The relationship between the mean (a) and the variance (b) of the cover area S of the two methods FI and FII at different m. It can be seen that when r increases, the variance of S increases first and then decreases; when r increases, the variance obtained by the sequential method FII is smaller than FI, and when m is greater, the effect is more obvious. Based on the simulation data, we can also use the Beta distribution to approximate the St/π obtained by FII. When ri is unequal, or σ2i in the two-dimensional normal distribution (1) is different, there are similar results, which are not detailed here.
This shows that FII is better than FI, that is, in order to increase the coverage area S, we recommend the method of sequentially determining the mean value of the random center of the circle. Weighted Unit Circle In the previous discussion, it was assumed that each position of the unit circle K is equally important. In practice, the weights of the different positions of the unit circle K may be different. The weight function may be sub-region or continuous. For example, the weight W obeys the conditional two-dimensional normal distribution at K, that is, the density function of W is f(w)=g(w)∫Kg(w)dw,w=(w1,w2)∈K,(9) Where g(w) is the density function of the two-dimensional normal distribution N2(θ, ηI2), where θ=(θ1, θ2) is the position of the mean, ie g(w)=12πηexp{12η[(w1θ1)2 ( W2θ2)2]}. (10) If θ is in the unit circle K, then θ has a larger weight around; if θ is far away from the unit circle K, the weights on K can be considered to be the same. Without loss of generality, the following assumes that the weight obeys the two-dimensional normal distribution of conditions. At this time, the objective function is changed from the area S of the coverage area Sm of equation (2) to the integration of the weight function on Sm, and is denoted as SW, ie, SW=∫Smf(w)dw, and weighted by the two methods FI and FIII. The relationship between the mean (a) and the variance (b) of the unit circle SW and σ2. Where f(w). Easy to know, 0 ≤ SW ≤ 1. In order to increase SW, we can see from the study of 2 sections that the sequential method is more effective. Therefore, we consider the sequential determination of the center Oi of the random circle, and set Oi to the two-dimensional normal distribution of (6), where the two coordinates of μi become μi1=Diw1f(w)dw1dw2,μi2=Diw2f ( w) dw1dw2, Di denotes an area that has not yet been covered by a random circle in the unit circle K, and f(w) = f(w1, w2) is a density function as shown. Due to the complexity of the theoretical calculations, we can use a uniformly designed test point approximation. We note that this sequential method is FIII. Test 1: Let θ=(0,0), and η=1 in equation (10). At this time, the origin O and the nearby concentric circles have a great weight.
Consider the condition (8), and give the mean and variance of the SW of the different σ2 in the methods FI and FIII when the number of random circles is m=2, 3, 4, and 5 when r=0.8, and each method in each case Repeat 50000 times. The result is similar, that is, compared with FI, the sequential method can increase the mean value of SW, and the smaller the σ2, the more obvious the improvement; for the same method, when the random number m is increased, the SW can be increased. Mean, reduce SW's variance. On the other hand, the empirical distribution of experimental data in each case can be approximated by the Beta distribution. Due to space limitations, it will not be given here. Test 2: Let θ = (θ1, θ2) ∈ K, and η ∈ [0.1, 3].
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