摘要:
摆镜平台是大口径空间天文望远镜图像稳定控制系统中的一个活动部件,为评估其寿命,进行了摆镜平台寿命试验。摆镜平台寿命试验为单子样寿命试验,先验概率信息很少,采用Bayes方法进行寿命估计有一定的困难。Bootstrap方法是一种非参数统计方法,它对未知分布不做任何假设,仅利用计算机对原始样本数据进行再抽样模拟未知分布,通过再生抽样将小样本问题转化成大样本问题。Bootstrap方法要求子样数要大于等于5。采用半经验虚拟增广子样和Bootstrap相结合的方法进行摆镜平台寿命的区间估计。在摆镜平台寿命服从威布尔(Weibull)分布的假定下,通过以往类似产品寿命试验数据推断出威布尔分布形状参数,然后用寿命试验值和威布尔分布形状参数推导出摆镜平台寿命的方差,在此基础上进行子样虚拟增广。虚拟增广子样需要满足两个条件:(1)增广子样的均值等于寿命试验值;(2)虚拟增广子样的方差等于类似产品寿命的经验方差。根据这两个条件建立一个方程组,用数值计算法求解方程组得到增广子样。这些作为Bootstrap区间估计原始样本的增广子样用于计算样本的经验分布和样本重抽样。通过重抽样得到10 000个样本,计算每个样本的均值,然后对10 000个均值进行排序,得到均值μ1≤μ2≤…≤μ10 000,采用分位数法计算置信水平为1-α的Bootstrap置信区间,得到置信区间(μk1,μk2),其中k1=[10 000×α/2], k2=[10 000×(1-α/2)]。寿命试验的加速因子为5.5,目前摆镜平台已正常工作1.25年,按上述方法计算0.95置信水平的Bootstrap置信下限为3.59年,如果正常工作两年,0.95置信水平的Bootstrap置信区间为(5.93年,16.81年)。
Abstract:
A tip-tilt platform is a movable component of the system for maintaining imaging stability in a large-aperture space telescope. There is an ongoing test to estimate the lifetime of a tip-tilt platform to be onboard a Chinese space solar telescope. This test has only a single sampling data point by nature. The prior probability distribution of the lifetime of a tip-tilt platform needed for any Bayesian lifetime estimation is also not available due to the scarcity of relevant lifetime data. A bootstrap method is a statistical method without parameterization, and it does not require any prior distribution. A bootstrap method, which can be employed with a computer, uses resampling to simulate the statistical distribution of interest. The resampling generates samples from a template sample which can be rather small (and is usually an actual sample). This effectively creates a large-size sample from a small-size sample. Generally, a bootstrap method requires the number of data points in its template sample to be no less than five. In this paper, we combine a semi-empirical sample virtual-augmentation method into a bootstrap method to estimate confidence intervals of the lifetime of the tip-tilt platform in the ongoing test. We take the lifetime of the tip-tilt platform to statistically follow a Weibull distribution,with the shape-parameter values of the Weibull distribution inferred from previous experimental data of similar products. The variance of lifetimes of similar tip-tilt platforms can be expressed by the shape-parameter values of the Weibull distribution and the recorded time of normally working of the platform in the test. Based on the expression the sample virtual-augmentation method numerically generates more data points to meet two statistical conditions, i.e., (1) their mean is equal to the recorded time of normally working of the platform, and (2) their variance is equal to the empirical variance of lifetimes of similar products. These data points constitute a template sample in our bootsrap estimation, in which 10000 samples are generated through resampling of the template sample. The mean of each generated sample is a statistical realization of lifetime. By sorting the means of the 10000 samples as μ1≤μ2≤…≤μ10000 we can find the confidence interval for a confidence level 1-α as (μk1, μk2), where k1=[10000×α/2] and k2=[10000×(1-α/2)]. We have accelerated the lifetime test by a factor of 5.5. Now that the tip-tilt platform has been working normally for more than 1.25 years (in actual time), our method yields a confidence interval at the 95% confidence level, (3.59 years, 10.18 years). If the tip-tilt platform can work normally for 2 years (in actual time), the confidence interval at the 95% level will be (5.93 years, 16.81 years).
下载:
