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[DOC] docstring with mathematical description for QPD_Empirical #255

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Apr 19, 2024
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[DOC] docstring with mathematical description for QPD_Empirical #255

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2ff7a4e
Update qpd_empirical.py
fkiraly Apr 18, 2024
5da33e4
Update qpd_empirical.py
fkiraly Apr 18, 2024
7d9bab3
fix formula and formatting
fkiraly Apr 18, 2024
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39 changes: 34 additions & 5 deletions skpro/distributions/qpd_empirical.py
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class QPD_Empirical(Empirical):
"""Empirical quantile parametrized distribution.

This distribution is parameterized by a set of quantile points.

todo: add docstr
r"""Empirical quantile parametrized distribution.

This distribution is parameterized by a set of quantile points and quantiles,
quantiles :math:`q_1, q_2, \dots, q_N`
at quantile points :math:`p_1, p_2, \dots, p_N`,
with :math:`0 \le p_1 < p_2 < \dots < p_N \le 1`.

It represents a distribution with piecewise constant CDF and quantile function,
the unique distribution satisfying:

* the support is :math:`[q_1, q_N]`
* for any quantile point :math:`p \in [p_1, p_N]`, it holds that
:math:`\mbox{ppf}(p)` = :math:`\mbox{ppf}(p_i)`,
where :math:`i` is the index minimizing :math:`|p_i - p|`,
in all cases where this minimizer is unique.

In vernacular terms, the quantile function agrees with the quantiles prescribed by
:math:`q_i` at the quantile points :math:`p_i`, and for other quantile points
agrees with the value at the nearest quantile point.

In explicit terms, the distribution is an empirical distribution (sum-of-diracs),
supported at the quantiles :math:`q_1, q_2, \dots, q_N`,
with weights :math:`w_1, w_2, \dots, w_N`
such that :math:`w_i = (p_{i+1} - p_{i-1})/2` for :math:`1 = 1, \dots, N`,
where we define :math:`p_0 = -p_1` and :math:`p_{N+1} = 2 - p_N`.
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Just one question:
you say: such that w_i = (p_{i+1} - p_{i-1})/2 for 1 = 1, \dots, N and then you define p_0 and p_{N+1}. Why you define them? How are these two values playing a role here?

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@fkiraly fkiraly Apr 18, 2024 •
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$w_i$ is defined in terms of $p_{i+1}$ and $p_{i-1}$ for $i= 1\dots N$, and $p_i$ have been defined above also only for $i= 1\dots N$.

So we need to treat $w_1$, $w_N$, either by writing them down explicitly, or by writing equivalent values for $p_0$, $p_{N+1}$. I thought the latter was clearer, in terms how the boundary treatment looks like?

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Ah sorry now I see ;-)

thanks for the explanation. I didn't check that there it was a p_{i-1}) and we have to define p_0 (same for p_{i+1}

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@fkiraly fkiraly Apr 18, 2024 •
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the $w_i$ should sum to 1, hopefully. The weights should make it that the steps in the ppf are exactly in the middle between two adjacent $p_i$.


Formally, the distribution is parametrized by the quantiles :math:`q_i`
and the quantile points :math:`p_i`, not by the quantiles and weights :math:`w_i`,
so it is distinct from the empirical distribution (``skpro`` ``Empirical``),
as a parameterized distribution,
by being quantile parameterized and not sample parameterized.

However, it is equivalent, as an unparameterized distribution,
to an ``Empirical`` distribution with weights and nodes given as above.

Parameters
----------
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