===================================================
Advanced Confidence Intervals and Chi-square maps
===================================================

Having good estimates for uncertainties in fitted parameters is important
for any scientific analysis.  One of principle attractions to using the
Levenberg-Marquardt algorithm that is the default fitting mechanism with
the :func:`minimize` function is that it will automatically calculate
estimates of parameter uncertainties and correlations.  This is very
convenient and generally reliable, but it should be made clear that the
basic assumptions made when these uncertainties are estimated are not
always perfect.  Unfortunately, it is sometimes difficult to tell when this
is the case.

It is therefore fairly common to see analyses that include explicit
exploration of Parameter values away from their best-fit solution, in order
to determine the degree of confidence in the best-fit values. Larch
provides two main functions to help explore such cases.  To be sure, they
are much slower than the automatic estimation of the uncertainties.  For
many (perhaps most) cases, they do not provide much better insight than the
automatic method.

.. function:: confidence_intervals(minimizer, sigmas=(1, 2, 3), prob_func=None, maxiter=200)

    Calculate confidence intervals for the parameters from a given fit.

    :param minimizer: the minimizer object returned by :func:`minimize`.
    :param sigmas: list of sigma-levels to find parameter values for.
    :param prob_func: ``None`` or callable function to calculate the
    	       probability from the optimized chi-square. By default,
	       :func:`f_compare`, the standard F-test, is used.
    :returns:  a dictionary of parameter names, with each value containing
               a list of (sigma, value) pairs.

This function will adjust the value for each parameter, re-optimizing the
other parameters until it finds the parameter values that increase sigma by
the levels indicated.

.. function:: confidence_report(conf_values)

    Convert the output of :func:`confidence_intervals` into a printable report.

    :param conf_values: confidence values returned by :func:`confidence_intervals`.
    :returns:  a string containing the report, which can be printed or stored.

.. function:: f_test(ndata, nparams, chisquare, chisquare0, nfix=1)

    Returns the standard F-test value for the probability that one fit is
    better than another.


Confidence interval, Example 1
-------------------------------

Let's begin with a shortened version of the first example from the previous
section.

.. literalinclude:: ../../examples/fitting/doc_example_conf1.lar

The printed output from ``fit_report(params)`` will include this::

    [[Variables]]
       amp            =  12.129867 +/- 0.124722 (init=  5.000000)
       cen            =  1.476822 +/- 0.017266 (init=  2.000000)
       off            =  0.998814 +/- 0.007131 (init=  0.000000)
       wid            =  2.022301 +/- 0.016938 (init=  1.000000)

    [[Correlations]]     (unreported correlations are <  0.100)
       amp, off             = -0.861
       amp, wid             =  0.808
       off, wid             = -0.692
    =======================================================

while the output from the much more explicit search done in
:func:`confidence_intervals` and reported by :func:`confidence_report` will be::


    # Confidence Interval Report
    # Sigmas:          -3         -2         -1          0          1          2          3
    # Percentiles:  -99.730    -95.450    -68.269      0.000     68.269     95.450     99.730
    #==========================================================================================
     amp             11.755      11.88     12.005      12.13     12.256     12.385     12.517
        -best      -0.37459   -0.24967   -0.12455          0    0.12604    0.25555    0.38759
     wid             1.9707     1.9885     2.0053     2.0223     2.0395     2.0569     2.0749
        -best     -0.051552  -0.033784  -0.016975          0   0.017156   0.034567   0.052626
     off             0.9766    0.98468     0.9912    0.99881     1.0064     1.0128     1.0209
        -best     -0.022212  -0.014138  -0.0076156          0  0.0075667   0.013987   0.022049
     cen             1.4248     1.4423     1.4596     1.4768     1.4941     1.5113     1.5289
        -best     -0.052057   -0.03452  -0.017248          0   0.017249   0.034524   0.052068

The automatic error estimates given from :func:`minimize` are meant to be
1-:math:`\sigma` uncertainties.   Comparing the two methods we find:

    ============= =============== =================================== ===============================
     Parameter      Best Value      Automatic 1-:math:`\sigma` value  Explicit 1-:math:`\sigma` value
    ============= =============== =================================== ===============================
      amp           12.1299         +/- 0.1247                         +0.1267, -0.1246
      cen            1.4768         +/- 0.0173                         +0.0172, -0.0172
      off            0.9988         +/- 0.0071                         +0.0076, -0.0076
      cen            2.0223         +/- 0.0169                         +0.0172, -0.0170
    ============= =============== =================================== ===============================

which seems to justify the use of the automated method.  The uncertainties
found from the more thorough exploration shows symmetric uncertainties,
even out to the 3-:math:`\sigma` level, and of the 4 1-:math:`\sigma`
uncertainties, 3 are within 2%, and the worst agreement, for the smallest
uncertainty is within 7%.    It also shows that the scaling of
uncertainties is fairly linear with :math:`\sigma`:  the 3-:math:`\sigma`
values are approximately 3 times the 1-:math:`\sigma` values.


Confidence interval, Example 2
-------------------------------

Of course, there are more challenging cases than the one above.  A double
exponential function is one such example, so we start with a fit to mock
data

.. literalinclude:: ../../examples/fitting/doc_example_conf2.lar


The resulting statistics report with the automated uncertainties is::

    ===================== FIT RESULTS =====================
    [[Statistics]]    Fit succeeded,  method = 'leastsq'.
       Message from fit    = Fit succeeded.
       npts, nvarys, nfree = 101, 4, 97
       nfev (func calls)   = 36
       chi_square          = 0.191322
       reduced chi_square  = 0.001972

    [[Variables]]
       a1             =  2.828857 +/- 0.149776 (init=  3.500000)
       a2             = -4.819553 +/- 0.159495 (init= -9.500000)
       t1             =  1.878519 +/- 0.100212 (init=  3.000000)
       t2             =  9.270866 +/- 0.309035 (init=  15.000000)

    [[Correlations]]     (unreported correlations are <  0.100)
       a2, t2               =  0.991
       a1, a2               = -0.991
       a1, t2               = -0.988
       a2, t1               = -0.968
       t1, t2               = -0.937
       a1, t1               =  0.935
    =======================================================

You can see that the correlations between **all 6 pairs of variables** is
above 90%.  The resulting plot of the best-fit looks fairly reasonable:

.. _fit_conf1:

.. figure::  ../_images/fit_example_conf2.png
    :target: ../_images/fit_example_conf2.png
    :width: 65%
    :align: center

    Fit to double exponential function.


But if we we ask for the more thorough investigation of the confidence
intervals in these parameters with::

    conf_int = confidence_intervals(minout)
    print confidence_report(conf_int)

the resulting report is::

    # Confidence Interval Report
    # Sigmas:          -3         -2         -1          0          1          2          3
    # Percentiles:  -99.730    -95.450    -68.269      0.000     68.269     95.450     99.730
    #==========================================================================================
     a1              2.4622     2.5704      2.691     2.8289     2.9936     3.1985      3.467
        -best      -0.36665   -0.25842   -0.13785          0    0.16472    0.36968    0.63812
     a2             -5.4926    -5.2111    -4.9945    -4.8196    -4.6726     -4.544     -4.429
        -best      -0.67309   -0.39152   -0.17499          0    0.14698    0.27553    0.39058
     t2              8.2604     8.6221     8.9556     9.2709     9.5761     9.8776     10.182
        -best       -1.0105   -0.64882   -0.31531          0    0.30526    0.60676    0.91132
     t1              1.6107     1.6942     1.7827     1.8785     1.9841     2.1049     2.2456
        -best      -0.26785   -0.18436   -0.09583          0    0.10553    0.22637    0.36709


Now can see more asymmetric uncertainty values, specifically that the
-n-:math:`\sigma` and +n-:math:`\sigma` are different, and don't seem to be
linear in n.  Comparing the 1-:math:`\sigma` levels between the automated
and explicit methods as we did above, we now have

    ============= =============== =================================== ===============================
     Parameter      Best Value      Automatic 1-:math:`\sigma` value  Explicit 1-:math:`\sigma` value
    ============= =============== =================================== ===============================
      a1            2.8289         +/- 0.1498                           +0.1647, -0.1379
      a2           -4.8196         +/- 0.1595                           +0.1470, -0.1750
      t1            9.2709         +/- 0.1002                           +0.1055, -0.0958
      t2            1.8785         +/- 0.3090                           +0.3053, -0.3153
    ============= =============== =================================== ===============================

In fairness, the automated values don't look too bad, given that they
cannot reflect asymmetric uncertainties.  But, like the reported
correlations, the full report above hints at a less than ideal case.

Chi-square Maps
------------------

We can further explore the correlation between pairs of variables by making
and visualizing a map of the chi-square (:math:`\chi^2`) statistic.  This
can be helpful to determine if the automatically calculated uncertainties
and correlation are reasonable, and to look for pathological cases.  The
:func:`chi2_map` function will calculate a map of :math:`\chi^2` for a pair
of variable parameters by brute force.


.. function:: chi2_map(minimizer, xname, yname, nx=11, ny=11, sigmas=5, xrange=None, yrange=None)

    Return a chi-square map for two parameters in a fit

    :param minimizer: the minimizer object returned by :func:`minimize`.
    :param xname:    name of variable for x-axis
    :param yname:    name of variable for y-axis
    :param nx:       number of steps in x [11]
    :param ny:       number of steps in y [11]
    :param sigmas:   extent of x, y values to calculate, in :math:`\sigma`
    :param xrange:   explicit x range of calculations [x.best +/- ``sigmas`` * x.stderr]
    :param yrange:   explicit y range of calculations [y.best +/- ``sigmas`` * y.stderr]

    :returns:        a tuple of (x_vals, y_vals, chi2_map)

    The ``xrange`` and ``yrange`` arguments can be used to fully dictate the x
    and y values to use.  By default, the x and y values are automatically
    determined from ``nx``, ``ny``, and ``sigmas``, with the ``sigmas`` argument
    sets how far from the best value to extend the ranges.


As an example usage, we return to the first example of the "well-behaved"
fit, and run :func:`chi2_map` on a pair of variables with low correlation
and a pair with high correlation::

     x1, y1, chi2_ampcen = chi2_map(mout, 'amp', 'cen', nx=21, ny=21)
     x2, y2, chi2_ampwid = chi2_map(mout, 'amp', 'wid', nx=21, ny=21)
     contour(chi2_ampcen, x=x1, y=y1, title='Correlation(amp, cen)')
     contour(chi2_ampwid, x=x2, y=y2, win=2, title='Correlation(amp, wid)')


with the resulting Chi-square maps looking like this:

.. subfigstart::

.. _fig-chi2map1a:

.. figure::  ../_images/fit_example_conf3a.png
    :target: ../_images/fit_example_conf3a.png
    :width: 100%
    :align: center

.. _fig-chi2map1b:

.. figure::  ../_images/fit_example_conf3b.png
    :target: ../_images/fit_example_conf3b.png
    :width: 100%
    :align: center

.. subfigend::
    :width: 0.45
    :label: fig-chi2map1

    Chi-square maps for two pairs of variables for the fit to Gaussian
    data.  With the best-fit chi-square value of 0.5134, the contour map
    for relatively uncorrelated parameters *amp* and *cen* is shown on the
    left, while that for the more highlycorrelated parameters *amp* and
    *wid* is shown on the right.


The circular map for the uncorrelated parameters *amp* and *cen*
and the elliptical map for the highly correlated parameters *amp* and *wid*
are exactly as would be expected, and what the automated estimate of
uncertainties and correlations assumes.

But now, if we turn to the more pathological case of the double
exponential, we calculate the chi-square maps as::

   x1, y1, c1 = chi2_map(minout, 'a1', 'a2', nx=25, ny=25)
   x2, y2, c2 = chi2_map(minout, 'a1', 't1', nx=25, ny=25)
   contour(c1, x=x1, y=y1,        title='Correlation(a1, a2)')
   contour(c2, x=x2, y=y2, win=2, title='Correlation(a1, t1)')


with the resulting contour plots:



.. subfigstart::

.. _fig-chi2map2a:

.. figure::  ../_images/fit_example_conf4a.png
    :target: ../_images/fit_example_conf4a.png
    :width: 100%
    :align: center

.. _fig-chi2map2b:

.. figure::  ../_images/fit_example_conf4b.png
    :target: ../_images/fit_example_conf4b.png
    :width: 100%
    :align: center

.. subfigend::
    :width: 0.45
    :label: fig-chi2map2

    Chi-square maps for two pairs of variables for the fit to Double
    Exponential.  With the best-fit chi-square value of 0.1913, the contour
    map for *a1 and *a2* is shown on the left, while that for *a1* and *t2*
    is shown on the right.



Here, the values of chi-square quickly grow very large away from the ideal
fit.  More importantly, the maps are not remotely elliptical.

Finally, it should be emphasized that while all the tests in this section
are informative, they are also fairly slow, re-running the fits many times.
It is probably safe to rely on the automatic calculations of uncertainties
and correlations, and use these methods on occasions of extremely high
correlation, or when nearing a final analysis.
