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lines 5-78 of file: example/user/ill_condition.py
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# {xrst_begin user_ill_condition.py}
# {xrst_comment_ch #}
#
# An Ill Conditioned Example Where Re-Scaling is Helpful
# ######################################################
#
# Notation
# ********
#
# .. csv-table::
#     :widths: auto
#
#     :math:`\bar{\chi}`,measured value for excess mortality at age zero
#     :math:`\sigma`,std for the measurement noise
#     :math:`\delta`,std for age difference for :math:`\chi`
#     :math:`\chi_0`,estimate for :math:`\chi` at age zero
#     :math:`\chi_1`,estimate for :math:`\chi` at age 100
#
# Objective
# *********
# For this case the negative log likelihood is, not counting terms
# that are constant w.r.t :math:`( \chi_0, \chi_1 )`,
#
# .. math::
#
#  f( \chi_0 , \chi_1 )
#  =
#  \frac{1}{2} \left( \frac{\chi_0 - \bar{\chi}}{\sigma} \right)^2
#  +
#  \frac{1}{2} \left( \frac{\chi_1 - \chi_0}{\delta} \right)^2
#
# Derivative
# **********
# The partial of the objective w.r.t. :math:`\chi_0` is
#
# .. math::
#
#  \partial_0 f ( \chi_0 , \chi_1 )
#  =
#  \frac{\chi_0 - \bar{\chi}}{ \sigma^2 }
#  -
#  \frac{\chi_1 - \chi_0}{\delta^2}
#
# The partial of the objective w.r.t. :math:`\chi_1` is
#
# .. math::
#
#  \partial_1 f ( \chi_0 , \chi_1 )
#  =
#  \frac{\chi_1 - \chi_0}{\delta^2}
#
# Ill-Conditioning
# ****************
# If :math:`\sigma` is much smaller than :math:`\delta`,
# then the optimization problem will weight the measurement equation
# :math:`\chi_0 - \bar{\chi} = 0` much more than the difference equation
# :math:`\chi_1 - \chi_0 = 0`.
# If :math:`\sigma` is much larger than :math:`\delta`,
# then the optimization problem will weight the difference equation
# much more than the measurement equation.
#
# Scaling
# *******
# Note that the initial scaling, in the example below, uses the
# mean values for :math:`\chi_0`, :math:`\chi_1` which are zero.
#
# Source Code
# ***********
# {xrst_literal
#     BEGIN PYTHON
#     END PYTHON
# }
#
# {xrst_end user_ill_condition.py}