I have a set of 2-dimensional vector observations (like to wind vectors), and I have separated them into subsamples based on whether an event that is independent of the observations occurred or not. I know how to test whether one (scalar) sample is statistically different from another, but I haven't been able to figure out how to test wind vectors. How do I test whether the winds in subsample 1 are statistically different from those in subsample two? Any help is greatly appreciated.
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1$\begingroup$ Perhaps you could explain how you know how to test univariate samples: that would show us what you mean by "statistically different." Or else you could describe what "statistically different" is intended to mean. $\endgroup$– whuber ♦Commented Jul 12, 2016 at 18:26
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1$\begingroup$ Univariate example: Two people forecast temperature 2 days in advance every day for a year. The two sets of forecasts are compared to reality (forecast-observations) to get forecast errors. One can use a paired t-test, assuming the correct distribution, to test the hypothesis that the mean errors of the two are different (with an accounting of serial correlation of the samples). In my current case, the individual components of the samples are approximately normally distributed, and the samples are not paired. I would like to test whether the means of the two vector samples are different. $\endgroup$– Sim AbersonCommented Jul 12, 2016 at 19:36
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$\begingroup$ Permutation test for this task described here $\endgroup$– zlonCommented Feb 11, 2017 at 22:58
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2 Answers
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Based on the additional comment you gave to @whuber, it seems that you are looking for an extension of t-test/ANOVA type methods to situations where you have a vector of outcomes per individual (i.e. multiple outcomes per individual, and in particular in the wind vector example, you have a vector with two components for each individual observation). If interest lies in comparing whether a vector of means from one sample is different from a vector of means from another sample, you can use MANOVA which assumes multivariate normality for the outcomes. Further details on assumptions, mathematical formulation, examples and interpretation can be found here.
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$\begingroup$ In the case of wind, it could be helpful to transform to direction/velocity, that is, polar coordinates. $\endgroup$ Commented Apr 30, 2019 at 10:05
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In general, it can be difficult to test for significance for high-d vectors. Depending on your application, it may be enough that some statistic of the data is significant. For example, you could apply labels to the subsamples (subsample A and subsample B), then apply a classifier with cross-validation to see how separable the subsamples are. In a permutation-test-like fashion, you can then shuffle the labels across all datapoints, and re-apply the classifier, which should give you chance performance. Do this for many shuffles to get a distribution of the cross-validated accuracies, and then see if the actual CV accuracy is statistically different from the shuffled distribution. Other possibilities include taking the angles between the vectors, etc.
