﻿ 鱼雷装备寿命分布拟合方法比较研究 Contrast Study on the Fitting Method of Life Distribution of Torpedo Equipment

Modeling and Simulation
Vol. 08  No. 01 ( 2019 ), Article ID: 28549 , 8 pages
10.12677/MOS.2019.81003

Contrast Study on the Fitting Method of Life Distribution of Torpedo Equipment

Xinpeng Wang1, Jingyuan Zhang1, Peng Wang1*, Xinyong Ying2

1College of Weaponry Engineering, Naval University of Engineering, Wuhan Hubei

2No. 91115 Unit of PLA, Zhoushan Zhejiang

Received: Dec. 31st, 2018; accepted: Jan. 14th, 2019; published: Jan. 21st, 2019

ABSTRACT

The life distribution fitting is the precondition of the torpedo equipment reliability research. Based on the failure data of torpedo with long service life, the maximum likelihood estimation method and the minimum χ2 estimation fitting method are respectively used for parameter estimation and goodness of fit test to determine the best life distribution function. Through the example comparison, the minimum χ2 estimation fitting method fits better and can better reflect the variation rule of equipment reliability, and has better engineering practice value.

Keywords:Torpedo, Life Distribution, Maximum Likelihood Estimation, Minimum χ2 Estimation, Parameter Estimation, Goodness of Fit Test

1海军工程大学兵器工程学院，湖北 武汉

291115部队，浙江 舟山

1. 引言

2. 可靠性数据结构

$\left({n}_{i},{s}_{i},{t}_{i}\right),i=1,2,\cdots ,k$ (1)

$F\left(t\right)=P\left(T\le t\right)=1-R\left(t\right)$ (2)

$R\left({t}_{i}\right)={\stackrel{^}{p}}_{i}={s}_{i}/{n}_{i},i=1,2,\cdots ,k$ (3)

3. 参数估计

3.1. 常见的分布类型

1) 指数分布： $\left\{F\left(t;\lambda \right)=1-{\text{e}}^{-\lambda t},\lambda \in R\right\}$

2) 威布尔分布： $\left\{F\left(t\right)=1-\mathrm{exp}\left(-{\left(t/\eta \right)}^{m}\right),m,\eta \in R\right\}$

3) 极值分布： $\left\{F\left(t\right)=1-\mathrm{exp}\left(-{\text{e}}^{-\lambda \left(t-\mu \right)}\right),\lambda ,\mu \in R\right\}$

4) 对数正态分布： $\left\{F\left(t\right)=1-\Phi \left(\frac{\mathrm{lg}x-\mu }{\sigma }\right),\mu >0\right\}$

3.2. 极大似然估计

$L=\underset{i=1}{\overset{k}{\prod }}F{\left({t}_{i}\right)}^{{r}_{i}}{\left(1-F\left({t}_{i}\right)\right)}^{{s}_{i}}$ (4)

$F\left(t;\lambda \right)=1-{\text{e}}^{-\lambda t}$

$\mathrm{ln}L\left({t}_{1},{t}_{2},\cdots ,{t}_{r};\lambda \right)=r\mathrm{ln}\lambda -\lambda \left[\underset{i=1}{\overset{r}{\sum }}{t}_{i}+\left(n-r\right){t}_{r}\right]$

$\frac{r}{\lambda }-\underset{i=1}{\overset{r}{\sum }}{t}_{i}-\left(n-r\right){t}_{r}=0$ (5)

$\mathrm{ln}L=r\mathrm{ln}m-mr\mathrm{ln}\eta +\left(m-1\right)\underset{i=1}{\overset{r}{\sum }}\mathrm{ln}{t}_{i}-\underset{i=1}{\overset{r}{\sum }}{\left(\frac{{t}_{i}}{\eta }\right)}^{m}-\left(n-r\right){\left(\frac{{t}_{r}}{\eta }\right)}^{m}$

$\left\{\begin{array}{l}{\eta }^{m}=\frac{1}{r}\left[\underset{i=1}{\overset{r}{\sum }}{t}_{i}^{m}+\left(n-r\right){t}_{r}^{m}\right]\\ \frac{\underset{i=1}{\overset{r}{\sum }}{t}_{i}^{m}\mathrm{ln}{t}_{i}+\left(n-r\right){t}_{r}^{m}\mathrm{ln}{t}_{r}}{\underset{i=1}{\overset{r}{\sum }}{t}_{i}^{m}+\left(n-r\right){t}_{r}^{m}}=\frac{1}{m}+\frac{1}{r}\underset{i=1}{\overset{r}{\sum }}\mathrm{ln}{t}_{i}\end{array}$ (6)

$\mathrm{ln}L=\underset{i=1}{\overset{k}{\sum }}{r}_{i}\mathrm{ln}\left[1-\mathrm{exp}\left(-\mathrm{exp}\left(a{t}_{i}+b\right)\right)\right]-\underset{i=1}{\overset{k}{\sum }}{s}_{i}{\text{e}}^{a{t}_{i}+b}$

$\left\{\begin{array}{l}\underset{i=1}{\overset{k}{\sum }}{r}_{i}\frac{-{t}_{i}{\text{e}}^{a{t}_{i}+b}{\text{e}}^{-{\text{e}}^{a{t}_{i+b}}}}{1-{\text{e}}^{-{\text{e}}^{a{t}_{i+b}}}}-\underset{i=1}{\overset{k}{\sum }}{s}_{i}{t}_{i}{\text{e}}^{a{t}_{i}+b}=0\\ \underset{i=1}{\overset{k}{\sum }}{r}_{i}\frac{-{\text{e}}^{a{t}_{i}+b}{\text{e}}^{-{\text{e}}^{a{t}_{i+b}}}}{1-{\text{e}}^{-{\text{e}}^{a{t}_{i+b}}}}-\underset{i=1}{\overset{k}{\sum }}{s}_{i}{\text{e}}^{a{t}_{i}+b}=0\end{array}$ (7)

$L\left(\mu ,{\sigma }^{2}\right)=\underset{i=1}{\overset{k}{\prod }}\frac{1}{\sqrt{2\text{π}}\sigma {t}_{i}}{\text{e}}^{-\frac{{\left(\mathrm{ln}{t}_{i}-\mu \right)}^{2}}{2{\sigma }^{2}}}={\left(2\text{π}{\sigma }^{2}\right)}^{-\frac{k}{2}}{\left(\underset{i=1}{\overset{k}{\prod }}{t}_{i}\right)}^{-1}\mathrm{exp}\left(-\frac{1}{2{\sigma }^{2}}\underset{i=1}{\overset{k}{\sum }}{\left(\mathrm{ln}{t}_{i}-\mu \right)}^{2}\right)$

$\mathrm{ln}L\left(\mu ,{\sigma }^{2}\right)=-\frac{k}{2}\mathrm{ln}\left(2\text{π}{\sigma }^{2}\right)-\mathrm{ln}\underset{i=1}{\overset{k}{\prod }}{t}_{i}-\frac{1}{2{\sigma }^{2}}\underset{i=1}{\overset{k}{\sum }}{\left(\mathrm{ln}{t}_{i}-\mu \right)}^{2}$

$\left\{\begin{array}{l}\frac{\text{d}\mathrm{ln}L\left(\mu ,{\sigma }^{2}\right)}{\text{d}\mu }=\frac{1}{{\sigma }^{2}}\underset{i=1}{\overset{k}{\sum }}\left(\mathrm{ln}{t}_{i}-\mu \right)=0\\ \frac{\text{d}\mathrm{ln}L\left(\mu ,{\sigma }^{2}\right)}{\text{d}{\sigma }^{2}}=-\frac{k}{2{\sigma }^{2}}+\frac{1}{2{\sigma }^{4}}\underset{i=1}{\overset{k}{\sum }}{\left(\mathrm{ln}{t}_{i}-\mu \right)}^{2}=0\end{array}$ (8)

3.3. 极小卡方估计法

1) 构造χ2统计量

${s}_{i}=\left\{\begin{array}{l}1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{p}_{i}\left(\theta \right)\\ 0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}1-{p}_{i}\left(\theta \right)\end{array}$ (9)

${n}_{i}\to \infty$ 时，根据中心极限定理得：

$\frac{\underset{i=1}{\overset{k}{\sum }}{s}_{i}-{n}_{i}{p}_{i}\left(\theta \right)}{\sqrt{{n}_{i}{p}_{i}\left(\theta \right)\left(1-{p}_{i}\left(\theta \right)\right)}}\sim N\left(0,1\right)$ (10)

$\frac{{n}_{i}{\left({\stackrel{^}{p}}_{i}-{p}_{i}\left(\theta \right)\right)}^{2}}{{p}_{i}\left(\theta \right)\left(1-{p}_{i}\left(\theta \right)\right)}\sim {\chi }^{2}\left(1\right)$ (11)

${\chi }^{2}\left(\theta \right)=\underset{i=1}{\overset{k}{\sum }}\frac{{n}_{i}{\left({\stackrel{^}{p}}_{i}-{p}_{i}\left(\theta \right)\right)}^{2}}{{p}_{i}\left(\theta \right)\left(1-{p}_{i}\left(\theta \right)\right)}$ (12)

${n}_{i}\to \infty$ 时， ${\chi }^{2}\left(\theta \right)$ 统计量的渐近分布是自由度为k的χ2分布。

2) 参数估计

(13)

$\stackrel{^}{\theta }$ 为参数 $\theta$ 的极小χ2估计，则对 ${\chi }^{2}\left(\theta \right)$ 统计量求关于 $\theta$ 的最小值，就可得到参数 $\theta$ 的估计

4. 拟合优度检验

$p=P\left({\chi }^{2}\left(k-m\right)>{\chi }^{2}\left(\stackrel{^}{\theta }\right)\right)$ (14)

5. 拟合方法的评判

${E}_{i}=|\frac{{p}_{i}\left(\stackrel{^}{\theta }\right)-{\stackrel{^}{p}}_{i}}{{\stackrel{^}{p}}_{i}}|×100%$ (15)

 (16)

6. 实例计算与分析

6.1. 数据来源及结构

Table 1. Storage life data of torpedo equipment

6.2. 参数估计和拟合优度检验

1) 极大似然估计

Table 2. The calculated results of maximum likelihood estimation and goodness of fit test

$F\left(t\right)=1-{\text{e}}^{-\text{e}\frac{t-28.982}{5.633}}$ (17)

Figure 1. The effect of maximum likelihood fitting estimation

2) 极小χ2估计拟合方法

$F\left(t\right)=1-{\text{e}}^{-\text{e}\frac{t-29.68\text{1}3}{5.9\text{211}}}$ (18)

Table 3. The calculated results of minimum χ2 estimation fitting method

Figure 2. The effect of minimum χ2 estimation fitting method

6.3. 拟合方法比较分析

1) 相对误差标准差计算及分布函数曲线

2) 拟合方法比较分析

Figure 3. Comparison of the two estimation methods

7. 结论

“十三五”水中兵器预先研究项目(3020605030302)。

Contrast Study on the Fitting Method of Life Distribution of Torpedo Equipment[J]. 建模与仿真, 2019, 08(01): 14-21. https://doi.org/10.12677/MOS.2019.81003

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NOTES

*通讯作者。