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4. Circular statistics

Tobias Stephan

2024-07-26

Source: vignettes/D_statistics.Rmd
D_statistics.Rmd

This vignette teaches you how to retrieve the mean direction of stress datasets.

library(tectonicr)
library(ggplot2) # load ggplot library

Mean direction

Directional data is π\pi-periodical. Thus, for the calculation of mean, the average of 35 and 355∘^{\circ} should be 15 instead of 195∘^{\circ}. tectonicr provides the circular mean (circular_mean()) and the quasi-median (circular_median()) as metrics to describe average direction:

data("san_andreas")
circular_mean(san_andreas$azi)
#> [1] 5.900587
circular_median(san_andreas$azi)
#> [1] 54

Quality weighted mean direction

Because the stress data is heteroscedastic, the data with less precise direction should have less impact on the final mean direction The weighted mean or quasi-median uses the reported measurements weighted by the inverse of the uncertainties:

circular_mean(san_andreas$azi, 1 / san_andreas$unc)
#> [1] 9.961752
circular_median(san_andreas$azi, 1 / san_andreas$unc)
#> [1] 54

The spread of directional data can be expressed by the standard deviation (for the mean) or the quasi-interquartile range (for the median):

circular_sd(san_andreas$azi, 1 / san_andreas$unc) # standard deviation
#> [1] 23.53222
circular_IQR(san_andreas$azi, 1 / san_andreas$unc) # interquartile range
#> [1] 27

Statistics in the Pole of Rotation (PoR) reference frame

NOTE: Because the σSHmax\sigma_{SHmax} orientations are subjected to angular distortions in the geographical coordinate system, it is recommended to express statistical parameters using the transformed orientations of the PoR reference frame.

data("cpm_models")
por <- subset(cpm_models, model == "NNR-MORVEL56") |>
  equivalent_rotation("na", "pa")
san_andreas.por <- PoR_shmax(san_andreas, por, type = "right")
circular_mean(san_andreas.por$azi.PoR, 1 / san_andreas$unc)
#> [1] 140.9378
circular_sd(san_andreas.por$azi.PoR, 1 / san_andreas$unc)
#> [1] 20.70855

circular_median(san_andreas.por$azi.PoR, 1 / san_andreas$unc)
#> [1] 134.3514
circular_IQR(san_andreas.por$azi.PoR, 1 / san_andreas$unc)
#> [1] 22.1692

The collected summary statistics can be quickly obtained by circular_summary():

circular_summary(san_andreas.por$azi.PoR, 1 / san_andreas$unc)
#>            n         mean           sd          var          25% quasi-median 
#>  407.0000000  140.9378309   20.7085472    0.2299245  124.4211789  134.3513545 
#>          75%       median        95%CI     skewness     kurtosis            R 
#>  146.5903801  135.2559405    3.8666117   -0.4975826    1.1926891    0.7782013

The summary statistics include also the circular variance, skewness, kurtosis, and the 95% confidence angle.

Rose diagram

tectonicr provides a rose diagram, i.e. histogram for angular data.

rose(san_andreas$azi, weights = 1 / san_andreas$unc, main = "North pole",
     dots = TRUE, stack = TRUE, dot_cex = 0.5, dot_pch = 21)

# add the density curve
plot_density(san_andreas$azi, kappa = 10, col = "dodgerblue", shrink = 1.5)

The diagram shows the uncertainty-weighted frequencies (equal area rose fans), the von Mises density distribution (blue curve), and the circular mean (red line) incl. its 95% confidence interval (transparent red).

Showing the distribution of the transformed data gives the better representation of the angle distribution as there is no angle distortion due to the arbitrarily chosen geographic coordinate system.

rose(san_andreas.por$azi, weights = 1 / san_andreas$unc, main = "PoR",
     dots = TRUE, stack = TRUE, dot_cex = 0.5, dot_pch = 21)
plot_density(san_andreas.por$azi, kappa = 10, col = "dodgerblue", shrink = 1.5)

# show the predicted direction
rose_line(135, radius = 1.1, col = "#009E73")

The green line shows the predicted direction.

Statistical tests

Test for random distribution

Uniformly distributed orientation can be described by the von Mises distribution (Mardia and Jupp, 1999). If the directions are distributed randomly can be tested with the Rayleigh Test:

rayleigh_test(san_andreas.por$azi.PoR)
#> Reject Null Hypothesis
#> $R
#> [1] 0.7782013
#> 
#> $statistic
#> [1] 246.4781
#> 
#> $p.value
#> [1] 9.034746e-108

Here, the test rejects the Null Hypothesis (statistic > p.value). Thus the σSHmax\sigma_{SHmax} directions have a preferred orientation.

Alternative statistical tests for circular uniformity are kuiper_test() and watson_test(). Read help() for more details…

Test for goodness-of-fit

Assuming a von Mises Distribution (circular normal distribution) of the orientation data, a (1−α%)/100(1-\alpha \%)/100confidence interval can be calculated (Mardia and Jupp, 1999):

confidence_interval(san_andreas.por$azi.PoR, conf.level = 0.95, w = 1 / san_andreas$unc)
#> $mu
#> [1] 140.9378
#> 
#> $conf.angle
#> [1] 1.461317
#> 
#> $conf.interval
#> [1] 139.4765 142.3991

The prediction for the σSHmax\sigma_{SHmax} orientation is 135∘135^{\circ}. Since the prediction lies within the confidence interval, it can be concluded with 95% confidence that the orientations follow the predicted trend of σSHmax\sigma_{SHmax}.

The (weighted) circular dispersion of the orientation angles around the prediction is another way of assessing the significance of a normal distribution around a specified direction. It can be measured by:

circular_dispersion(san_andreas.por$azi.PoR, y = 135, w = 1 / san_andreas$unc)
#> [1] 0.1232034

The value of the dispersion ranges between 0 and 2.

The standard error and the confidence interval of the calculated circular dispersion can be estimated by bootstrapping via:

circular_dispersion_boot(san_andreas.por$azi.PoR, y = 135, w = 1 / san_andreas$unc, R = 1000)
#> $MLE
#> [1] 0.2274201
#> 
#> $sde
#> [1] 0.02080516
#> 
#> $CI
#> [1] 0.1882919 0.2719524

The statistical test for the goodness-of-fit is the (weighted) Rayleigh Test with a specified mean direction (here the predicted direction of 135∘135^{\circ}:

weighted_rayleigh(san_andreas.por$azi.PoR, mu = 135, w = 1 / san_andreas$unc)
#> Reject Null Hypothesis
#> $C
#> [1] 0.7535932
#> 
#> $statistic
#> [1] 21.50053
#> 
#> $p.value
#> [1] 3.777096e-100

Here, the Null Hypothesis is rejected, and thus, the alternative, that is a uniform distribution around the predicted direction, cannot be excluded.

References

Mardia, K. V., and Jupp, P. E. (Eds.). (1999). “Directional Statistics” Hoboken, NJ, USA: John Wiley & Sons, Inc.  doi: 10.1002/9780470316979.

Ziegler, Moritz O., and Oliver Heidbach. 2017. “Manual of the Matlab Script Stress2Grid” GFZ German Research Centre for Geosciences; World Stress Map Technical Report 17-02. doi: 10.5880/wsm.2017.002.