The discrimination of significant earthquake precursors from background noise is treated as a multistep problem of pattern recognition. Statistical characteristics of helium-content recorded in short time intervals are used as informative parameters. The set of calculated characteristics includes estimations of the mean, the variance, and the results of spectral analysis of the investigated time series. The selection of significant parameters and the rigorous estimations of time shifts between geochemical and seismic series are carried out by analyzing their cross-covariance function. It is established that the most informative characteristics of a hydrothermal system are related to the dynamic fluctuations of the geochemical parameters. The final phase of prediction is based on the application of a method of statistical discovery of images. A method of earthquake-time prediction is suggested. By using this method, we may determine the 10-day interval during which an earthquake may occur two months in advance. The prediction may be improved by increasing the frequency of sampling and by improving the precision of analytical measurements, both of which can be achieved by automation of monitoring devices. Deployment of uniform monitoring networks is needed in regions designated for special prediction tasks.