2.1. Local Test for Single Alternative Hypothesis

In the case where there are faulty pseudorange measurements: E(l)≠Ax. Consequently, the least-squares estimator of the position becomes biased: $E({\hat x}) \ne {x}$. In order to detect a biased position a fault detection procedure is applied. When a biased position is detected, it can then be corrected by excluding the faulty pseudorange. If it is assumed that the i ^th pseudorange is faulty, then the correct model is given by:

(3)$${l} = {A\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}} {x}} + {c}_i \hat \nabla _i + {\tilde v}$$

where $\hat \nabla _i $ is the fault in the i ^th pseudorange, and c_i=[0, …, 0, 1, 0, …, 0]^T is a unit vector with the i ^th element equal to one. Solving this for the fault then leads to:

(4)$$\hat \nabla _i = ({c}_i^{\rm T} {PQ}_v {Pc}_i )^{ - 1} {c}_i^{\rm T} {PQ}_v {Pl\;} $$

which has the variance:

(5)$$\sigma _{\hat \nabla _i} ^2 = \sigma _0^2 Q_{\hat \nabla _i} = \sigma _0^2 ({c}_i^{\rm T} {PQ}_v {Pc}_i )^{ - 1} $$

where Q_v = Q−A(A^TPA)⁻¹A^T is the co-factor matrix of the estimated residuals (from the original Gauss-Markov model).

The outlier test statistic for the i ^th pseudorange can then be formed as (Baarda, Reference Baarda1968; Kok, Reference Kok1984):

(6)$$w_i = \displaystyle{{\hat \nabla _i} \over {\sigma _0 \sqrt {Q_{\hat \nabla _i}}}} = \displaystyle{{{c}_i^T {PQ}_v {Pl}} \over {\sigma _0 \sqrt {{c}_i^T {PQ}_v {Pc}_i}}} $$

The correlation coefficient between a pair of outlier statistics is given by:

(7)$$\rho _{ij} = \displaystyle{{{c}_i^{\rm T} {PQ}_v {Pc}_j} \over {\sqrt {{c}_i^{\rm T} {PQ}_v {Pc}_i} \sqrt {{c}_j^{\rm T} {PQ}_v {Pc}_j}}}. $$

Based on Equation (3) the null hypothesis corresponding to the assumption that there are no faulty pseudorange measurements is:

(8)$$H_0 :\left\{ {\matrix{ {E({l}) = {Ax}} \cr {w_i \sim N(0,1)} \cr}} \right.$$

Under the null hypothesis, Equations (1) and (3) are equivalent. Otherwise, the alternative hypothesis H _i means that the i ^th pseudorange is faulty:

(9)$$H_i :\left\{ {\matrix{ {E\left( {l} \right) = {Ax} + {c}_i \nabla _i} \cr {w_i \sim N(\delta, 1)\; \; \; \; \; \; \; \; \; \; \;}\cr}} \right.$$

Taking the expectation of the outlier statistic in Equation (6), the non-centrality parameter can be obtained as:

(10)$$\delta (\hat \nabla _i ) = \displaystyle{{\nabla _i} \over {\sigma _0 \sqrt {Q_{\hat \nabla _i}}}} $$

2.2. Fault Detection

In the fault detection phase, the overall validity of the null hypothesis is tested:

(11)$$H_0 :\; {\it \Omega} = \displaystyle{{{v}^{\rm T} {Pv}} \over {\sigma _0^2}} \sim {\rm \chi} ^2 (n - t,\; 0)$$

If this leads to a rejection of the null hypothesis, then it is concluded that a fault is present. Ideally, the probability of a false alert in the i ^th pseudorange is set such that the probability of any one of the outlier tests failing, when the null hypothesis is true, is equal to the probability of a false alert, P _FA. Due to the difficulty in achieving this exactly, the Type I error of the outlier test is conservatively set as (Kelly, Reference Kelly1998):

(12)$$\alpha _0 = 1 - \root n \of {1 - P_{{\rm FA}}} $$

The outlier test statistic in Equation (6) follows a standard normal distribution under H ₀. The evidence on whether the model error as specified by Equation (3) did, or did not, occur is based on the test:

(13)$$|w_i | \gt N_{\displaystyle{{\alpha _0} \over 2}} (0,1)$$

By letting i run from one up to and including n, all of the pseudorange measurements can be screened for the presence of a fault. If one or more of the outlier tests fails then it is concluded that a fault exists.

Besides the possibility of making a Type I error, there is also a possibility that the null hypothesis is accepted when in fact it is false. This error, denoted as β ₀, is a Type II error. Thus, it can be seen that when the null hypothesis is accepted there will be a possibility of making a Type II error. When accepting the alternative hypothesis, there is also a possibility of making a Type I error. Therefore, no matter what decision is made there is always the possibility of making an error. However, steps may be taken to control the possibility of making errors and guarantee that the probability of making a correct decision can be estimated.

By setting the threshold based on the probability of a false alert, the Type I error can be controlled. To control the Type II error, protection levels are formulated and compared with the alert limit. If the protection level is contained within the alert limit then the probability of making a Type II error is acceptable. Otherwise, it is not. When formulating the protection levels it is desired to set the size of Type II error for each test such that the probability of a fault going undetected by all of the tests is equal to the probability of a missed detection, P _MD. However, due to the difficulty in achieving this, Kelly (Reference Kelly1998) uses the approximation:

(14)$$\beta _0 \le P_{{\rm MD}}. $$

2.3. Fault Exclusion

When the fault detection procedure has detected a fault, the next step is to attempt to identify and remove the faulty pseudorange. Since the null hypothesis has been rejected, the pseudorange measurements conform to one of the alternative hypotheses:

(15)$$H_i :{\rm E}({l}) = {Ax} + {c}_i \nabla _i \quad i = 1, \cdots, n$$

To determine which alternate hypothesis, the largest outlier statistic, in absolute value, is found. The corresponding pseudorange is then deduced to be faulty. Mathematically this can be expressed as the j ^th pseudorange is faulty when:

(16)$$|w_j | \gt |w_i |\; \; \forall i\; \quad {\rm and}\quad |w_j | \gt {\rm N}_{\displaystyle{{\alpha _0} \over 2}} (0,1)$$

Once the faulty pseudorange has been identified, corrective action must be taken to mitigate its influence on the navigation solution. Here, the identified pseudorange is excluded from the model such that Equation (1) now has one fewer pseudorange measurements. Since the incorrect pseudorange can at times be identified due to the correlation, a FDE procedure would normally be reapplied to the updated model, until the null hypothesis is accepted.

3.1. Three Types of Error, Based on Two Alternative Hypotheses

In Förstner's pioneering studies, the decisions that can be made with two alternative hypotheses are described in Table 1 (Förstner, Reference Förstner1983; Li, Reference Li1986).

Table 1.

Decisions when testing two alternative hypotheses.

From Table 1, it can be seen that α ₀₀=α _0i+α _0j, and because of the symmetry of w _i and w _j, $\alpha _{0i} = \alpha _{0j} = {\textstyle{1 \over 2}}\alpha _{00} $. In addition, the following is satisfied:

(17)$$\beta _{ii} = \beta _{i0} + \gamma _{ij} $$

and:

(18)$$\beta _{\,jj} = \beta _{\,j0} + \gamma _{\,ji} $$

The estimation of parameters shown in Table 1 is based on the distribution of test statistics. It is assumed that there is an outlier in the i ^th observation, causing the expectation of w _i to become δ, which is the non-centrality parameter. The bias also causes the expectation of w _j to become δρ because of the correlation between w _i and w _j, which can be computed from Equation (6). Successful identification then actually means accepting the alternative hypothesis H _i rather than H _j. The joint distribution of w _i and w _j is:

(19)$${w} = (w_i {\kern 1pt} w_j )^{\rm T} \sim N({\mu}, {\kern 1pt} {D})$$

The expectation of the joint distribution of w _i and w _j in Equation (13) is then given by:

(20)$${\mu} = (\delta {\kern 1pt} \delta \rho )^{\rm T}, \,\; {\rm and}\, \; {D} = \left( {\matrix{ 1 & \rho \cr \rho & 1 \cr}} \right)$$

The probability function of the two outlier statistics is given by:

(21)$$f\,(w_i, {\kern 1pt} w_j ) = f_1 $$

If the critical value c _α and the distribution of w are known, the probability of successful identification, denoted as (1−β _ii), can be obtained from:

(22)$$1 - \beta _{ii} = \int_{|w_i | \gt c_\alpha, \; |w_i | \gt |w_j |} {\,f_1 dw_i dw_j} $$

The sizes of Type II and Type III errors can be obtained from:

(23)$$\beta _{i0} = \int_{|w_i | \le \; c_\alpha, \; |w_j | \le \; c_\alpha} {\,f_1 dw_i dw_j} $$

and:

(24)$$\gamma _{ij} = \int_{|w_j | \gt c_\alpha, \; |w_j | \gt |w_i |} {\,f_1 dw_i dw_j} $$

In Förstner (Reference Förstner1983) and Li (Reference Li1986) if α ₀, ρ, β _i0 and γ _ij are given, then the non-centrality parameter δ is obtained from:

(25)$$\delta _1 = \varphi _1 \left( {c_\alpha, \rho, \beta _{i0} \;} \right) = \varphi _1 \left( {\alpha _0, \rho, \beta _{i0} \;} \right)$$

or:

(26)$$\delta _2 = \varphi _2 (c_\alpha, \rho, \gamma _{ij} ) = \varphi _2 (\alpha _0, \rho, \gamma _{ij} )$$

By setting the values of β _i0 and γ _ij with the same preset values of α ₀ and ρ, the non-centrality parameters may be different. In this case, the greater value of δ is chosen to satisfy the requirements that the probability of a Type II error is not greater than β _i0 and that the probability of a Type III error is not greater than γ _ij (Förstner, Reference Förstner1983; Li, Reference Li1986). In this paper the non-centrality parameter is calculated from:

(27)$$\delta = \varphi (c_\alpha, \rho, \beta _{ii} ) = \varphi (\alpha _0, \rho, \beta _{ii} )$$

This is because the probability of making errors, β _ii, remains unchanged with different correlation coefficients. Nonetheless, correlation coefficients do determine the ratio between β _i0 and γ _ij. Consequently, given preset values for α ₀₀ and β _ii the non-centrality parameter will change along with the correlation coefficient. For this non-centrality parameter the probability of making Type II and Type III errors will not be greater than β _ii, and in addition their sum will be equal to β _ii.

3.2. Relationships between Different Parameters

Although there are many parameter factors that control the probabilities of making different types of error with two alternative hypotheses, there are only three that are fundamentally independent. They are α ₀, ρ and β _ii. Any other parameters can be obtained as shown in Table 2.

Table 2.

The relationship among different parameters.

From Table 2 it can be seen that once α ₀ and ρ are given, the threshold c _α, and the size of the Type I errors α ₀₀, α _0i and α _0j can be estimated. Also by setting β _ii as well, δ, β _i0 and γ _ij can be calculated.

The changes within β _i0 and γ _ij are shown in Figure 1 when β _ii is equal to 20% and α ₀ is set to values of 0·1%, 0·3%, 0·5%, and 1%. This illustrates the fact that the Type II error β _i0 and the Type III error γ _ij have opposite tendencies as the correlation coefficient increases. For small correlation coefficients β _i0 is large and γ _ij is small, whereas for large correlation coefficients γ _ij is large and β _i0 is small. When ρ is close to zero, β _i0 is around 20% and γ _ij is approximately zero. This is irrespective of the size of the Type I error. Conversely, β _i0 reduces quickly to about zero when ρ approaches 0·98 and γ _ij increases rapidly to 20%.

Figure 1.

Type II and III errors.

The above analyses are based on the preset values of α ₀ and β _ii in order to determine the dependence of the other parameters on ρ. In the following analysis, α ₀ is preset to 1% and changes in δ and ρ are compared to β _ii, β_{i 0} and γ _ij.

The value of β _ii, changing with δ and ρ, is shown in Figure 2, which demonstrates that a larger correlation coefficient leads to higher value of β _ii when δ is kept constant. In addition, larger δ results in smaller β _ii for the same value of ρ. This means that higher δ and smaller ρ will enhance the probability of correct identification. When δ becomes larger, the impact of the correlation coefficient on β _ii becomes much more significant. When ρ is zero, β _ii decreases quickly from around one to zero as δ increases. This means that when the outlier statistics are independent from each other the probability of committing errors can be controlled to near zero, once δ is large enough. However, as ρ approaches 1, even increasing the non-centrality parameter to 20 still does not reduce β _ii below 40%. This indicates that when the correlation coefficient is approximately 1 the non-centrality parameter only has a small effect in decreasing β _ii. Consequently, to control the probability of making errors, the goal should be to keep the correlation coefficients, between outlier statistics, to minimum values.

Figure 2.

The sum probability of making errors β _ii=β _i0+γ _ij.

The values for β _i0 and γ _ij are shown in Figure 3 and Figure 4 respectively. Figure 3 shows that β _i0 decreases from about one to zero as δ increases. The decrease within the β _i0 curves for different ρ are similar. Thus, the correlation coefficient plays a relatively minor role in influencing β _i0. Figure 4 shows that the correlation coefficient significantly impacts γ _ij. All of the curves quickly increase to a peak and then decrease slowly. The larger the correlation coefficient values the higher the peak and the slower the decrease. For large correlation coefficients, the peak also occurs at larger values of the non-centrality parameters. When ρ is zero, γ _ij is always close to zero no matter what the value of δ. When the correlation coefficient is close to 1, γ _ij increases dramatically from about zero to nearly 50% and then decreases rather slowly. Therefore, the dominant challenge in successful identification is avoiding making a Type III error, when the correlation coefficient is high. This is because it is difficult to control the probability of so doing by increasing δ.

Figure 3.

Type II error β _i0.

Figure 4.

Type III error γ _ij.

Based on the above analysis, it can be deduced that the probabilities of making Type I, II and III errors can be accurately estimated based on their correlation coefficients and their non-centrality parameters. These parameters can also be used to provide an accurate control for making correct decisions in a FDE procedure. For instance, a large correlation coefficient implies a greater chance of incorrect identification. Under these circumstances, a larger non-centrality parameter is required to control the probability of committing a Type III error.

3.3. Fault Verification

As can be seen, the test statistic is a function of all observation errors, and the correlation coefficient between each test statistic contributes to missed detection and wrong exclusion. This section presents a practical procedure to estimate the probability of a missed detection and the probability of a wrong exclusion. The calculation is based on the assumption: that if one considers only the bias error on one of the satellites, at a time, and neglects the range errors on all the other satellites, then the position estimation error and the test statistic become linearly proportional, and the slope varies depending on which satellite has the bias error. The satellite that is the most difficult to detect is the one with the maximum correlation coefficient and with the highest probability of a wrong exclusion. The probability of a missed detection is highest for the failure of that satellite (Lee, Reference Lee1995; Reference Lee, Van Dyke, Decleene, Studenny and Beckmann1996).

From Equation (6), when only the bias error on the i ^th observation is taken into consideration and the random errors on other satellites are neglected, the test statistic simplifies to:

(28)$$\bar w_i = \displaystyle{{{c}_i^{\rm T} {PQ}_v {Pc}_i \varepsilon _i} \over {\sigma _0 \sqrt {{c}_i^{\rm T} {PQ}_v {Pc}_i}}} $$

The influence of this bias error on another test statistic is:

(29)$$\bar w_j = \displaystyle{{{c}_j^{\rm T} {PQ}_v {Pc}_i \varepsilon _i} \over {\sigma _0 \sqrt {{c}_j^{\rm T} {PQ}_v {Pc}_j}}} = \rho _{\,ji} \bar w_i $$

However, as the real circumstance is never known, the hypothesis test is an inverse procedure, which actually uses the estimated statistic to deduce the unknown fault. According to Table 1, the second column of the test result shows that during the classical FDE procedure, when the greatest absolute value is pointed to the i ^th test statistic, there are two types of possibility, either resulting in a successful identification or a wrong exclusion. A wrong exclusion means that the outlier occurring on the j ^th observation impacts on the i ^th test statistic so that w _i becomes greater than the critical value. Consequently, the expectation shift of w _i may originate from:

(30)$$\delta _i \approx E(\bar w_i )$$

which will lead to a successful identification, or:

(31)$$\delta _j \approx \displaystyle{{E(\bar w_i )} \over {\rho _{ij}}} $$

which means a wrong exclusion will be committed after the test.

Based on Equations (30) and (31), the non-centrality parameter corresponding to the largest test statistic can be obtained. Then, using each non-centrality parameter and the correlation coefficient, the probability of successful identification and wrong exclusion can be calculated from the relationship in Table 2. There is:

(32)$$1 - \beta _{ii} = \varphi (\alpha _0, \rho, \delta _i )$$

or:

(33)$$\gamma _{\,ji} = \varphi (\alpha _0, \rho, \delta _j )$$

As it is complicated and time consuming to exactly calculate β _ii and γ _ji via numerical integration, the approximate solution by interpolation could be obtained via the grid data illustrated in Figures 2, 3 and 4, once the grid data is accurate enough. Comparing the estimated values of β _ii and γ _ji with the corresponding preset thresholds, decisions about the successful identification and wrong exclusion can be made.