Cross-correlation effect of ARAIM test statistic on false alarm risk

Abstract

The requirement for ARAIM continuity risk due to the monitor false alarm has been outlined in earlier works for ARAIM development (WG-C in ARAIM technical subgroup milestone 3 report, 2016). However, the expected continuity risk comes from an underlying conservative assumption that the correlation between multiple monitors for fault detection is negligible. Thus, we investigate the effect of the cross-correlation across ARAIM solution separation tests on the monitor false alarm probability ($P_{\text{FA}}$) by presenting a higher fidelity methodology to evaluate the $P_{\text{FA}}$ based on highly correlated fault detection tests. We carry out a preliminary assessment of ARAIM false alarm performance by using the proposed method. It was found that considering the cross-correlation among monitor test statistics reduces the predicted $P_{\text{FA}}$ by up to approximately 50% of the predefined requirement (e.g., $10^{ - 6}$) when triple satellite faults were considered. Despite such improvement, the baseline ARAIM implementation does not appear to be overly conservative.

Appendix: Standard integration form for the PFA evaluation

This section gives some elements for the development of (13) and (15). The derivation of the equations is based on the method in Genz and Kwong (2000). First, the reduction of the integration dimension is discussed. In (12), the integration region is represented as $- {\mathbf{T}} < {\mathbf{Qy}} = {\mathbf{Q}}\left[ {y_{1} ,y_{2} , \ldots ,y_{h} } \right]^{\text{T}} < {\mathbf{T}}$, and the matrix ${\mathbf{Q}}$ has the lower triangular form whose elements $q_{ij} = 0$ for all $j > r$. Here, let us assume that a rearrangement of the inequalities has been completed such that the matrix ${\mathbf{Q}}$ has the following form in (22):

$${\mathbf{Q}} = \left[ {{\mathbf{Q}}_{h \times r}^{{\prime }} \varvec{ }\mathbf{0}_{h \times h - r} } \right] = \left[ {\begin{array}{*{20}c} {q_{1,1} } & 0 & 0 & \cdots & \cdots & 0 & \cdots & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ {q_{{k_{1} ,1}} } & 0 & 0 & \cdots & \cdots & 0 & \cdots & 0 \\ \varvec{*} & {q_{1,2} } & 0 & \cdots & \cdots & 0 & \cdots & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ \varvec{*} & {q_{{k_{2} ,2}} } & 0 & \cdots & \cdots & 0 & \cdots & 0 \\ \vdots & \ddots & \ddots & \ddots & \ddots & 0 & \cdots & 0 \\ \varvec{*} & \varvec{*} & \cdots & \varvec{*} & {q_{1,r} } & 0 & \cdots & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ \varvec{*} & \varvec{*} & \cdots & \varvec{*} & {q_{{k_{r} ,r}} } & 0 & \cdots & 0 \\ \end{array} } \right]$$

(22)

where ${\mathbf{Q}}^{{\prime }}$ is a h × r submatrix of Q with permuted columns in terms of $y_{i}$ for $i = 1, \ldots ,r$, and the subscription $k_{i}$ for $i = 1, \ldots ,r$ denotes the number of inequalities in each group of $y_{i}$ (i.e., $\sum\nolimits_{j = 1}^{r} {k_{j} = h}$). Only $y_{1} ,y_{2} , \ldots ,y_{\text{r}}$ variables have constraints while the remaining h - r variables $y_{{{\text{r}} + 1}} ,y_{{{\text{r}} + 2}} , \ldots ,y_{\text{h}}$ are not constrained, so integrations in (12) which are related to those unconstrained variables should be all equal to 1 and those terms thus are not included in the equation. Therefore, the integration is reduced to r dimensional one in (13).

Next, normalization can be performed such that the matrix ${\mathbf{Q}}$ has the following form:

$${\bar{\mathbf{Q}}} = \left[ {{\bar{\mathbf{Q}}}_{h \times r}^{{\prime }} \mathbf{0}_{h \times h - r} } \right] = \left[ {\begin{array}{*{20}c} {1_{1,1} } & 0 & 0 & \cdots & \cdots & 0 & \cdots & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ {1_{{k_{1} ,1}} } & 0 & 0 & \cdots & \cdots & 0 & \cdots & 0 \\ * & {1_{1,2} } & 0 & \cdots & \cdots & 0 & \cdots & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ * & {1_{{k_{2} ,2}} } & 0 & \cdots & \cdots & 0 & \cdots & 0 \\ \vdots & \ddots & \ddots & \ddots & \ddots & 0 & \cdots & 0 \\ * & * & \cdots & * & {1_{1,r} } & 0 & \cdots & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ * & * & \cdots & * & {1_{{k_{r} ,r}} } & 0 & \cdots & 0 \\ \end{array} } \right].$$

(23)

In (23), entries corresponding to each group of $y_{i}$ (e.g., $q_{1,2} , \ldots ,q_{{k_{2} ,2}}$ for $y_{2}$ in the light blue colored box in 23) are all ones and $*$ could be a zero or nonzero component. In the process, if an inequality needs to be divided by a negative number, the order of the inequality must be changed so that after division by a negative number a scaled lower limit becomes an upper limit, and a scaled upper limit becomes a lower limit.

L and U designate the new limit vectors after permutations, normalizations and interchanges of the original limit vectors – T and T (see (13)), such that the new set of constraints for the integration region takes the following form:

$$\mathbf{L} < {\bar{\mathbf{Q}}}^{{\prime }} \mathbf{y} = {\bar{\mathbf{Q}}}^{{\prime }} \left[ {y_{1} ,y_{2} , \ldots ,y_{r} } \right]^{\text{T}} < \mathbf{U}.$$

(24)

We can now produce explicit expressions for the limits of the continuous integration variables. For instance, let $m_{i} = \sum\nolimits_{j = 1}^{i} {k_{j} }$, then the revised limits for $y_{i}$ in (24) can be determined by $k_{i}$ constraints for $y_{i}$ as shown in,

$$L_{i} \left( {y_{1} ,y_{2} , \ldots ,y_{i - 1} } \right) = \mathop {\hbox{max} }\limits_{{m_{i - 1} < n \le m_{i} }} \left( {l_{n} - \mathop \sum \limits_{j = 1}^{i - 1} q_{n,j} y_{j} } \right)$$

(25)

$$\begin{array}{*{20}c} {U_{i} \left( {y_{1} ,y_{2} , \ldots ,y_{i - 1} } \right) = \mathop {\hbox{min} }\limits_{{m_{i - 1} < n \le m_{i} }} \left( {u_{n} - \mathop \sum \limits_{j = 1}^{i - 1} q_{n,j} y_{j} } \right)} \\ \end{array}.$$

(26)

Here, $n$ is the index for the row of the matrix ${\bar{\mathbf{Q}}}^{{\prime }}$. By applying these bounds, the resulting expression of the integration in (12) becomes the expression in (13).

Second, the form of integration in (13) is further simplified using the transformation in

$$z_{\text{i}} = F\left( {y_{i} } \right)\quad {\text{for}}\quad i = 1, \ldots ,r$$

(27)

such that standard numerical integration methods are easily applied to the evaluation of the probability. $F( \cdot )$ designates the Gaussian cumulative distribution function (CDF). By definition, $\phi \left( {y_{i} } \right){\text{d}}y_{i} = {\text{d}}z_{i}$ where $\phi$ indicates the Gaussian probability density function (PDF) and (13) becomes

$$\begin{array}{*{20}c} {\int \limits_{{\bar{L}_{1} }}^{{\bar{U}_{1} }} \int \limits_{{\bar{L}_{2} }}^{{\bar{U}_{2} \left( {z_{1} } \right)}} \cdots \int \limits_{{\bar{L}_{r} \left( {z_{1} ,z_{2} , \ldots ,z_{r - 1} } \right)}}^{{\bar{U}_{r} \left( {z_{1} ,z_{2} , \ldots ,z_{r - 1} } \right)}} {\text{d}}\varvec{z}} \\ \end{array}$$

(28)

where new limits for $z_{i}$, $\bar{L}$ and $\bar{U}$ are derived based on (27), shown as in (29) and (30)

$$\bar{L}_{i} \left( {z_{1} , \ldots ,z_{i - 1} } \right) = F\left( {L_{i} \left( {F^{ - 1} \left( {z_{1} } \right), \ldots ,F^{ - 1} \left( {z_{i - 1} } \right)} \right)} \right)$$

(29)

$$\bar{U}_{i} \left( {z_{1} , \ldots ,z_{i - 1} } \right) = F\left( {U_{i} \left( {F^{ - 1} \left( {z_{1} } \right), \ldots ,F^{ - 1} \left( {z_{i - 1} } \right)} \right)} \right).$$

(30)

If we set $z_{i} = \bar{L}_{i} + \left( {\bar{U}_{i} - \bar{L}_{i} } \right)u_{i}$ for $i = 1, \ldots ,r$ so that integration limits all have an interval $\left[ {0,1} \right]$, from ${\text{d}}z_{i} = \left( {\bar{U}_{i} - \bar{L}_{i} } \right){\text{d}}u_{i}$, Eq. (28) can be expressed in the form of (14) in the previous section:

$$\left( {\bar{U}_{1} - \bar{L}_{1} } \right) \int \limits_{0}^{1} \left( {\bar{U}_{2} \left( {u_{1} } \right) - \bar{L}_{2} \left( {u_{1} } \right)} \right) \cdots \int \limits_{0}^{1} \left( {\bar{U}_{r} \left( {u_{1} , \ldots ,u_{r - 1} } \right) - \bar{L}_{r} \left( {u_{1} , \ldots ,u_{r - 1} } \right)} \right) \int \limits_{0}^{1} {\text{d}}\varvec{u}.$$

(31)

In this section, the simplified standard integration form in (14) and (15) for the $P_{\text{fa}}$ assessment was derived.