Evaluating Density Forecasts

We propose methods for evaluating density forecasts. We focus primarily on methods that are applicable regardless of the particular user's loss function. We illustrate the methods with a detailed simulation example, and then we present an application to density forecasting of daily stock market returns. We discuss extensions for improving suboptimal density forecasts, multi-step-ahead density forecast evaluation, multivariate density forecast evaluation, monitoring for structural change and its relationship to density forecasting, and density forecast evaluation with known loss function.

Liouvillian exceptional points in continuous variable system

The Liouvillian exceptional points for a quantum Markovian master equation of an oscillator in a generic environment are obtained. They occur at the points when the modified frequency of the oscillator vanishes, whereby the eigenvalues of the Liouvillian become real. In a generic system there are two parameters that modify the oscillator's natural frequency. One of the parameters can be the damping rate. The exceptional point then corresponds to critical damping of the oscillator. This situation is illustrated by the Caldeira--Leggett (CL) equation and the Markovian limit of the Hu--Paz--Zhang (HPZ) equation. The other parameter changes the oscillator's effective mass whereby the exceptional point is reached in the limit of extremely heavy oscillator. This situation is illustrated by a modified form of the Kossakowski--Lindblad (KL) equation. The eigenfunctions coalesce at the exceptional points and break into subspaces labelled by a natural number $N$. In each of the $N$-subspace, there is a $(N+1)$-fold degeneracy and the Liouvillian has a Jordan block structure of order-$(N+1)$. We obtain the explicit form of the generalized eigenvectors for a few Liouvillians. Because of the degeneracies, there is a freedom of choice in the generalized eigenfunctions. This freedom manifests itself as an invariance in the Jordan block structure under a similarity transformation whose form is obtained. We compare the relaxation of the first excited state of an oscillator in the underdamped region, critically damped region which corresponds to the exceptional point, and overdamped region using the generalized eigenvectors of the CL equation.Comment: To appear in Physica A (2023

General symmetry in the reduced dynamics of two-level system

We study general transformation on the density matrix of two-level system that keeps the expectation value of observable invariant. We introduce a set of generators that yields hermiticity and trace preserving general transformation which casts the transformation into simple form. The general transformation is in general not factorized and not completely positive. Consequently, either the parameter of transformation or the density matrix it acts on needs to be restricted. It can transform the system in the forward and backward direction with regard to its parameter, not as a semigroup in the time translation symmetry of dynamical maps. The general transformation can rotate the Bloch vector circularly or hyperbolically, dilate it or translate it. We apply the general transformation to study the general symmetry of amplitude damping and phase damping in two-level system. We generalize the generators to higher level systems.Comment: Accepted by European Physical Journal

Two distinct topological phases in the mixed valence compound YbB6 and its differences from SmB6

We discuss the evolution of topological states and their orbital textures in the mixed valence compounds SmB6 and YbB6 within the framework of the generalized gradient approximation plus onsite Coulomb interaction (GGA+U) scheme for a wide range of values of U. In SmB6, the topological Kondo insulator (TKI) gap is found to be insensitive to the value of U, but in sharp contrast, Kondo physics in isostructural YbB6 displays a surprising sensitivity to U. In particular, as U is increased in YbB6, the correlated TKI state in the weak-coupling regime transforms into a d-p-type topological insulator phase with a band inversion between Yb-5d and B-2p orbitals in the intermediate coupling range, without closing the insulating energy gap throughout this process. Our theoretical predictions related to the TKI and non-TKI phases in SmB6 and YbB6 are in substantial accord with recent angle-resolved photoemission spectroscopy (ARPES) experiments.Comment: 6 pages, 4 figures URL: http://link.aps.org/doi/10.1103/PhysRevB.91.15515

Evaluating density forecasts

The authors propose methods for evaluating and improving density forecasts. They focus primarily on methods that are applicable regardless of the particular user's loss function, though they take explicit account of the relationships between density forecasts, action choices, and the corresponding expected loss throughout. They illustrate the methods with a detailed series of examples, and they discuss extensions to improving and combining suboptimal density forecasts, multistep-ahead density forecast evaluation, multivariate density forecast evaluation, monitoring for structural change and its relationship to density forecasting, and density forecast evaluation with known loss function.Forecasting