It's Time To Fail
Published on Nov 19, 2015
No Description
MORE DECKS TO EXPLORE
PRESENTATION OUTLINE
It's Time to Fail
When thinking about a presentation related to statistics which would be topical I thought to reference a personal experience.
Photo by James Loesch
Brainstorm
I had a bit of a brainstorm wanting to pull together a real-life experience within the context of product development which may be applicable to the Fisher & Paykel environment.
During the previous interview I referenced a failure of my own stove. It is from this I chose to draw upon.
During the previous interview I referenced a failure of my own stove. It is from this I chose to draw upon.
Photo by M I T C H Ǝ L L
The Problem
At the time, the digital display on my stove became inoperable. The electronics still worked, however something had caused the unit to dislodge from the stove and I was unable to press the display because there was no force resistance.
In short, my stove was a fail. I decided to investigate why and removed the back plating to discover that the bracket holding the unit in place is made of plastic and had broken, presumably from heat stress. I eventually solved the problem using strong wire.
However, from a statistics viewpoint it raises the question of failure in terms of the plastic component. I decided to think about this within the context of statistical analysis.
In short, my stove was a fail. I decided to investigate why and removed the back plating to discover that the bracket holding the unit in place is made of plastic and had broken, presumably from heat stress. I eventually solved the problem using strong wire.
However, from a statistics viewpoint it raises the question of failure in terms of the plastic component. I decided to think about this within the context of statistical analysis.
Probability Distributions
Not to insult anyone's knowledge of statistics but I have encountered a mis-use of language and understanding of probability distributions and wish to specify for a moment.
People, bunnies, crops, machines -- do not have distributions, probabilities do, and the probability of an event occurring has a distribution.
So within the context of this problem I thought about the time to fail of the plastic bracket.
People, bunnies, crops, machines -- do not have distributions, probabilities do, and the probability of an event occurring has a distribution.
So within the context of this problem I thought about the time to fail of the plastic bracket.
Photo by Great Beyond
Consider Components
- A: mean time to fail of 4645 hours
- B: mean time to fail of 300 hours
Which distribution best describes the probabilities associated with these events?
Photo by particlem
Weibull
b is the shape parameter, q is the scale parameter, d is the location parameter
The Weibull distribution is one of the most commonly used distributions in reliability. It is commonly used to model time to fail, time to repair and material strength.
beta (b) is the shape parameter
theta (q) is the scale parameter
delta (d) is the location parameter
beta (b) is the shape parameter
theta (q) is the scale parameter
delta (d) is the location parameter
Flexibility
Weibull distribution approximates the log-normal distribution
The shape parameter (beta) is what gives the Weibull distribution its flexibility. By changing the value of the shape parameter, the Weibull distribution can model a wide variety of data.
If b (beta) = 1 the Weibull distribution is identical to the exponential distribution
If b (beta) = 2, the Weibull distribution is identical to the Rayleigh distribution
If b (beta) is between 3 and 4 the Weibull distribution approximates the normal distribution.
Basically Weibull distribution approximates the lognormal distribution for several values of b (beta).
If b (beta) = 1 the Weibull distribution is identical to the exponential distribution
If b (beta) = 2, the Weibull distribution is identical to the Rayleigh distribution
If b (beta) is between 3 and 4 the Weibull distribution approximates the normal distribution.
Basically Weibull distribution approximates the lognormal distribution for several values of b (beta).
Photo by khalid almasoud
Untitled Slide
Examples of the function with different shape parameters (beta)
With q (theta, scale) = 100 and d (delta, location) = 0.
y-axis is probability density
With q (theta, scale) = 100 and d (delta, location) = 0.
y-axis is probability density
Untitled Slide
This is one of the most important aspects of the effect of β on the Weibull distribution. As is indicated by the plot, Weibull distributions with β
Weibull distributions with β close to or equal to 1 have a fairly constant failure rate, indicative of useful life or random failures.
Weibull distributions with β > 1 have a failure rate that increases with time, also known as wear-out failures.
These comprise the three sections of the classic "bathtub curve." A mixed Weibull distribution with one subpopulation with β 1 would have a failure rate plot that was identical to the bathtub curve.
Weibull distributions with β close to or equal to 1 have a fairly constant failure rate, indicative of useful life or random failures.
Weibull distributions with β > 1 have a failure rate that increases with time, also known as wear-out failures.
These comprise the three sections of the classic "bathtub curve." A mixed Weibull distribution with one subpopulation with β
Range
The scale parameter determines the range
The scale parameter (theta) determines the range of the distribution. The scale parameter is also known as the characteristic life when the location (delta) parameter is equal to zero.
If the location (delta) parameter does not equal zero the the characteristic life is equal to q (theta, scale) + d (delta, location).
And 63.2% of all values fall below the characteristic life regardless of the value of the shape (beta) parameter.
If the location (delta) parameter does not equal zero the the characteristic life is equal to q (theta, scale) + d (delta, location).
And 63.2% of all values fall below the characteristic life regardless of the value of the shape (beta) parameter.
Photo by minnepixel
Untitled Slide
If the location (delta) parameter does not equal zero the the characteristic life is equal to q (theta, scale) + d (delta, location).
And 63.2% of all values fall below the characteristic life regardless of the value of the shape (beta) parameter.
And 63.2% of all values fall below the characteristic life regardless of the value of the shape (beta) parameter.
Location
Defines a failure-free zone
The location (delta) parameter is used to define a failure-free zone.
Photo by Stuart Herbert
Untitled Slide
The probability of failure when x (our event) is less than d (delta) is zero.
When d>0, there is a period when no failures can occur.
When d
At first this seems a little strange, but sometimes you encounter a negative location parameter. For example, failed units, failures during transportation, and shelf life failures.
Generally, the location parameter is assumed to be zero. The effect of the location parameter is shown in the figure below.
When d>0, there is a period when no failures can occur.
When d
At first this seems a little strange, but sometimes you encounter a negative location parameter. For example, failed units, failures during transportation, and shelf life failures.
Generally, the location parameter is assumed to be zero. The effect of the location parameter is shown in the figure below.
F(x) = Q(x) = 1 - R(x)
If we go back to our probability distribution f(x) and rewrite it in terms of it's cumulative distribution or cdf F(t). This function returns the probability of a failure occurring before a certain time.
We also throw in a function which provides the probability of a failure occurring after a certain time.
It is worth noting that the cdf measures the area under the pdf curve up to a given time, and that the area under the pdf curve is always equal to 1. Given these concepts, subtracting the cdf from 1 would result in the probability of a failure occurring after a given time. This is the widely-used reliability function. And Q(t) is our un-reliability function.
We also throw in a function which provides the probability of a failure occurring after a certain time.
It is worth noting that the cdf measures the area under the pdf curve up to a given time, and that the area under the pdf curve is always equal to 1. Given these concepts, subtracting the cdf from 1 would result in the probability of a failure occurring after a given time. This is the widely-used reliability function. And Q(t) is our un-reliability function.
Photo by Simply Boaz
Reliability function
We have our reliability function in terms of our shape, scale and location parameters.
Estimating the Shape
- Graphical
- Analytical
Need to know the shape parameter beta.
Graphical:
Weibull probability plotting
Hazard plotting
Analytical:
Maximum likliehood estimator
Method of Moments
Least squares method
Now, the question is which method do we use? The answer depends on whether one needs a quick or an accurate estimation.
Graphical:
Weibull probability plotting
Hazard plotting
Analytical:
Maximum likliehood estimator
Method of Moments
Least squares method
Now, the question is which method do we use? The answer depends on whether one needs a quick or an accurate estimation.
Photo by Neal.
Speedy
- Any graphical method
- Least squares method
- Maximum likelihood estimator
- Method of moments
If you want quick: the order based on computing time is:
1. Any graphical method.
2. Least Squares Method.
3 Maximum Likelihood Estimator.
4 Method of Moments
1. Any graphical method.
2. Least Squares Method.
3 Maximum Likelihood Estimator.
4 Method of Moments
Photo by Andreas-photography
Accurate
- Any of the three methods
- Select one with minimum MSE
Applying the three analytical methods (MLE,MOM and LSM)
and selecting the best one which gives the minimum mean squared
error.
and selecting the best one which gives the minimum mean squared
error.
Photo by 7D-Kenny
Obtaining Scale Paramter
- Use our mean time(s) to fail (4645, 300)
- Mean of the Weibull distribution
- E[A] = E[4645] = 4100
- E[B] = E[300] = 100
Use our mean times to fail and the mean or 1st expectation of the Weibull distribution.
(I'm not going to show the calculations for this)
(I'm not going to show the calculations for this)
Photo by StuartWebster
Maximise Reliability at 100 hours
Plug in:
Our event = 100 hours
location (delta) = 0
scale (theta1) = 4100 (from our mean of the Weibull)
scale (theta2) = 336 (from our mean of the Weibull)
shape (beta1) = 0.8 (from our estimation method)
shape (beta2) = 3 (from our estimation method)
Our event = 100 hours
location (delta) = 0
scale (theta1) = 4100 (from our mean of the Weibull)
scale (theta2) = 336 (from our mean of the Weibull)
shape (beta1) = 0.8 (from our estimation method)
shape (beta2) = 3 (from our estimation method)
Conclusion
Component B is the winner!
Although the mean of component A (4645) is more than 10 times as large as the mean of component B (300), the reliability of component B is greater than the reliability of component A at 100 hours
Photo by Greg Marshall
Untitled Slide
Photo by MightyBoyBrian
