In this talk we present new results of principal component analysis of the solar background magnetic field and sunspot magnetic field measured in the cycles 21-24 by Wilcox Solar Observatory and SOHO/MDI. We report a pair of principal components (PCs) of magnetic field waves covering more than 30% of the data variance and attribute these components to dynamo waves generated in two layer dynamo model. We derive mathematical laws describing these dynamo waves and describe their link to the solar activity index of sunspot numbers. Using the derived laws we predict the solar activity backward and forward for two millenniums and reveal close fit to all the observed activity features and the presence of a long-term activity cycle of 320-400 years in addition to the regular 22 year cycle. Preliminary interpretation of the PCA results with the modified Parker's two layer dynamo model accounting for both cycles (22 and 350 years) is also discussed.