Time Series Overview
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Minitab's time series procedures can be used to analyze data collected over time, commonly called a time series. These procedures include simple forecasting and smoothing methods, correlation analysis methods, and ARIMA modeling. Although correlation analysis may be performed separately from ARIMA modeling, we present the correlation methods as part of ARIMA modeling.

Simple forecasting and smoothing methods are based on the idea that reliable forecasts can be achieved by modeling patterns in the data that are usually visible in a time series plot, and then extrapolating those patterns to the future. Your choice of method should be based upon whether the patterns are static (constant in time) or dynamic (changes in time), the nature of the trend and seasonal components, and how far ahead that you wish to forecast. These methods are generally easy and quick to apply.

ARIMA modeling also makes use of patterns in the data, but these patterns may not be easily visible in a plot of the data. Instead, ARIMA modeling uses differencing and the autocorrelation and partial autocorrelation functions to help identify an acceptable model. ARIMA stands for Autoregressive Integrated Moving Average, which represent the filtering steps taken in constructing the ARIMA model until only random noise remains. While ARIMA models are valuable for modeling temporal processes and are also used for forecasting, fitting a model is an iterative approach that may not lend itself to application speed and volume.

Simple forecasting and smoothing methods

The simple forecasting and smoothing methods model components in a series that are usually easy to see in a time series plot of the data. This approach decomposes the data into its component parts, and then extends the estimates of the components into the future to provide forecasts. You can choose from the static methods of trend analysis and decomposition, or the dynamic methods of moving average, single and double exponential smoothing, and Winters' method. Static methods have patterns that do not change over time; dynamic methods have patterns that do change over time and estimates are updated using neighboring values.

You may use two methods in combination. That is, you may choose a static method to model one component and a dynamic method to model another component. For example, you may fit a static trend using trend analysis and dynamically model the seasonal component in the residuals using Winters' method. Or, you may fit a static seasonal model using decomposition and dynamically model the trend component in the residuals using double exponential smoothing. You might also apply a trend analysis and decomposition together so that you can use the wider selection of trend models offered by trend analysis (see Example of trend analysis and Example of decomposition). A disadvantage of combining methods is that the confidence intervals for forecasts are not valid.

For each of the methods, the following table provides a summary and a graph of fits and forecasts of typical data.

Correlation analysis and ARIMA modeling

Examining correlation patterns within a time series or between two time series is an important step in many statistical analyses. The correlation analysis tools of differencing, autocorrelation, and partial autocorrelation are often used in ARIMA modeling to help identify an appropriate model.

ARIMA modeling can be used to model many different time series, with or without trend or seasonal components, and to provide forecasts. The forecast profile depends upon the model that is fit. The advantage of ARIMA modeling compared to the simple forecasting and smoothing methods is that it is more flexible in fitting the data. However, identifying and fitting a model may be time-consuming, and ARIMA modeling is not easily automated.

·    Differences computes and stores the differences between data values of a time series. If you wish to fit an ARIMA model but there is trend or seasonality present in your data, differencing data is a common step in assessing likely ARIMA models. Differencing is used to simplify the correlation structure and to reveal any underlying pattern.

·    Lag computes and stores the lags of a time series. When you lag a time series, Minitab moves the original values down the column, and inserts missing values at the top of the column. The number of missing values inserted depends on the length of the lag.

·    Autocorrelation computes and plots the autocorrelations of a time series. Autocorrelation is the correlation between observations of a time series separated by k time units. The plot of autocorrelations is called the autocorrelation function or acf. View the acf to guide your choice of terms to include in an ARIMA model.

·    Partial Autocorrelation computes and plots the partial autocorrelations of a time series. Partial autocorrelations, like autocorrelations, are correlations between sets of ordered data pairs of a time series. As with partial correlations in the regression case, partial autocorrelations measure the strength of relationship with other terms being accounted for. The partial autocorrelation at a lag of k is the correlation between residuals at time t from an autoregressive model and observations at lag k with terms for all intervening lags present in the autoregressive model. The plot of partial autocorrelations is called the partial autocorrelation function or pacf. View the pacf to guide your choice of terms to include in an ARIMA model.

·    Cross Correlation computes and graphs correlations between two time series.

·    ARIMA fits a Box-Jenkins ARIMA model to a time series. ARIMA stands for Autoregressive Integrated Moving Average. The terms in the name-Autoregressive, Integrated, and Moving Average-represent filtering steps taken in constructing the ARIMA model until only random noise remains. Use ARIMA to model time series behavior and to generate forecasts.