I'm familiar with linear interpolation. I tried to search some details about piece-wise linear interpolation but failed to understand this method. Can someone explain the difference between these two interpolation methods or perhaps offer some references to learn?
piece-wise models, at least in my field, mean a regression model with where the slope is "broken" into different pieces or sections. This is commonly done in models dealing with repeated measures across time. For example, imagine that you have a time series model with 10 data points. You administered your treatment at data point 4. You would have two "sections" of your regression slope. One before treatment and one after. Comparing the two slopes lets you know things like the difference between the two treatments and the difference in rate of change.
My suggestion is read one of the very handy books on time series modeling. I have no suggestions since the resources I would suggest would assume a background in R which I do not know if you have.