************************
Detrending Light Curves
************************
In some cases, there may appear non-echoed long-term trends in light curves. ``mica2`` can account for those non-echoed 
trends by using a polynomial to do detrending. The order of the polynomial is specified in the parameter file via::
  
  FlagLongtermTrend         0                # Longterm trend in light curves, use a polynomial to fit 
                                             # input the order of the polynomial, e.g.,
                                             # 0, constant  (default)
                                             # 1, linear line 
                                             # 2, conic line
                                             # Use the default if you do not know this.


The default is using a constant and if there are visible non-echoed trends, usually a linear polynomial is sufficient.
``mica2`` outputs the values of the polynomial coefficients in a file ``data/trend.txt_xx``, in which two columns 
represent the best value and its uncertainties. The orders of rows in this file are: first the polynomial coefficients for 
the continuum, followed by those for lines. Then there is a blank row and goes for next dataset.
With these values, the polynomial can be expressed as 

.. math::
  
  f = \sum_i a_i t_i,

where :math:`a_i` is the coefficient and :math:`t_i` is the time node of the light curve data.
Below see an example for detrending light curves of 3C 273 from Zhang et al. (2019, ApJ, 876, 49).

.. figure:: _static/fig_3c273_detrend.jpg
  :scale: 50 %
  :align: center

  Time lag analysis of light curves of 3C 273 with long-term detrending (``FlagLongtermTrend=1``). Dashed 
  lines represent the long-term trends.