Reproducing kernel Hilbert space method for cox proportional hazard model - Abdul Manaf, Nur'azah (2016)
Numerous researchers are enthusiastic about statistical modeling to estimate the survival for patients. Usually, the information obtained from the survival data in biomedical sciences includes the age, race, health conditions, disease free time and the survival times of patients. Apart from developing the survival data models, estimations on the hazard functions are being done to estimate the chance of survival or the time from diagnosis to failure or death of the patients. It is expected from the result of analysis that using the proposed reproducing kernel method to develop survival models will be helpful in predicting the relapse time or death of patients and will contribute to intense understanding on the connection between reproducing kernels and survival data, which on exploitation will lead to more applications especially in solving related problems in statistics of several areas. Reproducing kernel Hilbert spaces (RKHS) has been used in the statistics literature for many years. This research explores the mathematical aspects and properties of reproducing kernel Hilbert space. The purpose of this research is to review the basic facts and the importance of RKHS that contribute to the kernel method and its application in statistics by analyzing the effect of kernel method on survival data. We propose a new reproducing kernel Hilbert space (RKHS) and prove that the kernel obtained satisfy the properties of RKHS. The task is to extend the Cox proportional hazard model by using the new reproducing kernel obtained and apply the kernel method to randomly selected survival data sets. The new kernel we construct will be used in the score function ƒ(x) of the representer theorem for the hazard model. As for the methodology, we obtain the partial differentials of the risk or loss function to fit the hazard model. We find optimal values of parameters of the score function ƒ(x) by using the Newton-Raphson method, which requires setting up the related function to be minimized. Then, we apply the kernel method to the survival data. Finally, we propose an algorithm of minimization of the loss function in the general Cox model. This algorithm is used to determine the vector i a that enables us to find the optimal parameters of ƒ(x)which is simplified as F(x)= ∑aᵢK(x,xᵢ) . The survival of patients is estimated through the observation of the exponential values, exp(ƒ(x)) the of model. The ƒ (x) values will affect the riskor failures of the patients. Simulations with different number of covariates will be performed using the proposed kernel K(Ax, By) = <Ax, By>. The simulations are done to investigate the effects of different number of covariates on the prediction of overall survival of patients. We have constructed a new reproducing kernel RKHS and obtained partial differentials of the loss function. The kernel method is expected to be efficient for problems involving data with a large number of covariates. The findings of this research will encourage future exploration of the use of kernel method in the prediction of survival times or failures in many areas such as science, engineering and economics.