Hi, the representation of a measurement with the help of a vector, in your case, direction and intensity of the wind, could still be analized in the classical way. You could draw the scalar values like absolute value and argument of your wind data into a 2d diagramm with the probability in z direction and work from there.

I see, do you mean project the vector to an axis by calculating the COS value? however, I don't want to transfer vector into scalar data. What I want to find may be a direct indicator of the set of vector...Do you have any idea?

Well, you can use every component of your vector in a multidimensional statistical analysis of course. The interpretation afterwards could perhaps be somewhat unintuitive. Yet using a vector or the abs and arg does not "destroy" any information of your measurement. Another question would be, what is the next step of analysis after you evaluated the vector statistics?

Well, I want to estimate the flux of wind between two points in a wind field. Now I have calculate the direction variance along the direct line between these two points, but this variance does not explain the flux of air pollution of these two points very well. Since variance is considered to be the optimal indicator of flux, and the wind data is a vector data instead of scalar data, using only direction variance waste the velocity attribute right? So I think there may exist a vector variance which can indicate the deviance of vector dataset. Any idea about this?

So you want to evaluate the mean and variance of wind speed along a fixed line between two points? And again i would try to evaluate the scalarproduct of the vector in any given point of the fixed line with the orientation of the line. And then use simple statistics on those scalar values. The information of wind direction is fixed due to the choice of your line betweend the points.

Well, I have already done this, but this variance doesn't work very well in explaining the difference of air quality between two points, my assumption is that the transmission of air pollution is determined by the motion of wind, the larger the variance of wind, the larger the difference of air quality between these two points, right?maybe because the temporal scale is two large( daily average)...or something else.

This sounds more like a spatial temporal correlation analysis in the first place. Aswell as the windspeeds are dependend on many more variables that can change with time. Do you have a pollution source or are you assuming a measured value of pollution at two points ?

Quite right, this is a spatial interpolation work. Instead of using Euclidean distance as the indicator of relation, as the Tobler's First Law says. Some other distance or function of direct distance can also be used. I want to use simulated wind field to change the direct distance, improving the correlation between this "distance" and air quality. So there is no pollution source, only measured data or station data.

Wind direction is not simple vector data. For vector data Mahalanobis distance would work, but not for wind data because there is no simple correlation. To estimate the variance of the wind direction you can't use variance of the wind angle (btw: the math reason for this is, because variance requires a metric statistical scale and the wind angle is not metric, because the difference between 1 and 359 degrees is not the same as the difference between 1 and 3 degrees). Even worse, you can't estimate wind variance by calculating variance of the x- and y-direction because variance calculation needs a meaningful mean (what is the mean of west wind and east wind? is it north or south?). Maybe the temporal variance could provide the information you need. Just calculate the daily difference of the wind angle and make sure that it is in the range between -180 and +180 degrees. You won't have a normal distribution, but the variance of the temporal difference gives an indication of wind variance in time. Please note that probably the daily resolution is not sufficient (how is the daily mean calculated??). Please note also that you have to exclude wind direction data if wind speed is low, because then the wind direction measurement is too erroneous.

Thank you for the insights! What would happen if we neglect the fact, that the question concerns wind data in the first place? Only measurement of some direction and intensity? And could the angle data be transformed onto the metric scale to calculate the deviation?

Sorry, perhaps I do not correctly understand what you mean with the first two questions. If you do not analyze wind, but another direction and intensity: The same applies as I said about wind.

Third question: the answer is no, because angle data are cyclic and metric scales are not.

Well i was thinking about that common rose diagramm which is like a circular histogram of wind directions measured at a weather station. What do i miss there?

Well, thank you for your insights! The temporal scale may not be a problem. What I want to calculate is the wind angle deviation along the path between two points, now this is a direct line. I want to treat this as an indicator of the difficulty that air pollution, now this is particulate matter, is exposed to when transmitting from one point to another. The higher the difficulty, the lower the correlation. Then use this correlation to interpolate the air pollution. The angle range along this path may be quite limited, because the research area is not very large. The third viewpoints is very useful, I may exclude the angles when speed is very low.

Look up Parallel Coordinates or get in touch with its inventor Professor Alfred Inselberg