# 4. Circular statistics

#### Tobias Stephan

#### 2025-05-22

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 \(\pi\)-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$azi)
    #> [1] 10.64134
    circular_median(san_andreas$azi)
    #> [1] 35.5

## 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] 9.98401
    circular_median(san_andreas$azi, 1 / san_andreas$unc)
    #> [1] 35.62501

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] 19.23893
    circular_IQR(san_andreas$azi, 1 / san_andreas$unc) # interquartile range
    #> [1] 35

## Statistics in the Pole of Rotation (PoR) reference frame

NOTE: Because the \(\sigma_{SHmax}\) 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 <- cpm_models[["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] 137.996
    circular_sd(san_andreas.por$azi.PoR, 1 / san_andreas$unc)
    #> [1] 18.671

    circular_median(san_andreas.por$azi.PoR, 1 / san_andreas$unc)
    #> [1] 135.6499
    circular_IQR(san_andreas.por$azi.PoR, 1 / san_andreas$unc)
    #> [1] 25.93367

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

    circular_summary(san_andreas.por$azi.PoR, 1 / san_andreas$unc, mode = TRUE)
    #>            n         mean           sd          var          25% quasi-median
    #> 1126.0000000  137.9960403   18.6710044    0.1913452  123.4943787  135.6498736
    #>          75%       median         mode           CI     skewness     kurtosis
    #>  149.4280453  137.4843360  138.0821918    5.1715084   -0.3386996    2.7427489
    #>            R
    #>    0.8086548

The summary statistics additionally include the circular quasi-quantiles, the variance, the skewness, the kurtosis, the mode, the 95% confidence angle, and the mean resultant length (R).

## 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 = 20, col = "#51127CFF", 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 = 20, col = "#51127CFF", shrink = 1.5)

    # show the predicted direction
    rose_line(135, radius = 1.1, col = "#FB8861FF")

The green line shows the predicted direction.

## QQ Plot

The (linearised) circular QQ-Plot (`circular_qqplot()`) can be used to visually assess whether our stress sample is drawn from an uniform distribution or has a preferred orientation.

    circular_qqplot(san_andreas.por$azi.PoR)

Our data clearly deviates from the diagonal line, indicating the data is not randomly distributed and has a strong preferred orientation around the 50% quantile.

## 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] 0.7118209
    #>
    #> $statistic
    #> [1] 570.5317
    #>
    #> $p.value
    #> [1] 1.664245e-248

Here, the test rejects the Null Hypothesis (`statistic > p.value`). Thus the \(\sigma_{SHmax}\) 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 \(100(1-\alpha)\%\) confidence 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] 137.996
    #>
    #> $conf.angle
    #> [1] 5.171508
    #>
    #> $conf.interval
    #> [1] 132.8245 143.1675

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

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] 0.09788172

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] 0.1917114
    #>
    #> $sde
    #> [1] 0.009597689
    #>
    #> $CI
    #> [1] 0.1739953 0.2118351

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

    weighted_rayleigh(san_andreas.por$azi.PoR, mu = 135, w = 1 / san_andreas$unc)
    #> Reject Null Hypothesis
    #> $C
    #> [1] 0.8042366
    #>
    #> $statistic
    #> [1] 38.16524
    #>
    #> $p.value
    #> [1] 3.589557e-315

Here, the Null Hypothesis is rejected, and thus, the alternative, i.e. an non-uniform distribution with the predicted direction as the mean 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.