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 IJO JOURNALS Volume 07 | Issue 08 | August 2024 | https://ijojournals.com/index.php/m/index 1 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. IJO JOURNALS Volume 07 | Issue 08 | August 2024 | https://ijojournals.com/index.php/m/index 2 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 IJO JOURNALS Volume 07 | Issue 08 | August 2024 | https://ijojournals.com/index.php/m/index 3 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           )( 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 IJO JOURNALS Volume 07 | Issue 08 | August 2024 | https://ijojournals.com/index.php/m/index 4 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). IJO JOURNALS Volume 07 | Issue 08 | August 2024 | https://ijojournals.com/index.php/m/index 5 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 IJO JOURNALS Volume 07 | Issue 08 | August 2024 | https://ijojournals.com/index.php/m/index 6 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 IJO JOURNALS Volume 07 | Issue 08 | August 2024 | https://ijojournals.com/index.php/m/index 7 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. IJO JOURNALS Volume 07 | Issue 08 | August 2024 | https://ijojournals.com/index.php/m/index 8 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 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 IJO JOURNALS Volume 07 | Issue 08 | August 2024 | https://ijojournals.com/index.php/m/index 9 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 IJO JOURNALS Volume 07 | Issue 08 | August 2024 | https://ijojournals.com/index.php/m/index 10 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 IJO JOURNALS Volume 07 | Issue 08 | August 2024 | https://ijojournals.com/index.php/m/index 11 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. References El-Morshedy, M., Alshammari, F.S., Hamed, Y. S., Eliwa, M.S. &Yousof, H.M. (2021).A New Family of Continuous Probability Distributions.Entropy, 23(194), 1–24. García-Valdés, R., Sánchez, A. M., Fernández-Palacios, J. M., Padrón, R. P., & Rodríguez- Rodríguez, M. A. (2020). 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FEMS Microbiology Ecology, 93(12), fix106.doi: 10.1093/femsec/fix106 Sumair, M., Aized, T., Gardezi, S.A.R., Bhutta, M.M.A., Rehman, S.M.S. &Rehman, S.U. (2021). Application of five continuous distributions and evaluation of wind potential at five stations using normal distribution. Energy Exploration & Exploitation, 39(6), 2214– 2239. Székely, E., Botta-Dukát, Z., &Lengyel, A. (2020).Redundancy analysis as a tool for identifying drivers of community composition in vegetation ecology.Journal of Vegetation Science, 31(3), 537-546. doi: 10.1111/jvs.12854 Van de Velden. M. (2011).On generalized canonical correlation analysis.Proc.58th World Statistical Congress. Dublin, 758–765. Wang, H., Zhang, Y., & Singh, R. (2022). Climate-crop yield relationships: a canonical correlation analysis. Agricultural and Forest Meteorology, 313, 108702. Wang, Y., & Liu, X. (2022).Canonical correlation analysis for identifying relationships between climate variables and crop yields.Journal of Agricultural Science, 160(3), 257-265. IJO JOURNALS Volume 07 | Issue 08 | August 2024 | https://ijojournals.com/index.php/m/index 13