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lines 5-107 of file: example/user/sim_log.py
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# {xrst_begin user_sim_log.py}
# {xrst_comment_ch #}
#
# Simulating Data with Log Transformed Distribution
# #################################################
#
# See Also
# ********
# :ref:`user_data_sim.py-name`
#
# Example Parameters
# ******************
# The following values are used to simulate the data
# {xrst_spell_off}
# {xrst_code py}
number_simulate   = 2000
iota_true         = 0.01
meas_value_global = iota_true * 1.5
eta_global        = iota_true * 1e-3
meas_std_global   = meas_value_global * 0.25
gamma_global      = meas_value_global * 0.25
# {xrst_code}
# {xrst_spell_on}
# Model
# *****
# The only non-zero model variable for this example is
# the rate of incidence for the world which is constant in age and time.
#
# Data
# ****
# There is only one data point for this example and it's integrand is
# :ref:`avg_integrand@Integrand, I_i(a,t)@Sincidence` .
# This data has a log transformed distribution with mean *iota_true* ,
# offset *eta_global* , and standard deviation
# *meas_std_global* .
#
# Covariate Multiplier
# ********************
# For this example there is one covariate multiplier.
# It is a :ref:`mulcov_table@mulcov_type@meas_noise` multiplier
# and the corresponding covariate value is one.
#
# Notation
# ********
#
# .. list-table::
#       :widths: auto
#
#       * - :math:`y`
#         - is the measurement value, *meas_value_global*
#       * - :math:`\mu`
#         - mean of the data, *iota_true*
#       * - :math:`\eta`
#         - offset in log transform, *eta_global*
#       * - :math:`\Delta`
#         - data measurement error, *meas_std_global*
#       * - :math:`\gamma`
#         - meta regression error, *gamma_global*
#       * - :math:`n`
#         - number of simulated data values,
#            :ref:`simulate_command@number_simulate`
#       * - :math:`z_i`
#         - *i*-th simulate data for :math:`i = 1, \ldots , n`
#
# delta
# *****
# The transformed standard deviation
# :ref:`delta<data_like@Transformed Standard Deviation, delta_i(theta)>`
# is given by
#
# .. math::
#
#   \delta = \log( y + \eta + \sigma ) - \log(y + \eta)
#
# sigma
# *****
# For this example we use the
# :ref:`data_like@Adjusted Standard Deviation, sigma_i(theta)@add_std_scale_none`
# option in the definition of the adjusted standard deviation
# :ref:`sigma<data_like@Adjusted Standard Deviation, sigma_i(theta)>`
# :math:`\sigma`; i.e.,
#
# .. math::
#
#   \sigma = \Delta + \gamma
#
# Simulations
# ***********
# The offset log transform of each simulated measurement :math:`z_i` has
# the following Gaussian distribution:
#
# .. math::
#
#   \log( z_i + \eta ) - \log( \mu + \eta ) \sim N(0, \delta^2 )
#
# Source Code
# ***********
# {xrst_literal
#       BEGIN PYTHON
#       END PYTHON
# }
#
# {xrst_end user_sim_log.py}