SAP HANA Predictive Analysis Library (PAL)

3.5.8 Single Exponential Smoothing
Single Exponential Smoothing model is suitable to model the time series without trend and seasonality. In the
model, the smoothed value is the weighted sum of previous smoothed value and previous observed value.
PAL provides two simple exponential smoothing algorithms: single exponential smoothing and adaptiveresponse-rate
simple exponential smoothing. The adaptive-response-rate single exponential smoothing
algorithm may have an advantage over single exponential smoothing in that it allows the value of alpha to be
modified.
For single exponential smoothing, let S t be the smoothed value for the t-th time period. Mathematically:
S 1 = x 0
S t = αx t−1 + (1−a)S t−1
Where α∈(0,1) is a user specified parameter. Forecast is made by:
S T+1 = αx T + (1−a)S T
For adaptive-response-rate single exponential smoothing, let S t be the smoothed value for the t-th time period.
Initialize for adaptive-response-rate single exponential smoothing as follows:
S 1 = x 0
α1 = α2 = α3 = δ = 0.2
A 0 = M 0 = 0
Update the parameter of α as follows:
E t = X t − S t
A t = δE t + (1 − δ)A t-1
M t = δ|E t | + (1 − δ)M t-1
The calculation of the smoothed value as follows:
S t+1 = α t x t + (1 − α t )S t
Where α, δ ∈(0,1) is a user specified parameter, and | | denotes absolute values.
It is worth nothing that when t ≥ T+2, the smoothed value S t , that is, the forecast value, is always S T+1 (x t−1 is
not available and S t−1 is used instead).
PAL calculates the prediction interval to get the idea of likely variation. Assume that the forecast data is
normally distributed. The mean value is S t and the variance is σ 2 . Let U t be the upper bound of prediction
interval for S t and L t be the lower bound. Then they are calculated as follows:
U t = S t + zσ
L t = S t - zσ
Here z is the one-tailed value of a standard normal distribution. It is derived from the input parameters
PREDICTION_CONFIDENCE_1 and PREDICTION_CONFIDENCE_2.
394 P U B L I C
SAP HANA Predictive Analysis Library (PAL)
PAL Functions