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.

Diagram elaborated on the basis of: A. Pradera, G. Beliakov, H. Bustince, B. De Baets, “A review of the relationships between implication, negation and aggregation functions from the point of view of material implication”, Information Sciences 329 (2016) 357–380, DOI: 10.1016/j.ins.2015.09.033.

An interactive version of this diagram is available at: https://wgalka.github.io/Means/klasyfikacja_agregacji.html

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

An aggregation function on the unit interval [1] is an increasing mapping $A:[0,1]^n\to[0,1]$, $n \in \mathbb{N}$, $n\ge 2$, satisfying $A(0,\dots,0) = 0$ and $A(1,\dots,1) = 1$. For inputs on another interval $[a,b]\subset\mathbb{R}$, an $[a,b]$-aggregation is an increasing mapping $A:[a,b]^n\to[a,b]$ satisfying $A(a,\dots,a)=a$ and $A(b,\dots,b)=b$.

The following aggregation functions are implemented in this library. Tags describe properties of each formula. averaging means the result lies between the minimum and maximum inputs; conjunctive means it does not exceed the minimum; symmetric means permuting the inputs does not change the result. aggregation interval states the intervals on which the formula is an aggregation function: an increasing map $A:[a,b]^n\to[a,b]$ satisfying $A(a,\dots,a)=a$ and $A(b,\dots,b)=b$. any [a,b] ⊂ ℝ includes intervals with negative values; any [a,b] ⊆ [0,∞) is restricted to nonnegative intervals. [0,1] means only the unit interval. These are mathematical properties; most callable objects do not validate input bounds at runtime. Some tags depend on parameters or the fuzzy measure.

$\mathbf{A_{ch}^{(\mu)}}$ - Choquet integral [1]

Tags: averaging aggregation interval: [0,1] symmetric if μ is symmetric

For an input vector $(x_1,\dots,x_n)\in[0,1]^n$, let $\sigma$ sort its indices so that $x_{\sigma(1)} \leq \dots \leq x_{\sigma(n)}$. Define $A_{\sigma(i)}=\{\sigma(i),\dots,\sigma(n)\}$ and $x_{\sigma(0)}=0$:

\begin{equation*} A_{ch}^{(\mu)}(x_1,\dots,x_n)=\sum_{i=1}^{n}(x_{\sigma(i)}-x_{\sigma(i-1)})\,\mu(A_{\sigma(i)}) \end{equation*}

Available fuzzy measures for the Choquet integral:

Each constructor returns a monotone fuzzy measure with $\mu(\emptyset)=0$ and $\mu(N)=1$.

Uniform/cardinality (uniform_measure(n)): $\mu(A)=|A|/n$; every position has equal singleton weight.
Relative (relative_measure(n)): $\mu(A)=\frac{\sum_{i\in A}i}{\sum_{j=1}^{n}j}$; singleton weights are proportional to 1, 2, ..., n, so later positions receive larger weights.
k-fuzzy/Sugeno (k_fuzzy_measure(weights)): for k ≠ 0, $\mu(A)=\frac{\prod_{i\in A}(1+k w_i)-1}{k}$; for k = 0, $\mu(A)=\sum_{i\in A}w_i$. The parameter k is chosen so $\mu(N)=1$; incompatible singleton weights raise ValueError.
Weighted mean (weighted_measure(weights)): $\mu(A)=\sum_{i\in A}w_i$. The weights must be in [0, 1] and sum to 1; with the Choquet integral this gives a weighted arithmetic mean.
OWA (owa_measure(weights)): $\mu(A)=\sum_{j=1}^{|A|}w_j$. The normalized weights are applied by descending value rank, so the first belongs to the largest value; with the Choquet integral this capacity gives the OWA aggregation.
Exponential cardinality: $\mu(A)=(|A|/n)^q$ is not implemented yet; choosing q is future work.

The constructors are in aggregationslib.aggregation; Decimal equivalents are available in aggregationslib.decimalaggregations.

With the default cardinality measure $\mu(A)=|A|/n$, this is the arithmetic mean, $A_{ch}^{(\mu)}(x_1,\dots,x_n)=\frac{1}{n}\sum_{i=1}^{n}x_i$.

For the quasi-overlap generalization from Equation (9), pass a two-argument function O(x, mu) as quasi_overlap to choquet_integral or A_ch. The default remains the classical product-based Choquet integral.

$\mathbf{A_{sg}^{(\mu)}}$ - Sugeno integral [1]

Tags: averaging aggregation interval: [0,1] symmetric if μ is symmetric

For an input vector $(x_1,\dots,x_n)\in[0,1]^n$, let $\sigma$ sort its indices so that $x_{\sigma(1)} \leq \dots \leq x_{\sigma(n)}$, and define $A_{\sigma(i)}=\{\sigma(i),\dots,\sigma(n)\}$:

\begin{equation*} A_{sg}^{(\mu)}(x_1,\dots,x_n)=\max_{1\leq i\leq n}\min(x_{\sigma(i)},\mu(A_{\sigma(i)})) \end{equation*}

Available fuzzy measures for the Sugeno integral:

Each constructor returns a monotone fuzzy measure with $\mu(\emptyset)=0$ and $\mu(N)=1$. The same constructors are available in aggregationslib.aggregation and aggregationslib.decimalaggregations.

Uniform/cardinality (uniform_measure(n)): $\mu(A)=|A|/n$; every position has equal singleton weight.
Relative (relative_measure(n)): $\mu(A)=\frac{\sum_{i\in A}i}{\sum_{j=1}^{n}j}$; singleton weights are proportional to 1, 2, ..., n, so later positions receive larger weights.
k-fuzzy/Sugeno (k_fuzzy_measure(weights)): for k ≠ 0, $\mu(A)=\frac{\prod_{i\in A}(1+k w_i)-1}{k}$; for k = 0, $\mu(A)=\sum_{i\in A}w_i$. The parameter k is chosen so $\mu(N)=1$; incompatible singleton weights raise ValueError.
Weighted mean (weighted_measure(weights)): $\mu(A)=\sum_{i\in A}w_i$. The weights must be in [0, 1] and sum to 1. In the Sugeno integral this measure is used in the max–min formula; it does not turn Sugeno into a weighted arithmetic mean.
OWA (owa_measure(weights)): $\mu(A)=\sum_{j=1}^{|A|}w_j$. The normalized weights are applied by descending value rank, so the first belongs to the largest value. In the Sugeno integral this capacity is used in the max–min formula; the result is not the OWA aggregation.
Exponential cardinality: $\mu(A)=(|A|/n)^q$ is not implemented yet; choosing q is future work.

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

Tags: averaging symmetric aggregation interval: any [a,b] ⊂ ℝ

\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

Tags: averaging symmetric aggregation interval: any [a,b] ⊆ [0,∞)

\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

Tags: conjunctive symmetric aggregation interval: [0,1]

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

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

Tags: averaging symmetric aggregation interval: any [a,b] ⊆ [0,∞)

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

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

Tags: averaging symmetric aggregation interval: any [a,b] ⊆ [0,∞)

\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

Tags: averaging symmetric aggregation interval: any [a,b] ⊆ [0,∞) r ≠ 0

\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

Tags: averaging symmetric aggregation interval: any [a,b] ⊂ ℝ r ≠ 0

\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

Tags: averaging symmetric aggregation interval: any [a,b] ⊆ [0,∞) if 0 ≤ r ≤ 1

\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

Tags: averaging symmetric conjunctive if p = 0 aggregation interval: any [a,b] ⊂ ℝ p ∈ [0,1]

\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

Tags: averaging symmetric disjunctive if p = 0 aggregation interval: any [a,b] ⊂ ℝ p ∈ [0,1]

\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

Tags: averaging symmetric aggregation interval: any [a,b] ⊂ ℝ

\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

Tags: averaging symmetric aggregation interval: any [a,b] ⊂ ℝ

\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

Tags: averaging symmetric aggregation interval: any [a,b] ⊂ ℝ p ∈ ℕ, p ≥ 1

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]

Tags: averaging (continuous formula) symmetric aggregation interval (after continuous extension): any [a,b] ⊂ (0,∞)

\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*}

The formula has a continuous averaging extension when input values coincide, but the current implementation does not handle ties: a zero denominator makes it return 0.

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

Tags: aggregation interval: common interval of A₁ and A₂ averaging if both inputs are averaging p ∈ [0,1]

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.

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Means

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Example2:

Aggregation functions

OWA

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

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