(defun square (x) (* x x))
(defun expect (x p)
  (let ((e 0))
    (mapc #'(lambda (xk pk)
          (setf e (+ e (* xk pk))))
      x p)
    e))

(defun variance (x p)
  (let ((e (expect x p))
    (e2 (expect (mapcar #'square x) p)))
    (- e2 (square e))))

(defun sigma (x p) "standard deviation"
  (sqrt (variance x p)))

(defun transpose (m) "transpose arbitrary MxN 2D list"
  (labels ((first-column-vector (m)
         (if (null m) nil
         (cons (caar m) (first-column-vector (cdr m)))))
       (delete-first-column (m)
         (if (null m) nil
         (cons (cdar m) (delete-first-column (cdr m))))))
    (if (null (car m)) nil
    (cons (first-column-vector m)
          (transpose (delete-first-column m))))))

(defun marginal-expect (xm pm &key (by :row)) "xm: 1D, pm: 2D"
  (expect xm (mapcar #'(lambda (p) (apply #'+ p))
             (case by (:row pm) (:col (transpose pm))))))

(defun covariance (pm &key row col) "pm: 2D"
  (labels ((sum2d (m)
         (apply #'+
            (mapcar (lambda (i) (apply #'+ i))
                m)))
       (conv (xm pm)
         (if (null xm) nil
         (cons (mapcar #'* (car xm) (car pm))
               (conv (cdr xm) (cdr pm)))))
       (cartesian (row col)
         (mapcar #'(lambda (ritem)
             (mapcar #'(lambda (citem)
                     (* ritem citem))
                 col))
             row)))
    (let ((e-rc (sum2d (conv (cartesian row col) pm)))
      (e-r (marginal-expect row pm :by :row))
      (e-c (marginal-expect col pm :by :col)))
      (- e-rc (* e-r e-c)))))

(defun linear-comb (xs ys &key (comb-method #'+) (mult-method #'*))
  (apply comb-method (mapcar mult-method xs ys)))


;; Probability table 작성
;; row:0<여행횟수<4, col:0<=소지 카드 수<3 일 때
(defvar *cards* '(0 1 2))
(defvar *travs* '(1 2 3))
(defvar *pt*'((0.32 0.18 0.04)
          (0.12 0.14 0.06)
          (0.01 0.03 0.10)))

;; (주변확률분포) 여행횟수의 기댓값 구하기
(marginal-expect *travs* *pt* :by :row) ;=> 1.6

;; 소지 카드 수의 기댓값
(marginal-expect *cards* *pt* :by :col) ;=> 0.75

;; 여행횟수와 카드 수 사이의 공분산 구하기
(covariance *pt* :row *travs* :col *cards*) ;=> 0.27

;; linear combination 할일이 많아서 만듬
(linear-comb '(0.4 0.3 0.2 0.1) '(10 20 30 40)) ;=> 20 (가중평균)
(linear-comb '(3 2 1) '(x y z) :comb-method #'list :mult-method #'cons)
;;=> ((3 . X) (2 . Y) (1 . Z))