---------------------------------------------
lines 5-143 of file: example/user/residual.py
---------------------------------------------

# {xrst_begin user_residual.py}
# {xrst_comment_ch #}
#
# Weighted Residuals
# ##################
#
# Example Parameters
# ******************
# The following values are used to simulate the data and define the priors:
# {xrst_spell_off}
# {xrst_code py}
omega_true      = [ 0.01, 0.02 ]
omega_mean      = 0.15
minimum_meas_cv = 0.10
# {xrst_code}
# {xrst_spell_on}
#
# Model
# *****
# The only non-zero model variable for this example is
# other cause mortality, :math:`\omega`, for the parent area.
# The parent omega grid has two points,
# one at age zero and the other at age 100.
# The corresponding true values for :math:`\omega` are
# ``omega_true[0]`` at age zero and ``omega_true[1]`` at age 100.
#
# Data
# ****
# The integrand for this data is
# :ref:`avg_integrand@Integrand, I_i(a,t)@mtother` ; i.e.,
# a direct measurement of :math:`\omega`.
# Both ages are included in the data for this example.
# Both the Gaussian and Log-Gaussian :ref:`densities<density_table-name>`
# are included.
# In addition, both the case where the
# :ref:`data_table@meas_std` is above and below the bound specified
# by :ref:`integrand_table@minimum_meas_cv` are included.
#
# Capital Delta
# =============
# We use the notation
# :ref:`Delta<data_like@Notation@Minimum CV Standard Deviation, Delta_i>` for
# the standard deviation adjusted by the minimum measurement cv.
#
# Lower delta
# ===========
# We use the notation
# :ref:`delta<data_like@Transformed Standard Deviation, delta_i(theta)>`
# for the transformed standard deviation.
#
# Gaussian Residuals
# ==================
# There are no measurement noise covariate multipliers, so
# :math:`\delta = \Delta` and the residual is
#
# .. math::
#
#   (y - \mu) / \delta
#
# where :math:`y` is the measured value and :math:`\mu` is the
# model value for the
# :ref:`average integrand<avg_integrand@Average Integrand, A_i>` .
#
# Log-Gaussian Residuals
# ======================
# There are no measurement noise covariate multipliers, so
#
# .. math::
#
#   \delta = \log ( y + \eta + \Delta ) - \log( y + \eta )
#
# where :math:`y` is the measured value
# and :math:`\eta` is the offset in the log transform.
# The residual is
#
# .. math::
#
#   \frac{ \log ( y + \eta ) - \log ( \mu + \eta ) } { \delta }
#
# where :math:`\mu` is the model value for the average integrand.
#
# Priors
# ******
#
# Value Residual
# ==============
# There are two value residuals, one for :math:`\omega` at age zero
# and the other at age 100.
# The density used for the value residuals is Log-Gaussian.
# The mean value used in the prior for the value residuals :math:`\mu` is
# ``omega_mean`` .
# The standard deviation used for the value residuals in
# ``omega_mean * 0.1``
# The log transformed standard deviation is
#
# .. math::
#
#   \delta = \log ( \mu + \eta + \sigma ) - \log( \mu + \eta )
#
# The residual is
#
# .. math::
#
#   \frac{ \log ( y + \eta ) - \log ( \mu + \eta ) } { \delta }
#
# where :math:`y` is the :ref:`fit_var_table@fit_var_value`
# for the model variable.
#
# Difference Residual
# ===================
# There is one difference residuals for the difference of
# :math:`\omega` at age zero and age 100.
# The density used for the value residuals is Log-Gaussian.
# The mean value used in the prior for the difference residual
# :math:`\mu = 0`.
# The standard deviation used for the difference residual is ``0.1`` .
# (This corresponds to a coefficient of variation of :math:`e^{0.1} - 1`.
# which is approximately equal to ``0.1`` ; i.e., 10 percent.)
# The age difference smoothing multiplier prior id
# :ref:`mulstd_dage_prior_id<smooth_table-name>` for this example is null,
# so :math:`\delta` is equal to the standard deviation ``0.1`` .
# The residual is
#
# .. math::
#
#   \frac{ \log ( z + \eta ) - \log ( y + \eta ) - \mu } { \delta }
#
# where :math:`y` (:math:`z`)
# is the :ref:`fit_var_table@fit_var_value`
# at age zero (age 100).
#
# Source Code
# ***********
# {xrst_literal
#       BEGIN PYTHON
#       END PYTHON
# }
#
# {xrst_end user_residual.py}