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Merge pull request #19 from circstat/median
refactor: Median
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‎docs/docs/feature-checklist.md‎

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| **Measures of Central Tendency** | | | | | |
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| Circular Mean | `circ_mean` | `mean(alpha)` | `circ_mean(alpha)` | `circ.mean` | `mean.circular` |
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| Circular Mean CI | `circ_mean_ci` | `mean(alpha, ci=95)` | `circ_confmean` | - | `mle.vonmises.bootstrap.ci` |
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| Circular Median | `circ_median` | `median` | `circ_median` | - | `median.circular`/`medianHL.circular` |
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| Circular Median (Fisher/Mardia) | `circ_median` (method=`"deviation"`/`"count"`) [^median-grouped] | `median` | `circ_median` | - | `median.circular` |
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| Hodges-Lehmann Median | `circ_median` (method=`"HL1"`/`"HL2"`/`"HL3"`) | - | - | - | `medianHL.circular`[^medianHL-broken] |
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| Circular Median CI | `circ_median_ci` | - | - | - | - |
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| Circular Quantile | `circ_quantile` | - | - | - | `quantile.circular` |
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| **Measures of Spread & Dispersion** | | | | | |
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| Circular Variance | `circ_var` | `var` | `circ_var` | `circ.disp` | `var.circular` |
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| Circular Standard Deviation | `circ_std` | `std` | `circ_std` | - | `sd.circular` |
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| Circular Dispersion | `circ_dispersion` | - | - | - | - |
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| Circular Range | `circ_range` | - | - | `circ.range` | `range.circular` |
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| Concentration Parameter (κ) | `circ_kappa` | `kappa` | `circ_kappa` | `est.kappa` | `A1inv` |
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| **Higher-Order Statistics** | | | | | |
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| Circular Moment | `circ_moment` | `moment` | `circ_moment` | `tri.moment` | `trigonometric.moment` |
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| Circular Skewness | `circ_skewness` | `skewness` | `circ_skewness` | - | - |
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| Circular Kurtosis | `circ_kurtosis` | `kurtoisis` | `circ_kurtosis` | - | - |
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| **Distance & Pairwise Comparisons** | | | | | |
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| Mean deviation | `circ_mean_deviation` | - | - | - | `meandeviation` |
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| Mean deviation [^mean-dev] | `circ_mean_deviation` | - | - | - | `meandeviation` |
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| Circular Distance | `circ_dist` | `cdist` | `circ_dist` | - | - |
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| Pairwise Circular Distance | `circ_pairdist` | `pairwise_cdiff` | `circ_dist2` | - | `dist.circular` |
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| Feature | H0 | PyCircStat2 | PyCircStat | CircStat (MATLAB) | CircStats (R) | circular (R) |
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|---------------------|------------|--------------------|--------------|-------------------|---------------|--------------------|
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| Kuiper’s Test | $\rho = 0$ | `circ_kuiper_test` | `kupier` | `circ_kuipertest` | `kuiper` | `kuiper.test` |
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| Kuiper’s Test | $\rho = 0$ | `kuiper_test` | `kuiper` | `circ_kuipertest` | `kuiper` | `kuiper.test` |
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| Rao’s Spacing Test | $\rho = 0$ | `rao_spacing_test` | `raospacing` | `circ_raotest` | `rao.spacing` | `rao.spacing.test` |
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| Watson's Test | $\rho = 0$ | `watson_test` | - | - | `watson` | `watson.test` |
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| Circular Range Test | $\rho = 0$ | `circ_range_test` | - | - | `circ_range` | `range.circular` |
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### 3. Correlation & Regression
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| Feature | PyCircStat2 | PyCircStat | CircStat (MATLAB) | CircStats (R) | circular (R) |
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|-------------------------------|----------------|------------|-------------------|---------------|---------------------------|
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| Circular-Circular Correlation | `circ_corrcc` | `corrcc` | `circ_corrcc` | `circ.cor` | `cor.circular` |
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| Circular-Linear Correlation | `circ_corrcl` | `corrcl` | `circ_corrcl` | - | - |
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| Circular-Circular Regression | `CCRegression` | - | - | `circ.reg` | `lm.circular(type="c-c")` |
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| Circular-Linear Regression | `CLRegression` | - | - | - | `lm.circular(type="c-l")` |
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| Feature | PyCircStat2 | PyCircStat | CircStat (MATLAB) | CircStats (R) | circular (R) |
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|-----------------------------------------------|----------------|------------|-------------------|---------------|---------------------------|
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| Circular-Circular Correlation | `circ_corrcc` | `corrcc` | `circ_corrcc` | `circ.cor` | `cor.circular` |
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| Circular-Linear Correlation | `circ_corrcl` | `corrcl` | `circ_corrcl` | - | - |
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| Circular-Circular Regression | `CCRegression` | - | - | `circ.reg` | `lm.circular(type="c-c")` |
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| Circular-Linear Regression [^cl-resp] | `CLRegression` | - | - | - | `lm.circular(type="c-l")` |
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| Linear-Circular Regression (harmonic) [^lc-resp] | `LCRegression` | - | - | - | - |
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| | PPF | `wrapstable.ppf` | - | - | - | - |
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| | RVS | `wrapstable.rvs` | - | - | `rwrpstab` | - |
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| | Fit | `wrapstable.fit` | - | - | - | - |
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| Asymmetric Trangular | PDF | - | - | - | - | `dasytriangular` |
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| Asymmetric Triangular | PDF | - | - | - | - | `dasytriangular` |
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| Projected Normal | PDF | - | - | - | - | `dpnorm` |
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| | RVS | - | - | - | - | `rpnorm` |
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[^uniform]: $\rho=0$ stands for uniform distributed.
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[^median]: $\theta$ stands for median.
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[^F]: $F$ stands for distributions.
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[^one-way]: Yet anothr one-way ANOVA.
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[^one-way]: Yet another one-way ANOVA.
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[^two-way]: Two-way ANOVA.
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[^median-grouped]: For grouped data (non-uniform `w`), `circ_median` uses the
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Mardia (1972) interpolation; the `method` argument is ignored on that path.
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[^medianHL-broken]: As of `circular` 0.5-2 (CRAN, 2025-09-24), `medianHL.circular`
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builds the pair-mean array but its C primitive calls the deviation median on
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the original `x` instead of on the pair-means, so HL1/HL2/HL3 all return the
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regular `median.circular` value. See Otieno (2002) §3.4 for the intended
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algorithm.
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[^mean-dev]: Different signatures: pycircstat2's `circ_mean_deviation(α, β)`
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evaluates Fisher (1993) eq. 2.32 — `d(β) = π − (1/n)Σ|π−|αᵢ−β||` — at every
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reference angle `β` (vector output). R's `meandeviation(x)` is the scalar
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`d(median(x))`. Same formula, different evaluation points.
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[^cl-resp]: Circular response, linear predictor.
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[^lc-resp]: Linear response, circular predictor (harmonic regression à la
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Pewsey et al. 2014, §8.4).

‎examples/B1-Fisher-1993.ipynb‎

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