# Methods and formulas for Winters' Method

Select the method or formula of your choice.

## Multiplicative

### Formula

The multiplicative model is:

• Lt = α (Yt / St–p ) + (1 – α) [Lt–1 + Tt–1 ]
• Tt = γ [Lt Lt–1 ] + (1 – γ) Tt–1
• St = δ (Yt / Lt ) + (1 – δ) St–p
• = (Lt–1 + Tt–1 ) St–p

### Notation

TermDescription
Lt level at time t, α is the weight for the level
Tt trend at time t,
γ weight for the trend
St seasonal component at time t
δ weight for the seasonal component
p seasonal period
Yt data value at time t
fitted value, or one-period-ahead forecast, at time t

### Formula

• Lt = α (Yt St–p ) + (1 – α) [Lt–1 + Tt–1 ]
• Tt = γ [Lt Lt–1 ] + (1 – γ) Tt–1
• St = δ (Yt Lt ) + (1 – δ) St–p
• = Lt–1 + Tt–1 + St–p

### Notation

TermDescription
Lt level at time t, α is the weight for the level
Tt trend at time t,
γ weight for the trend
St seasonal component at time t
δ weight for the seasonal component
p seasonal period
Yt data value at time t
fitted value, or one-period-ahead forecast, at time t

## Model fitting

Winters' method employs a level component, a trend component, and a seasonal component at each period. It uses three weights, or smoothing parameters, to update the components at each period. Initial values for the level and trend components are obtained from a linear regression on time. Initial values for the seasonal component are obtained from a dummy-variable regression using detrended data.

## Forecasting

Winters' Method uses the level, trend, and seasonal components to generate forecasts. Winters' Method also uses data up to the forecast origin time to generate the forecasts.

### Formula

The forecast for m periods ahead from a point at time t is:
• Multiplicative method: (Lt + mTt) * St + mp
• Additive method: Lt + mTt +St + mp

### Notation

TermDescription
Lt level
Tt trend at time t
TermDescription
St + mpseasonal component for the same period from the previous year

## MAPE

Mean absolute percentage error (MAPE) measures the accuracy of fitted time series values. MAPE expresses accuracy as a percentage.

### Notation

TermDescription
yt actual value at time t
fitted value
n number of observations

Mean absolute deviation (MAD) measures the accuracy of fitted time series values. MAD expresses accuracy in the same units as the data, which helps conceptualize the amount of error.

### Notation

TermDescription
yt actual value at time t
fitted value
n number of observations

## MSD

Mean squared deviation (MSD) is always computed using the same denominator, n, regardless of the model. MSD is a more sensitive measure of an unusually large forecast error than MAD.

### Notation

TermDescription
yt actual value at time t
fitted value
n number of observations
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