An underdetermined linear algebraic equation system (Formula presented.), where Φ is an (m00D7n)m003Cn rectangular constant matrix with rank (Formula presented,) and (Formula presented.), has an infinite number of solutions. Diffeomorphic modulation under observable response preserving homotopy (D-MORPH) regression seeks a solution satisfying the extra requirement of minimizing a chosen cost function, K. A wide variety of choices of the cost function makes it possible to achieve diverse goals, and hence D-MORPH regression has been successfully applied to solve a range of problems. In this paper, D-MORPH regression is extended to determine a sparse or a nonnegative sparse solution of the vector x. For this purpose, recursive reweighted least-squares (RRLS) minimization is adopted and modified to construct the cost function K for D-MORPH regression. The advantage of sparse and nonnegative sparse D-MORPH regression is that the matrix Φ does not need to have row-full rank, thereby enabling flexibility to search for sparse solutions x with ancillary properties in practical applications. These tools are applied to (a) simulation data for quantum-control-mechanism identification utilizing high dimensional model representation (HDMR) modeling and (b) experimental mass spectral data for determining the composition of an unknown mixture of chemical species.
All Science Journal Classification (ASJC) codes
- Applied Mathematics
- D-MORPH regression
- Least-squares regression
- Mass spectrum analysis
- Quantum-control-mechanism identification
- Underdetermined system