.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_examples/programmers/plot_models.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_auto_examples_programmers_plot_models.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_auto_examples_programmers_plot_models.py:


biogeme.models
==============

Examples of use of several functions.

This is designed for programmers who need examples of use of the
functions of the module. The examples are designed to illustrate the
syntax. They do not correspond to any meaningful model.

:author: Michel Bierlaire
:date: Wed Nov 22 15:24:34 2023

.. GENERATED FROM PYTHON SOURCE LINES 15-30

.. code-block:: Python


    import numpy as np
    import pandas as pd
    import matplotlib.pyplot as plt
    from biogeme.version import get_text
    import biogeme.database as db
    from biogeme import models
    from biogeme.nests import (
        OneNestForNestedLogit,
        OneNestForCrossNestedLogit,
        NestsForNestedLogit,
        NestsForCrossNestedLogit,
    )
    from biogeme.expressions import Variable, Beta








.. GENERATED FROM PYTHON SOURCE LINES 31-32

Version of Biogeme.

.. GENERATED FROM PYTHON SOURCE LINES 32-35

.. code-block:: Python

    print(get_text())






.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    biogeme 3.2.14 [2024-08-05]
    Home page: http://biogeme.epfl.ch
    Submit questions to https://groups.google.com/d/forum/biogeme
    Michel Bierlaire, Transport and Mobility Laboratory, Ecole Polytechnique Fédérale de Lausanne (EPFL)





.. GENERATED FROM PYTHON SOURCE LINES 36-38

Definition of a database
++++++++++++++++++++++++

.. GENERATED FROM PYTHON SOURCE LINES 38-52

.. code-block:: Python

    df = pd.DataFrame(
        {
            'Person': [1, 1, 1, 2, 2],
            'Exclude': [0, 0, 1, 0, 1],
            'Variable1': [1, 2, 3, 4, 5],
            'Variable2': [10, 20, 30, 40, 50],
            'Choice': [1, 2, 3, 1, 2],
            'Av1': [0, 1, 1, 1, 1],
            'Av2': [1, 1, 0, 1, 1],
            'Av3': [0, 1, 1, 1, 0],
        }
    )
    df






.. raw:: html

    <div class="output_subarea output_html rendered_html output_result">
    <div>
    <style scoped>
        .dataframe tbody tr th:only-of-type {
            vertical-align: middle;
        }

        .dataframe tbody tr th {
            vertical-align: top;
        }

        .dataframe thead th {
            text-align: right;
        }
    </style>
    <table border="1" class="dataframe">
      <thead>
        <tr style="text-align: right;">
          <th></th>
          <th>Person</th>
          <th>Exclude</th>
          <th>Variable1</th>
          <th>Variable2</th>
          <th>Choice</th>
          <th>Av1</th>
          <th>Av2</th>
          <th>Av3</th>
        </tr>
      </thead>
      <tbody>
        <tr>
          <th>0</th>
          <td>1</td>
          <td>0</td>
          <td>1</td>
          <td>10</td>
          <td>1</td>
          <td>0</td>
          <td>1</td>
          <td>0</td>
        </tr>
        <tr>
          <th>1</th>
          <td>1</td>
          <td>0</td>
          <td>2</td>
          <td>20</td>
          <td>2</td>
          <td>1</td>
          <td>1</td>
          <td>1</td>
        </tr>
        <tr>
          <th>2</th>
          <td>1</td>
          <td>1</td>
          <td>3</td>
          <td>30</td>
          <td>3</td>
          <td>1</td>
          <td>0</td>
          <td>1</td>
        </tr>
        <tr>
          <th>3</th>
          <td>2</td>
          <td>0</td>
          <td>4</td>
          <td>40</td>
          <td>1</td>
          <td>1</td>
          <td>1</td>
          <td>1</td>
        </tr>
        <tr>
          <th>4</th>
          <td>2</td>
          <td>1</td>
          <td>5</td>
          <td>50</td>
          <td>2</td>
          <td>1</td>
          <td>1</td>
          <td>0</td>
        </tr>
      </tbody>
    </table>
    </div>
    </div>
    <br />
    <br />

.. GENERATED FROM PYTHON SOURCE LINES 53-55

.. code-block:: Python

    my_data = db.Database('test', df)








.. GENERATED FROM PYTHON SOURCE LINES 56-58

Piecewise linear specification
++++++++++++++++++++++++++++++

.. GENERATED FROM PYTHON SOURCE LINES 60-103

A piecewise linear specification (sometimes called 'spline') is a
continuous but not differentiable function of the variable. It is
defined based on thresholds. Between two thresholds, the function is
linear. And the slope is changing after each threshold.

Consider a variable :math:`t` and an interval :math:`[a, a+b]`. We
define a new variable

.. math:: x_{[a,b]}(t) = \max(0,\min(t-a,b)) =
    \left\{
       \begin{array}{ll}
           0 & \text{if } t < a, \\
           t-a & \text{if } a \leq t < a+b, \\
           b  & \text{otherwise}.
       \end{array}
    \right.

For each interval :math:`]-\infty,a]`, we have

.. math:: x_{]-\infty,a]}(t) = \min(t,a) = \left\{
    \begin{array}{ll}
        t & \text{if } t < a, \\
        a  & \text{otherwise}.
     \end{array}
     \right..

For each interval :math:`[a,+\infty[`, we have

.. math:: x_{]-\infty,a]}(t) = \max(0,t-a) = \left\{
    \begin{array}{ll} 0& \text{if } t < a, \\ t-a &
    \text{otherwise}.  \end{array} \right..

If we consider a series of threshold

.. math:: \alpha_1 < \alpha_2 < \ldots <\alpha_K,

the piecewise linear transform of variable :math:`t` is

.. math:: \sum_{k=1}^{K-1} \beta_k x_{[\alpha_k,\alpha_{k+1}]},

where :math:`\beta_k` is the slope of the linear function in interval
:math:`[\alpha_k,\alpha_{k+1}]`.


.. GENERATED FROM PYTHON SOURCE LINES 105-108

The next statement generates the variables, given the thresholds. A
`None` is equivalent to :math:`\infty`, and can only appear first (and it
means :math:`-\infty`) or last (and it means :math:`+\infty`).

.. GENERATED FROM PYTHON SOURCE LINES 108-113

.. code-block:: Python

    x = Variable('x')
    thresholds = [None, 90, 180, 270, None]
    variables = models.piecewise_variables(x, thresholds)
    print(variables)





.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    [bioMin(x, `90.0`), bioMax(`0.0`, bioMin((x - `90.0`), `90.0`)), bioMax(`0.0`, bioMin((x - `180.0`), `90.0`)), bioMax(`0.0`, (x - `270.0`))]




.. GENERATED FROM PYTHON SOURCE LINES 114-116

The next statement automatically generates the formula, including
the Beta parameters, that are initialized to zero.

.. GENERATED FROM PYTHON SOURCE LINES 116-119

.. code-block:: Python

    formula = models.piecewise_formula('x', thresholds)
    print(formula)





.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    bioMultSum((Beta('beta_x_minus_inf_90', 0, None, None, 0) * bioMin(x, `90.0`)), (Beta('beta_x_90_180', 0, None, None, 0) * bioMax(`0.0`, bioMin((x - `90.0`), `90.0`))), (Beta('beta_x_180_270', 0, None, None, 0) * bioMax(`0.0`, bioMin((x - `180.0`), `90.0`))), (Beta('beta_x_270_inf', 0, None, None, 0) * bioMax(`0.0`, (x - `270.0`))))




.. GENERATED FROM PYTHON SOURCE LINES 120-123

It is also possible to initialize the Beta parameters with other
values. Note also that the first argument can be either the name of
the variable (as in the previous call) or the variable itself.

.. GENERATED FROM PYTHON SOURCE LINES 123-127

.. code-block:: Python

    betas = [-0.016806308, -0.010491137, -0.002012234, -0.020051303]
    formula = models.piecewise_formula(x, thresholds, betas)
    print(formula)





.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    bioMultSum((`-0.016806308` * bioMin(x, `90.0`)), (`-0.010491137` * bioMax(`0.0`, bioMin((x - `90.0`), `90.0`))), (`-0.002012234` * bioMax(`0.0`, bioMin((x - `180.0`), `90.0`))), (`-0.020051303` * bioMax(`0.0`, (x - `270.0`))))




.. GENERATED FROM PYTHON SOURCE LINES 128-129

We provide a plot of a piecewise linear specification.

.. GENERATED FROM PYTHON SOURCE LINES 131-140

.. code-block:: Python

    X = np.arange(0, 300, 0.1)
    Y = [
        models.piecewise_function(
            x, thresholds, [-0.016806308, -0.010491137, -0.002012234, -0.020051303]
        )
        for x in X
    ]
    plt.plot(X, Y)




.. image-sg:: /auto_examples/programmers/images/sphx_glr_plot_models_001.png
   :alt: plot models
   :srcset: /auto_examples/programmers/images/sphx_glr_plot_models_001.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    [<matplotlib.lines.Line2D object at 0x1337f5e20>]



.. GENERATED FROM PYTHON SOURCE LINES 141-143

Logit
+++++

.. GENERATED FROM PYTHON SOURCE LINES 145-148

.. code-block:: Python

    V = {1: Variable('Variable1'), 2: 0.1, 3: -0.1}
    av = {1: Variable('Av1'), 2: Variable('Av2'), 3: Variable('Av3')}








.. GENERATED FROM PYTHON SOURCE LINES 149-151

Calculation of the (log of the) logit for the three alternatives,
based on their availability.

.. GENERATED FROM PYTHON SOURCE LINES 153-154

Alternative 1

.. GENERATED FROM PYTHON SOURCE LINES 154-157

.. code-block:: Python

    p1 = models.logit(V, av, 1)
    p1.get_value_c(my_data, prepare_ids=True)





.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    array([0.        , 0.78614804, 0.95689275, 0.9644926 , 0.99260846])



.. GENERATED FROM PYTHON SOURCE LINES 158-161

.. code-block:: Python

    p1 = models.loglogit(V, av, 1)
    p1.get_value_c(my_data, prepare_ids=True)





.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    array([       -inf, -0.24061016, -0.04406397, -0.03615312, -0.00741899])



.. GENERATED FROM PYTHON SOURCE LINES 162-163

Alternative 2

.. GENERATED FROM PYTHON SOURCE LINES 163-166

.. code-block:: Python

    p2 = models.logit(V, av, 2)
    p2.get_value_c(my_data, prepare_ids=True)





.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    array([1.        , 0.11758308, 0.        , 0.01952317, 0.00739154])



.. GENERATED FROM PYTHON SOURCE LINES 167-170

.. code-block:: Python

    p2 = models.loglogit(V, av, 2)
    p2.get_value_c(my_data, prepare_ids=True)





.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    array([ 0.        , -2.14061016,        -inf, -3.93615312, -4.90741899])



.. GENERATED FROM PYTHON SOURCE LINES 171-172

Alternative 3

.. GENERATED FROM PYTHON SOURCE LINES 172-175

.. code-block:: Python

    p3 = models.logit(V, av, 3)
    p3.get_value_c(my_data, prepare_ids=True)





.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    array([0.        , 0.09626888, 0.04310725, 0.01598422, 0.        ])



.. GENERATED FROM PYTHON SOURCE LINES 176-179

.. code-block:: Python

    p3 = models.loglogit(V, av, 3)
    p3.get_value_c(my_data, prepare_ids=True)





.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    array([       -inf, -2.34061016, -3.14406397, -4.13615312,        -inf])



.. GENERATED FROM PYTHON SOURCE LINES 180-182

Calculation of the log of the logit for the three alternatives,
**assuming that they are all available**.

.. GENERATED FROM PYTHON SOURCE LINES 184-185

Alternative 1

.. GENERATED FROM PYTHON SOURCE LINES 185-188

.. code-block:: Python

    pa1 = models.logit(V, av=None, i=1)
    pa1.get_value_c(my_data, prepare_ids=True)





.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    array([0.57489742, 0.78614804, 0.90903106, 0.9644926 , 0.98663764])



.. GENERATED FROM PYTHON SOURCE LINES 189-192

.. code-block:: Python

    pa1 = models.loglogit(V, av=None, i=1)
    pa1.get_value_c(my_data, prepare_ids=True)





.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    array([-0.55356365, -0.24061016, -0.09537602, -0.03615312, -0.01345244])



.. GENERATED FROM PYTHON SOURCE LINES 193-194

Alternative 2

.. GENERATED FROM PYTHON SOURCE LINES 194-197

.. code-block:: Python

    pa2 = models.logit(V, av=None, i=2)
    pa2.get_value_c(my_data, prepare_ids=True)





.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    array([0.23373585, 0.11758308, 0.05001782, 0.01952317, 0.00734708])



.. GENERATED FROM PYTHON SOURCE LINES 198-201

.. code-block:: Python

    pa2 = models.loglogit(V, av=None, i=2)
    pa2.get_value_c(my_data, prepare_ids=True)





.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    array([-1.45356365, -2.14061016, -2.99537602, -3.93615312, -4.91345244])



.. GENERATED FROM PYTHON SOURCE LINES 202-203

Alternative 3

.. GENERATED FROM PYTHON SOURCE LINES 203-206

.. code-block:: Python

    pa3 = models.logit(V, av=None, i=3)
    pa3.get_value_c(my_data, prepare_ids=True)





.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    array([0.19136673, 0.09626888, 0.04095112, 0.01598422, 0.00601528])



.. GENERATED FROM PYTHON SOURCE LINES 207-210

.. code-block:: Python

    pa3 = models.loglogit(V, av=None, i=3)
    pa3.get_value_c(my_data, prepare_ids=True)





.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    array([-1.65356365, -2.34061016, -3.19537602, -4.13615312, -5.11345244])



.. GENERATED FROM PYTHON SOURCE LINES 211-213

Boxcox transform
++++++++++++++++

.. GENERATED FROM PYTHON SOURCE LINES 215-223

The Box-Cox transform of a variable :math:`x` is defined as

.. math:: B(x,\ell) =\frac{x^{\ell}-1}{\ell},

where :math:`\ell` is a parameter that can be estimated from data.  It has
the property that

.. math:: \lim_{\ell \to 0} B(x,\ell)=\log(x).

.. GENERATED FROM PYTHON SOURCE LINES 223-226

.. code-block:: Python

    x = Variable('Variable1')
    models.boxcox(x, 4)





.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    {{0:{{0:((Variable1**4.0 - `1.0`) / `4.0`), 1:(((log(Variable1) + (`4.0` * log(Variable1)**2.0)) + ((`16.0` * log(Variable1)**3.0) / `6.0`)) + ((`64.0` * log(Variable1)**4.0) / `24.0`))}[((`1e-05` > `4.0`) * ((-`1e-05`) < `4.0`))], 1:`0.0`}[(Variable1 == `0.0`)]



.. GENERATED FROM PYTHON SOURCE LINES 227-230

.. code-block:: Python

    x = Variable('Variable1')
    models.boxcox(x, 0)





.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    {{0:{{0:((Variable1**0.0 - `1.0`) / `0.0`), 1:(((log(Variable1) + (`0.0` * log(Variable1)**2.0)) + ((`0.0` * log(Variable1)**3.0) / `6.0`)) + ((`0.0` * log(Variable1)**4.0) / `24.0`))}[((`1e-05` > `0.0`) * ((-`1e-05`) < `0.0`))], 1:`0.0`}[(Variable1 == `0.0`)]



.. GENERATED FROM PYTHON SOURCE LINES 231-235

.. code-block:: Python

    ell = Variable('Variable2')
    e = models.boxcox(x, ell)
    print(e)





.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    {{0:{{0:(((Variable1 ** Variable2) - `1.0`) / Variable2), 1:(((log(Variable1) + (Variable2 * log(Variable1)**2.0)) + ((Variable2**2.0 * log(Variable1)**3.0) / `6.0`)) + ((Variable2**3.0 * log(Variable1)**4.0) / `24.0`))}[((Variable2 < `1e-05`) * (Variable2 > (-`1e-05`)))], 1:`0.0`}[(Variable1 == `0.0`)]




.. GENERATED FROM PYTHON SOURCE LINES 236-238

.. code-block:: Python

    e.get_value_c(my_data, prepare_ids=True)





.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    array([0.00000000e+00, 5.24287500e+04, 6.86303774e+12, 3.02231455e+22,
           1.77635684e+33])



.. GENERATED FROM PYTHON SOURCE LINES 239-241

We numerically illustrate that, when :math:`\lambda` goes to 0, the BoxCox
transform of :math:`x` converges to the log of :math:`x`.

.. GENERATED FROM PYTHON SOURCE LINES 243-249

.. code-block:: Python

    for ell in range(1, 16):
        x = 3
        bc = models.boxcox(x, 10**-ell).get_value()
        print(f'ell=l0^(-{ell}): {bc:.6g} - {np.log(x):.6g} = {bc - np.log(x):.6g}')






.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    ell=l0^(-1): 1.16123 - 1.09861 = 0.0626195
    ell=l0^(-2): 1.10467 - 1.09861 = 0.00605691
    ell=l0^(-3): 1.09922 - 1.09861 = 0.000603696
    ell=l0^(-4): 1.09867 - 1.09861 = 6.03497e-05
    ell=l0^(-5): 1.09862 - 1.09861 = 6.03476e-06
    ell=l0^(-6): 1.09861 - 1.09861 = 1.20695e-06
    ell=l0^(-7): 1.09861 - 1.09861 = 1.20695e-07
    ell=l0^(-8): 1.09861 - 1.09861 = 1.20695e-08
    ell=l0^(-9): 1.09861 - 1.09861 = 1.20695e-09
    ell=l0^(-10): 1.09861 - 1.09861 = 1.20695e-10
    ell=l0^(-11): 1.09861 - 1.09861 = 1.20695e-11
    ell=l0^(-12): 1.09861 - 1.09861 = 1.20703e-12
    ell=l0^(-13): 1.09861 - 1.09861 = 1.20792e-13
    ell=l0^(-14): 1.09861 - 1.09861 = 1.19904e-14
    ell=l0^(-15): 1.09861 - 1.09861 = 1.11022e-15




.. GENERATED FROM PYTHON SOURCE LINES 250-252

MEV models
++++++++++

.. GENERATED FROM PYTHON SOURCE LINES 254-262

MEV models are defined as

.. math:: \frac{e^{V_i + \ln G_i(e^{V_1},\ldots,e^{V_J})}}{\sum_j
    e^{V_j + \ln G_j(e^{V_1},\ldots,e^{V_J})}},

where :math:`G` is a generating function, and

.. math:: G_i=\frac{\partial G}{\partial y_i}(e^{V_1},\ldots,e^{V_J}).

.. GENERATED FROM PYTHON SOURCE LINES 264-276

**Nested logit model**:
the :math:`G` function for the nested logit model is defined such that

.. math:: G_i=\frac{\partial G}{\partial
    y_i}(e^{V_1},\ldots,e^{V_J}) = \mu e^{(\mu_m-1)V_i}
    \left(\sum_{i=1}^{J_m} e^{\mu_m
    V_i}\right)^{\frac{\mu}{\mu_m}-1},

where the choice set is partitioned into :math:`J_m` nests, each
associated with a parameter :math:`\mu_m`, and :math:`\mu` is the scale
parameter. The condition is :math:`0 \leq \mu \leq \mu_m` must be
verified. In general, :math:`\mu` is normalized to 1.0.

.. GENERATED FROM PYTHON SOURCE LINES 278-282

This is an example with 5 alternatives. Nest A contains alternatives
1, 2 and 4, and is associated with a scale parameter
:math:`\mu_A=1.2`. Nest B contains alternatives 3 and 5, and is associated
with a scale parameter :math:`\mu_B=2.3`.

.. GENERATED FROM PYTHON SOURCE LINES 284-287

.. code-block:: Python

    V = {1: Variable('Variable1'), 2: 0.1, 3: -0.1, 4: -0.2, 5: 0.2}
    av = {1: 1, 2: 0, 3: 1, 4: 1, 5: 1}








.. GENERATED FROM PYTHON SOURCE LINES 288-289

Definition of the nests.

.. GENERATED FROM PYTHON SOURCE LINES 289-298

.. code-block:: Python

    nest_a = OneNestForNestedLogit(
        nest_param=1.2, list_of_alternatives=[1, 2, 4], name='nest_a'
    )
    nest_b = OneNestForNestedLogit(
        nest_param=2.3, list_of_alternatives=[3, 5], name='name_b'
    )

    nests = NestsForNestedLogit(choice_set=list(V), tuple_of_nests=(nest_a, nest_b))








.. GENERATED FROM PYTHON SOURCE LINES 299-302

.. code-block:: Python

    p1 = models.nested(V, availability=av, nests=nests, choice=1)
    p1.get_value_c(my_data, prepare_ids=True)





.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    array([0.55789028, 0.78684631, 0.9138115 , 0.96786857, 0.98836318])



.. GENERATED FROM PYTHON SOURCE LINES 303-304

If all the alternatives are available, define the availability dictionary as None.

.. GENERATED FROM PYTHON SOURCE LINES 304-307

.. code-block:: Python

    p1 = models.nested(V, availability=None, nests=nests, choice=1)
    p1.get_value_c(my_data, prepare_ids=True)





.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    array([0.46404048, 0.72661956, 0.88850463, 0.959216  , 0.98563615])



.. GENERATED FROM PYTHON SOURCE LINES 308-309

The syntax is similar to obtain the log of the probability.

.. GENERATED FROM PYTHON SOURCE LINES 309-312

.. code-block:: Python

    p2 = models.lognested(V, availability=av, nests=nests, choice=1)
    p2.get_value_c(my_data, prepare_ids=True)





.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    array([-0.58359296, -0.23972233, -0.09013096, -0.03265898, -0.01170506])



.. GENERATED FROM PYTHON SOURCE LINES 313-316

.. code-block:: Python

    p2 = models.lognested(V, availability=None, nests=nests, choice=1)
    p2.get_value_c(my_data, prepare_ids=True)





.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    array([-0.7677835 , -0.31935224, -0.11821542, -0.04163899, -0.01446801])



.. GENERATED FROM PYTHON SOURCE LINES 317-321

If the value of the parameter :math:`\mu` is not 1, there is another
function to call. Note that, for the sake of computational
efficiency, it is not verified by the code if the condition :math:`0 \leq
\mu \leq \mu_m` is valid.

.. GENERATED FROM PYTHON SOURCE LINES 321-324

.. code-block:: Python

    p1 = models.nested_mev_mu(V, availability=av, nests=nests, choice=1, mu=1.1)
    p1.get_value_c(my_data, prepare_ids=True)





.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    array([0.57151598, 0.80643395, 0.92814743, 0.9755475 , 0.9919295 ])



.. GENERATED FROM PYTHON SOURCE LINES 325-328

.. code-block:: Python

    p1 = models.nested_mev_mu(V, availability=None, nests=nests, choice=1, mu=1.1)
    p1.get_value_c(my_data, prepare_ids=True)





.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    array([0.47623579, 0.74427029, 0.90223305, 0.96678265, 0.98918584])



.. GENERATED FROM PYTHON SOURCE LINES 329-332

.. code-block:: Python

    p1 = models.lognested_mev_mu(V, availability=av, nests=nests, choice=1, mu=1.1)
    p1.get_value_c(my_data, prepare_ids=True)





.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    array([-0.55946283, -0.21513329, -0.07456469, -0.02475643, -0.00810324])



.. GENERATED FROM PYTHON SOURCE LINES 333-336

.. code-block:: Python

    p1 = models.lognested_mev_mu(V, availability=None, nests=nests, choice=1, mu=1.1)
    p1.get_value_c(my_data, prepare_ids=True)





.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    array([-0.74184219, -0.29535101, -0.10288243, -0.03378157, -0.01087306])



.. GENERATED FROM PYTHON SOURCE LINES 337-338

The validity of the nested structure can be verified.

.. GENERATED FROM PYTHON SOURCE LINES 338-343

.. code-block:: Python

    nest_c = OneNestForNestedLogit(nest_param=2.3, list_of_alternatives=[3], name='name_c')
    nests = NestsForNestedLogit(choice_set=list(V), tuple_of_nests=(nest_a, nest_b))

    is_valid, msg = nests.check_partition()








.. GENERATED FROM PYTHON SOURCE LINES 344-346

.. code-block:: Python

    is_valid





.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    True



.. GENERATED FROM PYTHON SOURCE LINES 347-349

.. code-block:: Python

    print(msg)





.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    ; 




.. GENERATED FROM PYTHON SOURCE LINES 350-351

If an alternative belongs to two nests

.. GENERATED FROM PYTHON SOURCE LINES 353-358

.. code-block:: Python

    nest_a = OneNestForNestedLogit(nest_param=1.2, list_of_alternatives=[1, 2, 3, 4])
    nest_b = OneNestForNestedLogit(nest_param=2.3, list_of_alternatives=[3, 5])
    nests = NestsForNestedLogit(choice_set=list(V), tuple_of_nests=(nest_a, nest_b))
    is_valid, msg = nests.check_partition()








.. GENERATED FROM PYTHON SOURCE LINES 359-361

.. code-block:: Python

    is_valid





.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    False



.. GENERATED FROM PYTHON SOURCE LINES 362-364

.. code-block:: Python

    print(msg)





.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    ; The following alternatives appear both in nests nest_1 and nest_2: {3}.




.. GENERATED FROM PYTHON SOURCE LINES 365-381

**Cross-nested logit model**: the :math:`G` function for the cross
nested logit model is defined such that

.. math:: G_i=\frac{\partial G}{\partial
    y_i}(e^{V_1},\ldots,e^{V_J}) = \mu \sum_{m=1}^{M}
    \alpha_{im}^{\frac{\mu_m}{\mu}} e^{(\mu_m-1) V_i}\left(
    \sum_{j=1}^{J} \alpha_{jm}^{\frac{\mu_m}{\mu}} e^{\mu_m V_j}
    \right)^{\frac{\mu}{\mu_m}-1},

where each nest :math:`m`  is associated with a parameter :math:`\mu_m` and, for
each alternative :math:`i`, a parameter :math:`\alpha_{im} \geq 0` that captures
the degree of membership of alternative :math:`i` to nest :math:`m`. :math:`\mu` is
the scale parameter. For each alternative :math:`i`, there must be at
least one nest :math:`m` such that :math:`\alpha_{im}>0`. The condition :math:`0
\leq \mu \leq \mu_m` must be also verified. In general, :math:`\mu` is
normalized to 1.0.

.. GENERATED FROM PYTHON SOURCE LINES 383-390

This is an example with 5 alternatives and two nests.

- Alt. 1 belongs to nest A.
- Alt. 2 belongs to nest A.
- Alt. 3 belongs to both nest A (50%) and nest B (50%).
- Alt. 4 belongs to nest B.
- Alt. 5 belongs to nest B.

.. GENERATED FROM PYTHON SOURCE LINES 392-404

.. code-block:: Python

    V = {1: Variable('Variable1'), 2: 0.1, 3: -0.1, 4: -0.2, 5: 0.2}
    av = {1: 1, 2: 0, 3: 1, 4: 1, 5: 1}
    alpha_a = {1: 1, 2: 1, 3: 0.5, 4: 0, 5: 0}
    alpha_b = {1: 0, 2: 0, 3: 0.5, 4: 1, 5: 1}
    nest_a = OneNestForCrossNestedLogit(
        nest_param=1.2, dict_of_alpha=alpha_a, name='Nest a'
    )
    nest_b = OneNestForCrossNestedLogit(
        nest_param=2.3, dict_of_alpha=alpha_b, name='Nest b'
    )
    nests = NestsForCrossNestedLogit(choice_set=list(V), tuple_of_nests=(nest_a, nest_b))








.. GENERATED FROM PYTHON SOURCE LINES 405-408

.. code-block:: Python

    p1 = models.cnl(V, availability=av, nests=nests, choice=1)
    p1.get_value_c(my_data, prepare_ids=True)





.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    array([0.60161076, 0.81080413, 0.92317655, 0.97098919, 0.98933903])



.. GENERATED FROM PYTHON SOURCE LINES 409-410

If all the alternatives are available, define the availability dictionary as None.

.. GENERATED FROM PYTHON SOURCE LINES 410-413

.. code-block:: Python

    p1 = models.cnl(V, availability=None, nests=nests, choice=1)
    p1.get_value_c(my_data, prepare_ids=True)





.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    array([0.49345928, 0.74695583, 0.89735316, 0.9622809 , 0.98660661])



.. GENERATED FROM PYTHON SOURCE LINES 414-418

If the value of the parameter :math:`\mu` is not 1, there is another
function to call. Note that, for the sake of computational
efficiency, it is not verified by the code if the condition :math:`0 \leq
\mu \leq \mu_m` is verified.

.. GENERATED FROM PYTHON SOURCE LINES 418-421

.. code-block:: Python

    p1 = models.cnlmu(V, availability=av, nests=nests, choice=1, mu=1.1)
    p1.get_value_c(my_data, prepare_ids=True)





.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    array([0.6110354 , 0.828654  , 0.93675704, 0.97837752, 0.99280536])



.. GENERATED FROM PYTHON SOURCE LINES 422-425

.. code-block:: Python

    p1 = models.cnlmu(V, availability=None, nests=nests, choice=1, mu=1.1)
    p1.get_value_c(my_data, prepare_ids=True)





.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    array([0.50313975, 0.76313184, 0.91036491, 0.96956188, 0.99005685])



.. GENERATED FROM PYTHON SOURCE LINES 426-431

If the sample is endogenous, a correction must be included in the
model, as proposed by `Bierlaire, Bolduc and McFadden (2008)
<http://dx.doi.org/10.1016/j.trb.2007.09.003>`_.
In this case, the generating function must first be defined, and the
MEV model with correction is then called.

.. GENERATED FROM PYTHON SOURCE LINES 431-434

.. code-block:: Python

    logGi = models.get_mev_for_cross_nested(V, availability=av, nests=nests)
    logGi





.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    {1: logzero(bioMultSum(((`1.0`**1.2 * exp((`0.19999999999999996` * Variable1))) * bioMultSum(((`1.0` * `1.0`**1.2) * exp((`1.2` * Variable1))), ((`0.0` * `1.0`**1.2) * exp(`0.12`)), ((`1.0` * `0.5`**1.2) * exp(`-0.12`)), ((`1.0` * `0.0`**1.2) * exp(`-0.24`)), ((`1.0` * `0.0`**1.2) * exp(`0.24`)))**-0.16666666666666663), ((`0.0`**2.3 * exp((`1.2999999999999998` * Variable1))) * bioMultSum(((`1.0` * `0.0`**2.3) * exp((`2.3` * Variable1))), ((`0.0` * `0.0`**2.3) * exp(`0.22999999999999998`)), ((`1.0` * `0.5`**2.3) * exp(`-0.22999999999999998`)), ((`1.0` * `1.0`**2.3) * exp(`-0.45999999999999996`)), ((`1.0` * `1.0`**2.3) * exp(`0.45999999999999996`)))**-0.5652173913043478))), 2: logzero(bioMultSum(((`1.0`**1.2 * exp(`0.019999999999999997`)) * bioMultSum(((`1.0` * `1.0`**1.2) * exp((`1.2` * Variable1))), ((`0.0` * `1.0`**1.2) * exp(`0.12`)), ((`1.0` * `0.5`**1.2) * exp(`-0.12`)), ((`1.0` * `0.0`**1.2) * exp(`-0.24`)), ((`1.0` * `0.0`**1.2) * exp(`0.24`)))**-0.16666666666666663), ((`0.0`**2.3 * exp(`0.12999999999999998`)) * bioMultSum(((`1.0` * `0.0`**2.3) * exp((`2.3` * Variable1))), ((`0.0` * `0.0`**2.3) * exp(`0.22999999999999998`)), ((`1.0` * `0.5`**2.3) * exp(`-0.22999999999999998`)), ((`1.0` * `1.0`**2.3) * exp(`-0.45999999999999996`)), ((`1.0` * `1.0`**2.3) * exp(`0.45999999999999996`)))**-0.5652173913043478))), 3: logzero(bioMultSum(((`0.5`**1.2 * exp(`-0.019999999999999997`)) * bioMultSum(((`1.0` * `1.0`**1.2) * exp((`1.2` * Variable1))), ((`0.0` * `1.0`**1.2) * exp(`0.12`)), ((`1.0` * `0.5`**1.2) * exp(`-0.12`)), ((`1.0` * `0.0`**1.2) * exp(`-0.24`)), ((`1.0` * `0.0`**1.2) * exp(`0.24`)))**-0.16666666666666663), ((`0.5`**2.3 * exp(`-0.12999999999999998`)) * bioMultSum(((`1.0` * `0.0`**2.3) * exp((`2.3` * Variable1))), ((`0.0` * `0.0`**2.3) * exp(`0.22999999999999998`)), ((`1.0` * `0.5`**2.3) * exp(`-0.22999999999999998`)), ((`1.0` * `1.0`**2.3) * exp(`-0.45999999999999996`)), ((`1.0` * `1.0`**2.3) * exp(`0.45999999999999996`)))**-0.5652173913043478))), 4: logzero(bioMultSum(((`0.0`**1.2 * exp(`-0.039999999999999994`)) * bioMultSum(((`1.0` * `1.0`**1.2) * exp((`1.2` * Variable1))), ((`0.0` * `1.0`**1.2) * exp(`0.12`)), ((`1.0` * `0.5`**1.2) * exp(`-0.12`)), ((`1.0` * `0.0`**1.2) * exp(`-0.24`)), ((`1.0` * `0.0`**1.2) * exp(`0.24`)))**-0.16666666666666663), ((`1.0`**2.3 * exp(`-0.25999999999999995`)) * bioMultSum(((`1.0` * `0.0`**2.3) * exp((`2.3` * Variable1))), ((`0.0` * `0.0`**2.3) * exp(`0.22999999999999998`)), ((`1.0` * `0.5`**2.3) * exp(`-0.22999999999999998`)), ((`1.0` * `1.0`**2.3) * exp(`-0.45999999999999996`)), ((`1.0` * `1.0`**2.3) * exp(`0.45999999999999996`)))**-0.5652173913043478))), 5: logzero(bioMultSum(((`0.0`**1.2 * exp(`0.039999999999999994`)) * bioMultSum(((`1.0` * `1.0`**1.2) * exp((`1.2` * Variable1))), ((`0.0` * `1.0`**1.2) * exp(`0.12`)), ((`1.0` * `0.5`**1.2) * exp(`-0.12`)), ((`1.0` * `0.0`**1.2) * exp(`-0.24`)), ((`1.0` * `0.0`**1.2) * exp(`0.24`)))**-0.16666666666666663), ((`1.0`**2.3 * exp(`0.25999999999999995`)) * bioMultSum(((`1.0` * `0.0`**2.3) * exp((`2.3` * Variable1))), ((`0.0` * `0.0`**2.3) * exp(`0.22999999999999998`)), ((`1.0` * `0.5`**2.3) * exp(`-0.22999999999999998`)), ((`1.0` * `1.0`**2.3) * exp(`-0.45999999999999996`)), ((`1.0` * `1.0`**2.3) * exp(`0.45999999999999996`)))**-0.5652173913043478)))}



.. GENERATED FROM PYTHON SOURCE LINES 435-439

.. code-block:: Python

    correction = {1: -0.1, 2: 0.1, 3: 0.2, 4: -0.2, 5: 0}
    p1 = models.mev_endogenous_sampling(V, logGi, av, correction, choice=1)
    p1.get_value_c(my_data, prepare_ids=True)





.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    array([0.57460723, 0.79415806, 0.91584339, 0.96822294, 0.98835331])



.. GENERATED FROM PYTHON SOURCE LINES 440-444

.. code-block:: Python

    correction = {1: -0.1, 2: 0.1, 3: 0.2, 4: -0.2, 5: 0}
    p1 = models.logmev_endogenous_sampling(V, logGi, av, correction, choice=1)
    p1.get_value_c(my_data, prepare_ids=True)





.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    array([-0.55406855, -0.23047277, -0.0879099 , -0.03229291, -0.01171504])



.. GENERATED FROM PYTHON SOURCE LINES 445-449

.. code-block:: Python

    correction = {1: -0.1, 2: 0.1, 3: 0.2, 4: -0.2, 5: 0}
    p1 = models.mev_endogenous_sampling(V, logGi, av=None, correction=correction, choice=1)
    p1.get_value_c(my_data, prepare_ids=True)





.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    array([0.46401513, 0.72247789, 0.8853334 , 0.95771369, 0.98503002])



.. GENERATED FROM PYTHON SOURCE LINES 450-456

.. code-block:: Python

    correction = {1: -0.1, 2: 0.1, 3: 0.2, 4: -0.2, 5: 0}
    p1 = models.logmev_endogenous_sampling(
        V, logGi, av=None, correction=correction, choice=1
    )
    p1.get_value_c(my_data, prepare_ids=True)





.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    array([-0.76783811, -0.32506845, -0.12179098, -0.04320641, -0.01508316])



.. GENERATED FROM PYTHON SOURCE LINES 457-458

The MEV generating function for the following models are available.

.. GENERATED FROM PYTHON SOURCE LINES 460-461

Nested logit model

.. GENERATED FROM PYTHON SOURCE LINES 461-470

.. code-block:: Python

    V = {1: Variable('Variable1'), 2: 0.1, 3: -0.1, 4: -0.2, 5: 0.2}
    nest_a = OneNestForNestedLogit(
        nest_param=Beta('muA', 1.2, 1.0, None, 0), list_of_alternatives=[1, 2, 4]
    )
    nest_b = OneNestForNestedLogit(
        nest_param=Beta('muB', 2.3, 1.0, None, 0), list_of_alternatives=[3, 5]
    )
    nests = NestsForNestedLogit(choice_set=list(V), tuple_of_nests=(nest_a, nest_b))








.. GENERATED FROM PYTHON SOURCE LINES 471-474

.. code-block:: Python

    logGi = models.get_mev_for_nested(V, availability=None, nests=nests)
    logGi





.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    {1: (((Beta('muA', 1.2, 1.0, None, 0) - `1.0`) * Variable1) + (((`1.0` / Beta('muA', 1.2, 1.0, None, 0)) - `1.0`) * log(bioMultSum(exp((Beta('muA', 1.2, 1.0, None, 0) * Variable1)), exp((Beta('muA', 1.2, 1.0, None, 0) * `0.1`)), exp((Beta('muA', 1.2, 1.0, None, 0) * `-0.2`)))))), 2: (((Beta('muA', 1.2, 1.0, None, 0) - `1.0`) * `0.1`) + (((`1.0` / Beta('muA', 1.2, 1.0, None, 0)) - `1.0`) * log(bioMultSum(exp((Beta('muA', 1.2, 1.0, None, 0) * Variable1)), exp((Beta('muA', 1.2, 1.0, None, 0) * `0.1`)), exp((Beta('muA', 1.2, 1.0, None, 0) * `-0.2`)))))), 4: (((Beta('muA', 1.2, 1.0, None, 0) - `1.0`) * `-0.2`) + (((`1.0` / Beta('muA', 1.2, 1.0, None, 0)) - `1.0`) * log(bioMultSum(exp((Beta('muA', 1.2, 1.0, None, 0) * Variable1)), exp((Beta('muA', 1.2, 1.0, None, 0) * `0.1`)), exp((Beta('muA', 1.2, 1.0, None, 0) * `-0.2`)))))), 3: (((Beta('muB', 2.3, 1.0, None, 0) - `1.0`) * `-0.1`) + (((`1.0` / Beta('muB', 2.3, 1.0, None, 0)) - `1.0`) * log(bioMultSum(exp((Beta('muB', 2.3, 1.0, None, 0) * `-0.1`)), exp((Beta('muB', 2.3, 1.0, None, 0) * `0.2`)))))), 5: (((Beta('muB', 2.3, 1.0, None, 0) - `1.0`) * `0.2`) + (((`1.0` / Beta('muB', 2.3, 1.0, None, 0)) - `1.0`) * log(bioMultSum(exp((Beta('muB', 2.3, 1.0, None, 0) * `-0.1`)), exp((Beta('muB', 2.3, 1.0, None, 0) * `0.2`))))))}



.. GENERATED FROM PYTHON SOURCE LINES 475-476

And with the :math:`\mu` parameter.

.. GENERATED FROM PYTHON SOURCE LINES 478-481

.. code-block:: Python

    logGi = models.get_mev_for_nested_mu(V, availability=None, nests=nests, mu=1.1)
    logGi





.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    {1: ((log(`1.1`) + ((Beta('muA', 1.2, 1.0, None, 0) - `1.0`) * Variable1)) + (((`1.1` / Beta('muA', 1.2, 1.0, None, 0)) - `1.0`) * log(bioMultSum(exp((Beta('muA', 1.2, 1.0, None, 0) * Variable1)), exp((Beta('muA', 1.2, 1.0, None, 0) * `0.1`)), exp((Beta('muA', 1.2, 1.0, None, 0) * `-0.2`)))))), 2: ((log(`1.1`) + ((Beta('muA', 1.2, 1.0, None, 0) - `1.0`) * `0.1`)) + (((`1.1` / Beta('muA', 1.2, 1.0, None, 0)) - `1.0`) * log(bioMultSum(exp((Beta('muA', 1.2, 1.0, None, 0) * Variable1)), exp((Beta('muA', 1.2, 1.0, None, 0) * `0.1`)), exp((Beta('muA', 1.2, 1.0, None, 0) * `-0.2`)))))), 4: ((log(`1.1`) + ((Beta('muA', 1.2, 1.0, None, 0) - `1.0`) * `-0.2`)) + (((`1.1` / Beta('muA', 1.2, 1.0, None, 0)) - `1.0`) * log(bioMultSum(exp((Beta('muA', 1.2, 1.0, None, 0) * Variable1)), exp((Beta('muA', 1.2, 1.0, None, 0) * `0.1`)), exp((Beta('muA', 1.2, 1.0, None, 0) * `-0.2`)))))), 3: ((log(`1.1`) + ((Beta('muB', 2.3, 1.0, None, 0) - `1.0`) * `-0.1`)) + (((`1.1` / Beta('muB', 2.3, 1.0, None, 0)) - `1.0`) * log(bioMultSum(exp((Beta('muB', 2.3, 1.0, None, 0) * `-0.1`)), exp((Beta('muB', 2.3, 1.0, None, 0) * `0.2`)))))), 5: ((log(`1.1`) + ((Beta('muB', 2.3, 1.0, None, 0) - `1.0`) * `0.2`)) + (((`1.1` / Beta('muB', 2.3, 1.0, None, 0)) - `1.0`) * log(bioMultSum(exp((Beta('muB', 2.3, 1.0, None, 0) * `-0.1`)), exp((Beta('muB', 2.3, 1.0, None, 0) * `0.2`))))))}



.. GENERATED FROM PYTHON SOURCE LINES 482-483

Cross nested logit model

.. GENERATED FROM PYTHON SOURCE LINES 485-496

.. code-block:: Python

    V = {1: Variable('Variable1'), 2: 0.1, 3: -0.1, 4: -0.2, 5: 0.2}
    av = {1: 1, 2: 0, 3: 1, 4: 1, 5: 1}
    alpha_a = {1: 1, 2: 1, 3: 0.5, 4: 0, 5: 0}
    alpha_b = {1: 0, 2: 0, 3: 0.5, 4: 1, 5: 1}
    nest_a = OneNestForCrossNestedLogit(
        nest_param=Beta('muA', 1.2, 1.0, None, 0), dict_of_alpha=alpha_a
    )
    nest_b = OneNestForCrossNestedLogit(
        nest_param=Beta('muB', 2.3, 1.0, None, 0), dict_of_alpha=alpha_b
    )
    nests = NestsForCrossNestedLogit(choice_set=list(V), tuple_of_nests=(nest_a, nest_b))







.. GENERATED FROM PYTHON SOURCE LINES 497-500

.. code-block:: Python

    logGi = models.get_mev_for_cross_nested(V, availability=None, nests=nests)
    logGi





.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    {1: logzero(bioMultSum((((`1.0` ** Beta('muA', 1.2, 1.0, None, 0)) * exp(((Beta('muA', 1.2, 1.0, None, 0) - `1.0`) * Variable1))) * (bioMultSum(((`1.0` ** Beta('muA', 1.2, 1.0, None, 0)) * exp((Beta('muA', 1.2, 1.0, None, 0) * Variable1))), ((`1.0` ** Beta('muA', 1.2, 1.0, None, 0)) * exp((Beta('muA', 1.2, 1.0, None, 0) * `0.1`))), ((`0.5` ** Beta('muA', 1.2, 1.0, None, 0)) * exp((Beta('muA', 1.2, 1.0, None, 0) * `-0.1`))), ((`0.0` ** Beta('muA', 1.2, 1.0, None, 0)) * exp((Beta('muA', 1.2, 1.0, None, 0) * `-0.2`))), ((`0.0` ** Beta('muA', 1.2, 1.0, None, 0)) * exp((Beta('muA', 1.2, 1.0, None, 0) * `0.2`)))) ** ((`1.0` - Beta('muA', 1.2, 1.0, None, 0)) / Beta('muA', 1.2, 1.0, None, 0)))), (((`0.0` ** Beta('muB', 2.3, 1.0, None, 0)) * exp(((Beta('muB', 2.3, 1.0, None, 0) - `1.0`) * Variable1))) * (bioMultSum(((`0.0` ** Beta('muB', 2.3, 1.0, None, 0)) * exp((Beta('muB', 2.3, 1.0, None, 0) * Variable1))), ((`0.0` ** Beta('muB', 2.3, 1.0, None, 0)) * exp((Beta('muB', 2.3, 1.0, None, 0) * `0.1`))), ((`0.5` ** Beta('muB', 2.3, 1.0, None, 0)) * exp((Beta('muB', 2.3, 1.0, None, 0) * `-0.1`))), ((`1.0` ** Beta('muB', 2.3, 1.0, None, 0)) * exp((Beta('muB', 2.3, 1.0, None, 0) * `-0.2`))), ((`1.0` ** Beta('muB', 2.3, 1.0, None, 0)) * exp((Beta('muB', 2.3, 1.0, None, 0) * `0.2`)))) ** ((`1.0` - Beta('muB', 2.3, 1.0, None, 0)) / Beta('muB', 2.3, 1.0, None, 0)))))), 2: logzero(bioMultSum((((`1.0` ** Beta('muA', 1.2, 1.0, None, 0)) * exp(((Beta('muA', 1.2, 1.0, None, 0) - `1.0`) * `0.1`))) * (bioMultSum(((`1.0` ** Beta('muA', 1.2, 1.0, None, 0)) * exp((Beta('muA', 1.2, 1.0, None, 0) * Variable1))), ((`1.0` ** Beta('muA', 1.2, 1.0, None, 0)) * exp((Beta('muA', 1.2, 1.0, None, 0) * `0.1`))), ((`0.5` ** Beta('muA', 1.2, 1.0, None, 0)) * exp((Beta('muA', 1.2, 1.0, None, 0) * `-0.1`))), ((`0.0` ** Beta('muA', 1.2, 1.0, None, 0)) * exp((Beta('muA', 1.2, 1.0, None, 0) * `-0.2`))), ((`0.0` ** Beta('muA', 1.2, 1.0, None, 0)) * exp((Beta('muA', 1.2, 1.0, None, 0) * `0.2`)))) ** ((`1.0` - Beta('muA', 1.2, 1.0, None, 0)) / Beta('muA', 1.2, 1.0, None, 0)))), (((`0.0` ** Beta('muB', 2.3, 1.0, None, 0)) * exp(((Beta('muB', 2.3, 1.0, None, 0) - `1.0`) * `0.1`))) * (bioMultSum(((`0.0` ** Beta('muB', 2.3, 1.0, None, 0)) * exp((Beta('muB', 2.3, 1.0, None, 0) * Variable1))), ((`0.0` ** Beta('muB', 2.3, 1.0, None, 0)) * exp((Beta('muB', 2.3, 1.0, None, 0) * `0.1`))), ((`0.5` ** Beta('muB', 2.3, 1.0, None, 0)) * exp((Beta('muB', 2.3, 1.0, None, 0) * `-0.1`))), ((`1.0` ** Beta('muB', 2.3, 1.0, None, 0)) * exp((Beta('muB', 2.3, 1.0, None, 0) * `-0.2`))), ((`1.0` ** Beta('muB', 2.3, 1.0, None, 0)) * exp((Beta('muB', 2.3, 1.0, None, 0) * `0.2`)))) ** ((`1.0` - Beta('muB', 2.3, 1.0, None, 0)) / Beta('muB', 2.3, 1.0, None, 0)))))), 3: logzero(bioMultSum((((`0.5` ** Beta('muA', 1.2, 1.0, None, 0)) * exp(((Beta('muA', 1.2, 1.0, None, 0) - `1.0`) * `-0.1`))) * (bioMultSum(((`1.0` ** Beta('muA', 1.2, 1.0, None, 0)) * exp((Beta('muA', 1.2, 1.0, None, 0) * Variable1))), ((`1.0` ** Beta('muA', 1.2, 1.0, None, 0)) * exp((Beta('muA', 1.2, 1.0, None, 0) * `0.1`))), ((`0.5` ** Beta('muA', 1.2, 1.0, None, 0)) * exp((Beta('muA', 1.2, 1.0, None, 0) * `-0.1`))), ((`0.0` ** Beta('muA', 1.2, 1.0, None, 0)) * exp((Beta('muA', 1.2, 1.0, None, 0) * `-0.2`))), ((`0.0` ** Beta('muA', 1.2, 1.0, None, 0)) * exp((Beta('muA', 1.2, 1.0, None, 0) * `0.2`)))) ** ((`1.0` - Beta('muA', 1.2, 1.0, None, 0)) / Beta('muA', 1.2, 1.0, None, 0)))), (((`0.5` ** Beta('muB', 2.3, 1.0, None, 0)) * exp(((Beta('muB', 2.3, 1.0, None, 0) - `1.0`) * `-0.1`))) * (bioMultSum(((`0.0` ** Beta('muB', 2.3, 1.0, None, 0)) * exp((Beta('muB', 2.3, 1.0, None, 0) * Variable1))), ((`0.0` ** Beta('muB', 2.3, 1.0, None, 0)) * exp((Beta('muB', 2.3, 1.0, None, 0) * `0.1`))), ((`0.5` ** Beta('muB', 2.3, 1.0, None, 0)) * exp((Beta('muB', 2.3, 1.0, None, 0) * `-0.1`))), ((`1.0` ** Beta('muB', 2.3, 1.0, None, 0)) * exp((Beta('muB', 2.3, 1.0, None, 0) * `-0.2`))), ((`1.0` ** Beta('muB', 2.3, 1.0, None, 0)) * exp((Beta('muB', 2.3, 1.0, None, 0) * `0.2`)))) ** ((`1.0` - Beta('muB', 2.3, 1.0, None, 0)) / Beta('muB', 2.3, 1.0, None, 0)))))), 4: logzero(bioMultSum((((`0.0` ** Beta('muA', 1.2, 1.0, None, 0)) * exp(((Beta('muA', 1.2, 1.0, None, 0) - `1.0`) * `-0.2`))) * (bioMultSum(((`1.0` ** Beta('muA', 1.2, 1.0, None, 0)) * exp((Beta('muA', 1.2, 1.0, None, 0) * Variable1))), ((`1.0` ** Beta('muA', 1.2, 1.0, None, 0)) * exp((Beta('muA', 1.2, 1.0, None, 0) * `0.1`))), ((`0.5` ** Beta('muA', 1.2, 1.0, None, 0)) * exp((Beta('muA', 1.2, 1.0, None, 0) * `-0.1`))), ((`0.0` ** Beta('muA', 1.2, 1.0, None, 0)) * exp((Beta('muA', 1.2, 1.0, None, 0) * `-0.2`))), ((`0.0` ** Beta('muA', 1.2, 1.0, None, 0)) * exp((Beta('muA', 1.2, 1.0, None, 0) * `0.2`)))) ** ((`1.0` - Beta('muA', 1.2, 1.0, None, 0)) / Beta('muA', 1.2, 1.0, None, 0)))), (((`1.0` ** Beta('muB', 2.3, 1.0, None, 0)) * exp(((Beta('muB', 2.3, 1.0, None, 0) - `1.0`) * `-0.2`))) * (bioMultSum(((`0.0` ** Beta('muB', 2.3, 1.0, None, 0)) * exp((Beta('muB', 2.3, 1.0, None, 0) * Variable1))), ((`0.0` ** Beta('muB', 2.3, 1.0, None, 0)) * exp((Beta('muB', 2.3, 1.0, None, 0) * `0.1`))), ((`0.5` ** Beta('muB', 2.3, 1.0, None, 0)) * exp((Beta('muB', 2.3, 1.0, None, 0) * `-0.1`))), ((`1.0` ** Beta('muB', 2.3, 1.0, None, 0)) * exp((Beta('muB', 2.3, 1.0, None, 0) * `-0.2`))), ((`1.0` ** Beta('muB', 2.3, 1.0, None, 0)) * exp((Beta('muB', 2.3, 1.0, None, 0) * `0.2`)))) ** ((`1.0` - Beta('muB', 2.3, 1.0, None, 0)) / Beta('muB', 2.3, 1.0, None, 0)))))), 5: logzero(bioMultSum((((`0.0` ** Beta('muA', 1.2, 1.0, None, 0)) * exp(((Beta('muA', 1.2, 1.0, None, 0) - `1.0`) * `0.2`))) * (bioMultSum(((`1.0` ** Beta('muA', 1.2, 1.0, None, 0)) * exp((Beta('muA', 1.2, 1.0, None, 0) * Variable1))), ((`1.0` ** Beta('muA', 1.2, 1.0, None, 0)) * exp((Beta('muA', 1.2, 1.0, None, 0) * `0.1`))), ((`0.5` ** Beta('muA', 1.2, 1.0, None, 0)) * exp((Beta('muA', 1.2, 1.0, None, 0) * `-0.1`))), ((`0.0` ** Beta('muA', 1.2, 1.0, None, 0)) * exp((Beta('muA', 1.2, 1.0, None, 0) * `-0.2`))), ((`0.0` ** Beta('muA', 1.2, 1.0, None, 0)) * exp((Beta('muA', 1.2, 1.0, None, 0) * `0.2`)))) ** ((`1.0` - Beta('muA', 1.2, 1.0, None, 0)) / Beta('muA', 1.2, 1.0, None, 0)))), (((`1.0` ** Beta('muB', 2.3, 1.0, None, 0)) * exp(((Beta('muB', 2.3, 1.0, None, 0) - `1.0`) * `0.2`))) * (bioMultSum(((`0.0` ** Beta('muB', 2.3, 1.0, None, 0)) * exp((Beta('muB', 2.3, 1.0, None, 0) * Variable1))), ((`0.0` ** Beta('muB', 2.3, 1.0, None, 0)) * exp((Beta('muB', 2.3, 1.0, None, 0) * `0.1`))), ((`0.5` ** Beta('muB', 2.3, 1.0, None, 0)) * exp((Beta('muB', 2.3, 1.0, None, 0) * `-0.1`))), ((`1.0` ** Beta('muB', 2.3, 1.0, None, 0)) * exp((Beta('muB', 2.3, 1.0, None, 0) * `-0.2`))), ((`1.0` ** Beta('muB', 2.3, 1.0, None, 0)) * exp((Beta('muB', 2.3, 1.0, None, 0) * `0.2`)))) ** ((`1.0` - Beta('muB', 2.3, 1.0, None, 0)) / Beta('muB', 2.3, 1.0, None, 0))))))}



.. GENERATED FROM PYTHON SOURCE LINES 501-502

Cross nested logit model with :math:`\mu` parameter.

.. GENERATED FROM PYTHON SOURCE LINES 504-506

.. code-block:: Python

    logGi = models.get_mev_for_cross_nested_mu(V, availability=None, nests=nests, mu=1.1)
    logGi




.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    {1: log((`1.1` * bioMultSum((((`1.0` ** (Beta('muA', 1.2, 1.0, None, 0) / `1.1`)) * exp(((Beta('muA', 1.2, 1.0, None, 0) - `1.0`) * Variable1))) * (bioMultSum(((`1.0` ** (Beta('muA', 1.2, 1.0, None, 0) / `1.1`)) * exp((Beta('muA', 1.2, 1.0, None, 0) * Variable1))), ((`1.0` ** (Beta('muA', 1.2, 1.0, None, 0) / `1.1`)) * exp((Beta('muA', 1.2, 1.0, None, 0) * `0.1`))), ((`0.5` ** (Beta('muA', 1.2, 1.0, None, 0) / `1.1`)) * exp((Beta('muA', 1.2, 1.0, None, 0) * `-0.1`))), ((`0.0` ** (Beta('muA', 1.2, 1.0, None, 0) / `1.1`)) * exp((Beta('muA', 1.2, 1.0, None, 0) * `-0.2`))), ((`0.0` ** (Beta('muA', 1.2, 1.0, None, 0) / `1.1`)) * exp((Beta('muA', 1.2, 1.0, None, 0) * `0.2`)))) ** ((`1.1` / Beta('muA', 1.2, 1.0, None, 0)) - `1.0`))), (((`0.0` ** (Beta('muB', 2.3, 1.0, None, 0) / `1.1`)) * exp(((Beta('muB', 2.3, 1.0, None, 0) - `1.0`) * Variable1))) * (bioMultSum(((`0.0` ** (Beta('muB', 2.3, 1.0, None, 0) / `1.1`)) * exp((Beta('muB', 2.3, 1.0, None, 0) * Variable1))), ((`0.0` ** (Beta('muB', 2.3, 1.0, None, 0) / `1.1`)) * exp((Beta('muB', 2.3, 1.0, None, 0) * `0.1`))), ((`0.5` ** (Beta('muB', 2.3, 1.0, None, 0) / `1.1`)) * exp((Beta('muB', 2.3, 1.0, None, 0) * `-0.1`))), ((`1.0` ** (Beta('muB', 2.3, 1.0, None, 0) / `1.1`)) * exp((Beta('muB', 2.3, 1.0, None, 0) * `-0.2`))), ((`1.0` ** (Beta('muB', 2.3, 1.0, None, 0) / `1.1`)) * exp((Beta('muB', 2.3, 1.0, None, 0) * `0.2`)))) ** ((`1.1` / Beta('muB', 2.3, 1.0, None, 0)) - `1.0`)))))), 2: log((`1.1` * bioMultSum((((`1.0` ** (Beta('muA', 1.2, 1.0, None, 0) / `1.1`)) * exp(((Beta('muA', 1.2, 1.0, None, 0) - `1.0`) * `0.1`))) * (bioMultSum(((`1.0` ** (Beta('muA', 1.2, 1.0, None, 0) / `1.1`)) * exp((Beta('muA', 1.2, 1.0, None, 0) * Variable1))), ((`1.0` ** (Beta('muA', 1.2, 1.0, None, 0) / `1.1`)) * exp((Beta('muA', 1.2, 1.0, None, 0) * `0.1`))), ((`0.5` ** (Beta('muA', 1.2, 1.0, None, 0) / `1.1`)) * exp((Beta('muA', 1.2, 1.0, None, 0) * `-0.1`))), ((`0.0` ** (Beta('muA', 1.2, 1.0, None, 0) / `1.1`)) * exp((Beta('muA', 1.2, 1.0, None, 0) * `-0.2`))), ((`0.0` ** (Beta('muA', 1.2, 1.0, None, 0) / `1.1`)) * exp((Beta('muA', 1.2, 1.0, None, 0) * `0.2`)))) ** ((`1.1` / Beta('muA', 1.2, 1.0, None, 0)) - `1.0`))), (((`0.0` ** (Beta('muB', 2.3, 1.0, None, 0) / `1.1`)) * exp(((Beta('muB', 2.3, 1.0, None, 0) - `1.0`) * `0.1`))) * (bioMultSum(((`0.0` ** (Beta('muB', 2.3, 1.0, None, 0) / `1.1`)) * exp((Beta('muB', 2.3, 1.0, None, 0) * Variable1))), ((`0.0` ** (Beta('muB', 2.3, 1.0, None, 0) / `1.1`)) * exp((Beta('muB', 2.3, 1.0, None, 0) * `0.1`))), ((`0.5` ** (Beta('muB', 2.3, 1.0, None, 0) / `1.1`)) * exp((Beta('muB', 2.3, 1.0, None, 0) * `-0.1`))), ((`1.0` ** (Beta('muB', 2.3, 1.0, None, 0) / `1.1`)) * exp((Beta('muB', 2.3, 1.0, None, 0) * `-0.2`))), ((`1.0` ** (Beta('muB', 2.3, 1.0, None, 0) / `1.1`)) * exp((Beta('muB', 2.3, 1.0, None, 0) * `0.2`)))) ** ((`1.1` / Beta('muB', 2.3, 1.0, None, 0)) - `1.0`)))))), 3: log((`1.1` * bioMultSum((((`0.5` ** (Beta('muA', 1.2, 1.0, None, 0) / `1.1`)) * exp(((Beta('muA', 1.2, 1.0, None, 0) - `1.0`) * `-0.1`))) * (bioMultSum(((`1.0` ** (Beta('muA', 1.2, 1.0, None, 0) / `1.1`)) * exp((Beta('muA', 1.2, 1.0, None, 0) * Variable1))), ((`1.0` ** (Beta('muA', 1.2, 1.0, None, 0) / `1.1`)) * exp((Beta('muA', 1.2, 1.0, None, 0) * `0.1`))), ((`0.5` ** (Beta('muA', 1.2, 1.0, None, 0) / `1.1`)) * exp((Beta('muA', 1.2, 1.0, None, 0) * `-0.1`))), ((`0.0` ** (Beta('muA', 1.2, 1.0, None, 0) / `1.1`)) * exp((Beta('muA', 1.2, 1.0, None, 0) * `-0.2`))), ((`0.0` ** (Beta('muA', 1.2, 1.0, None, 0) / `1.1`)) * exp((Beta('muA', 1.2, 1.0, None, 0) * `0.2`)))) ** ((`1.1` / Beta('muA', 1.2, 1.0, None, 0)) - `1.0`))), (((`0.5` ** (Beta('muB', 2.3, 1.0, None, 0) / `1.1`)) * exp(((Beta('muB', 2.3, 1.0, None, 0) - `1.0`) * `-0.1`))) * (bioMultSum(((`0.0` ** (Beta('muB', 2.3, 1.0, None, 0) / `1.1`)) * exp((Beta('muB', 2.3, 1.0, None, 0) * Variable1))), ((`0.0` ** (Beta('muB', 2.3, 1.0, None, 0) / `1.1`)) * exp((Beta('muB', 2.3, 1.0, None, 0) * `0.1`))), ((`0.5` ** (Beta('muB', 2.3, 1.0, None, 0) / `1.1`)) * exp((Beta('muB', 2.3, 1.0, None, 0) * `-0.1`))), ((`1.0` ** (Beta('muB', 2.3, 1.0, None, 0) / `1.1`)) * exp((Beta('muB', 2.3, 1.0, None, 0) * `-0.2`))), ((`1.0` ** (Beta('muB', 2.3, 1.0, None, 0) / `1.1`)) * exp((Beta('muB', 2.3, 1.0, None, 0) * `0.2`)))) ** ((`1.1` / Beta('muB', 2.3, 1.0, None, 0)) - `1.0`)))))), 4: log((`1.1` * bioMultSum((((`0.0` ** (Beta('muA', 1.2, 1.0, None, 0) / `1.1`)) * exp(((Beta('muA', 1.2, 1.0, None, 0) - `1.0`) * `-0.2`))) * (bioMultSum(((`1.0` ** (Beta('muA', 1.2, 1.0, None, 0) / `1.1`)) * exp((Beta('muA', 1.2, 1.0, None, 0) * Variable1))), ((`1.0` ** (Beta('muA', 1.2, 1.0, None, 0) / `1.1`)) * exp((Beta('muA', 1.2, 1.0, None, 0) * `0.1`))), ((`0.5` ** (Beta('muA', 1.2, 1.0, None, 0) / `1.1`)) * exp((Beta('muA', 1.2, 1.0, None, 0) * `-0.1`))), ((`0.0` ** (Beta('muA', 1.2, 1.0, None, 0) / `1.1`)) * exp((Beta('muA', 1.2, 1.0, None, 0) * `-0.2`))), ((`0.0` ** (Beta('muA', 1.2, 1.0, None, 0) / `1.1`)) * exp((Beta('muA', 1.2, 1.0, None, 0) * `0.2`)))) ** ((`1.1` / Beta('muA', 1.2, 1.0, None, 0)) - `1.0`))), (((`1.0` ** (Beta('muB', 2.3, 1.0, None, 0) / `1.1`)) * exp(((Beta('muB', 2.3, 1.0, None, 0) - `1.0`) * `-0.2`))) * (bioMultSum(((`0.0` ** (Beta('muB', 2.3, 1.0, None, 0) / `1.1`)) * exp((Beta('muB', 2.3, 1.0, None, 0) * Variable1))), ((`0.0` ** (Beta('muB', 2.3, 1.0, None, 0) / `1.1`)) * exp((Beta('muB', 2.3, 1.0, None, 0) * `0.1`))), ((`0.5` ** (Beta('muB', 2.3, 1.0, None, 0) / `1.1`)) * exp((Beta('muB', 2.3, 1.0, None, 0) * `-0.1`))), ((`1.0` ** (Beta('muB', 2.3, 1.0, None, 0) / `1.1`)) * exp((Beta('muB', 2.3, 1.0, None, 0) * `-0.2`))), ((`1.0` ** (Beta('muB', 2.3, 1.0, None, 0) / `1.1`)) * exp((Beta('muB', 2.3, 1.0, None, 0) * `0.2`)))) ** ((`1.1` / Beta('muB', 2.3, 1.0, None, 0)) - `1.0`)))))), 5: log((`1.1` * bioMultSum((((`0.0` ** (Beta('muA', 1.2, 1.0, None, 0) / `1.1`)) * exp(((Beta('muA', 1.2, 1.0, None, 0) - `1.0`) * `0.2`))) * (bioMultSum(((`1.0` ** (Beta('muA', 1.2, 1.0, None, 0) / `1.1`)) * exp((Beta('muA', 1.2, 1.0, None, 0) * Variable1))), ((`1.0` ** (Beta('muA', 1.2, 1.0, None, 0) / `1.1`)) * exp((Beta('muA', 1.2, 1.0, None, 0) * `0.1`))), ((`0.5` ** (Beta('muA', 1.2, 1.0, None, 0) / `1.1`)) * exp((Beta('muA', 1.2, 1.0, None, 0) * `-0.1`))), ((`0.0` ** (Beta('muA', 1.2, 1.0, None, 0) / `1.1`)) * exp((Beta('muA', 1.2, 1.0, None, 0) * `-0.2`))), ((`0.0` ** (Beta('muA', 1.2, 1.0, None, 0) / `1.1`)) * exp((Beta('muA', 1.2, 1.0, None, 0) * `0.2`)))) ** ((`1.1` / Beta('muA', 1.2, 1.0, None, 0)) - `1.0`))), (((`1.0` ** (Beta('muB', 2.3, 1.0, None, 0) / `1.1`)) * exp(((Beta('muB', 2.3, 1.0, None, 0) - `1.0`) * `0.2`))) * (bioMultSum(((`0.0` ** (Beta('muB', 2.3, 1.0, None, 0) / `1.1`)) * exp((Beta('muB', 2.3, 1.0, None, 0) * Variable1))), ((`0.0` ** (Beta('muB', 2.3, 1.0, None, 0) / `1.1`)) * exp((Beta('muB', 2.3, 1.0, None, 0) * `0.1`))), ((`0.5` ** (Beta('muB', 2.3, 1.0, None, 0) / `1.1`)) * exp((Beta('muB', 2.3, 1.0, None, 0) * `-0.1`))), ((`1.0` ** (Beta('muB', 2.3, 1.0, None, 0) / `1.1`)) * exp((Beta('muB', 2.3, 1.0, None, 0) * `-0.2`))), ((`1.0` ** (Beta('muB', 2.3, 1.0, None, 0) / `1.1`)) * exp((Beta('muB', 2.3, 1.0, None, 0) * `0.2`)))) ** ((`1.1` / Beta('muB', 2.3, 1.0, None, 0)) - `1.0`))))))}




.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.111 seconds)


.. _sphx_glr_download_auto_examples_programmers_plot_models.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_models.ipynb <plot_models.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_models.py <plot_models.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_models.zip <plot_models.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_