The data situation:
Supose in different forests data have been collected. These data include the temperature at different time points, the hours of rain, the type of forest and whether the event (a forest fire) occured. So the data could be
forest_id timepoint temperature rain type event 1 11 20 4 A 0 1 12 30 1 A 1 2 5 25 3 A 0 3 6 29 0 B 0 3 7 26 1 B 0 3 8 28 0 B 1
The analysis goal:
As mentioned in this SO-question, my focus is not the inductive direction (finding out what makes a fire more likely), but the prediction of future events (what is the expected residual time until a fire appears).
My idea for the analysis:
Currently I think the desired model would provide a constant baseline hazard, and parameters that increase/lower this baseline hazard $\lambda_0$ dependent on the value of the covariates.
eg. it could be
baseline = -1.5,
temperature = 0.1,
rain = -0.2,
typeA = 0.05.
In formula, my hazard rate would be $h(t|x(t)) = \lambda_0 \cdot exp(\sum \beta\cdot x(t))$.
I think this a special case of Cox’s time varying proportional hazard model with $b_0(t) = \lambda_0 \forall t$. But in the comments to my SO-question, it is said that the Cox proportional hazards models does not allow a specification of the baseline hazard.
Petersen (1986) Fitting Parametric Survival Models with Time-Dependent Covariates describes such a method, without giving it a specific name. Blossfeld et. al. (2019) Event History Analysis With Stata has a Chapter 6 called Exponential Models with Time-Dependent Covariates (p. 128 ff.)
How is a survival model with constant (time invariant) baseline hazard and time varying covariates called? Is there a denser/shorter/more common name than Exponential Models with Time-Dependent Covariates?
When I discard the the timepoints before the event, I can run a prediction (based on accelerated failure times) as desired, but this approach discards a lot of information and thus seems be non-optimal.