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lines 6-253 of file: example/no_ode_xam.py
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{xrst_begin_parent no_ode_xam}
{xrst_spell
  avg
  dage
  dtime
  mtexcess
  sincidence
  smoothings
}

Example Using no_ode_fit To Initialize Optimization
###################################################
This example uses :ref:`glossary@mtexcess` data during a :ref:`no_ode_fit-name`
and then holds it out during the actual estimation.
This is meant to simulate the case where mtexcess is obtain
form other data to help initialize the optimization
(and without this smart initialization the optimization would fail).
For this example everything is constant in time so the
functions below do not depend on time.

Nodes
*****
The following is a diagram of the node tree for this example.
The :ref:`glossary@root_node` is n0,
the :ref:`glossary@fit_goal_set` and the leaf node set
are both {n2, n3} for this example::

                     n0
             /-----/\-----\
          n1              n2

fit_goal_set
============
{xrst_literal
    # BEGIN fit_goal_set
    # END fit_goal_set
}

Rates
*****
The non-zero dismod_at rates for this example are
:ref:`glossary@iota`, :ref:`glossary@chi`, and :ref:`glossary@omega`.
We use *iota(a, n, I)* , *chi(a, n, I)*
to denote the value for iota and chi
as a function of age *a*, node number *n*, and income *I*.
We use *omega(a, n)* to denote the value for omega
as a function of age *a* and node *n*.


Covariate
*********
There is one covariate for this example, income.
The reference value for income is the average income corresponding
to the :ref:`glossary@root_node`.

I_n
===
We use *I_n* for the reference value of income at node *n*.
The code below sets this reference using the name avg_income:
{xrst_literal
    # BEGIN avg_income
    # END avg_income
}

alpha
=====
We use *alpha_iota* ( *alpha_chi* )
for the :ref:`glossary@rate_value` covariate multiplier
which multiplies income and affects iota (chi).
The true value for these multipliers
(used which simulating the data) is
{xrst_literal
    # BEGIN alpha_true
    # END alpha_true
}

Random Effects
**************
For each node, there is a random effect on iota and chi that is constant
in age and time. Note that the leaf nodes have random effect for the node
above them as well as their own random effect.

R_n
===
We use *R_n* to denote the random effects for node *n*.
The code below sets this value:
{xrst_literal
    # BEGIN random_effect_true
    # END random_effect_true
}

Simulated Data
**************

Random Seed
===========
The random seed can be used to reproduce results.
If the original value of this setting is zero, the clock is used get
a random seed. The actual value or *random_seed* is always printed.
{xrst_literal
    # BEGIN random_seed
    # END random_seed
}

rate_true(rate, a, t, n, c)
===========================
For *rate* equal to iota, chi, and omega,
this is the true value for *rate*
in node *n* at age *a*, time *t*, and covariate values *c*.
The covariate values are a list in the
same order as the covariate table.
The values *t* and *c[1]* are not used by this function for this example.
{xrst_literal
    # BEGIN rate_true
    # END rate_true
}
The true model for chi is constant below the second age grid point
because it is not possible to determine chi at age zero from Sincidence
and prevalence measurements.

y_i
===
The simulated integrands for this example are
:ref:`glossary@mtexcess`,
:ref:`glossary@Sincidence`, and
:ref:`glossary@prevalence`.
The data is simulated without any noise
but it is modeled as having noise.
In addition, the mtexcess data is only used for the
no_ode_fit and is held out during actual fits.
This simulates the case where the mtexcess data was constructed from
the other data in order to help initialize the optimization.

n_i
===
Data is only simulated for the leaf nodes; i.e.,
each *n_i* is in the set { n3, n4, n5, n6 }.
Since the data does not have any nose, the data residuals are a measure
of how good the fit is for the nodes in the fit_goal_set.

a_i
===
For each leaf node, data is generated on the following *age_grid*:
{xrst_literal
    # BEGIN age_grid
    # END age_grid
}

I_i
===
For each leaf node and each age in *age_grid*,
data is generated for the following *income_grid*:
{xrst_literal
    # BEGIN income_grid
    # END income_grid
}
Note that the check of the fit for the nodes in the fit_goal_set
expects much more accuracy when the income grid is not chosen randomly.

Omega Constraints
*****************
The :ref:`omega_constraint-name` routine is used
to set the value of omega in the parent and child nodes.

Parent Rate Smoothing
*********************
The parent smoothings use the true value of iota and chi at age 50
and in the root_node:
{xrst_literal
    # BEGIN iota_chi_50
    # END iota_chi_50
}

iota and chi
============
This is the smoothing used in the root_node model for the rates.
Note that the value part of this smoothing is only used for the *root_node*.
This smoothing uses the *age_gird* and one time point.
There are no :ref:`glossary@dtime` priors because there is only one time point.
The smoothing for chi does not use the first age grid point, age zero,
because it is not possible to determine chi at age zero from Sincidence
and prevalence measurements.

Value Prior
===========
The following is the value prior used for the root_node
{xrst_literal
    # BEGIN parent_iota_value_prior
    # END parent_iota_value_prior
}
{xrst_literal
    # BEGIN parent_chi_value_prior
    # END parent_chi_value_prior
}
The mean and standard deviation are only used for the root_node.
The :ref:`create_shift_db-name`
routine replaces them for other nodes.

dage Prior
==========
The following is the dage prior used for the root_node:
{xrst_literal
    # BEGIN parent_dage_prior
    # END parent_dage_prior
}

Child Rate Smoothing
********************
This is the smoothing used for the
random effect for each child of the root_node.
There are no :ref:`glossary@dage` or dtime
priors because there is only one age and one time point in this smoothing.

Value Prior
===========
The following is the value prior used for the children of the root_node:
{xrst_literal
    # BEGIN child_value_prior
    # END child_value_prior
}

Alpha Smoothing
***************
This is the smoothing used for *alpha* which multiplies the income covariate.
There is only one age and one time point in this smoothing
so it does not have dage or dtime priors.

Value Prior
===========
The following is the value prior used for this smoothing:
{xrst_literal
    # BEGIN alpha_value_prior
    # END alpha_value_prior
}
The mean and standard deviation are only used for the root_node.
The create_shift_db
routine replaces them for other nodes.

Checking The Fit
****************
The results of the fit are checked by check_cascade_node
using the :ref:`check_cascade_node@avgint_table`
that was created by the root_node_db routine.
The node_id for each row is replaced by the node_id for the
fit being checked.
routine uses these tables to check that fit against the truth.

{xrst_end no_ode_xam}