Question

Write a JavaScript program to implement the Least-Squares Linear Regression algorithm shown below, for an arbitrary set of ( x, y ) data points entered by the user.

Least-Squares Linear Regression: • Linear regression basically means finding the ( equation for the ) straight line that is the closest “fit” to a set of ( x, y ) data points.

• Least-squares is a means of identifying the “best” fit between the line and the data. For each data point, the distance between the point and the line is calculated and squared, and the total of these squared distances is added up over all the data points. The straight line that yields the minimum value for this sum of squares is defined as the best fit.

• calculate the correlation coefficient, r, which is a metric for how well the line fits the data, and if the correlation is positive or negative. This value always lies between -1.0 and 1.0 inclusive. The closer it is to 1.0, the stronger the positive correlation; The closer to -1.0 the stronger the negative correlation; and if the coefficient is close to zero, then it shows a weak correlation ( if any ) between x and y.

The Algorithm: 1. Read in a series of ( x, y ) data pairs. While doing so, calculate the following sums:

• N = number of data pairs.

• Sx = ∑ x = Summation ( total ) of all X values.

• Sy = ∑ y = Summation ( total ) of all Y values.

• Sxx = ∑ x2 = Summation ( total ) of all X2 values.

• Syy = ∑ y2 = Summation ( total ) of all Y2 values.

• Sxy = ∑ xy = Summation ( total ) of all X∙Y values.

2. Then calculate: • Slope, m = ( N ∙ Sxy – Sx ∙ Sy ) / ( N ∙ Sxx – Sx ∙ Sx ) • Intercept, b = ( Sy – m ∙ Sx ) / N

3. Report the best-fit line as y = m x + b

4. Finally calculate and report the correlation coefficient, r, as:

• ? = ? ∙ ???−?? ∙ ?? /√( ? ∙ ???−?? ∙ ?? ) ∙ ( ? ∙ ???−?? ∙ ?? )

Answer 1

Please find the working commented code below :

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function leastSquareLinearRegression(x_array, y_array) {
    var sum_x = 0;
    var sum_y = 0;
    var sum_xy = 0;
    var sum_yy = 0;
    var sum_xx = 0;
    var count = 0;

    var x = 0;
    var y = 0;
    var input_length = x_array.length;

// find required sums
    for (let i = 0; i< input_length; i++) {
        x = x_array[i];
        y = y_array[i];
        sum_x+= x;
        sum_y+= y;
        sum_xx += x*x;
        sum_yy += y*y;
        sum_xy += x*y;
        count++;
    }

  
    // Find m and b:
   
    var m = (count*sum_xy - sum_x*sum_y) / (count*sum_xx - sum_x*sum_x);
    var b = (sum_y/count) - (m*sum_x)/count;

    document.write("Best fit line ( in the format y = mx+c ) is : <br>");
    document.write("y = " + m + "x + " + b + "<br>");

    // ? ∙ ???−?? ∙ ?? /√( ? ∙ ???−?? ∙ ?? ) ∙ ( ? ∙ ???−?? ∙ ?? )
var corr_coeff = input_length*(sum_xy)-(sum_x*sum_y);
corr_coeff = corr_coeff / Math.sqrt((input_length*sum_xx) - (sum_x*sum_x));
corr_coeff = corr_coeff / Math.sqrt((input_length*sum_yy) - (sum_y*sum_y));

    document.write("The correlation coefficient is : <br>");
    document.write(corr_coeff);
  
    return [ [], [] ];
}

var x = [1,2,3];
var y = [3,5,6];
document.write(leastSquareLinearRegression(x,y));

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Please find the screenshot of working code below :