Stochastic numerical analysis for Brownian motion on SO(3)

Piggott, Marc J.; Solo, Victor

doi:10.1109/cdc.2014.7039919

Cited by 9 publications

(6 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Remark 5: The algorithms presented in [5], [9] are both SLIE with the E-M method that considers N = 1. Other algorithms given for numerically solving the SDE (8) are found in [10], [11].…”

Section: Euler-maruyama (E-m) Algorithmmentioning

confidence: 98%

“…Obtaining Ω requires obtaining dX using (10), and equating it with (9). This leads to a differential equation in Ω that can be solved.…”

Section: B Motivating the Stochastic With The Deterministicmentioning

confidence: 99%

“…It plays a major role in engineering applications, e.g. see [4], [5], [7], [9], [13]. The corresponding Lie algebra is denoted by so(n).…”

Section: Matrix Lie Groupsmentioning

confidence: 99%

See 2 more Smart Citations

A simple approach to numerical methods for stochastic differential equations in Lie groups

Marjanovic

Piggott

Solo

2015

2015 54th IEEE Conference on Decision and Control (CDC)

Self Cite

View full text Add to dashboard Cite

A large number of important engineering applications require solving stochastic differential equations (SDEs) so that geometrical constraints are satisfied, e.g. the solution has to lie in a matrix Lie group. But the few papers that properly derive numerical schemes for SDEs evolving in Lie groups are in the mathematics literature and not accessible to engineers. The engineering literature is also small but plagued with problems. With this in mind we give a direct accessible derivation of numerical schemes for solving such equations. We do not rely on differential geometry or advanced random process theory. We give simulations in a simple case to show how geometric structure is preserved.

show abstract

“…Remark 5: The algorithms presented in [5], [9] are both SLIE with the E-M method that considers N = 1. Other algorithms given for numerically solving the SDE (8) are found in [10], [11].…”

Section: Euler-maruyama (E-m) Algorithmmentioning

confidence: 98%

“…Obtaining Ω requires obtaining dX using (10), and equating it with (9). This leads to a differential equation in Ω that can be solved.…”

Section: B Motivating the Stochastic With The Deterministicmentioning

confidence: 99%

See 1 more Smart Citation

A simple approach to numerical methods for stochastic differential equations in Lie groups

Marjanovic

Piggott

Solo

2015

2015 54th IEEE Conference on Decision and Control (CDC)

Self Cite

View full text Add to dashboard Cite

show abstract

“…Examples include computer vision [1][2][3][4], array signal processing [5], satellite attitude and pose estimation [6][7][8], robotics [6,9,10], etc. Particle Filtering (PF) methods have become a very popular class of algorithms to numerically solve these estimation problems in a recursive fashion as observations become available.…”

Section: Introductionmentioning

confidence: 99%

“…However, such filters cannot be considered for filtering problems in other manifolds such as matrix Lie groups G because the update schemes tend to immediately leave the manifold, e.g. see the example in [8]. Hence, these schemes are unstable.…”

Section: Introductionmentioning

confidence: 99%

An engineer's guide to particle filtering on matrix Lie groups

Marjanovic

Solo

2016

2016 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

Self Cite

View full text Add to dashboard Cite

In many important engineering applications the state dynamics of a system are modelled by Stochastic Differential Equations (SDEs) evolving in non-Euclidean spaces such as matrix Lie groups. Due to the advances in computing power, the problem of state estimation can be efficiently addressed by the particle filtering method. This requires dealing with both the geometry and the stochastics of the problem. However, the very few papers that properly deal with either are in the mathematics literature and not accessible. The engineering literature is also small but plagued with problems. With this in mind, we give a direct accessible derivation of the particle filter algorithm for state estimation in matrix Lie groups. We do not rely on differential geometry or advanced stochastic calculus. Simulation examples are provided.

show abstract

Deep Neural Network Architectures for Momentary Forecasting in Dry Bulk Markets: Robustness to the Impact of COVID-19

Lee

2023

IEEE Access

View full text Add to dashboard Cite

The COVID-19 pandemic has severely affected various global markets, increasing the need for new forecasting models for the dry bulk market. Therefore, this study proposes deep neural network (abbreviated DNN) architectures to build a model for momentary forecasting that does not affect accuracy in the case of economic shocks (i.e., COVID-19) and elucidates the strategy for obtaining DNNs. First, since momentary and short-term forecastings are fundamentally different, they might use independent methods; as such, I apply DNN for the time series classification to momentary forecasting. Second, the proposed architecture is constructed by considering sparsity, because designing DNN architectures robust to any impacts is a type of overfitting prevention for deep neural networks. Finally, this study proposes indices for quantitatively evaluating the DNN architectures that represent the realized forecasting performance of various deep neural networks. Using these indices, I demonstrate that optimal architectures may need to have model sparsity in the DNN (i.e., sparsity independent of the input data). The importance of this issue has been demonstrated experimentally. As a result, the architectures achieved target performances of 88%, 91%, and 79% accuracy and had stability for Panamax, Supramax, and Capesize vessels, respectively from February 2016 to September 2021 (i.e., five years and eight months). It is difficult to identify a correlation between model performance and volatility. Furthermore, before and after the COVID-19 shock, the performance of the proposed models compared to the optimal one exceeds that of other four recent models, namely "Facebook Prophet," "DARTS," "SKTIME," and "AutoTS".

show abstract

Stochastic numerical analysis for Brownian motion on SO(3)

Cited by 9 publications

References 19 publications

A simple approach to numerical methods for stochastic differential equations in Lie groups

A simple approach to numerical methods for stochastic differential equations in Lie groups

An engineer's guide to particle filtering on matrix Lie groups

Deep Neural Network Architectures for Momentary Forecasting in Dry Bulk Markets: Robustness to the Impact of COVID-19

Contact Info

Product

Resources

About