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{xrst_begin smooth_grid_table}

The Smooth Grid Table
#####################

Purpose
*******
Each row of the ``smooth_grid`` table specifies an
age, time, value prior, age difference prior, and time difference prior
for a single :ref:`model_variable<model_variables-name>` .
(The difference priors represent the smoothing in a mathematical sense.)
A set of rows with the same *smooth_id* ,
defines a statistical prior for a function of age and time.

smooth_grid_id
**************
This column has type ``integer`` and is the primary key for
the ``smooth_grid`` table.
Its initial value is zero, and it increments by one for each row.

smooth_id
*********
This column has type ``integer`` and is the primary key for
the :ref:`smooth_table-name` .
Each *smooth_id* value corresponds to a different
statistical prior for a function of age and time.

null
====
If *smooth_id* is null, the corresponding row of the
smooth grid table is not used.
This can be useful for deleting a row or column corresponding to a
*smooth_id* .

age_id
******
This column has type ``integer`` and is a
:ref:`age_table@age_id` in the age table.
The *age_id* value corresponds to a specific age.

time_id
*******
This column has type ``integer`` and is a
:ref:`time_table@time_id` in the time table.
Each *time_id* value corresponds to a specific time.

value_prior_id
**************
This column has type ``integer`` and is the
:ref:`prior_table@prior_id` for the function's values
at the corresponding age and time.

null
====
If *value_prior_id* is null,
:ref:`smooth_grid_table@const_value`
must not be null and is used for the value prior
at the corresponding age and time.

dage_prior_id
*************
This column has type ``integer`` and is the
:ref:`prior_table@prior_id` for the function's
forward difference w.r.t age
at the corresponding age and time; see age difference function
:ref:`smooth_dage@A^s` .
The corresponding
:ref:`prior_table@density_id`
cannot specify a
:ref:`censored density<density_table@Notation@Censored>` .

Maximum Age
===========
For each time grid point in the smoothing,
the *dage_prior_id* at the maximum age is not used.

null
====
If *dage_prior_id* is null, the corresponding prior is
a uniform from minus to plus infinity.

dtime_prior_id
**************
This column has type ``integer`` and is the
:ref:`prior_table@prior_id` for the function's
forward difference w.r.t time
at the corresponding age and time; see time difference function
:ref:`smooth_dtime@T^s` .
The corresponding
:ref:`prior_table@density_id`
cannot specify a
:ref:`censored density<density_table@Notation@Censored>` .

Maximum Age
===========
For each age grid point in the smoothing,
the *dtime_prior_id* at the maximum time is not used.

null
====
If *dtime_prior_id* is null, the corresponding prior is
a uniform from minus to plus infinity.

const_value
***********
This column has type ``real``
and specifies a specific value for a function using this smoothing
at this age and time (when it is not null).
This case is equivalent to
:ref:`prior_table@lower` ,
:ref:`prior_table@upper` , and
:ref:`prior_table@mean` , equal to *const_value*
and
:ref:`density<prior_table@density_id>` is ``uniform`` .

null
====
If *const_value* is null,
:ref:`value_prior_id<smooth_grid_table-name>` must not be null.
If *const_value* is not null,
*value_prior_id* must be null.

Rectangular Grid
****************
For each fixed value of *smooth_id*
the set of *age_id* , *time_id* pairs is rectangular.
In other words,
if an *age_id* value appears,
it appears once with each *time_id*
that appears with this *smooth_id* .
This also means,
if a *time_id* value appears,
it appears once with each *age_id*
that appears with this *smooth_id* .
{xrst_toc_hidden
    example/table/smooth_grid_table.py
}
Example
*******
The file :ref:`smooth_grid_table.py-name`
contains an example ``smooth`` table.

{xrst_end smooth_grid_table}