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lines 5-140 of file: example/user/lasso_covariate.py
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# {xrst_begin user_lasso_covariate.py}
# {xrst_spell
#     exp
# }
# {xrst_comment_ch #}
#
# Using Lasso on Covariate Multiplier
# ###################################
#
# See Also
# ********
# :ref:`user_group_mulcov.py-name`
#
# Purpose
# *******
# This example uses a :ref:`density_table@density_name@laplace`
# prior distribution on two covariate multipliers
# to detect which covariate is present in the data.
# It then refits the data with a uniform on the multiplier that is present
# and the other multiplier constrained to be zero.
#
# Problem Parameters
# ******************
# The following values are used to simulate the data and set the priors
# for fitting the data:
# {xrst_literal
#     begin problem parameters
#     end problem parameters
# }
#
# Age and Time Values
# *******************
# The age and time values do not matter for this problem
# because all the functions are constant in age and time.
# This can be seen by the fact that all of the smoothings have one age
# and one time point.
#
# Variables
# *********
# The are three model variables in this example:
#
# .. list-table::
#     :widths: auto
#
#     * - *iota_reference*
#       - :ref:`iota<avg_integrand@Rate Functions@iota_i(a,t)>`
#         corresponding to reference value of covariates
#     * - *alpha_income*
#       - :ref:`covariate multiplier<mulcov_table-name>`
#         corresponding to income.
#     * - *alpha_sex*
#       - covariate multiplier corresponding to sex.
#
# Covariate Table
# ***************
# There are two :ref:`covariates<covariate_table-name>` ,
# ``income`` and ``sex`` , in this example.
# Income is a linear function of :ref:`data_table@data_id` .
# It starts with 0.0 and ends with 1.0.
# To be specific,
#
#     *income* = *data_id* / ( *n_data* ``- 1`` )
#
# Sex cycles between the values -0.5, 0.0, +0.5
# as a function of *data_id* .
# Note that income and sex have range 1.0.
# This makes the scaling of the covariate multipliers simpler.
# The reference value for each covariate has been set to the midpoint
# of its range.
# We use *x_income* ( *x_sex* ) to denote the difference
# of income (sex) minus its reference value.
#
# Mulcov Table
# ************
# The :ref:`covariate multiplier table<mulcov_table-name>` has two rows,
# one for each covariate multiplier.
# Both multipliers affects the value of
# :ref:`iota<avg_integrand@Rate Functions@iota_i(a,t)>` ,
# but one multiplies income and the other sex.
# We use *alpha_income* ( *alpha_sex* ) to denote the
# covariate multiplier corresponding to income (sex).
# The model for the value of *iota* is
#
#     *iota_reference* * ``exp`` ( *alpha_income* * *x_income* + *alpha_sex* * *x_sex*  )
#
# Rate Table
# **********
# The :ref:`rate_table-name` only specifies that *iota* for the parent
# is the only nonzero rate for this example.
# In addition, it specifies the smoothing for that rate which has only
# one grid point. Hence there is only one model variable corresponding to
# the rates.
#
# Data Table
# **********
# For this example, all the data is
# :ref:`avg_integrand@Integrand, I_i(a,t)@Sincidence` ,
# with a Gaussian density, with mean value
#
#     *iota_reference* * ``exp`` ( *alpha_income* * ``x_income`` )
#
# and with standard deviation equal 10% of the mean.
#
# Prior Table
# ***********
# There are three priors in the :ref:`prior_table-name` , one for each
# model variable.
#
# iota
# ====
# The prior for fitting the reference value of *iota*
# is uniform with lower limit 0.001, and upper limit 1.
# Its mean value 0.1 is only used as a starting and scaling point for the
# fitting process; see :ref:`start_var_table-name` and :ref:`scale_var_table-name` .
#
# alpha
# =====
# There is a separate prior for *alpha_income* and *alpha_sex* .
# For the first fit, they are both a Laplace density with
# mean zero and standard deviation *laplace_std* .
# The first fit is used to decide that *alpha_income* is nonzero
# and *alpha_sex* is zero.
# The prior for *alpha_income* is changed to a uniform
# and the prior for *alpha_sex* is change to have upper and lower
# limit zero.
# A second fit is done to recover the value of *alpha_income*
# without shrinkage due to the Laplace prior.
#
# Source Code
# ***********
# {xrst_literal
#     BEGIN PYTHON
#     END PYTHON
# }
#
# {xrst_end user_lasso_covariate.py}