# Methods and formulas for Decomposition

Select the method or formula of your choice.

## Multiplicative

### Formula

Yt = Trend × Seasonal × Error

### Notation

TermDescription
Yt observation at time t

### Formula

Yt = Trend + Seasonal + Error

### Notation

TermDescription
Yt observation at time t

## Model fitting

Decomposition involves the following steps:
1. Minitab smoothes the data using a centered moving average with a length equal to the length of the seasonal cycle. When the seasonal cycle length is an even number, a two-step moving average is required to synchronize the moving average correctly.
2. Minitab either divides the moving average into (multiplicative model) or subtracts it from (additive model) the data to obtain what are often referred to as raw seasonal values.
3. For corresponding time periods in the seasonal cycles, Minitab determines the median of the raw seasonal values. For example, if you have 60 consecutive months of data (5 years), Minitab determines the median of the 4 raw seasonal values corresponding to January, to February, and so on.
4. Minitab adjusts the medians of the raw seasonal values so that their average is one (multiplicative model) or zero (additive model). These adjusted medians constitute the seasonal indices.
5. Minitab uses the seasonal indices to seasonally adjust the data.
6. Minitab fits a trend line to the seasonally adjusted data using least squares regression.

The data can be detrended by either dividing the data by the trend component (multiplicative model) or subtracting the trend component from the data (additive model).

## Forecasting

Decomposition calculates the forecast as the linear regression line multiplied by (multiplicative model) or added to (additive model) the seasonal indices. Data prior to the forecast origin are used for the decomposition.

## 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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