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lines 5-124 of file: example/user/jump_at_age.py
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# {xrst_begin user_jump_at_age.py}
# {xrst_spell
#     apx
#     def
#     exp
# }
# {xrst_comment_ch #}
#
# Zero Rate Until a Jump at a Known Age
# #####################################
#
# Purpose
# *******
# Usually the prior for the rate is smooth.
# This requires lots of data, at a fine age spacing,
# to resolve a jump in a rate at an unknown age.
# If the age at which a jump occurs is known, it is possible to resolve
# the jump with much less data by specifying a prior that has this knowledge.
#
# Parameters
# **********
# The following values are used to simulate the data and define the priors
# and can be changed:
# {xrst_spell_off}
# {xrst_code py}
iota_near_zero  = 1e-10
iota_after_20   = 1e-2
iota_eta        = 1e-5
time_table      = [ 2020.0, 2000.0 ]
age_table       = [ 100.0, 70.0, 40.0, 20.01, 19.99, 0.0 ]
# {xrst_code}
# {xrst_spell_on}
#
# iota_near_zero
# ==============
# This is the true value of *iota* up to and including age 20.
# Note that it is close to zero, but not equal to zero, so that we can
# use the rate case
# :ref:`option_table@rate_case@iota_pos_rho_zero` .
#
# iota_after_20
# =============
# This is the true value of *iota* for ages greater than 20.
#
# iota_eta
# ========
# Offset in log transformation used for values of eta.
#
# age_table
# =========
# The :ref:`age_table-name` does not need to be monotone increasing.
# For this example, it is the same as the table of ages at which
# *iota* is modeled .
# You can changed the order of ``age_table``
# and it will not affect the results.
# Note that the age table has a point just before and after the jump
# because iota is interpolated linear between age points.
#
# time_table
# ==========
# The :ref:`time_table-name` does not need to be monotone increasing.
# You can changed the order of ``time_table``
# and it will not affect the results.
#
# Model Variables
# ***************
# This example's variables are all
# :ref:`model_variables@Fixed Effects, theta@Parent Rates`
# for :ref:`rate_table@rate_name@iota` .
# The value of *iota* is modeled at each age in the ``age_table`` .
# The prior for the value of *iota* up to age 20 is a constant equal to
# ``iota_near_zero`` .
# After age 20 it is uniform with lower limit ``iota_near_zero`` ,
# upper limit 1 and mean ``iota_after_20 / 4.0``
# (The mean is only used for the initial value and scaling the optimization.)
# The prior for the forward age differences in *iota* at age 20
# is uniform, and above age 20 it is a Log-Gaussian with mean 0 and
# standard deviation 0.1 (about 10 percent coefficient of variation).
#
# Truth
# *****
# For this example the rate *iota* is constant
# with value ``iota_near_zero`` for ages less than or equal 20,
# and ``iota_after_20`` for ages greater than 20.
#
# Prevalence
# **********
# There is no remission or other cause mortality in this example; i.e.,
# *rho* and *omega* are zero.
# We approximation prevalence for ages less that 20 as zero.
# {xrst_code py}
def prevalence_apx(age) :
    import math
    if age < 20.0 :
        return 0.0
    result = 1.0 - math.exp( - iota_after_20 * (age - 20 ) )
    return result
# {xrst_code}
#
# Simulated Data
# **************
# For this example, the simulated data is all
# :ref:`avg_integrand@Integrand, I_i(a,t)@prevalence` .
# There is no noise simulated with the data; i.e., it is equal to
# the approximation described above.
# On the other hand, its is modeled as if it had standard deviation
# {xrst_code py}
meas_std = 0.01
# {xrst_code}
# There is a measured value for each age in the ``age_table``
# that is greater than 20.
#
# Source Code
# ***********
# {xrst_literal
#     BEGIN PYTHON
#     END PYTHON
# }
#
# {xrst_end user_jump_at_age.py}