Kalman  filtering					Kalman smoothing		
	時点	観測値	観測方程式の誤差項の分散	状態方程式の誤差項の分散	カルマンゲイン	フィルター化推定値	フィルター化推定値の分散	1期先の予測値	１期先の予測値の分散			平滑化推定値
	time	observation	variance of observation error	variance of process noise	Kalman gain	filtered estimates	variance　of filtered estimates	1-step-ahead forecasts	variance of 1-steo -ahead forecasts	t=T-1,T-2,T-3,....3,.2.,1  とTから過去に遡る逐次計算(backward recursion)		smoothed estimates
KFAS		$y				$att	$Ptt	$a	$P			$alphahat
	t	y[t]	Ht	Qt	Kt	a{t|t]	p[t|t]	a[t+1|t]	p[t+1|t]	a[t+1|T]-a[t|t]	p[t|t]/p[t+1|t]	a[t|T]
	0							200	10000			
	1	1120	12000	1300	=J7/(J7+D7)	=I7+(C7-I7)*F7	=J7-J7*F7	=I6	=J6+E7	=M8-G7	=H7/J8	=G7+K7*L7
	2	1160	12000	1300	=J8/(J8+D8)	=I8+(C8-I8)*F8	=J8-J8*F8	=G7	=H7+E8	=M9-G8	=H8/J9	=G8+K8*L8
	3	963	12000	1300	=J9/(J9+D9)	=I9+(C9-I9)*F9	=J9-J9*F9	=G8	=H8+E9	=M10-G9	=H9/J10	=G9+K9*L9
	4	1210	12000	1300	=J10/(J10+D10)	=I10+(C10-I10)*F10	=J10-J10*F10	=G9	=H9+E10	=M11-G10	=H10/J11	=G10+K10*L10
	5	1160	12000	1300	=J11/(J11+D11)	=I11+(C11-I11)*F11	=J11-J11*F11	=G10	=H10+E11	=M12-G11	=H11/J12	=G11+K11*L11
	6	1160	12000	1300	=J12/(J12+D12)	=I12+(C12-I12)*F12	=J12-J12*F12	=G11	=H11+E12	=M13-G12	=H12/J13	=G12+K12*L12
	7	813	12000	1300	=J13/(J13+D13)	=I13+(C13-I13)*F13	=J13-J13*F13	=G12	=H12+E13	=M14-G13	=H13/J14	=G13+K13*L13
	8	1230	12000	1300	=J14/(J14+D14)	=I14+(C14-I14)*F14	=J14-J14*F14	=G13	=H13+E14	=M15-G14	=H14/J15	=G14+K14*L14
	9	1370	12000	1300	=J15/(J15+D15)	=I15+(C15-I15)*F15	=J15-J15*F15	=G14	=H14+E15	=M16-G15	=H15/J16	=G15+K15*L15
	10	1140	12000	1300	=J16/(J16+D16)	=I16+(C16-I16)*F16	=J16-J16*F16	=G15	=H15+E16			=G16
	11	?					t=11 の時の予測値	=G16