Bayesian methods for outliers detection in GNSS time series

Abstract

This article is concerned with the problem of detecting outliers in GNSS time series based on Bayesian statistical theory. Firstly, a new model is proposed to simultaneously detect different types of outliers based on the conception of introducing different types of classification variables corresponding to the different types of outliers; the problem of outlier detection is converted into the computation of the corresponding posterior probabilities, and the algorithm for computing the posterior probabilities based on standard Gibbs sampler is designed. Secondly, we analyze the reasons of masking and swamping about detecting patches of additive outliers intensively; an unmasking Bayesian method for detecting additive outlier patches is proposed based on an adaptive Gibbs sampler. Thirdly, the correctness of the theories and methods proposed above is illustrated by simulated data and then by analyzing real GNSS observations, such as cycle slips detection in carrier phase data. Examples illustrate that the Bayesian methods for outliers detection in GNSS time series proposed by this paper are not only capable of detecting isolated outliers but also capable of detecting additive outlier patches. Furthermore, it can be successfully used to process cycle slips in phase data, which solves the problem of small cycle slips.

Appendix

The derivations of the conditional posterior distributions are as follows.

• (1) According to Bayesian statistical theory (Koch 1990, 2007; Box and Tiao 1973),

  $$\begin{aligned}&p\left( \phi |X,\sigma ^{2},\delta ^\mathrm{AO},\delta ^\mathrm{IO},W^\mathrm{AO},W^\mathrm{IO}\right) \\&\quad \propto p\left( X|\phi ,\sigma ^{2},\delta ^\mathrm{AO},\delta ^\mathrm{IO},W^\mathrm{AO},W^\mathrm{IO}\right) \cdot p(\phi )\\&\quad \propto \exp \left\{ {-\frac{1}{2\sigma ^{2}}\sum _{t=p+1}^n {\left( {x_t\!-\!\sum _{i=1}^p {\phi _i y_{t-i} }\!-\!w_t^\mathrm{AO} \delta _t^\mathrm{AO}\!-\!w_t^\mathrm{IO} \delta _t^\mathrm{IO} } \right) ^{2}} } \right\} \cdot \\&\quad \exp \left\{ {-\frac{1}{2}(\phi -\phi _0 )^{T}V(\phi -\phi _0 )} \right\} \\&\quad \propto \exp \left\{ -\frac{1}{2}\left[ \phi ^{T}\left( {V+\frac{1}{\sigma ^{2}}\sum _{t=p+1}^n {Y_{t-1} Y_{t-1}^T } } \right) \phi \right. \right. \\&\qquad -\left. \left. 2\phi ^{T}\left( {\frac{1}{\sigma ^{2}}\sum _{t=p+1}^n {Y_{t-1} (x_t -w_t^\mathrm{AO} \delta _t^\mathrm{AO} -w_t^\mathrm{IO} \delta _t^\mathrm{IO} )} +V\phi _0 } \right) \right] \right\} \end{aligned}$$

  where $Y_{t-1} =(y_{t-1} ,\ldots ,y_{t-p} )^{T}$. Then, the conditional posterior distribution of $\phi$ given $X,\sigma ^{2},\delta ^\mathrm{AO},\delta ^\mathrm{IO},W^\mathrm{AO},W^\mathrm{IO}$ is

  $$\begin{aligned} \phi |X,\sigma ^{2},\delta ^\mathrm{AO},\delta ^\mathrm{IO},W^\mathrm{AO},W^\mathrm{IO}\sim N_p \left( \hat{{\phi }}_0 ,\hat{{V}}^{-1}\right) \end{aligned}$$

  (34)

  where $\hat{{\phi }}_0 =\hat{{V}}^{-1}( \frac{1}{\sigma ^{2}}\sum _{t=p+1}^n Y_{t-1} (x_t -w_t^\mathrm{AO} \delta _t^\mathrm{AO}-w_t^\mathrm{IO} \delta _t^\mathrm{IO} ) +V\phi _0 ),\,\hat{{V}}=V+\frac{1}{\sigma ^{2}}\sum _{t=p+1}^n {Y_{t-1} Y_{t-1}^T }$ Similarly, the following conditional posterior distributions can be obtained.

• (2) The conditional posterior distribution of $\sigma ^{2}$ given $X,\phi ,\delta ^\mathrm{AO},\delta ^\mathrm{IO},W^\mathrm{AO},W^\mathrm{IO}$ is

  $$\begin{aligned} \sigma ^{2}|X,\phi ,\delta ^\mathrm{AO},\delta ^\mathrm{IO},W^\mathrm{AO},W^\mathrm{IO}\sim IG\left( {\frac{\upsilon _1 }{2},\frac{\upsilon _1 \lambda _1 }{2}} \right) \end{aligned}$$

  (35)

  where $\upsilon _1 =n-p+\upsilon ,\,\lambda _1 =\frac{1}{n-p+\upsilon }[\sum \nolimits _{t=p+1}^n {(x_t -\sum \nolimits _{i=1}^p {\phi _i y_{t-i} } -w_t^\mathrm{AO} \delta _t^\mathrm{AO} -w_t^\mathrm{IO} \delta _t^\mathrm{IO} )^{2}} +\upsilon \lambda ]$.

• (3) The conditional posterior distribution of $\delta _j^\mathrm{AO} ,\delta _j^\mathrm{IO}$ given $X,\phi ,\sigma ^{2},\delta _{(-j)}^\mathrm{AO} ,\delta _{(-j)}^\mathrm{IO} \delta ^\mathrm{IO},W^\mathrm{AO},W^\mathrm{IO}$ is

  $$\begin{aligned} \delta _j^\mathrm{AO} ,\delta _j^\mathrm{IO} |X,\phi ,\sigma ^{2},\delta _{(\!-\!j)}^\mathrm{AO} ,\delta _{(\!-\!j)}^\mathrm{IO} ,W^\mathrm{AO},W^\mathrm{IO}\!\sim \! MN(1,P_j )\nonumber \\ \end{aligned}$$

  (36)

where $P_j =(P_{1j} ,P_{2j} ,P_{3j} ,P_{4j} )^{T}$ and $MN(1,P_j )$ denotes the multinomial distribution. In fact,

$$\begin{aligned}&P\left( \delta _j^\mathrm{AO} ,\delta _j^\mathrm{IO} |X,\phi ,\sigma ^{2},\delta _{(-j)}^\mathrm{AO} ,\delta _{(-j)}^\mathrm{IO} ,W^\mathrm{AO},W^\mathrm{IO}\right) \\&\propto \exp \left\{ {-\frac{1}{2\sigma ^{2}}\sum _{t=p+1}^n {\left( {x_t -\sum _{i=1}^p {\phi _i y_{t-i} } -w_t^\mathrm{AO} \delta _t^\mathrm{AO} -w_t^\mathrm{IO} \delta _t^\mathrm{IO} } \right) ^{2}} } \right\} \cdot \\&\alpha _1^{\delta _j^\mathrm{AO} } (1-\alpha _1 )^{1-\delta _j^\mathrm{AO} }\cdot \alpha _2^{\delta _j^\mathrm{IO} } (1-\alpha _2 )^{1-\delta _j^\mathrm{IO} }\\&\propto \exp \left\{ {-\frac{1}{2\sigma ^{2}}\sum _{t=j}^T {\left( {x_t -\sum _{i=1}^p {\phi _i y_{t-i} } -w_t^\mathrm{AO} \delta _t^\mathrm{AO} -w_t^\mathrm{IO} \delta _t^\mathrm{IO} } \right) ^{2}} } \right\} \cdot \\&\alpha ^{\delta _j^\mathrm{AO} }(1-\alpha )^{1-\delta _j^\mathrm{AO} }\cdot \alpha ^{\delta _j^\mathrm{IO} }(1-\alpha )^{1-\delta _j^\mathrm{IO} } \end{aligned}$$

where $\delta _{(-j)}^\mathrm{AO} =(\delta _1^\mathrm{AO} ,\ldots ,\delta _{j-1}^\mathrm{AO} ,\delta _{j+1}^\mathrm{AO} ,\ldots ,\delta _n^\mathrm{AO} )^{T},\,\delta _{(-j)}^\mathrm{IO} =(\delta _1^\mathrm{IO} ,\ldots ,\delta _{j-1}^\mathrm{IO} ,\delta _{j+1}^\mathrm{IO} ,\ldots ,\delta _n^\mathrm{IO} )^{T},\,T=\min (n,p+j)$. So

$$\begin{aligned}&P\left( \delta _j^\mathrm{AO} =1,\delta _j^\mathrm{IO} =0|X,\phi ,\sigma ^{2},\delta _{(-j)}^\mathrm{AO} ,\delta _{(-j)}^\mathrm{IO} ,W^\mathrm{AO},W^\mathrm{IO}\right) \\&\quad \propto \alpha _1 (1\!-\!\alpha _2 )\exp \left\{ \!-\!\frac{1}{2\sigma ^{2}}\left[ \left( {x_j \!-\!\sum _{i=1}^p {\phi _i } y_{j-i} \!-\!w_j^\mathrm{AO} } \right) \right. \right. \\&\qquad +\left. \left. \sum _{t=j+1}^T {\left( {x_t^{*} \!-\!\sum _{i=1}^p {\phi _i x_{t-i}^{*} } \!-\!w_t^\mathrm{IO} \delta _t^\mathrm{IO} \!+\!C_{t-j} w_j^\mathrm{AO} } \right) ^{2}} \right] \right\} \\&P(\delta _j^\mathrm{AO} =0,\delta _j^\mathrm{IO} =1|X,\phi ,\sigma ^{2},\delta _{(-j)}^\mathrm{AO} ,\delta _{(-j)}^\mathrm{IO} ,W^\mathrm{AO},W^\mathrm{IO}) \\&\quad \propto \alpha _2 (1\!-\!\alpha _1 )\exp \left\{ \!-\!\frac{1}{2\sigma ^{2}}\left[ \left( x_j \!-\!\sum _{i=1}^p {\phi _i } y_{j-i} \!-\!w_j^\mathrm{IO} \right) \right. \right. \\&\quad +\left. \left. \sum _{t=j+1}^T \left( x_t^{*} \!-\!\sum _{i=1}^p {\phi _i x_{t-i}^{*} } \!-\!w_t^\mathrm{IO} \delta _t^\mathrm{IO} \right) ^{2}\right] \right\} \\&P\left( \delta _j^\mathrm{AO} =1,\delta _j^\mathrm{IO} =1|X,\phi ,\sigma ^{2},\delta _{(-j)}^\mathrm{AO} ,\delta _{(-j)}^\mathrm{IO} ,W^\mathrm{AO},W^\mathrm{IO}\right) \\&\quad \propto \alpha _1 \alpha _2 \exp \left\{ \!-\!\frac{1}{2\sigma ^{2}}\left[ \left( x_j \!-\!\sum _{i=1}^p {\phi _i } y_{j-i} \!-\!w_j^\mathrm{AO} \!-\!w_j^\mathrm{IO}\right) \right. \right. \\&\quad +\left. \left. \sum _{t=j+1}^T \left( x_t^{*} \!-\!\sum _{i=1}^p {\phi _i x_{t-i}^{*} } \!-\!w_t^\mathrm{IO} \delta _t^\mathrm{IO} \!+\!C_{t-j} w_j^\mathrm{AO} \right) ^{2}\right] \right\} \\&P\left( \delta _j^\mathrm{AO} =0,\delta _j^\mathrm{IO} =0|X,\phi ,\sigma ^{2},\delta _{(-j)}^\mathrm{AO} ,\delta _{(-j)}^\mathrm{IO} ,W^\mathrm{AO},W^\mathrm{IO}\right) \\&\quad \propto (1\!-\!\alpha _1 )(1\!-\!\alpha _2 )\exp \left\{ \!-\!\frac{1}{2\sigma ^{2}}\left[ \left( x_j \!-\!\sum _{i=1}^p {\phi _i } y_{j-i} \right) \right. \right. \\&\qquad +\left. \left. \sum _{t=j+1}^T\left( {x_t^{*} \!-\!\sum _{i=1}^p {\phi _i x_{t-i}^{*}}} \!-\!w_t^\mathrm{IO}\delta _t^\mathrm{IO} \right) ^{2}\right] \right\} \\ \end{aligned}$$

where

$$\begin{aligned} x_t^{*} =\left\{ {\begin{array}{ll} x_t , t=j \\ y_t , t\ne j \\ \end{array}} \right. ,&C_{t-j} =\left\{ {\begin{array}{ll} \phi _{t-j}, &{} t-j=1,\ldots ,p \\ 0, &{} t-j>p \\ \end{array}} \right. \end{aligned}$$

Therefore, according to the characters of the multinomial distribution, we can obtain that

$$\begin{aligned} P_{1j}&= P\left( \delta _j^\mathrm{AO} =1,\delta _j^\mathrm{IO} =0|X,\phi ,\sigma ^{2},\delta _{(-j)}^\mathrm{AO} ,\delta _{(-j)}^\mathrm{IO} ,,W^\mathrm{AO},W^\mathrm{IO}\right) \\&= \frac{q_{1j} }{q_{1j} +q_{2j} +q_{3j} +q_{4j} }\\ P_{2j}&= P\left( \delta _j^\mathrm{AO} =0,\delta _j^\mathrm{IO} =1|X,\phi ,\sigma ^{2},\delta _{(-j)}^\mathrm{AO} ,\delta _{(-j)}^\mathrm{IO} ,,W^\mathrm{AO},W^\mathrm{IO}\right) \\&= \frac{q_{2j} }{q_{1j} +q_{2j} +q_{3j} +q_{4j} }\\ P_{3j}&= P\left( \delta _j^\mathrm{AO} =1,\delta _j^\mathrm{IO} =1|X,\phi ,\sigma ^{2},\delta _{(-j)}^\mathrm{AO} ,\delta _{(-j)}^\mathrm{IO} ,,W^\mathrm{AO},W^\mathrm{IO}\right) \\&= \frac{q_{3j} }{q_{1j} +q_{2j} +q_{3j} +q_{4j} }\\ P_{4j}&= P\left( \delta _j^\mathrm{AO} =0,\delta _j^\mathrm{IO} =0|X,\phi ,\sigma ^{2},\delta _{(-j)}^\mathrm{AO} ,\delta _{(-j)}^\mathrm{IO} ,,W^\mathrm{AO},W^\mathrm{IO}\right) \\&= \frac{q_{4j} }{q_{1j} +q_{2j} +q_{3j} +q_{4j} }\\ \end{aligned}$$

where

$$\begin{aligned}&q_{1j} =\alpha _1 (1\!-\!\alpha _2 )\exp \left\{ -\frac{1}{2\sigma ^{2}}\left[ \left( x_j -\sum _{i=1}^p {\phi _i } y_{j-i} \!-\!w_j^\mathrm{AO} \right) \right. \right. \\&\qquad \quad \!+\left. \left. \sum _{t=j+1}^T \!\left( {x_t^{*} \!-\!\sum _{i=1}^p {\phi _i x_{t-i}^{*} } } \!-\!w_t^\mathrm{IO} \delta _t^\mathrm{IO} +C_{t\!-\!j} w_j^\mathrm{AO} \right) ^{\!2}\right] \right\} \\&q_{2j} =\alpha _2 (1\!-\!\alpha _1 )\exp \left\{ -\frac{1}{2\sigma ^{2}}\left[ \left( x_j -\sum _{i=1}^p {\phi _i } y_{j-i} \!-\!w_j^\mathrm{IO} \right) \right. \right. \\&\qquad \quad +\left. \left. \sum _{t=j+1}^T \left( {x_t^{*} -\sum _{i=1}^p {\phi _i x_{t-i}^{*} } } -w_t^\mathrm{IO} \delta _t^\mathrm{IO} \right) ^{2}\right] \right\} \\&q_{3j} \!=\!\alpha _1 \alpha _2 \exp \left\{ \!-\!\frac{1}{2\sigma ^{2}}\left[ \left( x_j -\sum _{i\!=\!1}^p {\phi _i } y_{j\!-\!i} -w_j^\mathrm{AO} -w_j^\mathrm{IO} \right) \right. \right. \\&\qquad \quad +\left. \left. \sum _{t=j+1}^T \left( {x_t^{*} \!-\!\sum _{i=1}^p {\phi _i x_{t\!-\!i}^{*} } } \!-\!w_t^\mathrm{IO} \delta _t^\mathrm{IO} \!+\!C_{t-j} w_j^\mathrm{AO} \right) ^{2}\right] \right\} \\&q_{4j} \!=\!(1-\alpha _1 )(1-\alpha _2 )\exp \left\{ -\frac{1}{2\sigma ^{2}}\left[ \left( x_j \!-\!\sum _{i=1}^p {\phi _i } y_{j\!-\!i} \right) \right. \right. \\&\qquad \quad +\left. \left. \sum _{t=j+1}^T \left( {x_t^{*} -\sum _{i=1}^p {\phi _i x_{t-i}^{*} } } -w_t^\mathrm{IO} \delta _t^\mathrm{IO} \right) ^{2}\right] \right\} \end{aligned}$$

• (4) The conditional posterior distribution of $w_j^\mathrm{AO}$ given $X,\phi ,\sigma ^{2},\delta ^\mathrm{AO},\delta ^\mathrm{IO},W_{(-j)}^\mathrm{AO} ,W^\mathrm{IO}$ is

  $$\begin{aligned}&w_j^\mathrm{AO} |X,\phi ,\sigma ^{2},\delta ^\mathrm{AO},\delta ^\mathrm{IO},W_{(-j)}^\mathrm{AO}, W^\mathrm{IO}\nonumber \\&\quad \sim N \left( \hat{{w}}_j^\mathrm{AO} ,(\sigma _j^2 )^\mathrm{AO}\right) \end{aligned}$$

  (37)

  where

  $$\begin{aligned}&\hat{{w}}_j^\mathrm{AO} \!=\!\left( \sigma _j^2\right) ^\mathrm{AO}\cdot \left\{ \delta _j^\mathrm{AO} \left[ \left( x_j \!-\!\sum _{i=1}^p {\phi _i y_{j-i} } \!-\!w_j^\mathrm{IO} \delta _j^\mathrm{IO}\right) \right. \right. \\&\quad +\left. \left. \sum _{t=j+1}^T {\phi _{t-j} \left( \sum _{i=1}^p {\phi _i }x_{t-i}^{*} +w_t^\mathrm{IO} \delta _t^\mathrm{IO} -x_t^{*} \right) } \right] +\mu _1 \right\} ,\\&\quad \times \left( \sigma _j^2\right) ^\mathrm{AO}=\left[ \frac{\left( \delta _j^\mathrm{AO}\right) ^{2}}{\sigma ^{2}}\left( 1+\sum _{i=1}^p {\phi _i^2 } \right) +\frac{1}{\sigma ^{2}}\right] ^{-1} \end{aligned}$$

• (5) The conditional posterior distribution of $w_j^\mathrm{IO}$ given $X,\phi ,\sigma ^{2},\delta ^\mathrm{AO},\delta ^\mathrm{IO},W^\mathrm{AO},W_{(-j)}^\mathrm{IO}$ is

  $$\begin{aligned}&w_j^\mathrm{IO} |X,\phi ,\sigma ^{2},\delta ^\mathrm{AO},\delta ^\mathrm{IO},W^\mathrm{AO},W_{(-j)}^\mathrm{IO}\nonumber \\&\quad \sim N\left( \hat{{w}}_j^\mathrm{IO} ,\left( \sigma _j^2\right) ^\mathrm{IO}\right) \end{aligned}$$

  (38)

  where $\hat{{w}}_j^\mathrm{IO} =(\sigma _j^2 )^\mathrm{IO}\cdot [\delta _j^\mathrm{IO} (y_j -\sum \limits _{i=1}^p {\phi _i y_{j-i} } )+\mu _2 ], (\sigma _j^2 )^\mathrm{IO}\qquad =[\frac{(\delta _j^\mathrm{IO} )^{2}}{\sigma ^{2}}+\frac{1}{\sigma ^{2}}]^{-1}$.