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Nested Logit

nested_logit module

The components of the derivative of the entropy for a two-layer nested logit model. One nest on each side must consist of the 0 option. The other nests are specified as nested lists. E.g. [[1, 3], [2,4]] describes two nests, one with types 1 and 3, and the other with types 2 and 4. On each side, the nests are the same for each type, with the same parameters.

e0_derivative_mu_nested_logit(muhat, additional_parameters=None)

Returns the derivatives of the parameter-independent part \(e_0\) wrt \(\mu\) for the nested logit.

Parameters:

Name Type Description Default
muhat Matching

a Matching

 required
additional_parameters list | None

a list with the nest structure

 None

Returns:

Type Description
ThreeArrays

the parameter-independent part of the hessian of the entropy
ThreeArrays

wrt \((\mu,\mu)\).
Source code in cupid_matching/nested_logit.py 
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def e0_derivative_mu_nested_logit(
    muhat: Matching, additional_parameters: list | None = None
) -> ThreeArrays:
    """Returns the derivatives of the parameter-independent part $e_0$
    wrt $\\mu$ for the nested logit.

    Args:
        muhat: a Matching
        additional_parameters: a list with the nest structure

    Returns:
        the parameter-independent part of the hessian of the entropy
        wrt $(\\mu,\\mu)$.
    """
    nests_for_each_x, nests_for_each_y = _get_params(additional_parameters)
    nests_x = change_indices(nests_for_each_x)
    nests_y = change_indices(nests_for_each_y)
    muxy, mux0, mu0y, *_ = muhat.unpack()
    X, Y = muxy.shape

    hess_x = np.zeros((X, Y, Y))
    hess_y = np.zeros((X, Y, X))
    hess_xy = np.zeros((X, Y))
    der_logx0 = 1.0 / mux0
    der_log0y = 1.0 / mu0y

    for x in range(X):
        dlogx0 = der_logx0[x]
        for nest in nests_x:
            mu_xn = np.sum(muxy[x, nest])
            der_logxn = 1.0 / mu_xn
            for y in nest:
                hess_x[x, y, :] = -dlogx0
                hess_x[x, y, nest] -= der_logxn
    for y in range(Y):
        dlog0y = der_log0y[y]
        for nest in nests_y:
            mu_ny = np.sum(muxy[nest, y])
            der_logny = 1.0 / mu_ny
            for x in nest:
                hess_y[x, y, :] = -dlog0y
                hess_y[x, y, nest] -= der_logny
    for x in range(X):
        for y in range(Y):
            hess_xy[x, y] = hess_x[x, y, y] + hess_y[x, y, x]

    return hess_x, hess_y, hess_xy

e0_derivative_r_nested_logit(muhat, additional_parameters=None)

Returns the derivatives of the parameter-independent part \(e_0\) wrt \(r\) for the nested logit.

Parameters:

Name Type Description Default
muhat Matching

a Matching

 required
additional_parameters list | None

a list with the nest structure

 None

Returns:

Type Description
TwoArrays

the parameter-independent part of the hessian of the entropy
TwoArrays

wrt \((\mu,r)\).
Source code in cupid_matching/nested_logit.py 
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def e0_derivative_r_nested_logit(
    muhat: Matching, additional_parameters: list | None = None
) -> TwoArrays:
    """Returns the derivatives of the parameter-independent part $e_0$
    wrt $r$ for the nested logit.

    Args:
        muhat: a Matching
        additional_parameters: a list with the nest structure

    Returns:
        the parameter-independent part of the hessian of the entropy
        wrt $(\\mu,r)$.
    """
    nests_for_each_x, nests_for_each_y = _get_params(additional_parameters)
    nests_x = change_indices(nests_for_each_x)
    nests_y = change_indices(nests_for_each_y)
    muxy, mux0, mu0y, n, m = muhat.unpack()
    X, Y = muxy.shape

    hess_n = np.zeros((X, Y))
    hess_m = np.zeros((X, Y))
    der_logx0 = 1.0 / mux0
    der_log0y = 1.0 / mu0y

    for x in range(X):
        dlogx0 = der_logx0[x]
        for nest in nests_x:
            for y in nest:
                hess_n[x, y] = dlogx0
    for y in range(Y):
        dlog0y = der_log0y[y]
        for nest in nests_y:
            for x in nest:
                hess_m[x, y] = dlog0y

    return hess_n, hess_m

e0_nested_logit(muhat, additional_parameters=None)

Returns the values of the parameter-independent part \(e_0\) for the nested logit.

Parameters:

Name Type Description Default
muhat Matching

a Matching

 required
additional_parameters list | None

a list with the nest structure

 None

Returns:

Type Description
np.ndarray

the (X,Y) matrix of the parameter-independent part
np.ndarray

of the first derivative of the entropy.
Source code in cupid_matching/nested_logit.py 
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def e0_nested_logit(
    muhat: Matching, additional_parameters: list | None = None
) -> np.ndarray:
    """Returns the values of the parameter-independent part $e_0$
    for the nested logit.

    Args:
        muhat: a Matching
        additional_parameters: a list with the nest structure

    Returns:
        the (X,Y) matrix of the parameter-independent part
        of the first derivative of the entropy.
    """
    nests_for_each_x, nests_for_each_y = _get_params(additional_parameters)
    nests_x = change_indices(nests_for_each_x)
    nests_y = change_indices(nests_for_each_y)
    muxy, mux0, mu0y, *_ = muhat.unpack()
    X, Y = muxy.shape
    e0_vals = np.zeros((X, Y))

    for x in range(X):
        mux0_x = mux0[x]
        for nest in nests_x:
            mu_xn = np.sum(muxy[x, nest])
            e0_vals[x, nest] = -log(mu_xn / mux0_x)
    for y in range(Y):
        mu0y_y = mu0y[y]
        for nest in nests_y:
            mu_ny = np.sum(muxy[nest, y])
            e0_vals[nest, y] -= log(mu_ny / mu0y_y)
    return e0_vals

e_derivative_mu_nested_logit(muhat, additional_parameters=None)

Returns the derivatives of the parameter-dependent part \(e\) wrt \(\mu\) for the nested logit.

Parameters:

Name Type Description Default
muhat Matching

a Matching

 required
additional_parameters list | None

a list with the nest structure

 None

Returns:

Type Description
ThreeArrays

the parameter-dependent part of the hessian of the entropy
ThreeArrays

wrt \((\mu,\mu)\).
Source code in cupid_matching/nested_logit.py 
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def e_derivative_mu_nested_logit(
    muhat: Matching, additional_parameters: list | None = None
) -> ThreeArrays:
    """Returns the derivatives of the parameter-dependent part $e$
     wrt $\\mu$ for the nested logit.

    Args:
        muhat: a Matching
        additional_parameters: a list with the nest structure

    Returns:
        the parameter-dependent part of the hessian of the entropy
        wrt $(\\mu,\\mu)$.
    """
    nests_for_each_x, nests_for_each_y = _get_params(additional_parameters)
    nests_x = change_indices(nests_for_each_x)
    nests_y = change_indices(nests_for_each_y)
    n_rhos = len(nests_for_each_x)
    n_deltas = len(nests_for_each_y)
    n_alpha = n_rhos + n_deltas

    muxy, *_ = muhat.unpack()
    X, Y = muxy.shape

    hess_x = np.zeros((X, Y, Y, n_alpha))
    hess_y = np.zeros((X, Y, X, n_alpha))
    hess_xy = np.zeros((X, Y, n_alpha))
    der_logxy = 1.0 / muxy

    for x in range(X):
        for i_n, nest in enumerate(nests_x):
            mux_nest_n = muxy[x, nest]
            mu_xn = np.sum(mux_nest_n)
            der_logxn = 1.0 / mu_xn
            for t in nest:
                hess_x[x, nest, t, i_n] = der_logxn
            hess_xy[x, nest, i_n] = der_logxn - der_logxy[x, nest]

    for y in range(Y):
        for i_n, nest in enumerate(nests_y):
            muy_nest_n = muxy[nest, y]
            mu_ny = np.sum(muy_nest_n)
            der_logny = 1.0 / mu_ny
            i_n2 = i_n + n_rhos
            for z in nest:
                hess_y[nest, y, z, i_n2] = der_logny
            hess_xy[nest, y, i_n2] = der_logny - der_logxy[nest, y]

    return hess_x, hess_y, hess_xy

e_derivative_r_nested_logit(muhat, additional_parameters=None)

Returns the derivatives of the parameter-dependent part \(e\) wrt \(r\) for the nested logit.

Parameters:

Name Type Description Default
muhat Matching

a Matching

 required
additional_parameters list | None

a list with the nest structure

 None

Returns:

Type Description
TwoArrays

the parameter-dependent part of the hessian of the entropy
TwoArrays

wrt \((\mu,r)\).
Source code in cupid_matching/nested_logit.py 
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def e_derivative_r_nested_logit(
    muhat: Matching, additional_parameters: list | None = None
) -> TwoArrays:
    """Returns the derivatives of the parameter-dependent part $e$
     wrt $r$ for the nested logit.

    Args:
        muhat: a Matching
        additional_parameters: a list with the nest structure

    Returns:
        the parameter-dependent part of the hessian of the entropy
        wrt $(\\mu,r)$.
    """
    nests_for_each_x, nests_for_each_y = _get_params(additional_parameters)
    n_rhos = len(nests_for_each_x)
    n_deltas = len(nests_for_each_y)
    n_alpha = n_rhos + n_deltas

    muxy, *_ = muhat.unpack()
    X, Y = muxy.shape

    hess_n = np.zeros((X, Y, n_alpha))
    hess_m = np.zeros((X, Y, n_alpha))

    return hess_n, hess_m

e_nested_logit(muhat, additional_parameters=None)

Returns the values of the parameter-dependent part \(e\) for the nested logit.

Parameters:

Name Type Description Default
muhat Matching

a Matching

 required
additional_parameters list | None

a list with the nest structure

 None

Returns:

Type Description
np.ndarray

the (X,Y,n_alpha) array of the parameter-dependent part
np.ndarray

of the first derivative of the entropy.
Source code in cupid_matching/nested_logit.py 
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def e_nested_logit(
    muhat: Matching, additional_parameters: list | None = None
) -> np.ndarray:
    """Returns the values of the parameter-dependent part  $e$
    for the nested logit.

    Args:
        muhat: a Matching
        additional_parameters: a list with the nest structure

    Returns:
        the (X,Y,n_alpha) array of the parameter-dependent part
        of the first derivative of the entropy.
    """
    nests_for_each_x, nests_for_each_y = _get_params(additional_parameters)
    nests_x = change_indices(nests_for_each_x)
    nests_y = change_indices(nests_for_each_y)
    n_rhos = len(nests_for_each_x)
    n_deltas = len(nests_for_each_y)
    n_alpha = n_rhos + n_deltas

    muxy, *_ = muhat.unpack()
    X, Y = muxy.shape

    e_vals = np.zeros((X, Y, n_alpha))

    for x in range(X):
        for i_n, nest in enumerate(nests_x):
            mux_nest_n = muxy[x, nest]
            mu_xn = np.sum(mux_nest_n)
            e_vals[x, nest, i_n] = -np.log(mux_nest_n / mu_xn)

    for y in range(Y):
        for i_n, nest in enumerate(nests_y):
            muy_nest_n = muxy[nest, y]
            mu_ny = np.sum(muy_nest_n)
            e_vals[nest, y, (i_n + n_rhos)] -= np.log(muy_nest_n / mu_ny)

    return e_vals