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Tomasz M. Łapiński

Publications and source records attributed to Tomasz M. Łapiński.

4 recordsLinked to original sources

Ensembled Direct Multi-Step forecasting methodology with comparison on macroeconomic and financial data

Accurate forecasts of macroeconomic and financial data, such as GDP, CPI, unemployment rates, and stock indices, are crucial for the success of countries, businesses, and investors, resulting in a constant demand for reliable forecasting models. This research introduces a novel methodology for time series forecasting that combines Ensemble technique with a Direct Multi-Step (DMS) forecasting procedure. This Ensembled Direct Multi-Step (EDMS) approach not only leverages the strengths of both techniques but also capitalizes on their synergy. The ensemble models in the methodology were selected based on performance, complexity, and computational resource requirements, encompassing a full spectrum of model complexities, from simple Linear and Polynomial Regression to medium-complexity ETS and complex LSTM model. Models were weighted based on their performances. In the DMS procedure we limit retraining to one- and five- year/month forecasts for economic and financial data respectively. Standard Iterative Multi-Step (IMS) procedure is employed for other horizons, effectively reducing computational demands while maintaining satisfactory results. The proposed methodology is benchmarked against Ensemble technique conventionally applied to IMS-generated forecasts, utilizing several publicly available macroeconomic and financial datasets. Results demonstrate a significant performance improvement with EDMS methodology, averaging a 33.32% enhancement across the analysed datasets, and sometimes improvement reaching above 60%.

stat.ME↗

Multivariate Laplace's approximation with estimated error and application to limit theorems

In this paper we obtain an approximation for the multivariate Laplace's integral with a large parameter and estimate error term for two cases, when the maximum of the exponent is in the interior of the domain and on the boundary. We are specifically interested in the situation when the function in the exponent depends on the large parameter. As an application we prove weak law of large numbers and central limit theorem. The second result gives different limiting distributions for two cases mentioned above. When the maximum of the exponent is in the interior of the domain it is Normal distribution and if it is on the boundary, it is Exponential in one direction of integration and Normal in other directions.

math.PR↗

Law of large numbers unifying Maxwell-Boltzmann, Bose-Einstein and Zipf-Mandelbort distributions, and related fluctuations

We consider a system composed of a fixed number of particles with total energy smaller or equal to some prescribed value. The particles are non-interacting, indistinguishable and distributed over fixed number of energy levels. The energy levels are degenerate and degeneracy is a function of the number of particles. Three cases of the degeneracy function is considered. It can increase with either the same rate as the number of particles or slower, or faster. We find useful properties of the entropy of the system and solve related entropy optimization problem. It turned out, there are several solutions. Depending on the magnitude of total energy, the maximum of the entropy can be in the interior of system's state space or on the boundary. On the boundary it can have further three cases depending on the degeneracy function. The main result, Law of Large Numbers yields the most probable state of the system, which equals to the point of maximum of the entropy. This point can be either Maxwell-Boltzmann statistics or Bose-Einstein statistics, or Zipf-Mandelbort law. We also find the limiting laws for the fluctuations. These laws are different for different cases of the entropy's maximum. They can be mixture of Normal, Exponential and Discrete distributions. Explicit rates of convergence of moment generating functions are provided for all the theorems. The overview of possible applications is provided in the last section.

math.PR↗

Approximations of the Sum of States by Laplace's Method for a System of Particles with a Finite Number of Energy Levels and Application to Limit Theorems

We consider a generic system composed of a fixed number of particles distributed over a finite number of energy levels. We make only general assumptions about system's properties and the entropy. System's constraints other than fixed number of particles can be included by appropriate reduction of system's state space. For the entropy we consider three generic cases. It can have a maximum in the interior of system's state space or on the boundary. On the boundary we can have another two cases. There the entropy can increase linearly with increase of the number of particles and in the another case grows slower than linearly. The main results are approximations of system's sum of states using Laplace's method. Estimates of the error terms are also included. As an application, we prove the law of large numbers which yields the most probable state of the system. This state is the one with the maximal entropy. We also find limiting laws for the fluctuations. These laws are different for the considered cases of the entropy. They can be mixtures of Normal, Exponential and Discrete distributions. Explicit rates of convergence are provided for all the theorems.

math.PR↗