Hi Faheem. You can google search for materials related to your question. But from a conceptual point of view, you just need a distance measure for functions, not scalars. Suppose you have a set of N one-step-ahead functional forecasts {f_i(x)}, i \in {1,2,...,N} and the corresponding set of realized "functional values" is {y_i(x)}, with a "common support" x \in A. If x is just a scalar real variable taking values in the interval A, then the forecast accuracy will be a functional version of the "Mean Squared Forecast Error" (MSFE) whose formula can be written as sum from {i=1} to {N} integral over {A} [y_i(x) - f_i(x) ]^2 dx. BUT since we have NOT defined the "squared value of a function", then just suppose you are computationally representing a function with quite a lot of points x_j, j \in {1,2....,B} (a grid of the interval A), then the functional MSFE is approximately given by sum from {i=1} to {N} [ sum from {j=1} to {B} [y_i(x_j) - f_i(x_j) ]^2 ] . Do you agree? Clearly, if x is a vector of Q real scalars, then the grid of the set A as a subset of the Reals in Q dimensions is not so trivial. Finally, you can also modify this formula to get the functional version of the "root MSFE", which is also pervasively used in the forecasting literature.

thanks Carlos R. Barrera-Chaupis for your response, but can use the inner product in the functional context for calculating the forecasting error?

That's OK, the L^2 inner product is more general than the L^2 norm, provided that we have carefully specified the interval (i.e., the "function space") on which they are being evaluated. But this is tangential to you question, indeed.

Dear Faheem.

I will still employ you to Google additional materials onliine in recommendation with Carlos. R. Barrera-Chaupis.

Regards

Thanks for your interest, I am searching online but fail, can you suggest a link which could solve my problem

Faheem Jan , I would suggest these two R links: https://otexts.com/fpp2/, and https://otexts.com/fpp3/ .

This guide to measuring forecasting accuracy describes appropriate accuracy measures as well as additional visual tools to make sure the data is collected correctly:

Chapter Forecast Evaluation Techniques for I4.0 Systems

The framework presented relates to the setup where you want to evaluate both point forecasts and interval predictions across many time series, but it can be easily adjusted for a single time series as well.