A Functional Inspired by Min-Max Normalization

For a function $F: \mathbb{R}^n \mapsto \mathbb{R}$ we can consider the functional

\[\frac{F(x) - F(a)}{F(b) - F(a)}\]

where \(a = \arg\min_x F(x)\) and \(b = \arg\max_x F(x)\). You may recognize this to be the form of min-max normalization. If we restrict $F$ to be a cumulative distribution function then this is also the form of a truncated probability distribution. Both of these notions are useful for certain kinds of modelling problems. For recreation rather than utility, I generalize this functional below.

We can consider a more general functional

\[g \left( f\left( F(x), F(a) \right), f\left( F(b), F(a) \right) \right)\]

where $g$ and $h$ are suitably-defined binary operators.

Taking common operations including addition, subtraction, multiplication, division, exponentiation, and logarithms, I’ve tabulated various choices below.

f	g	Functional
add	add	$2 F{\left(a \right)} + F{\left(b \right)} + F{\left(x \right)}$
add	subtract	$- F{\left(b \right)} + F{\left(x \right)}$
add	mult	$\left(F{\left(a \right)} + F{\left(b \right)}\right) \left(F{\left(a \right)} + F{\left(x \right)}\right)$
add	divide	$\frac{F{\left(a \right)} + F{\left(x \right)}}{F{\left(a \right)} + F{\left(b \right)}}$
add	exp	$\left(F{\left(a \right)} + F{\left(x \right)}\right)^{F{\left(a \right)} + F{\left(b \right)}}$
add	log	$\frac{\log{\left(F{\left(a \right)} + F{\left(b \right)} \right)}}{\log{\left(F{\left(a \right)} + F{\left(x \right)} \right)}}$
subtract	add	$- 2 F{\left(a \right)} + F{\left(b \right)} + F{\left(x \right)}$
subtract	subtract	$- F{\left(b \right)} + F{\left(x \right)}$
subtract	mult	$\left(F{\left(a \right)} - F{\left(b \right)}\right) \left(F{\left(a \right)} - F{\left(x \right)}\right)$
subtract	divide	$\frac{F{\left(a \right)} - F{\left(x \right)}}{F{\left(a \right)} - F{\left(b \right)}}$
subtract	exp	$\left(- F{\left(a \right)} + F{\left(x \right)}\right)^{- F{\left(a \right)} + F{\left(b \right)}}$
subtract	log	$\frac{\log{\left(- F{\left(a \right)} + F{\left(b \right)} \right)}}{\log{\left(- F{\left(a \right)} + F{\left(x \right)} \right)}}$
mult	add	$\left(F{\left(b \right)} + F{\left(x \right)}\right) F{\left(a \right)}$
mult	subtract	$\left(- F{\left(b \right)} + F{\left(x \right)}\right) F{\left(a \right)}$
mult	mult	$F^{2}{\left(a \right)} F{\left(b \right)} F{\left(x \right)}$
mult	divide	$\frac{F{\left(x \right)}}{F{\left(b \right)}}$
mult	exp	$\left(F{\left(a \right)} F{\left(x \right)}\right)^{F{\left(a \right)} F{\left(b \right)}}$
mult	log	$\frac{\log{\left(F{\left(a \right)} F{\left(b \right)} \right)}}{\log{\left(F{\left(a \right)} F{\left(x \right)} \right)}}$
divide	add	$\frac{F{\left(b \right)} + F{\left(x \right)}}{F{\left(a \right)}}$
divide	subtract	$\frac{- F{\left(b \right)} + F{\left(x \right)}}{F{\left(a \right)}}$
divide	mult	$\frac{F{\left(b \right)} F{\left(x \right)}}{F^{2}{\left(a \right)}}$
divide	divide	$\frac{F{\left(x \right)}}{F{\left(b \right)}}$
divide	exp	$\left(\frac{F{\left(x \right)}}{F{\left(a \right)}}\right)^{\frac{F{\left(b \right)}}{F{\left(a \right)}}}$
divide	log	$\frac{\log{\left(\frac{F{\left(b \right)}}{F{\left(a \right)}} \right)}}{\log{\left(\frac{F{\left(x \right)}}{F{\left(a \right)}} \right)}}$
exp	add	$F^{F{\left(a \right)}}{\left(b \right)} + F^{F{\left(a \right)}}{\left(x \right)}$
exp	subtract	$- F^{F{\left(a \right)}}{\left(b \right)} + F^{F{\left(a \right)}}{\left(x \right)}$
exp	mult	$F^{F{\left(a \right)}}{\left(b \right)} F^{F{\left(a \right)}}{\left(x \right)}$
exp	divide	$F^{- F{\left(a \right)}}{\left(b \right)} F^{F{\left(a \right)}}{\left(x \right)}$
exp	exp	$\left(F^{F{\left(a \right)}}{\left(x \right)}\right)^{F^{F{\left(a \right)}}{\left(b \right)}}$
exp	log	$\frac{\log{\left(F^{F{\left(a \right)}}{\left(b \right)} \right)}}{\log{\left(F^{F{\left(a \right)}}{\left(x \right)} \right)}}$
log	add	$\frac{\log{\left(F{\left(a \right)} \right)}}{\log{\left(F{\left(x \right)} \right)}} + \frac{\log{\left(F{\left(a \right)} \right)}}{\log{\left(F{\left(b \right)} \right)}}$
log	subtract	$\frac{\log{\left(F{\left(a \right)} \right)}}{\log{\left(F{\left(x \right)} \right)}} - \frac{\log{\left(F{\left(a \right)} \right)}}{\log{\left(F{\left(b \right)} \right)}}$
log	mult	$\frac{\log{\left(F{\left(a \right)} \right)}^{2}}{\log{\left(F{\left(b \right)} \right)} \log{\left(F{\left(x \right)} \right)}}$
log	divide	$\frac{\log{\left(F{\left(b \right)} \right)}}{\log{\left(F{\left(x \right)} \right)}}$
log	exp	$\left(\frac{\log{\left(F{\left(a \right)} \right)}}{\log{\left(F{\left(x \right)} \right)}}\right)^{\frac{\log{\left(F{\left(a \right)} \right)}}{\log{\left(F{\left(b \right)} \right)}}}$
log	log	$\frac{\log{\left(\frac{\log{\left(F{\left(a \right)} \right)}}{\log{\left(F{\left(b \right)} \right)}} \right)}}{\log{\left(\frac{\log{\left(F{\left(a \right)} \right)}}{\log{\left(F{\left(x \right)} \right)}} \right)}}$

Note that notation such as $F^2(a)$ and $F^{F(a)}(b)$ denote exponentiation, not powers of function composition.

If you are interested in tinkering with your own choice of functions, here is the source code to generate the above table.

from itertools import product

import sympy as sp
import pandas as pd

x = sp.Symbol('x')
b = sp.Symbol('b')
a = sp.Symbol('a')
F = sp.Function('F')
F_x = F(x)
F_b = F(b)
F_a = F(a)

ops = {
    
    'add' : lambda x,y: x + y,
    'subtract': lambda x,y: x - y,
    'mult' : lambda x,y: x * y,
    'divide': lambda x,y: x / y,
    'exp': lambda x,y: x ** y,
    'log': lambda x,y: sp.log(y, x)
    }

results = []
for f,g in product(ops, ops):
    result = ops[g](ops[f](F_x, F_a), ops[f](F_b, F_a))
    result = result.simplify()
    results.append([f, g, f'${sp.latex(result)}$'])

df = pd.DataFrame(results, columns=['f', 'g', 'Functional'])
print(df.to_markdown(index=False))

At a glance it may appear that our original functional is absent in the table, but noting that

\[\frac{F{\left(a \right)} - F{\left(x \right)}}{F{\left(a \right)} - F{\left(b \right)}} = \frac{-1}{-1} \frac{F(x) - F(a)}{F(b) - F(a)}\]

reveals that SymPy produced the same result but with a distinct expression.

It is easy to confirm that not all of the functionals I have tabulated above will provide a normalization onto the interval $[0,1]$. Because this and other properties may not be preserved, one might regard this as an “unnatural generalization”.