The document explains the max() and argmax() equations used in statistics to identify the maximum value from a set of values and the points that maximize a function, respectively. It outlines the conditions under which these functions operate and provides examples, including the use of Python's numpy library for finding maximum values. The argmax function, although discussed in a mathematical context, highlights that it does not evaluate functions directly but identifies maximums within provided sets.

1. Argmax Equation
Statistics
Portland Data Science Group
Created by Andrew Ferlitsch
Community Outreach Officer
July, 2017

2. Max Equation
• The max() equation returns the largest value from a set of values.
max( x1, x2, x3, x4, x5 )
x ∈ S
S : Set of Discrete Values
R: Set of Continuous Real Values
∈ : Symbol for Element of a Set
xi : An Instance of an Element of a Set
≥ : Greater than or equal to for all elements in a set
• Condition is met where element xj is the maximum in x ∈ S, when:
xj ≥ xi , x ∈ S
Enumerated set of values (S)
For all x that are elements of set x
xj is greater than or equal to all elements xi in set S

3. ArgMax Equation
• The argmax() equation returns the largest values (points) that
maximize a function.
argmax( f(x) )
x ∈ S
f(x) : function that takes x as an input
• Condition is met when the output of the function for at least
one element xj that is greater than or equal to the output of
for all x ∈ S.
f(xj) ≥ f(xi), x ∈ S
Which values of x maximize the output of f(x)
For all x that are elements of set x
The output for xj is greater than or equal to the output for all elements xi in set S

4. Examples:
argmax( x( 10 –x ) ) = 5
x ∈ R
x maybe continuous or discrete
x = 1 -> 1 * 9 = 9
x = 2 -> 2 * 8 = 16
x = 3 -> 3 * 7 = 21
x = 4 -> 4 * 6 = 24
x = 5 -> 5 * 5 = 25
x = 6 -> 6 * 4 = 24
x = 7 -> 7 * 3 = 21
x = 8 -> 8 * 2 = 16
x = 9 -> 1 * 9 = 9
x = 5 maximizes f(x)
argmax( cos(x) ) = { …, -4∏, -2∏, 0 , 2∏, 4∏, … }
x
The maximum values of cos(x) occur at 0 and intervals of 2∏.
Returns a set of values
Examples from Wikipedia

5. Python – Numpy.Argmax
data = np.array( [ 1, 2, 3 ] )
x = np.argmax( data )
Numpy function to create a numpy arrayValue is: array( [ 1, 2, 3 ] )
import numpy as np
Value is: 3
Numpy function to calculate argmax
Keyword to refer to library by an alias (shortcut) name
data = np.array( [ [ 1, 2, 3 ], [ 4, 5, 6 ], [ 0, 2, 3 ] )
x = np.argmax( data )
Numpy function to create a numpy matrix
Value is: 6
Note: numpy.argmax does not evaluate a function for the maximum output, but takes a set (which may have
bBeen generated by a function), and finds the maximum value in the set.