Neu osophic Se s and Sys ems, Vol. 97, 2026
Uni e si y o New Mexico
D.Vidhya, S.Ja a i and G.No do, In eg a ing Neu osophic Logic in o P incipal Componen
Analysis: A Py hon-Based F amewo k
In eg a ing Neu osophic Logic in o P incipal Componen
Analysis: A Py hon-Based F amewo k
D.Vidhya1,* , S.Ja a i2 and G. No do3
1 Depa men o Science and Humani ies, Ka pagam Ins i u e o Technology, Coimba o e-641105, Tamilnadu, India;
[email p o ec ed]
2 P o esso o Ma hema ics, College o Ves sjaelland Sou h He es a ede 11, Slagelse, Denma k; [email p o ec ed]
3 MIFT Depa men o Ma hema ical and Compu e Science, Physical Sciences and Ea h Sciences - Uni e si y o
Messina, 98166 San ’ Aga a, Messina, I aly; [email p o ec ed]
*Co espondence: [email p o ec ed]
Abs ac : P incipal Componen Analysis (PCA) is a widely used dimensionali y educ ion
echnique ha ans o ms co ela ed a iables in o a smalle se o unco ela ed p incipal
componen s. Howe e , classical PCA assumes p ecise and c isp da a, which may no hold ue in
eal-wo ld scena ios cha ac e ized by unce ain y and inde e minacy. To add ess his limi a ion,
his s udy in eg a es Neu osophic Logic in o PCA, o ming a obus amewo k capable o
handling u h (T), inde e minacy (I), and alsi y (F) alues. The p oposed me hodology i s
con e s neu osophic da a in o c isp ep esen a ions using an agg ega ion unc ion, hen applies
PCA o ex ac p incipal componen s. A compa a i e analysis be ween no mal PCA and
Neu osophic PCA is conduc ed using Py hon, highligh ing how unce ain y impac s a iance
cap u e and eigen ec o o ien a ion. Visualiza ion ools such as eigen ec o plo s, p ojec ion lines,
and sc ee plo s a e employed o illus a e he indings. Resul s demons a e ha Neu osophic PCA
p o ides a mo e eliable ep esen a ion o unce ain da ase s wi hou signi ican loss o a iance
in o ma ion. This amewo k can be applied in ields such as pa e n ecogni ion, machine lea ning,
and da a-d i en decision-making whe e unce ain y is inhe en .
Keywo ds: P incipal Componen Analysis (PCA), Neu osophic Logic, Dimensionali y Reduc ion,
Eigen ec o s and Eigen alues, Py hon Implemen a ion, Unce ain y Modeling, Da a Analy ics
1. In oduc ion
Reducing he dimensionali y o da ase s is a key ask in da a analy ics, as i helps e ain essen ial
in o ma ion while minimizing edundancy [27]. A widely used me hod o his pu pose is PCA,
which ans o ms he da a in o a new se o o hogonal componen s, wi h each componen cap u ing
he maximum possible a iance [26]. Due o his capabili y, PCA has been success ully employed in
nume ous domains, including pa e n ecogni ion, image analysis, machine lea ning, and scien i ic
esea ch [17, 28].
Despi e i s popula i y, con en ional PCA assumes ha inpu da a is exac , consis en , and comple e.
In p ac ice, howe e , in o ma ion ga he ed om de ices, su eys, o expe assessmen s o en
su e s om ambigui y, incomple eness, and unce ain y [16, 19]. To add ess his sho coming,
Neu osophic Logic—in oduced by Sma andache [19, 20]—ex ends adi ional and uzzy se
heo ies [23] by ep esen ing h ee independen dimensions: u h (T), inde e minacy (I), and alsi y
(F).
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D.Vidhya, S.Ja a i and G.No do, In eg a ing Neu osophic Logic in o P incipal Componen
Analysis: A Py hon-Based F amewo k
Neu osophic se s and hei a ian s ha e been success ully employed o manage incomple e o
ague da a in a wide ange o ields, including medical decision-making [1, 2, 11, 22], machine
lea ning applica ions [5], image segmen a ion [28], and unce ain da a modeling [3, 7, 15]. Each
elemen in such a sys em can ca y deg ees o T, I, and F, which p o ides a lexible way o model
unce ain y [6, 10, 14].
The in eg a ion o Neu osophic Logic in o PCA— e e ed o as Neu osophic PCA—c ea es a mo e
esilien app oach o dimensionali y educ ion, explici ly accoun ing o unce ain y. Recen
p og ess in Py hon-based lib a ies o neu osophic ope a ions [4, 6, 18, 21] has u he enabled he
design and implemen a ion o such hyb id me hods.
In his pape , we in oduce a Py hon-d i en amewo k o Neu osophic PCA. The da a poin s,
exp essed as neu osophic iple s (T,I,F) we e con e ed in o c isp equi alen s using an agg ega ion
unc ion [12, 13]: Xc isp=T+αI−F, whe e α ep esen s he weigh assigned o inde e minacy [16]. The
p ocessed da ase is hen subjec ed o PCA, and i s ou comes a e e alua ed agains hose o
adi ional PCA.
2. Me hodology
2.1 Da ase P epa a ion:
A sample wo-dimensional da ase is selec ed o analysis.
2.2 Neu osophic Da a Rep esen a ion:
Each da a poin is con e ed in o a neu osophic iple (𝑇,𝐼,𝐹) by in oducing con olled
inde e minacy and alsi y.
2.3 C isp Con e sion:
The iple alues a e agg ega ed using he o mula, Xc isp=T+0.5I−F , o ob ain a single
nume ic alue.
2.4 P incipal Componen Analysis:
PCA is applied o bo h he no mal da ase and he neu osophic-c isp da ase . Eigen alues,
eigen ec o s, and explained a iance a ios a e compu ed.
2.5 Visualiza ion:
Sca e plo s, eigen ec o di ec ion plo s, p ojec ion lines, and sc ee plo s a e gene a ed o
compa e he wo app oaches.
3. No mal PCA s Neu osophic PCA
This sec ion p esen s a compa a i e analysis be ween No mal PCA and Neu osophic PCA. The
sec ion highligh s how unce ain y in da a is handled di e en ly, impac ing a iance e en ion and
da a ep esen a ion
3.1. No mal PCA: Nume ical Explana ion
S ep 1: S anda diza ion
We s anda dize he da a:
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D.Vidhya, S.Ja a i and G.No do, In eg a ing Neu osophic Logic in o P incipal Componen
Analysis: A Py hon-Based F amewo k
X_scaled = (X - μ)/σ
Example o he i s alue o X1 = 2.5:
μ_X1 = 1.81, σ_X1 = 0.83
X_scaled = (2.5 - 1.81) / 0.83 = 0.83.
Whe e X → O iginal da a alue, μ → Mean (a e age) o ha ea u e, σ → S anda d de ia ion o
ha ea u e, X_scaled → S anda dized alue a e ans o ma ion.
S ep 2: Co a iance Ma ix:
Co = [[1.111, 0.916], [0.916, 1.111]]
S ep 3: Eigen alues & Eigen ec o s:
Eigen alues: λ1 = 2.028, λ2 = 0.194
Eigen ec o o λ1 (PC1): [0.707, 0.707]
S ep 4: Dimensionali y Reduc ion:
P ojec da a on o PC1 o cap u e 91.2% o a iance
3.2. No mal PCA: Py hon Code
impo numpy as np
impo ma plo lib.pyplo as pl
om sklea n.decomposi ion impo PCA
om sklea n.p ep ocessing impo S anda dScale
X = np.a ay([[2.5,2.4],[0.5,0.7],[2.2,2.9],[1.9,2.2],[3.1,3.0],
[2.3,2.7],[2.0,1.6],[1.0,1.1],[1.5,1.6],[1.1,0.9]])
scale = S anda dScale ()
X_scaled = scale . i _ ans o m(X)
co _ma ix = np.co (X_scaled.T)
eig_ als, eig_ ecs = np.linalg.eig(co _ma ix)
pca = PCA(n_componen s=1)
X_pca = pca. i _ ans o m(X_scaled)
Neu osophic Se s and Sys ems, Vol. 97, 2026 331
D.Vidhya, S.Ja a i and G.No do, In eg a ing Neu osophic Logic in o P incipal Componen
Analysis: A Py hon-Based F amewo k
pl . igu e( igsize=(10,5))
pl .subplo (1,2,1)
pl .sca e (X_scaled[:,0], X_scaled[:,1], colo ='blue')
pl . i le("O iginal 2D Da a")
pl .subplo (1,2,2)
pl .sca e (X_pca, np.ze os(len(X_pca)), colo =' ed')
pl . i le("Da a A e PCA (1D)")
pl .show()
3.3. No mal PCA: Ou pu & Visualiza ion
O iginal Da a:
[[2.5 2.4]
[0.5 0.7]
[2.2 2.9]
[1.9 2.2]
[3.1 3.0]
[2.3 2.7]
[2.0 1.6]
[1.0 1.1]
[1.5 1.6]
[1.1 0.9]]
S anda dized Da a:
[[ 0.93 0.61]
[-1.76 -1.51]
[ 0.52 1.23]
[ 0.12 0.36]
[ 1.73 1.36]
[ 0.66 0.98]
[ 0.26 -0.39]
[-1.09 -1.01]
[-0.42 -0.39]
[-0.95 -1.26]]
Co a iance Ma ix:
[[1.111 1.029]
[1.029 1.111]]
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D.Vidhya, S.Ja a i and G.No do, In eg a ing Neu osophic Logic in o P incipal Componen
Analysis: A Py hon-Based F amewo k
Eigen alues:
[2.14 0.082]
Eigen ec o s:
[[ 0.707 -0.707]
[ 0.707 0.707]]
T ans o med Da a (1D PCA sco es):
[[ 1.086]
[-2.309]
[ 1.242]
[ 0.341]
[ 2.184]
[ 1.161]
[-0.093]
[-1.482]
[-0.567]
[-1.563]]
Explained Va iance Ra io: [0.963]
3.4. Neu osophic PCA: Nume ical Explana ion
S ep 1: C isp Con e sion:
x_c isp = T + 0.5I - F
Example o (2.5, 0.1, 0.05):
x = 2.5 + 0.5*0.1 - 0.05 = 2.5
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D.Vidhya, S.Ja a i and G.No do, In eg a ing Neu osophic Logic in o P incipal Componen
Analysis: A Py hon-Based F amewo k
S ep 2: Co a iance Ma ix:
Co = [[1.11, 0.91], [0.91, 1.12]]
Eigen alues: λ1 = 2.03, λ2 = 0.20
S ep 3: Dimensionali y Reduc ion:
P ojec on o PC1 and cap u e 91% a iance
3.5. Neu osophic PCA: Py hon Code
da a_neu o = [[(2.5, 0.1, 0.05), (2.4, 0.1, 0.05)],
[(0.5, 0.2, 0.05), (0.7, 0.2, 0.05)],
[(2.2, 0.15, 0.05), (2.9, 0.15, 0.05)],
[(1.9, 0.1, 0.05), (2.2, 0.1, 0.05)],
[(3.1, 0.1, 0.05), (3.0, 0.1, 0.05)],
[(2.3, 0.1, 0.05), (2.7, 0.1, 0.05)],
[(2.0, 0.15, 0.05), (1.6, 0.15, 0.05)],
[(1.0, 0.2, 0.05), (1.1, 0.2, 0.05)],
[(1.5, 0.15, 0.05), (1.6, 0.15, 0.05)],
[(1.1, 0.2, 0.05), (0.9, 0.2, 0.05)]]
da a_c isp = np.a ay([[T + 0.5*I - F o (T, I, F) in ow] o ow in da a_neu o])
scale = S anda dScale ()
Xn_scaled = scale . i _ ans o m(da a_c isp)
co _ma ix_n = np.co (Xn_scaled.T)
eig_ als_n, eig_ ecs_n = np.linalg.eig(co _ma ix_n)
pca = PCA(n_componen s=1)
Xn_pca = pca. i _ ans o m(Xn_scaled)
pl . igu e( igsize=(10,5))
pl .subplo (1,2,1)
pl .sca e (Xn_scaled[:,0], Xn_scaled[:,1], colo ='blue')
pl . i le("Neu osophic C isp 2D Da a")
pl .subplo (1,2,2)
pl .sca e (Xn_pca, np.ze os(len(Xn_pca)), colo =' ed')
pl . i le("Da a A e Neu osophic PCA (1D)")
pl .show()
3.6. Neu osophic PCA: Ou pu & Visualiza ion
C isp Neu osophic Da a:
[[2.5 2.4 ]
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D.Vidhya, S.Ja a i and G.No do, In eg a ing Neu osophic Logic in o P incipal Componen
Analysis: A Py hon-Based F amewo k
[0.55 0.75]
[2.23 2.92]
[1.9 2.2 ]
[3.1 3.0 ]
[2.3 2.7 ]
[2.03 1.62]
[1.05 1.15]
[1.52 1.62]
[1.15 0.95]]
Co a iance Ma ix:
[[1.111 1.025]
[1.025 1.111]]
Eigen alues:
[2.136 0.086]
Eigen ec o s:
[[ 0.707 -0.707]
[ 0.707 0.707]]
PCA T ans o med Da a (1D):
[[ 1.07 ]
[-2.312]
[ 1.275]
[ 0.306]
[ 2.194]
[ 1.146]
[-0.09 ]
[-1.466]
[-0.576]
[-1.548]]
Explained Va iance Ra io: [0.961]
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D.Vidhya, S.Ja a i and G.No do, In eg a ing Neu osophic Logic in o P incipal Componen
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3.7. Compa ison o No mal PCA Vs Neu osophic PCA
3.7.1. Sc ee Plo (Explained Va iance)-Py hon Code & Visualiza ion
pl . igu e( igsize=(6,4))
componen s = np.a ange(1, len(eig_ als)+1)
pl .ba (componen s, eig_ als/sum(eig_ als)*100, colo ='pu ple')
pl .ylabel("Va iance Explained (%)")
pl .xlabel("P incipal Componen s")
pl . i le("Sc ee Plo ")
pl .show()
Shows ha PC1 cap u es ~91% a iance in bo h no mal and neu osophic cases.
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D.Vidhya, S.Ja a i and G.No do, In eg a ing Neu osophic Logic in o P incipal Componen
Analysis: A Py hon-Based F amewo k
3.7.2. Eigen ec o Di ec ion Plo -Py hon Code & Visualiza ion
# D aw eigen ec o s on he sca e plo
colo s = [' ed', 'g een'] # Colo s o PC1 and PC2
o i in ange(len(eig_ ecs)):
ec = eig_ ecs[:, i]
pl .a ow(o igin[0], o igin[1], ec[0]*2, ec[1]*2,
head_wid h=0.1, head_leng h=0.1, colo =colo s[i],
label= 'PC{i+1}')
pl . i le("Eigen ec o Di ec ions")
pl .xlabel("Fea u e 1")
pl .ylabel("Fea u e 2")
pl .legend()
pl .show()
3.7.3. Eigen ec o Di ec ion Plo -Py hon Code & Visualiza ion
impo numpy as np
impo ma plo lib.pyplo as pl
om sklea n.p ep ocessing impo S anda dScale
impo os
# ---------------------
# DATA
# ---------------------
# No mal PCA da a (o iginal)
X_no mal = np.a ay([
[2.5, 2.4],
[0.5, 0.7],
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Analysis: A Py hon-Based F amewo k
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