▁and ▁a ▁criter ion ▁variable ▁by ▁the ▁coefficient ▁of ▁correlation ▁r , ▁coefficient ▁of ▁determ ination ▁r 2 , ▁and ▁the ▁standard ▁regression ▁coefficient ▁ β . ▁Multiple ▁regression ▁analysis ▁express es ▁a ▁relationship ▁between ▁a ▁set ▁of ▁predict or ▁variables ▁and ▁a ▁single ▁criter ion ▁variable ▁by ▁the ▁multiple ▁correlation   R , ▁multiple ▁coefficient ▁of ▁determ ination ▁R ², ▁and ▁a ▁set ▁of ▁standard ▁partial ▁regression ▁weights ▁ β 1 , ▁ β 2 , ▁etc . ▁Can on ical ▁vari ate ▁analysis ▁capt ures ▁a ▁relationship ▁between ▁a ▁set ▁of ▁predict or ▁variables ▁and ▁a ▁set ▁of ▁criter ion ▁variables ▁by ▁the ▁canonical ▁correl ations ▁ ρ 1 , ▁ ρ 2 , ▁... , ▁and ▁by ▁the ▁sets ▁of ▁canonical ▁weights ▁C ▁and ▁D . ▁ ▁Can on ical ▁analysis ▁ ▁Can on ical ▁analysis ▁belongs ▁to ▁a ▁group ▁of ▁methods ▁which ▁involve ▁solving ▁the ▁characteristic ▁equation ▁for ▁its ▁lat ent ▁roots ▁and ▁vectors . ▁It ▁describes ▁formal ▁structures ▁in ▁hyp ers pace ▁invariant ▁with ▁respect ▁to ▁the ▁rotation ▁of ▁their ▁coordinates . ▁In ▁this ▁type ▁of ▁solution , ▁rotation ▁leaves ▁many ▁optim izing ▁properties ▁preserved , ▁provided ▁it ▁takes ▁place ▁in ▁certain ▁ways ▁and ▁in ▁a ▁sub space ▁of ▁its ▁corresponding ▁hyp ers pace . ▁This ▁rotation ▁from ▁the ▁maximum ▁inter vari ate ▁correlation ▁structure ▁into ▁a ▁different , ▁simpler ▁and ▁more ▁meaning ful ▁structure ▁increases ▁the ▁interpret ability ▁of ▁the ▁canonical ▁weights ▁C ▁and ▁D . ▁In ▁this ▁the ▁canonical ▁analysis ▁diff ers ▁from ▁Harold ▁Hot elling