# Decomposition of time series into weighted (weights known) stretched exponentials with unknown offset function

Looking for solution to decomposing function $$f(t)$$ into stretched exponential functions all with same meta-exponent and decay constants, but with different amplitudes $$p$$ (known) and offsets $$\tau$$ (unknown).

$$f(t)=\int_{-\infty}^{t}dt' \:p(t')e^{-\left(\frac{\left( t-\tau(t')\right)}{\alpha}\right)^\beta}$$

where $$f(t)>0$$ and $$p(t)>0$$ for all $$t$$, and both $$\alpha>0$$ and $$\beta>0$$. Want a closed-form solution for the delay function $$\tau(t)$$

Also interested in solving for $$\tau(t')$$ where the argument of the exponent contains an additional constant positive offset $$t_0>0$$ as well: $$e^{-\left(\frac{\left(t+t_0-\tau(t')\right)}{\alpha}\right)^\beta}$$