(defun variance (lst)                                                                                                                               
            (let ((mean1 (mean lst))                                                                                                                       
                 (n (length lst)))                                                                                                               
  (/ (reduce #'+ (map 'list #'(lambda (x) (square (- mean1 x))) lst))                                                                     
              n)))          

defmacro square (x)
  `(* ,x ,x))

(defun mean (lst)                                                                                                                                   
  (/ (reduce #'+ lst) (length lst)))   

;funtion to calculate groveidentity(when G,T,A,S are considered)
(defun groveidentity(pa pt x y sd mu tau)                                                                                                          
                (let*((l1 (- (/ 1 pa) 1))                                                                                                                   
                           (l2 (grovelr x y sd mu tau))                                                                                                     
                               (l3 (* l1 l2))                                                                                                               
                                   (factor (+ 1 (* pt  (- l3 1)))))                                                                                         
                                    factor)) 
;function to calculate grove LR
(defun grovelr(x y sd mu tau)                                                                                                                 
          (let* ((u (/ tau sd))                                                                                                                       
         (v (/ (* (- x y)(- x y) )(* 4 sd sd) ))                                                                                                            (w (/ (* (- y mu)(- y mu) )(* 2 tau tau)))                                                                                                             
           (lr (* u (+ (exp v) (exp w)))))                                                                                                                 
                              lr))       

(defun grove(&optional (x-values #(1.518255 1.518199 1.518273 1.518292)) 
                       (y-values #(1.518273  1.518347  1.518119  1.518255))
		       (mu 1.5182599)
		       (tau 0.00367)
	               (pa 0.01)
                       (pb 0.9))
       (let((X (mean x-values))
       (Y (mean y-values))
       (sd (sqrt (variance x-values))))
      (format t "grove lr is ~A   " (grovelr X Y sd mu tau))
     (format t "grove identity is ~A" (groveidentity pa pb X Y sd mu tau))))