# How to verify the correctness of forecast?

I would like to forecast the car rental (count time series). Given hourly integer valued car rentals for a month's period from 24th september to 24th October. I need to forecast car rental demand from 25th october to 31st October.

The given data is shown below:

The Ground Truth forecast data is given as below:

I will describe my approach below. Please correct me and give your suggestions.

I am using the tscount package from R. https://cran.r-project.org/web/packages/tscount/index.html to model this data using a negative binomial distribution having a conditional mean (time varying).

It is a linear combination of logs of past means and past observations as shown below.

Note: The shape parameter, $$\phi$$ controls the dispersion/variance of the neg-binomial distribution but it is NOT time varying.

First I computed the ACF of the given data and it is shown below:

I save the lags of the 10 largest and 10 smallest ACF values.

Similarly, I plot the PACF of the given data and extract the lags due to 10 largest and smallest PACF values. Then I take the unique lags from the combination of ACF and PACF lags. It represents p in the above equation. I am setting p = q.

Then I estimate the parameters of the model using

ts_fit_175_mon1 = tscount::tsglm(ts=ts175_mon1, link='log', model =
list(past_obs = reg_lags, past_mean = reg_lags),distr = 'nbinom')

summary(ts_fit_175_mon1)
tscount::scoring(ts_fit_175_mon1)


The resulting fit on the given data is given in orange and scoring rules are given below.

The forecast is given below in orange

I would like to understand what are the problems with this forecast and how to improve it? Why is the predicted peak much higher? Some of the peaks in the ground truth are not covered. What is being missed in the forecast?

I tried to increase the number of parameters by including 20 largest and smallest lags from ACF and PACF plots leads to increased peaking and worse scoring values. How to better make use of ACF and PACF plots?

When should I be satisfied with my forecast? When I achieve the least scoring value for a model?

Is there any step that I am missing before modelling?