Generating synthetic demand time series from real time series data.

Question

I want to recreate/reproduce a demand time series from real demand data given by a car rental fleet operator. Our contract with the operator has ended so we cannot use the real data as it is so I would like to synthesize a new demand time series that has similar characteristics to the real one.

The real demand time series is stochastic but has strong seasonal components. When plot the distribution of the entire demand time series, it follows a negative binomial distribution. However, the real demand time series cannot be just random samples from this distribution because the real demand time series has strong daily and weekly serial correlations. The FFT also reveals large magnitude periods of 24 hours and 168 hours.

1. Can I perform discrete wavelet transform to extract only the large magnitude frequencies localised in time? Then take the inverse and recreate the seasonal part of the real demand time series?

2. Then, subtract the seasonal part from the real demand time series to get the residuals. Now, the problem is how to model these residual distribution? The residuals can contain positive and negative numbers so negative binomial distribution is not suitable right?

3. can I shift the residuals to the positive domain and still model them using negative binomial distribution? Then after sampling shift the values back to thier respective value (including negative values)?

Or is the residuals coming from another distribution like normal? should I find this out by try MLE of different families of distributions on the residuals and then taking the maximum liklihood distribution?

I would like to add randomly sampled residuals back to the seasonal time series to recreate the real demand time series as closely as possible.

Thanks.