Dynamic time-warping is a method of matching by computing similarity using the elastic distance between two time series. This method was developed for speech recognition and it has been applied to many areas.

This matching method makes one time series to be compressed and stretched nonlinearly in the time axis direction based on the dynamic programming to minimize the elastic distance between two time series of different lengths.

In the literature, matching is implemented by changing the scale of time axis in the time series.

Jong Sung Il, a lecturer at the Faculty of Applied Mathematics, has proposed a time-phase scale varying DTW method of implementing matching by varying the scale of phase as well as that of time axis. Then, he applied it to a sine function with varying amplitude to verify its effectiveness.

For further details, please refer to his paper “Pattern Detection Method using Phase Varying Dynamic Time Warping in Time Series” in “Proceedings of KUTIC-2025”.

In recent years, with the rapid popularization of intelligent terminals and sensors, massive data have been accumulated rapidly, and the processing technology of big data has attracted more and more attention. Among them, large information data with functional characteristics such as physiological data, growth curve data, temperature data, etc. are also attracting significant attention. In fact, these data are discrete samples of continuous functions, and these data are usually known as functional data.

The main problem in functional data analysis is dimensionality reduction, and one of the common ways to reduce dimension is to take only a few principal components by performing functional principal component analysis (PCA) on given functional data. The conventional functional principal component analysis method uses the L2-norm, which has the advantage of being computationally convenient, but it is not efficient for data with outliers because the L2-norm is highly affected by outliers. It can be expected that the PCA method using the L1-norm instead of the L2-norm will have better robustness. Without considering the computational complexity when using the L1-norm, high computational cost of functional data analysis cannot be avoided.

Paek Hyon I, a researcher at the Faculty of Applied Mathematics, proposed a method for improving robustness by using L1-norm instead of L2-norm in the functional principal component analysis for functional data with vertical outliers, and constructed an efficient computational algorithm to reduce computational complexity.

This algorithm can achieve more efficient functional principal components with high robustness from data contaminated by vertical outliers while reducing the computational complexity of functional principal component analysis.

It was demonstrated through modeling studies that this method is more robust to outliers than L2-norm functional principal component analysis.

You can find details in his paper “L1-norm Principal Component Analysis Method for Functional Data with Vertical Outliers” in “Proceedings of KUTIC-2025”.

Kim Ung, a section head at the Faculty of Applied Mathematics, has studied a deterministic infinite horizon, a mixed continuous and impulse control problem with unbounded cost functional.

Major difficulties arise from the unboundedness of value functions.

He proved the comparison result of viscosity solution by converting unbounded value functions into bounded ones by suitable transformation and proved that the value function is the unique viscosity solution of the related first order Hamilton-Jacobi quasi-variational inequality (QVI). He constructed a time-discretization scheme for the QVI and proved that an approximate value function exists and it is the unique solution of the approximate QVI.

He also proved that the solution of the time discrete scheme converges to the viscosity solution of the QVI, when the discretization step goes to zero, and that the optimal control of the time discrete system determined by the corresponding dynamic programming is a minimizing sequence of the optimal feedback control of the impulse control problem.

You can find more information in his paper “Approximation Schemes for Impulse Control Systems with Unbounded Cost Functional” in “Proceedings of KUTIC-2025”.