数学代写|组合学代写Combinatorics代考|CS519

2022年9月27日

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• Statistical Computing 统计计算
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• Foundations of Data Science 数据科学基础
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数学代写|组合学代写Combinatorics代考|JPDA/Res with Parallel Object Tracks

In this section, the two objects move in close proximity to one another for an extended period of time. The object motion is quite similar to that in the IPDA numerical example in Sect. 2.8, and tracking results are shown in Figs. $3.3$ and 3.4.

Both objects move through $\mathcal{R}$ at a constant speed of $50 \mathrm{~m} / \mathrm{s}$. Object one begins in state $\mathbf{x}0^1=\left(\begin{array}{llll}-1700 & 0 & 1830 & -50\end{array}\right)^T$ at time $t_0=0$. It moves at this constant velocity for $15 \mathrm{~s}$, turns at $3^{\circ}$ per second counterclockwise for the next $30 \mathrm{~s}$, and then moves in the positive $x$-direction for $75 \mathrm{~s}$. Object two mirrors this process. For $k=0,1, \ldots, K$, if $\mathbf{x}_k^1=\left(\begin{array}{llll}x_k^1 & \dot{x}_k^1 & y_k^1 & \dot{y}_k^1\end{array}\right)^T$ is the state of ohject one at scan $k$, then $\mathbf{x}_k^2=\left(x_k^1 \quad \dot{x}_k^1-y_k^1-\dot{y}_k^1\right)^T$ is the state of object two at scan $k$. Thus, the two objects remain at a constant distance of $6 \sigma_M=300$ from one another for the final $75 \mathrm{~s}$. The region of interest spans $\mathcal{R}=[-2000,3000] \times[-3000,3000]$. The mean number of clutter measurements $\lambda_k^c \equiv \lambda^c=100$ is constant over all scans. Thus, in each scan, there is an average of approximately $0.24$ clutter measurements in a $3 \sigma_M$ measurement window. The tracker process noise standard deviation is set to $\sigma_p=4$, and the resolution parameter is chosen to be $\sigma{\mathrm{res}}=182.6$. This value for $\sigma_{\mathrm{res}}$ gives a resolution probability of approximately $0.75$ when the objects are moving in parallel (i.e., $300 \mathrm{~m}$ apart).

Figures $3.3$ and $3.4$ represent two different scenarios. The annotations in the figures are the same as in Fig. 3.2. In the first scenario, the unresolved weighting factor is $w_r=10 / 11 \approx 0.91$. Under the relative signal return strength interpretation, the signal return strength of object one is $10 \log _{10}\left(\frac{w_r}{1-w_r}\right)=10 \mathrm{~dB}$ higher than that of object two and, thus, most of the green unresolved measurements are close to object two, as seen in Fig. 3.3.

In the second scenario, the unresolved weighting factor is $w_r=0.5$, i.e., the signal return strengths of both objects are equal-neither of the objects is favored. Tracking results for both the standard JPDA and JPDA/Res trackers are displayed in Fig. 3.4.

数学代写|组合学代写Combinatorics代考|Discussion of Results

The standard JPDA filter inherently assumes that a given measurement is either clutter, or it originates from exactly one of the objects of interest. Thus, in the unresolved measurement case where an object’s return is buried under the return of another object or where objects are close enough so that they fall into a single resolution cell, the standard JPDA filter model is hopelessly mismatched to the reality of the data-it lacks model fidelity.

The mismatch is clearly seen in the upper subplots of Figs. $3.2$ and 3.3. In both the crossing and parallel scenarios, since the red object produces stronger returns than the blue, the unresolved measurement more often “favors” the red object than the blue; thus, the blue object is left without a measurement. Therefore, the existing unresolved measurement is, for all practical purposes, assigned to the red object when performing the measurement update, whereas the blue object’s estimate is extrapolated based on its motion model and the previous measurement update.

In contrast, the JPDA/Res filter allows for the possibility that only one signal return (measurement) could mean that the two objects are unresolved. The various ways this can happen are accounted for by the six terms (3.60)-(3.65). Thus, despite increased estimation error due to inflated covariance, both objects are successfully tracked as shown in the bottom subplots of Figs. $3.2$ and 3.3. It is also seen in the bottom subplot of Fig. $3.2$ that the blue object is extrapolated without being drawn off, or seduced, by clutter, as it was in the upper plot. This is rather remarkable, and somewhat unexpected. The explanation is the improved fidelity of the model.

JPDA/Res also improves tracking performance when both objects produce equal strength returns and, hence, neither object is favored. As illustrated in the top subplot of Fig. 3.4, although standard JPDA maintains two tracks in the presence of unresolved measurements, the tracks switch at the start of the parallel motion and then switch back later. This is an undesirable situation in, e.g., radar tracking where maintaining track IDs is of utmost importance. Such an outcome is not observed with JPDA/Res, however, as can be seen in the lower subplot of Fig. 3.4. The reason, again, is that the likelihood function has six terms (3.60)-(3.65) to account for the way point measurements can be generated by the sensor.

组合学代考

数学代写|组合学代写Combinatorics代考|结果讨论

JPDA/Res也提高了跟踪性能，当两个对象产生相同的强度回报，因此，两个对象都不受青睐。如图3.4的顶部子图所示，尽管标准JPDA在存在未解测量时保持两条轨迹，但在平行运动开始时轨迹切换，随后又切换回来。这是一个不希望出现的情况，例如，在雷达跟踪中，保持航迹标识是极其重要的。然而，在JPDA/Res中没有观察到这样的结果，从图3.4的下副图中可以看到。同样，原因是似然函数有6项(3.60)-(3.65)来解释传感器产生的点测量的方式

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MATLAB代写

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