/// <summary>
/// Fits an AR(1) model to the time series.
/// </summary>
/// <returns>The fit with parameters lag-1 coefficient, mean, and standard deviation.</returns>
public FitResult FitToAR1()
{
// AR1 model is
// (x_t - \mu) = \alpha (x_{t-1} - \mu) + u_{t}
// where u_{t} \sim N(0, \sigma) are IID
// It's easy to show
// m = E(x_t) = \mu
// c_0 = V(x_t) = E((x_t - m)^2) = \frac{\sigma^2}{1 - \alpha^2}
// c_1 = V(x_t, x_t-1) = E((x_t - m)(x_{t-1} - m)) = \alpha c_0
// which gives a way to get paramters via the method of moments. In particular,
// \alpha = c_1 / c_0
// For maximum likelyhood estimation (MLE), we need
// \log L = -\frac{1}{2} \sum_i \left[ \log (2 \pi \sigma^2)
// + \left[\frac{(x_i - \mu) - \alpha (x_{i-1} - \mu)}{\sigma}\right]^2
// Treatment of the 1st value a bit subtle. We could treat it as normally distributed as
// per m and c_0, but then it enters differently than all other values, which significantly
// complicates the equations. Or we could regard it as given and compute log likelihood
// conditional on it; then all values enter in the same way, but sum begins with the second
// value. We do the latter, which is called the "conditional MLE" in the literature.
// Differentiating once
// \frac{\partial L}{\partial \alpha} = \frac{1}{\sigma^2} \sum_i ( x_i - \mu - \alpha x_{i-1} ) x_{i-1}
// \frac{\partial L}{\partial \mu} = \frac{1}{\sigma^2} \sum_i ( x_i - \mu - \alpha x_{i-1} )
// \frac{\partial L}{\partial \sigma} = \sum_i \left[ \frac{(x_i - \mu - \alpha x_{i-1})^2}{\sigma^3} - \frac{1}{\sigma} \right]
// Set equal to zero to get equations for \mu, \alpha, \sigma. First two give a 2X2 system
// that can be solved for \mu and \alpha, then thrid for \sigma. The third equation just says
// \sum_i (x_i - \mu - \alpha x_{i-1})^2 = \sum_i \sigma^2
// that \sigma is the rms of residuals. If we play a little fast and loose with index ranges
// (e.g. ingoring difference between quantities computed over first n-1 and last n-1 values),
// then the other two give the same results as from the method of moments.
// Differentiating twice
// \frac{\partial^2 L}{\partial \alpha^2} = \frac{-1}{\sigma^2} \sum_i x_{i-1} x_{i-1} = \frac{-n (c_0 + m^2)}{\sigma^2}
// \frac{\partial^2 L}{\partial \mu^2} = \frac{-1}{\sigma^2} \sum_i 1 = \frac{-n}{\sigma^2}
// \frac{\partial^2 L}{\partial \sigma^2} = \frac{-2 n}{\sigma^2}
// Mixed derivatives vanish because of the first derivative conditions.
if (data.Count < 4)
{
throw new InsufficientDataException();
}
int n = data.Count;
// compute mean, variance, lag-1 autocorrelation
double m = data.Mean;
double c0 = 0.0;
for (int i = 1; i < data.Count; i++)
{
c0 += MoreMath.Sqr(data[i] - m);
}
double c1 = 0.0;
for (int i = 1; i < data.Count; i++)
{
c1 += (data[i] - m) * (data[i - 1] - m);
}
double alpha = c1 / c0;
// This expression for alpha is guaranteed to be asymptotically unbiased by MLE,
// but it is known to be biased at finite n, and in fact the bias is known.
// See http://www.alexchinco.com/bias-in-time-series-regressions/ for simulations and explanation.
// He cites Kendall, "Note on Bias in the Estimation of Autocorrelation", Biometrika (1954) 41 (3-4) 403-404
// See Shaman and Stine, "The Bias of Autogregressive Coefficient Estimators",
// Journal of the American Statistical Association (1988) Vol. 83, No. 403, pp. 842-848
// (http://www-stat.wharton.upenn.edu/~steele/Courses/956/Resource/YWSourceFiles/ShamanStine88.pdf)
// for derivation and formulas for AR(1)-AR(6).
// For AR(1), MLE systematically underestimates alpha:
// \hat{\alpha} = \alpha - \frac{1 + 3 \alpha}{n}
// I have confrimed the accuracy of this formula via my own simulations.
alpha = alpha + (1.0 + 3.0 * alpha) / n;
double sigma2 = 0.0;
TimeSeries residuals = new TimeSeries();
for (int i = 1; i < data.Count; i++)
{
double r = (data[i] - m) - alpha * (data[i - 1] - m);
residuals.Add(r);
sigma2 += MoreMath.Sqr(r);
}
sigma2 = sigma2 / (data.Count - 3);
// Solution to MLE says denominator is n-1, but (i) Fuller says to use n-3,
// (ii) simulations show n-3 is a better estimate, (iii) n-3 makes intuitive
// sense because there are 3 parameters. I would prefer a more rigorous
// argument, but that's good enough for now.
// The formulas for the variances of alpha and sigma follow straightforwardly from
// the second derivatives of the likelyhood function. For the variance of the mean
// we use the exact formula with the \gamma_k for an AR(1) model with the fitted
// alpha. After quite a bit of manipulation, that is
// v = \frac{\sigma^2}{(1-\alpha)^2} \left[ 1 -
// \frac{2\alpha}{n} \frac{1 - \alpha^n}{1 - \alpha^2} \right]
// which gives a finite-n correction to the MLE result. Near \alpha \approx \pm 1,
// we should use a series expansion to preserve accuracy.
double[] parameters = new double[] { alpha, m, Math.Sqrt(sigma2) };
SymmetricMatrix covariances = new SymmetricMatrix(3);
covariances[0, 0] = (1.0 - alpha * alpha) / n;
covariances[1, 1] = sigma2 / MoreMath.Sqr(1.0 - alpha) * (1.0 - 2.0 * alpha * (1.0 - MoreMath.Pow(alpha, n)) / (1.0 - alpha * alpha) / n) / n;
covariances[2, 2] = sigma2 / 2.0 / n;
TestResult test = residuals.LjungBoxTest();
return(new FitResult(
parameters,
covariances,
test
));
}
/// <summary>
/// Fits an MA(1) model to the time series.
/// </summary>
/// <returns>The fit with parameters lag-1 coefficient, mean, and standard deviation.</returns>
public FitResult FitToMA1()
{
if (data.Count < 4)
{
throw new InsufficientDataException();
}
// MA(1) model is
// y_t - \mu = u_t + \beta u_{t-1}
// where u_t ~ N(0, \sigma) are IID.
// It's easy to show that
// m = E(y_t) = \mu
// c_0 = V(y_t) = E((y_t - m)^2) = (1 + \beta^2) \sigma^2
// c_1 = V(y_t, y_{t-1}) = \beta \sigma^2
// So the method of moments gives
// \beta = \frac{1 - \sqrt{1 - (2 g_1)^2}}{2}
// \mu = m
// \sigma^2 = \frac{c_0}{1 + \beta^2}
// It turns out these are very poor (high bias, high variance) estimators,
// but they illustrate a basic requirement that g_1 < 1/2.
// The MLE estimator
int n = data.Count;
double m = data.Mean;
Func <double, double> fnc = (double theta) => {
double s = 0.0;
double uPrevious = 0.0;
for (int i = 0; i < data.Count; i++)
{
double u = data[i] - m - theta * uPrevious;
s += u * u;
uPrevious = u;
}
;
return(s);
};
Extremum minimum = FunctionMath.FindMinimum(fnc, Interval.FromEndpoints(-1.0, 1.0));
double beta = minimum.Location;
double sigma2 = minimum.Value / (data.Count - 3);
// While there is significant evidence that the MLE value for \beta is biased
// for small-n, I know of no anlytic correction.
double[] parameters = new double[] { beta, m, Math.Sqrt(sigma2) };
// The calculation of the variance for \mu can be improved over the MLE
// result by plugging the values for \gamma_0 and \gamma_1 into the
// exact formula.
SymmetricMatrix covariances = new SymmetricMatrix(3);
if (minimum.Curvature > 0.0)
{
covariances[0, 0] = sigma2 / minimum.Curvature;
}
else
{
covariances[0, 0] = MoreMath.Sqr(1.0 - beta * beta) / n;
}
covariances[1, 1] = sigma2 * (MoreMath.Sqr(1.0 + beta) - 2.0 * beta / n) / n;
covariances[2, 2] = sigma2 / 2.0 / n;
TimeSeries residuals = new TimeSeries();
double u1 = 0.0;
for (int i = 0; i < data.Count; i++)
{
double u0 = data[i] - m - beta * u1;
residuals.Add(u0);
u1 = u0;
}
;
TestResult test = residuals.LjungBoxTest();
FitResult result = new FitResult(
parameters, covariances, test
);
return(result);
}
