148 lines
4.0 KiB
Markdown
148 lines
4.0 KiB
Markdown
---
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tags:
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- destiny/uncertain
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- topic/estimating
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- topic/math
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- type/idea
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- authorship/llm
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title: Functional Estimating
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---
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# Functional Estimating
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Suppose a function
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$$
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f: ℝ^n \to ℝ
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$$
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That is, $f$ takes $n$ real parameters (an $n$-tuple) and returns a single real number.
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Suppose we want to determine the number of specified parameters necessary to
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reduce uncertainty of the output to some acceptable range.
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There are multiple possible approaches with varying levels of appropriateness.
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### [Multisets](https://en.wikipedia.org/wiki/Multiset)
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Focused on statistics, we determine we want a _population_ of all values of $f$.
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A true _set_ is insufficient for this use case as duplicate values are not respected.
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$$
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g: ℝ^m \to (\text{multisets of real numbers}) \\
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g(x_1,...,x_m) = \{\!\{ f(x_1,...,x_m,y_m+1,...y_n) | (y_m+1,...y_n) \in ℝ^{n-m} \}\!\}
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$$
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$g(x_1,...,x_m)$ is the uncountably infinite length _multiset_ of all possible
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outputs of $f(x_1,...,x_n)$
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### [Partial Application](https://en.wikipedia.org/wiki/Partial_application)
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It seems that the more common way of doing what I'm trying to would be to
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partially apply $(x_1,...,x_m)$ to $f(x_1,...,x_n)$
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$$
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f: ( X × Y × Z ) \to N \\
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\text{partial}(f): (Y × Z) \to N
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$$
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then work with the level set of $f_{\text{partial}}$
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<https://en.wikipedia.org/wiki/Currying>
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### [Pushforward Measure](https://en.wikipedia.org/wiki/Pushforward_measure)
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Start with a function $f:X \to Y$.
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Think of $X$ as the domain and $Y$ as the codomain.
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Pick a [measure](https://en.wikipedia.org/wiki/Measure_(mathematics)) $\mu$ on $X$.
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In general, a measure $\mu$ is a systematic way of assigning a "weight" to subsets of $X$.
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In continuous scenarios,
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$\mu$ might be something like the length/area/volume measure (Lebesgue measure)
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or a probability distribution
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(which also assigns a total measure of 1 to the entire domain).
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In discrete scenarios, $\mu$ is a count measure:
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each point in $X$ has weight 1,
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so the measure of a finite set is just how many points it has.
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Let $f : X \to Y$ be a measurable function and $\mu$ a measure on $X$.
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The pushforward measure $f_{\ast}\mu$ is the measure on $Y$ defined by
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$$
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\forall B \subseteq Y \text{(measurable)}, \quad
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(f_{\ast}\mu)(B) = \mu(\{x \in X : f(x) \in B\})
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$$
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In other words, "The measure of a subset $B$ of the codomain ($Y$)
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equals the measure (in $X$) of all points that map into $B$."
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$\mu(f^{−1}(\{y\})) =$ the weight (in this case count) assigned to $Y$
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If the domain $X$ is finite and $\mu$ is just counting measure:
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The pushforward measure $f_{\ast}\mu$ on the codomain $Y$ literally says something like:
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$(f_{\ast}\mu)(\{y\}) =$ the number of times $Y$ appears as an output.
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Equivalent to the multiset $\{\!\{ f(x): x \in X \}\!\}$
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When $X$ is infinite (countably or uncountably so),
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we move beyond plain
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"multisets" to a more general concept of "measure on $Y$."
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But it's the same intuitive idea:
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Instead of a simple integer count of duplicates, we have a "weight"
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(which could be infinite or fractional, depending on the measure).
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$μ(f−1({y}))$ is how "large" or "frequent" the set of points mapping to $Y$ is,
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according to $\mu$.
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## Abstractions
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There are two competing abstractions for measure analysis of the project price function:
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### Ignoring Aleatory Uncertainty
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We assume that price is certain for any defined scope, that _any_ variation in
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price reflects influence of a measurable parameter.
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```none
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f(a_1,...,*,...,a_n)
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_ ├
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│ ├ /
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█ ├ ______/
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█ ├ __/
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│ ├ _/
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▔ ├
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└──┴──┴──┴──┴──┴─x_k
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```
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$$
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f: ℝ^n \to ℝ
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$$
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**Pros:**
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* In theory far simpler.
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### Accounting for Aleatory Uncertainty
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We assume that price is subject to some inherent randomness
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```none
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f(a_1,...,*,...,a_n)
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┬ ├ ┬ ┬
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│ ├ ┬ │ █
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█ ├ ┬ █ █ │
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█ ├ ▁ ┿ │ ┴ ┴
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│ ├ ┿ ┴ ┴
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▔ ├ ▔
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└──┴──┴──┴──┴──┴─x_k
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```
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**Pros:**
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* Likely more accurate
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