# About modelling transient effect on time series

I'm working on a time series model which predicts daily sales. Below is a simplified depiction of the time series I'm trying to model.

It is a stationary series and it takes certain 'shock'(drastic increase in daily sales) and the effect of this 'shock' fades away gradually(almost reciprocally). My model was basically based on seasonal trend decomposition by Loess, but since this 'shock' happens not in a periodic way, the seasonality component couldn't capture the effect of it. I'm trying to add a new regressor to take these effects into account. Are there any good solution to encode this variable? I can identify the initial dates when these effects start, but the way they fade away has to be modeled, I think. Sorry about not including the actual data or model summary, etc. It is corporate data, so it's proprietary.

Well this looks extremely like neuronal spiking...

What I can recommend is to construct a kernel that sketches the shape of each 'shock'.

Then you can transform your data into an array of discrete 'shock' events.

Equivalently, your original signal is the convolution of event markers and kernel.

$Sig(t)=(Shock*kernel)(t)=\int_{-\infty}^{+\infty}Shock(\tau)kernel(t-\tau)d\tau$

since you said you can already identify the initial dates, getting kernel is essentially just taking the average of the signal pieces aligned to each initial date.