DSpace Community: KAIST Dept. of Mathematical SciencesKAIST Dept. of Mathematical Scienceshttp://hdl.handle.net/10203/5272023-06-03T18:06:24Z2023-06-03T18:06:24ZA prediction model for healthcare time-series data with a mixture of deep mixed effect models using Gaussian processesHong, JaehyoungChun, Hyonhohttp://hdl.handle.net/10203/3060702023-04-10T08:00:15Z2023-07-01T00:00:00ZTitle: A prediction model for healthcare time-series data with a mixture of deep mixed effect models using Gaussian processes
Authors: Hong, Jaehyoung; Chun, Hyonho
Abstract: Healthcare outcomes such as blood pressure and heart rate are commonly tracked across time owing to technological advances in wearable devices. This advance then makes it possible to predict health risks and to practice personalized medicine. For this type of healthcare data, it is important to reflect huge variation among subjects where the subject becomes an experimental unit. The person-specific model becomes critical for accurate prediction, but it is not optimal due to the noisy nature of the data. It has been demonstrated that sharing information across subjects via a mixed effect model can improve the prediction of individual responses compared to a completely personalized model. However, sharing information across all patients can dilute signals when there are several different patterns present in the data. That is, subjects may form groups and each group behaves differently. To reflect this feature, we extend a deep mixed effect model via a mixture of deep mixed effect models. Our mixed effect model is based on Gaussian processes where the mean adopts the deep neural networks to capture flexible time trends. Our model finds a highly nonlinear trend shared among segments of patients while clustering patients with similar trends into groups. Our approach shows great performance in simulation studies as well as real data analysis, emphasizing the importance of modeling group-specific trends when making accurate predictions from healthcare time-series data.2023-07-01T00:00:00ZConstruction of Blow-Up Manifolds to the Equivariant Self-dual Chern-Simons-Schrodinger EquationKim, KihyunKwon, Soonsikhttp://hdl.handle.net/10203/3060632023-04-10T07:00:11Z2023-06-01T00:00:00ZTitle: Construction of Blow-Up Manifolds to the Equivariant Self-dual Chern-Simons-Schrodinger Equation
Authors: Kim, Kihyun; Kwon, Soonsik
Abstract: We consider the self-dual Chern-Simons-Schrodinger equation (CSS) under equivariance symmetry. Among others, (CSS) has a static solution Q and the pseudoconformal symmetry. We study the quantitative description of pseudoconformal blow-up solutions u such that u(t, r) - e(i gamma)*/T-t Q(r/T-t) -> u* as t -> T-. When the equivariance index m >= 1, we construct a set of initial data (under a codimension one condition) yielding pseudoconformal blow-up solutions. Moreover, when m >= 3, we establish the codimension one property and Lipschitz regularity of the initial data set, which we call the blow-up manifold. This is a forward construction of blow-up solutions, as opposed to authors' previous work [25], which is a backward construction of blow-up solutions with prescribed asymptotic profiles. In view of the instability result of [25], the codimension one condition established in this paper is expected to be optimal. We perform the modulation analysis with a robust energy method developed by Merle, Raphael, Rodnianski, and others. One of our crucial inputs is a remarkable conjugation identity, which (with self-duality) enables the method of supersymmetric conjugates as like Schrodinger maps and wave maps. It suggests how we proceed to higher order derivatives while keeping the Hamiltonian form and construct adapted function spaces with their coercivity relations. More interestingly, it shows a deep connection with the Schrodinger maps at the linearized level and allows us to find a repulsivity structure for higher order derivatives.2023-06-01T00:00:00ZSymplectic Coordinates on the Deformation Spaces of Convex Projective Structures on 2-OrbifoldsChoi, SuhyoungJung, Hongtaekhttp://hdl.handle.net/10203/3070042023-06-02T01:00:12Z2023-06-01T00:00:00ZTitle: Symplectic Coordinates on the Deformation Spaces of Convex Projective Structures on 2-Orbifolds
Authors: Choi, Suhyoung; Jung, Hongtaek
Abstract: Let 𝒪 be a closed orientable 2-orbifold of negative Euler characteristic. Huebschmann constructed the Atiyah-Bott-Goldman type symplectic form ω on the deformation space C(𝒪) of convex projective structures on 𝒪. We show that the deformation space C(𝒪) of convex projective structures on 𝒪 admits a global Darboux coordinate system with respect to ω. To this end, we show that C(𝒪) can be decomposed into smaller symplectic spaces. In the course of the proof, we also study the deformation space C(𝒪) for an orbifold 𝒪 with boundary and construct the symplectic form on the deformation space of convex projective structures on 𝒪 with fixed boundary holonomy.2023-06-01T00:00:00ZNonlinear Diffusion for Bacterial Traveling Wave PhenomenonKim, Yong-JungMimura, MasayasuYoon, Changwookhttp://hdl.handle.net/10203/3060752023-04-10T08:00:59Z2023-05-01T00:00:00ZTitle: Nonlinear Diffusion for Bacterial Traveling Wave Phenomenon
Authors: Kim, Yong-Jung; Mimura, Masayasu; Yoon, Changwook
Abstract: The bacterial traveling waves observed in experiments are of pulse type which is different from the monotone traveling waves of the Fisher-KPP equation. For this reason, the Keller-Segel equations are widely used for bacterial waves. Note that the Keller-Segel equations do not contain the population dynamics of bacteria, but the population of bacteria multiplies and plays a crucial role in wave propagation. In this paper, we consider the singular limits of a linear system with active and inactive cells together with bacterial population dynamics. Eventually, we see that if there are no chemotactic dynamics in the system, we only obtain a monotone traveling wave. This is evidence that chemotaxis dynamics are needed even if population growth is included in the system.2023-05-01T00:00:00Z