Monotone Data Modelling Using Rational Cubic Fractal Interpolation Function

Geometric modelling of several intricate and complex structures such as trees, mountains, clouds, ferns, geographic topography, and coastlines is challenging in computer graphics. Traditional splines such as trigonometric, polynomial, exponential, and rational fail to simulate this significant class of complex structures, which are highly irregular in nature. For this purpose, this research develops a novel cutting-edge method for synthesizing and modelling structures. The proposed technique; C 1 fractal interpolation function (FIF) builds an iterated function system (IFS) by integrating fractal calculus and rational cubic polynomial functions. Appropriate conditions on scaling and shape parameters are derived to help maintain the inherited shape qualities of the data. Experiments in numerous scientific domains, such as the pharmaceutical and chemical industries have been presented as an example, to confirm the usefulness of the suggested model. Moreover, the graphic results demonstrated that the developed monotone hybrid model (MHM) offers a heterogeneous method for gathering data with a monotone structure.


INTRODUCTION
Interpolation is an indispensable tool for estimating the unknown function for data modelling. There are numerous interpolation techniques, which use various function families, including polynomial, exponential, rational, trigonometric, and spline. However, these traditional non-recursive approaches resulted in interpolants, which were differentiable several times, with the possible exception of a limited number of points and restricted model smooth structures. Many signals in the real world and in School of Science Volume 7 Issue 1, 2023 experimental data are complex and their composition frequently demonstrates irregularities and hardly ever appears to be smooth. Thereby, interpolants that could picture non-smooth signals, as well as smooth structures, were highly required. The prime purpose of this research is to provide an innovative framework for the approximate representation of non-smooth data from natural (feathers, trees, leaves, clouds, feathers, flowers, glaciers, landscapes, and galaxies), engineering or scientific phenomena together with the collection of smooth data. A novel form of fractal interpolation function is created utilizing the rational spline containing a cubic polynomial in the numerator as well as the denominator.
One of the pivotal characteristics of data generated from countless physical phenomena, scientific problems, and engineering applications is monotonicity where entities only have a meaning when their values are monotone. For instance, in the field of engineering, Tensile Strength, Ultimate Tensile Strength or Ultimate Strength is not an uncommon concept. It refers to the maximum level of stress that a structure or material can tolerate while being pulled or stretched before getting breaking. This phenomenon gave rise to monotone data. The applied forces, commonly called stress and the stretch, referred to as strain, are always in a monotonic relationship where one varies with the change in the value of the other [1]. Certain devices, such as; digital-to-analog (DAC used in audio video devices) and analog-to-digital (ADC used in music recording and digital signal processing), have monotonicity an essential characteristic used in feedback control loops [1]. In CAD and DAC, the digital output code always increases as the analog voltage input increase and vice versa. Monotonicity can be seen in Newton's law of cooling in which the rate of heat loss is directly proportional to the differences in temperature between the bodies [2].
Monotonicity also appears in other phenomena, such as the content of atrazine and nitrate in shallow ground waters, the link between the partial pressure of oxygen, and the percentage dissociation of hemoglobin percentage dissociation and oxygen partial pressure, erythrocyte sedimentation rate in patients with cancer and concentration of uric acid level in a patient affected from gout [3]. It is a fundamental function behind the renowned 'Probability theory' that uses the monotonically increasing function of a random variable in its cumulative distribution.
Natural phenomena can produce monotonic positive data as well. The relationship between a child's height and their intake of food and energy can be considered as an example. For instance, height is a positive quantity and always goes up. There are several other examples in everyday experiences where objects only create meaning when their values are monotonous. Examples are surfaces and the dose-response curve in biochemistry and pharmacology; consumption function that describes the relationship between a household's consumption and its available income in economics, statistical approximations of copulas and quasi-copulas, [4] and empirical option pricing models in finance [4].
Using different types of spline functions, some other useful techniques and algorithms were supplied to address the problem of monotonicity [1,3,7]. Typically, data that comes from experiments (for instance, information drawn from signals in the actual world, including seismic information, financial series, and bioelectric recordings are complex and immensely irregular. As a result, interpolating these data using traditional interpolation techniques became an inappropriate challenge. Barnsley [8] made an attempt by introducing a new interpolation method called Fractal Interpolation using a special type of iterated function system. It provided a powerful framework to deal with highly irregular data. However, consistently, when it is a prerequisite that the interpolant should reflect the inherited shape features (convexity, positivity, monotonicity) of the irregular data set, fractal interpolation functions failed to secure the shape properties of data. Later, Barnsley and Harrington [9] noticed that in case the problem is of shape preservation type, then the parameters of the IFS may be chosen suitably so that the corresponding FIF preserve the inherited properties of the data. This observation initiated a striking relationship School of Science Volume 7 Issue 1, 2023 between the classical splines interpolation functions and the fractal interpolation functions.
Recently, with the help of Barnsley and Harrington's results, various spline functions have been generalized in the form of fractal spline functions for instance [4,10]. Fractal spline [11] though interpolating both regular and irregular data does not guarantee that the model satisfies the monotone trend lies in their data. Non-monotone visual models of the aforementioned experiments misguide us about the actual scenario (which might help scientists to recognize and evaluate various patterns and artifacts in their data) in these experiments. Hence, the visual model must maintain the geometric property of the experimental data in order to achieve precise and proficient results.
The goal of the underlying research is to build a fractal geometry-based function to simulate hidden patterns and future behaviour in many disciplines. A monotone hybrid model (MHM) which is a combination of fractal interpolation function (FIF) and spline interpolation model (SIM) was presented for this purpose. The shape-preserving model is then created with the help of the proposed function to handle the non-linearity problem that the traditional interpolation model was unable to solve. Additionally, the proposed class surpasses its conventional non-recursive counterpart in approximating monotone functions with varied irregularities/fractality in their first derivatives (smooth to nowhere differentiable). The proposed hybrid model offered a variety of shape control parameters that can be sufficient and highly useful to achieve desired results. To meet the objective, a general piecewise rational cubic fractal interpolation function (GPRC FIF) [11] is used in the construction of MHM.
The proposed model has the following useful features in comparison with the existing models: • The proposed scheme converts the hybrid model [11] to a monotone hybrid model.
• The proposed hybrid model MHM is the generalization of the model presented in [7].
• Unlike [12], no derivative constraints are imposed in proposed model as MHM works for both data and data with derivatives proficiently.
Scientific Inquiry and Review Volume 7 Issue 1, 2023 • Once, the shape parameters are selected, the curve representation becomes unique in its description.
• Existing methods have no degree of freedom [1,3,12,13], whereas the proposed model gives two degrees of freedom.
• The proposed model works for larger as well as smaller data, whereas in [14], constraints are developed on the subinterval lengths for data modelling.
Four shape parameters are used in the proposed model, whereas in [3,15] one and two shape parameters are used respectively. The remaining sections are structured as follows. Section 2 focuses on a brief introduction to the basic methodology of the hybrid model developed in [11]. In Section 3, a new scheme is developed, which converts the hybrid model to a monotone hybrid model, for solving the problem of monotonicity. Whereas the proposed model is tested on different real data sets to verify their usefulness in Section 4. Comparative analysis has been presented in Section 5 and finally, Section 6 concludes with the significance of the proposed study.

METHODOLOGY
In this section, the hybrid model (GPRC FIF) [11] based on the fractal framework and spline function was briefly reviewed. The fractal framework was built on an iterative function system (IFS), which is defined below comprehensively.

2.2.Hybrid Model
The hybrid model [11] is termed as GPRC FIF corresponding to the iterative function system as: Here, < 1 are the scaling factors, is the derivative value at the knot points and , , , are the non-negative shape parameters.

A MONOTONE HYBRID MODEL
This section sets up a monotone hybrid model (MHM) to solve the monotonicity problem present in the hybrid model as defined in [11]. In setting up the MHM, certain conditions were imposed to shape the parameters and scaling factors, to acquire the desired model. Let, the monotone data set {( ,̂) ∈ × : = 1,2, … , } be distributed such that 1 ≤ 2 ≤ ⋯ ≤ with ̂+ 1 ≥̂, = 1,2, … , − 1.

RESULTS AND DISCUSSION
This section presents two experiments, which demonstrated the validity and adaptability of the proposed MHM. In these experiments, the derivatives values were calculated through geometric mean method. The monotone data set taken in Table 1 presents the experimental result concerning the watering the great northern beans. Distilled water with pH of 8.5 and Potassium Hydroxide was combined in the chemical solution, which was later used to watering the beans. After 40 days, the beans plant was taken out from vat to weight for the effect of chemical solution. It can be easily noticed that resulting data is monotone. The r-values represented the days and the v-values showed the height of beans. The data demonstrated in Table 1 is monotone; therefore, the resulting curve must be monotone. However, from Figure 1, which is constructed through hybrid model [11] it can be observed that the arbitrary value ( =7, =4, =8.2, =99, =0.9) does not preserve the inherited monotonicity of the curve. On the other hand, Figure 2 displays multiple curves, which are produced through the proposed MHM. In Figure 2, it is found that the curve at different iterations preserve the essential characteristic of the data, which is monotonicity, proficiently. Figures 3 and Figure 4 highlighted the effect of diverse values of and ,rescpectively. The data presented in these figures showed that with different choices of parameter values through MHM, monotone curve of any particular requirement may be easily gained.
The monotone data set in Table 2 reports the varying degree of saturation of hemoglobin with partial pressure of oxygen. It was observed that saturation of hemoglobin was directly proportional to the partial pressure of oxygen. The r-values represented the partial pressure of oxygen and z-values indicated the percentage of saturation.  The data demonstrated in Table 2 shows the monotone relationship, therefore, the curve must exhibit the same characteristic. Figure 5 is generated through the random values to the shape parameters and scaling factors ( =2.9, =3, =7, =15, =0.7) in the description of hybrid model as defined in Eq. (2.4). Figure 5 depicts non-monotone hemoglobin dissociation curve (HDC) which does not make any sense, as the behaviour of the curve should retain monotonicity. This flaw is removed in Figure 6 by using the proposed method presented in Section 3. It can easily be observed that Figure 6 conserve monotonicity at each level of iterations. Figures 6 and 7 showed the profile of HDC with diverse values of and . It is worth noticing that although various parameters may lead to different results but all conserve monotonicity.

Comparative Analysis
The proposed method guarantees to produce monotone model as compared to the hybrid model [11], which does not ensure to exhibit monotone model when data is monotone. All the existing models [3,12,13], failed to provide even a single degree of freedom, so MHM was a noteworthy addition as it offered two degree of freedom to modify and enhance the model. Unlike, [14] and [12], it is easy in the applicability, as it does not constrain the interval length or impose any conditions on derivatives, respectively. The scheme developed in [7] only works for regular data but the proposed approach, which is generalization of [7] is applicable for both regular and irregular datasets.

Conclusion
General Piecewise Rational Cubic Fractal Interpolation Function (GPRCFIF) was developed to provide a deeper understanding of the data, which displays rough, irregular, and fragmented configuration. Generally, it originates from complex functions or experiments. The GPRCFIF offers the family of four parameters and one scaling factor to modify, upgrade, and change the visual images. One of the most prominent characteristics of the GPRCFIF is its capacity to visualize data with smooth curves by setting all scaling factors to zero. The current study, in particular addressed the problem of monotonicity through the novel proposed model MHM. The experimental results illustrated that the performance of the proposed MHM was very efficient and heterogeneous as it contains the ability (due to recursive nature) to produce multiple monotone curves for the same monotone data according to the user's desire.