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**Correlation of Time Series**

**Applies to:** @RISK 6.x/7.x, Industrial Edition

How does time series correlation work? Is it different from correlating regular distributions?

Short answer: You can correlate between different time series, but that correlation may not be visible in the displayed numbers.

Let's clarify the time series process, to help in understanding correlation of time series. You create a time series by fitting **real-world data**. However, the fitting process usually involves **transforms,** such as differencing or taking the logarithm. @RISK actually fits the **transformed data,** not your real-world data. The time series is a series of transformed data, not real-world data. After fitting a series, @RISK projects the series forward into the future. Each time period's prediction has two components: a **formula** that uses transformed data from one or more previous time periods, plus a randomly generated **noise term** or error value. Remember, all of the predictions are done with the **underlying time series values**, not with real-world data. After computing the formula and noise term for a time period, @RISK reverses the transforms that were used in the fit, and the result is the **displayed numbers** that you see in your Excel worksheet. The displayed numbers differ from the underlying time series values when there are transforms in that particular time series.

You can correlate two or more time series functions using the @RISK *Define Correlations* window, or manually using RiskCorrmat( ) property functions, just as you would correlate regular @RISK distribution functions. However, **correlation between time series is fundamentally different from the correlation of standard distributions.** In correlating regular distributions, all the values for all iterations form one array per distribution, and the correlation is applied to those two arrays. **For regular distributions, correlation is an attribute of the whole simulation,** not of any particular iteration. By contrast, **when two time series functions are correlated, the correlation is reapplied, from scratch, in each iteration.** The two arrays that get correlated are the noise terms within the projected time periods of the two time series for that particular iteration. Again, this happens within each iteration, without reference to any other iteration.

This explains why **the displayed values won't match your correlation coefficients.** The displayed values aren't correlated; only the ** noise term parts of each underlying time series value are correlated.** The formula parts can't be correlated because they are generated by the time series function, so the underlying time series values as a whole are not correlated. What you see is the real-world data, computed by reversing the transforms of that particular time series. But even if there are no transforms, so that the displayed numbers equal the underlying time series values, still the correlated noise term is just part of each displayed number. Correlations are honored but are effectively buried, so

Can I correlate the successive periods of a given time series?

There is no way to correlate the noise terms between time periods of a given time series. Time series correlation applies only between one time series and another.

And when the time series were produced by batch fit?

When you generate sets of correlated time series using the** time series batch fit** command, it constructs a correlation matrix as part of its output. (See Correlated Time Series in Batch Fit.) The generated coefficients apply to the historical data that you supplied. You're free to alter the correlation coefficients, or add or remove time series in the matrix.

Forward projections use the coefficients in that matrix just like any other correlations for time series. These correlations are **applied in the same way as described above**: the noise terms of the underlying time series values can be correlated, but those noise terms are not displayed separately.

Can I define a copula for two or more time series?

Copulas cannot be defined for Time Series array functions, only for regular @RISK input distributions.

Last edited: 2016-09-19

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