Pressure-map information matrix ($\mathcal{I}=H^{\mathsf{T}}{R}^{-1}H$)

A pressure-map information matrix quantifies which coefficients of a modeled contact pressure field are constrained by a sensor array, relative to its measurement noise. For $y=Hp+\eta$ with fixed positive-definite error covariance $R$, defineHere $A$ is the noise-whitened pressure sensing matrix. Whitening removes the output-noise covariance from the residual metric; the primary SciPy covariance interface documents this transformation and demonstrates near-identity covariance on simulated observations.^[ref]

The matrix is positive semidefinite because $v^{\mathsf{T}}\mathcal{I}v=\Vert Av{\Vert }^2\ge 0$. Its null space is exactly the null space of $H$: an unobserved pressure change remains unobserved after invertible row whitening. With independent Gaussian errors of known covariance, the log likelihood is a constant minus $(y-Hp{)}^{\mathsf{T}}{R}^{-1}(y-Hp)/2$. Differentiating twice with respect to $p$ gives negative Hessian $\mathcal{I}$, identifying it as the Fisher information for this stated model.

If $H$ has full column rank, weighted least squares givesThis covariance follows by substituting $y=Hp+\eta$ and multiplying the error matrices. It concerns the unconstrained linear estimator with known response and covariance, not every reconstruction algorithm. Nonnegativity constraints, regularization and uncertain sensitivity kernels require additional treatment.

An original two-cell example begins with one average-pressure measurement, $H=(1/2,1/2)$, with $1\mathrm{kPa}$ standard error. In numerical kilopascal coordinates,Its eigenvalues are $1/2$ and zero: the mean is observed, but the pressure difference is not. A second independent copy doubles the information matrix while leaving the difference unobserved. More repeats of the same mixture improve precision only in the already measured direction.

Instead add a direct measurement of the first cell with the same independent error. NowThe inferred first-cell pressure has standard deviation $1\mathrm{kPa}$ and the second $\sqrt{5}\approx 2.236\mathrm{kPa}$. Their errors are negatively correlated because the second pressure is reconstructed by subtracting the first from twice the measured mean.

A rank-one update of pressure-map information follows when an independent new channel has sensitivity row $a^{\mathsf{T}}$ and error variance ${\sigma }_a^2$: its added information is $a{a}^{\mathsf{T}}/{\sigma }_a^2$. A channel aligned with a weak pressure direction can therefore be more useful than another precise copy of a strong direction. Correlated new errors cannot be added by that independent-channel formula.

A proposed characterization uses repeatable known loading patterns to estimate $H$, independent repeats to estimate $R$, and withheld patterns to test predicted reconstruction covariance. State the pressure basis and units: rescaling a coefficient changes the numerical information entries. A sharp reconstructed image or a large matrix determinant alone does not establish that the underlying response model or noise assumptions are correct.

A rank-one update of pressure-map information describes the extra constraint supplied by one independent scalar pressure measurement. Suppose the existing pressure-map information matrix is $\mathcal{I}=H^{\mathsf{T}}{R}^{-1}H$, and a new sensor reports $z=a^{\mathsf{T}}p+{\eta }_a$ with known variance ${\sigma }_a^2>0$, independent of the existing errors. Adding its squared residual to the weighted least squares objective givesThe added matrix has rank one when $a\ne 0$. It is positive semidefinite because its quadratic form is $(a^{\mathsf{T}}v{)}^2/{\sigma }_a^2$. Thus the new observation cannot reduce information in any direction under this fixed linear map and noise model. The noise normalization is the same whitening transformation used to express a correlated residual metric as an ordinary squared norm.^[ref]

The update can remove a missing direction only if the new channel actually sees it. More precisely,This follows because a sum of two nonnegative quadratic forms is zero exactly when both are zero. Repeating an existing mixture improves its precision but cannot identify a pressure contrast that all mixtures annihilate.

When $\mathcal{I}$ is positive definite, write $C={\mathcal{I}}^{-1}$. The Sherman–Morrison formula yields the updated estimator covariance matrixMultiplication by ${\mathcal{I}}_{+}$ verifies the identity directly. For a particular quantity $v^{\mathsf{T}}p$, its variance falls by $(v^{\mathsf{T}}Ca{)}^2/({\sigma }_a^2+a^{\mathsf{T}}Ca)$. A small noise level is valuable only together with a useful sensitivity direction.

For an original calculation in kilopascal coordinates, let $C=\mathrm{diag}(4,1)$, $a=(1,1{)}^{\mathsf{T}}$ and ${\sigma }_a^2=1$. ThenThe variance of the sum drops from $5$ to $5/6\mathrm{kP}{\mathrm{a}}^2$. The negative off-diagonal entry means the two inferred coefficients now have negatively correlated errors; it does not mean either individual variance is negative.

A correlated-channel update of pressure-map information needs the new channel's residual after predicting it from old channel noise. Simply adding $a{a}^{\mathsf{T}}/{\sigma }_a^2$ double-counts information when a common amplifier or reference contributes correlated errors. A proposed array experiment uses repeated known loads to estimate sensitivity and noise separately, adds one channel, and compares withheld-load errors with $C_{+}$. Agreement tests the full assumed model, including independence and unchanged sensor response, rather than the matrix identity alone.

A correlated-channel update of pressure-map information measures what a new pressure channel adds after accounting for noise that it shares with the old channels. The rank-one update of pressure-map information for independent noise is a special case. An extra channel may be a precise-looking duplicate of an existing noisy signal and contribute almost no new information.

Write the old and new observations as $y=Hp+e$ and $z=a^{\mathsf{T}}p+\eta$. Suppose their known joint error covariance matrix is positive definite,Subtract the part predicted from old noise by definingDirect multiplication gives $\mathrm{Cov}(e,\nu )=0$ and ${\tau }^2=\mathrm{Var}\nu =s-r^{\mathsf{T}}{R}^{-1}r>0$, a Schur complement. Thus the invertible residual transformation separates the weighted residual objective into its old term and one scalar innovation term:For jointly normal random variables, uncorrelatedness here also gives independence, and the matrix is the parameter Fisher information. For more general zero-mean errors with this fixed covariance, the displayed algebra still describes the generalized least squares information matrix. This is a partial whitening transformation; the primary covariance interface documents whitening and its identity-covariance check.^[ref]

An original scalar example observes the same pressure twice, with unit error variances and error correlation $\rho \in (-1,1)$. Then $H=a=R=s=1$, $r=\rho$, soAt $\rho =0.8$, the true estimator variance is $0.9$, whereas an independent-noise calculation would claim $0.5$. At $\rho =-0.8$, the variance is $0.1$: opposite noise contributions can cancel. Correlation can reduce or increase the benefit relative to the independent case; it does not make the correctly computed added information negative.

The effective sensitivity is $b$, not merely the original $a$. If $b=0$ and ${\tau }^2>0$, the new residual carries no additional pressure information. A zero-innovation-variance pressure channel requires separate treatment: division by ${\tau }^2=0$ is invalid, and the limiting relation may be a redundant duplicate or an exact constraint, depending on $b$.

A proposed array characterization measures simultaneous repeated readings under fixed known loads, estimates the complete cross-channel covariance, and tests reconstruction on withheld repetitions. A common reference offset can survive indefinitely despite more channels. The derivation assumes known, unchanged sensitivities and covariance; estimating them from few samples adds uncertainty not included by this update.