Running and Weighted Averages

Both running and weighted averages are important filtering methods for statistical analysis.

Running Average

Often used to illustrate climatic trends by temporally smoothing data.
Calculated by finding a number of successive means, each mean incorporating the same number of observations.
Each successive mean will drop first value of the mean interval and add the next value in the dataset to the next mean interval.
Tends to damp out extreme values and highlights the movement of data with time.
Also known as a moving average.

Example: Calculate a 3-year running average of gridded temperature anomaly data.

Locate Dataset and Variable	Select the "Datasets by Catagory" link in the blue banner on the Data Library page. Click on the "Atmosphere" link. Select the NOAA NCEP CPC CAMS dataset. Select the "anomaly" link under the Datasets and Variables subheading. Choose the "temperature anomaly" link, again located under the Datasets and Variables subheading. CHECK
Select Spatial Domain	Click on the "Data Selection" link in the function bar. Enter the text 130W to 30W and 70S to 70N, in the appropriate text boxes. Press the Restrict Ranges button and then the Stop Selecting button. CHECK
Calculate Running Average	Click on the "Expert Mode" link in the function bar. Enter the following text below the text already there: T 3 runningAverage Press the OK button. CHECK The command above will compute the 3-month running average.
View Running Average	To see your results, choose the viewer with coasts drawn. CHECK The image depicts the running average of the first three months of the dataset, Jan-Mar 1950. The anomlies may be easier to see if we change the color scale. Click on the right most link in the blue source bar to exit the viewer. Enter the following text in the Expert Mode text box below the text already there: DATA 1. STEP startcolormap DATA -10. 10. RANGE white purple purple -10. VALUE cyan -1. VALUE white white 1. bandmax yellow 1. VALUE red 10. VALUE firebrick endcolormap Press the OK button. CHECK These commands format the color scale so that anomalies are easier to observe. Replace the range with a three month period during an El Niño event: Jun-Aug 1983. Click the Redraw button. CHECK Running Average of Gridded Temperature Anomaly Data at 130W-30W, 70S-70N The high positive anomalies off the western coast of South America are assocaiated with the El Niño event that summer.

Example: Calculate the 20-year running average of April precipitation for a location in the Pampas Region of Southern South America.

Locate Dataset and Variable	Select the "Datasets by Catagory" link in the blue banner on the Data Library page. Click on the "Atmosphere" link. Select the NOAA NCEP CPC CAMS dataset. Select the "mean" link under the Datasets and Variables subheading. Choose the "precipitation" link, again located under the Datasets and Variables subheading. CHECK
Select Spatial Domain	Click on the "Data Selection" link in the function bar. Enter the text 60W and 25S, in the appropriate text boxes. Press the Restrict Ranges button and then the Stop Selecting button. CHECK
Select Temporal Domain	Click on the "Expert Mode" link in the function bar. Enter the following lines below the text already there: T 12 splitstreamgrid> T (Apr) VALUES Press the OK button. CHECK The splitstreamgrid function 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 is unperiodic. This grid represents the years from the beginning of the dataset to the end of the dataset. The next command, T (Apr) VALUES, will retain only the April values from the T grid.
View April Precipitation	To see the results of this operation, choose the time series viewer. CHECK Precipitation is labeled on the Y-axis in mm/month and time is labeled on the X-axis in years. Each X-axis value represents mean April precipitation for that year. Mean April Precipitation at 60W, 25S for 1950 to 2000 Without smoothing the data, it may be difficult to recognize any trends over the 50-year span pictured above. Applying a running average, however, will often make any trend in the data more distinguishable.
Calculate Running Average	Click on the right most link in the blue source bar to exit the viewer. Scroll down to the Grids subheading. Notice under the Grids subheading that the new time grid created, T2, represents months since 1950 ordered from 1950 to 2000 by 12. Every 12 grid points in T2 correspond to 1 year. The 20-year running average is calculated over the T2 variable, and must be evaluated in months, not years. In the Expert Mode text box, enter the following line below the text already there: T2 240 runningAverage Press the OK button. CHECK This command computes the 20-year (12*20 = 240 months) running average over T2.
View Running Average	To see the results of this operation, choose the time series viewer. CHECK Running Mean of April Precipitation at 60W, 25S for 1950 to 2000. The increasing trend in April precipitation from 1950 to 2000 becomes visible after running averages are employed. Note that the time grid extends from 1960 - 1990. This is due to the fact that each successive mean in the running average is labeled according to its midpoint. For example, the first mean in the running average includes the interval April 1950 - April 1969, and is labled as April 1960.

Weighted Average

Differs from a regular average in that each value in the dataset is not represented equally.
The degree of "importance" of each data value, which determines how one value will be included in the average relative to the other values, is called the weight.
Determined by weighting each value in the series, adding together the weighted values, and then dividing by the total weight.
Often used to account for area changes between meridians at varying latitudes by using the cosine of the latitude as the weights.

Example: Find spatially weighted averages of monthly solar radiation.

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 OSUSFC dataset. Select the "Data" link under the Datasets and Variables subheading. Choose the "solar radiation" link, again located under the Datasets and Variables subheading. CHECK Notice under the Grids subheading that the time variable is periodic from January to December. The OSUSFC dataset consists of monthly climatologies for each variable.
Select Domains	No ranges will be adjusted in this example. The dataset will be analyzed over its entire temporal and spatial grids.
Calculate Weighted Average	Click on the "Expert Mode" link in the function bar. Enter the following text below the text already there: {Y cosd} [X Y]weighted-average Press the OK button. CHECK The above command calculates a spatial average, weighted with the cosine of the latitude. The resulting dataset is a time series from Jan to Dec of average solar radiation in W/m².
View Weighted Average	To see the results of this operation, choose the time series viewer. CHECK Weighted Spatial Average of Monthly Climatological Solar Radiation Averaged over the world, solar radiation is at a minimum near July and at a maximum near January.
View Differences Between Weighted Average and Non-weighted Average	Click on the right most link in the blue source bar to exit the viewer. In the Expert Mode text box, enter the following text below the text already there: SOURCES .OSUSFC .DATA .solr [X Y]average sub Press the OK button. CHECK The commands above subtract the non-weighted average from the weighted average. To see the results of this operation, choose the time series viewer. CHECK Difference Between Weighted Average and Non-weighted Average of Monthly Climatological Solar Radiation The largest difference between the weighted average and the regular average occurs sometime near April and October. Refering to the previous graph of the weighted average, these are times when solar radiation is changing most rapidly.