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Lasso on Fixed Effects: Example and Test
########################################

Model
*****
We are given a set of times
:math:`\{ t_i \W{:} i = 0 , \ldots , N-1 \}` and

.. math::
    :nowrap:

    \begin{eqnarray}
        q( \theta, s )
        & = &
        \theta_0 (s / N)
        + \theta_1 \sin ( 2 \pi s  )
        + \theta_2 \cos ( 2 \pi s )
        \\
        z_i & = & q( \theta , t_i ) + e_i
        \\
        \B{p} ( e_i | \theta ) & \sim & \B{N} ( 0, \sigma )
    \end{eqnarray}

The idea in Lasso is that one or more of the components of
:math:`\theta` are zero and using the Laplace prior
we can recover this fact.
We use :math:`\B{L} ( \mu , \sigma )` to denote the Laplace distribution
with mean :math:`\mu` and standard deviation :math:`\sigma`.

.. math::

    \B{p} ( \theta ) \sim \B{L} ( 0 , \delta )

The corresponding fixed likelihood
:ref:`g(theta)<theory@Fixed Likelihood, g(theta)>`
is

.. math::

    g( \theta ) =
    \sum_{i=0}^{N-1} \left[
        \log ( \sigma \sqrt{2 \pi} )
        +
        \left( \frac{ z_i - q( \theta , t_i ) }{2 \sigma} \right)^2
    \right]
    +
    \sum_{j=0}^2 \left[
        \log \left( \delta \sqrt{2} \right)
        +
        \sqrt{2} \; \left| \frac{\theta_j}{\delta} \right|
    \right]

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