Vertical Scaling Factors Research Articles

Predict the prevalence and incidence of Parkinson's disease using fractal interpolation-LSTM model.

The investigation of the prediction of disease population is a noticeable exploration topic in the field of sciences. As a type of neurological disease, the incidence and prevalence of Parkinson's disease are still difficult to accurately study. In this paper, a method is proposed to forecast the number of incident cases (NumIn), incidence rate (InRa), the number of prevalent cases (NumPr), and prevalence rate (PrRa) of Parkinson's disease in ten countries selected. Using past data on the incidence rate, the number of prevalent cases, and the prevalence rate from 1990 to 2019, three types of fractal interpolations with different fractal dimensions are constructed for reconstructing the past data, where the vertical scaling factors are determined by the method proposed in this article. Then, the Long Short-Term Memory (LSTM) model is employed to forecast the values of NumIn, InRa, NumPr, and PrRa with Parkinson's disease in 2020. Meanwhile, the autoregressive integrated moving average model is used to predict the values compared with the LSTM model. The evaluation metrics employed for error calculation include the root mean square error and the coefficient of determination (R2). According to the proposed optimal criteria, the best predicted results are the average of three types of prediction values based on the LSTM model by analyzing and comparing eight predicted results.

Improve Fuzzy Inventory Model of Fractal Interpolation with Vertical Scaling Factor

The inventory model is used to determine the optimal inventory of a product. In certain cases, parameters in the inventory model are uncertain. Fractal interpolation techniques can be used to overcome parameter with uncertainty. Fractal interpolation results are affected by the fractal interpolation function and the vertical scaling factor. The vertical scaling factor is positive and less than 1. In this study, fractal interpolation techniques are introduced with variations in vertical scaling factor to overcome the uncertainty of demand data in inventory models. Furthermore, the interpolation results are used in fuzzy inventory models and expressed by Trapezoidal Fuzzy Number. This paper considers an inventory model with varying demand to optimize rice inventory. Based on the data obtained, the accuracy level will increase for the vertical scaling factor values close to 1. Optimal rice inventory of each successive fuzzy parameter is 1447963, 1013914, 504950, 215312. If the cost parameter is increased, then the amount of inventory is decreases.

Bivariate Fractal Interpolation Functions on Triangular Domain for Numerical Integration and Approximation

The primary objectives of this paper are to present the construction of bivariate fractal interpolation functions over triangular interpolating domain using the concept of vertex coloring and to propose a double integration formula for the constructed interpolation functions. Unlike the conventional constructions, each vertex in the partition of the triangular region has been assigned a color such that the chromatic number of the partition is 3. A new method for the partitioning of the triangle is proposed with a result concerning the chromatic number of its graph. Following the construction, a formula determining the vertical scaling factor is provided such that the actual double integral coincides with the integral value calculated using fractal theory. Convergence of the proposed method to the actual integral value is proven with sufficient lemmas and theorems. Adequate examples are presented to illustrate the method of construction and to verify the value of double integration.

ON THE CLASSICAL INTEGRAL OF FRACTAL FUNCTIONS

In this paper, the integral of classical fractal interpolation function (FIF) and A-fractal function is explored for both the cases of constant and variable scaling factors. The definite integral for the classical FIF in the closed interval of [Formula: see text] is estimated. The novel notion of affine-quadratic FIF is introduced and integrated for both constant and variable scaling factors. It is demonstrated that its integral is not an affine-quadratic FIF, however it is a FIF. Similarly, by choosing the vertical scaling factors as constants and variables, A-fractal function is integrated. Further, by assuming certain condition on the block matrix, it is shown that like the original A-fractal function its integral is also an attractor for the iterated function system.

FRACTAL INTERPOLATION ALGORITHMS FOR COMPLEX CURVES AND SURFACES

This paper provides a new idea for interpolating complex curves and surfaces by fractal interpolation. A type of bivariate fractal interpolation functions with function vertical scaling factors for the special interpolation data set is presented. The interpolation algorithm is proposed for complex curves by contour lines of two types of fractal interpolation functions which are proposed by this paper and Navascués et al. [Construction of fractal surfaces, Fractals 28(2) (2020) 2050033], respectively. Meanwhile, some properties of the interpolation algorithm are introduced. Then, the two classes of fractal interpolation functions are extended to the cases of trivariate functions. We also introduce the interpolation algorithm for complex surfaces based on iso-surfaces of fractal interpolation functions constructed. Finally, several examples are provided.

On Approximation Properties of Fractional Integral for A-Fractal Function

In this paper, the Riemann–Liouville fractional integral of an A-fractal function is explored by taking its vertical scaling factors in the block matrix as continuous functions from 0,1 to ℝ . As the scaling factors play a significant role in the generation of fractal functions, the necessary condition for the scaling factors in the block matrix is outlined for the newly obtained function. The resultant function of the fractional integral is demonstrated as an A-fractal function if the scaling factors obey the necessary conditions. Furthermore, this article proposes a fractional operator which defines the Riemann–Liouville fractional integral of an A-fractal function for each continuous function on C I , ℝ 2 , where C I , ℝ 2 is the space of all continuous functions from closed interval I ⊂ ℝ to ℝ 2 . In addition, the approximation properties such as linearity, boundedness, and semigroup property of the proposed fractional operator are investigated.

AFFINE FRACTAL LEAST SQUARES REGRESSION MODEL

This paper develops a method to find fractal curves to fit real data. With a formulation for fractal functions through a type of affine systems of iterative functional equations, we apply the procedure of minimizing the sum of square residuals that is used in the classical linear regression. We develop formulas for approximation and exact fractal functions for various fractal levels and number of equations. This method gives estimates for the parameters of the equations and corresponding functions, namely the fractal coefficients (vertical scaling factors) and directional coefficients. As a consequence, it is possible to upper estimate the Hausdorff dimension of the fitted curve. We provide worked examples, including estimating a fractal curve for a time series real data set of exchange rates between USD and EUR currencies.

Riemann-Liouville fractional derivatives of hidden variable recurrent fractal interpolation functions with function scaling factors and box dimension

In [M.G. Ri and C.H. Yun, Smoothness and fractional integral of hidden variable recurrent fractal interpolation function with function vertical scaling factors, Fractals 29(6) (2021) 2150136], the authors proved that the partial and mixed Riemann-Liouville fractional integrals of bivariable hidden variable recurrent fractal interpolation function(HVRFIF) with function scaling factors are HVRFIFs. As its continuation, in this paper, we show that Riemann-Liouville fractional derivatives of one variable and bivariable HVRFIFs are HVRFIFs under certain conditions. We also derive the relationship between the order of fractional calculus and the upper box dimension of its graph. To do it, firstly, we prove that Riemann-Liouville fractional derivative of one variable HVRFIF is HVRFIF. Secondly, we obtain estimation of the upper box dimension of the graph of one variable HVRFIF and derive the relationship between the upper box dimension and the order of fractional calculus. Finally, in the similar way to one variable, we show that the partial and mixed Riemann-Liouville fractional derivatives of bivarible HVRFIF are HVRFIFs.

A remarkable fact for the box dimensions of fractal interpolation curves of R3

In this paper, we give the fractal interpolation curves generated on the Koch curve by Hölder continuous functions and function vertical scaling factors, and at the same time, we present the estimation method of box dimensions of curves of R3 through the estimation of lower and upper box dimensions of obtained curves, where a remarkable fact is that the box dimensions are dependent not only on function vertical scaling factors but also on Hölder exponent of harmonic functions on the Koch curve.

SMOOTHNESS AND FRACTIONAL INTEGRAL OF HIDDEN VARIABLE RECURRENT FRACTAL INTERPOLATION FUNCTION WITH FUNCTION VERTICAL SCALING FACTORS

In this paper, we analyze the smoothness of hidden variable recurrent fractal interpolation function (HVRFIF) with function vertical scaling factors introduced in Yun [Hidden variable recurrent fractal interpolation function with four function contractivity factors, Fractals 27(7) (2019) 950113] and study on fractional integral of bivariable HVRFIF by using its smoothness. To do it, firstly, we analyze the smoothness of one variable HVRFIF and give the result of smoothness of bivariable HVRFIF. Next, we show that partial and mixed Riemann–Liouville fractional integrals of the bivariable HVRFIF are HVRFIFs.

Graph-directed random fractal interpolation function

"Barnsley introduced in [1] the notion of fractal interpolation function (FIF). He said that a fractal function is a (FIF) if it possess some interpolation properties. It has the advantage that it can be also combined with the classical methods or real data interpolation. Hutchinson and Ruschendorf [7] gave the stochastic version of fractal interpolation function. In order to obtain fractal interpolation functions with more exibility, Wang and Yu [9] used instead of a constant scaling parameter a variable vertical scaling factor. Also the notion of fractal interpolation can be generalized to the graph-directed case introduced by Deniz and  Ozdemir in [5]. In this paper we study the case of a stochastic fractal interpolation function with graph-directed fractal function."

Construction and box dimension of recurrent fractal interpolation surfaces

In this paper, we present a general framework to construct recurrent fractal interpolation surfaces (RFISs) on rectangular grids. Then we introduce bilinear RFISs, which are easy to be generated while there are no restrictions on interpolation points and vertical scaling factors. We also obtain the box dimension of bilinear RFISs under certain constraints, where the main assumption is that vertical scaling factors have uniform sums under a compatible partition.

The second and third waves in India: when will the pandemic be culminated?

An unprecedented upsurge of COVID-19-positive cases and deaths is currently being witnessed across India. According to WHO, India reported an average of 3.9 lakhs of new cases during the first week of May 2021 which equals 47% of new cases reported globally and 276 daily cases per million population. In this letter, the concept of SIR and fractal interpolation models is applied to predict the number of positive cases in India by approximating the epidemic curve, where the epidemic curve denotes the two-dimensional graphical representation of COVID-19-positive cases in which the abscissa denotes the time, while the ordinate provides the number of positive cases. In order to estimate the epidemic curve, the fractal interpolation method is implemented on the prescribed data set. In particular, the vertical scaling factors of the fractal function are selected from the SIR model. The proposed fractal and SIR model can also be explored for the assessment and modeling of other epidemics to predict the transmission rate. This letter investigates the duration of the second and third waves in India, since the positive cases and death cases of COVID-19 in India have been highly increasing for the past few weeks, and India is in a midst of a catastrophizing second wave. The nation is recording more than 120 million cases of COVID-19, but pandemics are still concentrated in most states. In order to predict the forthcoming trend of the outbreaks, this study implements the SIR and fractal models on daily positive cases of COVID-19 in India and its provinces, namely Delhi, Karnataka, Tamil Nadu, Kerala and Maharashtra.

Vertical scaling optimization algorithm for a fractal interpolation model of time offsets prediction in navigation systems

System time offsets between Global Navigation Satellite Systems is a key issue in multi-system positioning. A high-precision time offsets prediction model is required in actual navigation applications. In this study, the fractal characteristics of time offsets in American and Russian navigation systems are directly confirmed through fractal dimension calculations. Then, the influence of the vertical scaling factor on the prediction accuracy of the fractal interpolation prediction model is confirmed. It is found that perturbation on the estimated vertical scaling factor causes a minimal attractor bias, which is positively correlated with the prediction error. After proving the existence of the estimated vertical scaling factor of the minimal attractor biases, an optimization method of the fractal interpolation prediction model is proposed, and the related algorithms are given. The numerical results show that the time offsets prediction errors using the new model are less than 1 ns for the American navigation system and close to 2 ns for the Russian navigation system. The prediction accuracy of this model has an improvement of at least 60% over the commonly used quadratic and grey models, and an improvement of more than 13% over the standard fractal interpolation prediction model.

NONLINEAR RECURRENT HIDDEN VARIABLE FRACTAL INTERPOLATION CURVES WITH FUNCTION VERTICAL SCALING FACTORS

In this paper, we present a construction of new nonlinear recurrent hidden variable fractal interpolation curves. In order to get new fractal curves, we use Rakotch’s fixed point theorem. We construct recurrent hidden variable iterated function systems with function vertical scaling factors to generate more flexible fractal interpolation curves. We also give an illustrative example to demonstrate the effectiveness of our results.

Riemann Liouville fractional integral of hidden variable fractal interpolation function

Abstract In this paper, we study Riemann Liouville fractional integral of hidden variable fractal interpolation function (HVFIF) constructed by functions whose Lipschitz exponents are in (0, 1]. Firstly, we present a construction of HVFIF using functions of which Lipschitz exponents are in (0, 1], so that the Riemann Liouville fractional integral of the HVFIF becomes a fractal interpolation function, and give an example where Lipschitz exponents of functions of IFS are in (0, 1]. Secondly, we prove that the Riemann Liouville fractional integral is also a HVFIF with function vertical scaling factors defined newly. Finally, we give the graphs of 0.8- and 0.2-order fractional integrals of the HVFIFs constructed in the above example.

Nonlinear fractal interpolation curves with function vertical scaling factors

In this paper we present a method to generate new fractal interpolation curves. We ensure that attractors of nonlinear iterated function systems (IFSs) constructed by Geraghty contractions are graphs of some continuous functions which interpolate the given data. In particular, we give an explicit illustrative example to demonstrate the effectiveness of obtained results. Concerning IFSs, our methods and results extend known results from the literature.

NEW NONLINEAR RECURRENT HIDDEN VARIABLE FRACTAL INTERPOLATION SURFACES

In this paper, we present the construction of new nonlinear recurrent hidden variable fractal interpolation surfaces (RHVFISs) with function vertical scaling factors. We use Rakotch’s fixed point theorem which is a generalization of Banach’s fixed point theorem to get new nonlinear fractal surfaces. We construct recurrent vector-valued iterated function systems (IFSs) with function vertical scaling factors on rectangular grids and generate flexible and diverse RHVFISs which are attractors of the IFSs. We also give an explicit example to show the effectiveness of obtained results.

A FRACTAL INTERPOLATION FUNCTION THROUGH SHEAR FACTOR WITH PERTURBATION ERROR ANALYSIS

Fractal interpolation (FI) is widely used in nonlinear issues of natural and social sciences as it provides a constructive way to describe an irregular structure of data. A new approach of FI function (FIF) is proposed in this paper to represent uncertainty of irregular data. The proposed method is based on vertical shear factor instead of vertical scaling factor which is used in existing FI. Experiments were performed using existing datasets which confirmed the practical usefulness of the proposed method. A comparison with existing methods was also made to further verify the effectiveness of the proposed method. Furthermore, the criteria of the perturbed iterated function system (IFS) is proposed in order to satisfy the FI conditions and was explained with the help of an example. In addition, the error estimation between two IFS using the proposed refined equation is also presented.

CONSTRUCTION OF NONLINEAR HIDDEN VARIABLE FRACTAL INTERPOLATION FUNCTIONS AND THEIR STABILITY

This paper presents a method to construct nonlinear hidden variable fractal interpolation functions (FIFs) and their stability results. We ensure that the projections of attractors of vector-valued nonlinear iterated function systems (IFSs) constructed by Rakotch contractions and function vertical scaling factors are graphs of some continuous functions interpolating the given data. We also give an explicit example illustrating obtained results. Then, we get the stability results of the constructed FIFs in the case of the generalized interpolation data having small perturbations.