KendallTau

Kendall's tau-b — a robust rank correlation between two series measured by the balance of concordant and discordant pairs, with a tie correction.

Quick reference

Field	Value
Family	Price Statistics
Input type	(f64, f64) (pairwise)
Output type	f64
Output range	[−1, +1]
Default parameters	(period = 20) (Python)
Warmup period	period
Interpretation	+1 concordant, −1 discordant, 0 no monotonic link.

Formula

over all pairs (i < j) in the window:
  concordant if (x_j − x_i) and (y_j − y_i) share a sign
  discordant if opposite signs
  tie_x / tie_y if the respective difference is zero
n0    = N(N−1)/2
tau_b = (n_concordant − n_discordant) / sqrt((n0 − tie_x)(n0 − tie_y))

Kendall's tau counts, over every pair of observations, how often the two series move in the same direction. It is the most outlier-robust of the common correlation measures and captures any monotonic relationship, not just a linear one. The tau-b variant divides by a tie-corrected denominator so repeated values do not depress the score. Source: crates/wickra-core/src/indicators/kendall_tau.rs.

Parameters

Name	Type	Default	Valid range	Source	Description
period	usize	20 (Python)	>= 2	kendall_tau.rs:60	Rolling window of pairs. < 2 errors with Error::InvalidPeriod.

The period getter returns the window; value returns the current output if ready.

Inputs / Outputs

From crates/wickra-core/src/indicators/kendall_tau.rs:

use wickra::{Indicator, KendallTau};
// KendallTau: Input = (f64, f64), Output = f64
const _: fn(&mut KendallTau, (f64, f64)) -> Option<f64> = <KendallTau as Indicator>::update;

A (f64, f64) pair in, an Option<f64> out. The Python binding takes update(x, y) and batch(x[], y[]) (NaN warmup); Node takes update(x, y) and batch(x[], y[]).

Warmup

warmup_period() == period. The first value lands once the window holds period pairs (first_emission_at_warmup_period pins this).

Edge cases

• Monotone increasing → +1. A strictly co-increasing pair gives +1 (monotone_increasing_is_one pins this).
• Monotone decreasing → −1. A strictly anti-monotone pair gives −1 (monotone_decreasing_is_minus_one pins this).
• Constant channel → 0. A flat y makes every y-difference a tie; the tie-corrected denominator is 0 and the result is 0 (constant_channel_yields_zero pins this).
• Bounded. The reading stays within [−1, +1] (output_in_range pins this).
• Reset. k.reset() clears the window and the last value (reset_clears_state).

Examples

Rust

use wickra::{BatchExt, Indicator, KendallTau};

fn main() -> Result<(), Box<dyn std::error::Error>> {
    let mut k = KendallTau::new(10)?;
    let pairs: Vec<(f64, f64)> = (0..20).map(|i| (f64::from(i), 2.0 * f64::from(i))).collect();
    println!("{:?}", k.batch(&pairs).last().unwrap()); // Some(1.0)
    Ok(())
}

Output:

Some(1.0)

Python

import numpy as np
import wickra as ta

k = ta.KendallTau(20)
x = np.arange(40, dtype=float)
y = 2.0 * x + 1.0
print(k.batch(x, y)[-1])   # 1.0

Node

const ta = require('wickra');

const k = new ta.KendallTau(20);
console.log('warmupPeriod:', k.warmupPeriod()); // 20

Streaming

use wickra::{Indicator, KendallTau};

let mut k = KendallTau::new(20).unwrap();
let mut last = None;
for i in 0..40 {
    let x = f64::from(i);
    last = k.update((x, 2.0 * x));
}
println!("{last:?}"); // Some(1.0)

Streaming update and batch are equivalent tick-for-tick (batch_equals_streaming pins this).

Interpretation

1. Robust pairs trading. Use tau to screen pair candidates: it flags monotonic co-movement even when a few outliers would wreck a Pearson estimate.
2. Regime monitoring. A tau that decays from near +1 toward 0 warns that a historically reliable relationship is breaking down.
3. Non-linear links. Because it is rank-based, tau detects monotonic but curved relationships that linear correlation understates.

Common pitfalls

• Cost. Each evaluation is O(period²); keep period modest at tick rates.
• Not Pearson. Tau values are systematically smaller in magnitude than Pearson for the same data (roughly tau ≈ (2/π) · arcsin(r)); do not compare them directly.
• Levels, not returns. This correlates the raw inputs; difference them first if you want return-association.

References

Kendall, M. G. (1938), "A New Measure of Rank Correlation", Biometrika. The tau-b tie correction is the standard form for data with ties.

See also

• Indicator-PearsonCorrelation — linear correlation.
• Indicator-SpearmanCorrelation — rank-difference correlation.
• Indicator-RollingCorrelation — rolling correlation of returns.
• Indicators-Overview — the full taxonomy.