Gaussian Process를 이용한 Stochastic Volatility Modeling

안명호 ( Myong Ho An ),유미현 ( Mi Hyeon Ryoo ) 한국금융공학회 2016 금융공학산학연구 Vol.2 No.-

금융시장에서 시간에 따라 변화하는 주가나 수익률 혹은 변동성등을 예측하는 것은 오랜 시간동안 많은 사람들의 관심을 받아왔다. 본 논문에서는 Bayesian Regression 모델에 기반한 Gaussian Process를 이용해 Stochastic Volatility Model을 개발하려고 한다. Gaussian Process는 Bayesian 통계학에 기반을 두고 있기 때문에 명확한 모델을 알지 못해도, 관측값들을 이용해 변동성을 추정해볼 수 있고, 학습된 모델을 재활용해 지속적으로 학습을 시킬 수 있는 장점이 있다. 이렇게 학습된 모델을 이용해 변동성을 예측할 수 있고, 그 예측의 신뢰구간을 확인함으로써 예측에 대한 확률을 추정하는 것도 가능하다. 제시된 Gaussian Process를 이용한 Stochastic Volatility Model은 Non-parametric이기 때문에 특별한 유의사항 없이 편리하게 활용할 수 있다는 것도 중요한 장점이라고 할 수 있다. Gaussian Process의 유연성과 적용성은 변동성 모델링뿐만 아니라 다양한 분야에 적용할 수 있는 가능성을 보여준다. 개발된 모델을 KOSPI와 신한지주에 적용해보고 해당 종목들의 변동성을 비교적 잘 반영한다는 것을 확인 할 수 있었다. Predicting volatility, stock price or daily return have been interesting issues in financial market for decades. In this paper, we are going to present a way of creating a Stochastic volatility model by gaussian process. Gaussian process is based on bayesian statistics, so it does not demand a specific model but request observed data to complete the model. and, it allows to retrain the model with newly observed data without training the model from the scratch. Predicting volatility is also possible with the trained model. The model also provides critical interval on the predicted volatility, so it can compute probability of the predicted volatility. One of the explained model`s beauty is it has no special restriction on it to use. The flexibility and adaptability of Gaussian Process might show expansion to various kinds of application. To demonstrate effectiveness of the model, we apply it to model volatility of KOSPI and Shinhan holdings, and found it working well

Gaussian Process Regression and Classification for Probabilistic Damage Assessment of Spatially Distributed Systems

Matteo Pozzi,Qiaochu Wang 대한토목학회 2018 KSCE Journal of Civil Engineering Vol.22 No.3

We illustrate how methods for non-parametric regression and classification based on Gaussian Processes can be adapted for inferring the condition state of infrastructure components under spatially distributed stressors. When the stressor is modeled by a random field, observations collected in one location can reduce the uncertainty about the stressor intensity also in other locations. Exact inference is possible when the field is Gaussian and observations are perfect or affected by Gaussian noise. However, often the available observations are binary, as those related to the failure or survival of components, and indicate whether the local field is above or below a threshold whose value may also be uncertain. While no efficient scheme for exact inference is available in that setting, we can perform efficient approximate inference when the field is Gaussian and so is the uncertainty on the threshold value. The mathematical formulation for this problem is analogous to that of classification in machine learning, that can be based on latent Gaussian processes. We show how to formulate the problem and how to adapt deterministic methods, as Laplace’s method and Expectation Propagation, and methods based on random number generation, as Monte Carlo uniform sampling and importance sampling, to perform approximate inference. Our illustrative application is the condition assessment of assets exposed to a seismic event. Under specific assumptions, the seismic demand can be modeled as a Gaussian random field, and measures about the demand and about the survival and failure of assets can be processed globally, to update the risk assessment. Specifically, we evaluate methods for approximate inference and discuss their merits.

Superior and Inferior Limits on the Increments of Gaussian Processes

Park, Yong-Kab,Hwang, Kyo-Shin,Park, Soon-Kyu The Korean Statistical Society 1997 Journal of the Korean Statistical Society Vol.26 No.1

Csorgo-Revesz type theorems for Wiener process are developed to those for Gaussian process. In particular, some results of superior and inferior limits for the increments of a Gaussian process are differently obtained under mild conditions, via estimating probability inequalities on the suprema of a Gaussian process.

CONDITIONAL TRANSFORM WITH RESPECT TO THE GAUSSIAN PROCESS INVOLVING THE CONDITIONAL CONVOLUTION PRODUCT AND THE FIRST VARIATION

Chung, Hyun Soo,Lee, Il Yong,Chang, Seung Jun Korean Mathematical Society 2014 대한수학회보 Vol.51 No.6

In this paper, we define a conditional transform with respect to the Gaussian process, the conditional convolution product and the first variation of functionals via the Gaussian process. We then examine various relationships of the conditional transform with respect to the Gaussian process, the conditional convolution product and the first variation for functionals F in $S_{\alpha}$ [5, 8].

THE LIMITING LOG GAUSSIANITY FOR AN EVOLVING BINOMIAL RANDOM FIELD

Kim, Sung-Yeun,Kim, Won-Bae,Bae, Jong-Sig Korean Mathematical Society 2010 대한수학회논문집 Vol.25 No.2

This paper consists of two main parts. Firstly, we introduce an evolving binomial process from a binomial stock model and consider various types of limiting behavior of the logarithm of the evolving binomial process. Among others we find that the logarithm of the binomial process converges weakly to a Gaussian process. Secondly, we provide new approaches for proving the limit theorems for an integral process motivated by the evolving binomial process. We provide a new proof for the uniform strong LLN for the integral process. We also provide a simple proof of the functional CLT by using a restriction of Bernstein inequality and a restricted chaining argument. We apply the functional CLT to derive the LIL for the IID random variables from that for Gaussian.

Conditional transform with respect to the Gaussian process involving the conditional convolution product and the first variation

정현수,이일용,장승준 대한수학회 2014 대한수학회보 Vol.51 No.6

In this paper, we define a conditional transform with respect to the Gaussian process, the conditional convolution product and the first variation of functionals via the Gaussian process. We then examine various relationships of the conditional transform with respect to the Gaussian process, the conditional convolution product and the first variation for functionals F in Sα [5, 8].

적응적 역전파를 이용한 신경망 기반의 가우시안 프로세스 회귀 처리 및 해의 안정성 모델링

양정연 한국정보기술학회 2017 한국정보기술학회논문지 Vol.15 No.12

This paper describes a study on a modified neural network model to cover the Gaussian process regression method, which supports for probabilistic confidence interval of functional distribution and noisy signals. Discrimination function is redesigned to handle deviated errors under normal gaussian distribution. This modified error function is combined with typical back propagation process. Thus, the results become the superposition of weighted Gaussian functions and the decision boundaries, as in the case of conventional Gaussian process regression methods. Owing to the discontinuity of discrimination function, the stability of the proposed error function becomes deteriorated. The concept of low pass filtering is applied to improve the unstable problems of weight updating process. 본 논문은 공간 내 확률적 분포를 가진 함수 또는 잡음 신호를 확률 기반의 신뢰 구간으로 표현하기 위해, 기존의 역전파 신경망 기법을 변형하여 가우시안 프로세스의 회귀에 의한 결정 공간의 추종방식을 제안하고자 한다. 신경망 기법의 역전파 과정에 사용되는 오차 함수를 부호에 따라 가변적 가중치를 가지도록 설계하고, 이를 가우시안 함수에 기반한 표준 편차 처리에 적합하도록 변형함에 따라, 가우시안 확률 분포 함수의 조합에 의한 결정 경계의 추종 및 신경망 노드의 합성에 의한 가우시안 프로세스의 회귀 처리 과정을 수행하고자 한다. 수정된 오차 함수의 경우, 불연속성에 기인한 해 탐색 과정의 불안정성이 존재하기에 이를 회피하기 위해 저역 통과 필터의 개념을 이용한 안정성 방법에 대해 논하고자 한다.

Controlled kinetic Monte Carlo simulation of laser improved nano particle deposition process

Song, Ji-Hyeon,Choi, Kweon-Hoon,Dai, Ruonan,Choi, Jung-Oh,Ahn, Sung-Hoon,Wang, Yan Elsevier Sequoia 2018 Powder Technology Vol. No.

<P><B>Abstract</B></P> <P>Thin film coating is important in many applications such as electrodes, sensors, and energy devices. Nano particle deposition is one of the most used additive manufacturing processes for coating. It has advantages of efficiency, cost effectiveness, and ease of controlling film properties. Recent experimental studies showed that laser can enhance the efficiency of the deposition process. However, there is still a lack of fundamental understanding of the laser treatment effect on nano particles, which makes process control difficult and ad hoc. In this research, the effect of laser treatment on morphological change of films in the nano particle deposition system is studied with controlled kinetic Monte Carlo (cKMC) simulation. cKMC is a generalized version of classical kinetic Monte Carlo, which can be used to simulate both controlled and self-assembly processes at atomistic level with larger sizes and longer time scales than molecular dynamics. In this work, a coarse-grained cKMC model is constructed to simulate diffusion, laser irradiation, and deposition processes simultaneously. The simulation model is calibrated with experimental data. Different laser irradiation conditions on alumina particles are studied, which result in different thickness and porosity of the deposited layers. A Gaussian process regression modeling approach is also developed for model validation with the consideration of observation bias and discrepancy. Simulation results are in good agreement with the experimental results.</P> <P><B>Highlights</B></P> <P> <UL> <LI> Alumina nano particle deposition in aerosol flow is simulated with controlled kinetic Monte Carlo. </LI> <LI> Laser irradiation, diffusion, and deposition processes are simulated efficiently. </LI> <LI> A Gaussian process regression approach is proposed for model validation when observation bias is present. </LI> </UL> </P> <P><B>Graphical abstract</B></P> <P>[DISPLAY OMISSION]</P>

물리적 구배 정보를 이용한 공력계수 모형화를 위한 GE 크리깅의 적용

강신성(Shinseong Kang),이경훈(Kyunghoon Lee) 한국항공우주학회 2020 韓國航空宇宙學會誌 Vol.48 No.3

유도무기는 원통형 형상에서 기인한 기하학적 특성으로 6자유도 공력계수에 물리적 구배 조건을 내포하게 된다. 본 연구는 부가적으로 주어진 물리적 구배 정보를 공력계수 모형화에서 효과적으로 이용할 목적으로 구배 보강 가우스 과정을 사용하였다. 물리적 구배 정보를 활용한 공력계수 예측의 정확성을 살펴보기 위해, 가우스 과정에 기초한 공력계수 예측 모형을 구배 정보의 유무에 따라 각각 구성한 후 서로의 예측 정확도를 비교․분석하였다. 그 결과, 물리적 구배 정보를 고려한 공력계수 예측은 부여된 구배 조건을 정확히 만족하였을 뿐만 아니라 그렇지 않은 모형에 비해 예측 정확도가 더 우수함을 확인하였다. 다만, 구배 보강 가우스 과정으로는 물리적 구배 정보를 연속적으로 부여할 수 없으며 추가된 구배 정보로 인해 공력계수 예측 모형 구성에 요구되는 표본수가 증가하는 단점도 확인하였다. The six-DOF aerodynamic coefficients of a missile entail inherent physical gradient constraints originated from the geometric characteristics of a cylindrical fuselage. To effectively adopt the freely available gradient information in aerodynamic coefficients modeling, this research employed gradient-enhanced (GE) Gaussian process. To investigate the accuracy of aerodynamic coefficients predicted with gradients information, we compared two Gaussian-process-based models: ordinary and GE Gaussian process models with and without gradient information, respectively. As a result, we found that GE Gaussian process models were able to comply with imposed gradient information and more accurate than ordinary Gaussian process models. However, we also found that GE Gaussian process modeling cannot handle gradient information continuously and ends up with more samples due to additional gradient information.

LIMIT BEHAVIORS FOR THE INCREMENTS OF A d-DIMENSIONAL MULTI-PARAMETER GAUSSIAN PROCESS

CHOI YONG-KAB,LIN ZRENGYAN,SUNG HWA-SANG,HWANG KYO-SHIN,MOON HEE-JIN Korean Mathematical Society 2005 대한수학회지 Vol.42 No.6

In this paper, we establish limit theorems containing both the moduli of continuity and the large incremental results for finite dimensional Gaussian processes with N parameters, via estimating upper bounds of large deviation probabilities on suprema of the Gaussian processes.