Suppose s = f(x)*g(y)*h(z), where series data set {s, x, y} is known but {z} is unknown (with referential interval), meanwhile functions f(•) and g(•) have unknown parameters α and β respectively, but given structures; and h(•) is known.
By what means could we estimate the unknown parameters α and β along with the unknown series values {z}?
BTW, I prefer non-mainstream solutions to the trivial methods like deep learning etc.
Assuming that the independence of input variables is given (if x, y, z are independent, f(x), g(y), h(z) are also independent), there is no restrictions on the classifier choice and you could simply try, for example, SVM.
You'd better try Bayesian Analysis. In this case it is a problem of estimating parameters of a function (f*g parts) of data with a latent variable (z).
Specially look into E-M algorithm. Which is exactly this case. (E step to estimate the latent variable and M step for parameters.)
Given that your model is a product, I suggest starting by taking logs. Then log(h(z)) is an additive term that will be part of the residuals after you have fitted alpha and beta.
Can you assume that z is not correlated with x and y? Can you also scale the values so the arithmetic mean of h(z) is 1? In that case you could look for values of alpha and beta that minimise the squared error [log s - log f(x) - log g(y)]**2.
One caution: that loss function is chosen for mathematical convenience. It might not be a good measure of the real business loss for your application.
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