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

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

Version of Biogeme.

print(getText())

biogeme 3.2.13 [2023-12-23]
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)

Definition of a database

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

	Person	Exclude	Variable1	Variable2	Choice	Av1	Av2	Av3
0	1	0	1	10	1	0	1	0
1	1	0	2	20	2	1	1	1
2	1	1	3	30	3	1	0	1
3	2	0	4	40	1	1	1	1
4	2	1	5	50	2	1	1	0

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

Piecewise linear specification

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 \(t\) and an interval \([a, a+b]\). We define a new variable

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

For each interval \(]-\infty,a]\), we have

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

For each interval \([a,+\infty[\), we have

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

If we consider a series of threshold

\[\alpha_1 < \alpha_2 < \ldots <\alpha_K,\]

the piecewise linear transform of variable \(t\) is

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

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

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

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

[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`))]

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

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

bioMultSum((Beta('beta_x_minus_inf_90', 0, -1.3407807929942596e+154, 1.3407807929942596e+154, 0) * bioMin(x, `90.0`)), (Beta('beta_x_90_180', 0, -1.3407807929942596e+154, 1.3407807929942596e+154, 0) * bioMax(`0.0`, bioMin((x - `90.0`), `90.0`))), (Beta('beta_x_180_270', 0, -1.3407807929942596e+154, 1.3407807929942596e+154, 0) * bioMax(`0.0`, bioMin((x - `180.0`), `90.0`))), (Beta('beta_x_270_inf', 0, -1.3407807929942596e+154, 1.3407807929942596e+154, 0) * bioMax(`0.0`, (x - `270.0`))))

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.

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

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`))))

We provide a plot of a piecewise linear specification.

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

[<matplotlib.lines.Line2D object at 0x126cbe780>]

Logit

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

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

Alternative 1

p1 = models.logit(V, av, 1)
p1.getValue_c(my_data, prepareIds=True)

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

p1 = models.loglogit(V, av, 1)
p1.getValue_c(my_data, prepareIds=True)

array([-1.79769313e+308, -2.40610156e-001, -4.40639679e-002,
       -3.61531156e-002, -7.41899415e-003])

Alternative 2

p2 = models.logit(V, av, 2)
p2.getValue_c(my_data, prepareIds=True)

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

p2 = models.loglogit(V, av, 2)
p2.getValue_c(my_data, prepareIds=True)

array([ 0.00000000e+000, -2.14061016e+000, -1.79769313e+308,
       -3.93615312e+000, -4.90741899e+000])

Alternative 3

p3 = models.logit(V, av, 3)
p3.getValue_c(my_data, prepareIds=True)

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

p3 = models.loglogit(V, av, 3)
p3.getValue_c(my_data, prepareIds=True)

array([-1.79769313e+308, -2.34061016e+000, -3.14406397e+000,
       -4.13615312e+000, -1.79769313e+308])

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

Alternative 1

pa1 = models.logit(V, av=None, i=1)
pa1.getValue_c(my_data, prepareIds=True)

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

pa1 = models.loglogit(V, av=None, i=1)
pa1.getValue_c(my_data, prepareIds=True)

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

Alternative 2

pa2 = models.logit(V, av=None, i=2)
pa2.getValue_c(my_data, prepareIds=True)

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

pa2 = models.loglogit(V, av=None, i=2)
pa2.getValue_c(my_data, prepareIds=True)

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

Alternative 3

pa3 = models.logit(V, av=None, i=3)
pa3.getValue_c(my_data, prepareIds=True)

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

pa3 = models.loglogit(V, av=None, i=3)
pa3.getValue_c(my_data, prepareIds=True)

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

Boxcox transform

The Box-Cox transform of a variable \(x\) is defined as

\[B(x,\ell) =\frac{x^{\ell}-1}{\ell},\]

where \(\ell\) is a parameter that can be estimated from data. It has the property that

\[\lim_{\ell \to 0} B(x,\ell)=\log(x).\]

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

{{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`)]

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

{{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`)]

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

{{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`)]

e.getValue_c(my_data, prepareIds=True)

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

We numerically illustrate that, when \(\lambda\) goes to 0, the BoxCox transform of \(x\) converges to the log of \(x\).

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

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

MEV models

MEV models are defined as

\[\frac{e^{V_i + \ln G_i(e^{V_1},\ldots,e^{V_J})}}{\sum_j e^{V_j + \ln G_j(e^{V_1},\ldots,e^{V_J})}},\]

where \(G\) is a generating function, and

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

Nested logit model: the \(G\) function for the nested logit model is defined such that

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

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

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

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}

Definition of the nests.

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))

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

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

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

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

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

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

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

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

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

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

If the value of the parameter \(\mu\) 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 \(0 \leq \mu \leq \mu_m\) is valid.

p1 = models.nestedMevMu(V, availability=av, nests=nests, choice=1, mu=1.1)
p1.getValue_c(my_data, prepareIds=True)

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

p1 = models.nestedMevMu(V, availability=None, nests=nests, choice=1, mu=1.1)
p1.getValue_c(my_data, prepareIds=True)

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

p1 = models.lognestedMevMu(V, availability=av, nests=nests, choice=1, mu=1.1)
p1.getValue_c(my_data, prepareIds=True)

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

p1 = models.lognestedMevMu(V, availability=None, nests=nests, choice=1, mu=1.1)
p1.getValue_c(my_data, prepareIds=True)

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

The validity of the nested structure can be verified.

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()

is_valid

True

print(msg)

;

If an alternative belongs to two nests

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()

is_valid

False

print(msg)

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

Cross-nested logit model: the \(G\) function for the cross nested logit model is defined such that

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

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

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.

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))

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

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

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

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

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

If the value of the parameter \(\mu\) 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 \(0 \leq \mu \leq \mu_m\) is verified.

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

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

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

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

If the sample is endogenous, a correction must be included in the model, as proposed by Bierlaire, Bolduc and McFadden (2008). In this case, the generating function must first be defined, and the MEV model with correction is then called.

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

{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`))))}

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

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

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

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

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

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

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

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

The MEV generating function for the following models are available.

Nested logit model

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}
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))

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

{1: (((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) - `1.0`) * Variable1) + (((`1.0` / Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0)) - `1.0`) * log(bioMultSum(exp((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) * Variable1)), exp((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) * `0.1`)), exp((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) * `-0.2`)))))), 2: (((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) - `1.0`) * `0.1`) + (((`1.0` / Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0)) - `1.0`) * log(bioMultSum(exp((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) * Variable1)), exp((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) * `0.1`)), exp((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) * `-0.2`)))))), 4: (((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) - `1.0`) * `-0.2`) + (((`1.0` / Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0)) - `1.0`) * log(bioMultSum(exp((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) * Variable1)), exp((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) * `0.1`)), exp((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) * `-0.2`)))))), 3: (((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) - `1.0`) * `-0.1`) + (((`1.0` / Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0)) - `1.0`) * log(bioMultSum(exp((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) * `-0.1`)), exp((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) * `0.2`)))))), 5: (((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) - `1.0`) * `0.2`) + (((`1.0` / Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0)) - `1.0`) * log(bioMultSum(exp((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) * `-0.1`)), exp((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) * `0.2`))))))}

And with the \(\mu\) parameter.

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

{1: ((log(`1.1`) + ((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) - `1.0`) * Variable1)) + (((`1.1` / Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0)) - `1.0`) * log(bioMultSum(exp((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) * Variable1)), exp((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) * `0.1`)), exp((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) * `-0.2`)))))), 2: ((log(`1.1`) + ((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) - `1.0`) * `0.1`)) + (((`1.1` / Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0)) - `1.0`) * log(bioMultSum(exp((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) * Variable1)), exp((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) * `0.1`)), exp((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) * `-0.2`)))))), 4: ((log(`1.1`) + ((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) - `1.0`) * `-0.2`)) + (((`1.1` / Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0)) - `1.0`) * log(bioMultSum(exp((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) * Variable1)), exp((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) * `0.1`)), exp((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) * `-0.2`)))))), 3: ((log(`1.1`) + ((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) - `1.0`) * `-0.1`)) + (((`1.1` / Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0)) - `1.0`) * log(bioMultSum(exp((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) * `-0.1`)), exp((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) * `0.2`)))))), 5: ((log(`1.1`) + ((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) - `1.0`) * `0.2`)) + (((`1.1` / Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0)) - `1.0`) * log(bioMultSum(exp((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) * `-0.1`)), exp((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) * `0.2`))))))}

Cross nested logit model

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))

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

{1: logzero(bioMultSum((((`1.0` ** Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0)) * exp(((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) - `1.0`) * Variable1))) * (bioMultSum(((`1.0` ** Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0)) * exp((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) * Variable1))), ((`1.0` ** Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0)) * exp((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) * `0.1`))), ((`0.5` ** Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0)) * exp((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) * `-0.1`))), ((`0.0` ** Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0)) * exp((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) * `-0.2`))), ((`0.0` ** Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0)) * exp((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) * `0.2`)))) ** ((`1.0` / Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0)) - `1.0`))), (((`0.0` ** Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0)) * exp(((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) - `1.0`) * Variable1))) * (bioMultSum(((`0.0` ** Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0)) * exp((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) * Variable1))), ((`0.0` ** Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0)) * exp((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) * `0.1`))), ((`0.5` ** Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0)) * exp((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) * `-0.1`))), ((`1.0` ** Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0)) * exp((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) * `-0.2`))), ((`1.0` ** Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0)) * exp((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) * `0.2`)))) ** ((`1.0` / Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0)) - `1.0`))))), 2: logzero(bioMultSum((((`1.0` ** Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0)) * exp(((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) - `1.0`) * `0.1`))) * (bioMultSum(((`1.0` ** Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0)) * exp((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) * Variable1))), ((`1.0` ** Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0)) * exp((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) * `0.1`))), ((`0.5` ** Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0)) * exp((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) * `-0.1`))), ((`0.0` ** Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0)) * exp((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) * `-0.2`))), ((`0.0` ** Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0)) * exp((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) * `0.2`)))) ** ((`1.0` / Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0)) - `1.0`))), (((`0.0` ** Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0)) * exp(((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) - `1.0`) * `0.1`))) * (bioMultSum(((`0.0` ** Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0)) * exp((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) * Variable1))), ((`0.0` ** Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0)) * exp((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) * `0.1`))), ((`0.5` ** Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0)) * exp((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) * `-0.1`))), ((`1.0` ** Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0)) * exp((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) * `-0.2`))), ((`1.0` ** Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0)) * exp((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) * `0.2`)))) ** ((`1.0` / Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0)) - `1.0`))))), 3: logzero(bioMultSum((((`0.5` ** Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0)) * exp(((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) - `1.0`) * `-0.1`))) * (bioMultSum(((`1.0` ** Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0)) * exp((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) * Variable1))), ((`1.0` ** Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0)) * exp((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) * `0.1`))), ((`0.5` ** Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0)) * exp((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) * `-0.1`))), ((`0.0` ** Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0)) * exp((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) * `-0.2`))), ((`0.0` ** Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0)) * exp((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) * `0.2`)))) ** ((`1.0` / Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0)) - `1.0`))), (((`0.5` ** Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0)) * exp(((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) - `1.0`) * `-0.1`))) * (bioMultSum(((`0.0` ** Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0)) * exp((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) * Variable1))), ((`0.0` ** Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0)) * exp((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) * `0.1`))), ((`0.5` ** Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0)) * exp((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) * `-0.1`))), ((`1.0` ** Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0)) * exp((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) * `-0.2`))), ((`1.0` ** Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0)) * exp((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) * `0.2`)))) ** ((`1.0` / Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0)) - `1.0`))))), 4: logzero(bioMultSum((((`0.0` ** Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0)) * exp(((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) - `1.0`) * `-0.2`))) * (bioMultSum(((`1.0` ** Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0)) * exp((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) * Variable1))), ((`1.0` ** Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0)) * exp((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) * `0.1`))), ((`0.5` ** Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0)) * exp((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) * `-0.1`))), ((`0.0` ** Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0)) * exp((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) * `-0.2`))), ((`0.0` ** Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0)) * exp((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) * `0.2`)))) ** ((`1.0` / Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0)) - `1.0`))), (((`1.0` ** Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0)) * exp(((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) - `1.0`) * `-0.2`))) * (bioMultSum(((`0.0` ** Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0)) * exp((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) * Variable1))), ((`0.0` ** Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0)) * exp((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) * `0.1`))), ((`0.5` ** Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0)) * exp((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) * `-0.1`))), ((`1.0` ** Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0)) * exp((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) * `-0.2`))), ((`1.0` ** Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0)) * exp((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) * `0.2`)))) ** ((`1.0` / Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0)) - `1.0`))))), 5: logzero(bioMultSum((((`0.0` ** Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0)) * exp(((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) - `1.0`) * `0.2`))) * (bioMultSum(((`1.0` ** Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0)) * exp((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) * Variable1))), ((`1.0` ** Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0)) * exp((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) * `0.1`))), ((`0.5` ** Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0)) * exp((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) * `-0.1`))), ((`0.0` ** Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0)) * exp((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) * `-0.2`))), ((`0.0` ** Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0)) * exp((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) * `0.2`)))) ** ((`1.0` / Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0)) - `1.0`))), (((`1.0` ** Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0)) * exp(((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) - `1.0`) * `0.2`))) * (bioMultSum(((`0.0` ** Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0)) * exp((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) * Variable1))), ((`0.0` ** Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0)) * exp((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) * `0.1`))), ((`0.5` ** Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0)) * exp((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) * `-0.1`))), ((`1.0` ** Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0)) * exp((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) * `-0.2`))), ((`1.0` ** Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0)) * exp((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) * `0.2`)))) ** ((`1.0` / Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0)) - `1.0`)))))}

Cross nested logit model with \(\mu\) parameter.

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

{1: log((`1.1` * bioMultSum((((`1.0` ** (Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) / `1.1`)) * exp(((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) - `1.0`) * Variable1))) * (bioMultSum(((`1.0` ** (Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) / `1.1`)) * exp((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) * Variable1))), ((`1.0` ** (Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) / `1.1`)) * exp((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) * `0.1`))), ((`0.5` ** (Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) / `1.1`)) * exp((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) * `-0.1`))), ((`0.0` ** (Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) / `1.1`)) * exp((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) * `-0.2`))), ((`0.0` ** (Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) / `1.1`)) * exp((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) * `0.2`)))) ** ((`1.1` / Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0)) - `1.0`))), (((`0.0` ** (Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) / `1.1`)) * exp(((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) - `1.0`) * Variable1))) * (bioMultSum(((`0.0` ** (Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) / `1.1`)) * exp((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) * Variable1))), ((`0.0` ** (Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) / `1.1`)) * exp((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) * `0.1`))), ((`0.5` ** (Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) / `1.1`)) * exp((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) * `-0.1`))), ((`1.0` ** (Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) / `1.1`)) * exp((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) * `-0.2`))), ((`1.0` ** (Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) / `1.1`)) * exp((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) * `0.2`)))) ** ((`1.1` / Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0)) - `1.0`)))))), 2: log((`1.1` * bioMultSum((((`1.0` ** (Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) / `1.1`)) * exp(((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) - `1.0`) * `0.1`))) * (bioMultSum(((`1.0` ** (Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) / `1.1`)) * exp((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) * Variable1))), ((`1.0` ** (Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) / `1.1`)) * exp((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) * `0.1`))), ((`0.5` ** (Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) / `1.1`)) * exp((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) * `-0.1`))), ((`0.0` ** (Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) / `1.1`)) * exp((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) * `-0.2`))), ((`0.0` ** (Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) / `1.1`)) * exp((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) * `0.2`)))) ** ((`1.1` / Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0)) - `1.0`))), (((`0.0` ** (Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) / `1.1`)) * exp(((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) - `1.0`) * `0.1`))) * (bioMultSum(((`0.0` ** (Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) / `1.1`)) * exp((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) * Variable1))), ((`0.0` ** (Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) / `1.1`)) * exp((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) * `0.1`))), ((`0.5` ** (Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) / `1.1`)) * exp((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) * `-0.1`))), ((`1.0` ** (Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) / `1.1`)) * exp((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) * `-0.2`))), ((`1.0` ** (Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) / `1.1`)) * exp((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) * `0.2`)))) ** ((`1.1` / Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0)) - `1.0`)))))), 3: log((`1.1` * bioMultSum((((`0.5` ** (Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) / `1.1`)) * exp(((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) - `1.0`) * `-0.1`))) * (bioMultSum(((`1.0` ** (Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) / `1.1`)) * exp((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) * Variable1))), ((`1.0` ** (Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) / `1.1`)) * exp((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) * `0.1`))), ((`0.5` ** (Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) / `1.1`)) * exp((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) * `-0.1`))), ((`0.0` ** (Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) / `1.1`)) * exp((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) * `-0.2`))), ((`0.0` ** (Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) / `1.1`)) * exp((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) * `0.2`)))) ** ((`1.1` / Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0)) - `1.0`))), (((`0.5` ** (Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) / `1.1`)) * exp(((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) - `1.0`) * `-0.1`))) * (bioMultSum(((`0.0` ** (Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) / `1.1`)) * exp((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) * Variable1))), ((`0.0` ** (Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) / `1.1`)) * exp((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) * `0.1`))), ((`0.5` ** (Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) / `1.1`)) * exp((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) * `-0.1`))), ((`1.0` ** (Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) / `1.1`)) * exp((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) * `-0.2`))), ((`1.0` ** (Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) / `1.1`)) * exp((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) * `0.2`)))) ** ((`1.1` / Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0)) - `1.0`)))))), 4: log((`1.1` * bioMultSum((((`0.0` ** (Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) / `1.1`)) * exp(((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) - `1.0`) * `-0.2`))) * (bioMultSum(((`1.0` ** (Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) / `1.1`)) * exp((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) * Variable1))), ((`1.0` ** (Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) / `1.1`)) * exp((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) * `0.1`))), ((`0.5` ** (Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) / `1.1`)) * exp((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) * `-0.1`))), ((`0.0` ** (Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) / `1.1`)) * exp((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) * `-0.2`))), ((`0.0` ** (Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) / `1.1`)) * exp((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) * `0.2`)))) ** ((`1.1` / Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0)) - `1.0`))), (((`1.0` ** (Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) / `1.1`)) * exp(((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) - `1.0`) * `-0.2`))) * (bioMultSum(((`0.0` ** (Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) / `1.1`)) * exp((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) * Variable1))), ((`0.0` ** (Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) / `1.1`)) * exp((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) * `0.1`))), ((`0.5` ** (Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) / `1.1`)) * exp((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) * `-0.1`))), ((`1.0` ** (Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) / `1.1`)) * exp((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) * `-0.2`))), ((`1.0` ** (Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) / `1.1`)) * exp((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) * `0.2`)))) ** ((`1.1` / Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0)) - `1.0`)))))), 5: log((`1.1` * bioMultSum((((`0.0` ** (Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) / `1.1`)) * exp(((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) - `1.0`) * `0.2`))) * (bioMultSum(((`1.0` ** (Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) / `1.1`)) * exp((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) * Variable1))), ((`1.0` ** (Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) / `1.1`)) * exp((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) * `0.1`))), ((`0.5` ** (Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) / `1.1`)) * exp((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) * `-0.1`))), ((`0.0` ** (Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) / `1.1`)) * exp((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) * `-0.2`))), ((`0.0` ** (Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) / `1.1`)) * exp((Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0) * `0.2`)))) ** ((`1.1` / Beta('muA', 1.2, 1.0, 1.3407807929942596e+154, 0)) - `1.0`))), (((`1.0` ** (Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) / `1.1`)) * exp(((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) - `1.0`) * `0.2`))) * (bioMultSum(((`0.0` ** (Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) / `1.1`)) * exp((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) * Variable1))), ((`0.0` ** (Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) / `1.1`)) * exp((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) * `0.1`))), ((`0.5` ** (Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) / `1.1`)) * exp((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) * `-0.1`))), ((`1.0` ** (Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) / `1.1`)) * exp((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) * `-0.2`))), ((`1.0` ** (Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) / `1.1`)) * exp((Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0) * `0.2`)))) ** ((`1.1` / Beta('muB', 2.3, 1.0, 1.3407807929942596e+154, 0)) - `1.0`))))))}