It is generated using number theory methods (such as good lattice points) and is described in detail in [1,3]. The steps are as follows: N number of uniformly distributed points (ui, θi) are generated on the unit square [0,1] using the number theory method, i=1, N, and then by transforming xi=√uicos(2πθi), yi=√uisin(2πθi) ), 1 ≤ i ≤ N, (3) then {(xi, yi), 1 ≤ i ≤ N} is evenly distributed over K and can be used as a representative point of K. In previous studies, it was assumed that the mean value of the center of the random circle was the origin. This is obviously not the best strategy in practice because the actual problem is that the bigger the S, the better. If the random circles are covered sequentially, their center of the circle design takes into account the area covered by the preceding random circle. In practical problems, the importance of different fixed circles may not be the same. Important areas must be covered first. Therefore, fixed circles with unequal weights have more practical significance. The above two issues are the focus of this study. In the first section, we use a uniform design to consider the effect of changes in the parameters {r1, rm, σ21, σ2m} of m random circles on the mean, variance, and distribution of S, and estimate the distribution of S. The second section proposes a sequential covering method for random circles and compares them with traditional models. In the third section, it is assumed that different areas in the unit circle have different weights, and the sequential method is considered to cover the unit circle.
Coverage Area Distribution Study
This section considers the distribution of S under the classical assumptions (Oi~N2(0, σ2iI2) and its relationship with the parameters {r1, rm, σ21, σ2m}. Take 0.5 for this, we are in the parameters r1, r2, σ21, Random sampling of 2000 combinations within the range of σ2, and 2000 averages and variances; and then using the model to predict Mn and Vn, so as to obtain its prediction mean square error.
The results show that the prediction model has a mean square error of 0.0045, while the model (5) has a predicted mean squared error of 0.0013. Both have achieved good results. From the model, we know that the mean value of S is positively related to the radius ri of the random circle, and is negatively related to the logarithm of the variance σ2i. That is, when the radius is larger, the mean value may be larger. From the model, we can see that the variance of S and ri Both σ2i are positively correlated. From the estimation of 10,000 S for each parameter combination, the distribution of S can be further estimated, or the distribution of S/π can be estimated equivalently. We found that they can all be fitted with a Beta distribution. The qq of the Beta distribution of the 10000 simulation values ​​of S/π is given for the 4, 7, 18, and 20 parameter combinations. It is easy to see that the Beta distribution can better fit the St/π and the variance at the test point. When the σ21 or σ2 is small, the Beta distribution fits better, for example, the fourth test point. However, the two ends of the Beta distribution are often not well fitted, for example, the variances σ21 and σ2 are large, and when r1 and r2 are small, the coverage area is possible.
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