銆�銆�棣栧厛锛屽紩鍏ヤ竴鍊�(g猫)闆㈡暎鎺у埗閬庣▼鐨勭郴绲�(t菕ng)銆傝┎绯荤当(t菕ng)鍙�鐢ㄤ竴鍊�(g猫)绶氭�ч毃姗�(j墨)寰�鍒嗘柟绋嬶紙Linear Stochastic Difference equation锛変締鎻忚堪锛�

銆�銆�X(k)=A X(k-1)+B U(k)+W(k)

銆�銆�鍐嶅姞涓婄郴绲�(t菕ng)鐨勬脯閲忓�硷細

銆�銆�Z(k)=H X(k)+V(k)

銆�銆�涓婂叐寮忓瓙涓�锛孹(k)鏄痥鏅�(sh铆)鍒荤殑绯荤当(t菕ng)鐙�鎱�(t脿i)锛孶(k)鏄痥鏅�(sh铆)鍒诲皪绯荤当(t菕ng)鐨勬帶鍒堕噺銆侫鍜孊鏄�绯荤��(t菕ng)鍙冩暩(sh霉)锛屽皪浜庡�氭ā鍨嬬郴绲�(t菕ng)锛屼粬鍊�?y艒u)姒�浠�?銆俍(k)鏄痥鏅�(sh铆)鍒荤殑娓�閲忓�硷紝H鏄�娓�閲忕郴绲�(t菕ng)鐨勫弮鏁�(sh霉)锛屽皪浜庡�氭脯閲忕郴绲�(t菕ng)锛孒鐐虹煩闄�銆俉(k)鍜孷(k)鍒嗗垾琛ㄧず閬庣▼鍜屾脯閲忕殑鍣�鑱�銆備粬鍊戣��鍋囪ō(sh猫)鎴愰珮鏂�鐧藉櫔鑱�(White Gaussian Noise)锛屼粬鍊戠殑covariance 鍒嗗垾鏄疩锛孯锛堥�欓噷鍋囪ō(sh猫)涓嶉毃绯荤当(t菕ng)鐙�鎱�(t脿i)璁婂寲鑰岃畩鍖栵級銆�

銆�銆�灏嶄簬婊胯冻涓婇潰鐨勬�濅��(绶氭�ч毃姗�(j墨)寰�鍒嗙郴绲�(t菕ng)锛岄亷绋嬪拰娓�閲忛兘鏄�楂樻柉鐧藉櫔鑱�)锛�鍗＄埦鏇兼烤娉㈠櫒鏄�鐨勪俊鎭�铏曠悊鍣�銆備笅闈㈡垜鍊戜締鐢ㄤ粬鍊戠祼(ji茅)鍚堜粬鍊戠殑covariances 渚嗕及绠楃郴绲�(t菕ng)鐨勫寲杓稿嚭锛堥�炰技涓婁竴绡�(ji茅)閭ｅ��(g猫)婧�搴︾殑渚嬪瓙锛夈��

銆�銆�棣栧厛鍒╃敤绯荤当(t菕ng)鐨勯亷绋嬫ā鍨�锛屼締闋�(y霉)娓�涓嬩竴鐙�鎱�(t脿i)鐨勭郴绲�(t菕ng)銆傚亣瑷�(sh猫)鐝�(xi脿n)鍦ㄧ殑绯荤当(t菕ng)鐙�鎱�(t脿i)鏄痥锛屾牴鎿�(j霉)绯荤当(t菕ng)鐨勬ā鍨�锛屽彲浠ュ熀浜庣郴绲�(t菕ng)鐨勪笂涓�鐙�鎱�(t脿i)鑰岄爯(y霉)娓�鍑虹��(xi脿n)鍦ㄧ媭鎱�(t脿i)锛�

銆�銆�X(k|k-1)=A X(k-1|k-1)+B U(k) 鈥︹�︹��.. (1)

銆�銆�寮�(1)涓�锛孹(k|k-1)鏄�鍒╃敤涓婁竴鐙�鎱�(t脿i)闋�(y霉)娓�鐨勭��(ji茅)鏋滐紝X(k-1|k-1)鏄�涓婁竴鐙�鎱�(t脿i)鐨勭祼(ji茅)鏋�锛孶(k)鐐虹従(xi脿n)鍦ㄧ媭鎱�(t脿i)鐨勬帶鍒堕噺锛屽�傛灉娌掓湁鎺у埗閲�锛屽畠鍙�浠ョ��0銆�

銆�銆�鍒扮従(xi脿n)鍦ㄧ偤姝�锛屾垜鍊戠殑绯荤当(t菕ng)绲�(ji茅)鏋滃凡缍�(j墨ng)鏇存柊浜嗭紝鍙�鏄�锛屽皪鎳�(y墨ng)浜嶺(k|k-1)鐨刢ovariance閭勬矑鏇存柊銆傛垜鍊戠敤P琛ㄧずcovariance锛�

銆�銆�P(k|k-1)=A P(k-1|k-1) A鈥�+Q 鈥︹�︹�� (2)

銆�銆�寮�(2)涓�锛孭(k|k-1)鏄疿(k|k-1)灏嶆噳(y墨ng)鐨刢ovariance锛孭(k-1|k-1)鏄疿(k-1|k-1)灏嶆噳(y墨ng)鐨刢ovariance锛孉鈥欒〃绀篈鐨勮綁(zhu菐n)缃�鐭╅櫍锛孮鏄�绯荤��(t菕ng)閬庣▼鐨刢ovariance銆傚紡瀛�1锛�2灏辨槸鍗＄埦鏇兼烤娉㈠櫒5鍊�(g猫)鍏�寮忕��(d膩ng)涓�鐨勫墠鍏╁��(g猫)锛屼篃灏辨槸灏嶇郴绲�(t菕ng)鐨勯爯(y霉)娓�銆�

銆�銆�鐝�(xi脿n)鍦ㄦ垜鍊戞湁浜嗙従(xi脿n)鍦ㄧ媭鎱�(t脿i)鐨勯爯(y霉)娓�绲�(ji茅)鏋�锛岀劧鍚庢垜鍊戝啀鏀堕泦鐝�(xi脿n)鍦ㄧ媭鎱�(t脿i)鐨勬脯閲忓��銆傜祼(ji茅)鍚堥爯(y霉)娓�鍊煎拰娓�閲忓�硷紝鎴戝�戝彲浠ュ緱鍒扮従(xi脿n)鍦ㄧ媭鎱�(t脿i)(k)鐨勫寲浼扮畻鍊糥(k|k)锛�

銆�銆�X(k|k)= X(k|k-1)+Kg(k) (Z(k)-H X(k|k-1)) 鈥︹�︹�� (3)

銆�銆�鍏朵腑Kg鐐哄崱鐖炬浖澧炵泭(Kalman Gain)锛�

銆�銆�Kg(k)= P(k|k-1) H鈥� / (H P(k|k-1) H鈥� + R) 鈥︹�︹�� (4)

銆�銆�鍒扮従(xi脿n)鍦ㄧ偤姝�锛屾垜鍊戝凡缍�(j墨ng)寰楀埌浜唊鐙�鎱�(t脿i)涓嬬殑浼扮畻鍊糥(k|k)銆備絾鏄�鐐轰簡瑕佸彟鍗＄埦鏇兼烤娉㈠櫒涓嶆柗鐨勯��(y霉n)琛屼笅鍘荤洿鍒扮郴绲�(t菕ng)閬庣▼绲�(ji茅)鏉燂紝鎴戝�戦倓瑕佹洿鏂発鐙�鎱�(t脿i)涓媂(k|k)鐨刢ovariance锛�

銆�銆�P(k|k)=锛圛-Kg(k) H锛塒(k|k-1) 鈥︹�︹�� (5)

銆�銆�鍏朵腑I 鐐�1鐨勭煩闄ｏ紝灏嶄簬鍠�妯″瀷鍠�娓�閲�锛孖=1銆傜暥(d膩ng)绯荤当(t菕ng)閫�(j矛n)鍏�k+1鐙�鎱�(t脿i)鏅�(sh铆)锛孭(k|k)灏辨槸寮忓瓙(2)鐨凱(k-1|k-1)銆傞�欐ǎ锛岀畻娉曞氨鍙�浠ヨ嚜鍥炴�哥殑閬�(y霉n)绠椾笅鍘�銆�

銆�銆�鍗＄埦鏇兼烤娉㈠櫒鐨勫師鐞嗗熀鏈�鎻忚堪浜�锛屽紡瀛�1锛�2锛�3锛�4鍜�5灏辨槸浠栫殑5 鍊�(g猫)鍩烘湰鍏�寮忋�傛牴鎿�(j霉)閫�5鍊�(g猫)鍏�寮�锛屽彲浠ュ緢瀹规槗鐨勫��(sh铆)鐝�(xi脿n)瑷�(j矛)绠楁��(j墨)鐨勭▼搴�銆�

銆�銆�鑸変緥瀛愪締瑾�鏄庡崱鐖炬浖婵炬尝鍣ㄧ殑宸ヤ綔閬庣��锛岄厤浠ョ▼搴忔ā鎿�绲�(ji茅)鏋�銆�

銆�銆�鍏堟妸鎴块枔鐪嬫垚涓�鍊�(g猫)绯荤当(t菕ng)锛岀劧鍚庡皪閫欏��(g猫)绯荤当(t菕ng)寤烘ā銆傜暥(d膩ng)鐒讹紝鎴戝�戣�嬬殑妯″瀷涓嶉渶瑕侀潪甯稿湴绮剧⒑銆傛垜鍊戞墍鐭ラ亾鐨勯�欏��(g猫)鎴块枔鐨勬韩搴︽槸璺熷墠涓�鏅�(sh铆)鍒荤殑婧�搴︾浉鍚岀殑锛屾墍浠�A=1銆傛矑鏈夋帶鍒堕噺锛屾墍浠�U(k)=0銆傚洜姝ゅ緱鍑猴細

銆�銆�X(k|k-1)=X(k-1|k-1) 鈥︹�︹��.. (6)

銆�銆�寮忓瓙锛�2锛夊彲浠ユ敼鎴愶細

銆�銆�P(k|k-1)=P(k-1|k-1) +Q 鈥︹�︹�� (7)

銆�銆�鍥�?y脿n)闂噛閲忕殑鍊兼槸婧�搴﹁��(j矛)鐨�锛岃窡婧�搴︾洿鎺ュ皪鎳�(y墨ng)锛屾墍浠�H=1銆傚紡瀛�3锛�4锛�5鍙�浠ユ敼鎴愪互涓嬶��

銆�銆�X(k|k)= X(k|k-1)+Kg(k) (Z(k)-X(k|k-1)) 鈥︹�︹�� (8)

銆�銆�Kg(k)= P(k|k-1) / (P(k|k-1) + R) 鈥︹�︹�� (9)

銆�銆�P(k|k)=锛�1-Kg(k)锛塒(k|k-1) 鈥︹�︹�� (10)

銆�銆�鐝�(xi脿n)鍦ㄦ垜鍊戞ā鎿�涓�绲勬脯閲忓�间綔鐐鸿几鍏�銆傚亣瑷�(sh猫)鎴块枔鐨勭湡瀵�(sh铆)婧�搴︾��25搴�锛屾垜妯℃摤浜�200鍊�(g猫)娓�閲忓��锛岄�欎簺娓�閲忓�肩殑骞冲潎鍊肩偤25搴�锛屼絾鏄�鍔犲叆浜嗘��(bi膩o)婧�(zh菙n)鍋忓樊鐐哄咕搴︾殑楂樻柉鐧藉櫔鑱诧紙鍦ㄥ湒涓�鐐鸿��(l谩n)绶氾級銆�

銆�銆�鐐轰簡浠ゅ崱鐖炬浖婵炬尝鍣ㄩ枊濮嬪伐浣滐紝鎴戝�戦渶瑕佸憡瑷村崱鐖炬浖鍏╁��(g猫)闆舵檪(sh铆)鍒荤殑鍒濆�嬪��锛屾槸X(0|0)鍜孭(0|0)銆備粬鍊戠殑鍊间笉鐢ㄥお鍦ㄦ剰锛岄毃渚跨郸涓�鍊�(g猫)灏卞彲浠ヤ簡锛屽洜?y脿n)娈�S钁楀崱鐖炬浖鐨勫伐浣滐紝X鏈冮�愭几鐨勬敹鏂�銆備絾鏄�灏嶄簬P锛屼竴鑸�涓嶈�佸彇0锛屽洜?y脿n)妾獲妯ｅ彲鑳芥渻浠ゅ崱鐖炬浖瀹屽叏鐩镐俊浣犵郸瀹氱殑X(0|0)鏄�绯荤��(t菕ng)鐨�锛屽緸鑰屼娇绠楁硶涓嶈兘鏀舵杺銆傛垜閬镐簡X(0|0)=1搴�锛孭(0|0)=10銆�

銆�銆�瑭茬郴绲�(t菕ng)鐨勭湡瀵�(sh铆)婧�搴︾��25搴�锛屽湒涓�鐢ㄩ粦绶氳〃绀�銆傚湒涓�绱呯窔鏄�鍗＄埦鏇兼烤娉㈠櫒杓稿嚭鐨勫寲绲�(ji茅)鏋滐紙瑭茬祼(ji茅)鏋滃湪绠楁硶涓�瑷�(sh猫)缃�浜哘=1e-6锛孯=1e-1锛�

銆�銆�Kalman_filter

銆�銆�Kalman_smoother - implements the RTS equations

銆�銆�Learn_kalman - finds maximum likelihood estimates of the parameters using EM

銆�銆�Sample_lds - generate random samples

銆�銆�AR_to_SS - convert Auto Regressive model of order k to State Space form

銆�銆�SS_to_AR

銆�銆�Learn_AR - finds maximum likelihood estimates of the parameters using least squares