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The Correlation Coefficient
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Although the regression line in Figure 8-1 seems to indicate an upward trend, we don t know how well this line actually fits the data. If the line and data have only a slight correlation, the regression line is of little interest. If the line fits the data well, it is of much greater value. The most common way to determine the correlation of the data to the regression line is to compute the correlation coefficient, which is a number between 1 and 1. The correlation coefficient represents the amount of deviation from the mean that is explained by the line. This may sound confusing, but it really isn t. The correlation coefficient is related to the
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Figure 8-1
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A graph of the average share price and regression line
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distance each data point is from the line. If the correlation coefficient is 1, the data corresponds perfectly to the line. A coefficient of 0 means that there is no correlation between the line and the data. (In this case, any line would be about as good!) The sign of the correlation coefficient must be set according to the sign of the slope of the regression line, which is b. If the correlation coefficient is positive, it means that there is a direct relationship between the dependent variable and the independent variable. If the correlation coefficient is negative, then an inverse relationship exists. The formula to find the correlation coefficient is
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where Mx is the mean of X and My is the mean of Y. The sign is set based on the sign of the slope of the regression line. Generally, an absolute value of 0.81 or greater is considered a strong correlation. It means that about 66 percent of the data fits the regression line. To convert any correlation coefficient into a percentage, simply square it. This squared value is called the coefficient of determination. The regress( ) method, shown next, computes the regression equation and the correlation coefficient:
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/* Compute the regression equation and coefficient of correlation for a set of values. The values represent the Y coordinate. The X coordinate is time (i.e., ascending increments of 1). */
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8: Statistics, Graphing, and Java
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public static RegData regress(double[] vals) { double a, b, yAvg, xAvg, temp, temp2, cor; double vals2[] = new double[vals.length]; // Create number format with 2 decimal digits. NumberFormat nf = NumberFormat.getInstance(); nf.setMaximumFractionDigits(2); // Find mean of Y values. yAvg = mean(vals); // Find mean of X component. xAvg = 0.0; for(int i=0; i < vals.length; i++) xAvg += i; xAvg /= vals.length; // Find b. temp = temp2 = 0.0; for(int i=0; i < vals.length; i++) { temp += (vals[i]-yAvg) * (i-xAvg); temp2 += (i-xAvg) * (i-xAvg); } b = temp/temp2; // Find a. a = yAvg - (b*xAvg); // Compute the coefficient of correlation. for(int i=0; i < vals.length; i++) vals2[i] = i+1; cor = temp/vals.length; cor /= stdDev(vals) * stdDev(vals2); return new RegData(a, b, cor, "Y = " + nf.format(a) + " + " + nf.format(b) + " * X"); } }
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It is important to point out that regress( ) assumes that the independent variable (X) is time that is, the stepwise progression of events represented in units of one. The mean of the X values is computed using this sequence:
// Find mean of X component. xAvg = 0.0;
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for(int i=0; i < vals.length; i++) xAvg += i; xAvg /= vals.length;
Here, the values from 0 to the number of elements in the set are summed and then divided by the number of the elements. This yields the average of X. Because time is used for the X axis, regress( ) performs what is sometimes called a timeseries analysis. This is why only a single array of values need be passed. It would be possible to modify regress( ) to accept two arrays, one containing the Y values and one containing the X values, but this was not necessary for the purposes of this chapter. The regress( ) method returns the values of a and b, a string representation of the regression equation, and the correlation coefficient, contained within a RegData object. RegData is shown here:
// This class holds the regression analysis data. class RegData { public double a, b; public double cor; public String equation; public RegData(double i, double j, double k, String str) { a = i; b = j; cor = k; equation = str; } }