Advanced Graphical Analysis

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Detecting Trends using Standardized Time Series

Using a standardized time series (STS) plot, it is possible to detect trends in the output that might not otherwise be visible. The STS plot should appear to be symmetric about zero if there is no trend in the data (adequate initial truncation). When there is a trend, the STS plots will be pulled either above (increasing trend) or below (decreasing trend) the zero line. STS can also be used for other types of inference such as confidence interval estimation. The STS plots in SIGMA are the unscaled standardized time series for the selected output measurement. The best way to learn how to use STS plots is to look at a few simple output series. Try generating a sequence of independent random variables, say, with the one state change


Look at the line plot of the successive values of X and the corresponding STS plot. Now add a trend to the data, say,


and look at the difference in the same two plots. Do this for some more interesting and subtle trends (exponential decay, quadratic, sudden shift, etc.) and see how the STS behaves. You will find that familiarity with the behaviors of STS plots is a valuable visual tool for output analysis.

Standardizing a time series is similar to the familiar procedure of standardizing or normalizing a scalar statistic. Standardizing a scalar statistic, such as a sample mean, involves centering the statistic to have a zero mean and scaling its magnitude to generic units of measurement called standard deviations. Limit theorems can be applied that give us the asymptotic (large sample) probabilistic behavior of correctly standardized statistics under certain hypotheses. This limiting model for scalar statistics is typically the standard normal probability distribution. This model can be used for statistical inference such as testing hypotheses or constructing confidence intervals. Here we extend this concept to the standardization of an entire time series.

The value of standardizing time series comes from the fact that the same mathematical analysis can be applied to series from a variety of sources. Thus, the technique of standardization serves as a mathematical surrogate for experience with the data under study. No matter what the original time series looks like, the standardized time series will be familiar if certain hypotheses are correct. An unusual appearance of a standardized time series can be used to conclude that these hypotheses are not valid. The statistical significance of these conclusions can be computed in the same manner as with standardized scalar statistics.

To illustrate using STS plots to detect trends, consider first the STS plot in the next figure. Since this plot is mostly negative, a clear downward trend in the data is evident.

STS Plot Indicating a Downward Trend in the Data
STS Plot Indicating a Downward Trend in the Data

A line plot of the actual data is shown in the next figure, where the downward trend indicated by the STS plot is at best only marginally apparent.

Plot of the Raw Data Series for the STS Plot Above
Plot of the Raw Data Series for the STS Plot Above

The next figure is an STS plot that indicates the presence of a strong increasing trend in the data. The STS plot is pulled in the positive direction by this positive trend in the data. The above figure should be compared to the below figure, where a negative trend was indicated.

An STS Plot indicating an Increasing Trend in the Data
An STS Plot indicating an Increasing Trend in the Data

The raw data for the STS plot in the above figure is plotted in the below figure, where the increasing trend in the data is again only slightly detectable.

Raw Data for the STS Plot Above
Raw Data for the STS Plot Above

The sensitivity of STS plots to trend has a down-side: they can indicate a trend which may disappear as more data is collected. However, given the potential seriousness of simulation initialization bias causing an artificial trend in the output, it seems better to be able to detect trends easily at the risk of falsely indicating a non-existent trend.


The autocorrelation function is a plot of the correlation between two observations in the same output series as a function of how far apart (lag) the observations are. For example, the lag 1 autocorrelation function is the sample correlation between two adjacent output measurements. The autocorrelation function should drop off sharply at a lag of 1 if the observations are not correlated. This would indicate that the batch size is large enough to remove correlations between successive batched means and initial transient output observations have been truncated. Crude 95% confidence bounds at Image:math1.png can be used as a very rough guide (Brockwell and Davis, 1987).

The standardized time series (STS) plots can also be used to visually assess whether or not there is significant positive serial correlation in an output series. The more jagged the STS plot appears, the less serial dependency in the output. If the STS plot is smoother than you are typically used to seeing, you can suspect either a serious trend in the data or significant positive serial dependency between successive observations of the output.

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