# Methods and formulas for Trend Analysis

Select the method or formula of your choice.

## Linear

### Formula

The linear trend model is:

Yt = β0 + β1 t + et

### Notation

TermDescription
β0 the constant
β1 average change from one period to the next
tvalue of the time unit
etthe error term

### Formula

The quadratic trend model, which can account for simple curvature in the data, is:

Yt = β0 + β1 t + β2 t2 + et

### Notation

TermDescription
β0 the constant
β1 and β2 the coefficients
tvalue of the time unit
etthe error term

## Exponential growth

### Formula

The exponential growth trend model accounts for exponential growth or decay. For example, a savings account might exhibit exponential growth.

Yt = β0 β1t + et

### Notation

TermDescription
β0 the constant
β1 the coefficient
tvalue of the time unit
etthe error term

## S-curve

### Formula

The data has an S-shape, which indicates that the direction of the change varies over time.

Yt = 10a / (β0 + β1 β2t )

### Notation

TermDescription
β0 the constant
β1 and β2 the coefficients
tvalue of the time unit

## Weights

If you supply coefficients from a previous trend analysis fit, Minitab performs a weighted trend analysis. If the weight for a particular coefficient is α, Minitab estimates the new coefficient by:

α p1 + (1 – α)p2

### Notation

TermDescription
p1 coefficient estimated from the current data
p2 prior coefficient

## Forecasts

Minitab uses the trend equation to calculate the forecast for specific time values. Data before the forecast origin are used to fit the trend.

## 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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