Source: https://jmrplens.github.io/phonometry/guides/miso-coherence/ # Multiple and partial coherence (Bendat & Piersol) When several partially correlated sources drive one response, the ordinary coherence of each source with the output is misleading: a source that only *correlates* with the true cause inherits a spurious coherence through it, so reading the ordinary coherences alone can credit the wrong source. Bendat & Piersol, *Random Data* (4th ed., 2010, Chapter 7), resolve this for a multiple-input/single-output (MISO) system with the **multiple** and **partial** coherence functions. `miso_coherence` computes them from the same Welch cross-spectral core as the rest of `phonometry.metrology`, for several correlated inputs and one output.

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```python import matplotlib.pyplot as plt import numpy as np from scipy import signal from phonometry import miso_coherence, noise_signal fs = 8192.0 # Input 1 drives a low-frequency path; input 2 = 0.7*x1 + independent noise # drives a high-frequency path, so input 2 is correlated with input 1. x1 = noise_signal(fs, 32.0, color="white", seed=1) x2 = 0.7 * x1 + noise_signal(fs, 32.0, color="white", seed=2) low = signal.butter(4, 400.0, fs=fs, output="sos") high = signal.butter(4, 1500.0, btype="high", fs=fs, output="sos") noise = noise_signal(fs, 32.0, color="white", rms=0.05, seed=3) y = signal.sosfilt(low, x1) + signal.sosfilt(high, x2) + noise res = miso_coherence([x1, x2], y, fs, nperseg=2048) f = res.frequencies band = (f >= 20.0) & (f <= 4000.0) fig, (ax_top, ax_bot) = plt.subplots(2, 1, figsize=(10, 7.4), sharex=True) db = lambda v: 10 * np.log10(v) ax_top.semilogx(f[band], db(res.output_psd[band]), color="gray", label="Measured output") for i, color in ((0, "#1f77b4"), (1, "#2ca02c")): ax_top.semilogx(f[band], db(res.coherent_output_spectra[i][band]), color=color, label=f"Input {i + 1} contribution") ax_top.semilogx(f[band], db(res.noise_psd[band]), "--", color="#d62728", label="Residual noise") ax_top.set_ylabel("Coherent output [dB re 1/Hz]") ax_top.legend() ax_bot.semilogx(f[band], res.ordinary_coherence[1][band], ":", color="#2ca02c", label="Input 2 ordinary (inflated by x1)") ax_bot.semilogx(f[band], res.partial_coherence[1][band], color="#2ca02c", label="Input 2 partial (x1 removed)") ax_bot.semilogx(f[band], res.multiple_coherence[band], color="black", label="Multiple") ax_bot.set_xlabel("Frequency [Hz]") ax_bot.set_ylabel("Coherence") ax_bot.set_ylim(0, 1.05) ax_bot.legend() plt.show() ```

The `.plot()` method draws the same two panels in one call (`res.plot()`, or `res.plot(language="es")` for Spanish labels). ## 1. Ordinary, multiple and partial coherence For a system with `q` inputs `x1..xq` and output `y`, `miso_coherence` estimates every auto- and cross-spectrum by Welch's method and reports three coherence functions. The **ordinary coherence** of input `i` with the output, on its own (Eq. 7.109), is the familiar single-input quantity `γ²iy = |Giy|²/(Gii·Gyy)`. The **multiple coherence** (Eq. 7.35) is the fraction of the output autospectrum linearly explained by *all* inputs jointly, `γ²y:x = GxyᴴGxx⁻¹Gxy/Gyy = 1 - Gnn/Gyy`, where `Gnn` is the residual output spectrum uncorrelated with every input. If the output carries additive uncorrelated noise of per-band signal-to-noise ratio `SNR`, this is exactly `SNR/(1+SNR)`, the closed-form oracle used to verify the implementation. The **partial coherence** of input `i` (Eq. 7.87), `γ²iy·(i-1)! = |Giy·(i-1)!|²/(Gii·(i-1)!·Gyy)`, is its coherence with the output once the linear effect of the inputs *before it in the conditioning order* has been removed. The 4th-edition definition keeps the *total* output `Gyy` in the denominator, so the partial coherences of the ordered inputs sum to the multiple coherence (Eq. 7.116) and reduce exactly to the ordinary coherences when the inputs are mutually uncorrelated (Eq. 7.117). ```python res = miso_coherence([x1, x2], y, fs) res.ordinary_coherence # shape (q, F): each input on its own res.multiple_coherence # shape (F,): all inputs jointly res.partial_coherence # shape (q, F): conditioned on the preceding inputs ``` ## 2. Conditioning: separating cause from correlation The partial coherences come from the **conditioned spectra** `Gij·r!`, computed by the Gaussian-elimination recursion of Section 7.3. Removing the linear effect of a pivot input `r` from every remaining record is one Schur complement step (Eq. 7.94), `Gij·r! = Gij·(r-1)! - Grj·(r-1)!·Gir·(r-1)!/Grr·(r-1)!`, applied for each input in the conditioning order until only the residual output spectrum `Gyy·q!` remains. In the figure above, input 2 is `0.7·x1 + independent noise`, so it correlates with input 1 but drives no low-frequency path. Its ordinary coherence with the output therefore reads about 0.3 across the low band, borrowed entirely through input 1, while its partial coherence collapses to zero there once input 1 is conditioned out. ```python low = (res.frequencies > 100.0) & (res.frequencies < 300.0) res.ordinary_coherence[1][low].mean() # ~0.32 (inflated through x1) res.partial_coherence[1][low].mean() # ~0.00 (x1 removed) ``` ## 3. Which source dominates each band The conditioning also splits the output power source by source. The **partial coherent output spectrum** of input `i` (Eq. 7.86), `Gvi = γ²iy·(i-1)!·Gyy`, is the share of the output autospectrum it contributes, and the shares plus the residual noise reconstruct the output exactly (Eqs. 7.88/7.121): `sum(Gvi) + Gnn = Gyy`. Comparing the shares band by band answers "which source dominates here?". `dominant_input()` returns, for every frequency, the index of the input with the largest share. ```python res.coherent_output_spectra # shape (q, F): Gvi per input dominant = res.dominant_input() # index of the strongest source per bin f = res.frequencies dominant[np.argmin(abs(f - 200.0))] # 0 (input 1 drives the low band) dominant[np.argmin(abs(f - 2500.0))] # 1 (input 2 drives the high band) ``` ## 4. Statistical quality and the conditioning order The random errors follow Section 9.3. Conditioning on the `i-1` preceding inputs costs `i-1` degrees of freedom, so the `i`-th ordered input carries `nd-(i-1)` effective averages (Eqs. 9.100/9.101) and the `q`-input multiple coherence carries `nd-(q-1)` (Eq. 9.98). The result exposes `multiple_coherence_random_error` and `coherent_output_random_error` alongside the effective average count `n_averages`. The ordinary and multiple coherences do not depend on the conditioning order, but the partial coherences and the coherent-output decomposition do: each input is conditioned on whatever precedes it. Absent a physical ordering, Bendat & Piersol (Section 7.2.4) recommend ordering the inputs by descending ordinary coherence with the output. Pass `order` to choose: ```python res = miso_coherence([x1, x2, x3], y, fs, order=(2, 0, 1)) res.order # (2, 0, 1): the order actually applied res.plot() # the two panels, recomputed in the applied order ``` Two practical pitfalls are worth naming. *Averaging*: every conditioning step spends a degree of freedom, so with few segments the partial coherences of the last-ordered inputs are the least trustworthy numbers on the page; average generously before reading a small partial coherence as zero. *Delay bias*: like the ordinary coherence, the Welch estimate is biased low when a bulk delay between an input and the output becomes a noticeable fraction of the segment length; remove known propagation delays first (see [Correlation and delay](https://jmrplens.github.io/phonometry/guides/correlation-delay/)) and keep `nperseg` well above the longest remaining delay. The estimators share the Welch core of the [calibrated spectral analysis](https://jmrplens.github.io/phonometry/guides/spectral-analysis/) page, so a MISO coherence and a `power_spectral_density` computed with the same segment length are consistent bin by bin. ## See also - [Spectral analysis](https://jmrplens.github.io/phonometry/guides/spectral-analysis/): the single-input coherent output spectrum this page generalizes, and the shared Welch core. - [Correlation and delay](https://jmrplens.github.io/phonometry/guides/correlation-delay/): estimating and removing the bulk delays that bias coherence low. - [Multichannel and Performance](https://jmrplens.github.io/phonometry/guides/multichannel/): the per-channel path this cross-channel analysis complements. ## References - Bendat, J. S., & Piersol, A. G. (2010). *Random Data: Analysis and Measurement Procedures* (4th ed.). Wiley. ISBN 978-0-470-24877-5. [doi:10.1002/9781118032428](https://doi.org/10.1002/9781118032428). Chapter 7 (multiple-input/output relationships: ordinary coherence Eq. 7.109, multiple coherence Eq. 7.35, partial coherence Eq. 7.87, the conditioned-spectrum recursion Eq. 7.94, the coherent output decomposition Eqs. 7.86/7.116/7.121) and Section 9.3 (statistical errors of the multiple and conditioned estimates, Eqs. 9.98-9.101).