1,721,041 research outputs found
Fractional Branching Processes
Full text linkLo studio di processi di tipo nascita-morte è molto importante nell'analisi di fenomeni fisici quali, ad esempio, la dinamica di popolazioni oppure la diffusione di epidemie, o ancora il susseguirsi delle scosse di assestamento nello studio dei terremoti. Modelli classici (eventualmente nella loro versione non lineare) sono per esempio il processo di pure nascite, di pure morti o di nascite-morti. Sfruttando l'apparato teorico del calcolo frazionario, notevolmente sviluppato negli ultimi anni, si è costruita un'intera classe di nuovi processi, sempre di tipo nascita-morte, la cui evoluzione è guidata da equazioni differenziali e alle differenze finite frazionarie (su questo punto si possono consultare. La struttura della tesi è la seguente: nel primo capitolo è delineata una panoramica dei modelli classici con una descrizione delle proprietà fondamentali e del comportamento dinamico. L'ultima sezione del capitolo è dedicata a una concisa descrizione del calcolo frazionario; tra le altre cose vengono definiti i concetti fondamentali di integrale frazionario e di derivata frazionaria, questa nelle due definizioni di Riemann-Liouville e Caputo. Utilizzando il calcolo frazionario, nei successivi capitoli 2, 3 e 4, vengono definiti e analizzati in dettaglio i processi frazionari di pure nascite non lineare, di pure morti non lineare, lineare e sublineare e il processo lineare di nascite-morti frazionario. Nel capitolo 5 viene proseguito lo studio della versione lineare del processo di pure nascite frazionario (Yule-Furry frazionario) con la determinazione della distribuzione dei tempi inter nascita, e di alcune convenienti rappresentazioni; sono inoltre presentati i risultati di alcune simulazioni per facilitare lo studio della dinamica del processo e alcune considerazioni di tipo inferenziale (stimatore dei momenti) per la stima del tasso di nascita e dell'ordine frazionario di differenziazione. Nell'ultimo capitolo vengono proposte alcune estensioni dei modelli studiati nei capitoli precedenti tramite subordinazione con tempi aleatori, permettendo quindi di introdurre un'ulteriore fonte di variabilità. Vengono inoltre derivate alcune rappresentazioni per i processi non lineari subordinati di pure nascite e derivate relazioni funzionali che coinvolgono alcune funzioni speciali quali, per esempio, quella di Mittag-Leffler
Fractional calculus modelling for unsteady unidirectional flow of incompressible fluids with time-dependent viscosity
No full textIn this note we analyze a model for a unidirectional unsteady flow of a viscous incompressible fluid with time dependent viscosity. A possible way to take into account such behaviour is to introduce a memory formalism, including thus the time dependent viscosity by using an integro-differential term and therefore generalizing the classical equation of a Newtonian viscous fluid. A possible useful choice, in this framework, is to use a rheology based on stress/strain relation generalized by fractional calculus modelling. This is a model that can be used in applied problems, taking into account a power law time variability of the viscosity coefficient. We find analytic solutions of initial value problems in an unbounded and bounded domain. Furthermore, we discuss the explicit solution in a meaningful particular case. © 2012 Elsevier B.V
Some results on time-varying fractional partial differential equations and birth-death processes
Full text linkAnalytic solutions of fractional differential equations by operational methods
No full textWe describe a general operational method that can be used in the analysis of fractional initial and boundary value problems with additional analytic conditions. As an example, we derive analytic solutions of some fractional generalisation of differential equations of mathematical physics. Fractionality is obtained by substituting the ordinary integer-order derivative with the Caputo fractional derivative. Furthermore, operational relations between ordinary and fractional differentiation are shown and discussed in detail. Finally, a last example concerns the application of the method to the study of a fractional Poisson process. © 2012 Elsevier Inc. All rights reserved
Bayesian adaptive designs in multi-arm multi-stage clinical trials
No full textClinical trial seek to investigate novel treatments, asses the relative benefits of competing therapies, and establish optimal treatment combinations. Statistical models provide an explicit way to models patients response to a treatment, and make inference about the clinical utility of therapies which guides clinical decision making.
Statistical designs for clinical trials are a formal procedure the aim to maximize the the quality of generated information on the performance of experimental treatment. We explore a particular class of clinical trial design called adaptive design, which allows modifications of one or more specified aspects of the design based on the analysis of information (usually interim data) collected from subjects in the study.
The interest in adaptive design studies arises from the belief that these methods provide a promising new venue in the task of improving drug development compared to conventional non-adaptive statistical methods for the design of clinical experiments. In particular, the approach of adaptive design may increase the likelihood of a patient to be treated with a successful drug and may reduce the uncertainty on the treatment effect. The class of adaptive designs includes adaptive randomization procedures, sample size re-estimations, and sequential or group-sequential interim analysis.
The Bayesian approach is ideally suited to dynamically adapt the design as information arises during a trial. Accumulated data can be used at any time to modify the design of the trial, for instance, by stopping treatment assignments to ineffective arms or unbalancing randomization towards arms with strong evidence of treatment superiority.
In this thesis we focus on two particular sub-classes of Bayesian adaptive designs:Two-stage designs for phase II clinical trials and Response-adaptive randomization designs for multi arm (multi-stage) clinical trial.
A bayesian approach for randomized two-stage designs: Two-stage designs are commonly used in phase II clinical trials, especially in cancer clinical trials. Standard two-stage designs, introduced in Chapter 1, involve one single experimental arm that is compared to a pre-fixed desired level. However, the rate of failure in phase III oncology trials is surprisingly high, partly owing to inadequate phase II studies. Recently, the use of randomized designs in phase II has been increasingly recommended to avoid such limitations.
With the supervision of Prof. Valeria Sambucini, we proposed a randomized version of a Bayesian two-stage design due to Tan and Machin [120] (see Chapter 2 of the thesis). The design selects the two-stage sample sizes by ensuring a large posterior probability that the true response rate of the experimental treatment exceeds that of the standard agent, assuming that the experimental treatment is actually more effective (see Cellamare et al. [35]). This optimistic assumption is realized by fixing virtual outcomes in favor of the experimental arm. However, the design does not account for the uncertainty about future data. Therefore, in Chapter 3 we propose a two-arm two-stage design based on a Bayesian predictive approach (see Cellamare and Sambucini [34]). The idea is to ensure a large probability, expressed in terms of the prior predictive probability of the data, of obtaining a substantial posterior evidence in favour of the experimental treatment under the assumption that it is actually more effective than the standard agent. This design is a randomized version of the two-stage design that has been proposed for single-arm phase II trials by Sambucini [104]. We examine the main features of our novel design as all the parameters involved vary and compare our approach with Jung’s minimax and optimal designs [76]. A potential limitation of the proposed design is that the second stage sample size is determined before observing the first stage data. It can produce some paradoxical situations in which a second stage analysis is performed and additional patients recruited, despite the first stage results were already sufficient to make a final decision. As also suggested by Sambucini[105], we solve this potential problem by using an adaptive version of the Bayesian predictive two-arm two-stage design, in which the
second stage sample size is selected after the first stage results have been observed.
Bayesian response-adaptive design for multi-arm clinical trials: In the planning of a clinical trial, the randomization of patients to either the experimental or control groups is among the most important advances in the history of medical research. Randomization prevents confounding due to latents factors that are correlated with the health outcome and control potential bias of the treatment effect estimates by balancing patients among the treatment arms . However, this property could be sometimes in conflict with ethical assumptions. As experiments on human subjects, clinical trials are characterized by the necessity of finding a balance between collective ethics and individual ethics. When the observation of a failure represents an extreme outcome (i.e. death), the traditional balanced randomization becomes ethically infeasible because of unjustifiable sacrifice of individual ethics. In this context, response-adaptive randomization designs represent a class of designs in which the probability of treatment assignment changes according to patient’s outcome and treating more patients with effective arms compared to fixed randomization. Response-adaptive designs have been widely studied in literature and we provide a review of them in either frequentist or Bayesian framework in Chapter 4. Under the supervision of Prof. Lorenzo Trippa and Prof. Steffen Ventz at the Harvard School of Public Health (and Dana Farber Cancer Institute), we studied the use of Bayesian adaptive randomization (BAR) design in the context of multi-arm clinical trials, in which multiple experimental arms are compared to a common control arm (see Chapter 5). In collaboration with Dr. Carole D. Mitnick and motivated by a multi-arm randomized clinical trial for fluoroquinolone-susceptible multi-drug resistant tuberculosis (MDR-TB)5 called endTb, we build a response-adaptive clinical trial in which the randomization procedure is updated using two preliminary outcomes. The primary study outcome is treatment success after 72 weeks from treatment and two preliminary responses are measured after 8 and 39 weeks (see Cellamare et al. [36]). We compared the proposed design with a standard multi-arm multi-stage design through hypothetical scenarios based on historical data.
Our simulations show how BAR may be more efficient than standard multi-arm multi-stage designs. In particular, when we compare the statistical power of BAR to that of non-adaptive designs under a variety of realistic hypothetical scenarios, we observe that our design requires less patients than non-adaptive designs to ensure a fixed predefine power. Moreover, BAR consistently allocates more participants to effective arm(s). In conclusion, given the objective of evaluating several new therapeutic regimens in a timely fashion, Bayesian response adaptive designs seem more appealing for MDR-TB trials. This approach offers the resource benefit of requiring fewer participants and tends to increase allocation to the effective regimens. Despite the attractive operating characteristics of response adaptive design in the multi arm settings, as shown in the case of the endTb trial, multi-arm clinical trials design presented in literature are generally based on the assumption that all experimental treatments are available at the enrollment of the first patient.
In several real situations, new drugs are rarely at the same stage of development and multi-arm designs may delay in the clinical evaluation of new treatments. These limitations motivate our study of statistical methods for adding new experimental arms after a clinical trial started enrolling patients (see Chapter 6). We consider both balanced and response-adaptive randomization for experimental designs that allow
investigators to add new arms during the course of the trial (see Ventz, Cellamare et al [134]). We discuss their application in the endTb context and we evaluate the proposed experimental designs using a set of realistic simulation scenarios. Our results showed that adding treatments to an ongoing trial yield substantial gain in efficacy compared to multiple independent two-arms trials. The use of standard response-adaptive algorithms can behave poorly in this setting and adjustments of the procedures are required. Moreover, we found that, despite the complexity and the computational burden, response-adaptive algorithms can potentially outperform the balanced algorithm
Simulation and estimation for the fractional Yule process
No full textIn this paper, we propose some representations of a generalized linear birth process called fractional Yule process (fYp). We also derive the probability distributions of the random birth and sojourn times. The inter-birth time distribution and the representations then yield algorithms on how to simulate sample paths of the fYp. We also attempt to estimate the model parameters in order for the fYp to be usable in practice. The estimation procedure is then tested using simulated data as well. We also illustrate some major characteristics of fYp which will be helpful for real applications
On a Fractional Binomial Process
No full textThe classical binomial process has been studied by Jakeman (J. Phys. A 23: 2815-2825, 1990) (and the references therein) and has been used to characterize a series of radiation states in quantum optics. In particular, he studied a classical birth-death process where the chance of birth is proportional to the difference between a larger fixed number and the number of individuals present. It is shown that at large times, an equilibrium is reached which follows a binomial process. In this paper, the classical binomial process is generalized using the techniques of fractional calculus and is called the fractional binomial process. The fractional binomial process is shown to preserve the binomial limit at large times while expanding the class of models that include non-binomial fluctuations (non-Markovian) at regular and small times. As a direct consequence, the generality of the fractional binomial model makes the proposed model more desirable than its classical counterpart in describing real physical processes. More statistical properties are also derived
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