

































IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )   

                                                                                                             Osuagwu, Chidimma Udo.* 
https://ijojournals.com/                                                    Volume 07 || Issue 08 || August, 2024 || 

PERFORMANCE EVALUATION OF CANONICAL CORRELATION ANALYSIS AND REDUNDANCY ANALYSISUSING GAUSSIAN, GAMMA, EXPONENTIAL AND BETA DISTRIBUTED DATA 

 
 

PERFORMANCE EVALUATION OF CANONICAL CORRELATION ANALYSIS AND REDUNDANCY 
ANALYSISUSING GAUSSIAN, GAMMA, EXPONENTIAL AND BETA DISTRIBUTED DATA 

 

 Osuagwu, Chidimma Udo 
Department of Statistics, Federal University of Technology, Owerri, Imo State Nigeria 
 

 
Okenwe Idochi 

Department of Statistics, School of Applied Sciences, Ken Saro Wiwa 
Polytechnic PMB 20, Bori, Rivers State Nigeria 

 

Abstract 
 

This study was embarked to examine the performance evaluation of canonical correlation and 

redundancy analysis with some continuous distributed data (Gaussian, Gamma, Exponential and 

Beta). The objectives of the study were to: obtain the relative efficiency of CCA and RDA 

techniques for four continuous distributed simulated data; and determine the model performance 

adequacy of CCA and RDA techniques. Three variates of the response variable (Y1, Y2, Y3) and 

three variates of independent variables (X1, X2, X3) were used for the simulation. The means 

used for response and independent variables for the Gaussian distribution were 80, 85 and 90, 

whereas their standard deviations were 10, 12 and 15. The alpha values used for response and 

independent variables for the Gamma distribution were 80, 85 and 90 whereas their theta values 

were 40, 43 and 45. The rates parameters used for response and independent variables for the 

Exponential distribution were 0.5. 0.7 and 0.9; whereas the shape parameters used for the Beta 

distribution were taking from 2 to 5 values. The adequacy of the CCA and RDA was evaluated 

with Wilcoxon rank sum test; and the study concluded thatRDA was more efficient than that of 

CCA for the Beta distributed data, while for Gaussian, Gamma and Exponential distributed data, 

the relative efficiency of the CCA and RDA was the same. The study also concluded that the X-

variates of the CCA and RDA did not differ. 

 
Keywords: Canonical correlation analysis,Redundancy analysis, Gaussian, Gamma, 
Exponential, Beta,Performance evaluation, Simulated data.  
 
 
1 Introduction 
 
Canonical Correlation Analysis (CCA) and Redundancy Analysis (RDA) are multivariate 

statistical techniques used to analyze the relationships between two or more sets of 

variables.CCA is a method for analyzing the relationships between two sets of variables, X and 

Y, by finding the linear combinations of variables that maximize the correlation between the two 

sets, which was developed by Hotelling in 1936 (Górecki et al, 2020). Canonical Correlation 

Analysis (CCA) involves finding a linear transformation that converts the original variables from 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )   

                                                                                                             Osuagwu, Chidimma Udo.* 
https://ijojournals.com/                                                    Volume 07 || Issue 08 || August, 2024 || 

PERFORMANCE EVALUATION OF CANONICAL CORRELATION ANALYSIS AND REDUNDANCY ANALYSISUSING GAUSSIAN, GAMMA, EXPONENTIAL AND BETA DISTRIBUTED DATA 

 
 

two sets into new sets of variables (Wang et al., 2022). These new variables have the property of 

being uncorrelated within each set, but maximally correlated between sets. The resulting pairs of 

new variables are called canonical variates, and the correlation coefficients between these pairs 

are known as canonical correlations. By identifying these canonical variates and correlations, 

CCA reveals the underlying relationships between the two sets of variables (Li et al., 2020). 

 

RDA is a statistical method that summarizes the linear relationships between two sets of 

variables, one set being the explanatory variables and the other being the response variables 

(Ramette, 2017). It's an extension of multiple linear regression; allowing for multiple response 

variables to be regressed on multiple explanatory variables. RDA produces an ordination that 

summarizes the main patterns of variation in the response matrix, which can be explained by a 

matrix of explanatory variables (Hui&Warton, 2022). 

 

The results of RDA include the total variance of the data set, partitioned into constrained and 

unconstrained variances, which shows how much variation in the response variables was 

redundant with the variation in the explanatory variables (Székely et al., 2020). RDA also 

produces scores for objects, response variables, and explanatory variables, which can be used to 

ordinate points and vectors. RDA is often used in ecological studies to relate environmental 

variables to species composition. For example, Ramette (2007) used RDA to analyze the 

relationships between microbial community composition and environmental variables in coastal 

sands 

 

This study therefore was aimed to: ascertain the relative efficiency of CCA and RDA techniques 

for four continuous distributed simulated data; and determine the model performance adequacy 

of CCA and RDA techniques. 

 

2 Review of Related Literature  
 

Makino (2022) explored the application of rotation in correspondence analysis (CA) from a 

canonical correlation perspective. CA is a statistical method used to visualize the relationship 

between two categorical variables, typically emphasizing graphical representations. Makino's 

study introduced a CA formulation based on canonical correlation analysis (CCA), where 

correlations within and between row/column categories in a reduced dimensional space can be 

expressed through canonical variables. However, existing CCA-based formulations only allowed 

for orthogonal rotation. Makino proposed an alternative CCA-based formulation that permits 

oblique rotation, defining the CA loss function as maximizing the generalized coefficient of 

determination, which measures the proximity between two variables. The study demonstrated the 

benefits of the proposed formulation through simulation studies and real data examples. 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )   

                                                                                                             Osuagwu, Chidimma Udo.* 
https://ijojournals.com/                                                    Volume 07 || Issue 08 || August, 2024 || 

PERFORMANCE EVALUATION OF CANONICAL CORRELATION ANALYSIS AND REDUNDANCY ANALYSISUSING GAUSSIAN, GAMMA, EXPONENTIAL AND BETA DISTRIBUTED DATA 

 
 

Nayir and Saridas (2022) investigated the relationship between culturally responsive teacher 

roles and innovative work behavior using canonical correlation analysis. The study aimed to 

identify the relationship between these two constructs based on teachers' views. The results 

showed that the first canonical function, which maximized the relationship between the two 

datasets, shared approximately 77% variance. Furthermore, the analysis revealed a positive 

relationship between the culturally regulating teacher (CRT) and culturally mediating teacher 

(CMT) variables in the culturally responsive teacher roles dataset and the GII and FSI variables 

in the innovative work behavior dataset. 

McKeague and Zhang (2021) investigated significance testing for canonical correlation analysis 

in high-dimensional settings. They addressed the challenge of testing for linear relationships 

between large sets of random variables using post-selection inference techniques. The authors 

developed a stabilized one-step estimator for the Euclidean norm of canonical correlations, 

which was shown to be consistent and asymptotically normal under certain conditions. They also 

proposed a greedy search algorithm for computing the estimator, leading to a computationally 

tractable omnibus test for the global null hypothesis. Additionally, they constructed a confidence 

interval that accounted for variable selection. 

García-Valdés et al. (2020) conducted a study using Redundancy Analysis (RDA) to examine the 

impacts of climate change on species distribution in a Mediterranean ecosystem, incorporating 

155 plant species and 15 environmental variables. The analysis revealed that climate variables, 

including temperature, precipitation, and drought, explained a significant portion of the variation 

in species distribution, accounting for 24.5% of the variation. Additionally, soil and topographic 

variables played important roles, explaining 20.1% and 15.4% of the variation, respectively. The 

study's findings suggested that climate change led to shifts in species distribution, resulting in 

some species expanding their ranges while others contract. The RDA framework provided a 

powerful tool for understanding the complex relationships between climate change and species 

distribution, with important implications for conservation and management efforts in the face of 

climate change. 

 

3 Materials and Methods 
 

3.1 Canonical Variates and Canonical Correlations 

The canonical correlations measure the strength of association between the two sets of variables 

(Wang & Liu, 2022).The first group of p variables is represented by the (p  1) random vector 

X(1), while the second group of q variables is represented by the (q  1) random vector X(2). It 

will be assumed, in the theoretical development, that X(1) represents the smaller set, so that p  q. 

For the random vectors X(1) and X(2), let 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )   

                                                                                                             Osuagwu, Chidimma Udo.* 
https://ijojournals.com/                                                    Volume 07 || Issue 08 || August, 2024 || 

PERFORMANCE EVALUATION OF CANONICAL CORRELATION ANALYSIS AND REDUNDANCY ANALYSISUSING GAUSSIAN, GAMMA, EXPONENTIAL AND BETA DISTRIBUTED DATA 

 
 

 














2211
)2()1(

22
)2()2()2(

11
)1()1()1(

),(

)(;)(

)(;)(

XX

XμX

XμX

Cov

CovE

CovE

     (1) 

It will be convenient to consider X(1) and X(2) jointly, so, the random vector 

 















































)2(

)2(
2

)2(
1

)1(

)1(
2

)1(
1

)2(

)1(

)1)((

q

p

qp

X

X

X

X

X

X





X

X
X        (2) 

has mean vector 

 



















)2(

)1(

)2(

)1(

)1)(( )(

)(
)(

μ

μ

X

X
Xμ

E

E
E

qp

      (3) 

and covariance matrix  

 
)()( qpqp 

Σ  = E(X– )E(X – ) 

 













))(())((

))(())((
)2()2()2()2()1()1()2()2(

)2()2()1()1()1()1()1()1(

μXμXμXμX

μXμXμXμX

EE

EE
 

 












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



)(
22

)(
21

)(
12

)(
11

qqpq

qppp

ΣΣ

ΣΣ

         (4) 

 

3.2 Matrices and Computational Procedures of Redundancy Analysis 

Let x be a 1p vector that includes p predictor variables in the first set and y be a 1q

vectorthat includes q criterion variables in the second set. According to Van Den Wollenberg in 

1977, all variables in x and y  should be standardized variableswith zero mean and unit variance 

(Gua et al., 2023). Thus, the )()( qpqp  covariance matrix of the 1)(  qp  vector )yx(  is 

a correlation matrix, denoted by R , which can be partitioned as 

 

 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )   

                                                                                                             Osuagwu, Chidimma Udo.* 
https://ijojournals.com/                                                    Volume 07 || Issue 08 || August, 2024 || 

PERFORMANCE EVALUATION OF CANONICAL CORRELATION ANALYSIS AND REDUNDANCY ANALYSISUSING GAUSSIAN, GAMMA, EXPONENTIAL AND BETA DISTRIBUTED DATA 

 
 

 ,











yyyx

xyxx

RR

RR
R          (4) 

where xxR  is the pp  correlation matrix of x , yyR  is the qq  correlation matrix of y , and

xyyx RR  is a pq  matrix that includes the inter-set correlations between x and y.  

To construct p redundancy variates, denoted by ),,,2,1( pii   with p predictor variables in 

x, the characteristic equation is evaluated as shown in Equation (5): 

 ,0)(  ixxyxxy i wRμRR         (5) 

where iμ  is the thi  eigenvalue and iw  is the thi eigen-vector. Then, one can employ the weight 

coefficients that are the elements of the scaled eigenvector iw  to construct ,i such that: 

(a) i is uncorrelated with ),( jij  and (b) i has unit variance ).,,2,1( pi   

 

3.3 Continuous Probability Distributions 

Four probability distributions known as the Gaussian, Gamma, Beta and Exponential are 

discussed in this study. 

 

3.3.1 The Gaussian Distribution 

A random variable (R.V.) in continuous form say X , choosing the whole real values in intervals 

  ,  is known to be a Gaussian (also known as normal) distribution with 2 and   as its 

parameters if the probability density function (pdf) is defined by  

0,,,

2

2
1

0
2

1
)( 2 


























 x

x

e

otherwise

xf     (6) 

Where the study used the notation );( 2N  to show that X  is normal with mean   and 

variance 2 (Sumair, et al., 2021). This pdf is bell-shaped, symmetrical, and centered at its mean 

value  .  

The entire area bounded by this function  f x  and axis-x  is 1 and therefore the area beneath 

the curve across two values of X , say,  and  with a b a b , constitutes the probability that the 

R.V X  lies across  and a b , which we write as  P a X b  . An example of a normal R.V is 

height of students at a specified age for a specified sex in a specified racial group even though 

heights must be positive (El-Morshedy et al., 2021). 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )   

                                                                                                             Osuagwu, Chidimma Udo.* 
https://ijojournals.com/                                                    Volume 07 || Issue 08 || August, 2024 || 

PERFORMANCE EVALUATION OF CANONICAL CORRELATION ANALYSIS AND REDUNDANCY ANALYSISUSING GAUSSIAN, GAMMA, EXPONENTIAL AND BETA DISTRIBUTED DATA 

 
 

The pdf of a standard normal distribution is  

 











otherwise

zezf

z

,0

,21 2

2

      (7) 

The respective mean and variance of a Gaussian distribution are respectively given as; 

  XE           (8) 

and 

  2XVar           (9) 

 
3.3.2 Gamma Distribution 

A R.V X is said to follow a gamma R.V with parameters and , if its pdf is given by: 

 















otherwise ,0

0,0,0,)(
1


 



x
ex

xf

x

      (10) 

where    is the gamma function defined as; 

  dtt
et




0

1         (11) 

The gamma probability density function as given in Equation (10) is a normal or legitimate pdf.  

 

The respective mean and variance of a Gamma distribution are respectively given as; 

)(XE           (12) 

and 

2)( XVar           (13) 
 

To obtain the scale (  ) and shape ( ) parameters of a gamma distribution, we have 

 )(XE           (14) 
22)(  XVar          (15) 

From Equation (14), put 



  into Equation (15) to obtain 

 2            (16) 

From Equation (16), the scale parameter is obtained as 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )   

                                                                                                             Osuagwu, Chidimma Udo.* 
https://ijojournals.com/                                                    Volume 07 || Issue 08 || August, 2024 || 

PERFORMANCE EVALUATION OF CANONICAL CORRELATION ANALYSIS AND REDUNDANCY ANALYSISUSING GAUSSIAN, GAMMA, EXPONENTIAL AND BETA DISTRIBUTED DATA 

 
 






2

            (17) 

Substitute Equation (17) into Equation (14) to obtain the shape parameter as 

2

2




             (18) 

Equations (17) and (18) were employed in the simulation of data for gamma distribution in this 

study. 

 
3.2.3 The Exponential Distribution  

A continuous random variable X , is said to have an exponential distribution with parameter 

0  if it has a probability density function defined by  












otherwise

xe
xf

x

0
)(

0,


         (19) 

The respective mean and variance of an Exponential distribution are respectively given as; 

 



1
)( XE          (20) 

and 

2

1
)(


XVar          (21) 

To obtain the rate parameter ( ) of an exponential distribution, we have 







11
)( XE          (22) 

Equations(22) was employed in the simulation of data for an exponential distribution in this 

study. 

 

3.2.4 Beta Distribution 

A standard beta distribution is a two-parameter family of distribution for a continuous random 

variableY , defined in a finite interval on a real line with its density function given by 

 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )   

                                                                                                             Osuagwu, Chidimma Udo.* 
https://ijojournals.com/                                                    Volume 07 || Issue 08 || August, 2024 || 

PERFORMANCE EVALUATION OF CANONICAL CORRELATION ANALYSIS AND REDUNDANCY ANALYSISUSING GAUSSIAN, GAMMA, EXPONENTIAL AND BETA DISTRIBUTED DATA 

 
 














otherwise ,0

10,
),(

)1(
)(

11

y
B

yy
yf 



      (23) 

where  , 0 and ( , ) is the beta function; its formula is given byB     

dtttB 1
1

0

1 )1(),(   
         (24) 

 

The mean and variance of Y are 







)(XE          (25) 

and 

2

2

))(1(
)(







XVar        (26) 

The scale (  ) and shape ( ) parameters of a beta distribution are obtained as; 

2

22 )(







          (27)  

and 

2

22 )1)((







         (28) 

 

4 Results  

4.1 Simulated Data of Different Sample Sizes for CCA and RDA 
 

Data were simulated on R-Studio command window, calling for the CCA and RDA function for 

Gaussian distribution, Gamma distribution, Exponential distribution and Beta distribution for 

samples of sizes 10, 20, 30, 40, 50, 60 and 70. Three variates of the response variable (Y1, Y2, 

Y3) and three variates of independent variables (X1, X2, X3) were used for the simulation. The 

means used for response and independent variables for the Gaussian distribution were 80, 85 and 

90,whereas their standard deviations were 10, 12 and 15. The alpha values used for response and 

independent variables for the Gamma distribution were 80, 85 and 90 whereas their theta values 

were 40, 43 and 45. The rates parameters used for response and independent variables for the 

Exponential distribution were 0.5. 0.7 and 0.9; whereas the shape parameters used for the Beta 

distribution were taking from values from 2 to 5 and the results obtained are summarized in 

Table 1. 

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                                                                                                             Osuagwu, Chidimma Udo.* 
https://ijojournals.com/                                                    Volume 07 || Issue 08 || August, 2024 || 

PERFORMANCE EVALUATION OF CANONICAL CORRELATION ANALYSIS AND REDUNDANCY ANALYSISUSING GAUSSIAN, GAMMA, EXPONENTIAL AND BETA DISTRIBUTED DATA 

 
 

 
Table 1: Summary Results from the Four Distributions for Different Sample Sizes 
Distribution  Correlation Eigen-value X-Mean Vector Y-Mean Vector 
 Sample CCA RDA CCA RDA CCA RDA 

 

 
 
 
 
 
 
 
 
 

Gaussian 

 
10 

0.9609 0.7180 81.6417 81.6417 78.5301 78.5301 
0.5303 0.2717 75.8878 75.8878 89.9215 89.9215 
0.4498 0.0020 89.3103 89.3103 84.7729 84.7729 

SD = 0.2748 SD = 0.3616     

 
20 

0.5343 0.5715 81.5175 81.5175 76.8798 76.8798 
0.3713 0.2015 82.4420 82.4420 87.3907 87.3907 
0.1763 0.0098 91.7642 91.7642 95.7642 95.7642 

      SD = 0.1792 SD = 0.2855     

 
30 

0.4399 0.3159 82.1643 82.1643 80.9557 80.9557 
0.2426 0.0852 85.5123 85.5123 86.2550 86.2550 
0.0318 0.0050 91.6238 91.6238 89.4687 89.4687 

SD = 0.2041 SD = 0.1614     

 
40 

0.2583 0.1442 80.8160 80.8160 78.6760 78.6760 
0.1231 0.1146 87.3596 87.3596 83.7756 83.7756 
0.0028 0.0044 90.5419 90.5419 89.0918 89.0918 

SD = 0.1278 SD = 0.0737     

 
50 

0.3583 0.0809 78.4572 78.4572 79.9256 79.9256 
0.1483 0.0211 85.5144 85.5144 84.7669 84.7669 
0.0103 0.0057 89.8407 89.8407 92.1156 92.1156 

SD = 0.1752 SD = 0.0397     

 
60 

0.3444 0.0599 81.1089 81.1089 80.7678 80.7678 
0.1502 0.0105 79.5728 79.5728 86.6178 86.6178 
0.031 0.0012 88.9582 88.9582 89.5820 89.5820 

SD = 0.158 SD = 0.0316     

 
70 

0.2150 0.1753 81.4539 81.4539 81.7998 81.7998 
0.0973 0.0724 85.5524 85.5524 82.6863 82.6863 
0.0012 0.0082 91.8743 91.8743 90.5477 90.5477 

SD = 0.1071 SD = 0.0843     

 
 
 
 
 
 
 
 
 
 
Gamma 

 
10 

0.8263 0.8768 3311.883 3311.883 3117.559 3117.559 
0.4987 0.5569 3691.902 3691.902 3520.850 3520.850 
0.2967 0.3712 3967.700 3967.700 4020.455 4020.455 

SD = 0.2673 SD = 0.2558     

 
20 

0.3916 0.2781 3145.036 3145.036 3329.500 3329.500 
0.1921 0.0870 3703.089 3703.089 3672.906 3672.906 
0.0523 0.0022 4019.466 4019.466 4122.456 4122.456 

SD = 0.1705 SD = 0.1413     

 
30 

0.5626 0.1461 3316.971 3316.971 3139.415 3139.415 
0.3175 0.0736 3601.826 3601.826 3687.641 3687.641 
0.0309 0.0065 4074.426 4074.426 4072.765 4072.765 

SD = 0.2661 SD = 0.0698     

 
40 

0.4502 0.0491 3151.013 3151.013 3127.473 3127.473 
0.2016 0.0176 3618.440 3618.440 3702.444 3702.444 
0.0370 0.0014 4011.102 4011.102 4057.979 4057.979 

SD = 0.2080 SD = 0.0243     

 
50 

0.3273 0.1721 3288.111 3288.111 3169.010 3169.010 
0.2901 0.0077 3565.821 3565.821 3641.781 3641.781 
0.0622 0.0005 4136.478 4136.478 4100.952 4100.952 

SD = 0.1435 SD = 0.0971     

 
60 

0.2258 0.0436 3158.815 3158.815 3195.705 3195.705 
0.1697 0.0121 3631.133 3631.133 3751.599 3751.599 
0.0145 0.0001 4115.701 4115.701 4001.559 4001.559 

SD = 0.1095 SD = 0.0225     

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )   

                                                                                                             Osuagwu, Chidimma Udo.* 
https://ijojournals.com/                                                    Volume 07 || Issue 08 || August, 2024 || 

PERFORMANCE EVALUATION OF CANONICAL CORRELATION ANALYSIS AND REDUNDANCY ANALYSISUSING GAUSSIAN, GAMMA, EXPONENTIAL AND BETA DISTRIBUTED DATA 

 
 

 
70 

0.2659 0.2709 3211.757 3211.757 3239.607 3239.607 
0.2170 0.1029 3666.475 3666.475 3686.072 3686.072 
0.1373 0.0066 4075.567 4075.567 3998.146 3998.146 

SD = 0.0649 SD = 0.1338     

 
 
 
 
 
 
 
 
 
 
 
Exponential 

 
10 

0.4361 0.8230 1.1191 1.1191 2.4481 2.4481 
0.3081 0.0420 2.8346 2.8346 1.5564 1.5564 
0.1447 0.0237 0.8268 0.8268 1.1561 1.1561 

SD = 0.1461 SD = 0.4563     

 
20 

0.4120 0.3147 2.2670 2.2670 1.9106 1.9106 
0.3517 0.1538 1.3586 1.3586 1.6637 1.6637 
0.0190 0.0013 1.6114 1.6114 1.1930 1.1930 

SD = 0.2117 SD = 0.1567     

 
30 

0.4655 0.2104 1.5786 1.5786 1.5809 1.5809 
0.2066 0.0256 1.2137 1.2137 1.1047 1.1047 
0.0284 0.0042 0.8006 0.8006 0.7525 0.7525 

SD = 0.2198 SD = 0.1134     

 
40 

0.4752 0.0573 1.4892 1.4892 1.8839 1.8839 
0.2934 0.0276 1.2863 1.2863 1.4824 1.4824 
0.1192 0.0041 0.9426 0.9426 1.0180 1.0180 

SD = 0.1780 SD = 0.0267     

 
50 

0.2732 0.2847 2.0737 2.0737 2.3757 2.3757 
0.2422 0.1221 1.5447 1.5447 1.5730 1.5730 
0.0173 0.0188 0.9702 0.9702 1.0287 1.0287 

SD = 0.1397 SD = 0.1340     

 
60 

0.3197 0.1642 2.0932 2.0932 1.6467 1.6467 
0.2416 0.0383 1.4325 1.4325 1.6291 1.6291 
0.1147 0.0150 1.0182 1.0182 1.1243 1.1243 

SD = 0.1035 SD = 0.0803     

 
70 

0.3419 0.0333 1.8292 1.8292 1.9513 1.9513 
0.2172 0.0152 1.2600 1.2600 1.2107 1.2107 
0.0537 0.0081 0.9883 0.9883 0.9481 0.9481 

SD = 0.1445 SD = 0.0130     

 
 
 
 
 
 
 
 
 
 
Beta 

 
10 

0.7963 0.2434 0.2932 0.2932 0.2610 0.2610 
0.4646 0.0627 0.4356 0.4356 0.3669 0.3669 
0.0710 0.0031 0.6326 0.6326 0.5026 0.5026 

SD = 0.3631 SD = 0.1251     

 
20 

0.5851 0.5324 0.3218 0.3218 0.2944 0.2944 
0.5173 0.1768 0.4216 0.4216 0.4652 0.4652 
0.3069 0.0034 0.6327 0.6327 0.5084 0.5084 

SD = 0.1451 SD = 0.2697     

 
30 

0.4148 0.3035 0.3263 0.3263 0.2616 0.2616 
0.1741 0.0262 0.4498 0.4498 0.4344 0.4344 
0.0256 0.0080 0.5727 0.5727 0.5483 0.5483 

SD = 0.1964 SD = 0.1656     

 
40 

0.4796 0.1522 0.3285 0.3285 0.2641 0.2641 
0.1904 0.0629 0.4468 0.4468 0.4209 0.4209 
0.1156 0.0511 0.6191 0.6191 0.6008 0.6008 

SD = 0.1922 SD = 0.0553     

 
50 

0.5701 0.0681 0.2805 0.2805 0.2751 0.2751 
0.2277 0.0133 0.4535 0.4535 0.4555 0.4555 
0.0613 0.0001 0.6037 0.6037 0.5730 0.5730 

SD = 0.2594 SD = 0.0361     

 
60 

0.5456 0.1719 0.2424 0.2424 0.2994 0.2994 
0.2209 0.0387 0.4367 0.4367 0.4373 0.4373 
0.1515 0.0067 0.5737 0.5737 0.5745 0.5745 

SD = 0.2104 SD = 0.0876     

 0.2990 0.0806 0.2953 0.2953 0.3030 0.3030 
0.0762 0.0192 0.4589 0.4589 0.4352 0.4352 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )   

                                                                                                             Osuagwu, Chidimma Udo.* 
https://ijojournals.com/                                                    Volume 07 || Issue 08 || August, 2024 || 

PERFORMANCE EVALUATION OF CANONICAL CORRELATION ANALYSIS AND REDUNDANCY ANALYSISUSING GAUSSIAN, GAMMA, EXPONENTIAL AND BETA DISTRIBUTED DATA 

 
 

70 0.0525 0.0003 0.6028 0.6028 0.5580 0.5580 
SD = 0.1360 SD = 0.0420     

 
Table 1 shows the standard deviation of the correlations and eigenvalues for CCA and RDA 

respectively.It can be observed that the standard deviation of the RDA is lower than that of CCA 

except for the cases of sample sizes 10 and 20 for Gaussian distribution; sample size 70 for 

Gamma distribution, sample size 10 for Exponential distribution and sample size 20 for Beta 

distribution, but there is need to examine if the differencesare significant. It is also observed that 

the X and Y-variates of the CCA and RDA do not differ. 

 

4.2 Model Performance Adequacy of CCA and RDA Techniques 

 

Table 2: Summary of Decision for Testing SD Values for CCA and RDA  

   SD Values Ranks Z p-value Decision 
Distribution Sample CCA       RDA CCA      

RDA 
   

 
 

Gaussian 

10 0.2748 0.3616 12 14  
 

 
0.958 

 

 
 
 

0.338 

 
 

 
Do not Reject H0 

20 0.1792 0.2855 10 13 

30 0.2041 0.1614 11 8 

40 0.1278 0.0737 6 3 

50 0.1752 0.0397 9 2 

60 0.158 0.0316 7 1 

70 0.1071 0.0843 5 4 

 
 
 

Gamma 

10 0.2673 0.2558 14 12  
 
 

1.725 
 
 
 

 
 
 

0.085 

 
 
 

Do not Reject H0 

20 0.1705 0.1413 10 8 

30 0.2661 0.0698 13 4 

40 0.2080 0.0243 11 2 

50 0.1435 0.0971 9 5 

60 0.1095 0.0225 6 1 

70 0.0649 0.1338 3 7 

 
 

Exponential 

10 0.1461 0.4563 9 14  
 
 

1.469 
 
 
 

 
 
 

0.142 

 
 
 

Do not Reject H0 

20 0.2117 0.1567 12 10 

30 0.2198 0.1134 13 5 

40 0.1780 0.0267 11 2 

50 0.1397 0.1340 7 6 

60 0.1035 0.0803 4 3 

70 0.1445 0.0130 8 1 

 
 

Beta 

10 0.3631 0.1251 14 5  
 
 

2.108 
 

 
 
 

0.035 

 
 
 

Reject H0 

20 0.1451 0.2697 7 13 

30 0.1964 0.1656 10 8 

40 0.1922 0.0553 9 3 

50 0.2594 0.0361 12 1 

60 0.2104 0.0876 11 4 

70 0.1360 0.0420 6 2 

 

Table 2 shows the Wilcoxon rank sum testsignificance difference result for the four continuous 

distributions employed in this study. The result reveals that there is no significant difference in 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )   

                                                                                                             Osuagwu, Chidimma Udo.* 
https://ijojournals.com/                                                    Volume 07 || Issue 08 || August, 2024 || 

PERFORMANCE EVALUATION OF CANONICAL CORRELATION ANALYSIS AND REDUNDANCY ANALYSISUSING GAUSSIAN, GAMMA, EXPONENTIAL AND BETA DISTRIBUTED DATA 

 
 

the standard deviation of the correlations and eigenvalues for the methods for Gaussian, Gamma 

and Exponential distributions. This implies that the relative efficiency of the CCA and RDA is 

the same for the Gaussian, Gamma and Exponential distributed data. On the other hand, the 

result reveals thatthere is significant difference in the standard deviation of the correlations and 

eigenvalues for the methods for Beta distribution. This implies that RDA is more efficient than 

that of CCA for the Beta distributed data. 

 

4 Conclusion 

This study used canonical correlation and redundancy analysis via four continuous distributions 

(Gaussian, Gamma, Exponential and Beta) in order to assess their performances. The adequacy 

of the CCA and RDA was evaluated with Wilcoxon rank sum test; and the study concluded 

thatRDA is more efficient than that of CCA for the Beta distributed data, while for Gaussian, 

Gamma and Exponential distributed data, the relative efficiency of the CCA and RDA is the 

same. The study also concluded that the X-variates of the CCA and RDA do not differ. 

 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )   

                                                                                                             Osuagwu, Chidimma Udo.* 
https://ijojournals.com/                                                    Volume 07 || Issue 08 || August, 2024 || 

PERFORMANCE EVALUATION OF CANONICAL CORRELATION ANALYSIS AND REDUNDANCY ANALYSISUSING GAUSSIAN, GAMMA, EXPONENTIAL AND BETA DISTRIBUTED DATA 

 
 

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