Benchmarking (DK §3.10.2)
=========================

Monthly survey values :math:`y_t`, subject to survey error, and annual
totals :math:`x_i` of the true values, known exactly (benchmarks). DK's
model (DK eq. 3.48) is

.. math::

   y_t = \mu_t + \gamma_t + \delta_t w_t + \varepsilon_t + \sigma^s_t \xi^s_t,

with an integrated random walk trend, a dummy seasonal, a calendar effect
whose coefficient changes each January, an irregular and an AR(1) survey
error with known standard deviation :math:`\sigma^s_t`.

The series is arranged as :math:`y_1, \dots, y_{12}, x_1, y_{13}, \dots`,
with the state

.. math::

   \alpha_t = (\mu_t, \dots, \mu_{t-11}, \gamma_t, \dots, \gamma_{t-11},
              \delta_t, \varepsilon_t, \dots, \varepsilon_{t-11}, \xi^s_t)',

so that both kinds of observation are exact linear functions of the state
(H = 0). From December to the benchmark point the transition is the
identity; from the benchmark point to January it is the monthly one with
the yearly change of :math:`\delta`. The system matrices vary over time.

DK give no data; the example simulates them from the model with known
parameters.

Results
-------

The smoothed true values add up to every annual total to within 1e-12. Their
root mean squared error against the simulated truth is 0.53 with the
benchmarks and 0.74 without, against 1.07 for the raw survey values.

Program
-------

.. literalinclude:: ../../example/dk_3_10_2_benchmarking.f90
   :language: fortran

Output
------

.. literalinclude:: output/dk_3_10_2_benchmarking.txt
   :language: text