a convex set
F.N. Grigoriev, N.A. Kuznetsov
The problem of testing statistical hypotheses: the main hypothesis H0 the patient is suffering from Parkinson's disease and the alternative hypothesis for it H1 – the patient has no this disease. It is believed that all the information for the testing of hypotheses containes in the EEG recording the patient. It was chosen as the criterion for testing is statistical hypotheses H0 and H1.
It is assumed that at some time interval EEG potentials findings with sufficient accuracy for a given problem can be described as stochastic processes fourth-order autoregression with unknown parameters.
Estimates of these parameters for each patient were determined by solving the sistem of equations of the Yule-Walker. Test statistic was given by an affine function of the parameters of an autoregressive process.
Coefficients of affinity function (statistics) were determined by the «training set» containing n – 5 EEG clearly ill patients, and m 5 EEG of normal ones. Autoregressive process parameters were determined for training set of each patient.
Then, in a four-parameter space have been constructed set, ie, convex hulls for the parameters of patients and healthy subjects. It was followed constructed a hyperplane separating these sets. The equation of the hyperplane obtained was taken as test statistic.
Parameter estimates an autoregressive process the corresponding EEG were determined for the diagnosis of the patient. And these estimates were substituted into the expression for the statistic (the equation of the hyperplane).
If this gives a value greater than zero, there is a statistically well-founded suspicion that the patient is sick, and vice versa, if the resulting value is less than or equal to zero, then the patient is healthy.
The problem was completely solved for a particular training sample.