Means

Means, Aggregation functions…

Documentation:

English: https://wgalka.github.io/Means/en/
Polski: https://wgalka.github.io/Means/pl/

Aggregations functions are: - Conjuctive - the final aggregated value will always be influenced by the smallest value among the inputs. - Disjnuctive - the aggregated value will always be influenced by the largest value among the inputs.

Example 1:

# example data
data = [0.2, 0.6, 0.7]
# configure function parameters
func1 = A_amn(p=0.5)
# use aggregation funciton
print(func1(data))

# Combine two aggregations - arithmetic mean and minimum
func2 = Combine2Aggregations(A_ar(), min)
# use combination of aggregation funciton
print(func2(data))

Example2:

To get information about aggregation function you can use __str__() or ‘repr()’ methods.

func1 = A_amn(p=0.5)
print(func1)
>>>A_amn(0.5)

func2 = Combine2Aggregations(A_ar(), A_md())
print(func2)
>>>A_armd

func3 = Combine2Aggregations(A_ar(), A_pw(r=3))
print(func3.__repr__()) # function parameters are printed in order: func1, func2
>>>A_arpw(r=3)

Aggregation functions

Aggregation function [1] is a mapping $A:[0,1]^n\to[0,1]$, $n \in \mathbb{N}$, $n\ge 2$ which is is increasing and fulfills two boundary conditions, namely $A(0,\dots,0) = 0$, $A(1,\dots,1) = 1$.

The floowoing aggregation functions are implemented in this library:

$\mathbf{A_{ar}}$ - Arithmetic mean

\begin{equation*} A_{ar}(x_1,...,x_n)=\frac{1}{n}\sum \limits_{k=1}^{n} x_k \end{equation*}

$\mathbf{A_{qd}}$ - Quadratic mean

\begin{equation*} A_{qd}(x_1,...,x_n)=\sqrt{\frac{1}{n}\sum \limits_{k=1}^{n} x_k^2} \end{equation*}

$\mathbf{A_{pr}}$ - Product aggregation

\begin{equation*} A_{pr}(x_1, x_2, \dots, x_n) = \prod_{k=1}^n x_k \end{equation*}

$\mathbf{A_{gm}}$ - Geometric mean

\begin{equation*} A_{gm}(x_1,...,x_n)=\sqrt[n]{\prod \limits_{k=1}^{n} x_k} \end{equation*}

$\mathbf{A_{hm}}$ - Harmonic mean

\begin{equation*} A_{hm}(x_1,...,x_n) = \begin{cases} 0, & \text{if } \exists k,\, 1 \leq k \leq n : x_k = 0 \\ \frac{n}{\sum\limits_{k=1}^{n} \frac{1}{x_k}}, & \text{otherwise} \end{cases} \end{equation*}

$\mathbf{A_{pw}^{(r)}}$ - Power mean

\begin{equation*} A_{pw}^{(r)}(x_1,...,x_n) = \begin{cases} 0, & r < 0, \; \exists k,\, 1 \leq k \leq n : x_k = 0 \\ \Big(\frac{1}{n} \sum\limits_{k=1}^{n} x_k^r \Big)^{\frac{1}{r}}, & \text{otherwise} \end{cases} \end{equation*}

special cases of power means $A_{\text{pw}}^r$ [1] are: - Arithmetic Mean where r=1; - Quadratic Mean where r=2; - Harmonic Mean where r=-1; - Geometric Mean obtained as the limit $\lim_{r \to 0} A_{\text{pw}}^r$.

$\mathbf{A_{ex}^{(r)}}$ - Exponential mean

\begin{equation*} A_{ex}^{(r)}(x_1,...,x_n)= \frac{1}{r}\ln \Big(\frac{1}{n} \sum \limits_{k=1}^{n} e^{rx_k}\Big), \text{where } r \in \mathbb{R}, r \neq 0 \end{equation*}

For computer implementations, to avoid numerical instability (overflow when $r x_k > 709.78$ or underflow when all $r x_k < -709.78$), we use a numerically stable Log-Sum-Exp formulation by shifting the exponents by $a_{max} = \max_{j} (r x_j)$:

\begin{equation*} A_{ex}^{(r)}(x_1,...,x_n) = \frac{1}{r} \left[ a_{max} + \ln \left( \sum_{i=1}^n e^{r x_i - a_{max}} \right) - \ln(n) \right] \end{equation*}

$\mathbf{A_{lm}}$ - Lehmer mean

\begin{equation*} A_{lm}(x_1,...,x_n)= \begin{cases} 0, & r\leq 1, \exists_{1 \leq k \leq n} x^{k} = 0 \\ \frac{\sum \limits_{k=1}^{n}x_k^r}{\sum \limits_{k=1}^{n}x_k^{r-1}}, & \text{otherwise} \end{cases} \end{equation*}

$\mathbf{A_{amn}^{(p)}}$ - Arithmetic minimum mean

\begin{equation*} A_{amn}^{(p)}(x_1,...,x_n)=\frac{p}{n}\sum \limits_{k=1}^{n} x_k+ (1-p) \min \limits_{1 \leq k \leq n}x_k, p \in [0, 1] \end{equation*}

$\mathbf{A_{amx}^{(p)}}$ - Arithmetic maximum mean

\begin{equation*} A_{amx}^{(p)}(x_1,...,x_n)=\frac{p}{n}\sum \limits_{k=1}^{n} x_k+ (1-p) \max \limits_{1\leq k \leq n}x_k, p \in [0, 1] \end{equation*}

OWA

In the case of n = 2 (when only two values are aggregated) the calculation of the median and the Olympic aggregation are reduced to the case of calculating the arithmetic mean of the given values.

$\mathbf{A_{md}}$ - Median - ordered weighted aggregation

\begin{equation*} A_{md}(x_1,\dots,x_n) = \begin{cases} y_{(n+1)/2},&\text{if $n$ is odd}\\ \frac{y_{n/2}+y_{(n/2)+1}}{2},&\text{if $n$ is even} \end{cases}, \end{equation*}

$\mathbf{A_{ol}}$ - Olimpic aggregation

\begin{equation*} A_{ol}(x_1,\dots,x_n) = \frac{1}{n-2} \sum \limits_{k=2}^{n-1} y_k, \quad \text{where } \{y_1, \dots, y_n\} = \{x_1, \dots, x_n\},\ y_1 \leq y_2 \leq \dots \leq y_n. \end{equation*}

$\mathbf{A_{oln}^{(p)}}$ - Olimpic aggregation

We can specify how many greatest and smallest records remove

\begin{equation*} A_{oln}^{(p)}(x_1,\dots,x_n) = \frac{1}{n-2p} \sum \limits_{k=p+1}^{n-p} y_k, \quad \text{where } \{y_1, \dots, y_n\} = \{x_1, \dots, x_n\},\ y_1 \leq y_2 \leq \dots \leq y_n, p \in \mathbb{N}, p\geq 1 \end{equation*}

$\mathbf{A_{ln}}$ - Logaritmic aggregation [2]

\begin{align*} A_{ln}\left(x_1, x_2, \ldots, x_n\right)=(n-1) ! \sum_{i=1}^n \frac{x_i}{\prod_{\substack{j=1 \\ j \neq i}}^n \log \left(x_i / x_j\right)} \end{align*}

$\mathbf{A^{(p)}_{A_1,A_2}}$ - Convex combinations of aggregation functions

We can construct covex combination of aggregation functions as $A^{(p)}_{A_1,A_2}=pA_1+(1-p)A_2$, where $A_1, A_2$ are aggregation functions and $p \in [0,1]$.

References

Beliakov, G., Bustince, H., and Calvo, T.: A practical Guide to Averaging Functions. Berlin: Springer Vol. 329, 2016.
Mustonen, Seppo. (2010). Logarithmic mean for several arguments.

Metadata

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