Depth-Based Detection

In order to formulate the hypothesis test, we need to first define the two populations that are to be compared. The first population is that of a general class of arbitrary distribution, representative of the background medium-i.e. Gaussian noise. The second population is a measured time-reversal operator consisting of a bounded rank signal perturbation with additive noise; in which the noise is scaled to simulate a set of signal-to-noise sample values. From these two populations, we define a new depth-based detection statistic, but first let us revisit the binary hypothesis test utilized for our example.

𝐻_"∶= 𝒏 𝑡 (4.5a)

𝐻_#∶= 𝒔 𝑡 + 𝒏 𝑡 (4.5b)

Typically, we measure a component of the received signal and compare this value to a pre-determined, or adaptive, threshold that allows us to transform equation (4.5a) and (4.5b) into

𝛿_ThresholdN×

Ø

O×N _(4.6)

with the null hypothesis indicating the absence of a signal. The depth-based detection method is also based on a threshold statistic, determined from a ratio of two dispersion values

P_Ð

P_Threshold = 𝛿 (4.7)

in which 𝑉_F and 𝑉Threshold are the differential dispersion values for the populations

of the measured time-reversal operator and noise matrix with arbitrary distribution, respectively. Differential dispersion values are derived from the difference of two dispersion values, as shown below

𝑉_Threshold = 𝑉_Threshold? − 𝑉_Threshold™ (4.8) The differential dispersion is the difference between the volume defined by the contour 𝛽, and that of the volume of a second contour 𝑝. Typically, we define 𝛽 ≃ 1, to ensure we incorporate all of the population values in our depth functional; the

second dispersion is found from a smaller contour defined by 𝑝 ∈ 0,1 . In this instance, we have defined 𝑝 = 0.5,0.75,0.9 . The difference between these two contours defines the volume of an annular region, 𝑉Threshold; with an increase in

the annular region being attributed to the presence of a signal. We compare the differential dispersion of the assume noise threshold, with that of the measured data. In this manner, the second differential dispersion value in the threshold statistic is given as

𝑉_F = 𝑉_F?− 𝑉_F™ (4.9)

The depth-based detection binary hypothesis test is now akin to 𝛿_N×O×_ØN_𝛿

Threshold (4.10)

where the 𝛿_Threshold is determined for a given class of measurement noise. For the purpose of this thesis, the threshold is found empirically through a Monte Carlo simulation; a large number of noise realizations were created, for two population groups of white Gaussian noise comprised of 124 singular values, to determine the empirical volume of the annular region bounded by the contour 𝐷™ _{and 𝐷}?_{; this}

Monte Carlo simulation was repeated several times to ensure a consistent estimator for the empirical mean 𝜇 and standard deviation 𝜎. For each instance, the empirical volume is calculated for the annulus by subtracting the volume of the 𝑝 = 0.5,0.75,0.9 contours from the 𝑝max = 1 contour. The mean and variance for

the volume of the annular region is listed in Table 2

Table 2: Empirical Thresholds

𝑽_𝒑𝒕𝒉 Annulus 𝝁 𝝈 𝑽_Empirical

𝟗𝟎% 0.0016 3.2𝑒=D _{𝜇 + 𝑘𝜎}

𝟕𝟓% 0.0011 3.2𝑒=D _{𝜇 + 𝑘𝜎}

𝟓𝟎% 4.3𝑒=D _2.3𝑒=D _{𝜇 + 𝑘𝜎}

From Chebyshev's Inequality, we know that for any distribution in which the standard deviation is defined, the variables that fall within a certain number of standard deviations from the defined mean, 𝑘𝜎, is at least as much as [55]

Table 3: Empirical Threshold Table

Minimum Population from the Mean

Number of Standard Deviations k 50% ₂ 75% 2 89% 3 94% 4 96% 5 97% 6 𝟏 − 𝟏 𝒌𝟐 k

From Table 3, we see that for an empirical false alarm rate of 6%, we would require the 𝑉ö€Ó[W€\-j, 𝜇 + 𝑘𝜎, to be equivalent to 𝜇 + 4𝜎; likewise, for an empirical false

alarm rate of 4% and 3% , we require the 𝑉_{ö€Ó[W€\-j} to be 𝜇 + 5𝜎 and 𝜇 + 6𝜎 , respectively. Further, the dispersion resulting from the addition of a signal in the measured noise is manifest from the outlying nature of the signal singular values, when compared to the body of the measured noise plus a priori noise distribution data depth functional [56]; which is true for nominal signal-to-noise ratios (SNR). This is due to the fact that the singular values associated with the signal exhibit a level of eigenvalue repulsion, allowing them to be separate from the body of the data depth functional, see Figure 15. As the SNR decreases, this 'eigenvalue' repulsive force becomes weaker and the signal singular values become distributed on the outer contour of the data depth functional. From this vantage point, we are not seeking the point of deepest depth for signal detection, but the values for which the singular values are most outlying, and result in an increase in volume for a given annular contour bounded by 𝑝h€ _{∈ 𝑝, 1 . Since we have defined the metric}

as the ratio of scale curves, and the false alarm rate is controlled by the empirical 𝑉_{ö€Ó[W€\-j}, in this manner the threshold is actually equivalent to unity, 𝛿_{ö€Ó[W€\-j} = 1. The Chebyshev Inequality represents a more severe constraint on the detection statistic, and should be more robust, though may result in lower 𝑃_¬ versus SNR for a given 𝑃_Q-.

Having introduced the depth-based detection algorithm, a summary of the procedures for implementing the algorithm are listed in the table below for

Algorithm 4.1: Depth-Based Detection Algorithm

1a: Calculate 𝑵 Versions of the target-free time-reversal operator from target-free training data, B_{𝑵;Threshold}

1b: Calculate the time-reversal operator under test, B

2: Set the desired values for contour 𝑫𝜷 _{and 𝑫}𝒑

3a: Calculate the differential volumes from the annulus of each of the 𝑵 target-free time-

reversal operator formed by the contours of 𝑫𝜷 _{and 𝑫}𝒑

𝑽_{𝑵;Threshold}= 𝑽_{𝑵;Threshold}𝜷 − 𝑽_{𝑵;Threshold}𝒑

3b: Calculate the differential volumes from the annulus of the time-reversal operator under test

formed by the contours of 𝑫𝜷 _{and 𝑫}𝒑

𝑽𝒊= 𝑽𝒊

𝜷_{− 𝑽} 𝒊 𝒑

4a: Calculate the mean, 𝝁, and standard deviation, 𝝈 of the target-free differential volumes, 𝑽Threshold

4b: Set the desired false alarm rate, 𝒌, to form 𝑽_Threshold= 𝝁 + 𝒌𝝈

4c: Calculate the nonparametric threshold value for the binary hypothesis test 𝛿_Threshold=𝑽𝑻𝒉𝒓𝒆𝒔𝒉𝒐𝒍𝒅; 𝝁a𝒌𝝈

𝑽𝑻𝒉𝒓𝒆𝒔𝒉𝒐𝒍𝒅; 𝝁a𝒌𝝈 = 1

5: Calculate the ratio of differential volumes for the threshold and time-reversal operator under test

𝛿 = 𝑉b

𝑉_{ö€Ó[W€\-j} 6: Determine the outcome of the binary hypothesis test

𝛿_N×O×_ØN_𝛿

Threshold

Having determined the empirical threshold statistic for the hypothesis testing, the proceeding section is dedicated to a demonstration of efficacy of the depth-based detection algorithm for a range-spread target comprised of three scattering centers corrupted by measurement noise.