## Stationarity

A random variable or random process is said to be stationary if all of its statistical parameters are independent of time. While most statistical techniques require that data is stationary, most atmospheric processes are visibly nonstationary.

Instead, climate data sampled more than once per year is usually cyclo-stationary, due to the seasonal nature of our climate. For example, long-term averages of monthly mean temperatures exhibit a very strong annual cycle in the mid- and high latitudes, with one maximum in the summer and one minimum in the winter. While monthly mean temperatures tend to be cyclostationary, average mean January temperatures at a fixed location are approximately stationary, since we may assume that past and future January datasets will possess a similiar mean and standard deviation.

## Approaches to Nonstationary Data

There are three approaches used to manage nonstationary data, two of which attempt to manipulate the data so that stationarity may be reasonably assumed.
1. Subtract the periodic mean function from the data, which creates a series of anomalies (The concept of anomalies are discussed in an alternate section).
• The dataset is called cyclo-stationary in the mean if it becomes stationary after the periodic cycle has been removed.
• Cyclo-stationary data may be analyzed using common statistical techniques.
• Does not account for variability in the dispersion of the data.
2. Transform the series of anomalies calculated in method 1 to standardized anomalies.
• Divide each anomaly value by the standard deviation.
• Referred to as "normalizing" the data.
• Removes the influences of location and variability.
3. Stratify the data.
• Analyze separate subsets of the data that are short enough to be considered stationary.
• For example, analyzing observations for all available January data for a given location, making the assumption that the 31-day record is uninfluenced by seasonal changes.

## Analyzing and Transforming Cyclo-Stationary Data

Example: Normalize Cyclo-Stationary Data: Sea Surface Temperatures at 50.5W, 29.5N.
 Locate Dataset and Variable Select the "Datasets by Catagory" link in the blue banner on the Data Library page. Click on the "Air-Sea Interface" link. Select the NOAA NCEP EMC CMB GLOBAL Reyn_Smith dataset. Click on the "Reyn_SmithOIv2" link under the Datasets and Variables subheading. Select the "monthly" link under the Datasets and Variables subheading. Choose the "Sea Surface Temperature" link again under the Datasets and Variables subheading. CHECK Select Temporal and Spatial Domains Click on the "Data Selection" link in the function bar. Enter the text 50.5 W, 29.5 N, and Jan 1982 to Dec 2003 in the appropriate text boxes. Press the Restrict Ranges button and then the Stop Selecting button. CHECK View Sea Surface Temperature Time Series To see the result of this operation, choose the time series viewer in the function bar. CHECK Time Series of Sea Surface Temperature at 50.5W, 29.5N Notice that sea surface temperatures are indeed cyclo-stationary. Even though sea surface temperatures do not vary seasonally as much as temperatures on land, there still appears to be a distinct maximum and minimum present each year. Depending on the statistical methods that will be applied to the data, it may be important to transform the data to be approximately stationary. Generate Periodic Mean Function Return to the dataset page by clicking on the right-most link in the blue source bar at the top of the page. Click on the "Expert Mode" link in the function bar. Type the following command under the text already there. ```T 12 splitstreamgrid ``` Press the OK button. CHECK The splitstreamgrid command splits the time grid into two new time grids. The T grid has a period of 12 months and a step of 1. This grid represents data from January, Februrary, March, etc. The T2 grid has a step of 12 and represents the years from the beginning of the dataset to the end of the dataset. Select the "Filters" link in the function bar. Choose the Average over "T2" command. CHECK EXPERT Taking the average over T2 will generate the mean SST for each month. In this case, the mean function is also the monthly climatology. Create Anomalies Click on the "Expert Mode" link in the function bar if the Expert Mode text box is not displayed. Enter the following lines under the text already there: ``` SOURCES .NOAA .NCEP .EMC .CMB .GLOBAL .Reyn_SmithOIv2 .monthly .sst X (50.5W) VALUES T (Jan 1982) (Dec 2003) RANGEEDGES Y (29.5N) VALUES ``` Click the OK button. CHECK The above commands add another variable to the interface: the original Reyn_Smith Monthly SST data. Again in the Expert Mode text box, enter the following lines under the text already there: ```sub -1 mul ``` Click the OK button. CHECK The sub command subtracts the Reyn_Smith Monthly SST dataset from the mean Monthly SST dataset. However, we want the mean to be subtracted from the original data so we multiply by -1 in the next command. In the atmospheric sciences, when an average is subtracted from a data value, it is called an anomaly. The concept of anomalies are discussed in further detail in the Climatologies and Standardized Anomalies section. To see the result of these operations, choose the time series viewer. CHECK Sea Surface Temperature Anomalies at 50.5W, 29.5N Negative anomalies represent below average temperatures while positive anomalies represent above average temperatures. Normalize Data Return to the dataset page by clicking on the right-most link in the blue source bar at the top of the page. In the Expert Mode text box, enter the following lines under the text already there: ```[T] standardize ``` Click the OK button. CHECK The function standardize removes the influences of dispersion. In the previous step, we subtracted the mean from each value. This produces a series of anomalies. The standardize function divides each anomaly by the RMSA, which is very similar to standard deviation. To learn more about the root mean square anomaly and standard deviation functions, refer to the Measures of Dispersion section. To see the result of these operations, choose the time series viewer. CHECK Standardized Sea Surface Temperature Anomalies at 50.5W, 29.5N The standardized anomaly dataset has a mean of 0 and standard deviation of 1. Note that standardized anomalies are dimensionless quantities.