public void ValidateCumulativeDistribution(double location, double scale, double dof, double x, double p)
{
var dist = new StudentT(location, scale, dof);
Assert.That(dist.CumulativeDistribution(x), Is.EqualTo(p).Within(1e-13));
Assert.That(StudentT.CDF(location, scale, dof, x), Is.EqualTo(p).Within(1e-13));
}
public Vector <double> PValues()
{
Vector <double> result = TStatistics();
int df = intercept ? N - k - 1 : N - k;
result.PointwiseAbs().Map(t => 2 * (1 - StudentT.CDF(0, 1, df, t)), result);
return(result);
}
//[TargetedPatchingOptOut("Performance critical to inline this type of method across NGen image boundaries")]
public static double TDist(double x, int degreesFreedom, int tails)
{
switch (tails)
{
case 1:
return 1d - StudentT.CDF(0d, 1d, degreesFreedom, x);
case 2:
return 1d - StudentT.CDF(0d, 1d, degreesFreedom, x) + StudentT.CDF(0d, 1d, degreesFreedom, -x);
default:
throw new ArgumentOutOfRangeException("tails");
}
}
public static double GetPValue(double tScore, double df)
{
return(2.0D * (1 - StudentT.CDF(0.0, 1.0, df, Math.Abs(tScore))));
}
// Perform the Augmented Dickey Fuller Test on a timeseries with a given number of lags
public float ADF(float[] values, int lag = 1)
{
// To keep the regression feasable in terms of the proportion of features and observations, the lag will be restricted to the squareroot of all observations
// This relation was arbitrarly choosen and could be changed -> however a decreasing growth rate for the lag in the number of observations, is desirable for run time efficiency and computational precision
if (lag > Math.Sqrt(values.Length))
{
lag = Convert.ToInt32(Math.Sqrt(values.Length));
} // A warning could be added
// The reasoning of the following code follows the ADF as described here: https://nwfsc-timeseries.github.io/atsa-labs/sec-boxjenkins-aug-dickey-fuller.html
// The ADF performs a regresssion where the Dependent variable y_delta is regressed on the lag one of y_t and multiple lags of the differenced timeseries
// First calculate the set of dependent variables
double[] getYDifferences()
{
double[] results = new double[values.Length - 1];
for (int i = 0; i < results.Length; i++)
{
results[i] = (double)values[i + 1] - values[i];
}
return(results);
};
double[] YDifferences = getYDifferences();
// Because the ADF requires YDifferences to be regressed on its own lags (p), the number dependent variables for the observation are actually YDifferences.Lenght - lag
int length = YDifferences.Length - lag;
// alpha will be generated by allowing the regression to have an intercept, the other parameters are defined in the lines below
// beta is defined as being the coefficient of t, thus the regression will require a count variable for the observatins
int[] t_values = Enumerable.Range(0, length).ToArray();
// The regression requires the last <length> observations of y_t with lag one, this is the parameter whose coefficient will be used to create the test statistic
float[] y_1 = new float[length];
Array.Copy(values, lag, y_1, 0, length); // lag does not need to be corrected by -1 as the YDifferences are calculated looking forward
// define a function to get the differences in yDifferences with the desired shift -> shift 0 yields the dependent variable
double[] YDifferencesWithLag(int shift)
{
double[] yDiff = new double[length];
Array.Copy(YDifferences, lag - shift, yDiff, 0, length);
return(yDiff);
}
// Y_t - Y_{t-1}-> the dependent variable of the regression
var y = DenseVector.OfArray(YDifferencesWithLag(0));
// define a matrix with the lagged YDifferences, as features, a constant to facilitate an intercept in the regresion is added by the regressionmodel
var x_values = new Vector <double> [lag + 2]; // plus 2 for t_values and y_1
x_values[0] = new DenseVector(t_values.Select(i => Convert.ToDouble(i)).ToArray());
x_values[1] = new DenseVector(y_1.Select(i => (double)i).ToArray());
// Fill the columns of the matrix with the lagged timeseries values (lag is increasing -> i)
for (int i = 0; i < lag; i++)
{
float[] ydiff_t_i = new float[length];
Array.Copy(values, lag - (i + 1), ydiff_t_i, 0, length);
x_values[i + 2] = new DenseVector(Array.ConvertAll(ydiff_t_i, v => (double)v));
}
var x = DenseMatrix.OfColumns(x_values);
// Perform a regression
var rm = new MultipleRegressionModel();
rm.Fit(y, x);
// return the p-Value for the Intercept
float testStatistic = (float)(((float)rm.betas.ToArray()[2]) / Math.Sqrt(rm.VarSEbyJ(2))); // cf. for ADF statistic with mutliple regressors (not shown in the link above) https://en.wikipedia.org/wiki/Augmented_Dickey%E2%80%93Fuller_test
return((float)(StudentT.CDF(0, 1, length - x.ColumnCount, testStatistic))); // The test is a onesided, testing for t-values beyond the critical threshold -> a sufficiently low CDF value here implies stationarity
}
static double tdistOneTail(double x, int degreesOfFreedom)
{
return(1 - StudentT.CDF(0, 1, degreesOfFreedom, x));
}