前面最小二乘法中数据是一次全部测量好，然后进行求解。但实际中有时存在测量数据是在线获得的，即时刻获得测量数据。比如无人驾驶车辆，需要实时判断前面车辆的运动状态，获取其加速度，所以每时每刻都需要进行测量。当每次获得新的测量数据后，需要进行最小二乘法以更新车辆状态。如果每次更新时，都采用公式
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\mathbf{\hat{x}} = (A^TA)^{-1}A^T\mathbf{b}
x ^ = ( A T A ) − 1 A T b
为了使记号简洁，我们令
x
^
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\mathbf{\hat{x}_m}
x ^ m
m
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A^TA
A T A
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A = \left[ \begin{matrix} \mathbf{a^T_{r1}} \\ \mathbf{a^T_{r2}} \\ \vdots \\ \mathbf{a^T_{rm}} \end{matrix} \right]
A = ⎣ ⎢ ⎢ ⎢ ⎡ a r 1 T a r 2 T ⋮ a r m T ⎦ ⎥ ⎥ ⎥ ⎤
m
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A^TA= \left[ \begin{matrix} \mathbf{a_{r1}} , \mathbf{a_{r2}}, \cdots , \mathbf{a_{rm}} \end{matrix} \right] \left[ \begin{matrix} \mathbf{a^T_{r1}} \\ \mathbf{a^T_{r2}} \\ \vdots \\ \mathbf{a^T_{rm}} \end{matrix} \right]= \mathbf{a_{r1}}\mathbf{a^T_{r1}} + \mathbf{a_{r2}}\mathbf{a^T_{r2}} + \cdots + \mathbf{a_{rm}}\mathbf{a^T_{rm}}
A T A = [ a r 1 , a r 2 , ⋯ , a r m ] ⎣ ⎢ ⎢ ⎢ ⎡ a r 1 T a r 2 T ⋮ a r m T ⎦ ⎥ ⎥ ⎥ ⎤ = a r 1 a r 1 T + a r 2 a r 2 T + ⋯ + a r m a r m T
令
P
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m
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P_m = (A^TA)_m^{-1}
P m = ( A T A ) m − 1
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P_{m} = (\mathbf{a_{r1}}\mathbf{a^T_{r1}} + \cdots + \mathbf{a_{r(m-1)}}\mathbf{a^T_{r(m-1)}} + \mathbf{a_{rm}}\mathbf{a^T_{rm}} )^{-1}=(P_{m-1}^{-1}+\mathbf{a_{rm}}\mathbf{a^T_{rm}})^{-1}
P m = ( a r 1 a r 1 T + ⋯ + a r ( m − 1 ) a r ( m − 1 ) T + a r m a r m T ) − 1 = ( P m − 1 − 1 + a r m a r m T ) − 1
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A^T\mathbf{b} = \left[ \begin{matrix} \mathbf{a_{r1}} , \mathbf{a_{r2}}, \cdots , \mathbf{a_{rm}} \end{matrix} \right] \left[ \begin{matrix} b_1 \\ b_2 \\ \vdots \\ b_m \end{matrix} \right]= b_1\mathbf{a_{r1}} + b_2\mathbf{a_{r2}} + \cdots + b_m\mathbf{a_{rm}}
A T b = [ a r 1 , a r 2 , ⋯ , a r m ] ⎣ ⎢ ⎢ ⎢ ⎡ b 1 b 2 ⋮ b m ⎦ ⎥ ⎥ ⎥ ⎤ = b 1 a r 1 + b 2 a r 2 + ⋯ + b m a r m
令
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B_m = (A^T\mathbf{b})_m
B m = ( A T b ) m
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B_{m} = b_1\mathbf{a_{r1}} + \cdots + b_{m-1}\mathbf{a_{r(m-1)}} + b_m\mathbf{a_{rm}} = B_{m-1} + b_m\mathbf{a_{rm}}
B m = b 1 a r 1 + ⋯ + b m − 1 a r ( m − 1 ) + b m a r m = B m − 1 + b m a r m
根据公式
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\mathbf{\hat{x}} = (A^TA)^{-1}A^T\mathbf{b}
x ^ = ( A T A ) − 1 A T b
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\mathbf{\hat{x}_{m}} = P_{m}B_{m} = (P_{m-1}^{-1}+\mathbf{a_{rm}}\mathbf{a^T_{rm}})^{-1}(B_{m-1} + b_{m}\mathbf{a_{rm}})
x ^ m = P m B m = ( P m − 1 − 1 + a r m a r m T ) − 1 ( B m − 1 + b m a r m )
这就是获得第
m
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P_{m-1}^{-1}
P m − 1 − 1
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B m − 1
根据
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\mathbf{\hat{x}_{m}} = P_{m}B_{m}
x ^ m = P m B m
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B_{m-1} = P_{m-1}^{-1}\mathbf{\hat{x}_{m-1}}
B m − 1 = P m − 1 − 1 x ^ m − 1
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P_{m}^{-1} =P_{m-1}^{-1}+\mathbf{a_{rm}}\mathbf{a^T_{rm}}
P m − 1 = P m − 1 − 1 + a r m a r m T
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\mathbf{\hat{x}_{m}} = P_{m}(P_{m-1}^{-1}\mathbf{\hat{x}_{m-1}} + b_{m}\mathbf{a_{rm}})\\=P_{m}( (P_{m}^{-1}-\mathbf{a_{rm}}\mathbf{a^T_{rm}})\mathbf{\hat{x}_{m-1}} + b_{m}\mathbf{a_{rm}} )\\=\mathbf{\hat{x}_{m-1}}+P_{m}\mathbf{a_{rm}}(b_{m}-\mathbf{a^T_{rm}}\mathbf{\hat{x}_{m-1}})
x ^ m = P m ( P m − 1 − 1 x ^ m − 1 + b m a r m ) = P m ( ( P m − 1 − a r m a r m T ) x ^ m − 1 + b m a r m ) = x ^ m − 1 + P m a r m ( b m − a r m T x ^ m − 1 )
这就是最优近似解的递归公式。
递归公式中
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P_{m} = (P_{m-1}^{-1}+\mathbf{a_{rm}}\mathbf{a^T_{rm}})^{-1}
P m = ( P m − 1 − 1 + a r m a r m T ) − 1
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(A+BCD)^{-1}=A^{-1}-A^{-1}B(C^{-1}+DA^{-1}B)^{-1}DA^{-1}
( A + B C D ) − 1 = A − 1 − A − 1 B ( C − 1 + D A − 1 B ) − 1 D A − 1
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\mathbf{\hat{x}_{m}} =\mathbf{\hat{x}_{m-1}}+K_{m}\epsilon_{m}\\ \epsilon_{m} = b_{m}-\mathbf{a^T_{rm}}\mathbf{\hat{x}_{m-1}}\\ K_{m} = P_{m}\mathbf{a_{rm}}\\ P_{m} = P_{m-1} - \frac {P_{m-1}\mathbf{a_{rm}}\mathbf{a^T_{rm}}P_{m-1}}{1+\mathbf{a^T_{rm}}P_{m-1}\mathbf{a_{rm}}}
x ^ m = x ^ m − 1 + K m ϵ m ϵ m = b m − a r m T x ^ m − 1 K m = P m a r m P m = P m − 1 − 1 + a r m T P m − 1 a r m P m − 1 a r m a r m T P m − 1
递归最小二乘法中所有测量数据重要性都是一样的，这样可能会带来一个问题。还是以无人驾驶车辆实时判断前面车辆的运动状态为例，我们假设前面车辆做匀加速运动，实际上车辆运动状态时刻会发生变化，时而加速时而减速（这称为机动性），故为了准确获取车辆当前状态，当前数据的重要性，显然要大于历史数据，不能同等对待。而递归最小二乘法却同等对待，故最优近似解更新速度慢，不能实时反映车辆运作状态的改变，会慢半拍。如何提高当前数据的重要性，同时又不导致计算复杂度的增加，本节提出一种方法。根据
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P_{m} = P_{m-1} - \frac {P_{m-1}\mathbf{a_{rm}}\mathbf{a^T_{rm}}P_{m-1}}{1+\mathbf{a^T_{rm}}P_{m-1}\mathbf{a_{rm}}}
P m = P m − 1 − 1 + a r m T P m − 1 a r m P m − 1 a r m a r m T P m − 1
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K_{m} = P_{m}\mathbf{a_{rm}} = \frac {P_{m-1}\mathbf{a_{rm}}}{1+\mathbf{a^T_{rm}}P_{m-1}\mathbf{a_{rm}}}
K m = P m a r m = 1 + a r m T P m − 1 a r m P m − 1 a r m
最优近似解的更新量为
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K_{m}\epsilon_{m}
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K_{m} = \frac {P_{m-1}\mathbf{a_{rm}}}{1/\kappa+\mathbf{a^T_{rm}}P_{m-1}\mathbf{a_{rm}}} ,\kappa \ge 1
K m = 1 / κ + a r m T P m − 1 a r m P m − 1 a r m , κ ≥ 1
修正系数
κ
\kappa
κ
κ
\kappa
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κ
\kappa
κ
κ
\kappa
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＝
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\kappa＝1
κ ＝ 1