Functions & Formulae

Each applied formula has its own function page, with a signature, implementations, and tests.

Matrix root of an annual transition matrix for a shorter model cycle

P^(1/n) = V * D^(1/n) * V^(-1); (P^(1/n))^n = P

Converts a transition matrix estimated over one interval into the matrix for a cycle n times shorter by taking the n-th root of the whole matrix, so that n short cycles reproduce the original matrix, including movements through intermediate states within the interval. Cohort propagation with a matrix is HE-FM-MM-001, the matrix exponential of a rate matrix HE-CF-CONT-001, competing exits HE-FM-TP-004 and cell-by-cell rescaling of a single probability HE-FM-TP-003. Notation follows the Transition Matrix article.

  • Six-month matrix as the square root of a progressive three-state annual transition matrix

    r_11 = sqrt(p_11); r_22 = sqrt(p_22); r_12 = p_12 / (r_11 + r_22); r_13 = 1 - r_11 - r_12; r_23 = 1 - r_22

    For a progressive model with states progression-free (1), progressed (2) and dead (3), no return from 2 to 1 and death absorbing, the annual matrix is upper triangular, so its eigenvalues are its diagonal entries and the square root has diagonal entries equal to their square roots. The progression entry follows from requiring the root squared to return the annual progression probability, the death entries are the row remainders, and two six-month cycles then reproduce every entry of the annual matrix.