The rigorous variancecovariance component estimation in Least Squares proposed by Helmert (1907) was simplified to a VCE algorithm based on measurement redundant contribution (Förstner 1979), which becomes popular in real applications (Cui et al. 2001; Bähr et al. 2007; etc.). Furthermore, Wang (1997) transplanted it into Kalman filter through an alternative derivation of the Kalman filtering algorithm by applying three groups of the independent random information: the purelypredicted state vector, the process noise vector and the real measurement vector. This derivation can project the system innovation vector into the residuals associated with these above used three groups of the measurements and also makes possible the calculation of their redundancy contribution. So, the VCE process as in (Förstner 1979) can be realized in Kalman filter. For the needs of further development, the relevant details will briefly be reviewed below.
VCE using redundancy contribution in LS
Let least square system is represented by
$$ L+v=B\delta \widehat{x}+F\left({x}^{(0)}\right) $$
(2.1)
where are

L
:

the measurement vector

v
:

the measurement residual vector

B
:

the design matrix

F
:

the nonlinear observation equations

x
^{(0)}
:

the approximate of the parameter vector x

\( \delta \widehat{x} \)
:

the correction vector for x
^{(0)}.
Assume that L consists of m statistically independent measurement types, (2.1) can be partitioned into
$$ \left[\begin{array}{c}\hfill {L}_1\hfill \\ {}\hfill \vdots \hfill \\ {}\hfill {L}_i\hfill \\ {}\hfill \vdots \hfill \\ {}\hfill {L}_m\hfill \end{array}\right]+\left[\begin{array}{c}\hfill {v}_1\hfill \\ {}\hfill \vdots \hfill \\ {}\hfill {v}_i\hfill \\ {}\hfill \vdots \hfill \\ {}\hfill {v}_m\hfill \end{array}\right]=\left[\begin{array}{c}\hfill {B}_1\hfill \\ {}\hfill \vdots \hfill \\ {}\hfill {B}_i\hfill \\ {}\hfill \vdots \hfill \\ {}\hfill {B}_m\hfill \end{array}\right]\delta \widehat{x}\left[\begin{array}{c}\hfill {f}_1\left({x}^{(0)}\right)\hfill \\ {}\hfill \vdots \hfill \\ {}\hfill {f}_i\left({x}^{(0)}\right)\hfill \\ {}\hfill \vdots \hfill \\ {}\hfill {f}_m\left({x}^{(0)}\right)\hfill \end{array}\right] $$
(2.2)
where are

L
_{
i
}
:

n
_{
i
} × 1 vector of the ith type of the measurements

v
_{
i
}
:

the residual vector of L
_{
i
}

B
_{
i
}
:

the design matrix associated with L
_{
i
}
The measurement weight matrix is also grouped into
$$ P= diag\left(\begin{array}{ccccc}\hfill {P}_1\hfill & \hfill \cdots \hfill & \hfill {P}_i\hfill & \hfill \cdots \hfill & \hfill {P}_m\hfill \end{array}\right) $$
(2.3)
with its corresponding covariance matrix
$$ \begin{array}{l}D= diag\left(\begin{array}{ccccc}\hfill {D}_1\hfill & \hfill \cdots \hfill & \hfill {D}_i\hfill & \hfill \cdots \hfill & \hfill {D}_m\hfill \end{array}\right)\\ {}\kern1em = diag\left(\begin{array}{ccccc}\hfill {\sigma}_{01}^2{P}_1^{1}\hfill & \hfill \cdots \hfill & \hfill {\sigma}_{0i}^2{P}_i^{1}\hfill & \hfill \cdots \hfill & \hfill {\sigma}_{0m}^2{P}_m^{1}\hfill \end{array}\right)\end{array} $$
(2.4)
where \( {\sigma}_{0i}^2\left(1, \dots,\ m\right) \) is the ith variance component (variance factor) of the unit weight to be estimated for the ith group of the measurements (Cui, et al. 2001).
Under the assumption of \( {v}_i\sim N\left(0,\ {D}_{v_i}\right) \), the expectation of the weighted residual sum of squares is (Cui, et al. 2001):
$$ \begin{array}{l}E\left({v}_i^T{P}_i{v}_i\right)=\left({n}_i2tr\left({N}^{1}{N}_i\right)+tr{\left({N}^{1}{N}_i\right)}^2\right){\sigma}_{0i}^2\\ {}\kern5.25em +{\displaystyle \sum_{j=1,j\ne i}^m\left\{tr\left({N}^{1}{N}_i{N}^{1}{N}_j\right){\sigma}_{0j}^2\right\}}\end{array} $$
(2.5)
with N = B
^{T}
PB, and \( {N}_i={B}_i^T{P}_i{B}_i \) (i = 1, 2, …, m). The rigorous solution for \( {\sigma}_{0i}^2\left(1, \dots,\ m\right) \) can be delivered by solving the m dimensional equation system. There have been multiple simplified algorithms, of which one is
$$ E\left({v}_i^T{P}_i{v}_i\right)={\sigma}_{0i}^2\left({n}_itr\left({N}^{1}{N}_i\right)\right) $$
(2.6)
by assuming that \( {\sigma}_{01}^2={\sigma}_{02}^2=\cdots ={\sigma}_{0m}^2={\sigma}_{0i}^2 \) in (2.5). Furthermore, provided that n
_{
i
} − tr(N
^{− 1}
N
_{
i
}) = r
_{
i
}, the practical estimation of \( {\sigma}_{0i}^2\left(1, \dots,\ m\right) \) is reduced to (Förstner 1979)
$$ {\widehat{\sigma}}_{0i}^2={v}_i^T{P}_i{v}_i/{r}_i $$
(2.7)
where r
_{
i
} is the total redundancy contribution that reflects the extent of the influence of L
_{
i
} on the parameter estimation. The bigger r
_{
i
} is, the less L
_{
i
} affects the parameter estimation. The number of the measurements in a group can be one or more. With a group of independent measurements, the redundant index of each measurement satisfies 0 < r
_{
i
} < 1. When r
_{
i
} = 1, the measurement is completely redundant. It becomes a high leverage measurement in case r
_{
i
} tends to zero.
The alternative derivation of Kalman filter
Let the linear or linearized system described by KF at time t
_{
k
} be
$$ \begin{array}{cc}\hfill {x}_k={\varPhi}_k{x}_{k1}+{\varLambda}_k{w}_k\hfill & \hfill \left(\mathrm{system}\ \mathrm{model}\right)\hfill \end{array} $$
(2.8)
$$ \begin{array}{cc}\hfill {z}_k={H}_k{x}_k+{\varepsilon}_k\hfill & \hfill \left(\mathrm{measurement}\ \mathrm{model}\right)\hfill \end{array} $$
(2.9)
where is

x
_{
k
}
:

the n
_{x} × 1 state vector

w
_{
k
}
:

the n
_{w} × 1 process noise vector

Λ
_{
k
}
:

the coefficient matrix of w
_{
k
}

z
_{
k
}
:

the n
_{z} × 1 measurement vector

ε
_{
k
}
:

the measurement noise vector

Φ
_{
k
}
:

the state transition matrix

H
_{
k
}
:

the design matrix
with w
_{
k
} ~ N(0, Q
_{
w
}), ε
_{
k
} ~ N(0, R
_{
k
}), where w
_{
k
} and ε
_{
k
} are uncorrelated with each other and themselves from epoch to epoch.
By considering three independent groups of the measurements and pseudomeasurements at an arbitrary epoch k (Wang 1997):

(1)
the raw measurement vector l
_{
z
} = z
_{
k
} with its variance matrix \( {D}_{l_z}={R}_k \),

(2)
the zero mean process noise vector w
_{
k
} as a pseudo measurement vector l
_{
w
} = w
_{
k
} with its variance matrix \( {D}_{l_w}={Q}_k \),

(3)
the predicted state vector x
_{
k/k − 1} from the previous epoch as another pseudomeasurement vector l
_{
w
} = x
_{
k/k − 1} = Φ
_{
k
}
x
_{
k − 1} with its variance matrix \( {D}_{l_x}={\varPhi}_k{D}_{x_{x1}}{\varPhi}_k^T \), where \( {D}_{x_{x1}} \) is the variance matrix of x
_{
k − 1},
a measurement equation system can be constructed for Kalman filter algorithm at epoch k as follow:
$$ \left[\begin{array}{c}\hfill {v}_z\hfill \\ {}\hfill {v}_w\hfill \\ {}\hfill {v}_x\hfill \end{array}\right]=\left[\begin{array}{cc}\hfill {H}_k\hfill & \hfill O\hfill \\ {}\hfill O\hfill & \hfill I\hfill \\ {}\hfill I\hfill & \hfill {\varLambda}_k\hfill \end{array}\right]\ \left[\begin{array}{c}\hfill \widehat{x}\hfill \\ {}\hfill \widehat{w}\hfill \end{array}\right]\left[\begin{array}{c}\hfill {l}_z\hfill \\ {}\hfill {l}_w\hfill \\ {}\hfill {l}_x\hfill \end{array}\right] $$
(2.10)
to which the Least Squares Principle can be applied to derive the identical solution for Kalman filter (Wang 1997; Caspary and Wang 1998).
One of the significant contributions made by this alternate derivation of KF was about to handle the process noise vector separately, which has made possible the simultaneous estimation of the variance components associated with the process noise vector w
_{
k
} and the measurement noise vector ε
_{
k
}.
Global VCE for Q and R in KF using redundancy contribution
Because of the solution equivalence between LS and KF as summarized in 2.2, we are intuitively inspired to realize the most popular simplified VCE technique (Forstner’s method) in our novel multisensor integrated kinematic positioning and navigation (Qian, et al. 2013, 2015; Wang et al. 2014). Wang (1997) proved that the measurement residual vectors for three independent measurement groups at epoch k as in (2.10) can be computed based on the same innovation vector as follow:
$$ \begin{array}{c}\hfill {v}_z=\left(I{H}_kK\right)\left({z}_k{H}_k{x}_{k/k1}\right)\hfill \\ {}\hfill {v}_w={Q}_k{\varLambda}_k^T{D}_{x_{k/k1}}^{1}K\left({z}_k{H}_k{x}_{k/k1}\right)\hfill \\ {}\hfill {v}_x={\varPhi}_k{D}_{k1}{\varPhi}_k^T{D}_{x_{k/k1}}^{1}K\left({z}_k{H}_k{x}_{k/k1}\right)\hfill \end{array} $$
(2.11)
where K is the Kalman gain matrix at epoch k, the covariance matrix of the predicted state x
_{
k/k − 1} is \( {D}_{x_{k/k1}}={\varPhi}_k{D}_{x_{x1}}{\varPhi}_k^T+{\varLambda}_k{Q}_k{\varLambda}_k^T \), and the innovation vector d = z
_{
k
} − H
_{
k
}
x
_{
k/k − 1}.
Consequently, three residual vectors (v
_{
z
}, v
_{
w
}, v
_{
x
}) for three measurement vectors (l
_{
z
}, l
_{
w
}, l
_{
x
}) are actually correlated with each other through the same innovation vector. In addition, the corresponding redundancy indices for each measurement group are (Wang 1997):
$$ \begin{array}{l}{r}_z=tr\left(I{H}_kK\right)\hfill \\ {}{r}_w=tr\left({Q}_k{\varLambda}_k^T{H}_k^T{D}_d^{1}{H}_k{\varLambda}_k\right)\hfill \\ {}{r}_x=tr\left({\varPhi}_k{D}_{x_{k1}}{\varPhi}_k^T{H}_k^T{D}_d^{1}{H}_K\right)\hfill \end{array} $$
(2.12)
wherein the total system redundancy n
_{z} = r
_{z} + r
_{w} + r
_{x}, and the covariance matrix of the innovation vector is denoted by \( {D}_d={R}_k+{H}_k{D}_{x_{k/k1}}{H}_k^T \). Thus, for an arbitrary epoch k, the variance factors of the three measurement groups defined in (2.10) can be estimated as follow:
$$ {\sigma}_{0j}^2(k)={v}_j^T{D}_{l_j}^{1}{v}_j/{r}_j\kern0.75em \left(j=z,w,x\right) $$
(2.13)
As for a global variance component estimate over a specific or the whole time duration, a simple accumulation can obtain a reliable estimate due to the crossepochorthogonal properties of the measurement residuals (Wang 1997). For example, the global variance component estimate up to epoch k is:
$$ {\sigma}_{0j}^2\left(k\Big1\dots k\right)=\frac{{\displaystyle \sum_{i=1}^k{v}_{j_i}^T{D}_{l_j}^{1}{v}_{j_i}}}{{\displaystyle \sum_{i=1}^k{r}_{j_i}}}\kern0.75em \left(j=z,w,x\right) $$
(2.14)
Commonly, the components in w
_{
k
} and ε
_{
k
} are modeled as uncorrelated so that Q
_{
k
} and R
_{
k
} become diagonal. As a result, the redundant index for each independent component in either w
_{
k
} or ε
_{
k
} is given by
$$ {r}_z^{i_z}=1{\left({H}_kK\right)}_{i_z{i}_z} $$
(2.15)
$$ {r}_w^{i_w}={\left({Q}_k{\varLambda}_k^T{H}_k^T{D}_d^{1}{H}_k{\varLambda}_k\right)}_{i_w{i}_w} $$
(2.16)
Accordingly, the individual variance component in vector w
_{
k
} or ε
_{
k
} can be estimated by analogy with (2.13) and (2.14).