Find definitions and interpretation guidance for every graph that is provided with ARIMA.

Time Series Plot

The time series plot displays the data in chronological order. When you generate forecasts, Minitab displays the forecasts and their 95% confidence limits on the plot.

Interpretation

Use the time series plot to determine whether the data are stationary. A stationary time series has a mean, variance, and autocorrelation function that are essentially constant through time. Examine the time series plot to determine the following:
  • Determine whether different variations are present in the data. If different variations are present, then you must transform the data so that the variance is constant.
  • Determine whether the data are centered around a constant mean. If the mean is not constant, you may need to difference the data to make the mean constant.

ACF of Residuals

The plot shows the autocorrelation function of the residuals. The autocorrelation function is a measure of the correlation between the observations of a time series that are separated by k time units (yt and yt–k).

Interpretation

Use the autocorrelation function of the residuals to determine whether the model meets the assumptions that the residuals are independent. If the assumption is not met, the model may not fit the data and you should use caution when you interpret the results. If no significant correlations are present, then you can conclude that the residuals are independent. However, you may see 1 or 2 significant correlations at higher order lags that are not seasonal lags. These lags are usually due to random error, and are not a sign that the assumption is not met. So, in this case, you can also conclude that the residuals are independent.

PACF of Residuals

The partial autocorrelation function is a measure of the correlation between the observations of a time series that are separated by k time units (yt and yt–k), after adjusting for the presence of all the other terms of shorter lag (yt–1, yt–2, ..., yt–k–1).

Interpretation

Use the partial autocorrelation function of the residuals to determine whether the model meets the assumptions that the residuals are independent. If the assumption is not met, the model may not fit the data and you should use caution when you interpret the results. If no significant correlations are present, then you can conclude that the residuals are independent.

Histogram of the residuals

The histogram of the residuals shows the distribution of the residuals for all observations. lf the model fits the data well, the residuals should be random with a mean of 0. So the histogram should be approximately symmetric around 0.

Normal probability plot of the residuals

The normal plot of the residuals displays the residuals versus their expected values when the distribution is normal.

Interpretation

Use the normal plot of the residuals to determine whether the residuals are normally distributed. However, this analysis does not require normally distributed residuals.

If the residuals are normally distributed, the normal probability plot of the residuals should approximately follow a straight line. The following patterns imply that the residuals are not normally distributed.

S-curve implies a distribution with long tails.

Inverted S-curve implies a distribution with short tails.

Downward curve implies a right-skewed distribution.

A few points lying away from the line implies a distribution with outliers.

Residuals versus fits

The residuals versus fits plot displays the residuals on the y-axis and the fitted values on the x-axis.

Interpretation

Use the residuals versus fits plot to determine whether the residuals are unbiased and have a constant variance. Ideally, the points should fall randomly on both sides of 0, with no recognizable patterns in the points.

The patterns in the following table may indicate that the residuals are biased and have a nonconstant variance.
Pattern What the pattern may indicate
Fanning or uneven spreading of residuals across fitted values Nonconstant variance
Curvilinear A missing higher-order term
A point that is far away from zero An outlier

If you see nonconstant variance or patterns in the residuals, your forecasts may not be accurate.

Residuals versus order

The residuals versus order plot displays the residuals in the order that the data were collected.

Interpretation

Use the residuals versus order plot to determine how accurate the fits are compared to the observed values during the observation period. Patterns in the points may indicate that model does not fit the data. Ideally, the residuals on the plot should fall randomly around the center line.

The following patterns may indicate that the model does not fit the data.
Pattern What the pattern may indicate
A consistent long-term trend The model does not fit the data
A short-term trend A shift or a change in pattern
A point that is far away from the other points An outlier
A sudden shift in the points The underlying pattern for the data has changed
The following examples show patterns that may indicate that the model does not fit the data.

Residuals systematically decrease as the order of the observations increases from left to right.

A sudden change in the values of the residuals occurs from low (left) to high (right).

Residuals versus variables

The residuals versus variables plot displays the residuals versus another variable.

Interpretation

Use the plot to determine whether the variable affects the response in a systematic way. If patterns are present in the residuals, the other variables are associated with the response. You can use this information as the basis for additional studies.

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