scholarly journals Modeling Spiky Functional Data With Derivatives of Smooth Functions in Function-on-Function Regression

2023 ◽  
Author(s):  
Ruiyan Luo ◽  
Xin Qi
Keyword(s):  
Functional Data ◽  
Smooth Functions ◽  
Derivatives Of

scholarly journals Application of functional data analysis to classify communities according to the pattern of malaria incidence, to guide targeted control strategies.

2020 ◽  
Author(s):  
Sokhna DIENG ◽  
Pierre Michel ◽  
Abdoulaye Guindo ◽  
Kankoe Sallah ◽  
El-hadj Ba ◽  
...  
Keyword(s):  
Time Series ◽  
Data Analysis ◽  
Functional Data Analysis ◽  
Functional Data ◽  
Temporal Pattern ◽  
Malaria Incidence ◽  
Smooth Functions ◽  
High Incidence ◽  
Incidence Pattern ◽  
Low Incidence

Abstract Background Effective targeting of malaria control in low transmission areas requires identification of transmission foci or hotspots. We investigated the use of functional data analysis to identify and describe spatio-temporal pattern of malaria incidence in an area with seasonal transmission in west-central Senegal. Method Malaria surveillance was maintained over 5 years from 2008 to 2012 at health facilities serving a population of 500,000 in 575 villages in two health districts in Senegal. Smooth functions were fitted from the time series of malaria incidence for each village, using cubic B-spline basis functions. The resulting smooth functions for each village were classified using hierarchical clustering (Ward’s method), using several different dissimilarity measures. The optimal number of clusters was then determined based on four cluster validity indices, to determine the main types of distinct temporal pattern of malaria incidence. Epidemiological indicators characterizing the resulting malaria incidence pattern in terms of the timing of seasonal outbreaks, were calculated based on the slope (velocity) and rate of change of the slope (acceleration) of the incidence over time. Results Three distinct patterns of malaria incidence were identified. A pattern characterized by high incidence, in 12/575 (2%) villages, with average incidence of 114 cases/1000 person-years over the 5 year study period; a pattern with intermediate incidence in 97 villages (17%), with average incidence of 13 cases/1000 person-years; and a pattern with low incidence in 466 (81%) villages, with average incidence 2.6 cases/1000 person-years. Epidemiological indicators characterizing the fluctuations in malaria incidence showed that seasonal outbreaks started later, and ended earlier, in the low incidence pattern. Conclusion Functional data analysis can be used to classify communities based on time series of malaria incidence, and to identify high incidence communities. Indicators can be derived from the fitted functions which characterize the timing of outbreaks. These tools may help to better target control measures.

scholarly journals Automatic choice of parameters at approximation of functions by rational splines

1987 ◽  
pp. 52
Author(s):  
A.D. Malysheva
Keyword(s):  
Sufficient Conditions ◽  
Order Of Approximation ◽  
Approximation Of Functions ◽  
Smooth Functions ◽  
Higher Derivatives ◽  
Necessary And Sufficient ◽  
Automatic Choice ◽  
Derivatives Of ◽  
Best Parameters ◽  
Approximation Of A Function

We obtain necessary and sufficient conditions put on the parameters of rational splines that provide given order of approximation of smooth functions. We point out the formulas of asymptotically the best parameters of rational splines that, while providing the best order of approximation of a function by rational splines, do not contain information about the values of higher derivatives of a function.

scholarly journals Outlier Detection for Pandemic-Related Data Using Compositional Functional Data Analysis

2021 ◽  
pp. 251-266
Author(s):  
Christopher Rieser ◽  
Peter Filzmoser
Keyword(s):  
Outlier Detection ◽  
Functional Data Analysis ◽  
Functional Data ◽  
Subject Area ◽  
Smooth Functions ◽  
Related Data ◽  
Relative Information ◽  
The Subject ◽  
Multivariate Functional Data ◽  
Events Data

AbstractWith accurate data, governments can make the most informed decisions to keep people safer through pandemics such as the COVID-19 coronavirus. In such events, data reliability is crucial and therefore outlier detection is an important and even unavoidable issue. Outliers are often considered as the most interesting observations, because the fact that they differ from the data majority may lead to relevant findings in the subject area. Outlier detection has also been addressed in the context of multivariate functional data, thus smooth functions of several characteristics, often derived from measurements at different time points (Hubert et al. in Stat Methods Appl 24(2):177–202, 2015b). Here the underlying data are regarded as compositions, with the compositional parts forming the multivariate information, and thus only relative information in terms of log-ratios between these parts is considered as relevant for the analysis. The multivariate functional data thus have to be derived as smooth functions by utilising this relative information. Subsequently, already established multivariate functional outlier detection procedures can be used, but for interpretation purposes, the functional data need to be presented in an appropriate space. The methodology is illustrated with publicly available data around the COVID-19 pandemic to find countries displaying outlying trends.

scholarly journals A verified method for solving piecewise smooth initial value problems

2013 ◽  
Vol 23 (4) ◽  
pp. 731-747 ◽  
Author(s):  
Ekaterina Auer ◽  
Stefan Kiel ◽  
Andreas Rauh
Keyword(s):  
Initial Value Problems ◽  
System Model ◽  
Future Research ◽  
Initial Value ◽  
Smooth Functions ◽  
Solution Processes ◽  
Right Hand ◽  
Future Research Directions ◽  
The Right ◽  
Derivatives Of

Abstract In many applications, there is a need to choose mathematical models that depend on non-smooth functions. The task of simulation becomes especially difficult if such functions appear on the right-hand side of an initial value problem. Moreover, solution processes from usual numerics are sensitive to roundoff errors so that verified analysis might be more useful if a guarantee of correctness is required or if the system model is influenced by uncertainty. In this paper, we provide a short overview of possibilities to formulate non-smooth problems and point out connections between the traditional non-smooth theory and interval analysis. Moreover, we summarize already existing verified methods for solving initial value problems with non-smooth (in fact, even not absolutely continuous) right-hand sides and propose a way of handling a certain practically relevant subclass of such systems. We implement the approach for the solver VALENCIA-IVP by introducing into it a specialized template for enclosing the first-order derivatives of non-smooth functions. We demonstrate the applicability of our technique using a mechanical system model with friction and hysteresis. We conclude the paper by giving a perspective on future research directions in this area.

scholarly journals On derivatives of smooth functions represented in multiwavelet bases

2019 ◽  
Vol 4 ◽  
pp. 100033 ◽  
Author(s):  
Joel Anderson ◽  
Robert J. Harrison ◽  
Hideo Sekino ◽  
Bryan Sundahl ◽  
Gregory Beylkin ◽  
...  
Keyword(s):  
Smooth Functions ◽  
Derivatives Of

scholarly journals Partial Derivative Estimation for Underlying Functional-Valued Process in a Unified Framework

2020 ◽  
Vol 2020 ◽  
pp. 1-17
Author(s):  
Yunbei Ma ◽  
Fanyin Zhou ◽  
Xuan Luo
Keyword(s):  
Data Analysis ◽  
Functional Data Analysis ◽  
Functional Data ◽  
Rates Of Convergence ◽  
Sample Function ◽  
Unified Framework ◽  
Partial Derivatives ◽  
Novel Approach ◽  
Consistency Results ◽  
Derivatives Of

We consider functional data analysis when the observations at each location are functional rather than scalar. When the dynamic of underlying functional-valued process at each location is of interest, it is desirable to recover partial derivatives of a sample function, especially from sparse and noise-contaminated measures. We propose a novel approach based on estimating derivatives of eigenfunctions of marginal kernels to obtain a representation for functional-valued process and its partial derivatives in a unified framework in which the number of locations and number of observations at each location for each individual can be any rate relative to the sample size. We derive almost sure rates of convergence for the procedures and further establish consistency results for recovered partial derivatives.

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