Exponential smoothing models backcasting and determining initial values python

Question

I have made python code for exponential smoothing (ES) that takes in about 15 different cases including:

• Simple Exponential Smoothing (SES)
• Simple Seasonal models (both multiplicative and additive)
• Brown's Linear Exponential Smoothing
• Holt's Double Exponential Smoothing
• Exponential trend method
• Damped-Trend Linear Exponential Smoothing
• Multiplicative damped trend (Taylor, 2003)
• Holt-Winters Exponential Smoothing: multiplicative trend, additive trend, multiplicative season, additive season, and damped models for all four variations

14 of these cases can be found in page 8 of Exponential smoothing: The state of the art Part II.

What values are commonly used for the initial values for the different ES models? What methods are used to determine points at time 0 and 1? (I am using Nan (not a number) to substitute in the code right now).

Answer 1

Some common choices for initial values are given at the bottom of https://www.otexts.org/fpp/7/6.

However, it is much better to optimize the initial values along with the smoothing parameters.

Answer 2

If the assumptions of the smoothing methods are met, the models are well initiated by first values of whatever their parameters use. That is either first value, first difference or first seasonal set and first seasonal differences for additive single, double and seasonal exponential smoothing methods respectively.