Computing survival functions of the sum of two independent Markov processes: an application to bladder carcinoma treatment

García-Mora, Belén; Santamaría, C.; Rubio, Gregorio; Pontones, J.L.

doi:10.1080/00207160.2013.765560

Cited by 8 publications

(14 citation statements)

References 6 publications

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“…However, sometimes, the real data do not include the complete evolution of the disease because this evolution is treated in different and independent units within the same department or hospital and consequently the real data are registered separately. For this reason, given the need to describe the complete evolution of the bladder cancer (from a primary tumor to the extirpation of the bladder), in a previous paper [16], we concatenated two Markov processes. Each one analyzed one part of the evolution of this chronic disease because in that study, we had two different and independent real databases belonging to different units of the La Fe University Hospital in Valencia (Spain).…”

Section: Introductionmentioning

confidence: 99%

“…Thinking about the practical application of our approach, for example, if we find large matrices because of dealing with many states, we aim to build approximations that facilitate the task. In Section 2, we present the distribution function of the previous paper [16]. In Section 3, we develop a first approximation of this distribution function to avoid the calculation of an inverse matrix in its expression due to the possibility of bad conditioning of the matrix.…”

Section: Introductionmentioning

confidence: 99%

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A Phase-Type Distribution for the Sum of Two Concatenated Markov Processes Application to the Analysis Survival in Bladder Cancer

2020

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Stochastic processes are useful and important for modeling the evolution of processes that take different states over time, a situation frequently found in fields such as medical research and engineering. In a previous paper and within this framework, we developed the sum of two independent phase-type (PH)-distributed variables, each of them being associated with a Markovian process of one absorbing state. In that analysis, we computed the distribution function, and its associated survival function, of the sum of both variables, also PH-distributed. In this work, in one more step, we have developed a first approximation of that distribution function in order to avoid the calculation of an inverse matrix for the possibility of a bad conditioning of the matrix, involved in the expression of the distribution function in the previous paper. Next, in a second step, we improve this result, giving a second, more accurate approximation. Two numerical applications, one with simulated data and the other one with bladder cancer data, are used to illustrate the two proposed approaches to the distribution function. We compare and argue the accuracy and precision of each one of them by means of their error bound and the application to real data of bladder cancer.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A Phase-Type Distribution for the Sum of Two Concatenated Markov Processes Application to the Analysis Survival in Bladder Cancer

2020

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“…Increasingly the Fréchet derivative is also required, with recent examples including computation of correlated choice probabilities [1], registration of MRI images [6], Markov models applied to cancer data [14], matrix geometric mean computation [24], and model reduction [33], [34]. Higher order Fréchet derivatives have been used to solve nonlinear equations on Banach spaces by generalizing the Halley method [4, sec.…”

Section: Introduction Matrix Functions F : Cmentioning

confidence: 99%

Higher Order Fréchet Derivatives of Matrix Functions and the Level-2 Condition Number

Higham¹,

Relton²

2014

SIAM J. Matrix Anal. & Appl.

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Abstract. The Fréchet derivative L f of a matrix function f : C n×n → C n×n controls the sensitivity of the function to small perturbations in the matrix. While much is known about the properties of L f and how to compute it, little attention has been given to higher order Fréchet derivatives. We derive sufficient conditions for the kth Fréchet derivative to exist and be continuous in its arguments and we develop algorithms for computing the kth derivative and its Kronecker form. We analyze the level-2 absolute condition number of a matrix function ("the condition number of the condition number") and bound it in terms of the second Fréchet derivative. For normal matrices and the exponential we show that in the 2-norm the level-1 and level-2 absolute condition numbers are equal and that the relative condition numbers are within a small constant factor of each other. We also obtain an exact relationship between the level-1 and level-2 absolute condition numbers for the matrix inverse and arbitrary nonsingular matrices, as well as a weaker connection for Hermitian matrices for a class of functions that includes the logarithm and square root. Finally, the relation between the level-1 and level-2 condition numbers is investigated more generally through numerical experiments.Key words. matrix function, Fréchet derivative, Gâteaux derivative, higher order derivative, matrix exponential, matrix logarithm, matrix square root, matrix inverse, matrix calculus, partial derivative, Kronecker form, level-2 condition number, expm, logm, sqrtm, MATLAB AMS subject classifications. 65F30, 65F60DOI. 10.1137/130945259 Introduction. Matrix functions f : Cn×n → C n×n such as the matrix exponential, the matrix logarithm, and matrix powers A t for t ∈ R are being used within a growing number of applications including model reduction [5] The Fréchet derivative of f at A ∈ C n×n is the unique function L f (A, · ) that is linear in its second argument and for all E ∈ C n×n satisfies

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“…The Fréchet derivative of a matrix function also arises as an object of interest in its own right in a variety of applications, of which some recent examples are the computation of correlated choice probabilities [1], computing linearized backward errors for matrix functions [6], analysis of complex networks [7], [8], Markov models of cancer [9], computing matrix geometric means [21], nonlinear optimization for model reduction [25], [26], and tensor-based morphometry [30]. Software for computing Fréchet derivatives of matrix functions is available in a variety of languages [16].…”

Section: Introductionmentioning

confidence: 99%

Estimating the Condition Number of the Fréchet Derivative of a Matrix Function

Higham¹,

Relton²

2014

SIAM J. Sci. Comput.

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Abstract. The Fréchet derivative L f of a matrix function f : C n×n → C n×n is used in a variety of applications and several algorithms are available for computing it. We define a condition number for the Fréchet derivative and derive upper and lower bounds for it that differ by at most a factor 2. For a wide class of functions we derive an algorithm for estimating the 1-norm condition number that requires O(n 3 ) flops given O(n 3 ) flops algorithms for evaluating f and L f ; in practice it produces estimates correct to within a factor 6n. Numerical experiments show the new algorithm to be much more reliable than a previous heuristic estimate of conditioning.

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Computing survival functions of the sum of two independent Markov processes: an application to bladder carcinoma treatment

Cited by 8 publications

References 6 publications

A Phase-Type Distribution for the Sum of Two Concatenated Markov Processes Application to the Analysis Survival in Bladder Cancer

A Phase-Type Distribution for the Sum of Two Concatenated Markov Processes Application to the Analysis Survival in Bladder Cancer

Higher Order Fréchet Derivatives of Matrix Functions and the Level-2 Condition Number

Estimating the Condition Number of the Fréchet Derivative of a Matrix Function

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