@(@\newcommand{\B}[1]{ {\bf #1} }
\newcommand{\R}[1]{ {\rm #1} }
\newcommand{\W}[1]{ \; #1 \; }@)@This is dismod_at-20221105 documentation: Here is a link to its
current documentation
.
Model
The only non-zero model variable for this example is
other cause mortality, @(@
\omega
@)@, for the parent area.
The parent omega grid has two points,
one at age zero and the other at age 100.
The corresponding true values for @(@
\omega
@)@ are
omega_true[0] at age zero and omega_true[1] at age 100.
Data
The integrand for this data is
mtother
; i.e.,
a direct measurement of @(@
\omega
@)@.
Both ages are included in the data for this example.
Both the Gaussian and Log-Gaussian densities
are included.
In addition, both the case where the
meas_std
is above and below the bound specified
by minimum_meas_cv
are included.
Delta
We use the notation
Delta
for
the standard deviation adjusted by the minimum measurement cv.
sigma
We use the notation
sigma
for
the transformed standard deviation.
There are no measurement noise covariate multipliers, so the
adjusted standard deviation is also equal to @(@
\sigma
@)@.
Gaussian Residuals
In the Gaussian case, @(@
\sigma = \Delta
@)@ and
@[@
(y - \mu) / \sigma
@]@,
where @(@
y
@)@ is the measured value and @(@
\mu
@)@ is the
model value for the
average integrand
.
Log-Gaussian Residuals
@[@
\sigma = \log ( y + \eta + \Delta ) - \log( y + \eta )
@]@where @(@
y
@)@ is the measured value
and @(@
\eta
@)@ is the offset in the log transform.
The residual is
@[@
\frac{ \log ( y + \eta ) - \log ( \mu + \eta ) } { \sigma }
@]@
where @(@
\mu
@)@ is the model value for the average integrand.
Value Residual
There are two value residuals, one for @(@
\omega
@)@ at age zero
and the other at age 100.
The density used for the value residuals is Log-Gaussian.
The mean value used in the prior for the value residuals @(@
\mu
@)@ is
omega_mean.
The standard deviation used for the value residuals in
omega_mean * 0.1
The log transformed standard deviation is
@[@
\sigma = \log ( \mu + \eta + \delta ) - \log( \mu + \eta )
@]@
The residual is
@[@
\frac{ \log ( y + \eta ) - \log ( \mu + \eta ) } { \sigma }
@]@
where @(@
y
@)@ is the fit_var_value
for the model variable.
Difference Residual
There is one difference residuals for the difference of
@(@
\omega
@)@ at age zero and age 100.
The density used for the value residuals is Log-Gaussian.
The mean value used in the prior for the difference residual
@(@
\mu = 0
@)@.
The standard deviation used for the difference residual is 0.1.
(This corresponds to a coefficient of variation of @(@
e^{0.1} - 1
@)@.
which is approximately equal to 0.1; i.e., 10 percent.)
The age difference smoothing multiplier prior id
mulstd_dage_prior_id
for this example is null,
so @(@
\delta
@)@ is equal to the standard deviation 0.1.
The residual is
@[@
\frac{ \log ( z + \eta ) - \log ( y + \eta ) - \mu } { \delta }
@]@
where @(@
y
@)@ (@(@
z
@)@)
is the fit_var_value
at age zero (age 100).