Introduction to Weibull Analysis

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# Introduction to Weibull Analysis
### 21 Jan 2022

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## Introduction

Weibull Analysis is a family of graphical and statistical techniques used estimate important life characteristics of a product by fitting a parameterized probability distribution to life data

This presentation introduces several aspects of Weibull analysis and how they are implemented on various types of reliability data

- The Weibull distribution

+ Background, properties, and importance
    + Distribution functions
    + Parameters 
        + Shape parameter `$\beta$` 
        + Scale parameter `$\eta$`
        + Location parameter `$\theta$`
    + Relationship to other distributions

- Graphical methods of estimating the Weibull shape and scale parameters

- Numerical methods of estimating the Weibull shape and scale parameters

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# The Weibull Distribution

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## Background of the Weibull Distribution

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The distribution is named after Ernst Hjalmar Waloddi Weibull (1887–1979), the <span class="explain">Swedish</span><span class="tooltip">As a Swede, his surname should be pronounced as "Vay-bull" not "Why-bull"</span> engineer, scientist, and mathematician

The Weibull distribution was actually discovered by the French mathematician Maurice Rene Fréchet in the course of deriving the Fréchet distribution (aka the inverse Weibull distribution)

Weibull popularized his namesake distribution after publishing the <span class="explain">landmark paper</span><span class="tooltip">Weibull, Waloddi. "*A Statistical Distribution Function of Wide Applicability.*" Journal of Applied Mechanics 18 (1951): 293-297.</span> in 1951

+ Weibull had hoped to publish in a prominent mathematics journal, but had to "settle" for an applied engineering journal
+ Initial reaction to the paper was sharply negative
+ In the 1970s, the U.S. Air Force and automotive industry began implementing the distribution and the methods described in the paper
+ Today, Weibull analysis is a foremost method for evaluating life data

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## Properties of the Weibull Distribution

The Weibull distribution is a member of a distribution family called "lifetime distributions"

- These distributions describe continuous random variables that are defined over strictly positive values - `$T \in \mathbb{R}^{+}$`
- There are many members of the lifetime distribution family (exponential, lognormal, loglogistic, gamma, Birmbaum-Saunders)
- Likewise, there are many distributions that are not members of the lifetime distribution family (normal, logistic, smallest extreme value)

Lifetime distributions have successfully served as population models for failure times arising from a wide range of products and failure mechanisms

- In some cases there are probabilistic arguments based on the physics of failure  that justify the choice of a model
- More often, however, the model chosen solely because of its demonstrated success in fitting failure data 
- <font color="red">**This is why the Weibull distribution is so popular**</font> - it's a flexible distribution, capable of fitting many different failure patterns

The Weibull is also a member of the extreme value distribution family

- These models describe the time to failure of the weakest (or strongest) link in a "chain" of components
+ The Weibull is directly related to the smallest extreme value distributionIf `$X \sim \text{WEIB}(\eta, \beta)$` then `$\log[X] \sim \text{SEV}(\mu, \sigma)$`

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## Weibull shape and scale parameters

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$$
`\begin{aligned}
F(t = \theta|\theta,\beta) &= 1 - \exp\bigg[-\bigg(\frac{\theta}{\theta}\bigg)^{\beta}\bigg]\\\\
 &= 1 - \exp\bigg[-\bigg(1\bigg)^{\beta}\bigg]\\\\
&=1-\exp\bigg[-1\bigg]\\\\
&=1-0.368\\\\
&=0.632
\end{aligned}`
$$

]

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$$
`\begin{aligned}
F(t| \theta, \beta=1) &= 1 - \exp\bigg[-\bigg(\frac{t}{\theta}\bigg)^{1}\bigg]
\end{aligned}`
$$

]

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## Probability Distribution Functions

- For continuous random variables with "lifetime" distributions `$(T\in \mathbb{R}^{+})$` analysts model failure using the functions below

| Formal Name                      | AKA                        | General Expression    | Weibull    |
|----------------------------------|----------------------------|-----------------------|------------|
| cumulative distribution function | distribution function, CDF | `$F(t) = \Pr(T \le t)$` | `$\displaystyle F(t) = 1-\exp\left[-\left(\frac{t}{\eta}\right)^{\beta}\right]$` |
| probability density function     | density function, PDF      | `$f(t) = \Pr(T = t)$`   | `$\displaystyle f(t) = \frac{\beta}{\eta}\left(\frac{t}{\eta}\right)^{\beta-1}\exp\left[-\left(\frac{t}{\eta}\right)^{\beta}\right]$` |
| reliability function             | survival function          | `$S(t) = \Pr(T > t)$`   | `$\displaystyle S(t) = \exp\left[-\left(\frac{t}{\eta}\right)^{\beta}\right]$` |
| hazard function                  | failure rate, hazard rate  | `$\displaystyle h(t) = \lim_{\Delta t \to 0} \frac{\Pr(t < T \le t+\Delta t\vert T \ge t)}{\Delta t}$`             | `$\displaystyle h(t) = \frac{\beta}{\eta}\left(\frac{t}{\eta}\right)^{\beta-1}$` |
| cumulative hazard function       | Cumulative hazard rate     | `$\displaystyle H(t) = \int_{0}^{t} h(u)du$`             | `$\displaystyle H(t) = \left(\frac{t}{\eta}\right)^{\beta}$` |
| quantile function                | inverse CDF, percent point | `$t(p) = F^{-1}(p)$`    | `$\displaystyle t(p) = \eta\left(\ln\left[\frac{1}{1 - p}\right]\right)^{1/\beta}$` |

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## Probability Distribution Functions - CDF & PDF

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The cumulative distribution function (CDF) defines the total probability that the random variable `$T$` is equal to or less than `$t$`

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]

.pull-right[

The cumulative distribution function (CDF) defines the total probability that the random variable `$T$` is equal to or less than `$t$`

]

]

.panel[.panel-name[CDF/PDF Relationship]

]

]

---
## Probability Function Table

Each cell in the table below shows the expressions used to transform from the function at the top to the function on the left

|      | `$F(t)$`                   | `$f(t)$`                              | `$S(t)$`                 | `$h(t)$`                               | `$H(t)$`|
|------|-------------------------|------------------------------------|---------------------- |-------------------------------------|-------|
| `$F(t)$`|  | `$\displaystyle\int_0^{t}f(u)du$`  | `$1-S(t)$`     | `$\displaystyle 1-\exp\left[-\int_0^{t} h(u)du\right]$`| `$\displaystyle 1-\exp\left[-H(t)\right]$`|
| `$f(t)$`| `$\displaystyle\frac{d}{dt}F(t)$` | | `$\displaystyle-\frac{d}{dt}S(t)$`| `$\displaystyle h(t)\cdot\exp\left[-\int_0^{t} h(u)du\right]$` | `$\displaystyle -\frac{dH(t)/dt}{\exp[H(t)]}$` |
| `$S(t)$`| `$1-F(t)$`  | `$\displaystyle \int_t^{\infty}f(u)du$` |  | `$\displaystyle\exp\left[-\int_0^{t} h(u)du\right]$`  | `$\displaystyle \exp[-H(t)]$`|
| `$h(t)$`| `$\displaystyle \frac{dF(t)/dt}{1-F(t)}$`| `$\displaystyle \frac{f(t)}{\int_t^{\infty}f(u)du}$`| `$\displaystyle -\frac{d}{dt}\ln \left[S(t)\right]$`|                              | `$\displaystyle \frac{d}{dt}H(t)$`|
| `$H(t)$`| `$\displaystyle-\ln[1-F(t)]$`           | `$\displaystyle -\ln\left[\int_t^{\infty}f(u)du\right]$`      | `$\displaystyle -\ln[S(t)]$`           | `$\displaystyle \int_0^t h(u)du$`                    |       |

---
class: inverse, center, middle

# Overview of Probability Plots

---
# Probability Plotting

Probability plotting is a graphical method of fitting data to a chosen univariate probability model

The cumulative distribution function of the chosen model is "linearized" or rearranged to the form of the equation of a line

$$
`\begin{aligned}
G(\widehat{F}) &= m\cdot g(t) +b\\\\ 
y &= m\cdot x + b
\end{aligned}`
$$
Where

- `$\widehat{F}$` is an estimate of the failure probability (unreliability)
- `$t$` are the times at which an event was observed (failure, suspension)
- `$G(\cdot)$` and `$g(\cdot)$` are transformations, specific to the chosen distribution

If these plotted points fall "roughly" on a straight line, we can conclude that the chosen distribution is adequate to describe the data

- There are several methods to determine which line best-fits the plotted points, the most commonly used are:
    + Least squares optimization
    + Maximum likelihood
- Once the best-fit line is determined, the slope and intercept are used to determine the parameter values

<span class="explain">**Note: a probability plot is not sufficient to conclude that the chosen distribution is the best fit!**</span><span class="tooltip">The Weibull distribution is a powerful tool, but is not always the best choice. In many cases the lognormal distribution offers similar - if not better performance.<br/><br/>Moreover, the Weibull is incapable of modeling multiple failure modes simultaneously. For these situations a more complex distribution is needed</span>

---
### How to linearize the Weibull CDF

.panelset[
.panel[.panel-name[Step 1]
$$
`\begin{aligned}
\redbf{F(t|\beta, \eta)} &\redbf{= 1 - \exp\bigg[-\bigg(\frac{t}{\eta}\bigg)^{\beta}\bigg]}&\redbf{\text{Start with CDF for a Weibull Distribution}}& \Longleftarrow\\[10pt]
\mbf{\widehat{F}} &\mbf{= 1 - \exp\bigg[-\bigg(\frac{t}{\eta}\bigg)^{\beta}\bigg]}&\mbf{\text{Substitute nonparametric estimate}\;\widehat{F} \text{ for } F(t|\beta, \eta)}&\\[10pt]
\mbf{1-\widehat{F}} &\mbf{= \exp\bigg[-\bigg(\frac{t}{\eta}\bigg)^{\beta}\bigg]}& \mbf{\text{Move}\;1\;\text{and negative sign}\;(-)\;\text{over}}&\\[10pt]
\mbf{\ln\bigg[1-\widehat{F}\bigg]}&\mbf{=-\bigg(\frac{t}{\eta}\bigg)^{\beta}}&\mbf{\text{Take}\;\ln[\cdot]\;\text{of both sides}}&\\[10pt]
\mbf{\ln\bigg[\frac{1}{1-\widehat{F}}\bigg]}&\mbf{=\bigg(\frac{t}{\eta}\bigg)^{\beta}}& \mbf{\text{Move negative sign}\;(-)\;\text{over}}&\\[10pt]
\mbf{\ln\bigg[\ln\bigg[\frac{1}{1-\widehat{F}}\bigg]\bigg]} &\mbf{=\beta\ln[t] - \beta\ln[\eta]}&\mbf{\text{Take}\;\ln[\cdot]\;\text{of both sides}}&
\end{aligned}`
$$
]

.panel[.panel-name[Step 2]

$$
`\begin{aligned}
\mbf{F(t|\beta, \eta)} &\mbf{= 1 - \exp\bigg[-\bigg(\frac{t}{\eta}\bigg)^{\beta}\bigg]}&\mbf{\text{Start with CDF for a Weibull Distribution}}&\\[10pt]
\redbf{\widehat{F}} &\redbf{= 1 - \exp\bigg[-\bigg(\frac{t}{\eta}\bigg)^{\beta}\bigg]}&\redbf{\text{Substitute nonparametric estimate}\;\widehat{F} \text{ for } F(t|\beta, \eta)}& \Longleftarrow\\[10pt]
\mbf{1-\widehat{F}} &\mbf{= \exp\bigg[-\bigg(\frac{t}{\eta}\bigg)^{\beta}\bigg]}& \mbf{\text{Move}\;1\;\text{and negative sign}\;(-)\;\text{over}}&\\[10pt]
\mbf{\ln\bigg[1-\widehat{F}\bigg]}&\mbf{=-\bigg(\frac{t}{\eta}\bigg)^{\beta}}&\mbf{\text{Take}\;\ln[\cdot]\;\text{of both sides}}&\\[10pt]
\mbf{\ln\bigg[\frac{1}{1-\widehat{F}}\bigg]}&\mbf{=\bigg(\frac{t}{\eta}\bigg)^{\beta}}& \mbf{\text{Move negative sign}\;(-)\;\text{over}}&\\[10pt]
\mbf{\ln\bigg[\ln\bigg[\frac{1}{1-\widehat{F}}\bigg]\bigg]} &\mbf{=\beta\ln[t] - \beta\ln[\eta]}&\mbf{\text{Take}\;\ln[\cdot]\;\text{of both sides}}&
\end{aligned}`
$$
]

.panel[.panel-name[Step 3]

$$
`\begin{aligned}
\mbf{F(t|\beta, \eta)} &\mbf{= 1 - \exp\bigg[-\bigg(\frac{t}{\eta}\bigg)^{\beta}\bigg]}&\mbf{\text{Start with CDF for a Weibull Distribution}}&\\[10pt]
\mbf{\widehat{F}} &\mbf{= 1 - \exp\bigg[-\bigg(\frac{t}{\eta}\bigg)^{\beta}\bigg]}&\mbf{\text{Substitute nonparametric estimate}\;\widehat{F} \text{ for } F(t|\beta, \eta)}&\\[10pt]
\redbf{1-\widehat{F}} &\redbf{= \exp\bigg[-\bigg(\frac{t}{\eta}\bigg)^{\beta}\bigg]}& \redbf{\text{Move}\;1\;\text{and negative sign}\;(-)\;\text{over}}& \Longleftarrow\\[10pt]
\mbf{\ln\bigg[1-\widehat{F}\bigg]}&\mbf{=-\bigg(\frac{t}{\eta}\bigg)^{\beta}}&\mbf{\text{Take}\;\ln[\cdot]\;\text{of both sides}}&\\[10pt]
\mbf{\ln\bigg[\frac{1}{1-\widehat{F}}\bigg]}&\mbf{=\bigg(\frac{t}{\eta}\bigg)^{\beta}}& \mbf{\text{Move negative sign}\;(-)\;\text{over}}&\\[10pt]
\mbf{\ln\bigg[\ln\bigg[\frac{1}{1-\widehat{F}}\bigg]\bigg]} &\mbf{=\beta\ln[t] - \beta\ln[\eta]}&\mbf{\text{Take}\;\ln[\cdot]\;\text{of both sides}}&
\end{aligned}`
$$
]

.panel[.panel-name[Step 4]

$$
`\begin{aligned}
\mbf{F(t|\beta, \eta)} &\mbf{= 1 - \exp\bigg[-\bigg(\frac{t}{\eta}\bigg)^{\beta}\bigg]}&\mbf{\text{Start with CDF for a Weibull Distribution}}&\\[10pt]
\mbf{\widehat{F}} &\mbf{= 1 - \exp\bigg[-\bigg(\frac{t}{\eta}\bigg)^{\beta}\bigg]}&\mbf{\text{Substitute nonparametric estimate}\;\widehat{F} \text{ for } F(t|\beta, \eta)}&\\[10pt]
\mbf{1-\widehat{F}} &\mbf{= \exp\bigg[-\bigg(\frac{t}{\eta}\bigg)^{\beta}\bigg]}& \mbf{\text{Move}\;1\;\text{and negative sign}\;(-)\;\text{over}}&\\[10pt]
\redbf{\ln\bigg[1-\widehat{F}\bigg]}&\redbf{=-\bigg(\frac{t}{\eta}\bigg)^{\beta}}&\redbf{\text{Take}\;\ln[\cdot]\;\text{of both sides}}& \Longleftarrow\\[10pt]
\mbf{\ln\bigg[\frac{1}{1-\widehat{F}}\bigg]}&\mbf{=\bigg(\frac{t}{\eta}\bigg)^{\beta}}& \mbf{\text{Move negative sign}\;(-)\;\text{over}}&\\[10pt]
\mbf{\ln\bigg[\ln\bigg[\frac{1}{1-\widehat{F}}\bigg]\bigg]}& \mbf{=\beta\ln[t] - \beta\ln[\eta]}&\mbf{\text{Take}\;\ln[\cdot]\;\text{of both sides}}&
\end{aligned}`
$$
]

.panel[.panel-name[Step 5]

$$
`\begin{aligned}
\mbf{F(t|\beta, \eta)} &\mbf{= 1 - \exp\bigg[-\bigg(\frac{t}{\eta}\bigg)^{\beta}\bigg]}&\mbf{\text{Start with CDF for a Weibull Distribution}}&\\[10pt]
\mbf{\widehat{F}} &\mbf{= 1 - \exp\bigg[-\bigg(\frac{t}{\eta}\bigg)^{\beta}\bigg]}&\mbf{\text{Substitute nonparametric estimate}\;\widehat{F} \text{ for } F(t|\beta, \eta)}&\\[10pt]
\mbf{1-\widehat{F}} &\mbf{= \exp\bigg[-\bigg(\frac{t}{\eta}\bigg)^{\beta}\bigg]}& \mbf{\text{Move}\;1\;\text{and negative sign}\;(-)\;\text{over}}&\\[10pt]
\mbf{\ln\bigg[1-\widehat{F}\bigg]}&\mbf{=-\bigg(\frac{t}{\eta}\bigg)^{\beta}}&\mbf{\text{Take}\;\ln[\cdot]\;\text{of both sides}}&\\[10pt]
\redbf{\ln\bigg[\frac{1}{1-\widehat{F}}\bigg]}&\redbf{=\bigg(\frac{t}{\eta}\bigg)^{\beta}}& \redbf{\text{Move negative sign}\;(-)\;\text{over}}& \Longleftarrow\\[10pt]
\mbf{\ln\bigg[\ln\bigg[\frac{1}{1-\widehat{F}}\bigg]\bigg]} &\mbf{=\beta\ln[t] - \beta\ln[\eta]}&\mbf{\text{Take}\;\ln[\cdot]\;\text{of both sides}}&
\end{aligned}`
$$
]

.panel[.panel-name[Step 6]

$$
`\begin{aligned}
\mbf{F(t|\beta, \eta)} &\mbf{= 1 - \exp\bigg[-\bigg(\frac{t}{\eta}\bigg)^{\beta}\bigg]}&\mbf{\text{Start with CDF for a Weibull Distribution}}&\\[10pt]
\mbf{\widehat{F}} &\mbf{= 1 - \exp\bigg[-\bigg(\frac{t}{\eta}\bigg)^{\beta}\bigg]}&\mbf{\text{Substitute nonparametric estimate}\;\widehat{F} \text{ for } F(t|\beta, \eta)}&\\[10pt]
\mbf{1-\widehat{F}} &\mbf{= \exp\bigg[-\bigg(\frac{t}{\eta}\bigg)^{\beta}\bigg]}& \mbf{\text{Move}\;1\;\text{and negative sign}\;(-)\;\text{over}}&\\[10pt]
\mbf{\ln\bigg[1-\widehat{F}\bigg]}&\mbf{=-\bigg(\frac{t}{\eta}\bigg)^{\beta}}&\mbf{\text{Take}\;\ln[\cdot]\;\text{of both sides}}&\\[10pt]
\mbf{\ln\bigg[\frac{1}{1-\widehat{F}}\bigg]}&\mbf{=\bigg(\frac{t}{\eta}\bigg)^{\beta}}& \mbf{\text{Move negative sign}\;(-)\;\text{over}}&\\[10pt]
\redbf{\ln\bigg[\ln\bigg[\frac{1}{1-\widehat{F}}\bigg]\bigg]} &\redbf{=\beta\ln[t] - \beta\ln[\eta]}&\redbf{\text{Take}\;\ln[\cdot]\;\text{of both sides}}&\Longleftarrow
\end{aligned}`
$$
]

.panel[.panel-name[Step 7]

$$
`\begin{aligned}
\mbf{F(t|\beta, \eta)} &\mbf{= 1 - \exp\bigg[-\bigg(\frac{t}{\eta}\bigg)^{\beta}\bigg]}&\mbf{\text{Start with CDF for a Weibull Distribution}}&\\[10pt]
\mbf{\widehat{F}} &\mbf{= 1 - \exp\bigg[-\bigg(\frac{t}{\eta}\bigg)^{\beta}\bigg]}&\mbf{\text{Substitute nonparametric estimate}\;\widehat{F} \text{ for } F(t|\beta, \eta)}&\\[10pt]
\mbf{1-\widehat{F}} &\mbf{= \exp\bigg[-\bigg(\frac{t}{\eta}\bigg)^{\beta}\bigg]}& \mbf{\text{Move}\;1\;\text{and negative sign}\;(-)\;\text{over}}&\\[10pt]
\mbf{\ln\bigg[1-\widehat{F}\bigg]}&\mbf{=-\bigg(\frac{t}{\eta}\bigg)^{\beta}}&\mbf{\text{Take}\;\ln[\cdot]\;\text{of both sides}}&\\[10pt]
\mbf{\ln\bigg[\frac{1}{1-\widehat{F}}\bigg]}&\mbf{=\bigg(\frac{t}{\eta}\bigg)^{\beta}}& \mbf{\text{Move negative sign}\;(-)\;\text{over}}&\\[10pt]
\bluebf{\ln\bigg[\ln\bigg[\frac{1}{1-\widehat{F}}\bigg]\bigg]} &\mbf{=}\;\greenbf{\beta} \redbf{\ln[t]} - \purplebf{\beta\ln[\eta]}&\mbf{\text{Take}\;\ln[\cdot]\;\text{of both sides}}&\\[10pt]
\bluebf{y}\hspace{30pt}&\mbf{=}\;\greenbf{m}\cdot \redbf{x} \hspace{5pt}+ \hspace{5pt}\purplebf{b}&\text{Equation of a line in slope/intercept form}& \Longleftarrow
\end{aligned}`
$$
]
]
---
class: inverse, center, middle

# Elements of Weibull Plots

---
## Elements of Weibull Plots

.pull-left[

]

.pull-right[
<br/>
<red>Plotted Points representing the "observed unreliabilities"</red>

- <red>X-axis:</red> `$\redbf{t_i, i = 1,2,\ldots,N}$`<br/> <red>ordered event times</red>
- <red>Y-axis:</red> `$\redbf{\hat{F}(t_i), i = 1,2,\ldots,N}$`<br/> <red>non-parametric estimate of the Weibull CDF</red>

]

---
## Elements of Weibull Plots

.pull-left[

]

.pull-right[
<br/><br/>
Plotted Points

- X-axis: `$t_i, i = 1,2,\ldots,N$`<br/> ordered event times
- Y-axis: `$\hat{F}(t_i), i = 1,2,\ldots,N$`<br/> non-parametric estimate of the Weibull CDF

<red>Axes: transformed according to a <u>linearized Weibull CDF</u></red>

]

---
## Elements of Weibull Plots

.pull-left[

]

.pull-right[
<br/><br/>
Plotted Points

- X-axis: `$t_i, i = 1,2,\ldots,N$`<br/> ordered event times
- Y-axis: `$\hat{F}(t_i), i = 1,2,\ldots,N$`<br/> non-parametric estimate of the Weibull CDF

Axes: transformed according to a linearized Weibull CDF

<red>Best fit line representing the predicted values from the Weibull model
</red>

- <red>Estimated using maximum likelihood</red> `$\redbf{\widehat{\beta_{_{MLE}}},\widehat{\eta_{_{MLE}}}}$`
<br/>
- <red>Estimated using ordinary least squares</red> `$\redbf{\widehat{\beta_{_{OLS}}},\widehat{\eta_{_{OLS}}}}$`

]

---
## Elements of Weibull Plots

.pull-left[

]

.pull-right[
<br/><br/>
Plotted Points

- X-axis: `$t_i, i = 1,2,\ldots,N$`<br/> ordered event times
- Y-axis: `$\hat{F}(t_i), i = 1,2,\ldots,N$`<br/> non-parametric estimate of the Weibull CDF

Axes: transformed according to a linearized Weibull CDF

Best fit line representing the predicted values from the Weibull model

- Estimated using maximum likelihood `$\widehat{\beta_{_{MLE}}},\widehat{\eta_{_{MLE}}}$`
<br/>
- Estimated using ordinary least squares `$\widehat{\beta_{_{OLS}}},\widehat{\eta_{_{OLS}}}$`

<red>Upper and lower</red> `$\redbf{100(1-\alpha)\%}$` <red>confidence intervals</red>

]

---
## Two methods for generating Weibull plots

.panelset[
.panel[.panel-name[Method 1: Transform the data (linearize the CDF)]

.pull-left[
<img src="data:image/png;base64,#C:/Users/Aubur/github/auburngrads/LMI/reliability_course/images/build_weibull/both_axes_2-crop.png" width="800" style="display: block; margin: auto;" />
]

.pull-right[

- In this method the nonparametric estimate of the CDF and the event times are transformed as shown below

$$
`\begin{aligned}
G(\widehat{F}) &=\ln\bigg[\ln\bigg[\frac{1}{1-\widehat{F}}\bigg]\bigg]\\\\
g(t) &= \ln[t]
\end{aligned}`
$$

- These transformed values are plotted on the "true", untransformed axes

- The advantage of the method is that the transformed values can be used to estimate the parameters `$\eta$` and `$\beta$`

- The disadvantage of this method is that isn't always user friendly

]
]

.panel[.panel-name[Method 2: Transforming the axes]

.pull-left[
<img src="data:image/png;base64,#C:/Users/Aubur/github/auburngrads/LMI/reliability_course/images/build_weibull/both_axes_1-crop.png" width="800" style="display: block; margin: auto;" />
]

.pull-right[
- In this method the axes are "untransformed" by replacing the "true" y-axis values with their corresponding unreliabilities

- The advantage of this method is that the raw failure times and nonparametric estimates can be plotted directly

- The disadvantage of this method is that the values cannot be used to estimate the parameters `$\eta$` and `$\beta$`

- This method is used often and has been popularized through the use of <a target="_blank" href="https://www.weibull.com/GPaper/">**Plotting papers <i class="fa fa-external-link-alt" role="presentation" aria-label="external-link-alt icon"></i>**</a> 
]
]
]
---
# Options for the

The nonparametric estimate of the CDF `$\widehat{F}$`, or the unreliability, plays a key role in probability plotting

However, one cannot observe reliability or unrealiability values -- only observe failures or suspensions

Reliability or unreliability values must be estimated from the data

The following slides discuss several estimators that are used to compute these values

- Plotting positions

- The Kaplan-Meier estimator (aka the product limit estimator)

- The Turnbull Estimator

The choice of which estimator to use is driven by the type censoring (suspensions) present in the data

---
# Probability Plotting Positions

Plotting positions express the non-exceedance probability `$F$` of the `$i^{th}$` ascending data value

Various formulae have been developed to compute these probabilities for use with a chosen distribution

The generic plotting position formula is expressed as

$$
\widehat{F(t_{i})}=\frac{i-a}{n+1-2a}
$$

- where

+ `$i$` is an index of the ordered observations (smallest `$\rightarrow$`largest)
    
    + `$n$` is the number of observations

+ `$a$` is the plotting position parameter

+ `$\widehat{F(t_{i})}$` is the nonparametric estimate of the CDF at time `$t_{i}$`

---
## Probability Plotting Positions

The value of `$a$` is chosen to produce approximately unbiased estimates of `$F(t_{i})$` for an assumed distribution

.pull-left[

Author     | Year | `$a$`   |   Formula
-----------|------|-------|---------------
Hazen      | 1914 | 0.50  | `$\displaystyle\frac{i-0.5}{n}$`
Weibull    | 1939 | 0     | `$\displaystyle\frac{i}{n+1}$`
Blom       | 1958 | 0.375 | `$\displaystyle\frac{i-0.375}{n+0.25}$`
Gringorten | 1963 | 0.44  | `$\displaystyle\frac{i-0.44}{n+0.12}$`
Chegodayev | 2000 | 0.30  | `$\displaystyle\frac{i-0.30}{n+0.4}$`
Cunnane    | 1977 | 0.40  | `$\displaystyle\frac{i-0.40}{n+0.2}$`
Median     | 1943 | 0.3175| `$\displaystyle\frac{i-0.3175}{n+0.365}$`

]

.pull-right[

<h4>The plotting positions used most often in practice are</h4>

$$
`\begin{aligned}
\text{Hazen}:\; &\widehat{F(t_{i})}=\frac{i-0.5}{n}\\\\
\text{Chegodayev}:\; &\widehat{F(t_{i})}=\frac{i-0.3}{n+0.4}\;\text{also known as "Median-Ranks"}\\\\
\text{Weibull}:\; &\widehat{F(t_{i})}=\frac{i}{n+1}\\\\
\end{aligned}`
$$

]

---
## Plotting Position Example - Complete Data

.panelset[

.panel[.panel-name[Overview]

The data used in this example is a complete data set

]

.panel[.panel-name[Data]

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class=\"display\">\n  <thead>\n    <tr>\n      <th> <\/th>\n      <th>Event time (t)<\/th>\n      <th>Hazen<\/th>\n      <th>Median Ranks<\/th>\n      <th>Weibull<\/th>\n    <\/tr>\n  <\/thead>\n<\/table>","options":{"pageLength":8,"columnDefs":[{"className":"dt-right","targets":[1,2,3,4]},{"orderable":false,"targets":0}],"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[8,10,25,50,100]}},"evals":[],"jsHooks":[]}</script>
]

.panel[.panel-name[Plot]

]
]
---
class: inverse, center, middle

# Maximum Likelihood Estimation

---
## The Likelihood Function and Its Maximum

- The value of the likelihood function `$\mathscr{L}(\underline{\theta}|\underline{t})$` depends on

1) The assumed parametric model

2) The observed data

- The total likelihood is comprised of the contributions from every observation

+ For observations `$t_i, i = 1,\cdots,n$`, the model with the highest joint probability is the model that is most likely to have generated the observations

+ For a single observation, the model providing the greatest contribution to the total likelihood may not be the correct model

+ As the number of observatons is increased, more information is obtained and it becomes easier to differentiate which model best-fits the data and best describes the underlying failure process

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## Likelihood Contributions For Reliability Data

- For failure data, each observation makes one of four contributions to the likelihood function

`$$\mathscr{L}_{i}(\underline{\theta}|t_{i})=\begin{cases} S(t_{i}) &\mbox{for a right censored observation}\\\\F(t_{i}) &\mbox{for a left censored observation}\\\\F(t_{i})-F(t_{i-1}) &\mbox{for an interval censored observation}\\\\\lim\limits_{\Delta_i\rightarrow 0} \frac{(F(t_{i})-\Delta_{i})-F(t_{i})}{\Delta_{i}} &\mbox{for an "exact" observation}\end{cases}$$`

- Thus, the total likelihood function may be expressed as

`$$\mathscr{L}(\underline{\theta}|\underline{t})=C\prod_{i=1}^{n} \mathscr{L}_{i}(\underline{\theta}|t_i)
=C\prod_{i=1}^{m+1}[F(t_{i})]^{l_{i}}[F(t_{i})-F(t_{i-1})]^{d_{i}}[1-F(t_{i})]^{r_{i}}$$`

- where

+ `$l_i=1$` if `$t_i$` is a left censored observation (0 otherwise)
    + `$d_i=1$` if `$t_i$` is an interval censored observation (0 otherwise)
    + `$r_i=1$` if `$t_i$` is a right censored observation (0 otherwise)
    + `$n = \sum_{j=1}^{m+1}(l_{j}+d_{j}+r_{j})$`