Our principal aim is to predict the class of vehicle and lane relative to the new observed functional data. In order to measure the performance of our functional nonparametric discrimination method, we set up two samples from the collected data sets. One is the learning sample, and the other is the testing sample. The learning sample allows us to estimate the posterior probabilities. The testing sam-ple is useful for assessing the discriminant power. We can get the predicted groups of testing sample. The misclassification rate can be evaluated by the proportion of predicted groups not equal to the observed groups. Repeating this procedure 5000 times, we will get 5000 misclassification rates. The distribution of these misclas-sification rates can give a good idea of the power of nonparametric discrimination of functional data with various semi-metrics. Three types of semi-metrics are con-sidered, and each of them has different forms being used. The different forms of three proximity measures are as follows:
(i) PLS-type semi-metrics with a number of factors taking its values in 2, 3, 4, 5, 6, 7, 8 and 9 successively.
(ii) PCA-type semi-metrics with a number of components taking its values in 2, 3, 4, 5, 6, 7 and 8 successively.
(iii) Derivative-type semi-metrics with a number of derivatives equal to zero (clas-sical L2-norm).
We first consider the recognition of traveling vehicles with fixed lane and speed.
In order to measure the performance of our functional nonparametric discrimina-tion method, we build randomly two samples from the original dataset. The learn-ing sample and testlearn-ing sample have 18 functional observations for each of classes of vehicles. Repeating the procedure 5000 times, we will get 5000 misclassification rates. The Figure 4 displays the box-plot of misclassification rates for three types of semi-metrics proximity. The Figure 5 displays the scatter plot of mean and vari-ance of misclassification rates for three types of semi-metrics proximity. Our goal is to predict the classes of traveling vehicles correctly. So we hope to find the robust methodology with lower misclassification rate. From the Figure 4 and Figure 5 we prefer PLS-type semi-metrics with 2 factors and PCA-type semi-metrics with 3 components to the others. The proportion that we can correctly recognize the class of traveling vehicle is about 84%. The Figure 6 displays the classification rate over 5000 runs with 2 classes of vehicles. The columns denote the real classes and the rows denote the predicted classes. The correct classification rates and misclas-sification rates are both exhibited in the Figure 6. The diagonal rates denote the correct classification rate over the 5000 runs. Others denote the misclassification rate over the 5000 runs.
plsr2 plsr4 plsr6 plsr8 pca2 pca4 pca6 pca8
0.00.10.20.30.40.5
SEMI−METRICS
MISCLSSIFICATION RATES
Figure 4: Dynamic vehicle recognition data discrimination over 5000 runs with 2 classes of vehicles.
0.0020 0.0025 0.0030 0.0035 0.0040 0.0045 0.0050 0.0055
0.160.180.200.220.240.260.28
Var of Misclassification Rate
Mean of Misclassification Rate
plsr2
plsr3 plsr4
plsr5
plsr6 plsr7
plsr8 plsr9
pca3pca2 pca5 pca4
pca6 pca8 pca7
deriv0
Figure 5: Scatter plot of mean and variance of misclassification rate over 5000 runs with 2 classes of vehicles.
Figure 6: The classification rate over 5000 runs with 2 classes of vehicles. The columns denote the real classes and the rows denote the predicted classes.
Our second aim is to classify the lanes and dynamic vehicles with fixed speed.
In order to measure the performance of our functional nonparametric discrimina-tion method, we also build randomly two samples from the original dataset. The numbers of functional observations of small vehicles on lane 1, 2, 3 and 4 take its value in 10, 10, 10 and 10. Then the numbers of functional observations of large vehicles on lane 1, 2, 3 and 4 take its value in 10, 10, 10 and 5. The size of learning sample and testing sample are the same. We also repeat 5000 times above procedure to get 5000 misclassification rates. The Figure 7 and 8 are analo-gous to the Figure 4 and Figure 5. We tend to regard PLS-type semi-metrics with 2 or 3 factors as the proximity for these data. The correct classification rate is about 42%, a dramatic drop due to increasing in number of classes. The Figure 9 displays the classification rate over 5000 runs with 8 classes of vehicles and lanes.
The columns denote the real classes and the rows denote the predicted classes.
The correct classification rates and misclassification rates are both exhibited in the Figure 9. The diagonal rates denote the correct classification rate over the 5000 runs. Others denote the misclassification rate over the 5000 runs.
plsr2 plsr4 plsr6 plsr8 pca2 pca4 pca6 pca8
0.40.50.60.70.8
Semi−Metrics
Misclassification Rates
Figure 7: Dynamic vehicle recognition data discrimination over 5000 runs with 8 classes of vehicles and lanes.
0.0022 0.0024 0.0026 0.0028
0.600.620.640.66
Var of Misclassification Rate
Mean of Misclassification Rate
plsr2 plsr3
plsr4 plsr5
plsr6 plsr7
plsr8 plsr9
pca2
pca3 pca4
pca6 pca8 pca5pca7 deriv0
Figure 8: Scatter plot of mean and variance of misclassification rate over 5000 runs with 8 classes of vehicles and lanes.
Figure 9: The classification rate over 5000 runs with 8 classes of vehicles and lanes.
The columns denote the real classes and the rows denote the predicted classes.
4 Conclusion
This research uses three semi-metrics proximity to deal with nonparametric dis-crimination of functional data. The result shows PLS-type semi-metrics proximity is more appropriate for the Radar microwave data set of the recognition of travel-ing vehicles. The vehicles functional data can be treated as multivariate data with 30 predictors, then PCA and PLS methodology implement data reduction with 2 or 3 dimension. But their recognition rate is higher than that with more factors.
The large vehicles contain cranes, buses, trucks, goods wagons, etc. These vehicles have distinct forms, but we classify them analog. This may cause the recognition rate to be not as high as we expect. No matter how we expect that the nonpara-metric discrimination of functional data with these three semi-nonpara-metrics proximity contribute to improve traffic jam and build the lane for a special purpose.
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