Extrapolation Problem Research Articles

Machine Learning and the End of Atmospheric Corrections: A Comparison between High-Resolution Sea Surface Salinity in Coastal Areas from Top and Bottom of Atmosphere Sentinel-2 Imagery

This paper introduces a discussion about the need for atmospheric corrections by comparing data-driven sea surface salinity (SSS) derived from Top- and Bottom-of-Atmosphere imagery. Atmospheric corrections are used to remove the effect of the atmosphere in reflectances acquired by satellite sensors. The Sentinel-2 Level-2A product provides atmospherically corrected Bottom-of-Atmosphere (BOA) imagery, derived from Level-1C Top-of-Atmosphere (TOA) tiles using the Sen2Cor processor. SSS at high resolution in coastal areas (100m) is derived from multispectral signatures using artificial neural networks. These obtain relationships between satellite band information and in situ SSS data. Four scenarios with different input variables are tested for both TOA and BOA imagery, for interpolation (previous information on all platforms is available in the training dataset) and extrapolation (certain platforms are isolated and the network does not have any previous information on these) problems. Results show that TOA always outperforms BOA in terms of higher coefficient of determination (R2), lower mean absolute error (MAE) and lower most common error (μe). The best TOA results are R2=0.99, MAE=0.4PSU and μe=0.2PSU. Moreover, the evaluation of the neural network in all the pixels of Sentinel-2 tiles shows that BOA results are accurate only far away from the coast, while TOA data provides useful information on nearshore mixing patterns, estuarine processes and is able to estimate freshwater salinity values. This suggests that land adjacency corrections could be a relevant source of error. Sun glint corrections appear to be another source of error. TOA imagery is more accurate than BOA imagery when using machine learning algorithms and big data, as there is a clear loss of information in the atmospheric correction process that affects the multispectral–in situ relationships. Finally, the time and computational resources gained by avoiding atmospheric corrections can make the use of TOA imagery interesting in future studies, such as the estimation of chlorophyll or coloured dissolved organic matter.

Nonparametric Bayesian approach to extrapolation problems in configuration interaction methods

We propose a non-parametric extrapolation method based on constrained Gaussian processes for configuration interaction methods. Our method has many advantages: (i) applicability to small data sets such as results of {\it ab initio} methods, (ii) flexibility to incorporate constraints, which are guided by physics, into the extrapolation model, (iii) providing predictions with quantified extrapolation uncertainty, etc. In the present study, we show an application to the extrapolation needed in full configuration interaction method as an example.

Continental-scale tree-ring-based projection of Douglas-fir growth: Testing the limits of space-for-time substitution.

A central challenge in global change research is the projection of the future behavior of a system based upon past observations. Tree-ring data have been used increasingly over the last decade to project tree growth and forest ecosystem vulnerability under future climate conditions. But how can the response of tree growth to past climate variation predict the future, when the future does not look like the past? Space-for-time substitution (SFTS) is one way to overcome the problem of extrapolation: the response at a given location in a warmer future is assumed to follow the response at a warmer location today. Here we evaluated an SFTS approach to projecting future growth of Douglas-fir (Pseudotsuga menziesii), a species that occupies an exceptionally large environmental space in North America. We fit a hierarchical mixed-effects model to capture ring-width variability in response to spatial and temporal variation in climate. We found opposing gradients for productivity and climate sensitivity with highest growth rates and weakest response to interannual climate variation in the mesic coastal part of Douglas-fir's range; narrower rings and stronger climate sensitivity occurred across the semi-arid interior. Ring-width response to spatial versus temporal temperature variation was opposite in sign, suggesting that spatial variation in productivity, caused by local adaptation and other slow processes, cannot be used to anticipate changes in productivity caused by rapid climate change. We thus substituted only climate sensitivities when projecting future tree growth. Growth declines were projected across much of Douglas-fir's distribution, with largest relative decreases in the semiarid U.S. Interior West and smallest in the mesic Pacific Northwest. We further highlight the strengths of mixed-effects modeling for reviving a conceptual cornerstone of dendroecology, Cook's 1987 aggregate growth model, and the great potential to use tree-ring networks and results as a calibration target for next-generation vegetation models.

On an extrapolation problem for characteristic functions

Let f be the characteristic function of a probability measure μf on ℝn, and let σ > 0. We study the following extrapolation problem: under what conditions on the neighborhood of infinity Vσ = {x ∈ ℝn : |xk| > σ, k = 1, …n} in ℝndoes there exist a characteristic function g on ℝn, such that g = f on Vσ but g ≢ f? Let μf have a nonzero absolutely continuous part with continuous density 𝜑. In this paper, we give certain sufficient conditions on 𝜑 and Vσ under which the latter question has an affirmative answer. We also address the optimality of these conditions. Our results indicate that not only does the size of both Vσ and the support supp 𝜑 matter, but also certain arithmetic properties of supp 𝜑.

Mixed-type stochastic differential equations driven by standard and fractional Brownian motions with Hurst indices greater than 1/3

In this paper we consider mixed-type stochastic differential equations driven by standard and fractional Brownian motions with Hurst indices greater than 1/3. There are proved theorems on the existence, uniqueness, and continuous dependence of solutions on the initial data. We provide an analog of the Ito formula to change variables. Asymptotic expansions of functionals on the solutions of mixed-type stochastic differential equations for small times are obtained. We receive analogs of the Kolmogorov equations for mathematical expectations and probability densities in the commutative case. Finally, we consider an application of mixed-type stochastic differential equations to solving the problem of macroeconomic variables extrapolation in credit risks models.

The Measurement Problem of Consciousness

This paper addresses what we consider to be the most pressing challenge for the emerging science of consciousness: the measurement problem of consciousness. That is, by what methods can we determine the presence of and properties of consciousness? Most methods are currently developed through evaluation of the presence of consciousness in humans and here we argue that there are particular problems in application of these methods to nonhuman cases—what we call the indicator validity problem and the extrapolation problem. The first is a problem with the application of indicators developed using the differences between conscious and unconscious processing in humans to the identification of other conscious vs. nonconscious organisms or systems. The second is a problem in extrapolating any indicators developed in humans or other organisms to artificial systems. However, while pressing ethical concerns add urgency to the attribution of consciousness and its attendant moral status to nonhuman animals and intelligent machines, we cannot wait for certainty and we advocate the use of a precautionary principle to avoid causing unintentional harm. We also intend that the considerations and limitations discussed in this paper can be used to further analyze and refine the methods of consciousness science with the hope that one day we may be able to solve the measurement problem of consciousness.

A Theoretical and Experimental Investigation on the Measurement of the Electromagnetic Field Level Radiated by 5G Base Stations

This paper presents some theoretical considerations and experimental results regarding the problem of maximum power extrapolation for the assessment of the exposure to electromagnetic fields radiated by 5G base stations. In particular the results of an extensive experimental campaign using an extrapolation procedure recently proposed for 5G signal is discussed and experimentally checked on a SU-MIMO signal. The results confirm the effectiveness of the extrapolation technique. Starting from an analysis (that represents a further novel contribution of this paper) on the impact of Spatial Division Multiple Access techniques used in 5G on the measurement of EMF level, some indications of possible extension of the technique to the highly complex MU-MIMO case are also given.

Application of Recent Developments in Deep Learning to ANN-based Automatic Berthing Systems

Previous studies on Artificial Neural Network (ANN)-based automatic berthing showed considerable increases in performance by training ANNs with a set of berthing datasets. However, the berthing performance deteriorated when an extrapolated initial position was given. To overcome the extrapolation problem and improve the training performance, recent developments in Deep Learning (DL) are adopted in this paper. Recent activation functions, weight initialization methods, input data-scaling methods, a higher number of hidden layers, and Batch Normalization (BN) are considered, and their effectiveness has been analyzed based on loss functions, berthing performance histories, and berthing trajectories. Finally, it is shown that the use of recent activation and weight initialization method results in faster training convergence and a higher number of hidden layers. This leads to a better berthing performance over the training dataset. It is found that application of the BN can overcome the extrapolated initial position problem.

Estimation method of Hurst exponent of fractional Brownian motion

The article contains fractional Brownian motion research, statistical estimations of Hurst exponent and their properties. This random process is used in model development, trend forecasting, in particular as special case of long-memory processes. The first recorded model with Hurst Exponent appeared in Harold Hursts research, published in 1951, that was dedicated to the analysis of the flow of Nil river. After that improved model of fractional Brownian motion was widely used in different financial market researches. Due to high relevance of such random processes, the problems of extrapolation of fractional Brownian motion and point estimation have become very important. This article is dedicated to the new approach of point estimation of Hurst Exponent.

Resurgent extrapolation: rebuilding a function from asymptotic data. Painlevé I

Extrapolation is a generic problem in physics and mathematics: how to use asymptotic data in one parametric regime to learn about the behavior of a function in another parametric regime. For example: extending weak coupling expansions to strong coupling, or high temperature expansions to low temperature, or vice versa. Such extrapolations are particularly interesting in systems possessing dualities. Here we study numerical procedures for performing such an extrapolation, combining ideas from resurgent asymptotics with well-known techniques of Borel summation, Padé approximants and conformal mapping. We illustrate the method with the concrete example of the Painlevé I equation, which has applications in many branches of physics and mathematics. Starting solely with a finite number of coefficients from asymptotic data at infinity on the positive real line, we obtain a high precision extrapolation of the function throughout the complex plane, even across the phase transition into the pole region. The precision far exceeds that of state-of-the-art numerical integration methods along the real axis. The methods used are both elementary and general, not relying on Painlevé integrability properties, and so are applicable to a wide class of extrapolation problems.

Multi-Channel Weather Radar Echo Extrapolation with Convolutional Recurrent Neural Networks

This article presents an investigation into the problem of 3D radar echo extrapolation in precipitation nowcasting, using recent AI advances, together with a viewpoint from Computer Vision. While Deep Learning methods, especially convolutional recurrent neural networks, have been developed to perform extrapolation, most works use 2D radar images rather than 3D images. In addition, the very few ones which try 3D data do not show a clear picture of results. Through this study, we found a potential problem in the convolution-based prediction of 3D data, which is similar to the cross-talk effect in multi-channel radar processing but has not been documented well in the literature, and discovered the root cause. The problem was that, when we generated different channels using one receptive field, some information in a channel, especially observation errors, might affect other channels unexpectedly. We found that, when using the early-stopping technique to avoid over-fitting, the receptive field did not learn enough to cancel unnecessary information. If we increased the number of training iterations, this effect could be reduced but that might worsen the over-fitting situation. We therefore proposed a new output generation block which generates each channel separately and showed the improvement. Moreover, we also found that common image augmentation techniques in Computer Vision can be helpful for radar echo extrapolation, improving testing mean squared error of employed models at least 20% in our experiments.

Robust Linear Interpolation and Extrapolation of Stationary Time Series in Lp

To deal with uncertainty of the spectral distribution, we consider minimax interpolation and extrapolation problems in Lp for stationary processes. The interpolation and extrapolation problems can be regarded as a linear approximation problem on the unit disk in the complex plane. Although the robust one‐step‐ahead predictor and interpolator has already been considered separately in the previous literature, we give two conditions for the uncertainty class to find the minimax interpolator and extrapolator in the general framework from both the point of view of the observation set and the point of view of evaluation on the interpolation and extrapolation error under the Lp‐norm. We show that there exists a minimax interpolator and extrapolator for the class of spectral densities ε‐contaminated by unknown spectral densities under our conditions. When the uncertainty class contains spectral distribution functions which are not absolutely continuous to the Lebesgue measure, we show that there exists an approximate interpolator and extrapolator in Lp such that its maximal interpolation and extrapolation error is arbitrarily close to the minimax error when the spectral distributions have densities. Our results are applicable to the stationary harmonizable stable processes.

The myth and fallacy of simple extrapolation in medicine

Simple extrapolation is the orthodox approach to extrapolating from clinical trials in evidence-based medicine: extrapolate the relative effect size (e.g. the relative risk) from the trial unless there is a compelling reason not to do so. I argue that this method relies on a myth and a fallacy. The myth of simple extrapolation is the idea that the relative risk is a ‘golden ratio’ that is usually transportable due to some special mathematical or theoretical property. The fallacy of simple extrapolation is an unjustified argument from ignorance: we conclude that the relative effect size is transportable in the absence of evidence to the contrary. In short, simple extrapolation is a deeply problematic solution to the problem of extrapolation.

Time domain load extrapolation method for CNC machine tools based on GRA-POT model

The load extrapolation is to obtain load data that may occur during the whole life cycle of product based on the limited measured load data, especially extreme load. Aiming at the two problems of traditional time domain extrapolation: without considering the interval time between extreme adjacent values and the high sensitivity of the peak over threshold (POT) model to the extreme threshold, a time domain load extrapolation method for computerized numerical control (CNC) machine tools based on Gray relational analysis-peak over threshold model (GRA-POT) is proposed. Firstly, for the first problem, an improved Markov chain Monte Carlo (MCMC) time domain load reconstruction method is proposed and analysis of reconstructed time domain load; secondly, according to the mean excess function (MEF) method, the extreme threshold range of POT model is determined and divided into multiple sets. The generalized Pareto distribution (GPD) fitting is performed for multiple sets of candidate load extreme samples. Finally, the Gray relational degrees (GRD) between each group of GPD and the median rank distribution are calculated by the Gray relational analysis (GRA), and the extreme threshold corresponding to the maximum GRD is selected as the optimal threshold of the POT model. The dynamic cutting force signal of CNC machine tool under constant speed cutting condition was reconstructed and extrapolated using the proposed method, and the dynamic cutting force spectrum of CNC machine tool was compiled. Case analysis results show that the POT extrapolation model with high fitting precision can be obtained using the proposed method. Furthermore, the precision of load spectrum of CNC machine tool is improved.

Estimation of Changes of Forest Structural Attributes at Three Different Spatial Aggregation Levels in Northern California using Multitemporal LiDAR

Accurate estimates of growth and structural changes are key for forest management tasks such as determination of optimal rotation times, optimal rotation times, site indices and for identifying areas experiencing difficulties to regenerate. Estimation of structural changes, especially for biomass, is also key to quantify greenhouse gas (GHG) emissions/sequestration. We compared two different modeling strategies to estimate changes in V, BA and B, at three different spatial aggregation levels using auxiliary information from two light detection and ranging (LiDAR) flights. The study area is Blacks Mountains Experimental Forest, a ponderosa pine dominated forest in Northern California for which two LiDAR acquisitions separated by six years were available. Analyzed strategies consisted of (1) directly modeling the observed changes as a function of the LiDAR auxiliary information ( δ -modeling method) and (2) modeling V, BA and B at two different points in time, including a term to account for the temporal correlation, and then computing the changes as the difference between the predicted values of V, BA and B for time two and time one. We analyzed predictions and measures of uncertainty at three different level of aggregation (i.e., pixels, stands or compartments and the entire study area). Results showed that changes were very weakly correlated with the LiDAR auxiliary information. Both modeling alternatives provided similar results with a better performance of the δ -modeling for the entire study area; however, this method also showed some inconsistencies and seemed to be very prone to extrapolation problems. The y -modeling method, which seems to be less prone to extrapolation problems, allows obtaining more outputs that are flexible and can outperform the δ -modeling method at the stand level. The weak correlation between changes in structural attributes and LiDAR auxiliary information indicates that pixel-level maps have very large uncertainties and estimation of change clearly requires some degree of spatial aggregation; additionally, in similar environments, it might be necessary to increase the time lapse between LiDAR acquisitions to obtain reliable estimates of change.

Extrapolation Problem for Multidimensional Stationary Sequences with Missing Observations

This paper focuses on the problem of the mean square optimal estimation of linear functionals which dependon the unknown values of a multidimensional stationary stochastic sequence. Estimates are based on observations of these quence with an additive stationary noise sequence. The aim of the paper is to develop methods of finding the optimal estimates of the functionals in the case of missing observations. The problem is investigated in the case of spectral certainty where the spectral densities of the sequences are exactly known. Formulas for calculating the mean-square errors and the spectral characteristics of the optimal linear estimates of functionals are derived under the condition of spectral certainty.The minimax (robust) method of estimation is applied in the case of spectral uncertainty, where spectral densities of the sequences are not known exactly while sets of admissible spectral densities are given. Formulas that determine the least favorable spectral densities and the minimax spectral characteristics of the optimal estimates of functionals are proposed for some special sets of admissible densities.

Matrices whose inverses are tridiagonal, band or block-tridiagonal and their relationship with the covariance matrices of a random Markov process

The article discusses the matrices of the form A1n, Amn, AmN, whose inverses are: tridiagonal matrix A-1n (n - dimension of the A-mn matrix), banded matrix A-mn (m is the half-width band of the matrix) or block-tridiagonal matrix A-m N (N = n x m - full dimension of the block matrix; m - the dimension of the blocks) and their relationships with the covariance matrices of measurements with ordinary (simple) Markov Random Processes (MRP), multiconnected MRP and vector MRP, respectively. Such covariance matrices frequently occur in the problems of optimal filtering, extrapolation and interpolation of MRP and Markov Random Fields (MRF). It is shown, that the structures of the matrices A1n, Amn, AmN have the same form, but the matrix elements in the first case are scalar quantities; in the second case matrix elements represent a product of vectors of dimension m; and in the third case, the off-diagonal elements are the product of matrices and vectors of dimension m. The properties of such matrices were investigated and a simple formulas of their inverses were found. Also computational efficiency in the storage and the inverse of such matrices have been considered. To illustrate the acquired results, an example on the covariance matrix inversions of two-dimensional MRP is given.

To the problem of verification of hypotheses in hydrology

The article considers the problem of assessing the truth of scientific knowledge and outlines three approaches to determining the truth of scientific hypotheses: practicality, consistency, and coherence with the provisions of a higher order. Each approach or concept has its own evaluation criteria. These three approaches or concepts are revealed on the example of verification of three hydrological problems: the use of statistical distribution functions for estimating extreme values, the parameterization of the Chézy coefficients and the efficiency of using arithmetic averages for characterizing the chemical composition of rivers. The study shows that the transfer of technologies well-tested on interpolation tasks to solving extrapolation problems is incorrect. The variety of calculated ratios for estimating the Chézy coefficient resulted from the complexity and multiparameter nature of hydraulic resistance formation processes in channel flows, as well as the stochasticity of the morphometric parameters of natural watercourses. The article also proves the inconsistency of arithmetic average evaluations for describing the processes of hydrochemical regime of water bodies. Parametric estimates of hydrochemical indicators of water quality are characterized by significant instability of statistical characteristics. The presence of a random parameter, that is the water discharge in the denominator, makes the distribution of hydrochemical parameters closer to the Cauchy distribution for which not only the second, but also the first statistical moment do not exist.

Numerical Solution for the Extrapolation Problem of Analytic Functions

In this work, a numerical solution for the extrapolation problem of a discrete set of n values of an unknown analytic function is developed. The proposed method is based on a novel numerical scheme for the rapid calculation of higher order derivatives, exhibiting high accuracy, with error magnitude of O(10−100) or less. A variety of integrated radial basis functions are utilized for the solution, as well as variable precision arithmetic for the calculations. Multiple alterations in the function's direction, with no curvature or periodicity information specified, are efficiently foreseen. Interestingly, the proposed procedure can be extended in multiple dimensions. The attained extrapolation spans are greater than two times the given domain length. The significance of the approximation errors is comprehensively analyzed and reported, for 5832 test cases.

Optimal Estimates in Problems of Extrapolation, Filtration, and Interpolation of Functionals of Random Processes with Values in a Hilbert Space

General formulas are obtained for efficient calculation of optimal estimates for functionals of random processes with values in a Hilbert space. In a special case where the process under study is a solution of a nonlinear evolutionary differential equation with a small nonlinearity, the optimal estimates are expanded in powers of a small parameter and the expansion coefficients are given in the form of algorithms and are calculated explicitly in terms of known quantities of the differential equation.