData entries must NaN, otherwise they will override the parameter)
edata.setObservedDataStdDev(rdata['t']*0+np.nan,
                            list(model.getObservableIds()).index('observable_x1withsigma'))

# enable sensitivities
solver.setSensitivityOrder(amici.SensitivityOrder.first) # First-order ...
solver.setSensitivityMethod(amici.SensitivityMethod.adjoint)        # ... adjoint sensitivities
model.requireSensitivitiesForAllParameters()              # ... w.r.t. all parameters

# compute adjoint sensitivities
rdata = amici.runAmiciSimulation(model, solver, edata)
#print(rdata['sigmay'])
print('Log-likelihood: %f\nGradient: %s' % (rdata['llh'], rdata['sllh']))

Log-likelihood: -1190.452734
Gradient: [-8.18063367e+01 -7.40378749e+01  1.87640047e+02  2.07890554e+01
  2.62573207e+02  1.77402064e-01  1.15646253e+01  2.11221869e+04]

Finite differences gradient check

Compare AMICI-computed gradient with finite differences

[19]:
from scipy.optimize import check_grad

def func(x0, symbol='llh', x0full=None, plist=[], verbose=False):
    p = x0[:]
    if len(plist):
        p = x0full[:]
        p[plist] = x0
    verbose and print('f: p=%s' % p)

    old_parameters = model.getParameters()
    solver.setSensitivityOrder(amici.SensitivityOrder.none)
    model.setParameters(p)
    rdata = amici.runAmiciSimulation(model, solver, edata)

    model.setParameters(old_parameters)

    res = np.sum(rdata[symbol])
    verbose and print(res)
    return res

def grad(x0, symbol='llh', x0full=None, plist=[], verbose=False):
    p = x0[:]
    if len(plist):
        model.setParameterList(plist)
        p = x0full[:]
        p[plist] = x0
    else:
        model.requireSensitivitiesForAllParameters()
    verbose and print('g: p=%s' % p)

    old_parameters = model.getParameters()
    solver.setSensitivityMethod(amici.SensitivityMethod.forward)
    solver.setSensitivityOrder(amici.SensitivityOrder.first)
    model.setParameters(p)
    rdata = amici.runAmiciSimulation(model, solver, edata)

    model.setParameters(old_parameters)

    res = rdata['s%s' % symbol]
    if not isinstance(res, float):
        if len(res.shape) == 3:
            res = np.sum(res, axis=(0, 2))
    verbose and print(res)
    return res

epsilon = 1e-4
err_norm = check_grad(func, grad, p_orig, 'llh', epsilon=epsilon)
print('sllh: |error|_2: %f' % err_norm)
# assert err_norm < 1e-6
print()

for ip in range(model.np()):
    plist = [ip]
    p = p_orig.copy()
    err_norm = check_grad(func, grad, p[plist], 'llh', p, [ip], epsilon=epsilon)
    print('sllh: p[%d]: |error|_2: %f' % (ip, err_norm))

print()
for ip in range(model.np()):
    plist = [ip]
    p = p_orig.copy()
    err_norm = check_grad(func, grad, p[plist], 'y', p, [ip], epsilon=epsilon)
    print('sy: p[%d]: |error|_2: %f' % (ip, err_norm))

print()
for ip in range(model.np()):
    plist = [ip]
    p = p_orig.copy()
    err_norm = check_grad(func, grad, p[plist], 'x', p, [ip], epsilon=epsilon)
    print('sx: p[%d]: |error|_2: %f' % (ip, err_norm))

print()
for ip in range(model.np()):
    plist = [ip]
    p = p_orig.copy()
    err_norm = check_grad(func, grad, p[plist], 'sigmay', p, [ip], epsilon=epsilon)
    print('ssigmay: p[%d]: |error|_2: %f' % (ip, err_norm))

sllh: |error|_2: 31.850873

sllh: p[0]: |error|_2: 0.006287
sllh: p[1]: |error|_2: 0.016510
sllh: p[2]: |error|_2: 0.017028
sllh: p[3]: |error|_2: 0.009608
sllh: p[4]: |error|_2: 0.083404
sllh: p[5]: |error|_2: 0.000280
sllh: p[6]: |error|_2: 0.001050
sllh: p[7]: |error|_2: 31.850739

sy: p[0]: |error|_2: 0.002974
sy: p[1]: |error|_2: 0.002717
sy: p[2]: |error|_2: 0.001308
sy: p[3]: |error|_2: 0.000939
sy: p[4]: |error|_2: 0.006106
sy: p[5]: |error|_2: 0.000000
sy: p[6]: |error|_2: 0.000000
sy: p[7]: |error|_2: 0.000000

sx: p[0]: |error|_2: 0.001033
sx: p[1]: |error|_2: 0.001076
sx: p[2]: |error|_2: 0.000121
sx: p[3]: |error|_2: 0.000439
sx: p[4]: |error|_2: 0.001569
sx: p[5]: |error|_2: 0.000000
sx: p[6]: |error|_2: 0.000000
sx: p[7]: |error|_2: 0.000000

ssigmay: p[0]: |error|_2: 0.000000
ssigmay: p[1]: |error|_2: 0.000000
ssigmay: p[2]: |error|_2: 0.000000
ssigmay: p[3]: |error|_2: 0.000000
ssigmay: p[4]: |error|_2: 0.000000
ssigmay: p[5]: |error|_2: 0.000000
ssigmay: p[6]: |error|_2: 0.000000
ssigmay: p[7]: |error|_2: 0.000000
[20]:
eps=1e-4
op=model.getParameters()


solver.setSensitivityMethod(amici.SensitivityMethod.forward) # forward sensitivity analysis
solver.setSensitivityOrder(amici.SensitivityOrder.first) # first-order sensitivities
model.requireSensitivitiesForAllParameters()
solver.setRelativeTolerance(1e-12)
rdata = amici.runAmiciSimulation(model, solver, edata)

def fd(x0, ip, eps, symbol='llh'):
    p = list(x0[:])
    old_parameters = model.getParameters()
    solver.setSensitivityOrder(amici.SensitivityOrder.none)
    p[ip]+=eps
    model.setParameters(p)
    rdata_f = amici.runAmiciSimulation(model, solver, edata)
    p[ip]-=2*eps
    model.setParameters(p)
    rdata_b = amici.runAmiciSimulation(model, solver, edata)

    model.setParameters(old_parameters)
    return (rdata_f[symbol]-rdata_b[symbol])/(2*eps)

def plot_sensitivities(symbol, eps):
    fig, axes = plt.subplots(4,2, figsize=(15,10))
    for ip in range(4):
        fd_approx = fd(model.getParameters(), ip, eps, symbol=symbol)

        axes[ip,0].plot(edata.getTimepoints(), rdata[f's{symbol}'][:,ip,:], 'r-')
        axes[ip,0].plot(edata.getTimepoints(), fd_approx, 'k--')
        axes[ip,0].set_ylabel(f'sensitivity {symbol}')
        axes[ip,0].set_xlabel('time')


        axes[ip,1].plot(edata.getTimepoints(), np.abs(rdata[f's{symbol}'][:,ip,:]-fd_approx), 'k-')
        axes[ip,1].set_ylabel('difference to fd')
        axes[ip,1].set_xlabel('time')
        axes[ip,1].set_yscale('log')

    plt.tight_layout()
    plt.show()
[21]:
plot_sensitivities('x', eps)
_images/ExampleSteadystate_39_0.png
[22]:
plot_sensitivities('y', eps)
_images/ExampleSteadystate_40_0.png

Export as DataFrame

Experimental data and simulation results can both be exported as pandas Dataframe to allow for an easier inspection of numeric values

[23]:
# run the simulation
rdata = amici.runAmiciSimulation(model, solver, edata)
[24]:
# look at the ExpData as DataFrame
df = amici.getDataObservablesAsDataFrame(model, [edata])
df
[24]:
time datatype t_presim k0 k0_preeq k0_presim observable_x1 observable_x2 observable_x3 observable_x1_scaled observable_x2_offsetted observable_x1withsigma observable_x1_std observable_x2_std observable_x3_std observable_x1_scaled_std observable_x2_offsetted_std observable_x1withsigma_std
0 0.0 data 0.0 1.0 NaN NaN -1.191094 0.281469 0.033354 1.645144 3.900680 -0.518642 1.0 1.0 1.0 1.0 1.0 NaN
1 0.5 data 0.0 1.0 NaN NaN -1.599410 1.140165 1.368902 1.535490 5.678167 0.694825 1.0 1.0 1.0 1.0 1.0 NaN
2 1.0 data 0.0 1.0 NaN NaN 1.522095 2.020846 1.109229 0.594295 3.230374 1.275482 1.0 1.0 1.0 1.0 1.0 NaN
3 1.5 data 0.0 1.0 NaN NaN 2.455856 0.961785 0.052925 1.839501 2.783796 0.085480 1.0 1.0 1.0 1.0 1.0 NaN
4 2.0 data 0.0 1.0 NaN NaN -0.600864 0.021845 0.220294 -0.110990 5.003351 2.020780 1.0 1.0 1.0 1.0 1.0 NaN
5 2.5 data 0.0 1.0 NaN NaN 1.422341 0.045064 1.361070 0.945899 4.589646 -0.593535 1.0 1.0 1.0 1.0 1.0 NaN
6 3.0 data 0.0 1.0 NaN NaN -0.672523 0.873615 1.560601 2.677974 2.187698 1.054870 1.0 1.0 1.0 1.0 1.0 NaN
7 3.5 data 0.0 1.0 NaN NaN 2.278515 -0.005105 0.799689 1.240916 4.268726 0.491755 1.0 1.0 1.0 1.0 1.0 NaN
8 4.0 data 0.0 1.0 NaN NaN 0.078411 0.979531 1.745795 0.858134 4.609529 0.620844 1.0 1.0 1.0 1.0 1.0 NaN
9 4.5 data 0.0 1.0 NaN NaN 0.074017 0.226267 0.604246 -0.091356 3.652345 0.261807 1.0 1.0 1.0 1.0 1.0 NaN
10 5.0 data 0.0 1.0 NaN NaN 0.537820 1.233766 0.194859 -0.345351 4.477322 3.378227 1.0 1.0 1.0 1.0 1.0 NaN
11 5.5 data 0.0 1.0 NaN NaN 0.498204 3.218945 0.190730 0.462838 5.946970 1.310100 1.0 1.0 1.0 1.0 1.0 NaN
12 6.0 data 0.0 1.0 NaN NaN 2.345616 0.956306 -1.770517 0.761754 3.894865 1.546055 1.0 1.0 1.0 1.0 1.0 NaN
13 6.5 data 0.0 1.0 NaN NaN 0.995109 -0.151091 0.805937 2.929645 4.348557 0.234915 1.0 1.0 1.0 1.0 1.0 NaN
14 7.0 data 0.0 1.0 NaN NaN 1.363276 1.622225 0.030980 -0.082327 3.761730 0.724608 1.0 1.0 1.0 1.0 1.0 NaN
15 7.5 data 0.0 1.0 NaN NaN 0.190180 0.962294 1.425918 1.106591 4.175984 0.245791 1.0 1.0 1.0 1.0 1.0 NaN
16 8.0 data 0.0 1.0 NaN NaN -0.362771 -0.251576 0.540826 1.507657 3.871633 2.260562 1.0 1.0 1.0 1.0 1.0 NaN
17 8.5 data 0.0 1.0 NaN NaN 0.884408 0.930074 0.510605 2.012371 2.593086 -0.654355 1.0 1.0 1.0 1.0 1.0 NaN
18 9.0 data 0.0 1.0 NaN NaN -1.554260 1.449906 -1.477985 0.871124 5.497123 0.232772 1.0 1.0 1.0 1.0 1.0 NaN
19 9.5 data 0.0 1.0 NaN NaN 0.492781 0.700873 0.242116 0.279648 4.236658 -0.889212 1.0 1.0 1.0 1.0 1.0 NaN
20 10.0 data 0.0 1.0 NaN NaN -0.964663 -0.485664 2.087815 2.328707 4.787617 0.395106 1.0 1.0 1.0 1.0 1.0 NaN
[25]:
# from the exported dataframe, we can actually reconstruct a copy of the ExpData instance
reconstructed_edata = amici.getEdataFromDataFrame(model, df)
[26]:
# look at the States in rdata as DataFrame
amici.getResidualsAsDataFrame(model, [edata], [rdata])
[26]:
time t_presim k0 k0_preeq k0_presim observable_x1 observable_x2 observable_x3 observable_x1_scaled observable_x2_offsetted observable_x1withsigma
0 0.0 0.0 1.0 NaN NaN 1.291094 0.118531 0.666646 1.445144 0.500680 6.186425
1 0.5 0.0 1.0 NaN NaN 2.138777 0.455486 1.177412 0.456756 1.993488 1.554579
2 1.0 0.0 1.0 NaN NaN 0.942023 1.287558 1.012805 0.565850 0.502913 6.954092
3 1.5 0.0 1.0 NaN NaN 1.885457 0.231133 0.023151 0.698703 0.946856 4.849190
4 2.0 0.0 1.0 NaN NaN 1.161398 0.693991 0.150600 1.232059 1.287515 14.602456
5 2.5 0.0 1.0 NaN NaN 0.869285 0.653687 1.294769 0.160213 0.890895 11.465911
6 3.0 0.0 1.0 NaN NaN 1.219394 0.191652 1.496868 1.584233 1.494264 5.079991
7 3.5 0.0 1.0 NaN NaN 1.737155 0.671214 0.738182 0.158196 0.602617 0.496051
8 4.0 0.0 1.0 NaN NaN 0.457870 0.328229 1.686301 0.214427 0.958227 0.845638
9 4.5 0.0 1.0 NaN NaN 0.457522 0.411248 0.546593 1.154432 0.014831 2.697316
10 5.0 0.0 1.0 NaN NaN 0.010728 0.609084 0.138899 1.399534 0.852641 28.511357
11 5.5 0.0 1.0 NaN NaN 0.024710 2.606212 0.136330 0.582991 2.334237 7.871853
12 6.0 0.0 1.0 NaN NaN 1.826626 0.354703 1.823477 0.276225 0.293262 10.270655
13 6.5 0.0 1.0 NaN NaN 0.479809 0.742320 0.754307 1.899046 0.757328 2.803840
14 7.0 0.0 1.0 NaN NaN 0.851446 1.040670 0.019418 1.105987 0.180175 2.127777
15 7.5 0.0 1.0 NaN NaN 0.318388 0.389765 1.376658 0.089456 0.603455 2.627767
16 8.0 0.0 1.0 NaN NaN 0.868271 0.815679 0.492622 0.496656 0.307529 17.550614
17 8.5 0.0 1.0 NaN NaN 0.381793 0.373840 0.463381 1.007140 0.963148 11.569706
18 9.0 0.0 1.0 NaN NaN 2.054162 0.901024 1.524300 0.128680 1.948242 2.671303
19 9.5 0.0 1.0 NaN NaN 0.004569 0.158864 0.196646 0.715052 0.694649 13.865618
20 10.0 0.0 1.0 NaN NaN 1.459612 1.021246 2.043129 1.338809 1.252036 0.998435
[27]:
# look at the Observables in rdata as DataFrame
amici.getSimulationObservablesAsDataFrame(model, [edata], [rdata])
[27]:
time datatype t_presim k0 k0_preeq k0_presim observable_x1 observable_x2 observable_x3 observable_x1_scaled observable_x2_offsetted observable_x1withsigma observable_x1_std observable_x2_std observable_x3_std observable_x1_scaled_std observable_x2_offsetted_std observable_x1withsigma_std
0 0.0 simulation 0.0 1.0 NaN NaN 0.100000 0.400000 0.700000 0.200000 3.400000 0.100000 1.0 1.0 1.0 1.0 1.0 0.1
1 0.5 simulation 0.0 1.0 NaN NaN 0.539367 0.684679 0.191491 1.078734 3.684679 0.539367 1.0 1.0 1.0 1.0 1.0 0.1
2 1.0 simulation 0.0 1.0 NaN NaN 0.580072 0.733287 0.096424 1.160145 3.733287 0.580072 1.0 1.0 1.0 1.0 1.0 0.1
3 1.5 simulation 0.0 1.0 NaN NaN 0.570399 0.730652 0.076076 1.140799 3.730652 0.570399 1.0 1.0 1.0 1.0 1.0 0.1
4 2.0 simulation 0.0 1.0 NaN NaN 0.560535 0.715836 0.069694 1.121069 3.715836 0.560535 1.0 1.0 1.0 1.0 1.0 0.1
5 2.5 simulation 0.0 1.0 NaN NaN 0.553056 0.698751 0.066301 1.106112 3.698751 0.553056 1.0 1.0 1.0 1.0 1.0 0.1
6 3.0 simulation 0.0 1.0 NaN NaN 0.546871 0.681963 0.063733 1.093741 3.681963 0.546871 1.0 1.0 1.0 1.0 1.0 0.1
7 3.5 simulation 0.0 1.0 NaN NaN 0.541360 0.666109 0.061506 1.082720 3.666109 0.541360 1.0 1.0 1.0 1.0 1.0 0.1
8 4.0 simulation 0.0 1.0 NaN NaN 0.536280 0.651302 0.059495 1.072561 3.651302 0.536280 1.0 1.0 1.0 1.0 1.0 0.1
9 4.5 simulation 0.0 1.0 NaN NaN 0.531538 0.637515 0.057653 1.063076 3.637515 0.531538 1.0 1.0 1.0 1.0 1.0 0.1
10 5.0 simulation 0.0 1.0 NaN NaN 0.527091 0.624681 0.055960 1.054183 3.624681 0.527091 1.0 1.0 1.0 1.0 1.0 0.1
11 5.5 simulation 0.0 1.0 NaN NaN 0.522914 0.612733 0.054400 1.045829 3.612733 0.522914 1.0 1.0 1.0 1.0 1.0 0.1
12 6.0 simulation 0.0 1.0 NaN NaN 0.518989 0.601603 0.052960 1.037978 3.601603 0.518989 1.0 1.0 1.0 1.0 1.0 0.1
13 6.5 simulation 0.0 1.0 NaN NaN 0.515299 0.591229 0.051629 1.030598 3.591229 0.515299 1.0 1.0 1.0 1.0 1.0 0.1
14 7.0 simulation 0.0 1.0 NaN NaN 0.511830 0.581555 0.050399 1.023660 3.581555 0.511830 1.0 1.0 1.0 1.0 1.0 0.1
15 7.5 simulation 0.0 1.0 NaN NaN 0.508568 0.572529 0.049259 1.017136 3.572529 0.508568 1.0 1.0 1.0 1.0 1.0 0.1
16 8.0 simulation 0.0 1.0 NaN NaN 0.505500 0.564103 0.048203 1.011000 3.564103 0.505500 1.0 1.0 1.0 1.0 1.0 0.1
17 8.5 simulation 0.0 1.0 NaN NaN 0.502615 0.556234 0.047224 1.005231 3.556234 0.502615 1.0 1.0 1.0 1.0 1.0 0.1
18 9.0 simulation 0.0 1.0 NaN NaN 0.499902 0.548881 0.046315 0.999804 3.548881 0.499902 1.0 1.0 1.0 1.0 1.0 0.1
19 9.5 simulation 0.0 1.0 NaN NaN 0.497350 0.542008 0.045471 0.994700 3.542008 0.497350 1.0 1.0 1.0 1.0 1.0 0.1
20 10.0 simulation 0.0 1.0 NaN NaN 0.494949 0.535581 0.044686 0.989898 3.535581 0.494949 1.0 1.0 1.0 1.0 1.0 0.1
[28]:
# look at the States in rdata as DataFrame
amici.getSimulationStatesAsDataFrame(model, [edata], [rdata])

[28]:
time t_presim k0 k0_preeq k0_presim x1 x2 x3
0 0.0 0.0 1.0 NaN NaN 0.100000 0.400000 0.700000
1 0.5 0.0 1.0 NaN NaN 0.539367 0.684679 0.191491
2 1.0 0.0 1.0 NaN NaN 0.580072 0.733287 0.096424
3 1.5 0.0 1.0 NaN NaN 0.570399 0.730652 0.076076
4 2.0 0.0 1.0 NaN NaN 0.560535 0.715836 0.069694
5 2.5 0.0 1.0 NaN NaN 0.553056 0.698751 0.066301
6 3.0 0.0 1.0 NaN NaN 0.546871 0.681963 0.063733
7 3.5 0.0 1.0 NaN NaN 0.541360 0.666109 0.061506
8 4.0 0.0 1.0 NaN NaN 0.536280 0.651302 0.059495
9 4.5 0.0 1.0 NaN NaN 0.531538 0.637515 0.057653
10 5.0 0.0 1.0 NaN NaN 0.527091 0.624681 0.055960
11 5.5 0.0 1.0 NaN NaN 0.522914 0.612733 0.054400
12 6.0 0.0 1.0 NaN NaN 0.518989 0.601603 0.052960
13 6.5 0.0 1.0 NaN NaN 0.515299 0.591229 0.051629
14 7.0 0.0 1.0 NaN NaN 0.511830 0.581555 0.050399
15 7.5 0.0 1.0 NaN NaN 0.508568 0.572529 0.049259
16 8.0 0.0 1.0 NaN NaN 0.505500 0.564103 0.048203
17 8.5 0.0 1.0 NaN NaN 0.502615 0.556234 0.047224
18 9.0 0.0 1.0 NaN NaN 0.499902 0.548881 0.046315
19 9.5 0.0 1.0 NaN NaN 0.497350 0.542008 0.045471
20 10.0 0.0 1.0 NaN NaN 0.494949 0.535581 0.044686