= Noise-whitened pressure sensing matrix
A <noise-whitened pressure sensing matrix> measures how a contact <pressure> pattern changes sensor outputs relative to their joint noise, rather than their raw numerical size. For a linear measurement model
$$
y=Hp+\eta,\qquad \mathbb E\eta=0,\qquad \operatorname{Cov}(\eta)=R,
$$
assume the error covariance $R$ is positive definite. Choose an invertible matrix $W$ satisfying $WRW^{\mathsf T}=I$. Then
$$
\boxed{\widetilde y=Wy,\qquad A=WH,\qquad \widetilde y=Ap+W\eta.}
$$
The transformed errors have identity covariance. This is the <whitening transformation> documented in the primary SciPy covariance interface; its example obtains approximately identity sample covariance from simulated observations.<SciPy 1.16.2 documentation>[Covariance.whiten, description and Examples, from_precision and np.cov]{ref}
A <Cholesky decomposition> $R=LL^{\mathsf T}$ supplies $W=L^{-1}$. Compute $Wy$ and $WH$ by solving triangular systems rather than forming an inverse unnecessarily. The least-squares residual becomes
$$
\|W(y-Hp)\|^2=(y-Hp)^{\mathsf T}R^{-1}(y-Hp).
$$
Thus an output change is significant according to its noise covariance, not merely the Euclidean length of the raw readings. The original <pressure> basis and its units must still be stated; whitening rows does not remove arbitrary column rescalings.
For an original two-channel example, use numerical <pressure> values in kilopascals and
$$
R=\begin{pmatrix}4&1.8\\1.8&1\end{pmatrix}\,\mathrm{kPa^2},
\qquad L=\begin{pmatrix}2&0\\0.9&\sqrt{0.19}\end{pmatrix}\,\mathrm{kPa}.
$$
The separate standard deviations are $2$ and $1\,\mathrm{kPa}$ with correlation $0.9$. A residual $(2,1)\,\mathrm{kPa}$ whitens to $(1,0.1/\sqrt{0.19})$, giving squared length about $1.0526$. The residual $(2,-1)\,\mathrm{kPa}$ instead whitens to $(1,-1.9/\sqrt{0.19})$, giving squared length $20$. Treating the channels as independent would assign both residuals squared standardized length two. The positively correlated noise makes opposite-sign discrepancies much less plausible than a shared shift.
A simple repeated-measurement model illustrates the information consequence. If two sensors measure the same scalar <pressure> with equal standard deviation $\sigma$ and correlation $\rho$, their mean has <variance> $\sigma^2(1+\rho)/2$. At $\rho=0.9$ the standard deviation is about $0.9747\sigma$, rather than the independent-channel value $0.7071\sigma$. Adding an almost duplicate channel therefore supplies little improvement for that common <pressure>.
The <pressure-map information matrix> is $A^{\mathsf T}A=H^{\mathsf T}R^{-1}H$. In a full-column-rank linear Gaussian model with known $R$, the unconstrained weighted least-squares estimate has covariance $(A^{\mathsf T}A)^{-1}$. Small singular values of $A$ identify <near-null directions of a pressure sensor array> after accounting for noise. Alternative whiteners differ by an orthogonal row transformation and give the same information matrix; an individual whitened row need not represent one physical sensing cell.
A proposed array characterization applies repeatable known <pressure> patterns, estimates the channel covariance from repeated residuals, and checks the covariance on independent repeats after whitening. Drift and response-model uncertainty must be included if they belong to the intended error model. If $R$ is singular, a simple inverse whitening is unavailable: exact zero-noise constraints and redundant stochastic directions need separate treatment. A visually smooth <pressure> reconstruction does not establish that its weak directions were measured precisely.
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