Quadratic Regression

				
					/// <summary>
/// Calculates the best fit quadratic regression of a set of data.
/// </summary>
/// <param name="y"> The y values of the input data. </param>
/// <returns> A list of the regression values from start to finish. </returns>
private List<double> QuaraticRegression(List<double> y)
{
	int n = y.Count;

	// set all needed lists
	List<double> x = new List<double>();
	List<double> x2 = new List<double>();
	List<double> x3 = new List<double>();
	List<double> x4 = new List<double>();
	List<double> xy = new List<double>();
	List<double> x2y = new List<double>();

	// centers x values based off of number of y inputs
	int zero = Convert.ToInt32(y.Count / 2);
	if (n % 2 == 0)
	{
		for (int i = 0; i < zero; i++)
			x.Add(i * -1);
		for (int i = 1; i <= zero; i++)
			x.Add(i);
		x.Sort();
	}
	else
	{
		for (int i = 0; i <= zero; i++)
			x.Add(i * -1);
		for (int i = 1; i <= zero; i++)
			x.Add(i);
		x.Sort();
	}

	// calculate each needed list
	for (int i = 0; i < n; i++)
	{
		double xVal = x[i];
		x2.Add(Math.Pow(xVal, 2));
		x3.Add(Math.Pow(xVal, 3));
		x4.Add(Math.Pow(xVal, 4));
		x2y.Add(x2[i] * y[i]);
		xy.Add(xVal * y[i]);
	}

	// build needed matricies for using Least Squares Method quadratic regression
	double[,] num = {
		{   x2y.Sum()   },
		{   xy.Sum()    },
		{   y.Sum()     }
	};

	double[,] den = {
		{   x4.Sum()   ,   x3.Sum()   ,   x2.Sum()   },
		{   x3.Sum()   ,   x2.Sum()   ,   x.Sum()    },
		{   x2.Sum()   ,   x.Sum()    ,   x.Count    }
	};

	// matrix of minors
	double[,] minors = {
		{   den[1,1] * den[2,2] - den[1,2] * den[2,1]    ,   den[1,0] * den[2,2] - den[1,2] * den[2,0]    ,   den[1,0] * den[2,1] - den[1,1] * den[2,0]     },
		{   den[0,1] * den[2,2] - den[0,2] * den[2,1]    ,   den[0,0] * den[2,2] - den[0,2] * den[2,0]    ,   den[0,0] * den[2,1] - den[0,1] * den[2,0]     },
		{   den[0,1] * den[1,2] - den[0,2] * den[1,1]    ,   den[0,0] * den[1,2] - den[0,2] * den[1,0]    ,   den[0,0] * den[1,1] - den[0,1] * den[1,0]     }
	};

	// matrix of coefitients
	double[,] cofs = {
		{   minors[0,0]        ,    minors[0,1] * -1   ,   minors[0,2]        },
		{   minors[1,0] * -1   ,    minors[1,1]        ,   minors[1,2] * -1   },
		{   minors[2,0]        ,    minors[2,1] * -1   ,   minors[2,2]        }
	};

	// adjusted matrix
	double[,] adj = {
		{   cofs[0,0]   ,   cofs[1,0]   ,   cofs[2,0]   },
		{   cofs[0,1]   ,   cofs[1,1]   ,   cofs[2,1]   },
		{   cofs[0,2]   ,   cofs[1,2]   ,   cofs[2,2]   }
	};

	// determinate
	double d = den[0, 0] * minors[0, 0] + den[0, 1] * minors[0, 1] + den[0, 2] * minors[0, 2];

	// inverted matrix
	double[,] inv = {
		{   adj[0,0] / d    ,   adj[0,1] / d   ,   adj[0,2] / d    },
		{   adj[1,0] / d    ,   adj[1,1] / d   ,   adj[1,2] / d    },
		{   adj[2,0] / d    ,   adj[2,1] / d   ,   adj[2,2] / d    }
	};

	// calculate constants
	double a = inv[0, 0] * num[0, 0] + inv[0, 1] * num[1, 0] + inv[0, 2] * num[2, 0];
	double b = inv[1, 0] * num[0, 0] + inv[1, 1] * num[1, 0] + inv[1, 2] * num[2, 0];
	double c = inv[2, 0] * num[0, 0] + inv[2, 1] * num[1, 0] + inv[2, 2] * num[2, 0];

	// set list of function values
	List<double> fx = new List<double>();
	for (int i = 0; i < n; i++)
		fx.Add(a * Math.Pow(x[i], 2) + (b * x[i]) + c);	

	/* find reliability of graph using R^2 method
	double sse = 0;
	double sst = 0;
	for (int i = 0; i < n; i++)
	{
		sse += Math.Pow(y[i] - (a * Math.Pow(x[i], 2) + b * x[i] + c), 2);
		sst += Math.Pow(y[i] - y.Average(), 2);
	}

	double r2 = 1 - sse / sst;
	*/

	return fx;
}