Stochastic Differential Equations With Applications to Biomedical Signal Processing
Bibliographic record
Abstract
Dynamic behavior of biological systems is often governed by complex physiological processes that are inherently stochastic.Therefore most physiological signals belong to the group of stochastic signals for which it is impossible to predict an exact future value even if we know its entire past history.That is there is always an aspect of a signal that is inherently random i.e. unknown.Commonly used biomedical signal processing techniques often assume that observed parameters and variables are deterministic in nature and model randomness through so called observation errors which do not influence the stochastic nature of underlying processes (e.g., metabolism, molecular kinetics, etc.).An alternative approach would be based on the assumption that the governing mechanisms are subject to instantaneous changes on a certain time scale.As an example fluctuations in the respiratory rate and/or concentration of oxygen (or equivalently partial pressures) in various compartments is strongly affected by a metabolic rate, which is inherently stochastic and therefore is not a smooth process.As a consequence one of the mathematical techniques that is quickly assuming an important role in modeling of biological signals is stochastic differential equations (SDE) modeling.These models are natural extensions of classic deterministic models and corresponding ordinary differential equations.In this chapter we will present computational framework necessary for successful application of SDE models to actual biomedical signals.To accomplish this task we will first start with mathematical theory behind SDE models.These models are used extensively in various fields such as financial engineering, population dynamics, hydrology, etc.Unfortunately, most of the literature about stochastic differential equations seems to place a large emphasis on rigor and completeness using strict mathematical formalism that may look intimidating to non-experts.In this chapter we will attempt to present answer to the following questions: in what situations the stochastic differential models may be applicable, what are the essential characteristics of these models, and what are some possible tools that can be used in solving them.We will first introduce mathematical theory necessary for understanding SDEs.Next, we will discuss both univariate and multivariate SDEs and discuss the corresponding computational issues.We will start with introducing the concept of stochastic integrals and illustrate the solution process using one univariate and one multivariate example.To address the computational complexity in realistic biomedical signal models we will further discuss the aforementioned biochemical transport model and derive the stochastic integral solution www.intechopen.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.006 | 0.002 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".