统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Some Continuous Survival Distributions

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我们提供的多元统计分析Multivariate Statistical Analysis及其相关学科的代写,服务范围广, 其中包括但不限于:

  • Statistical Inference 统计推断
  • Statistical Computing 统计计算
  • Advanced Probability Theory 高等概率论
  • Advanced Mathematical Statistics 高等数理统计学
  • (Generalized) Linear Models 广义线性模型
  • Statistical Machine Learning 统计机器学习
  • Longitudinal Data Analysis 纵向数据分析
  • Foundations of Data Science 数据科学基础
Continuous Lifetimes 的图像结果
统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Some Continuous Survival Distributions

统计代写|多元统计分析代写Multivariate Statistical Analysis代考|The Exponential Distribution

The distribution can be defined by its survivor function $F(t)=\exp (-t / \xi)$; the scale parameter $\xi>0$ is actually the mean lifetime $\xi=\mathrm{E}(T)$. The primary distinguishing feature of the distribution is arguably its constant hazard function: $h(t)=1 / \xi$. It is the only distribution with such a constant hazard (subject to the usual mathematical provisos). So, as time proceeds, there is no recognition of age: the probability of imminent failure remains at the same level throughout. This makes the distribution both discreditable (in most applications) and compelling (as a prototype model). The lack-of-memory property is expressed, if I recall, as
\mathrm{P}(T>a+b \mid T>a)=\mathrm{P}(T>b)
your mission, if you decide to accept it, is to prove this-see the Exercises.

A natural generalisation of the exponential survivor function is $F(t)=$ $\exp \left{-(t / \xi)^{v}\right}$, raising $t / \xi$ to a power $v>0$; the exponential distribution is regained when $v=1$. The corresponding hazard function is $h(t)=(v / \xi)(t / \xi)^{v-1}$, an increasing function of $t$ when the shape parameter $v>1$ and decreasing when $v<1$. The mean and variance are expressed in terms of the gamma function (see the Exercises), but with an explicit survivor function, quantiles are more accessible. Thus, the upper $q$ th quantile $t_{q}$, for which $\mathrm{P}\left(T>t_{q}\right)=q$, is given by $t_{q}=\xi(-\log q)^{1 / v}$.

The Weibull distribution is fairly ubiquitous in reliability, even boasting a must-have handbook, commonly referred to as the Weibull Bible. The reasons for this popularity probably have to do with its being an extreme-value distribution with the associated weakest-link interpretation, the variety of shapes that can be accommodated by the hazard function, and the simple form of $\bar{F}(t)$, which facilitates data plotting.

统计代写|多元统计分析代写Multivariate Statistical Analysis代考|The Pareto Distribution

The Pareto distribution can arise naturally in the survival context as follows. Begin with an exponential distribution for $T: F(t)=e^{-t / \xi}$. Suppose that, due to circumstances beyond our control (as they say, whenever I go by train), $\xi$ varies randomly over individual units. Specifically, say $\lambda=1 / \xi$ has a gamma distribution with density
f(\lambda)=\Gamma(\gamma)^{-1} \alpha^{\gamma} \lambda^{\gamma-1} \mathrm{e}^{-\alpha \lambda}
(This somewhat artificial distributional assumption oils the wheels: a choice always has to be made between hair-shirt realism and mathematical tractability.) Thus, the conditional survivor function of $T$ is $\bar{F}(t \mid \lambda)=\mathrm{e}^{-\lambda t}$ and the unconditional one is obtained as
\bar{F}(t)=\int_{0}^{\infty} \bar{F}(t \mid \lambda) f(\lambda) d \lambda=\Gamma(\gamma)^{-1} \alpha^{\gamma} \int_{0}^{\infty} \lambda^{\gamma-1} \mathrm{e}^{-\lambda(t+\alpha)} d \lambda=(1+t / \alpha)^{-\gamma} .
The parameters of this Pareto survivor function are $\alpha>0$ (scale) and $\gamma>0$ (shape).

The hazard function, $h(t)=\gamma /(\alpha+t)$, is decreasing in $t$ and tends to zero as $t \rightarrow \infty$. Such behaviour is fairly atypical but not altogether unknown in practice. It would be appropriate for units that become less failure-prone with age, ones having ever-decreasing probability of imminent failure, settling in as time goes on. (One might cite as an example humans, who become less gaffe-prone with age-if only that were true!) However, there will certainly be failure at some finite time since $\bar{F}(\infty)=0$, a reflection of the fact that $h(t)$ here does not decrease fast enough for $\int_{0}^{\infty} h(s) d s$ to be finite.

统计代写|多元统计分析代写Multivariate Statistical Analysis代考|The Shape of Hazard

A variety of alternative distributions has been applied in survival analysis and reliability. Essentially, any distribution on $(0, \infty)$ will serve. Thus, one can contemplate distributions such as the gamma, Gompertz, Burr, and inverse Gaussian. And then, when one has finished contemplating them, one can note that distributions on $(-\infty, \infty)$ can be converted, usually via a log-transform: this yields the log-normal and the log-logistic, for example. (And, yes, you can start a sentence with and: and one of the best-loved hymns in the English language starts with And.) Some properties of these will be set as exercises below.

For most systems, hazard functions increase with age as the system becomes increasingly prone to crack-up. (No laughing at the back-your professor isn’t gaga yet.) The trade description for this is IFR (increasing failure rate). Likewise, DFR stands for decreasing failure rate: this would apply to systems that wear in or learn from experience (of others, preferably), becoming less liable to failure with age. The bog-standard example is the Weibull hazard, IFR for $v>1$ and DFR for $v<1$.

In some cases the legendary bathtub hazard shape (down-along-up) crops up. In animal lifetimes this can reflect high infant mortality, followed by a lower and fairly constant hazard, eventually ending up with the ravages of old age. To model such a shape we might simply add together three Weibull possibilities, that is, take the hazard function as
h(t)=\left(v_{0} / \xi_{0}\right)+\left(v_{1} / \xi_{1}\right)\left(t / \xi_{1}\right)^{v_{1}-1}+\left(v_{2} / \xi_{2}\right)\left(t / \xi_{2}\right)^{v_{2}-1} .
The corresponding survivor function is
\bar{F}(t) &=\exp \left{-\int_{0}^{t} h(s) d s\right}=\exp \left{-t / \xi_{0}-\left(t / \xi_{1}\right)^{v_{1}}-\left(t / \xi_{2}\right)^{v_{2}}\right} \
&=G_{0}(t) G_{1}(t) G_{2}(t),
where the $G_{j}(j=0,1,2)$ are the survivor functions of the three contributing distributions. The representation in terms of random variables is $T=$ $\min \left(T_{0}, T_{1}, T_{2}\right)$, where the $T_{j}$ are independent with survivor functions $G_{j}$.
In reliability applications the bathtub hazard is relevant for manufactured units when there is relatively high risk of early failure (wear in), a high risk of late failure (wear out), and a lower hazard in between (where it is constant). Unfortunately, in my experience, it is often only the first two characteristics that are apparent in practice. In medical applications the bathtub hazard can reflect treatment with non-negligible operative mortality that is otherwise life preserving; as Boag (1949) put it, it is not the malady but the remedy that can prove fatal. The construction, whereby a specified hazard form is constructed by combining several contributions, is generally applicable, not just confined to the minimum of three independent Weibull variates.

Continuous Lifetimes 的图像结果
统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Some Continuous Survival Distributions


统计代写|多元统计分析代写Multivariate Statistical Analysis代考|The Exponential Distribution

分布可以由其幸存者函数定义F(吨)=经验⁡(−吨/X); 尺度参数X>0实际上是平均寿命X=和(吨). 该分布的主要区别特征可以说是其恒定的风险函数:H(吨)=1/X. 它是唯一具有这种恒定风险的分布(受通常的数学限制)。因此,随着时间的推移,年龄不再存在:即将失败的概率始终保持在同一水平。这使得分发既不可信(在大多数应用程序中)又引人注目(作为原型模型)。如果我记得,内存不足的属性表示为

指数幸存者函数的自然推广是F(吨)= \exp \left{-(t / \xi)^{v}\right}\exp \left{-(t / \xi)^{v}\right}, 提高吨/X对权力在>0; 当指数分布恢复时在=1. 对应的风险函数是H(吨)=(在/X)(吨/X)在−1, 的增函数吨当形状参数在>1并减少时在<1. 均值和方差用 gamma 函数表示(参见练习),但使用明确的幸存者函数,分位数更易于访问。因此,上q第分位数吨q, 为此磷(吨>吨q)=q, 是(谁)给的吨q=X(−日志⁡q)1/在.

Weibull 分布在可靠性方面相当普遍,甚至拥有一本必备手册,通常称为 Weibull 圣经。这种流行的原因可能与它是具有相关最弱链接解释的极值分布,风险函数可以容纳的各种形状以及简单的形式有关F¯(吨),这有助于数据绘图。

统计代写|多元统计分析代写Multivariate Statistical Analysis代考|The Pareto Distribution

帕累托分布可以在生存环境中自然出现,如下所示。从指数分布开始吨:F(吨)=和−吨/X. 假设由于我们无法控制的情况(正如他们所说,每当我坐火车时),X在各个单元上随机变化。具体来说,说λ=1/X具有密度的 gamma 分布

危险函数,H(吨)=C/(一种+吨), 正在减少吨并且趋向于零吨→∞. 这种行为是相当不典型的,但在实践中并非完全未知。这适用于随着时间的推移而变得不太容易发生故障的单元,那些即将发生故障的概率不断降低的单元,随着时间的推移而安顿下来。(人们可以举人类为例,随着年龄的增长,他们变得不那么容易失态了——如果这是真的!)然而,在某个有限的时间内肯定会失败,因为F¯(∞)=0, 反映了这样一个事实H(吨)这里的下降速度不够快∫0∞H(s)ds是有限的。

统计代写|多元统计分析代写Multivariate Statistical Analysis代考|The Shape of Hazard

各种替代分布已应用于生存分析和可靠性。本质上,任何分布在(0,∞)将服务。因此,可以考虑诸如伽马、Gompertz、Burr 和逆高斯分布。然后,当一个人完成了对它们的思考后,人们可以注意到分布在(−∞,∞)可以转换,通常通过对数变换:例如,这会产生对数正态和对数逻辑。(而且,是的,你可以用 and 开始一个句子:英语中最受欢迎的赞美诗之一以 And 开头。)这些属性的一些属性将设置为下面的练习。

对于大多数系统,随着系统变得越来越容易崩溃,危险函数会随着年龄的增长而增加。(不要在背后笑——你的教授还不是 gaga。)这个的商业描述是 IFR(增加失败率)。同样,DFR 代表降低故障率:这将适用于磨损或从经验中学习(最好是其他人的)的系统,随着年龄的增长变得不太容易出现故障。沼泽标准示例是 Weibull 危险,IFR 为在>1和 DFR 为在<1.

在某些情况下,传说中的浴缸危险形状(向下-向上)突然出现。在动物的一生中,这可以反映高婴儿死亡率,其次是较低且相当稳定的危害,最终以老年的蹂躏而告终。为了模拟这样的形状,我们可以简单地将三个 Weibull 可能性相加,也就是说,将风险函数设为
\begin{对齐} \bar{F}(t) &=\exp \left{-\int_{0}^{t} h(s) d s\right}=\exp \left{-t / \xi_{ 0}-\left(t / \xi_{1}\right)^{v_{1}}-\left(t / \xi_{2}\right)^{v_{2}}\right} \ &= G_{0}(t) G_{1}(t) G_{2}(t), \end{aligned}\begin{对齐} \bar{F}(t) &=\exp \left{-\int_{0}^{t} h(s) d s\right}=\exp \left{-t / \xi_{ 0}-\left(t / \xi_{1}\right)^{v_{1}}-\left(t / \xi_{2}\right)^{v_{2}}\right} \ &= G_{0}(t) G_{1}(t) G_{2}(t), \end{aligned}
在哪里Gj(j=0,1,2)是三个贡献分布的幸存者函数。随机变量的表示是吨= 分钟(吨0,吨1,吨2), 其中吨j与幸存者函数无关Gj.
在可靠性应用中,当早期故障风险相对较高(磨损)、晚期故障风险较高(磨损)和两者之间的风险较低(恒定时)时,浴缸危害与制造单元相关。不幸的是,根据我的经验,在实践中通常只有前两个特征是明显的。在医疗应用中,浴缸的危害可以反映治疗具有不可忽略的手术死亡率,否则可以挽救生命;正如 Boag (1949) 所说,可以证明致命的不是疾病,而是补救措施。通过组合几个贡献来构建特定危险形式的构造通常适用,而不仅限于三个独立的 Weibull 变量的最小值。

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术语 广义线性模型(GLM)通常是指给定连续和/或分类预测因素的连续响应变量的常规线性回归模型。它包括多元线性回归,以及方差分析和方差分析(仅含固定效应)。



有限元是一种通用的数值方法,用于解决两个或三个空间变量的偏微分方程(即一些边界值问题)。为了解决一个问题,有限元将一个大系统细分为更小、更简单的部分,称为有限元。这是通过在空间维度上的特定空间离散化来实现的,它是通过构建对象的网格来实现的:用于求解的数值域,它有有限数量的点。边界值问题的有限元方法表述最终导致一个代数方程组。该方法在域上对未知函数进行逼近。[1] 然后将模拟这些有限元的简单方程组合成一个更大的方程系统,以模拟整个问题。然后,有限元通过变化微积分使相关的误差函数最小化来逼近一个解决方案。





随机过程,是依赖于参数的一组随机变量的全体,参数通常是时间。 随机变量是随机现象的数量表现,其时间序列是一组按照时间发生先后顺序进行排列的数据点序列。通常一组时间序列的时间间隔为一恒定值(如1秒,5分钟,12小时,7天,1年),因此时间序列可以作为离散时间数据进行分析处理。研究时间序列数据的意义在于现实中,往往需要研究某个事物其随时间发展变化的规律。这就需要通过研究该事物过去发展的历史记录,以得到其自身发展的规律。


多元回归分析渐进(Multiple Regression Analysis Asymptotics)属于计量经济学领域,主要是一种数学上的统计分析方法,可以分析复杂情况下各影响因素的数学关系,在自然科学、社会和经济学等多个领域内应用广泛。


MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中,其中问题和解决方案以熟悉的数学符号表示。典型用途包括:数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发,包括图形用户界面构建MATLAB 是一个交互式系统,其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题,尤其是那些具有矩阵和向量公式的问题,而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问,这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展,得到了许多用户的投入。在大学环境中,它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域,MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要,工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数(M 文件)的综合集合,可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。



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