Differential equations in data analysis

Dattner, Itai

doi:10.1002/wics.1534

Cited by 13 publications

(10 citation statements)

References 227 publications

(243 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It is clear that ɛ ( k ), u 1 ( k ) and u 2 ( k ) in Equation (20), and ζ ( k ), v 1 ( k ) and v 2 ( k ) in Equation (21), explicitly containing the measurement error e ( k ), makes the resultant estimates biased and inconsistent and also indicates the main source of errors causing bias. The pseudo linear regression Equation (20) and Equation (21) are, in fact, error-in-variables models and thus can be solved by using advanced methods, such as the two-stage least squares (Dattner, 2020) and prediction error minimization-based non-linear least squares (Wei and Xie, 2021).…”

Section: Simulationsmentioning

confidence: 99%

Parameter estimation for grey system models: gradient matching versus integral matching

Wei

Xie²,

Yang³

2022

View full text Add to dashboard Cite

PurposeThe cumulative sum (Cusum) operator, also referred to as accumulating generation operator, is the fundamental of grey system models and proves to be successful in various real-world applications. This paper aims to uncover the advantages of the Cusum operator from a parameter estimation perspective, i.e. comparing integral matching with classical gradient matching.Design/methodology/approachGrey system models are represented as a state space form to investigate the effect of measurement errors on estimation performance; subsequently, gradient matching and integral matching are respectively formulated to estimate parameters from noisy observations and, then, their quantitative relationships are established by using matrix computation tricks.FindingsExtensive simulations, which are conducted on both linear and non-linear models under different sample size and noise level combinations, show that integral matching is superior to gradient matching, and, also the former is less sensitive to measurement error.Originality/valueThis paper explains why the Cusum operator is widely utilized in grey system models, thereby further solidifying the mathematical fundamentals of grey system models.

show abstract

Section: Simulationsmentioning

confidence: 99%

Parameter estimation for grey system models: gradient matching versus integral matching

Wei

Xie²,

Yang³

2022

View full text Add to dashboard Cite

show abstract

“…20 On the other hand, accounting for unnecessary age groups (over stratifying the population) inflates the number of parameters that need to be inferred and can lead to both identification problems and a heavy computational burden. [21][22][23] In addition, complex models are often too difficult to study using analytic techniques and are instead examined using numerical methods. Having a data-driven criterion for clustering incidence data into age groups is instrumental in constructing accurate, yet parsimonious age group models.…”

Section: Introductionmentioning

confidence: 99%

Data-driven clustering of infectious disease incidence into age groups

Yaari

Huppert

Dattner

2022

Stat Methods Med Res

Self Cite

View full text Add to dashboard Cite

Understanding the patterns of infectious diseases spread in the population is an important element of mitigation and vaccination programs. A major and common characteristic of most infectious diseases is age-related heterogeneity in the transmission, which potentially can affect the dynamics of an epidemic as manifested by the pattern of disease incidence in different age groups. Currently there are no statistical criteria of how to partition the disease incidence data into clusters. We develop the first data-driven methodology for deciding on the best partition of incidence data into age-groups, in a well defined statistical sense. The method employs a top-down hierarchical partitioning algorithm, with a stopping criteria based on multiple hypotheses significance testing controlling the family wise error rate. The type one error and statistical power of the method are tested using simulations. The method is then applied to Covid-19 incidence data in Israel, in order to extract the significant age-group clusters in each wave of the epidemic.

show abstract

Section: Introductionmentioning

confidence: 99%

“…Despite its compute advantages, traditional two-stage approaches suffer from limited accuracy for real systems and, at best, are used to provide an initial guess for parameter values. 40 This is because, especially when data is noisy or contains outliers, data-driven models used to interpolate data tend to yield low-quality derivative estimates, which reduces the quality of parameter estimates obtained when solving the NLP. 41 Furthermore, it is often the desire to experimentally explore a system using multiple experimental runs with varying conditions, yet none of the derivative estimation techniques currently proposed have a straightforward way to account for multiple batches of data with a single data-driven model.…”

Section: Introductionmentioning

confidence: 99%

Two-Stage Approach to Parameter Estimation of Differential Equations Using Neural ODEs

Bradley

Boukouvala

2021

Ind. Eng. Chem. Res.

View full text Add to dashboard Cite

Modeling physiochemical relationships using dynamic data is a common task in fields throughout science and engineering. A common step in developing generalizable, mechanistic models is to fit unmeasured parameters to measured data. However, fitting differential equation-based models can be computation-intensive and uncertain due to the presence of nonlinearity, noise, and sparsity in the data, which in turn causes convergence to local minima and divergence issues. This work proposes a merger of machine learning (ML) and mechanistic approaches by employing ML models as a means to fit nonlinear mechanistic ordinary differential equations (ODEs). Using a two-stage indirect approach, neural ODEs are used to estimate state derivatives, which are then used to estimate the parameters of a more interpretable mechanistic ODE model. In addition to its computational efficiency, the proposed method demonstrates the ability of neural ODEs to better estimate derivative information than interpolating methods based on algebraic data-driven models. Most notably, the proposed method is shown to yield accurate predictions even when little information is known about the parameters of the ODEs. The proposed parameter estimation approach is believed to be most advantageous when the ODE to be fit is strongly nonlinear with respect to its unknown parameters.

show abstract

Differential equations in data analysis

Cited by 13 publications

References 227 publications

Parameter estimation for grey system models: gradient matching versus integral matching

Parameter estimation for grey system models: gradient matching versus integral matching

Data-driven clustering of infectious disease incidence into age groups

Two-Stage Approach to Parameter Estimation of Differential Equations Using Neural ODEs

Contact Info

Product

Resources

About