Enhanced Forecasting of Alzheimer’s Disease Progression Using HigherOrder Circular Pythagorean Fuzzy Time Series
Abstract:
This study introduces an advanced forecasting method, utilizing a higherorder circular Pythagorean fuzzy time series (CPyFTSs) 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 (CPyFS) 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 higherorder 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
Decisionmaking, 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 problemsolving 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 decisionmaking 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 nonmembership. 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 nonmembership degree M(p), adhering to the constraint that 0 ≤ V(p) + M(p) ≤ 1. The utility of IFS in various realworld 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 decisionmaking contexts (Tan, 2011; Xu, 2011).
Addressing scenarios where the sum of membership and nonmembership 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 nonmembership 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 decisionmaking 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 COVID19 (Ashraf et al., 2020) and healthcare diagnostics (Mahmood et al., 2019), establishing itself as a valuable tool in decisionmaking. Further extending this concept, Ullah et al. (2018) introduced Tspherical fuzzy set (TSFS) for tackling multidimensional decisionmaking difficulties. The novel contribution of this paper lies in its exploration of CPyFS. Unlike PyFS, CPyFS incorporates a circular radius, enhancing the management of uncertainty in higherdimensional 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 timeseries 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 (CIFS), which replaces points with circles centered at (ȷA(x), ℓA(x)). Each element of CIFS 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 CIFS circle, offering a more comprehensive model for contradictory and ambiguous information. The CIFS 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 decisionmakers to construct grades as circular memberships within the CIFS framework. Subsequent research on CIFS 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 decisionmaking. 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 higherorder IFS. Despite these developments, a challenge persisted in determining the radius of a circle in PyFS, a crucial aspect for indepth analysis. This gap led to the development of CPyFS, representing a paradigm shift in prediction algorithms. CPyFS uniquely handles membership forms, including circular radius, which diverges from traditional member representations. Particularly useful in scenarios where the sum of membership and nonmembership is less than or equal to one with a circular radius, CPyFTSs have shown efficacy in time series forecasting. The present study focuses on CPyFTSs, aiming to reduce error rates in higherorder 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 decisionmaking, especially in healthcare management.
The structure of the remainder of this study is outlined as follows:
The application of fuzzy sets and CPyFS in bridging the subsequent sections of the article is discussed.
Definitions pertinent to the proposed method are provided, including those for circular Pythagorean membership, nonmembership, and radius values essential for score calculation.
Concepts pertaining to timevariant and timeinvariant CPyFTSs 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 higherorder 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, CPyFSs, 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, µ_{q}(o) is the membership function of the fuzzy set Q, mapping µ_{q}(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, µ_{ξ}(o), ν_{ξ}(o)⟩; o $\in$ Ψ}, wherein the membership and nonmembership degrees are determined by the functions µ_{ξ}(o), ν_{ξ}(o): → [0, 1], and for each element o $\in$ Ψ, it holds that 0 ≤ µ_{ξ}^{2}(o) + ν_{ξ}^{2}(o) ≤ 1.
Definition 2.3: Çakır et al. (2022): For a universal set Ψ, a CPyFS 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, µ_{ξ}: Ψ → [0, 1] and ν_{ξ}: Ψ → [0, 1] describe the degrees of membership and nonmembership, respectively, of the element o $\in$ Ψ. The distinctive feature of CPyFS, 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 CPyFS are defined as follows: For any two sets ˚A and Ø within CPyFS (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 CPyFS f_{k}(e) = ⟨µ_{ξ}(o), ν_{ξ}(o); r⟩ (k = 1, 2, ....,) are defined, then F(e) = f_{1}(o), f_{2}(o) is a collection of f_{k}(e) constructed to form CPyFTSs 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 timeinvariant circular Pythagorean time series, with L(e, e－1) = L(e －1, e－2) for all e. Conversely, V(e) is termed a timevariant circular Pythagorean time series when this condition is not met.
Definition 2.8: A circular Pythagorean logical relationship is defined as G_{a }→ G_{b}, where V(e－1) = G_{a} and (e) = G_{b}, with G_{a}, G_{b} denoting the current and future states of the circular Pythagorean logical relations (CPLRs). This set is represented as G_{a1}, G_{a2}, ......, G_{an}_{ }→ G_{b}, where V(e－n) = G_{a1}, V(e－n+1) = G_{a2}, since V(e) is influenced by multiple CPyFSs V(e－n), V(e－n+1), V(e－1), etc. Such relationships are termed higherorder 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 CPyFTSs. 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 = [A_{min }－ A_{1}, A_{max }－ A_{2}]. Here, A_{1} and A_{2} 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 ρ_{v}, the nth circular Pythagorean fuzzy membership and nonmembership, is determined based on the constructed intervals.
Step IV: The radius of a CPyFS is computed using Eqs. (5) and (6).
Let the Pythagorean fuzzy pairings in a PyFS N_{i} be {⟨c_{i,}_{1}, d_{i,}_{1}⟩⟨c_{i,}_{2}, d_{i,}_{2}⟩, ....}, where i is the number of PyFS N_{i}, each of which includes λ_{i}. 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 (CPyFLRs) are formulated. CPyFLRs are represented by ρ_{a }→ ρ_{b}, where ρ_{a} is the CPyFS of year y and ρ_{b} is the CPyFS of the subsequent year y+1. Moreover, ρ_{a} denotes the present state, and ρ_{b} denotes the state that occurs next.
Step VII: Circular Pythagorean fuzzy logical relationship groups (CPyFLRGs) are constructed based on the CPyFLRs.
The process for ascertaining the forecasted values in CPyFSs is described as follows:
In scenarios where the circular Pythagorean value of data ℘_{a} is not influenced by any other circular Pythagorean values, the CPyFLRGs 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 CPyFLRGs of ℘_{b}.
If the CPyFLRGs of ℘_{b} are vacuous (℘_{b }→ ℘_{b}), the forecasted value is identified as the center of ℘_{b}.
In situations where the CPyFLRGs of ℘_{b} are onetoone (℘_{b }→ ℘_{a}), the forecasted value of ℘_{a} is the median value.
For cases where the CPyFLRGs of ℘_{b} are not onetoone (℘_{b }→ ℘_{a}_{1}, ℘_{a}_{2}, ……℘_{an}), the forecasted value is the average of the median values of ℘_{a}_{1}, ℘_{a}_{2}, ..., ℘_{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 $=\sqrt{\frac{\sum_{i=1}^n\left(O_iF_i\right)^2}{\tau}}$
Forecasting percentage error $(œ)=\frac{\leftF_iO_i\right}{O_i} \times 100$
$\mathrm{AFE}=\frac{\sum(œ)}{\tau}$
In these formulations, F_{i} and O_{i} 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 neurologicallyfocused 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 (CPyF) values.
Date  True Value  CPyF Value  Date  True Value  CPyF Value 
01112001  3929.69  ℘_{1}  03122001  4646.61  ℘_{6} 
02112001  3998.48  ℘_{1}  04122001  4766.43  ℘_{7} 
05112001  4080.51  ℘_{1}  05122001  4924.56  ℘_{8} 
06112001  4082.92  ℘_{1}  06122001  5208.86  ℘_{11} 
07112001  4158.15  ℘_{2}  07122001  5333.93  ℘_{12} 
08112001  4135.03  ℘_{2}  10122001  5321.28  ℘_{12} 
09112001  4123.78  ℘_{2}  11122001  5273.97  ℘_{11} 
12112001  4172.63  ℘_{2}  12122001  5539.31  ℘_{13} 
13112001  4136.54  ℘_{2}  13122001  5407.54  ℘_{12} 
14112001  4277.70  ℘_{3}  14122001  5486.73  ℘_{13} 
15112001  4403.59  ℘_{4}  17122001  5456.15  ℘_{13} 
16112001  4446.62  ℘_{4}  18122001  5329.19  ℘_{12} 
19112001  4548.63  ℘_{5}  19122001  5221.96  ℘_{11} 
20112001  4455.80  ℘_{4}  20122001  5309.10  ℘_{12} 
21112001  4533.37  ℘_{5}  21122001  5109.24  ℘_{10} 
22112001  4450.02  ℘_{4}  24122001  5164.73  ℘_{10} 
23112001  4519.08  ℘_{5}  25122001  5372.81  ℘_{12} 
26112001  4608.32  ℘_{6}  26122001  5392.43  ℘_{12} 
27112001  4580.33  ℘_{6}  27122001  5332.98  ℘_{12} 
28112001  4447.58  ℘_{4}  28122001  5398.28  ℘_{12} 
29112001  4465.83  ℘_{5}  31122001  5551.24  ℘_{14} 
30112001  4441.12  ℘_{4} 



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 (A_{min}) and maximum (A_{max}) values from Table 1, adjusted by two chosen positive numbers A_{1} = 9.69 and A_{2} = 48.76.
Step II: Segmentation of the discourse universe
The universe Ψ is divided into 14 intervals, denoted as ħ_{v} = [3920 + (v − 1)p, 3920 + vp], v = 1, 2, 3,....14 and p = 120.
Step III: Establishment of CPyFTS
Fourteen CPyFTS, ℘_{v}(v = 1, 2, 3, ....12), are established within the discourse universe based on the interval ħ_{v}. The CPyFTS 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 nonmembership values to CPyFSs 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 CPyFSs
The radius for each CPyFS 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 CPyFS score value
The score value for each data point within the CPyFS is calculated using Eq. (7). For instance, the score degree for the actual value of 4080.51, located between the circular Pythagorean membership and nonmembership 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, nonmembership, and radii values for the CPyFS. 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.
$\zeta_{\wp_1}(4080.51)=\frac{1}{3}(0.660.34+\sqrt{2 \times 0.45}(2 \times 0.51))=0.11$
$\zeta_{\wp_1}(4080.51)=\frac{1}{3}(0.340.66+\sqrt{2 \times 0.39}(2 \times 0.51))=0.11$
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 (CPLRs)
CPLRs 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 CPyFLRGs
Building upon the CPLRs, CPyFLRGs are developed. These groups are displayed in Table 3.
℘_{1}→℘_{1}  ℘_{1}→℘_{1}  ℘_{1}→℘_{1}  ℘_{1}→℘_{2} 





℘_{2}→℘_{2}  ℘_{2}→℘_{2}  ℘_{2}→℘_{2}  ℘_{2}→℘_{2}  ℘_{2}→℘_{3} 




℘_{3}→℘_{4} 








℘_{4}→℘_{4}  ℘_{4}→℘_{5}  ℘_{4}→℘_{5}  ℘_{4}→℘_{5}  ℘_{4}→℘_{5}  ℘_{4}→℘_{6} 



℘_{5}→℘_{4}  ℘_{5}→℘_{4}  ℘_{5}→℘_{6}  ℘_{5}→℘_{4} 





℘_{6}→℘_{6}  ℘_{6}→℘_{4}  ℘_{6}→℘_{7} 






℘_{7}→℘_{8} 








℘_{8}→℘_{11} 








℘_{10}→℘_{10}  ℘_{10}→℘_{12} 







℘_{11}→℘_{12}  ℘_{11}→℘_{13}  ℘_{11}→℘_{12} 






℘_{12}→℘_{12}  ℘_{12}→℘_{11}  ℘_{12}→℘_{13}  ℘_{12}→℘_{11}  ℘_{12}→℘_{12}  ℘_{12}→℘_{12}  ℘_{12}→℘_{12}  ℘_{12}→℘_{12}  ℘_{12}→℘_{14} 
℘_{13}→℘_{12}  ℘_{13}→℘_{13}  ℘_{13}→℘_{12} 






Table 4 presents the forecasted values derived from the CPyFSs 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 
01112001  3929.69  −  03122001  4646.61  4520 
02112001  3998.48  4100  04122001  4766.43  4600 
05112001  4080.51  4100  05122001  4924.56  4880 
06112001  4082.92  4100  06122001  5208.86  5240 
07112001  4158.15  4100  07122001  5333.93  5360 
08112001  4135.03  4220  10122001  5321.28  5420 
09112001  4123.78  4220  11122001  5273.97  5420 
12112001  4172.63  4220  12122001  5539.31  5420 
13112001  4136.54  4220  13122001  5407.54  5420 
14112001  4277.70  4220  14122001  5486.73  5420 
15112001  4403.59  4400  17122001  5456.15  5420 
16112001  4446.62  4520  18122001  5329.19  5420 
19112001  4548.63  4520  19122001  5221.96  5420 
20112001  4455.80  4520  20122001  5309.10  5420 
21112001  4533.37  4520  21122001  5109.24  5420 
22112001  4450.02  4520  24122001  5164.73  5240 
23112001  4519.08  4520  25122001  5372.81  5240 
26112001  4608.32  4520  26122001  5392.43  5420 
27112001  4580.33  4600  27122001  5332.98  5420 
28112001  4447.58  4600  28122001  5398.28  5420 
29112001  4465.83  4520  31122001  5551.24  5420 
30112001  4441.12  4520 



Figure 1 depicts a graphical representation of both the actual and forecasted values related to Alzheimer’s disease cases.
5. Circular Pythagorean Logical Relationships (CPLRs) of Order II
In this section, the methodology extends to constructing the CPLRs and their corresponding groups for secondorder forecasting in Alzheimer’s disease. The Table 5 delineates the CPLRs 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} 

Based on the CPLRs, Table 6 presents the CPyFLRGs 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} 

 ℘_{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} 


 ℘_{6}, ℘_{6→}℘_{4} 


℘_{6}, ℘_{4→}℘_{5} 


 ℘_{4}, ℘_{6→}℘_{7} 


℘_{6}, ℘_{7→}℘_{8} 


 ℘_{7}, ℘_{8→}℘_{11} 


℘_{8}, ℘_{11→}℘_{12} 


 ℘_{11}, ℘_{11→}℘_{12}  ℘_{11}, ℘_{12→}℘_{10} 

℘_{12}, ℘_{12→}℘_{11}  ℘_{12}, ℘_{12→}℘_{12}  ℘_{12}, ℘_{12→}℘_{12}  ℘_{12}, ℘_{12→}℘_{14}  ℘_{12}, ℘_{11→}℘_{13}  ℘_{12}, ℘_{11→}℘_{12} 

℘_{11}, ℘_{13→}℘_{12} 


 ℘_{13}, ℘_{12→}℘_{13}  ℘_{13}, ℘_{12→}℘_{11} 

℘_{12}, ℘_{13→}℘_{13} 


 ℘_{13}, ℘_{13→}℘_{12} 


℘_{12}, ℘_{10→}℘_{10} 


 ℘_{10}, ℘_{10→}℘_{12} 


℘_{10}, ℘_{10→}℘_{12} 






Table 7 illustrates the forecasted values for Alzheimer's disease, computed using the secondorder CPyFSs as outlined in Section B.
Date  True Value  Forecasted Value  Years  True Value  Forecasted Value 
01112001  3929.69  −  03122001  4646.61  4580 
02112001  3998.48  −  04122001  4766.43  4760 
05112001  4080.51  4100  05122001  4924.56  4880 
06112001  4082.92  4100  06122001  5208.86  5240 
07112001  4158.15  4100  07122001  5333.93  5360 
08112001  4135.03  4160  10122001  5321.2  5240 
09112001  4123.78  4220  11122001  5273.97  5400 
12112001  4172.63  4220  12122001  5539.31  5420 
13112001  4136.54  4220  13122001  5407.54  5360 
14112001  4277.70  4220  14122001  5486.73  5360 
15112001  4403.59  4400  17122001  5456.15  5480 
16112001  4446.62  4400  18122001  5329.19  5360 
19112001  4548.63  4520  19122001  5221.96  5360 
20112001  4455.80  4520  20122001  5309.10  5420 
21112001  4533.37  4580  21122001  5109.24  5420 
22112001  4450.02  4520  24122001  5164.73  5120 
23112001  4519.08  4580  25122001  5372.81  5360 
26112001  4608.32  4520  26122001  5392.43  5360 
27112001  4580.33  4640  27122001  5332.98  5400 
28112001  4447.58  4400  28122001  5398.28  5400 
29112001  4465.83  4520  31122001  5551.24  5400 
30112001  4441.12  4520 



A graphical representation, Figure 2, compares the actual values with the forecasted values for Alzheimer’s disease using the secondorder CPyFS 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 secondorder CPyFTSs forecasting. It has been observed that the secondorder CPyFTS forecasting demonstrates a superior performance over the firstorder model, as evidenced by the calculated error rates using established error measurement formulas.
A notable trend is observed in the forecasting accuracy: higherorder CPyFTS models tend to yield lower error rates. This pattern holds for the thirdorder forecasting error, which is smaller than that of the secondorder. This indicates that, generally, as the order increases, the accuracy of the CPyFTS 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 daytoday operational efficiency. Anticipating patient inflow facilitated more effective resource allocation, ensuring optimal patient care.
The significance of the radius in CPyFS extends beyond its traditional role in membership and nonmembership determination. In CPyFS, 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 CPyFSs when dealing with scenarios where the sum of membership and nonmembership degrees is one or less. It has been discerned that traditional PyFSs are inadequate in addressing such cases, leading to the utilization of CPyFSs in instances where the aggregate of membership and nonmembership values equals one. The proposed approach utilizing CPyFSs 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 higherorder forecasts revealed that higherorder 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 CPyFSs across diverse timeseries 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 CPyFSs 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.