Time series modelling
First idea is to use, instead of an invalid data point, the last valid value. This is done in finance where some stocks are not quoted everyday (and when you want to publish the value of your fund you use the last available value for the stocks).

However, this can be an issue when you have too many missing values (especially for prediction).

You can try Bayesian data augmentation (works somewhat well with time series) but if you say that you have a lot of missing data you have to accept that anything that you do on rather poor quality data will be of limited quality.