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# Linear Estimation

## 1. Kalman Filter Tutorial (Cont'd...)

### 1.4 Properties of Kalman filter (cont'd...)

#### Innovation sequence (cont'd...)

In quadratic form, we know that $V^{T}FV=Tr[VV^{T}F]$, where $V$ is a vector and $F$ is a symmetric matrix. $Tr[. ]$ denotes the trace operation of square matrix. Apply quadratic form on the last expression of the previous page, we get $$\label{4h} G=E[Tr\lbrace v_{k}v_{k}^{T}(R_{k}+H_{k}P_{k|k-1}H_{k}^{T})^{-1}+H_{k}e_{k|k-1}e_{k|k-1}^{T}H_{k}^{T}(R_{k}+H_{k}P_{k|k-1}H_{k}^{T})^{-1} \rbrace ]$$ we know that $R_{k}$ is the covariance of $v_{k}$ and $P_{k|k-1}$ is the covariance of $e_{k|k-1}$. So, we apply the expectation on the above equation because expectation is a linear operator, i.e. $$\label{4i} G=Tr[R_{k}(R_{k}+H_{k}P_{k|k-1}H_{k}^{T})^{-1}+H_{k}P_{k|k-1}H_{k}^{T}(R_{k}+H_{k}P_{k|k-1}H_{k}^{T})^{-1}],$$ where, we take the trace of the matrix to get minimum value of $G$. If we assume the dimension of $R_{k}$ is $G\times G$, then the identity matrix has dimension $G \times G$. If the components of the innovation $Z_{k}$ are independent, white Gaussian and identically distributed, then its quadratic form will follow chi-square distribution with $G$ degrees of freedom. So, this quadratic form will be helpful for determining apriori test statistics of the Kalman filter model, and this powerful test statistics is commonly known as universally most powerful test statistics (UMPTS).