arXiv subjects

Ta-Hsin Li

Publications and source records attributed to Ta-Hsin Li.

11 recordsLinked to original sources

arXiv2026

Spline Quantile Regression with Cubic and Linear Smoothing Splines

Spline quantile regression (SQR) is a method introduced recently by Li and Megiddo (2026) for linear quantile regression where the regression coefficients are treated as smooth functions of the quantile level. With the coefficients represented by cubic splines with fixed knots on a given set of quantiles, the SQR method produces an estimate for the functional coefficients by solving a penalized quantile regression problem. The $\ell_1$-norm of the second derivatives of the coefficients is employed as the penalty for regulating the roughness of the functional coefficients. This extends the SQR method by introducing additional pairings of the functional representation for the regression coefficients and the penalty for their roughness. The resulting cubic and linear SQR solutions are shown to be smoothing splines which are optimal in a functional space larger than the respective spline space with fixed knots. It is shown that the cubic SQR can be reformulated and solved as a quadratic program and the linear SQR as a linear program. A simulation study demonstrates that the SQR solutions not only offer a concise functional representation of the regression coefficients with distinct smoothness characteristics, but also provide a capability of producing more accurate estimates of the regression coefficients when the underlying functions are suitably smooth. Application of the SQR solutions is demonstrated by real-data examples, including a Granger causality analysis of stock market indices.

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arXiv2025

Spline Quantile Regression

Quantile regression is a powerful tool capable of offering a richer view of the data as compared to least-squares regression. Quantile regression is typically performed individually on a few quantiles or a grid of quantiles without considering the similarity of the underlying regression coefficients at nearby quantiles. When needed, an ad hoc post-processing procedure such as kernel smoothing is employed to smooth the individually estimated coefficients across quantiles and thereby improve the performance of these estimates. This paper introduces a new method, called spline quantile regression (SQR), that unifies quantile regression with quantile smoothing and jointly estimates the regression coefficients across quantiles as smoothing splines. We discuss the computation of the SQR solution as a linear program (LP) using an interior-point algorithm. We also experiment with some gradient algorithms that require less memory than the LP algorithm. The performance of the SQR method and these algorithms is evaluated using simulated and real-world data.

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arXiv2024

Spline Autoregression Method for Estimation of Quantile Spectrum

The quantile spectrum was introduced in Li (2012; 2014) as an alternative tool for spectral analysis of time series. It has the capability of providing a richer view of time series data than that offered by the ordinary spectrum especially for nonlinear dynamics such as stochastic volatility. A novel method, called spline autoregression (SAR), is proposed in this paper for estimating the quantile spectrum as a bivaraite function of frequency and quantile level, under the assumption that the quantile spectrum varies smoothly with the quantile level. The SAR method is facilitated by the quantile discrete Fourier transform (QDFT) based on trigonometric quantile regression. It is enabled by the resulting time-domain quantile series (QSER) which represents properly scaled oscillatory characteristics of the original time series around a quantile. A functional autoregressive (AR) model is fitted to the QSER on a grid of quantile levels by penalized least-squares with the AR coefficients represented as smoothing splines of the quantile level. While the ordinary AR model is widely used for conventional spectral estimation, the proposed SAR method provides an effective way of estimating the quantile spectrum as a bivariate function in comparison with the alternatives. This is confirmed by a simulation study.

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arXiv2024

Quantile-Crossing Spectrum and Spline Autoregression Estimation

The quantile-crossing spectrum is the spectrum of quantile-crossing processes created from a time series by the indicator function that shows whether or not the time series lies above or below a given quantile at a given time. This bivariate function of frequency and quantile level provides a richer view of serial dependence than that offered by the ordinary spectrum. We propose a new method for estimating the quantile-crossing spectrum as a bivariate function of frequency and quantile level. The proposed method, called spline autoregression (SAR), jointly fits an AR model to the quantile-crossing series across multiple quantiles; the AR coefficients are represented as spline functions of the quantile level and penalized for their roughness. Numerical experiments show that when the underlying spectrum is smooth in quantile level the proposed method is able to produce more accurate estimates in comparison with the alternative that ignores the smoothness.

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arXiv2024

Expectile Periodograms

This paper introduces a novel periodogram-like function, called the expectile periodogram, for modeling spectral features of time series and detecting hidden periodicities. The expectile periodogram is constructed from trigonometric expectile regression, in which a specially designed check function is used to substitute the squared $l_2$ norm that leads to the ordinary periodogram. The expectile periodogram retains the key properties of the ordinary periodogram as a frequency-domain representation of serial dependence in time series, while offering a more comprehensive understanding by examining the data across the entire range of expectile levels. We establish the asymptotic theory and investigate the relationship between the expectile periodogram and the so called expectile spectrum. Simulations demonstrate the efficiency of the expectile periodogram in the presence of hidden periodicities. Finally, by leveraging the inherent two-dimensional nature of the expectile periodogram, we train a deep learning (DL) model to classify earthquake waveform data. Remarkably, our approach outperforms alternative periodogram-based methods in terms of classification accuracy.

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arXiv2024

A semi-parametric estimation method for quantile coherence with an application to bivariate financial time series clustering

In multivariate time series analysis, spectral coherence measures the linear dependency between two time series at different frequencies. However, real data applications often exhibit nonlinear dependency in the frequency domain. Conventional coherence analysis fails to capture such dependency. The quantile coherence, on the other hand, characterizes nonlinear dependency by defining the coherence at a set of quantile levels based on trigonometric quantile regression. This paper introduces a new estimation technique for quantile coherence. The proposed method is semi-parametric, which uses the parametric form of the spectrum of a vector autoregressive (VAR) model to approximate the quantile coherence, combined with nonparametric smoothing across quantiles. At a given quantile level, we compute the quantile autocovariance function (QACF) by performing the Fourier inverse transform of the quantile periodograms. Subsequently, we utilize the multivariate Durbin-Levinson algorithm to estimate the VAR parameters and derive the estimate of the quantile coherence. Finally, we smooth the preliminary estimate of quantile coherence across quantiles using a nonparametric smoother. Numerical results show that the proposed estimation method outperforms nonparametric methods. We show that quantile coherence-based bivariate time series clustering has advantages over the ordinary VAR coherence. For applications, the identified clusters of financial stocks by quantile coherence with a market benchmark are shown to have an intriguing and more informative structure of diversified investment portfolios that may be used by investors to make better decisions.

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arXiv2022

Quantile Fourier Transform, Quantile Series, and Nonparametric Estimation of Quantile Spectra

A nonparametric method is proposed for estimating the quantile spectra and cross-spectra introduced in Li (2012; 2014) as bivariate functions of frequency and quantile level. The method is based on the quantile discrete Fourier transform (QDFT) defined by trigonometric quantile regression and the quantile series (QSER) defined by the inverse Fourier transform of the QDFT. A nonparametric spectral estimator is constructed from the autocovariance function of the QSER using the lag-window (LW) approach. Smoothing techniques are also employed to reduce the statistical variability of the LW estimator across quantiles when the underlying spectrum varies smoothly with respect to the quantile level. The performance of the proposed estimation method is evaluated through a simulation study.

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arXiv2020

A Systematic Approach to Surveillance and Detection of Hierarchical Healthcare Cost Drivers and Utilization Offsets

There is strong interest among healthcare payers to identify emerging healthcare cost drivers to support early intervention. However, many challenges arise in analyzing large, high dimensional, and noisy healthcare data. In this paper, we propose a systematic approach that utilizes hierarchical search strategies and enhanced statistical process control (SPC) algorithms to surface high impact cost drivers. Our approach aims to provide interpretable, detailed, and actionable insights of detected change patterns attributing to multiple clinical factors. We also proposed an algorithm to identify comparable treatment offsets at the population level and quantify the cost impact on their utilization changes. To illustrate our approach, we apply it to the IBM Watson Health MarketScan Commercial Database and organized the detected emerging drivers into 5 categories for reporting. We also discuss some findings in this analysis and potential actions in mitigating the impact of the drivers.

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arXiv2019

A Semi-Parametric Estimation Method for the Quantile Spectrum with an Application to Earthquake Classification Using Convolutional Neural Network

In this paper, a new estimation method is introduced for the quantile spectrum, which uses a parametric form of the autoregressive (AR) spectrum coupled with nonparametric smoothing. The method begins with quantile periodograms which are constructed by trigonometric quantile regression at different quantile levels, to represent the serial dependence of time series at various quantiles. At each quantile level, we approximate the quantile spectrum by a function in the form of an ordinary AR spectrum. In this model, we first compute what we call the quantile autocovariance function (QACF) by the inverse Fourier transformation of the quantile periodogram at each quantile level. Then, we solve the Yule-Walker equations formed by the QACF to obtain the quantile partial autocorrelation function (QPACF) and the scale parameter. Finally, we smooth QPACF and the scale parameter across the quantile levels using a nonparametric smoother, convert the smoothed QPACF to AR coefficients, and obtain the AR spectral density function. Numerical results show that the proposed method outperforms other conventional smoothing techniques. We take advantage of the two-dimensional property of the estimators and train a convolutional neural network (CNN) to classify smoothed quantile periodogram of earthquake data and achieve a higher accuracy than a similar classifier using ordinary periodograms.

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arXiv2019

Quantile-Frequency Analysis and Spectral Measures for Diagnostic Checks of Time Series With Nonlinear Dynamics

Nonlinear dynamic volatility has been observed in many financial time series. The recently proposed quantile periodogram offers an alternative way to examine this phenomena in the frequency domain. The quantile periodogram is constructed from trigonometric quantile regression of time series data at different frequencies and quantile levels, enabling the quantile-frequency analysis (QFA) of nonlinear serial dependence. This paper introduces some spectral measures based on the quantile periodogram for diagnostic checks of financial time series models and for model-based discriminant analysis. A simulation-based parametric bootstrapping technique is employed to compute the $p$-values of the spectral measures. The usefulness of the proposed method is demonstrated by a simulation study and a motivating application using the daily log returns of the S\&P 500 index together with GARCH-type models. The results show that the QFA method is able to provide additional insights into the goodness of fit of these financial time series models that may have been missed by conventional tests. The results also show that the QFA method offers a more informative way of discriminant analysis for detecting regime changes in financial time series.

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arXiv2019

A Systematic Approach to Detect Hierarchical Healthcare Cost Drivers and Interpretable Change Patterns

There is strong interest among payers to identify emerging healthcare cost drivers to support early intervention. However, many challenges arise in analyzing large, high dimensional, and noisy healthcare data. In this paper, we propose a systematic approach that utilizes hierarchical and multi-resolution search strategies using enhanced statistical process control (SPC) algorithms to surface high impact cost drivers. Our approach aims to provide interpretable, detailed, and actionable insights of detected change patterns attributing to multiple demographic and clinical factors. We also proposed an algorithm to identify comparable treatment offsets at the population level and quantify the cost impact on their utilization changes.

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