Open AccessEnhanced Forecasting of Alzheimer’s Disease Progression Using Higher-Order Circular Pythagorean Fuzzy Time Series
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This study introduces an advanced forecasting method, utilizing a higher-order circular Pythagorean fuzzy time series (C-PyFTSs) approach, for the prediction of Alzheimer’s disease progression. Distinct from traditional forecasting methodologies, this novel approach is grounded in the principles of circular Pythagorean fuzzy set (C-PyFS) theory. It uniquely incorporates both positive and negative membership values, further augmented by a circular radius. This design is specifically tailored to address the inherent uncertainties and imprecisions prevalent in medical data. A key innovation of this method is its consideration of the circular nature of time series, which significantly enhances the accuracy and robustness of the forecasts. The higher-order aspect of this forecasting method facilitates a more comprehensive predictive model, surpassing the capabilities of existing techniques. The efficacy of this method has been rigorously evaluated through extensive experiments, benchmarked against conventional time series forecasting methods. The empirical results underscore the superiority of the proposed method in accurately predicting the trajectory of Alzheimer’s disease. This advancement holds substantial promise for improving prognostic assessments in clinical settings, offering a more nuanced understanding of disease progression.
1. Introduction
Decision-making, defined as the process of selecting the optimal choice from a range of alternatives to achieve organizational objectives, is a critical area of research in today's complex problem-solving environments (Attaullah et al., 2022). The field of Multicriteria Decision Making (MCDM) particularly addresses challenges encompassing multiple objectives or conditions. Numerous MCDM techniques have been developed to manage decisions involving diverse, competing criteria in scenarios characterized by ambiguity. Applications of these methods span various domains, including printer selection (Gündoğdu & Ashraf, 2021) and solar power plant development (Khan et al., 2020). Traditional MCDM algorithms, however, face limitations in handling imprecise or unclear verbal judgments, as they require exact numerical values. To address this gap, enhancements have been made to standard fuzzy sets in MCDM methodologies through the incorporation of Pythagorean fuzzy sets, neutrosophic sets, and spherical fuzzy sets (Chinram et al., 2020). These advancements have significantly improved the handling of ambiguity in data forecasting, which is increasingly relevant for MCDM.
The introduction of fuzzy set theory, developed by Zadeh et al. (1996), marked a significant advancement in decision-making processes plagued by statistical ambiguity. Fuzzy sets, defined for each element x in a domain set, assign a membership degree ranging from 0 to 1. However, fuzzy sets encounter limitations, notably their inability to represent non-membership. To overcome this limitation, Atanassov (1999) introduced the intuitionistic fuzzy set (IFS), which offers a more comprehensive understanding of membership degrees. IFS utilizes both membership degree V(p) and non-membership degree M(p), adhering to the constraint that 0 ≤ V(p) + M(p) ≤ 1. The utility of IFS in various real-world applications has been extensively researched and validated (Atanassov, 2007). Building upon the concept of IFS, Nayagam et al. (2011) explored an interval valued Pythagorean fuzzy set (IVIFS), which represents an extension of IFS and a modification of the standard fuzzy set. IVIFS has found wide application in decision-making contexts (Tan, 2011; Xu, 2011).
Addressing scenarios where the sum of membership and non-membership degrees exceeds one, Yager (2013) proposed the Pythagorean fuzzy set (PyFS). PyFS, based on the Pythagorean theorem, allows for a more nuanced representation of uncertainty compared to standard fuzzy sets. Cuong & Kreinovich (2013) introduced the picture fuzzy set (PFS) concept, which includes membership degree V(p), neutral membership degree K(p), and non-membership degree M(p), with the constraint that 0 ≤ V(p) + K(p) + M(p) ≤ 1. Garg (2017) further developed weighted averaging operations for PFS, and its applications in various decision-making fields have been extensively studied (Dutta, 2017). However, PFS encounters limitations in situations where V(p) + K(p) + M(p) ≥ 1, leading to inadequate outcomes. To address these challenges, Ashraf et al. (2019b) proposed the spherical fuzzy set (SFS), a variation of PFS, which enables more accurate and precise representation of uncertainty. SFS has been employed in diverse areas (Ashraf et al., 2019a), including COVID-19 (Ashraf et al., 2020) and healthcare diagnostics (Mahmood et al., 2019), establishing itself as a valuable tool in decision-making. Further extending this concept, Ullah et al. (2018) introduced T-spherical fuzzy set (T-SFS) for tackling multidimensional decision-making difficulties. The novel contribution of this paper lies in its exploration of C-PyFS. Unlike PyFS, C-PyFS incorporates a circular radius, enhancing the management of uncertainty in higher-dimensional spaces.
Prediction, defined as the process of deducing patterns or future occurrences from historical data, plays a crucial role in diverse fields such as marketing, economics, finance, and weather forecasting. The analysis of time-series data, which changes over time, is instrumental in addressing these predictive challenges. The concept of fuzzy time series, as delineated in Song & Chissom (1993b)’s definition , represents a significant advancement in this domain. Following their foundational work, Song & Chissom (1993a) and Song & Chissom (1994) utilized fuzzy sets for data projection, which was later refined by Joshi & Kumar (2012a). The exploration of fuzzy theory in data estimation has been pursued through various methodologies by numerous scholars, with notable contributions found in (Athar & Riaz, 2022; Farid & Riaz, 2023; Farid et al., 2023; Riaz & Farid, 2022; Riaz et al., 2022a; Riaz et al., 2022b) A majority of these studies have employed IFS, recognizing their utility in encapsulating uncertainty in fuzzy logic connections. However, only a select few forecasting models, notably those developed by Kumar & Gangwar (2015b) and Joshi & Kumar (2012b), have incorporated IFS (Gangwar & Kumar, 2014). The wind speed prediction model proposed by Jiang et al. (2019) has been widely adopted, with its effectiveness demonstrated using data from the University of Alabama (Cheng et al., 2008; Chou, 2011). In these studies, error comparisons between outcomes were conducted to identify the most effective forecasting strategy.
Building upon the IFS concept, Ashraf et al. (2023b) introduced the circular intuitionistic fuzzy set (C-IFS), which replaces points with circles centered at (ȷA(x), ℓA(x)). Each element of C-IFS is represented by a circle with a radius r ranging from 0 to 1 and centered at (ȷA(x), ℓA(x)). This innovative approach allows for a single total membership value within the C-IFS circle, offering a more comprehensive model for contradictory and ambiguous information. The C-IFS differentiates itself from regular IFS at r > 0, while at r = 0, it converges to a traditional IFS (Chen, 2023). This concept not only provides an enhanced understanding of membership but also enables decision-makers to construct grades as circular memberships within the C-IFS framework. Subsequent research on C-IFS has been applied to MCDM issues (Perçin, 2022; Khan et al., 2022), demonstrating its applicability and effectiveness. Building upon this, the concept of the circular and disc spherical fuzzy set emerged as a further evolution, encapsulating the advancements of previous methodologies (Ashraf et al., 2023a).
Historically, evidence assessment and rating have been fundamental in scientific decision-making. However, these methods have demonstrated limitations in projecting future values. Chen (1996) pioneered the use of time series analysis for enrollment forecasting, marking a significant shift in predictive methodologies. Following this, Kumar & Gangwar (2015a) introduced the concept of induced IFS to enhance forecasting capabilities. Further advancement was made by Abhishekh et al. (2018), who applied this technique to higher-order IFS. Despite these developments, a challenge persisted in determining the radius of a circle in PyFS, a crucial aspect for in-depth analysis. This gap led to the development of C-PyFS, representing a paradigm shift in prediction algorithms. C-PyFS uniquely handles membership forms, including circular radius, which diverges from traditional member representations. Particularly useful in scenarios where the sum of membership and non-membership is less than or equal to one with a circular radius, C-PyFTSs have shown efficacy in time series forecasting. The present study focuses on C-PyFTSs, aiming to reduce error rates in higher-order forecasting. This work exemplifies the application of the proposed method in forecasting Alzheimer’s disease indices. The study of these indices serves to deepen the understanding of the medical field, assisting in effective management and monitoring of patient conditions. Additionally, the findings offer governments valuable insights for informed decision-making, especially in healthcare management.
The structure of the remainder of this study is outlined as follows:
The application of fuzzy sets and C-PyFS in bridging the subsequent sections of the article is discussed.
Definitions pertinent to the proposed method are provided, including those for circular Pythagorean membership, non-membership, and radius values essential for score calculation.
Concepts pertaining to time-variant and time-invariant C-PyFTSs are introduced.
A detailed flowchart is presented, elucidating the proposed forecasting strategy and its application in data prediction.
The methodology is applied to Alzheimer’s disease data, with results tabulated for comprehensive analysis.
The study then extends to higher-order forecasting, building upon the initial findings.
The study concludes with a presentation of the overall findings and implications.
2. Preliminaries
This section succinctly delineates the foundational concepts of time series analysis, C-PyFSs, and fuzzy sets, which are instrumental in bridging to the subsequent section of the study.
Definition 2.1: The concept of Zadeh's fuzzy set is articulated as follows: Given a set Q, the fuzzy set Q within a universal set O is represented by:
$Q=\left\{\left\langle o, \mu_q(o)\right\rangle \mid \forall o \in O\right\}$
where, µq(o) is the membership function of the fuzzy set Q, mapping µq(o): Q → [0, 1]. This function quantifies the degree of membership of element o in Q.
Definition 2.2: Khan et al. (2023): Considering a nonempty set Ψ, a Pythagorean fuzzy set ξ within Ψ is defined as ξ = {⟨o, µξ(o), νξ(o)⟩; o $\in$ Ψ}, wherein the membership and non-membership degrees are determined by the functions µξ(o), νξ(o): → [0, 1], and for each element o $\in$ Ψ, it holds that 0 ≤ µξ2(o) + νξ2(o) ≤ 1.
Definition 2.3: Çakır et al. (2022): For a universal set Ψ, a C-PyFS ξ in Ψ is characterized as:
$\xi=\left\{\left\langle o, \mu_{\xi}(o), v_{\xi}(o) ; r\right\} \mid o \in \Psi\right\}$
where,
where, µξ: Ψ → [0, 1] and νξ: Ψ → [0, 1] describe the degrees of membership and non-membership, respectively, of the element o $\in$ Ψ. The distinctive feature of C-PyFS, denoted by r $\in$ [0,1], is the radius of a circle that encapsulates each component o $\in$ Ψ.
The degree of uncertainty in this context is computed using the formula:
Definition 2.4: Çakır et al. (2022): The operations constituting C-PyFS are defined as follows: For any two sets ˚A and Ø within C-PyFS (Ψ), it is established that:
$\stackrel{\circ}{A} \subseteq \emptyset\ \text{iff}\ o \in \Psi,\left(\mu_\stackrel{\circ}{A}(o) \leq \mu_{\emptyset}(o)\right.\ \text{and}\ \left.\nu_\stackrel{\circ}{A}(o) \geq \nu_{\emptyset}(o)\right)$;
$\stackrel{\circ}{A}=\emptyset\ \text{iff} \stackrel{\circ}{A} \subseteq \emptyset\ \text{and}\ \emptyset \subseteq\stackrel{\circ}{A}$;
$\stackrel{\circ}{A}^c=\left\{\left(o, \nu_\stackrel{\circ}{A}(o), \mu_\stackrel{\circ}{A}(o)\right)\right\}$;
$d(\stackrel{\circ}{A}, \emptyset)=\frac{1}{2}\left(\frac{r_{\stackrel{\circ}{A}}{-r_{\emptyset}}}{\sqrt{2}}+\sqrt{\frac{1}{2 k} \sum_{j=1}^k\left(\mu_{\stackrel{\circ}{A}}\left(o_j\right)-\mu_{\emptyset}\left(o_j\right)\right)^2+\left(\nu_{\stackrel{\circ}{A}}\left(o_j\right)-\nu_{\emptyset}\left(o_j\right)\right)^2+\left(\pi_{\stackrel{\circ}{A}}\left(o_j\right)-\pi_{\emptyset}\left(o_j\right)\right)^2}\right)$
where, d(˚A, Ø) is the standardized shortest distance between the sets ˚A and Ø.
Definition 2.5: If ϑ(e)(e = 0, 1, 2, ….,) is a subset of L and the universe of discourse upon which C-PyFS fk(e) = ⟨µξ(o), νξ(o); r⟩ (k = 1, 2, ....,) are defined, then F(e) = f1(o), f2(o) is a collection of fk(e) constructed to form C-PyFTSs on ϑ(e)(e = 0, 1, 2, ….,).
Definition 2.6: Given that L(e－1, e) represents a circular Pythagorean logical relationship, it is determined that V(e) = V(e－1)×L(e－1, e), where V(e) is influenced by V(e－1). This relationship is denoted as V(e－1) → V(e).
Definition 2.7: Assuming V(e) is influenced by V(e－1) and symbolized as V(e－1) → V(e), it follows that V(e) and V(e－1) share a circular Pythagorean relationship, expressed as V(e) = V(e－1)×L(e－1, e). If L(e－1, e) is independent of time e, V(e) is classified as a time-invariant circular Pythagorean time series, with L(e, e－1) = L(e －1, e－2) for all e. Conversely, V(e) is termed a time-variant circular Pythagorean time series when this condition is not met.
Definition 2.8: A circular Pythagorean logical relationship is defined as Ga → Gb, where V(e－1) = Ga and (e) = Gb, with Ga, Gb denoting the current and future states of the circular Pythagorean logical relations (C-PLRs). This set is represented as Ga1, Ga2, ......, Gan → Gb, where V(e－n) = Ga1, V(e－n+1) = Ga2, since V(e) is influenced by multiple C-PyFSs V(e－n), V(e－n+1), V(e－1), etc. Such relationships are termed higher-order circular Pythagorean time series.
3. An Algorithm of Handling Circular Pythagorean Time Series Forecasting
The proposed methodology encompasses three distinct segments (A, B, and C) for effectively addressing scenarios in C-PyFTSs. Initially, the establishment of circular Pythagorean logical relations and their groups is undertaken. Subsequently, the circular Pythagorean forecasting technique is applied to ascertain the anticipated value of the issue. Finally, the limitations of the approach are critically examined.
The following steps outline the process for constructing circular Pythagorean logical relations and their groups using the score formula:
Step I: The time series data are mapped to the specified range Ψ, defining the discourse universe as Ψ = [Amin － A1, Amax － A2]. Here, A1 and A2 are chosen positive values to accommodate the entire data time series, while Amin and Amax represent the smallest and largest data points in the time series, respectively.
Step II: The discourse universe Ψ is segmented into intervals of equal duration.
Step III: The value of ρv, the n-th circular Pythagorean fuzzy membership and non-membership, is determined based on the constructed intervals.
Step IV: The radius of a C-PyFS is computed using Eqs. (5) and (6).
Let the Pythagorean fuzzy pairings in a PyFS Ni be {⟨ci,1, di,1⟩⟨ci,2, di,2⟩, ....}, where i is the number of PyFS Ni, each of which includes λi. The arithmetic average of the Pythagorean fuzzy pairs is calculated as follows:
The radius is the greatest Euclidean distance in the set $\left\langle\mu_{\left(N_i\right)}, v_{\left(N_i\right)}\right\rangle$.
Step V: The score degree is calculated using the equation, and the highest value of score degree is selected:
where, p is a value between 0 and 1.
Step VI: The circular Pythagorean fuzzy logical relationships (C-PyFLRs) are formulated. C-PyFLRs are represented by ρa → ρb, where ρa is the C-PyFS of year y and ρb is the C-PyFS of the subsequent year y+1. Moreover, ρa denotes the present state, and ρb denotes the state that occurs next.
Step VII: Circular Pythagorean fuzzy logical relationship groups (C-PyFLRGs) are constructed based on the C-PyFLRs.
The process for ascertaining the forecasted values in C-PyFSs is described as follows:
In scenarios where the circular Pythagorean value of data ℘a is not influenced by any other circular Pythagorean values, the C-PyFLRGs of the corresponding value remain constant. In cases where the value dependent on ℘a cannot be determined, the circular Pythagorean value defaults to zero. If the circular Pythagorean value of data ℘a is derived from ℘b(℘b → ℘a), attention is directed to the C-PyFLRGs of ℘b.
If the C-PyFLRGs of ℘b are vacuous (℘b → ℘b), the forecasted value is identified as the center of ℘b.
In situations where the C-PyFLRGs of ℘b are one-to-one (℘b → ℘a), the forecasted value of ℘a is the median value.
For cases where the C-PyFLRGs of ℘b are not one-to-one (℘b → ℘a1, ℘a2, ……℘an), the forecasted value is the average of the median values of ℘a1, ℘a2, ..., ℘an.
The precision of time series forecasting is commonly evaluated using RMSE and AFE. The following definitions apply to these measures of forecasting accuracy:
RMSE $=\sqrt{\frac{\sum_{i=1}^n\left(O_i-F_i\right)^2}{\tau}}$
Forecasting percentage error $(œ)=\frac{\left|F_i-O_i\right|}{O_i} \times 100$
$\mathrm{AFE}=\frac{\sum(œ)}{\tau}$
In these formulations, Fi and Oi represent the forecasted and observed data points, respectively, within the time series. τ represents the total number of observations in the time series. A lower value of RMSE or AFE indicates enhanced accuracy in the forecasting method.
4. Implementation of the Proposed Method of Alzheimer’s Disease
This case study details the implementation of predictive analytics in a renowned medical department specializing in neurological disorders, with a focus on Alzheimer’s disease. The study demonstrates how the integration of advanced data analytics techniques has substantially improved the ability to predict daily patient numbers, providing insights into the disease and revolutionizing patient care and resource management.
Alzheimer's disease, a progressive neurodegenerative disorder, affects millions globally. In the context of a neurologically-focused medical department, the challenge was the efficient management of the influx of Alzheimer’s patients. The unpredictable nature of patient admissions complicated staff scheduling, resource allocation, and patient care planning. The application of predictive analytics was aimed at accurately forecasting the daily patient count.
The core aim of this case study is to illustrate how predictive analytics has transformed patient management approaches. By analyzing historical data and employing advanced modeling techniques, the study sought to forecast the daily number of Alzheimer’s patients. Table 1 presents a comparison between true patient numbers and forecasted values using circular Pythagorean fuzzy (C-PyF) values.
Date | True Value | C-PyF Value | Date | True Value | C-PyF Value |
01-11-2001 | 3929.69 | ℘1 | 03-12-2001 | 4646.61 | ℘6 |
02-11-2001 | 3998.48 | ℘1 | 04-12-2001 | 4766.43 | ℘7 |
05-11-2001 | 4080.51 | ℘1 | 05-12-2001 | 4924.56 | ℘8 |
06-11-2001 | 4082.92 | ℘1 | 06-12-2001 | 5208.86 | ℘11 |
07-11-2001 | 4158.15 | ℘2 | 07-12-2001 | 5333.93 | ℘12 |
08-11-2001 | 4135.03 | ℘2 | 10-12-2001 | 5321.28 | ℘12 |
09-11-2001 | 4123.78 | ℘2 | 11-12-2001 | 5273.97 | ℘11 |
12-11-2001 | 4172.63 | ℘2 | 12-12-2001 | 5539.31 | ℘13 |
13-11-2001 | 4136.54 | ℘2 | 13-12-2001 | 5407.54 | ℘12 |
14-11-2001 | 4277.70 | ℘3 | 14-12-2001 | 5486.73 | ℘13 |
15-11-2001 | 4403.59 | ℘4 | 17-12-2001 | 5456.15 | ℘13 |
16-11-2001 | 4446.62 | ℘4 | 18-12-2001 | 5329.19 | ℘12 |
19-11-2001 | 4548.63 | ℘5 | 19-12-2001 | 5221.96 | ℘11 |
20-11-2001 | 4455.80 | ℘4 | 20-12-2001 | 5309.10 | ℘12 |
21-11-2001 | 4533.37 | ℘5 | 21-12-2001 | 5109.24 | ℘10 |
22-11-2001 | 4450.02 | ℘4 | 24-12-2001 | 5164.73 | ℘10 |
23-11-2001 | 4519.08 | ℘5 | 25-12-2001 | 5372.81 | ℘12 |
26-11-2001 | 4608.32 | ℘6 | 26-12-2001 | 5392.43 | ℘12 |
27-11-2001 | 4580.33 | ℘6 | 27-12-2001 | 5332.98 | ℘12 |
28-11-2001 | 4447.58 | ℘4 | 28-12-2001 | 5398.28 | ℘12 |
29-11-2001 | 4465.83 | ℘5 | 31-12-2001 | 5551.24 | ℘14 |
30-11-2001 | 4441.12 | ℘4 |
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This segment delineates the application of the developed approach to Alzheimer's disease data from 2001, providing a systematic explanation of the results for easier interpretation and validation of the model. The methodology is outlined in the following steps:
Step I: Definition of the discourse universe
The discourse universe Ψ for the 2001 Alzheimer's patient data is defined as [3920, 5600]. This range is determined using the minimum (Amin) and maximum (Amax) values from Table 1, adjusted by two chosen positive numbers A1 = 9.69 and A2 = 48.76.
Step II: Segmentation of the discourse universe
The universe Ψ is divided into 14 intervals, denoted as ħv = [3920 + (v − 1)p, 3920 + vp], v = 1, 2, 3,....14 and p = 120.
Step III: Establishment of C-PyFTS
Fourteen C-PyFTS, ℘v(v = 1, 2, 3, ....12), are established within the discourse universe based on the interval ħv. The C-PyFTS are determined as follows:
℘v = [3920+(v−1)p, 3920+vp, 3920+(i+1)p] for v = 1, 2, 3……13 where p = 120
℘v = [3920+(v−1)p, 3920+vp , 3920+ip] for v = 14 where p = 120
Membership and non-membership values to C-PyFSs are calculated using Eqs. (3) and (4), assuming ϵ = 0.001.
℘1 = {(3929.69, 0.08, 0.92), (3998.48, 0.65, 0.35), (4080.51, 0.66, 0.34), (4082.92, 0.64, 0.36), (4158.15, 0.01, 0.99), (4135.03, 0.21, 0.79), (4123.78, 0.30, 0.70), (4136.54, 0.19, 0.81)}
℘2 = {(4080.51, 0.34, 0.66), (4082.92, 0.36, 0.64), (4158.15, 0.97, 0.03), (4135.03, 0.79, 0.21), (4123.78, 0.70, 0.30), (4172.63, 0.89, 0.11), (4136.54, 0.80, 0.20), (4277.70, 0.02, 0.98)}
℘3 = {(4172.63, 0.10, 0.90), (4277.70, 0.98, 0.02)}
℘4 = {(4403.59, 0.97, 0.03), (4446.62, 0.61, 0.39), (4455.80, 0.53, 0.47), (4450.02, 0.58, 0.42), (4159.08, 0.01, 0.99), (4447.58, 0.60, 0.40), (4465.83, 0.45, 0.55), (4441.12, 0.66, 0.34)}
℘5 = {(4403.59, 0.03, 0.97), (4446.62, 0.39, 0.61), (4548.63, 0.76, 0.53), (4455.80, 0.46, 0.54), (4533.37, 0.89, 0.11), (4450.02, 0.42, 0.58), (4519.08, 0.99, 0.01), (4608.32, 0.26, 0.74), (4580.33, 0.50, 0.50), (4447.58, 0.40, 0.60), (4465.83, 0.55, 0.45), (4441.61, 0.35, 0.65)}
℘6 = {(4548.63, 0.24, 0.76), (4533.37, 0.11, 0.89), (4608.32, 0.73, 0.27), (4580.33, 0.50, 0.50), (4646.61, 0.94, 0.06)}
℘7 = {(4646.61, 0.05, 0.95), (4766.43, 0.95, 0.05)}
℘8 = {(4766.43, 0.05, 0.95), (4924.56, 0.63, 0.37)}
℘9 = {(4924.56, 0.37, 0.63), (5109.24, 0.09, 0.91)}
℘10 = {(5208.86, 0.26, 0.74), (5221.96, 0.15, 0.85), (5109.24, 0.91, 0.09), (5164.73, 0.63, 0.37)
℘11 = {(5208.86, 0.74, 0.26), (5333.93, 0.22, 0.78), (5329.19, 0.26, 0.74), (5221.96, 0.85, 0.15), (5309.10, 0.42, 0.58), (5164.73, 0.37, 0.63), (5332.98, 0.22, 0.78), (5273.97, 0.72, 0.28), (5321.28, 0.32, 0.68)}
℘12 = {(5333.93, 0.78, 0.22), (5407.54, 0.60, 0.40), (5456.15, 0.20, 0.80), (5329.19, 0.74, 0.26), (5309.10, 0.57, 0.43), (5372.81, 0.89, 0.11), (5392.43, 0.73, 0.27), (5332.98, 0.77, 0.23), (5398.28, 0.68, 0.32), (5321.28, 0.68, 0.32), (5273.97, 0.28, 0.72)}
℘13 = {(5407.54, 0.40, 0.60), (5486.73, 0.94, 0.06), (5456.15, 0.80, 0.20), (5372.81, 0.11, 0.89), (5392.43, 0.27, 0.73), (5398.28, 0.32, 0.68), (5551.24, 0.41, 0.59), (5539.31, 0.50, 0.50)
℘14 = {(5486.73, 0.06, 0.94), (5551.24, 0.59, 0.41), (5539.31, 0.49, 0.51)
Step IV: Calculation of the radius of C-PyFSs
The radius for each C-PyFS is calculated utilizing Eqs. (5) and (6).
℘1 = {(3929.69 (0.08, 0.92; 0.37)), (3998.48 (0.65, 0.35; 0.44)), (4080.51 (0.66, 0.34; 0.45)), (4082.92 (0.64, 0.36; 0.42)), (4158.15 (0.01, 0.99; 0.47)), (4135.03 (0.21, 0.79; 0.19)), (4123.78 (0.30, 0.70; 0.06)), (4136.54 (0.19, 0.81; 0.21))}
℘2 = {(4080.51 (0.34, 0.66; 0.39)), (4082.92 (0.36, 0.64; 0.36)), (4158.15 (0.97, 0.03; 0.52)), (4135.03 (0.79, 0.21; 0.26)), (4123.78 (0.70, 0.30; 0.12)), (4172.63 (0.89, 0.11; 0.40)), (4136.54 (0.80, 0.20; 0.28)), (4277.70 (0.02, 0.98; 0.84))}
℘3 = {(4172.63 (0.10, 0.90; 0.62)), (4277.70 (0.98, 0.02, 0.62))}
℘4 = {(4403.59 (0.97, 0.03; 0.59)), (4446.62 (0.61, 0.39; 0.08)), (4455.80 (0.53, 0.47; 0.02)), (4450.02 (0.58, 0.42; 0.04)), (4159.08 (0.01, 0.99; 0.77)), (4447.58 (0.60, 0.40; 0.07)), (4465.83 (0.45, 0.55; 0.14)), (4441.12 (0.66, 0.34: 0.15))}
℘5 = {(4403.59 (0.03, 0.97; 0.65)), (4446.62 (0.39, 0.61; 0.14)), (4548.63 (0.76, 0.53; 0.26)), (4455.80 (0.46, 0.54; 0.04)), (4533.37 (0.89, 0.11; 0.57)), (4450.02 (0.42, 0.58; 0.10)), (4519.08 (0.99, 0.01; 0.71)), (4608.32 (0.26, 0.74; 0.32)), (4580.33 (0.50, 0.50; 0.02)), (4447.58 (0.40, 0.60 : 0.13)), (4465.83 (0.55, 0.45; 0.09)), (4441.61 (0.35, 0.65; 0.20))}
℘6 = {(4548.63 (0.24, 0.76; 0.38)), (4533.37 (0.11, 0.89; 0.56)), (4608.32 (0.73, 0.27; 0.32)), (4580.33 (0.50, 0.50; 0.01)), (4646.61 (0.94, 0.06; 0.62))}
℘7 = {(4646.61 (0.05, 0.95; 0.63)), (4766.43 (0.95, 0.05; 0.63))}
℘8 = {(4766.43 (0.05, 0.95; 0.41)), (4924.56 (0.63, 0.37; 0.41))}
℘9 = {(4924.56 (0.37, 0.63; 0.20)), (5109.24 (0.09, 0.91; 0.20))}
℘10 = {(5208.86 (0.26, 0.74; 0.32)), (5221.96 (0.15, 0.85; 0.48)), (5109.24 (0.91, 0.09; 0.60)), (5164.73 (0.63, 0.37; 0.20))
℘11 = {(5208.86 (0.74, 0.26; 0.40)), (5333.93 (0.22, 0.78; 0.34)), (5329.19 (0.26, 0.74; 0.29)), (5221.96 (0.85, 0.15; 0.55)), (5309.10 (0.42, 0.58; 0.05)), (5164.73 (0.37, 0.63; 0.12)), (5332.98 (0.22, 0.78; 0.33)), (5273.97 (0.72, 0.28; 0.37)), (5321.28 (0.32, 0.68; 0.19))}
℘12 = {(5333.93 (0.78, 0.22; 0.13)), (5407.54 (0.60, 0.40; 0.13)), (5456.15 (0.20, 0.80; 0.70)), (5329.19 (0.74, 0.26; 0.07)), (5309.10 (0.57, 0.43; 0.17)), (5372.81 (0.89, 0.11; 0.28)), (5392.43 (0.73, 0.27; 0.05)), (5332.98 (0.77, 0.23; 0.12)), (5398.28 (0.68, 0.32; 0.02)), (5321.28 (0.68, 0.32; 0.02)), (5273.97 (0.28, 0.72; 0.58))}
℘13 = {(5407.54 (0.40, 0.60; 0.10)), (5486.73 (0.94, 0.06; 0.67)), (5456.15 (0.80, 0.20; 0.47)), (5372.81 (0.11, 0.89; 0.51)), (5392.43 (0.27, 0.73; 0.28)), (5398.28 (0.32, 0.68; 0.21)), (5551.24 (0.41, 0.59; 0.09)), (5539.31 (0.50, 0.50; 0.05))
℘14 = {(5486.73 (0.06, 0.94; 0.46)), (5551.24 (0.59, 0.41; 0.30)), (5539.31 (0.49, 0.51; 0.16))
Step V: Computation of the C-PyFS score value
The score value for each data point within the C-PyFS is calculated using Eq. (7). For instance, the score degree for the actual value of 4080.51, located between the circular Pythagorean membership and non-membership values of ℘1 and ℘2, is ascertained.
In the process of determining the highest score degree for the specific data point of 4080.51, the methodological steps outlined in Step IV were employed. These steps involve the calculation of membership, non-membership, and radii values for the C-PyFS. For the purpose of this calculation, it is assumed that the parameter p has a value of 0.5
$\mu_{\wp_1}(4080.51)=0.66, \nu_{\wp_1}(4080.51)=0.34, r_{\wp_1}(4080.51)=0.45$
Similarly,
$\mu_{\wp_2}(4080.51)=0.34, \nu_{\wp_2}(4080.51)=0.66, r_{\wp_2}(4080.51)=0.39$
The score degree for the data point 4080.51 was then calculated for both ℘1 and ℘2. Then the larger value was selected.
$\zeta_{\wp_1}(4080.51)=\frac{1}{3}(0.66-0.34+\sqrt{2 \times 0.45}(2 \times 0.5-1))=0.11$
$\zeta_{\wp_1}(4080.51)=\frac{1}{3}(0.34-0.66+\sqrt{2 \times 0.39}(2 \times 0.5-1))=-0.11$
Given that ℘1 exhibited a higher score degree than ℘2, ℘1 was determined to be the circular Pythagorean value for the data point 4080.51. Subsequent calculations followed a similar procedure for the remaining data points.
Step VI: Formulation of circular Pythagorean logical relationships (C-PLRs)
C-PLRs are established and presented in Table 2.
℘1→℘1 | ℘1→℘1 | ℘1→℘1 | ℘1→℘2 | ℘2→℘2 | ℘2→℘2 | ℘2→℘2 |
℘2→℘2 | ℘2→℘3 | ℘3→℘4 | ℘4→℘4 | ℘4→℘5 | ℘5→℘4 | ℘4→℘5 |
℘5→℘4 | ℘4→℘5 | ℘5→℘6 | ℘6→℘6 | ℘6→℘4 | ℘4→℘5 | ℘5→℘4 |
℘4→℘6 | ℘6→℘7 | ℘7→℘8 | ℘8→℘11 | ℘11→℘12 | ℘12→℘12 | ℘12→℘11 |
℘11→℘13 | ℘13→℘12 | ℘12→℘13 | ℘13→℘13 | ℘13→℘12 | ℘12→℘11 | ℘11→℘12 |
℘12→℘10 | ℘10→℘10 | ℘10→℘12 | ℘12→℘12 | ℘12→℘12 | ℘12→℘12 | ℘12→℘14 |
Step VII: Creation of C-PyFLRGs
Building upon the C-PLRs, C-PyFLRGs are developed. These groups are displayed in Table 3.
℘1→℘1 | ℘1→℘1 | ℘1→℘1 | ℘1→℘2 |
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℘2→℘2 | ℘2→℘2 | ℘2→℘2 | ℘2→℘2 | ℘2→℘3 |
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℘3→℘4 |
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℘4→℘4 | ℘4→℘5 | ℘4→℘5 | ℘4→℘5 | ℘4→℘5 | ℘4→℘6 |
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℘5→℘4 | ℘5→℘4 | ℘5→℘6 | ℘5→℘4 |
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℘6→℘6 | ℘6→℘4 | ℘6→℘7 |
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℘7→℘8 |
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℘8→℘11 |
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℘10→℘10 | ℘10→℘12 |
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℘11→℘12 | ℘11→℘13 | ℘11→℘12 |
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℘12→℘12 | ℘12→℘11 | ℘12→℘13 | ℘12→℘11 | ℘12→℘12 | ℘12→℘12 | ℘12→℘12 | ℘12→℘12 | ℘12→℘14 |
℘13→℘12 | ℘13→℘13 | ℘13→℘12 |
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Table 4 presents the forecasted values derived from the C-PyFSs model. Due to the absence of an initial value on November 1, 2001, the model was unable to generate a forecast for that date. Subsequently, forecasted values for the following days were computed using the established methodology.
Date | True Value | Forecasted Value | Year | True Value | Forecasted Value |
01-11-2001 | 3929.69 | − | 03-12-2001 | 4646.61 | 4520 |
02-11-2001 | 3998.48 | 4100 | 04-12-2001 | 4766.43 | 4600 |
05-11-2001 | 4080.51 | 4100 | 05-12-2001 | 4924.56 | 4880 |
06-11-2001 | 4082.92 | 4100 | 06-12-2001 | 5208.86 | 5240 |
07-11-2001 | 4158.15 | 4100 | 07-12-2001 | 5333.93 | 5360 |
08-11-2001 | 4135.03 | 4220 | 10-12-2001 | 5321.28 | 5420 |
09-11-2001 | 4123.78 | 4220 | 11-12-2001 | 5273.97 | 5420 |
12-11-2001 | 4172.63 | 4220 | 12-12-2001 | 5539.31 | 5420 |
13-11-2001 | 4136.54 | 4220 | 13-12-2001 | 5407.54 | 5420 |
14-11-2001 | 4277.70 | 4220 | 14-12-2001 | 5486.73 | 5420 |
15-11-2001 | 4403.59 | 4400 | 17-12-2001 | 5456.15 | 5420 |
16-11-2001 | 4446.62 | 4520 | 18-12-2001 | 5329.19 | 5420 |
19-11-2001 | 4548.63 | 4520 | 19-12-2001 | 5221.96 | 5420 |
20-11-2001 | 4455.80 | 4520 | 20-12-2001 | 5309.10 | 5420 |
21-11-2001 | 4533.37 | 4520 | 21-12-2001 | 5109.24 | 5420 |
22-11-2001 | 4450.02 | 4520 | 24-12-2001 | 5164.73 | 5240 |
23-11-2001 | 4519.08 | 4520 | 25-12-2001 | 5372.81 | 5240 |
26-11-2001 | 4608.32 | 4520 | 26-12-2001 | 5392.43 | 5420 |
27-11-2001 | 4580.33 | 4600 | 27-12-2001 | 5332.98 | 5420 |
28-11-2001 | 4447.58 | 4600 | 28-12-2001 | 5398.28 | 5420 |
29-11-2001 | 4465.83 | 4520 | 31-12-2001 | 5551.24 | 5420 |
30-11-2001 | 4441.12 | 4520 |
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Figure 1 depicts a graphical representation of both the actual and forecasted values related to Alzheimer’s disease cases.
5. Circular Pythagorean Logical Relationships (C-PLRs) of Order II
In this section, the methodology extends to constructing the C-PLRs and their corresponding groups for second-order forecasting in Alzheimer’s disease. The Table 5 delineates the C-PLRs for order II Alzheimer’s disease forecasting.
℘1, ℘1→℘1 | ℘1, ℘1→℘1 | ℘1, ℘1→℘2 | ℘1, ℘2→℘2 | ℘2, ℘2→℘2 | ℘2, ℘2→℘2 |
℘2, ℘2→℘2 | ℘2, ℘2→℘3 | ℘2, ℘3→℘4 | ℘3, ℘4→℘4 | ℘4, ℘4→℘5 | ℘4, ℘5→℘4 |
℘5, ℘4→℘5 | ℘4, ℘5→℘4 | ℘5, ℘4→℘5 | ℘4, ℘5→℘6 | ℘5, ℘6→℘6 | ℘6, ℘6→℘4 |
℘6, ℘4→℘5 | ℘4, ℘5→℘4 | ℘5, ℘4→℘6 | ℘4, ℘6→℘7 | ℘6, ℘7→℘8 | ℘7, ℘8→℘11 |
℘8, ℘11→℘12 | ℘11, ℘12→℘12 | ℘12, ℘12→℘11 | ℘12, ℘11→℘13 | ℘11, ℘13→℘12 | ℘13, ℘12→℘13 |
℘12, ℘13→℘13 | ℘13, ℘13→℘12 | ℘13, ℘12→℘11 | ℘12, ℘11→℘12 | ℘11, ℘12→℘10 | ℘12, ℘10→℘10 |
℘10, ℘10→℘12 | ℘10, ℘12→℘12 | ℘12, ℘12→℘12 | ℘12, ℘12→℘12 | ℘12, ℘12→℘14 |
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Based on the C-PLRs, Table 6 presents the C-PyFLRGs for order II.
℘1, ℘1→℘1 | ℘1, ℘1→℘1 | ℘1, ℘1→℘2 | ℘1, ℘2→℘2 | |||
℘2, ℘2→℘2 | ℘2, ℘2→℘2 | ℘2, ℘2→℘2 | ℘2, ℘2→℘3 | ℘2, ℘3→℘4 | ||
℘3, ℘4→℘4 |
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| ℘4, ℘4→℘5 | |||
℘4, ℘5→℘4 | ℘4, ℘5→℘4 | ℘4, ℘5→℘6 | ℘4, ℘5→℘4 | ℘5, ℘4→℘5 | ℘5, ℘4→℘5 | ℘5, ℘4→℘6 |
℘5, ℘6→℘6 |
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| ℘6, ℘6→℘4 |
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℘6, ℘4→℘5 |
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| ℘4, ℘6→℘7 |
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℘6, ℘7→℘8 |
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| ℘7, ℘8→℘11 |
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℘8, ℘11→℘12 |
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| ℘11, ℘11→℘12 | ℘11, ℘12→℘10 |
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℘12, ℘12→℘11 | ℘12, ℘12→℘12 | ℘12, ℘12→℘12 | ℘12, ℘12→℘14 | ℘12, ℘11→℘13 | ℘12, ℘11→℘12 |
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℘11, ℘13→℘12 |
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| ℘13, ℘12→℘13 | ℘13, ℘12→℘11 |
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℘12, ℘13→℘13 |
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| ℘13, ℘13→℘12 |
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℘12, ℘10→℘10 |
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| ℘10, ℘10→℘12 |
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℘10, ℘10→℘12 |
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Table 7 illustrates the forecasted values for Alzheimer's disease, computed using the second-order C-PyFSs as outlined in Section B.
Date | True Value | Forecasted Value | Years | True Value | Forecasted Value |
01-11-2001 | 3929.69 | − | 03-12-2001 | 4646.61 | 4580 |
02-11-2001 | 3998.48 | − | 04-12-2001 | 4766.43 | 4760 |
05-11-2001 | 4080.51 | 4100 | 05-12-2001 | 4924.56 | 4880 |
06-11-2001 | 4082.92 | 4100 | 06-12-2001 | 5208.86 | 5240 |
07-11-2001 | 4158.15 | 4100 | 07-12-2001 | 5333.93 | 5360 |
08-11-2001 | 4135.03 | 4160 | 10-12-2001 | 5321.2 | 5240 |
09-11-2001 | 4123.78 | 4220 | 11-12-2001 | 5273.97 | 5400 |
12-11-2001 | 4172.63 | 4220 | 12-12-2001 | 5539.31 | 5420 |
13-11-2001 | 4136.54 | 4220 | 13-12-2001 | 5407.54 | 5360 |
14-11-2001 | 4277.70 | 4220 | 14-12-2001 | 5486.73 | 5360 |
15-11-2001 | 4403.59 | 4400 | 17-12-2001 | 5456.15 | 5480 |
16-11-2001 | 4446.62 | 4400 | 18-12-2001 | 5329.19 | 5360 |
19-11-2001 | 4548.63 | 4520 | 19-12-2001 | 5221.96 | 5360 |
20-11-2001 | 4455.80 | 4520 | 20-12-2001 | 5309.10 | 5420 |
21-11-2001 | 4533.37 | 4580 | 21-12-2001 | 5109.24 | 5420 |
22-11-2001 | 4450.02 | 4520 | 24-12-2001 | 5164.73 | 5120 |
23-11-2001 | 4519.08 | 4580 | 25-12-2001 | 5372.81 | 5360 |
26-11-2001 | 4608.32 | 4520 | 26-12-2001 | 5392.43 | 5360 |
27-11-2001 | 4580.33 | 4640 | 27-12-2001 | 5332.98 | 5400 |
28-11-2001 | 4447.58 | 4400 | 28-12-2001 | 5398.28 | 5400 |
29-11-2001 | 4465.83 | 4520 | 31-12-2001 | 5551.24 | 5400 |
30-11-2001 | 4441.12 | 4520 |
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A graphical representation, Figure 2, compares the actual values with the forecasted values for Alzheimer’s disease using the second-order C-PyFS approach.
To assess the accuracy of the forecasts, Table 8 presents the calculations for RMSE and AFE. These metrics are crucial for evaluating the precision of the forecasting method.
Tools | Proposed Method (Order I) | Proposed Method (Order II) |
MSE | 98.03 | 84.61 |
AFE | 1.61 | 1.34 |
6. Discussion
The results delineated in Table 8 articulate a comparative analysis between first- and second-order C-PyFTSs forecasting. It has been observed that the second-order C-PyFTS forecasting demonstrates a superior performance over the first-order model, as evidenced by the calculated error rates using established error measurement formulas.
A notable trend is observed in the forecasting accuracy: higher-order C-PyFTS models tend to yield lower error rates. This pattern holds for the third-order forecasting error, which is smaller than that of the second-order. This indicates that, generally, as the order increases, the accuracy of the C-PyFTS model improves, suggesting a more refined prediction capability. However, it is crucial to underscore that the quality and completeness of the data play a pivotal role in enhancing forecast accuracy, alongside the chosen forecasting methodology.
The RMSE value distinctly validates the efficacy of the proposed algorithm for addressing complex forecasting scenarios. The accuracy of the forecasting method, as manifested in the error metrics, has significant practical implications. Specifically, in the context of patient care within the neurological department, the application of the predictive analytics model enabled proactive patient management and optimized day-to-day operational efficiency. Anticipating patient inflow facilitated more effective resource allocation, ensuring optimal patient care.
The significance of the radius in C-PyFS extends beyond its traditional role in membership and non-membership determination. In C-PyFS, the radius is instrumental in influencing the overall dimensions and configuration of the fuzzy set. This, in turn, impacts the set’s ability to represent intricate and uncertain data comprehensively, enhancing the model's adaptability and interpretability in handling complex fuzzy logic problems.
7. Conclusions
The study presented herein demonstrates the increasing preference for C-PyFSs when dealing with scenarios where the sum of membership and non-membership degrees is one or less. It has been discerned that traditional PyFSs are inadequate in addressing such cases, leading to the utilization of C-PyFSs in instances where the aggregate of membership and non-membership values equals one. The proposed approach utilizing C-PyFSs has been identified as less complex and more straightforward, primarily due to the adoption of a simplified scoring formula. This methodology was applied to forecast the indices of Alzheimer’s disease, demonstrating its utility in predicting data using the established criteria. Furthermore, the extension of this approach to higher-order forecasts revealed that higher-order predictions are characterized by reduced errors, thereby enhancing their utility in future value estimations.
The application of the recommended strategy yielded predictions for the ensuing years, indicating its potential for extensive use in various forecasting scenarios. Future research avenues may explore the application of C-PyFSs across diverse time-series forecasting problems, comparing their efficacy against existing methodologies. Such investigations could offer additional insights and enhancements to the forecasting process, broadening the scope and applicability of C-PyFSs in diverse research and practical domains.
This article does not contain any studies with human participants or animals performed by any of the authors.
The data used to support the research findings are available from the corresponding author upon request.
The authors declare no conflict of interest.
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