Let's assume an analytical model predicts an epidemic trend over time, i.e. number of infections over time. We also have a computer simulation results over time to verify the performance of the model. The goal is to prove the simulation results and predicted values of the analytical model (which are both a time series) are statistically close or similar. By similarity I mean the model predicts the values close to what simulation is providing.
Background: Researching around this topic, I came across the following posts:
Both discussions suggest three approaches, where I am interested in two of them basically:
(1). Use of ARIMA; (2). Use of Granger test
For the first suggested solution, this is what has been written there in regards to ARIMA, in (1):
Run ARIMA on both data sets. (The basic idea here is to see if the same set of parameters (which make up the ARIMA model) can describe both your temp time series. If you run auto.arima() in forecast (R), then it will select the parameters p,d,q for your data, a great convenience.
I ran auto.arima on the simulation values and then ran forecast, here are the results:
ARIMA(2,0,0) with zero mean Coefficients: ar1 ar2 1.4848 -0.5619 s.e. 0.1876 0.1873 sigma^2 estimated as 121434: log likelihood=-110.64 AIC=227.27 AICc=229.46 BIC=229.4
I ran auto.arima on predicted model values and then forecast. This is the result of the predicted model:
ARIMA(2,0,0) with non-zero mean Coefficients: ar1 ar2 intercept 1.5170 -0.7996 1478.8843 s.e. 0.1329 0.1412 290.4144 sigma^2 estimated as 85627: log likelihood=-108.11 AIC=224.21 AICc=228.21 BIC=227.05
Question 1 What are the values that need to be compared to prove that the two series are similar especially the trend over time?
Regarding the second suggested option, I have read about it and found that Granger test is usually used to see if the values of series A at time t can predict the values of Series B at time t+1.
Question 2 Basically, in my case I want to compare the values of time series A and B at the same time, how this one is relevant to my case then?
Question 3 Is there any available method can be used to prove that the trend of two time-series over time is similar?
FYI. I saw another method which is using Pearson Correlation Coefficient and I could follow the reasoning there. Moreover, verifying analytical models with simulations has been widely used in the literature. see: