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Estimating the Covariance Matrix of Linear DC Networks

IP.com Disclosure Number: IPCOM000121887D
Original Publication Date: 1991-Oct-01
Included in the Prior Art Database: 2005-Apr-03
Document File: 5 page(s) / 162K

Publishing Venue

IBM

Related People

Chen, B: AUTHOR [+2]

Abstract

For any linear DC network, the covariance matrix of the responses, which are the branch voltages and the chord currents of the Reduced Tableau formation [1], can be calculated analytically knowing: (1) the variances of the conductances in the branches, (2) the variances of the resistances in chords, (3) the pair-wise correlation between them, and (4) the topology of the network. Knowing all this information, which is present in the Tableau matrix as implemented in ASTAP, then the variance-covariance matrix of the responses can be analytically estimated as the product of three matrices, which themselves may be represented as the product of other matrices [2,3,4].

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Estimating the Covariance Matrix of Linear DC Networks

      For any linear DC network, the covariance matrix of the
responses, which are the branch voltages and the chord currents of
the Reduced Tableau formation [1], can be calculated analytically
knowing: (1) the variances of the conductances in the branches, (2)
the variances of the resistances in chords, (3) the pair-wise
correlation between them, and (4) the topology of the network.
Knowing all this information, which is present in the Tableau matrix
as implemented in ASTAP, then the variance-covariance matrix of the
responses can be analytically estimated as the product of three
matrices, which themselves may be represented as the product of other
matrices [2,3,4].
 where:
      Y         is a vector of the responses of the network:
                     branch voltages and chord currents for Tableau
formulation
      VAR(Y)    is the variance-covariance matrix of the responses
                     diagonal entries are the variances
                     off-diagonal entries are the covariances
        J       is the Jacobian of the responses wrt elements in the
circuit
      Jij       is the partial derivative of Yi w.r.t. Xj
      Y         is a vector of the circuit elements: resistors and
conductors
      VAR(X)    is the variance-covariance matrix of the circuit
elements iagonal entries are the variances
                     off-diagonal entries are the covariances
      v(xii)    is the variance of Xi
      v(xij)    is the covariance of Xi and Xj

      The difficulty with this approach is determining the Jacobian
of the responses with respect to the circuit elements.  It is usually
determined by single-sided univariant perturbations which requires
reanalyzing the circuit many times.  For the Tableau formation, the
Jacobian can be easily calculated from (1) a single analysis and (2)
the coefficient matrix for the formulation of circuit equations.
Covariance Matrix for the Tableau Formulation

      The derivation of the Jacobian matrix for the Tableau
formulation is easier to illustrate for a simple example than to
derive theoretically.  Consider the following simple circuit (as
described as ASTAP) [5]:

      The Tableau matrix for the circuit is (blanks are equivalent to
zeroes):

      Eliminating the nonzero quantities below the diagonal generates
a smaller matrix in terms of the branch voltages and chord elements:

      Eliminating rows with only a single entry further reduces the
matrix so that only the resistors and conductors appear on the
diagonal:

      Differentiating implicitly with respect to G1, G3 and R2, these
equations produce the Jacobian matrix:

      The matrix of unknowns is the Jacobian. Pre-multiplying both
sides of the above equation by the inverse of the coe...