Prerequisite: Correlation Coefficient
Given two arrays X and Y. Find Spearman’s Rank Correlation. In Spearman rank correlation instead of working with the data values themselves (as discussed in Correlation coefficient), it works with the ranks of these values. The observations are first ranked and then these ranks are used in correlation. The Algorithm for this correlation is as follows
Rank each observation in X and store it in Rank_X Rank each observation in Y and store it in Rank_Y Obtain Pearson Correlation Coefficient for Rank_X and Rank_Y
The formula used to calculate Pearson’s Correlation Coefficient (r or rho) of sets X and Y is as follows:
Algorithm for calculating Pearson’s Coefficient of Sets X and Y
function correlationCoefficient(X, Y) n = X.size sigma_x = sigma_y = sigma_xy = 0 sigma_xsq = sigma_ysq = 0 for i in 0...N-1 sigma_x = sigma_x + X[i] sigma_y = sigma_y + Y[i] sigma_xy = sigma_xy + X[i] * Y[i] sigma_xsq = sigma_xsq + X[i] * X[i] sigma_ysq = sigma_ysq + Y[i] * Y[i] num =( n * sigma_xy - sigma_x * sigma_y) den = sqrt( [n*sigma_xsq - (sigma_x)^ 2]*[ n*sigma_ysq - (sigma_y) ^ 2] ) return num/den
While assigning ranks, it may encounter ties i.e two or more observations having the same rank. To resolve ties, this will use fractional ranking scheme. In this scheme, if n observations have the same rank then each observation gets a fractional rank given by:
fractional_rank = (rank) + (n-1)/2
The next rank that gets assigned is rank + n and not rank + 1. For instance, if the 3 items have same rank r, then each gets fractional_rank as given above. The next rank that can be given to another observation is r + 3. Note that fractional ranks need not be fractions. They are the arithmetic mean of n consecutive ranks ex r, r + 1, r + 2 … r + n-1.
(r + r+1 + r+2 + ... + r+n-1) / n = r + (n-1)/2
Some Examples :
Input : X = [15 18 19 20 21] Y = [25 26 28 27 29] Solution : Rank_X = [1 2 3 4 5] Rank_Y = [1 2 4 3 5 ] sigma_x = 1+2+3+4+5 = 15 sigma_y = 1+2+4+3+5 = 15 sigma_xy = 1*2+2*2+3*4+4*3+5*5 = 54 sigma_xsq = 1*1+2*2+3*3+4*4+5*5 = 55 sigma_ysq = 1*1+2*2+3*3+4*4+5*5 = 55 Substitute values in formula Coefficient = Pearson(Rank_X, Rank_Y) = 0.9 Input: X = [15 18 21 15 21 ] Y = [25 25 27 27 27 ] Solution: Rank_X = [1.5 3 4.5 1.5 4.5] Rank_Y = [1.5 1.5 4 4 4] Calculate and substitute values of sigma_x, sigma_y, sigma_xy, sigma_xsq, sigma_ysq. Coefficient = Pearson(Rank_X, Rank_Y) = 0.456435
The Algorithm for fractional ranking scheme is given below:
function rankify(X) N = X.size() // Vector to store ranks Rank_X(N) for i = 0 ... N-1 r = 1 and s = 1 // Count no of smaller elements in 0...i-1 for j = 0...i-1 if X[j] < X[i] r = r+1 if X[j] == X[i] s = s+1 // Count no of smaller elements in i+1...N-1 for j = i+1...N-1 if X[j] < X[i] r = r+1 if X[j] == X[i] s = s+1 //Assign Fractional Rank Rank_X[i] = r + (s-1) * 0.5 return Rank_X
Note:
There is a direct formula to calculate Spearman’s coefficient given by
A CPP Program to evaluate Spearman’s coefficient is given below
// Program to find correlation // coefficient #include <iostream> #include <vector> #include <cmath> using namespace std;
typedef vector< float > Vector;
// Utility Function to print // a Vector void printVector( const Vector &X)
{ for ( auto i: X)
cout << i << " " ;
cout << endl;
} // Function returns the rank vector // of the set of observations Vector rankify(Vector & X) { int N = X.size();
// Rank Vector
Vector Rank_X(N);
for ( int i = 0; i < N; i++)
{
int r = 1, s = 1;
// Count no of smaller elements
// in 0 to i-1
for ( int j = 0; j < i; j++) {
if (X[j] < X[i] ) r++;
if (X[j] == X[i] ) s++;
}
// Count no of smaller elements
// in i+1 to N-1
for ( int j = i+1; j < N; j++) {
if (X[j] < X[i] ) r++;
if (X[j] == X[i] ) s++;
}
// Use Fractional Rank formula
// fractional_rank = r + (n-1)/2
Rank_X[i] = r + (s-1) * 0.5;
}
// Return Rank Vector
return Rank_X;
} // function that returns // Pearson correlation coefficient. float correlationCoefficient
(Vector &X, Vector &Y)
{ int n = X.size();
float sum_X = 0, sum_Y = 0,
sum_XY = 0;
float squareSum_X = 0,
squareSum_Y = 0;
for ( int i = 0; i < n; i++)
{
// sum of elements of array X.
sum_X = sum_X + X[i];
// sum of elements of array Y.
sum_Y = sum_Y + Y[i];
// sum of X[i] * Y[i].
sum_XY = sum_XY + X[i] * Y[i];
// sum of square of array elements.
squareSum_X = squareSum_X +
X[i] * X[i];
squareSum_Y = squareSum_Y +
Y[i] * Y[i];
}
// use formula for calculating
// correlation coefficient.
float corr = ( float )(n * sum_XY -
sum_X * sum_Y) /
sqrt ((n * squareSum_X -
sum_X * sum_X) *
(n * squareSum_Y -
sum_Y * sum_Y));
return corr;
} // Driver function int main()
{ Vector X = {15,18,21, 15, 21};
Vector Y= {25,25,27,27,27};
// Get ranks of vector X
Vector rank_x = rankify(X);
// Get ranks of vector y
Vector rank_y = rankify(Y);
cout << "Vector X" << endl;
printVector(X);
// Print rank vector of X
cout << "Rankings of X" << endl;
printVector(rank_x);
// Print Vector Y
cout << "Vector Y" << endl;
printVector(Y);
// Print rank vector of Y
cout << "Rankings of Y" << endl;
printVector(rank_y);
// Print Spearmans coefficient
cout << "Spearman's Rank correlation: "
<< endl;
cout<<correlationCoefficient(rank_x,
rank_y);
return 0;
} |
// Java Program to find correlation // coefficient import java.util.*;
class GFG
{ // Utility Function to print
// a Vector
static void printVector(ArrayList<Double> X)
{
for ( double i : X)
System.out.print(i + " " );
System.out.println();
}
// Function returns the rank vector
// of the set of observations
static ArrayList<Double> rankify(ArrayList<Double> X)
{
int N = X.size();
// Rank Vector
ArrayList<Double> Rank_X = new ArrayList<Double>();
for ( int i = 0 ; i < N; i++) {
Rank_X.add(0d);
int r = 1 , s = 1 ;
// Count no of smaller elements
// in 0 to i-1
for ( int j = 0 ; j < i; j++) {
if (X.get(j) < X.get(i))
r++;
if (X.get(j) == X.get(i))
s++;
}
// Count no of smaller elements
// in i+1 to N-1
for ( int j = i + 1 ; j < N; j++) {
if (X.get(j) < X.get(i))
r++;
if (X.get(j) == X.get(i))
s++;
}
// Use Fractional Rank formula
// fractional_rank = r + (n-1)/2
Rank_X.set(i, (r + (s - 1 ) * 0.5 ));
}
// Return Rank Vector
return Rank_X;
}
// function that returns
// Pearson correlation coefficient.
static double
correlationCoefficient(ArrayList<Double> X,
ArrayList<Double> Y)
{
int n = X.size();
double sum_X = 0 , sum_Y = 0 , sum_XY = 0 ;
double squareSum_X = 0 , squareSum_Y = 0 ;
for ( int i = 0 ; i < n; i++) {
// sum of elements of array X.
sum_X = sum_X + X.get(i);
// sum of elements of array Y.
sum_Y = sum_Y + Y.get(i);
// sum of X[i] * Y[i].
sum_XY = sum_XY + X.get(i) * Y.get(i);
// sum of square of array elements.
squareSum_X = squareSum_X + X.get(i) * X.get(i);
squareSum_Y = squareSum_Y + Y.get(i) * Y.get(i);
}
// use formula for calculating
// correlation coefficient.
double corr
= (n * sum_XY - sum_X * sum_Y)
/ Math.sqrt(
(n * squareSum_X - sum_X * sum_X)
* (n * squareSum_Y - sum_Y * sum_Y));
return corr;
}
// Driver function
public static void main(String[] args)
{
ArrayList<Double> X = new ArrayList<Double>(
Arrays.asList(15d, 18d, 21d, 15d, 21d));
ArrayList<Double> Y = new ArrayList<Double>(
Arrays.asList(25d, 25d, 27d, 27d, 27d));
// Get ranks of vector X
ArrayList<Double> rank_x = rankify(X);
// Get ranks of vector y
ArrayList<Double> rank_y = rankify(Y);
System.out.println( "Vector X" );
printVector(X);
// Print rank vector of X
System.out.println( "Rankings of X" );
printVector(rank_x);
// Print Vector Y
System.out.println( "Vector Y" );
printVector(Y);
// Print rank vector of Y
System.out.println( "Rankings of Y" );
printVector(rank_y);
// Print Spearmans coefficient
System.out.println( "Spearman's Rank correlation: " );
System.out.println(
correlationCoefficient(rank_x, rank_y));
}
} // This code is contributed by phasing17 |
# Python3 Program to find correlation coefficient # Utility Function to print # a Vector def printVector(X):
print ( * X)
# Function returns the rank vector # of the set of observations def rankify(X):
N = len (X)
# Rank Vector
Rank_X = [ None for _ in range (N)]
for i in range (N):
r = 1
s = 1
# Count no of smaller elements
# in 0 to i-1
for j in range (i):
if (X[j] < X[i]):
r + = 1
if (X[j] = = X[i]):
s + = 1
# Count no of smaller elements
# in i+1 to N-1
for j in range (i + 1 , N):
if (X[j] < X[i]):
r + = 1
if (X[j] = = X[i]):
s + = 1
# Use Fractional Rank formula
# fractional_rank = r + (n-1)/2
Rank_X[i] = r + (s - 1 ) * 0.5
# Return Rank Vector
return Rank_X
# function that returns # Pearson correlation coefficient. def correlationCoefficient(X, Y):
n = len (X)
sum_X = 0
sum_Y = 0
sum_XY = 0
squareSum_X = 0
squareSum_Y = 0
for i in range (n):
# sum of elements of array X.
sum_X = sum_X + X[i]
# sum of elements of array Y.
sum_Y = sum_Y + Y[i]
# sum of X[i] * Y[i].
sum_XY = sum_XY + X[i] * Y[i]
# sum of square of array elements.
squareSum_X = squareSum_X + X[i] * X[i]
squareSum_Y = squareSum_Y + Y[i] * Y[i]
# use formula for calculating
# correlation coefficient.
corr = (n * sum_XY - sum_X * sum_Y) / ((n * squareSum_X -
sum_X * sum_X) * (n * squareSum_Y - sum_Y * sum_Y)) * * 0.5
return corr
# Driver function X = [ 15 , 18 , 21 , 15 , 21 ]
Y = [ 25 , 25 , 27 , 27 , 27 ]
# Get ranks of vector X rank_x = rankify(X)
# Get ranks of vector y rank_y = rankify(Y)
print ( "Vector X" )
printVector(X) # Print rank vector of X print ( "Rankings of X" )
printVector(rank_x) # Print Vector Y print ( "Vector Y" )
printVector(Y) # Print rank vector of Y print ( "Rankings of Y" )
printVector(rank_y) # Print Spearmans coefficient print ( "Spearman's Rank correlation: " )
print (correlationCoefficient(rank_x, rank_y))
# This code is contributed by phasing17 |
// Program to find correlation // coefficient using System;
using System.Collections.Generic;
class GFG {
// Utility Function to print
// a Vector
static void printVector(List< double > X)
{
foreach ( var i in X) Console.Write(i + " " );
Console.WriteLine();
}
// Function returns the rank vector
// of the set of observations
static List< double > rankify(List< double > X)
{
int N = X.Count;
// Rank Vector
List< double > Rank_X = new List< double >();
for ( int i = 0; i < N; i++) {
Rank_X.Add(0);
int r = 1, s = 1;
// Count no of smaller elements
// in 0 to i-1
for ( int j = 0; j < i; j++) {
if (X[j] < X[i])
r++;
if (X[j] == X[i])
s++;
}
// Count no of smaller elements
// in i+1 to N-1
for ( int j = i + 1; j < N; j++) {
if (X[j] < X[i])
r++;
if (X[j] == X[i])
s++;
}
// Use Fractional Rank formula
// fractional_rank = r + (n-1)/2
Rank_X[i] = (r + (s - 1) * 0.5);
}
// Return Rank Vector
return Rank_X;
}
// function that returns
// Pearson correlation coefficient.
static double correlationCoefficient(List< double > X,
List< double > Y)
{
int n = X.Count;
double sum_X = 0, sum_Y = 0, sum_XY = 0;
double squareSum_X = 0, squareSum_Y = 0;
for ( int i = 0; i < n; i++) {
// sum of elements of array X.
sum_X = sum_X + X[i];
// sum of elements of array Y.
sum_Y = sum_Y + Y[i];
// sum of X[i] * Y[i].
sum_XY = sum_XY + X[i] * Y[i];
// sum of square of array elements.
squareSum_X = squareSum_X + X[i] * X[i];
squareSum_Y = squareSum_Y + Y[i] * Y[i];
}
// use formula for calculating
// correlation coefficient.
double corr
= (n * sum_XY - sum_X * sum_Y)
/ Math.Sqrt(
(n * squareSum_X - sum_X * sum_X)
* (n * squareSum_Y - sum_Y * sum_Y));
return corr;
}
// Driver function
public static void Main( string [] args)
{
List< double > X = new List< double >(
new double [] { 15, 18, 21, 15, 21 });
List< double > Y = new List< double >(
new double [] { 25, 25, 27, 27, 27 });
// Get ranks of vector X
List< double > rank_x = rankify(X);
// Get ranks of vector y
List< double > rank_y = rankify(Y);
Console.WriteLine( "Vector X" );
printVector(X);
// Print rank vector of X
Console.WriteLine( "Rankings of X" );
printVector(rank_x);
// Print Vector Y
Console.WriteLine( "Vector Y" );
printVector(Y);
// Print rank vector of Y
Console.WriteLine( "Rankings of Y" );
printVector(rank_y);
// Print Spearmans coefficient
Console.WriteLine( "Spearman's Rank correlation: " );
Console.WriteLine(
correlationCoefficient(rank_x, rank_y));
}
} // This code is contributed by phasing17 |
// Program to find correlation // coefficient // Utility Function to print // a Vector function printVector(X)
{ for ( var i of X)
process.stdout.write(i + " " );
process.stdout.write( "\n" );
} // Function returns the rank vector // of the set of observations function rankify(X) {
let N = X.length;
// Rank Vector
let Rank_X = new Array(N);
for ( var i = 0; i < N; i++)
{
var r = 1, s = 1;
// Count no of smaller elements
// in 0 to i-1
for ( var j = 0; j < i; j++) {
if (X[j] < X[i] ) r++;
if (X[j] == X[i] ) s++;
}
// Count no of smaller elements
// in i+1 to N-1
for ( var j = i+1; j < N; j++) {
if (X[j] < X[i] ) r++;
if (X[j] == X[i] ) s++;
}
// Use Fractional Rank formula
// fractional_rank = r + (n-1)/2
Rank_X[i] = r + (s-1) * 0.5;
}
// Return Rank Vector
return Rank_X;
} // function that returns // Pearson correlation coefficient. function correlationCoefficient
(X, Y)
{ let n = X.length;
let sum_X = 0, sum_Y = 0,
sum_XY = 0;
let squareSum_X = 0,
squareSum_Y = 0;
for ( var i = 0; i < n; i++)
{
// sum of elements of array X.
sum_X = sum_X + X[i];
// sum of elements of array Y.
sum_Y = sum_Y + Y[i];
// sum of X[i] * Y[i].
sum_XY = sum_XY + X[i] * Y[i];
// sum of square of array elements.
squareSum_X = squareSum_X +
X[i] * X[i];
squareSum_Y = squareSum_Y +
Y[i] * Y[i];
}
// use formula for calculating
// correlation coefficient.
let corr = (n * sum_XY -
sum_X * sum_Y) /
Math.sqrt((n * squareSum_X -
sum_X * sum_X) *
(n * squareSum_Y -
sum_Y * sum_Y));
return corr;
} // Driver function let X = [15,18,21, 15, 21]; let Y= [25,25,27,27,27]; // Get ranks of vector X let rank_x = rankify(X); // Get ranks of vector y let rank_y = rankify(Y); console.log( "Vector X" );
printVector(X); // Print rank vector of X console.log( "Rankings of X" );
printVector(rank_x); // Print Vector Y console.log( "Vector Y" );
printVector(Y); // Print rank vector of Y console.log( "Rankings of Y" );
printVector(rank_y); // Print Spearmans coefficient console.log( "Spearman's Rank correlation: " );
console.log(correlationCoefficient(rank_x, rank_y));
// This code is contributed by phasing17 |
Output
Vector X 15 18 21 15 21 Rankings of X 1.5 3 4.5 1.5 4.5 Vector Y 25 25 27 27 27 Rankings of Y 1.5 1.5 4 4 4 Spearman's Rank correlation: 0.456435
Time Complexity: O(N*N)
Auxiliary Space: O(N)
Python code to calculate Spearman’s Correlation using Scipy Library
We can use scipy to calculate Spearman’s correlation coefficient. Scipy is one of the most used python library for mathematical calculations.
from scipy.stats import spearmanr
# sample data x = [ 1 , 2 , 3 , 4 , 5 ]
y = [ 5 , 4 , 3 , 2 , 1 ]
# calculate Spearman's correlation coefficient and p-value corr, pval = spearmanr(x, y)
# print the result print ( "Spearman's correlation coefficient:" , corr)
print ( "p-value:" , pval)
|
Output:
Spearman's correlation coefficient: -0.9999999999999999 p-value: 1.4042654220543672e-24