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.
Prediction always differs from observation , and extrapolation remains an open challenge  or even a hopelessly ill-conditioned  problem, in a vast variety of scientific disciplines. It relies on numerical methods which attempt to predict the future, unknown values of a studied phenomenon, given a limited set of observations. Its importance is reflected in the Scopus database , by a search with the term extrapolation, yielding nearly 150 thousands of research items, in a wide range of scientific fields (Table S1). The visual presentation of the corresponding keywords (Figure S1) in a bibliometric map  reveals that extrapolation is closely associated with modeling, simulation, uncertainty, and interpolation, which is the reciprocal problem to simulate the unknown mechanism which produced the given data, aiming to predict utilizing the constructed model. Contradictorily to interpolation and despite the significance of extrapolation and forecasting methods, they exhibit a decreasing pattern in literature since the early 1980s (Figure S2), indicating the futility to predict for extended time frames. The given data is often called time-series and their extrapolation forecasting, which usually depend on machine learning algorithms such as support vector machines [6–8] and artificial neural networks [2, 9, 10], which construct nonlinear regression models, to predict beyond the known domain. Prediction algorithms are useful in a wide range of scientific disciplines, such as earth sciences [7, 9], finance [8, 11], computer science [12, 13], and engineering [14–16]. Several methods have been proposed for time-series forecasting  and competitions regarding accuracy have been conducted, utilizing statistical [11, 17] and machine learning procedures [2, 10, 15]; however, the prediction horizon regards only a small percentage (~20-30%) of the given data extension. Similarly, for less uncertain problems as the extrapolation of curves defined by polynomials (splines) [18–20], extrapolation can cause unpredictable results, and their extension should be short . Apparently, prediction procedures are essential for a vast variety of scientific fields, always based on numerical interpolation methods.
More specifically, in the case of analytic functions, where the given data follow an unknown rigor mathematical law, any extrapolation results are too weak to be interesting . Previous theoretical works focus on the interpolation problem, such as the ability of neural networks with one hidden layer to approximate analytic functions , proofs regarding the degree of rational approximation of an analytic function , and analysis of the rate of decrease of the best approximations of analytic functions . Accuracy and convergence techniques are demonstrated in , yet it is shown theoretically  that the approximation of analytic functions cannot converge exponentially. Approximation of analytic functions is investigated utilizing a variety of approximators [27–29], such as Hermite [30, 31] and Airy functions . Computational works regard the approximation of a function  as well as its derivatives [34, 35], investigating the interpolation errors among the given nodal points. Such errors are higher than the approximation errors (Runge phenomenon) especially at the boundaries , affecting dramatically the extrapolation outside the given domain. Moreover, the round-off errors of computing machines follow some stochastic law  and recent works deal with the high impact of infinitesimal errors in the given data, utilizing extended arithmetic precision [37, 38]. Accordingly, the constitution of an accurate numerical method to approximate an analytic function and its derivatives, which is vital for the extrapolation problem, remains a challenge.
The purpose of this study was to provide a generic numerical solution for the extrapolation problem. It was attained for extended extrapolation horizons of even greater than 200% the given domain length if the set of data are derived from an unknown analytic function and their precision is high. The rationale of the proposed method adheres to the following three stages: (a) interpolation of the set of values at specified points using integrated radial basis functions (IRBFs), (b) computation of high order derivatives at any point of the closed domain with high accuracy based on a novel numerical scheme, and (c) successive application of the Taylor series formula to extrapolate at points outside the given domain. The method is capable of interpolating within the given data with high precision, avoiding the Runge phenomenon at the boundaries , as well as of computing the higher order derivatives with remarkable accuracy, which are fundamental problems in numerous applications. The effects of the method’s parameters on the prediction errors were extensively investigated by their analysis of variance (ANOVA) and feature extraction utilizing the Random Forest  method, for 5832 test cases. Illustrative examples of highly nonlinear analytic functions in two and three dimensions demonstrate the prediction extents attained by the proposed method.
2. Numerical Solution
2.1. Extrapolation of One Infinitesimal dx
Let be an analytic function, which is unknown. It is given that the function takes values at specified points as in Figure 1(a), for a generic analytic function. The extrapolation problem, that is, to predict the values which take the function outside the given domain , can be transformed into the computation of ordinary derivatives of , , at point . In particular, applying the Taylor formula from the point to the next point , the corresponding value of is given byfor an infinitesimal step . is the value of at point , depending on the ordinary derivatives of at . These derivatives should be computed, as they are unknown.
2.2. Numerical Evaluation of the Derivatives
Around the endpoint , we used the Taylor formula for computing the value of the function at the first predicted point . In order to calculate the derivatives, we need to consider the function values within the end interval from the point with abscissa to . Accordingly, the Taylor expansion from the endpoint to any previous point (Figure 1(b)) may also be written in the formBy interchanging the positions of and , we may writeand by applying it to points dx (Figure 1(b)) and writing the resulting system of equations in matrix form, we obtainIt is important to note that all the derivatives are considered at the endpoint . Hence, extracting the derivatives as a column vector, we haveThus, only the first column of the matrix in (5) varies as it refers to the values of f within the edge interval. As the objective is to compute the n ordinary derivatives of f, (5) is separated in submatrices, in order to rearrange the equations and develop a formula to calculate the vector D, containing the values of the n derivatives of f at point . The solid lines indicate the submatrices B, C, H, D, and ; hence, (5) can be written in matrix partitions formwhereSince (6) is a system of equations, in order to calculate the vector D (containing the values of the derivatives of at point ), by (6), we deducewhich is a system of n equations. Thus, the matrix of the n ordinary derivatives can be directly computed bywhere n, and hence the number of the calculated derivatives, can be arbitrarily selected. Equation (8) offers a direct computation of the n ordinary derivatives D. Since C is still unknown, we can compute it, by interpolating f with IRBFs as defined subsequently.
2.3. Function Approximation with IRBFs
Radial Basis Functions (RBFs) networks are universal approximators while Integrated RBFs are capable of attaining precise approximation for a function and its derivatives [34, 35] as well as for the solution of partial differential equations [37, 38]. The essential formulation of RBFs, adopted in this work, is as follows.
Let be a variable taking values in a domain . The known values of the function are given at the positions as per Figure 1, while the interval length between two adjacent points , is equal to . The given values of the unknown function can be approximated with RBFs , centered at the N points. The RBFs approximation is defined bywhere are the unknown weights for the approximation with RBFs, represents the distance of a given point in the domain from any other point , and is the radial basis kernels such as Gaussian, Shifted Multiquadrics, and Sigmoid or their integrals, as in , , , , , . Applying (9) at the given points results in the approximated values of , ; hence we deduce thatIf we write (10) times and define as a matrix containing the values of , with i indicating the rows and j the columns of , a vector containing the unknown approximation weights, and a vector comprising the values of at the nodal points , we may writeHence we deriveThe vector b is known, and the matrix can be calculated from the distances of the known points and , so the weights of the RBFs approximation of the function are computed by numerically inverting matrix and multiplying it with vector b. After the computation of vectors , the radial basis functions can be recomputed for each intermediate point , in order to approximate the values of the unknown function at these points. Accordingly, each approximation vector which corresponds to the points , , when multiplied with the calculated weights of (12), results in the interpolated values of , and sowhere are the approximation vectors for each hj and the RBFs weights. The vectors are computed utilizing , , , , , , resulting in the vector C, which is the function’s f approximation at points hj (Figure 1(b)).
The definitions of the radial basis functions φ utilized in (12), (13) are specified in the following group of , , , , , : the Gaussian and the Shifted Logarithmic, as well as integrations of them, for one and two times. Equations , , , , , are written concerning , in order to be generic for any and hence for points xi and hj.
The Gaussian RBF is defined bywhere c is a constant influencing the shape of φ [40–42]. By integration of the Gaussian kernel, we obtainand if we integrate for a second time, thenwhere erf is the error function, , exhibiting a sigmoid scheme, which is commonly used in ANNs .
Respectively, the Shifted Logarithmic RBF is defined byand by integration of the kernel we obtainand if we integrate for a second time, then
2.4. Iterative Implementation of Taylor Series Expansion
Exploiting the computed derivatives with high accuracy at the end of the domain , the predicted value of the function can be computed at the next step , by the Taylor series expansion of (1). This procedure is repeated iteratively, for each next point utilizing each time the shifted vectorsThe N points are initially the positions where the function values are given. This procedure is iteratively applied using at each step the previous values of the function (eliminating each time the first point) and one new value of the function, calculated by the procedure ( and ). Keeping constant the length dx between a predicted point and its subsequent, the matrix remains the same, as in the initial interpolation domain. The same stands for the inverted matrix as well as the matrices , and despite the sequential shifts outside the original domain, the relative distances r of , , , , , remain the same. The same stands for the matrices H and H-1. Hence, the interpolation of the function within the end increment dx each time is accomplished by matrix multiplication and not inversion, which is performed only one time for and one time for H. Utilizing this approach, during the iterations of the extrapolation procedure, the computational time is decreased dramatically, as these matrices are computed only once.
2.5. The Extrapolation Algorithm
3.1. Parameters Affecting the Method and Preparation of Dataset to Test Performance
The efficiency of the procedure was verified through numerical examples for a variety of highly nonlinear functions and their extrapolation spans. The extrapolation span is based on the accuracy of the interpolation method ((9), (10), (12), (13), , , , , , ) as well as the numerical differentiation ((1), (5), (6), , , , , , , (8)). As demonstrated in Figure 1(b), the intervals for the differentiation hj are selected near the node of the domain discretization, in order to achieve the highest possible accuracy, because this region is within the selected dx, and simultaneously, the interpolation error is minimized near this node, due to the Runge phenomenon . Hence we selectwhere l indicates the limit distance near the node . However, for extremely low values of h, the matrix H contains elements near zero, and its inversion grows into unstable one (high condition number and inversion errors).
The features which have an effect on the calculations, as well as their values (in parentheses) utilized in the parametric investigation, are the number of Taylor terms utilized, indicated as number of derivatives, (25, 50, 75); the number of computer digits used for the arbitrary precision calculations (500, 1000, 2000); the span of the given domain L (1/2, 1); the number of the divisions N (50, 100, 200); the limit l (5, 10, 20); the number of IRBFs integrations (0, 1, 2); the kernel of RBFs networks (Gaussian = 1, Shifted Logarithmic = 2); their shape parameter c (1/5, 1, 5); and the unknown function which is studied (, ). Accordingly, the resulting database consists of 5832 records of test cases. The output of each parametric execution of the proposed procedure was the condition number of the H and matrices, their inversion errors, and the error of the first and second derivatives according to the proposed procedure. Moreover, the error of the first step of the extrapolation was computed (ε), in order to investigate its dependence on the problem statement parameters (N, L, dx, f) as well as the method’s attributes (#derivatives, digits, kernel, c, #integrations).
3.2. Effects of Parameters on Extrapolation Accuracy and Computational Time
The was utilized as a measure for the extrapolation error, as the values of exhibit a variation within the domain to the IEEE® arithmetic representation for positive infinity . Given that the lower values of error are important, the analysis of variance was conducted for the cases with values of ; hence the resulting database consists of 2292 records. ANOVA found statistically significant differences between the means (MD) of for 25 and 50 derivatives (MD = 43.5803, p-value = 9.5606 10−10) and for 25 and 75 derivatives (MD = 54.2206, p-value = ) as demonstrated in Figure S3 and Table S2. Similarly, the number of digits (Figure S4 and Table S3) was found significant for the (MD = 20.3576, p-value = 9.6963 10−10 for 500 to 1000 digits and MD = 33.9628, p-value = 9.5606 10−10 for 500 to 2000 digits). The number of divisions exhibits a clear difference as well, in Figure S5 and Table S4 (MD = 22.7727, p-value = 9.5606 10−10 for 50 to 100 divisions and MD = 49.6978, p-value = 9.5606 10−10 for 50 to 200 divisions). The univariate linear correlation of the condition number of with (Figure S12) exhibits an R2 of 0.2559 (p-value = 3.1608 10−149). For the condition number of H as well as the inversion errors of H and , the correlations are even lower (Figures S13-S15); however, the majority of the values are 10323 for the condition of H and 10−323 for the inversion errors.
In order to further investigate more complex associations among the studied parameters and the extrapolation error , the random forests method  was utilized. The numerical dataset was divided into a train set (85% of observations) and a test set (15%) in order to constitute and investigate the reliability of the predictive model. The R2 for the predicted versus actual for the test set was 0.8954 (Figure S18), indicating a reliable model. The features significance was evaluated in terms of their contribution to the prediction error in the constituted model (Figure S19A), signifying high values for the studied function, the condition of , the number of derivatives, and the domain length. Similarly, considering the computation time as dependent variable, the R2 for the test set was 0.9658 and the highest predictive features were the number of divisions, the number of integrations, and number of digits (Figure S19B).
3.3. Illustrative Functions and Their Extrapolation Spans
In Figure 2 the extrapolation results are demonstrated by four test functions as indicated. The initial domain contains only the known given data for and , and after the vertical lines, each graph contains the exact function values, the predicted and the normalized extrapolation error , which take values in the . The text cases include a vast variety of analytic functions as well as their combinations. In Figures 2(a)–2(d) the given curvature or periodicity information is meager, compared to the predicted evolution of the function’s values. Interestingly, the logged error plot exhibits a logarithmic scheme, indicating a weak form of stability .
In Figure 3, the proposed procedure is implied firstly in the x-axis and then in the y-axis for the function . Given only the cyan region, the method can predict the highly nonlinear colored surface in the x, y, z space.
4.1. Numerical Differentiation
Numerical differentiation is highly sensitive to noise , especially for the higher order derivatives. After an extensive examination the literature utilizing IRBFs [35, 45, 46] for the derivatives approximation, or for the solution of specific problems [47, 48] as well as other differentiation methods [49, 50] and a variety of formulations for IRBFs (, , , , , ), the proposed procedure accomplished striking accuracy, with error magnitude ε of O(10−100) or less (Figures S3-S11 and supplementary database), for the derivatives’ computation. This finally permitted the implementation of the Taylor method for significant extrapolation extents (Figures 2 and 3 and Data File S1). The numerical calculation of the derivatives with variable precision arithmetic offers high accuracy [37, 38]. The inverse problem of numerical integration exhibits lower errors with ; however, the integration is less sensitive to—even small—errors in the given data . Cheng , similarly to this work, found that the errors of the derivatives approximation are of one or more order of magnitude higher than the function’s (Figures S16, S17, and Data File S1). However, the digits studied were 100-200, while the precise computation of the derivatives was vital for the extrapolation, and so for a higher number of digits. Similarly, Mai-Duy et al.  examined the compactly integrated radial basis functions, with errors for the derivatives for fifty digits accuracy.
4.2. RBFs’ Matrix Φ and Shape Parameter
The approximation scheme ((9), (10), (12), (13), , , , , , ) permits a precise interpolation, within the given domain, with or less, as found in the numerical experiments (supplementary database). Accordingly, the approximation of the function’s values is attainable with high accuracy, not only at the nodes xi, but for any intermediate point between two given points and, hence, for points hj (Figure 1(b)), permitting the precise calculation of vector C (13). Arbitrary precision found capable of avoiding the ill-conditioning of was also indicated by Cheng  and Huang . Using arbitrary precision in the calculations, the inversion errors in -I were found of O(<10−100) or even exact zero (supplementary database). The arbitrary precision has also been used for the robust computation of by Cheng , where the Gaussian RBFs and inversed multiquadrics exhibit accuracy of interpolation with (~10−20-10−30).
The association of the condition number of with exhibits low R2 (0.25595) with slightly negative slope (Figure S12), although the high condition number is considered to increase instability [37, 41, 52] indicating the need for high precision to the calculations; however the majority of the values are 10323, that is, the maximum real number considered by the software . Sensitivity analysis was selected instead of optimizing each method’s parameters such as times of RBFs integration, the kernel function, or its shape parameter [40–42], as the interpolation errors in a number of the test cases were equal to zero at the nodes, eliminating any relevant objective function. The number of integrations of RBF φ increased computational time, as it magnifies the and matrices, due to the more complex formulas (, , , ).
4.3. Summary of Findings and Limitations of the Proposed Solution
In brief, a method for the extrapolation of analytic functions is developed, accompanied by a systematic investigation of the involved parameters. The proposed scheme for the numerical differentiation exhibited low enough errors to permit adequate extrapolation spans. The constituted database of the numerical experiments highlights the fact that, for the same problem formulation (L, N, dx, f), the derivatives calculation exhibits a high variation, indicating the vagueness of the transference from the presumed theory to the actual calculations. The numerical investigation of the extrapolation errors suggests that only utilizing high accuracy and a precise approximation scheme for a function as well as its derivatives, the extrapolation is attainable. Thus, for real-world phenomena, with laboratory accuracies even of O(10−20), the predictions are limited to short length. Only if the measurements contain some hundreds of significant digits, the proposed solution is efficient. As this is difficult to be accomplished by laboratory instruments, this work’s findings provide strong evidence that we are far from lengthy predictions in physics, biology, and engineering and also even more far from phenomena studied in health and social sciences. However, the parametric investigation suggests that the precision in calculations and the utilized methods are vastly significant, as the extrapolation horizons achieved by the proposed numerical scheme are about an order of magnitude higher than those in the existing literature, highlighting the potentiality of predictions.
5. Materials and Methods
All the software operations ran on an Intel i7-6700 CPU @3.40GHz with 32GB memory and SSD hard disk, and the execution time was accordingly tracked. The errors of the derivatives in the relevant literature are reported in the Supplementary Data File S1, and they highlight the immense accuracy of numerical differentiation achieved in this work.
Conflicts of Interest
The author declares no conflicts of interest.
This study would not have been possible without professor’s John T. Katsikadelis helpful discussions, valuable suggestions, and clarifications regarding the integrated radial basis functions as well as his confidence that the problem is solvable. Special thanks are due to Dr. Natia R. Anastasi for reviewing the original manuscript. I would also like to thank Dr. Savvas Chatzichristofis, Director of Informatics at Neapolis University Pafos, Nikitas Giannakopoulos, Ass. Professor of Medicine, University of Würzburg, Germany, and Angeliki Sivitanidou and Thomas Dimopoulos, Lecturers at Neapolis University, for suggestions regarding language.
The supplementary materials are divided into two parts: (A) The supplementary figures and tables for the statistical analyses of parameters affecting extrapolation accuracy, and the bibliometric review figures. (B) The supplementary database: Data File S1. experiments.xlsx (attached): database for the parametric investigation of extrapolation errors. Figure S1: bibliometric map of papers with keyword extrapolation. Utilizing a bibliometric procedure  for a co-keywords based bibliometric analysis for the top papers in terms of their citations, containing extrapolation as keyword, the keywords highly associated with extrapolation—closer on the bibliometric map—are interpolation, risk assessment, prediction, uncertainty, modeling, etc. The almost coincident positions of extrapolation and interpolation indicate that the techniques aiming to extend a series of values beyond the observed domain are mainly based on interpolation procedures, which is an effort to mathematically formulate the phenomenon . Figure S2: Evolution of terms in Google Books. Commencing the eighties and on, the effort of the researchers is decreasing, reflecting the failure to predict time-series values. This is demonstrated through a search in over 25 million—as of October 2015—books , for the words forecasting and extrapolation and the closely related topic of interpolation . Figure S3: variance of for number of derivatives. Figure S4: variance of for number of digits. Figure S5: variance of for number of divisions. Figure S6: variance of for number of integrations. Nonsignificant differences (as per Table S5). Figure S7: variance of for domain length. Nonsignificant differences (as per Table S6). Figure S8: variance of for function. The function sin(x) exhibits less errors than the more complex (MD = 15.6422, p-value = 1.0597 10−10, from Table S7). Figure S9: variance of for kernel. The Gaussian kernel exhibits lower errors than the Shifted Logarithmic (MD = 9.9051, p-value = 3.3708 10−7 - Table S8). Figure S10: variance of for limit . Nonsignificant differences (as per Table S9). Figure S11: variance of for shape parameter. The shape parameter 0.2 exhibits higher errors than 5 (MD = 8.5855, p-value = 0.0065 - Table S10). Figure S12: scatter plot for condition number of and . Figure S13: scatter plot for condition number of and . Figure S14: scatter plot for inversion error of and . Figure S15: scatter plot for inversion error of and . Figure S16: scatter plot for error of first derivative and . Figure S17: scatter plot for error of second derivative and . Figure S18: scatter plot for actual versus predicted . Table S2: differences between the means of for groups with an uneven number of derivatives. Table S3: differences between the means of for groups with an uneven number of digits. Table S4: differences between the means of for groups with an uneven number of divisions. Table S5: differences between the means of for groups with an uneven number of integrations. Table S6: differences between the means of for groups with uneven domain lengths. Table S7: differences between the means of for groups with uneven functions f. Table S8: differences between the means of for groups with uneven RBF kernel. Table S9: differences between the means of for groups with uneven limit l. Table S10: differences between the means of for groups with uneven RBFs shape parameter. Figure S19: features significance. (Supplementary Materials)
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