An Application of Rank-order Statistics to the Joint Spatial and Temp Variations of Meteorological Elements

Principal components or empirical orthogonal functions are virtually the sole statistical tool used to date for investigations of space-time variability of meteorological elements. Maximum statistical efficiency and possible physical interpretation of empirical orthogonal functions derives from the assumptions of stationarity and homoscedasticity of the scalar variables in space and time. In this study, a rank table technique is given in which temporal data from a number of stations is ranked time-wise, and rank sums for each time obtained by summing ranks over the total number of stations. The technique offers some advantages for investigations of joint space-time variability. First, it is nonparametric; second, analysis of variance schemes are simplified; and third, a test of homoscedasticity can easily be performed. Networks of streamflow and precipitation data over the conterminous 48 States are used to illustrate the use of the technique. As a result, streamflow and precipitation data are shown t o be spatially heteroscedastic-dry periods are better correlated spatially than wet periods. A runs test on the temporally varying rank sums suggests that while precipitation is not temporally heteroscedastic (dry and wet periods are both essential randomly distributed), streamflow data might be. Apparently, years of deficient streamflow tend to be persistent while years of excessive streamflow are essentially randomly distributed in time. 1. lNTRODUCTlON functions that can be chosen-that is, it can be shown The joint space-time variability of meteorological elements seems to be a relatively unexplored subject. The most sophisticated method of analysis of such joint variability is the technique of empirical orthogonal functions or principal components (Lorenz 1956). Originating not because of intrinsic interest in space-time variations, but from the need for a statistically efficient and stable regression forecast technique, principal component analysis (PCA) has now.been applied in a wide variety of problems. Examples are: parameterization of the vertical structure of pressure and temperature fields for numerical forecasting models by Holmstrom (1963), climatological representation of precipitation fields by Stidd (1967) and Sellers (1968), the representation of joint (multivariate) climatological fields by Kutzbach (196?), space-filtering of meteorological fields by Grimmer (1963), and for reproducing the relevant space-time properties of hydrometeorological variables in stochastic streamflow modeling by Fiering (1964). An interesting example of the reversal of space and time coordinates in conventional PCA has been given by Brier (1968). All of these applications are basically concerned with quantitative descriptions of space-time variability. Such variability is represented in PCA by a partition of the variation of the element into a series of orthogonal spatial functions and corresponding amplitudes which vary only in time. According to the authors of the references cited, there are two major advantages of PCA over such other possible expansions. The first is that under given assumptions principal components are the most efficient orthogonal *The National Center for Atmospherie. Research is sponsored by the National Science Foundation. that no other set of orthogonal functions can be chosen which explain a greater portion of the combined spaceti me variance than the empirical orthogonal functions o r principal components. Second, since they are “natural” i n the sense that no analytic functions are involved, the possibility that they may be interpreted physica.lly is suggested (Sellers 1968, Stidd 1967). For purposes of this article, it must be pointed out that there are two assumpti ons inherent in the mathematical demonstration of the maximum efficiency of principal component analysis. The most basic assumption is that the statistical behavior of the physical quantity is stationary. This assumption is, of course, common in statistical theory, but because of changes in the behavior of the atmosphere over the globe and on the diurnal and annual time scales, it is probably not satisfied. A further assumption is that of the homoscedasticity of the error term. That is, the mean-square error is distributed over the entire sample (space and time) in the fashion of classical least squares; and therefore the error variance does not depend, in particular, upon the magnitude of the element represented. (The statistical term scedastic is synonymous with variance, and the term homoscedastic therefore means a constancy of variance-in common usage in regression theory it denotes constant error variance.) The relative variability of the element to be represented by a given spatial function is of fixed pattern but arbitrary sign, or sense, depending upon the sign of its amplitude. This “flip-flop” feature is a result of the fact that the spatial functions are the eigenvectors of the correlation or covariance matrix, and are therefore symmetric functions in the same sense that correlation fundion is February 1970 Paul R. Julian 143 symmetric. Although recognized by some workers as a disadvantage (Mitchell 1960), this flip-flop feature is seen by Sellers as an advantage--“. . . it allows the signs of the anomalies (of monthly total precipitation over an area) to go either way.” However, it should be recognized that the atmosphere may not behave in such a homoscedastic fashion and that anomalies of one sign for example (dry) may not behave in similar fashion to those of the other sign (wet). Indeed, that such might be the case is suggested by a study of Hoyt and Langbein (1944) on variations in streamflow. Based on a study of 32 yr of annual (wateryear) runoff from streams distributed over the 48 States, they point out that “. . . the extent of unbroken areas of defic.ient streamflow (below the 25 percentile) seems generally greater than that of unbroken areas of excessivc streamflow (abovc the 75 percentile).” Taking the 5 wettest years, roughly 60 percent of 48 States had runoff in the upper quartile, and in the 5 driest years, 75 percent of the region was in the driest quartile. The application of PCA to the joint space-time variations in streamflow would in this instance not be optimum in the sense that the principal component (spatial) functions would be, morphologically, a compromise between wet and dry patterns. The purpose of this study is to investigate the heteroscedastic behavior of some meteorological fieldsspecifically precipitation and streamflow-and to suggest nonparametric or distribution-free methodology capable of detecting such behavior.