Standardized Time Series

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As a guide to standardizing a time series, we will first review the procedure of standardizing a scalar statistic. We will use the familiar t-statistic as an example. The data will consist of n observations

that are independent and have identical normal distributions. We wish to make inferences about the unknown population mean, μ. The average of the data sample

will be the statistic used for these inferences. The population variance, Image:sts3.png, is an unknown nuisance parameter.

Standardization involves the following steps.

1. Center the Statistic: The population mean, μ, is subtracted from the sample mean giving the random variable, Image:sts4.png, which has an expected value of zero. (Strictly speaking, this difference between the average and the mean would not be called a "statistic" since it includes the parameter, μ.)
2. Scale the Statistic Magnitude: Since statistics can come in an almost endless variety of measurement units, we will need to express the statistic in a common unit of measurement called a standard deviation. The magnitude of the statistic is scaled by dividing by Image:sts5.png. Our statistic is now
which is our standardized statistic. Standardized sample means will all have the same first two moments. The unknown scaling parameter, σ, can be either estimated or cancelled out of a ratio statistic; the cancelling out of this parameter in a ratio statistic is the more common approach and is followed here.
3. Cancel the Scale Parameter: The data is aggregated or batched into b exclusive adjacent groups of size m (if necessary, discard data from the front of the run so that Image:sts7.png). The average of each batch is denoted as Image:sts8.png, i = 1,..., b. The usual unbiased estimator of the variance of the batched means is
Inferences about the parameter, µ, are based on the random t-ratio,
4. Apply Limit Theorems: The limiting distribution of Tb-1 is known. As Image:sts11.png (making Image:sts12.png since b is fixed), the distribution function of Image:sts13.png converges to that of a Image:sts14.png random variable with b-1 degrees of freedom. As Image:sts15.png, Image:sts16.png will converge to the constant μ from the law of large numbers. Also, from the Central Limit Theorem of statistics, the distribution function of Image:sts17.png will converge to that of a standard normal random variable. Thus, the distribution function of the ratio, Tb-1 (being a continuous mapping) will converge to that of a t random variable with b-1 degrees of freedom. The unknown scaling constant, σ, is cancelled out of the ratio.
5. Use the Limiting Probability Model for Inference: The limiting distribution of Tb-1 can be used for statistical inference and estimation.

The concept of standardization can be applied to an entire time series. The original series of observations is transformed into a standardized series of observations. We will hypothesize (and test) that the series is stationary. We also assume that there is some minimal amount of randomness in the process; however, we do not assume that the data is independent. The mathematical assumptions needed are given in a paper by Schruben (1983), where it is argued that many simulations on a computer will meet the imposed restrictions for applicability. Suppose we want to standardize the output from the ith run of a simulation, with i representing the run number. Let

denote m stationary but perhaps dependent observations in the ith output time series. We will standardize the sequence of cumulative means up to and including the kth observations, given by,

Similar steps to those in standardizing a scalar statistic are followed in standardizing a time series. The steps in standardization are as follows:

1. Center the Series: For run i, the sequence given by
will have a mean of zero if the series has a constant mean.
2. (a) Scale the Series Magnitude: The scaling constant for dependent sequences (independent of the run i) that we use is defined as
which is just the population variance in the special case of independent identically distributed data. Magnitude scaling is done by dividing Image:sts22.png by Image:sts23.png. The scaling constant is again unknown but will cancel out of our statistics as before.

There is one additional step required that was not necessary in the scalar standardization case. Different time series can be of different length, so we must also scale the length of the series. Thus, we have the additional step:

(b) Scale the Series Index: We will define the continuous index, Image:sts24.png. Our previous index is thus given by Image:sts25.png. We also add the starting point Image:sts26.png so that the standardized series is 0 at t = 0 and t = 1. The result is that all standardized time series have indices on the unit interval and start and end at zero. We now have what we will call a standardized time series given by
where Image:sts28.png is the greatest integer function.
3. Cancel the Scale Parameter: There are several functions that might be considered for the denominator of a ratio that cancels σ. We will consider here only one such function, the sum or limiting area under the function Image:sts29.png.
4. Apply Limit Theorems: The standardized series, Image:sts29.png, will converge in probability distribution to that of a Brownian Bridge stochastic process. Thus, the Brownian Bridge process plays the role in time series standardization that the normal random variable played in scalar standardization. An important feature of the standardized series, Image:sts29.png, is that it is constructed to be asymptotically independent of the sample mean, Image:sts31.png.
There are several functions of Image:sts29.png that will also be asymptotically Image:sts32.png distributed. The area, Image:sts33.png, will have a limiting Image:sts34.png normal distribution with zero mean and variance
will have a limiting Image:sts37.png distribution with one degree of freedom.
Now consider where each of b independent replications (or b batches of data) are standardized in the manner above. We can then add the resulting Image:sts37.png random variables, Image:sts36.png, for each replication or batch to obtain a Image:sts37.png random variable with b degrees of freedom.
5. Use the Limiting Probability Model for Inference: In a manner similar to the scalar case, the standardized (scalar) sample mean of all of the data is divided by the square root of Image:sts38.png over b to form a ratio where the unknown scale parameter, σ , cancels. For large values of m, the distribution of this ratio can be accurately modeled as having a t distribution with b degrees of freedom. The resulting (1-α)100% asymptotic confidence interval for the mean μ is
where Image:sts40.png is the "grand mean" of all of the data in all batches or replications. More complicated, but superior, confidence intervals can be obtained by weighting the standardized time series as in Goldsman and Schruben (1990). The SIGMA function STS{X} is equal to Image:sts41.png for the output time series of values of X; this function can be used with the other weightings given in the reference.
Also, each of the replication or batch means can be treated as a scalar random variable and standardized and squared, giving another Image:sts42.png random variable. Due to the independence of Image:sts43.png and the Image:sts44.png's, these Image:sts42.png random variables can be added, giving a Image:sts42.png random variable with 2b-1 degrees of freedom. This can be considered as a "pooled" estimator of Image:sts45.png, which we will denote as Q. The same types of inferences can be made for the dependent simulation output series as were applicable in the independent data case. The resulting "t variate" is given by

Theoretical properties of confidence intervals formed using standardized time series are presented by Goldsman and Schruben (1984), which compares the standardized time series approach to conventional methods.

Standardized time series has been implemented in several simulation analysis packages, most notably at IBM (Heidelberger and Welch, 1983), Bell Labs (Nozari, 1985), and General Electric (Duersch and Schruben, 1986). These packages typically control initialization bias (see also Schruben, Singh, and Tierney, 1983, and Schruben, 1982) and run duration as well as produce confidence intervals. Other applications of standardized time series have been to selection and ranking problems (Goldsman, 1983) and simulation model validation (Chen and Sargent, 1984).

The asymptotic arguments above require that the batch size, m, become large as n is increased. The common method is to allow the batch size to grow as the sample size increases and keep the number of batches, b, fixed. Fixing b at 10 or 20 seems reasonable in most applications as long as the sample size is large (see Schmeiser, 1983).

The limiting probability model of a t random variable has the virtue that it is widely tabulated and has been studied extensively. There exist other limiting models that might be used, but none have been developed to the extent of the t model.

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