# Least Square Regression Line

Given a set of coordinates in the form of (X, Y), the task is to find the least regression line that can be formed.

In statistics, Linear Regression is a linear approach to model the relationship between a scalar response (or dependent variable), say Y, and one or more explanatory variables (or independent variables), say X.
Regression Line: If our data shows a linear relationship between X and Y, then the straight line which best describes the relationship is the regression line. It is the straight line that covers the maximum points in the graph.

Examples:

Input: X = [95, 85, 80, 70, 60]
Y = [90, 80, 70, 65, 60]
Output: Y = 5.685 + 0.863*X
Explanation:
The graph of the data given below is:
X = [95, 85, 80, 70, 60]
Y = [90, 80, 70, 65, 60]
The regression line obtained is Y = 5.685 + 0.863*X The graph shows that the regression line is the line that covers the maximum of the points.

Input: X = [100, 95, 85, 80, 70, 60]
Y = [90, 95, 80, 70, 65, 60]
Output: Y = 4.007 + 0.89*X

Approach:

A regression line is given as Y = a + b*X where the formula of b and a are given as:
b = (nΣ(xiyi) – Σ(xi)Σ(yi)) ÷ (nΣ(xi2)-Σ(xi)2)
a = ȳ – b.x̄

where x̄ and ȳ are mean of x and y respectively.

1. To find regression line, we need to find a and b.
2. Calculate a, which is given by 3. Calculate b, which is given by 4. Put value of a and b in the equation of regression line.

Below is the implementation of the above approach.

## C++

 // C++ program to find the  // regression line  #include  using namespace std;     // Function to calculate b  double calculateB(int x[], int y[], int n)  {             // sum of array x      int sx = accumulate(x, x + n, 0);         // sum of array y      int sy = accumulate(y, y + n, 0);         // for sum of product of x and y      int sxsy = 0;         // sum of square of x      int sx2 = 0;      for(int i = 0; i < n; i++)       {          sxsy += x[i] * y[i];           sx2 += x[i] * x[i];      }      double b = (double)(n * sxsy - sx * sy) /                         (n * sx2 - sx * sx);         return b;  }     // Function to find the  // least regression line  void leastRegLine( int X[], int Y[], int n)  {         // Finding b      double b = calculateB(X, Y, n);         int meanX = accumulate(X, X + n, 0) / n;      int meanY = accumulate(Y, Y + n, 0) / n;         // Calculating a      double a = meanY - b * meanX;         // Printing regression line      cout << ("Regression line:") << endl;      cout << ("Y = ");      printf("%.3f + ", a);      printf("%.3f *X", b);  }     // Driver code  int main()  {             // Statistical data      int X[] = { 95, 85, 80, 70, 60 };      int Y[] = { 90, 80, 70, 65, 60 };             int n = sizeof(X) / sizeof(X);             leastRegLine(X, Y, n);  }     // This code is contributed by PrinciRaj1992

## Java

 // Java program to find the  // regression line     import java.util.Arrays;     public class GFG {         // Function to calculate b      private static double calculateB(          int[] x, int[] y)      {          int n = x.length;             // sum of array x          int sx = Arrays.stream(x).sum();             // sum of array y          int sy = Arrays.stream(y).sum();             // for sum of product of x and y          int sxsy = 0;             // sum of square of x          int sx2 = 0;          for (int i = 0; i < n; i++) {              sxsy += x[i] * y[i];              sx2 += x[i] * x[i];          }          double b = (double)(n * sxsy - sx * sy)                     / (n * sx2 - sx * sx);             return b;      }         // Function to find the      // least regression line      public static void leastRegLine(          int X[], int Y[])      {             // Finding b          double b = calculateB(X, Y);             int n = X.length;          int meanX = Arrays.stream(X).sum() / n;          int meanY = Arrays.stream(Y).sum() / n;             // calculating a          double a = meanY - b * meanX;             // Printing regression line          System.out.println("Regression line:");          System.out.print("Y = ");          System.out.printf("%.3f", a);          System.out.print(" + ");          System.out.printf("%.3f", b);          System.out.print("*X");      }         // Driver code      public static void main(String[] args)      {          // statistical data          int X[] = { 95, 85, 80, 70, 60 };          int Y[] = { 90, 80, 70, 65, 60 };             leastRegLine(X, Y);      }  }

## C#

 // C# program to find the  // regression line  using System;  using System.Linq;     class GFG{     // Function to calculate b  private static double calculateB(int[] x,                                    int[] y)  {      int n = x.Length;         // Sum of array x      int sx = x.Sum();         // Sum of array y      int sy = y.Sum();         // For sum of product of x and y      int sxsy = 0;         // Sum of square of x      int sx2 = 0;      for(int i = 0; i < n; i++)      {          sxsy += x[i] * y[i];           sx2 += x[i] * x[i];      }      double b = (double)(n * sxsy - sx * sy) /                          (n * sx2 - sx * sx);         return b;  }     // Function to find the  // least regression line  public static void leastRegLine(int []X, int []Y)  {             // Finding b      double b = calculateB(X, Y);         int n = X.Length;      int meanX = X.Sum() / n;      int meanY = Y.Sum() / n;         // Calculating a      double a = meanY - b * meanX;         // Printing regression line      Console.WriteLine("Regression line:");      Console.Write("Y = ");      Console.Write("{0:F3}",a );      Console.Write(" + ");      Console.Write("{0:F3}", b);      Console.Write("*X");  }     // Driver code  public static void Main(String[] args)  {             // Statistical data      int []X = { 95, 85, 80, 70, 60 };      int []Y = { 90, 80, 70, 65, 60 };         leastRegLine(X, Y);  }  }     // This code is contributed by gauravrajput1

Output:

Regression line:
Y = 5.685 + 0.863*X


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