Module: More on Canonical Correlations

Subject: Statistics

Paper: Multivariate Analysis

Module: More on Canonical Correlations

Development Team

Principal investigator: Dr. Bhaswati Ganguli, Professor, Department of Statistics, University of Calcutta

Paper co-ordinator: Dr. Sugata SenRoy,Professor, Department of Statistics, University of Calcutta

Content writer: Souvik Bandyopadhyay, Senior Lecturer, Indian Institute of Public Health, Hyderabad

Content reviewer: Dr. Kalyan Das,Professor, Department of Statistics, University of Calcutta

Development Team

Principal investigator: Dr. Bhaswati Ganguli, Professor, Department of Statistics, University of Calcutta

Paper co-ordinator: Dr. Sugata SenRoy,Professor, Department of Statistics, University of Calcutta

Content writer: Souvik Bandyopadhyay, Senior Lecturer, Indian Institute of Public Health, Hyderabad

Content reviewer: Dr. Kalyan Das,Professor, Department of Statistics, University of Calcutta

Development Team

Principal investigator: Dr. Bhaswati Ganguli, Professor, Department of Statistics, University of Calcutta

Paper co-ordinator: Dr. Sugata SenRoy,Professor, Department of Statistics, University of Calcutta

Content writer: Souvik Bandyopadhyay, Senior Lecturer, Indian Institute of Public Health, Hyderabad

Content reviewer: Dr. Kalyan Das,Professor, Department of Statistics, University of Calcutta

Development Team

Principal investigator: Dr. Bhaswati Ganguli, Professor, Department of Statistics, University of Calcutta

Paper co-ordinator: Dr. Sugata SenRoy,Professor, Department of Statistics, University of Calcutta

Content writer: Souvik Bandyopadhyay, Senior Lecturer, Indian Institute of Public Health, Hyderabad

Content reviewer: Dr. Kalyan Das,Professor, Department of Statistics, University of Calcutta

Standardized variables

I Let there ber variables in the1^st groupX₁= (X₁, . . . , X_r) ands variables in the2^nd groupX2 = (Xr+1, . . . , Xr+s), with r ≤sand

E(X₁) =µ₁ andE(X₂) =µ₂ V ar(X₁) = Σ₁₁,V ar(X₂) = Σ₂₂ andCov(X1,X2) = Σ12 (orΣ⁰₂₁)

I Let Z1 = (z1, . . . , Zr)and Z2 = (Zr+1, . . . , Zr+s), where Z_j = (X_j−µ_j)/σ_jj for j = 1, . . . , r+s.

with µ_j =E(X_j) andσ_jj =V ar(X_j).

I What is the effect of this standardization on the canonical correlation ?

Standardized variables

I Let there ber variables in the1^st groupX₁= (X₁, . . . , X_r) ands variables in the2^nd groupX2 = (Xr+1, . . . , Xr+s), with r ≤sand

E(X₁) =µ₁ andE(X₂) =µ₂ V ar(X₁) = Σ₁₁,V ar(X₂) = Σ₂₂ andCov(X1,X2) = Σ12 (orΣ⁰₂₁)

I Let Z1 = (z1, . . . , Zr)and Z2 = (Zr+1, . . . , Zr+s), where Z_j = (X_j−µ_j)/σ_jj for j = 1, . . . , r+s.

with µ_j =E(X_j) andσ_jj =V ar(X_j).

I What is the effect of this standardization on the canonical correlation ?

Standardized variables

I Let there ber variables in the1^st groupX₁= (X₁, . . . , X_r) ands variables in the2^nd groupX2 = (Xr+1, . . . , Xr+s), with r ≤sand

E(X₁) =µ₁ andE(X₂) =µ₂ V ar(X₁) = Σ₁₁,V ar(X₂) = Σ₂₂ andCov(X1,X2) = Σ12 (orΣ⁰₂₁)

I Let Z1 = (z1, . . . , Zr)and Z2 = (Zr+1, . . . , Zr+s), where Z_j = (X_j−µ_j)/σ_jj for j = 1, . . . , r+s.

with µ_j =E(X_j) andσ_jj =V ar(X_j).

I What is the effect of this standardization on the canonical correlation ?

Canonical corrletion for standardized variables

I The answer is of course

Corr(Z₁,Z₂) =Corr(X₁,X₂)

since correlations do not change with base and scale changes.

I However, the canonical variables will be different.

I In general, for standardized variables, with V ar(Z1) =ρ₁₁, V ar(Z₂) =ρ₂₂ andCov(Z₁,Z₂) =ρ₁₂,

U_j^∗ =p⁰_jρ^−1/2₁₁ Z1 and V_j^∗ =q⁰_jρ^−1/2₂₂ Z2,

with Corr(U_j^∗, V_j^∗) =p λj.

Canonical corrletion for standardized variables

I The answer is of course

Corr(Z₁,Z₂) =Corr(X₁,X₂)

since correlations do not change with base and scale changes.

I However, the canonical variables will be different.

I In general, for standardized variables, with V ar(Z1) =ρ₁₁, V ar(Z₂) =ρ₂₂ andCov(Z₁,Z₂) =ρ₁₂,

U_j^∗ =p⁰_jρ^−1/2₁₁ Z1 and V_j^∗ =q⁰_jρ^−1/2₂₂ Z2,

with Corr(U_j^∗, V_j^∗) =p λj.

Canonical corrletion for standardized variables

I The answer is of course

Corr(Z₁,Z₂) =Corr(X₁,X₂)

since correlations do not change with base and scale changes.

I However, the canonical variables will be different.

I In general, for standardized variables, with V ar(Z1) =ρ₁₁, V ar(Z₂) =ρ₂₂ andCov(Z₁,Z₂) =ρ₁₂,

U_j^∗ =p⁰_jρ^−1/2₁₁ Z1 and V_j^∗ =q⁰_jρ^−1/2₂₂ Z2, with Corr(U_j^∗, V_j^∗) =p

λj.

Justification

Observe that

a⁰_j(X1−µ₁) = aj1(X1−µ1) +. . .+ajr(Xr−µr)

= aj1

√σ₁₁(X₁−µ₁)

√σ₁₁ +. . .+ajr

√σ_rr(X_r−µ_r)

√σ_rr

= aj1

√σ11Z1+. . .+ajr

√σrrZr

I DefineV =diag((σ_kk))_k=1....,r.

I Thus ifa_j is the coefficient of thej^th canonical variateU_j, ajV^−1/2 is the coefficient of the corresponding canonical variate U_j^∗ of the standardized variables.

Justification

Observe that

a⁰_j(X1−µ₁) = aj1(X1−µ1) +. . .+ajr(Xr−µr)

= aj1

√σ₁₁(X₁−µ₁)

√σ₁₁ +. . .+ajr

√σ_rr(X_r−µ_r)

√σ_rr

= aj1

√σ11Z1+. . .+ajr

√σrrZr

I DefineV =diag((σ_kk))_k=1....,r.

I Thus ifa_j is the coefficient of thej^th canonical variateU_j, ajV^−1/2 is the coefficient of the corresponding canonical variate U_j^∗ of the standardized variables.

Results

I Let U= (U1, . . . , Ur) be the canonical variables for the first group with

U=AX1, where A= (p1,p2, . . . ,pr)

I Also let V= (V1, . . . , Vr)be the canonical variables for the second group with

V=BX2, where B= (q1,q2, . . . ,qr)

I Now Cov(U,X₁) =Cov(AX₁,X₁) =AΣ₁₁.

I Since V ar(U_j) = 1,

Corr(U_j, X_1k) =Cov(U_j, X_1k)/V ar(X_1k).

Results

I Let U= (U1, . . . , Ur) be the canonical variables for the first group with

U=AX1, where A= (p1,p2, . . . ,pr)

I Also let V= (V1, . . . , Vr)be the canonical variables for the second group with

V=BX2, where B= (q1,q2, . . . ,qr)

I Now Cov(U,X₁) =Cov(AX₁,X₁) =AΣ₁₁.

I Since V ar(U_j) = 1,

Corr(U_j, X_1k) =Cov(U_j, X_1k)/V ar(X_1k).

Results

I Let U= (U1, . . . , Ur) be the canonical variables for the first group with

U=AX1, where A= (p1,p2, . . . ,pr)

I Also let V= (V1, . . . , Vr)be the canonical variables for the second group with

V=BX2, where B= (q1,q2, . . . ,qr)

I Now Cov(U,X₁) =Cov(AX₁,X₁) =AΣ₁₁.

I Since V ar(U_j) = 1,

Corr(U_j, X_1k) =Cov(U_j, X_1k)/V ar(X_1k).

Results

I Let U= (U1, . . . , Ur) be the canonical variables for the first group with

U=AX1, where A= (p1,p2, . . . ,pr)

I Also let V= (V1, . . . , Vr)be the canonical variables for the second group with

V=BX2, where B= (q1,q2, . . . ,qr)

I Now Cov(U,X₁) =Cov(AX₁,X₁) =AΣ₁₁.

I Since V ar(U_j) = 1,

Corr(U_j, X_1k) =Cov(U_j, X_1k)/V ar(X_1k).

Results (contd.)

∴Corr(U,X₁) =Cov(AX₁, V₁₁^−1/2X₁) =AΣ₁₁V₁₁^−1/2.

I Summarizing,

Corr(U,X1) =AΣ11V₁₁^−1/2, Corr(V,X2) =BΣ22V₂₂^−1/2, Corr(U,X₂) =AΣ₁₂V₂₂^−1/2, Corr(V,X1) =BΣ21V₁₁^−1/2,

I These results give the relation between the canonical variables and the original variables.

Results (contd.)

∴Corr(U,X₁) =Cov(AX₁, V₁₁^−1/2X₁) =AΣ₁₁V₁₁^−1/2.

I Summarizing,

Corr(U,X1) =AΣ11V₁₁^−1/2, Corr(V,X2) =BΣ22V₂₂^−1/2, Corr(U,X₂) =AΣ₁₂V₂₂^−1/2, Corr(V,X1) =BΣ21V₁₁^−1/2,

I These results give the relation between the canonical variables and the original variables.

Special Cases

I We have r andsdimensional vectorsX₁ andX₂.

I Ifr =s= 1, i.e. one variable in each set (sayX1 andX2) then Corr(X₁, X₂) =√

λ₁.

I Ifr >1 ands= 1, then for any j= 1, . . . , rand a₀=e_j, Corr(X1j, X2) = Corr(a⁰₀X1, X2)

≤ max_aCorr(a⁰X₁, X₂) =p λ₁.

I In fact, observe that

maxaCorr(a⁰X1, X2) =p λ1

gives the multiple correlation ofX₂ andX₁ = (X₁₁, . . . , X_1r), i.e. the multiple correlation exceeds all simple correlations.

Special Cases

I We have r andsdimensional vectorsX₁ andX₂.

I Ifr =s= 1, i.e. one variable in each set (sayX1 andX2) then Corr(X₁, X₂) =√

λ₁.

I Ifr >1 ands= 1, then for any j= 1, . . . , rand a₀=e_j, Corr(X1j, X2) = Corr(a⁰₀X1, X2)

≤ max_aCorr(a⁰X₁, X₂) =p λ₁.

I In fact, observe that

maxaCorr(a⁰X1, X2) =p λ1

gives the multiple correlation ofX₂ andX₁ = (X₁₁, . . . , X_1r), i.e. the multiple correlation exceeds all simple correlations.

Special Cases

I We have r andsdimensional vectorsX₁ andX₂.

I Ifr =s= 1, i.e. one variable in each set (sayX1 andX2) then Corr(X₁, X₂) =√

λ₁.

I Ifr >1 ands= 1, then for any j= 1, . . . , rand a₀=e_j, Corr(X1j, X2) = Corr(a⁰₀X1, X2)

≤ max_aCorr(a⁰X₁, X₂) =p λ₁.

I In fact, observe that

maxaCorr(a⁰X1, X2) =p λ1

gives the multiple correlation ofX₂ andX₁ = (X₁₁, . . . , X_1r), i.e. the multiple correlation exceeds all simple correlations.

Special Cases

I We have r andsdimensional vectorsX₁ andX₂.

I Ifr =s= 1, i.e. one variable in each set (sayX1 andX2) then Corr(X₁, X₂) =√

λ₁.

I Ifr >1 ands= 1, then for any j= 1, . . . , rand a₀=e_j, Corr(X1j, X2) = Corr(a⁰₀X1, X2)

≤ max_aCorr(a⁰X₁, X₂) =p λ₁.

I In fact, observe that

maxaCorr(a⁰X1, X2) =p λ1

gives the multiple correlation ofX₂ andX₁ = (X₁₁, . . . , X_1r), i.e. the multiple correlation exceeds all simple correlations.

Special Cases

I In general, for anyr and s, with j= 1, . . . , rand k= 1, . . . , s andb₀ =e_k,

Corr(X1j, X_2k) = Corr(a⁰₀X1,b⁰₀X2)

≤ max_a,bCorr(a⁰X₁,b⁰X₂) =p λ₁.

I Observe that for any b0 =e_k,

maxaCorr(a⁰X1,b⁰₀X2) ≤ maxa,bCorr(a⁰X1,b⁰X2) =p λ1. Since the left hand side gives the multiple correlation of X₂ andX1= (X11, . . . , X1r), the first canonical correlation acts as an upper bound to all multiple correlations.

Special Cases

I In general, for anyr and s, with j= 1, . . . , rand k= 1, . . . , s andb₀ =e_k,

Corr(X1j, X_2k) = Corr(a⁰₀X1,b⁰₀X2)

≤ max_a,bCorr(a⁰X₁,b⁰X₂) =p λ₁.

I Observe that for any b0 =e_k,

maxaCorr(a⁰X1,b⁰₀X2) ≤ maxa,bCorr(a⁰X1,b⁰X2) =p λ1. Since the left hand side gives the multiple correlation of X₂ andX1= (X11, . . . , X1r), the first canonical correlation acts as an upper bound to all multiple correlations.

How to calculate canonical correlations

I To calculate the canonical correlations, suppose we have data (X1i,X2i), fori= 1, . . . , n i.e. we have a n×(r+s) order data matrix.

I DefineX= X1

X₂

andS =

S11 S12

S21 S22

, whereX_j = _n¹Pn

i=1X_ji,j= 1,2 andSjk = ¹_nPn

i=1(Xji−Xj)⁰(Xki−Xk),j, k= 1,2.

I The method is then similar to the population canonical correlation, except that now we work withS instead ofΣ.

I The following Result follows as for the population canonical correlations.

How to calculate canonical correlations

I To calculate the canonical correlations, suppose we have data (X1i,X2i), fori= 1, . . . , n i.e. we have a n×(r+s) order data matrix.

I DefineX= X1

X₂

andS =

S11 S12

S21 S22

, whereX_j = _n¹Pn

i=1X_ji,j= 1,2 andSjk = ¹_nPn

i=1(Xji−Xj)⁰(Xki−Xk),j, k= 1,2.

I The method is then similar to the population canonical correlation, except that now we work withS instead ofΣ.

I The following Result follows as for the population canonical correlations.

How to calculate canonical correlations

I To calculate the canonical correlations, suppose we have data (X1i,X2i), fori= 1, . . . , n i.e. we have a n×(r+s) order data matrix.

I DefineX= X1

X₂

andS =

S11 S12

S21 S22

, whereX_j = _n¹Pn

i=1X_ji,j= 1,2 andSjk = ¹_nPn

i=1(Xji−Xj)⁰(Xki−Xk),j, k= 1,2.

I The method is then similar to the population canonical correlation, except that now we work withS instead ofΣ.

I The following Result follows as for the population canonical correlations.

How to calculate canonical correlations

I To calculate the canonical correlations, suppose we have data (X1i,X2i), fori= 1, . . . , n i.e. we have a n×(r+s) order data matrix.

I DefineX= X1

X₂

andS =

S11 S12

S21 S22

, whereX_j = _n¹Pn

i=1X_ji,j= 1,2 andSjk = ¹_nPn

i=1(Xji−Xj)⁰(Xki−Xk),j, k= 1,2.

I The method is then similar to the population canonical correlation, except that now we work withS instead ofΣ.

I The following Result follows as for the population canonical correlations.

Result

I Let ˆλ₁ ≥λˆ₂≥. . .≥ˆλ_r be the eigenvalues of

S₁₁^−1/2S12S₂₂⁻¹S21S₁₁^−1/2 andpˆ1,pˆ2, . . . ,pˆr the corresponding eigen-vectors.

I In fact, ˆλ1 ≥λˆ2≥. . .≥ˆλr are also the r largest eigen-values of S₂₂^−1/2S21S₁₁⁻¹S12S₂₂^−1/2 withqˆ1,qˆ2, . . . ,qˆr the

corresponding eigen-vectors.

Result

Thus for any unit(X_1k,X_2k), thej^th sample canonical variate pair isUˆ1 = ˆp⁰₁S₁₁^−1/2X1k andVˆ2 = ˆq⁰₁S₂₂^−1/2X2k with

Corr( ˆU1,Vˆ1) = ˆρj = qλˆj.

Result

I Let ˆλ₁ ≥λˆ₂≥. . .≥ˆλ_r be the eigenvalues of

S₁₁^−1/2S12S₂₂⁻¹S21S₁₁^−1/2 andpˆ1,pˆ2, . . . ,pˆr the corresponding eigen-vectors.

I In fact, ˆλ1 ≥λˆ2≥. . .≥ˆλr are also the r largest eigen-values of S₂₂^−1/2S21S₁₁⁻¹S12S₂₂^−1/2 withqˆ1,qˆ2, . . . ,qˆr the

corresponding eigen-vectors.

Result

Thus for any unit(X_1k,X_2k), thej^th sample canonical variate pair isUˆ1 = ˆp⁰₁S₁₁^−1/2X1k andVˆ2 = ˆq⁰₁S₂₂^−1/2X2k with

Corr( ˆU1,Vˆ1) = ˆρj = qλˆj.

Result

I Let ˆλ₁ ≥λˆ₂≥. . .≥ˆλ_r be the eigenvalues of

S₁₁^−1/2S12S₂₂⁻¹S21S₁₁^−1/2 andpˆ1,pˆ2, . . . ,pˆr the corresponding eigen-vectors.

I In fact, ˆλ1 ≥λˆ2≥. . .≥ˆλr are also the r largest eigen-values of S₂₂^−1/2S21S₁₁⁻¹S12S₂₂^−1/2 withqˆ1,qˆ2, . . . ,qˆr the

corresponding eigen-vectors.

Result

Thus for any unit(X_1k,X_2k), thej^th sample canonical variate pair isUˆ1 = ˆp⁰₁S₁₁^−1/2X1k andVˆ2 = ˆq⁰₁S₂₂^−1/2X2k with

Corr( ˆU1,Vˆ1) = ˆρj = qλˆj.

Example

I The following problem (Johnson-Wichern, 2009) seeks to study the relationship between 5 job characteristics and7job satisfaction variables, i.e. r = 5 ands= 7.

I The data is based on interviews of 784executives.

I The following is the correlation matrix :



















1.00 0.49 0.53 049 0.51 0.33 0.32 0.20 0.19 0.30 0.37 0.21 0.49 1.00 0.57 0.46 0.53 0.30 0.21 0.16 0.08 0.27 0.35 0.20 0.53 0.57 1.00 0.48 0.57 0.31 0.23 0.14 0.07 0.24 0.37 0.18 0.49 0.46 0.48 1.00 0.57 0.24 0.22 0.12 0.1‘9 0.21 0.29 0.26 0.51 0.53 0.57 0.57 1.00 0.38 0.32 0.17 0.23 0.32 0.36 0.27 0.33 0.30 0.31 0.24 0.38 1.00 0.43 0.27 0.24 0.34 0.37 0.40 0.32 0.21 0.23 0.22 0.32 0.43 1.00 0.33 0.26 0.54 0.32 0.58 0.20 0.16 0.14 0.12 0.17 0.27 0.33 1.00 0.25 0.46 0.29 0.45 0.19 0.08 0.07 0.19 0.23 0.24 0.26 0.25 1.00 0.28 0.30 0.27 0.30 0.27 0.24 0.21 0.32 0.34 0.52 0.46 0.28 1.00 0.35 0.59 0.37 0.35 0.37 0.29 0.36 0.37 0.32 0.29 0.30 0.35 1.00 0.31 0.21 0.20 0.18 0.26 0.27 0.40 0.58 0.45 0.27 0.59 0.31 1.00



















Example

I The following problem (Johnson-Wichern, 2009) seeks to study the relationship between 5 job characteristics and7job satisfaction variables, i.e. r = 5 ands= 7.

I The data is based on interviews of 784executives.

I The following is the correlation matrix :



















1.00 0.49 0.53 049 0.51 0.33 0.32 0.20 0.19 0.30 0.37 0.21 0.49 1.00 0.57 0.46 0.53 0.30 0.21 0.16 0.08 0.27 0.35 0.20 0.53 0.57 1.00 0.48 0.57 0.31 0.23 0.14 0.07 0.24 0.37 0.18 0.49 0.46 0.48 1.00 0.57 0.24 0.22 0.12 0.1‘9 0.21 0.29 0.26 0.51 0.53 0.57 0.57 1.00 0.38 0.32 0.17 0.23 0.32 0.36 0.27 0.33 0.30 0.31 0.24 0.38 1.00 0.43 0.27 0.24 0.34 0.37 0.40 0.32 0.21 0.23 0.22 0.32 0.43 1.00 0.33 0.26 0.54 0.32 0.58 0.20 0.16 0.14 0.12 0.17 0.27 0.33 1.00 0.25 0.46 0.29 0.45 0.19 0.08 0.07 0.19 0.23 0.24 0.26 0.25 1.00 0.28 0.30 0.27 0.30 0.27 0.24 0.21 0.32 0.34 0.52 0.46 0.28 1.00 0.35 0.59 0.37 0.35 0.37 0.29 0.36 0.37 0.32 0.29 0.30 0.35 1.00 0.31 0.21 0.20 0.18 0.26 0.27 0.40 0.58 0.45 0.27 0.59 0.31 1.00



















Example

I The following problem (Johnson-Wichern, 2009) seeks to study the relationship between 5 job characteristics and7job satisfaction variables, i.e. r = 5 ands= 7.

I The data is based on interviews of 784executives.

I The following is the correlation matrix :



















1.00 0.49 0.53 049 0.51 0.33 0.32 0.20 0.19 0.30 0.37 0.21 0.49 1.00 0.57 0.46 0.53 0.30 0.21 0.16 0.08 0.27 0.35 0.20 0.53 0.57 1.00 0.48 0.57 0.31 0.23 0.14 0.07 0.24 0.37 0.18 0.49 0.46 0.48 1.00 0.57 0.24 0.22 0.12 0.1‘9 0.21 0.29 0.26 0.51 0.53 0.57 0.57 1.00 0.38 0.32 0.17 0.23 0.32 0.36 0.27 0.33 0.30 0.31 0.24 0.38 1.00 0.43 0.27 0.24 0.34 0.37 0.40 0.32 0.21 0.23 0.22 0.32 0.43 1.00 0.33 0.26 0.54 0.32 0.58 0.20 0.16 0.14 0.12 0.17 0.27 0.33 1.00 0.25 0.46 0.29 0.45 0.19 0.08 0.07 0.19 0.23 0.24 0.26 0.25 1.00 0.28 0.30 0.27 0.30 0.27 0.24 0.21 0.32 0.34 0.52 0.46 0.28 1.00 0.35 0.59 0.37 0.35 0.37 0.29 0.36 0.37 0.32 0.29 0.30 0.35 1.00 0.31 0.21 0.20 0.18 0.26 0.27 0.40 0.58 0.45 0.27 0.59 0.31 1.00



















Example (contd.)

I The correlation matrix of the 12 variables imply that we need to look at 35 correlations (the largest being0.59) to

understand the relationship between the two sets. This is difficult to interpret.

I Instead do a canonical correlation.

I The first canonical variables are

Uˆ1 = 0.42X11+ 0.21X12+ 0.17X13−0.02X14+ 0.44X15

Vˆ1 = 0.42X21+ 0.22X22−0.03X23+ 0.01X24+ 0.29X25

+0.52X₂₆−0.12X₂₇ with Corr( ˆU₁,Vˆ₁) = 0.55.

I The 1^st canonical correlation coefficient shows that there is reasonable relationship between the two groups.

Example (contd.)

I The correlation matrix of the 12 variables imply that we need to look at 35 correlations (the largest being0.59) to

understand the relationship between the two sets. This is difficult to interpret.

I Instead do a canonical correlation.

I The first canonical variables are

Uˆ1 = 0.42X11+ 0.21X12+ 0.17X13−0.02X14+ 0.44X15

Vˆ1 = 0.42X21+ 0.22X22−0.03X23+ 0.01X24+ 0.29X25

+0.52X₂₆−0.12X₂₇ with Corr( ˆU₁,Vˆ₁) = 0.55.

I The 1^st canonical correlation coefficient shows that there is reasonable relationship between the two groups.

Example (contd.)

I The correlation matrix of the 12 variables imply that we need to look at 35 correlations (the largest being0.59) to

understand the relationship between the two sets. This is difficult to interpret.

I Instead do a canonical correlation.

I The first canonical variables are

Uˆ1 = 0.42X11+ 0.21X12+ 0.17X13−0.02X14+ 0.44X15

Vˆ1 = 0.42X21+ 0.22X22−0.03X23+ 0.01X24+ 0.29X25

+0.52X₂₆−0.12X₂₇ with Corr( ˆU₁,Vˆ₁) = 0.55.

I The 1^st canonical correlation coefficient shows that there is reasonable relationship between the two groups.

Example (contd.)

I The correlation matrix of the 12 variables imply that we need to look at 35 correlations (the largest being0.59) to

understand the relationship between the two sets. This is difficult to interpret.

I Instead do a canonical correlation.

I The first canonical variables are

Uˆ1 = 0.42X11+ 0.21X12+ 0.17X13−0.02X14+ 0.44X15

Vˆ1 = 0.42X21+ 0.22X22−0.03X23+ 0.01X24+ 0.29X25

+0.52X₂₆−0.12X₂₇ with Corr( ˆU₁,Vˆ₁) = 0.55.

I The 1^st canonical correlation coefficient shows that there is reasonable relationship between the two groups.

Advantages of canonical correlations

I Firstly it reduces the number of correlations coefficients to be studied from rsto 1 or2. Particularly, convenient when r and sare large.

I Secondly, if the canonical correlations are high, then we can work with one group only instead of both (usually the smaller), i.e. it can act as a dimension reduction technique.

Advantages of canonical correlations

I Firstly it reduces the number of correlations coefficients to be studied from rsto 1 or2. Particularly, convenient when r and sare large.

I Secondly, if the canonical correlations are high, then we can work with one group only instead of both (usually the smaller), i.e. it can act as a dimension reduction technique.

Summary

I The canonical correlation of standardized variables is discussed.

I Results relating the original and canonical variables are derived.

I Some special cases are listed.

I A numerical illustration is given.

Summary

I The canonical correlation of standardized variables is discussed.

I Results relating the original and canonical variables are derived.

I Some special cases are listed.

I A numerical illustration is given.

Summary

I The canonical correlation of standardized variables is discussed.

I Results relating the original and canonical variables are derived.

I Some special cases are listed.

I A numerical illustration is given.

Summary

I The canonical correlation of standardized variables is discussed.

I Results relating the original and canonical variables are derived.

I Some special cases are listed.

I A numerical illustration is given.