2d:null 2e:["$","title",null,{"children":"The Weibull Distribution | Michael Deagen | Observable"}] 33:T43e,What we see above is commonly referred to as a **Weibull probability plot**. In this type of plot, a Weibull cumulative distribution function should appear as a *straight line*. This is same distribution as before, just with different x- and y-axis scales. Note how the shape parameter, ${tex`\beta`}, is much more evident via the *slope* of the line, and the scale parameter, ${tex`\eta`}, is the x-value where the probability crosses 63.2%. This probability, ${tex`p \approx 0.632`}, results in ${tex`ln(-ln(1-p))=0`}, where ${tex`E=\eta`}. How did we achieve this Weibull y-axis scaling, where consecutive integers are closer together in the middle and farther apart as you approach 0% and 100%? In short, the underlying y-axis scale *is* linear, but the labels you see along the y-axis come from a set of reference values embedded in the Vega-Lite spec. While the labels display the probability, ${tex`p`}, ranging from 1% to 99%, their underlying values are ${tex`ln(-ln(1-p))`} and range linearly from -4.6 to 1.527. Use the slider below to see this linear conversion in action.34:T481,weibull_pdf = ({ "$schema": "https://vega.github.io/schema/vega-lite/v5.json", "width": 600, "height": 400, "params": [ {"name": "beta", "bind": {"element": viewof beta}}, {"name": "eta", "bind": {"element": viewof eta}} ], "data": { "sequence": {"start": 0.1, "stop": 1201, "step": 2, "as": "seq_x"} }, "layer": [ { "transform": [ {"calculate": "beta / eta","as": "a"}, {"calculate": "beta -1","as": "b_exp"}, {"calculate": "pow((datum.seq_x / eta), (beta - 1))","as": "b"}, {"calculate": "exp(-1 * pow((datum.seq_x / eta), beta))","as": "c"}, {"calculate": "datum.a * datum.b * datum.c","as": "pdf"} ], "mark": {"type": "line", "strokeDash":[10,5], "clip":true}, "encoding": { "x": {"field": "seq_x", "type": "quantitative", "title": "Electric Field (kV/mm)","scale":{"domain":[0,1200]}}, "y": {"field": "pdf", "type": "quantitative", "title": "Probability of Failure", "axis":{"format":"p"},"scale":{"domain":[0,0.02]}} } } ] , "config": { "axis": {"titleFontSize": 20, "labelFontSize": 16, "labelFlush": false} } })35:T476,weibull_cdf = ({ "$schema": "https://vega.github.io/schema/vega-lite/v5.json", "width": 600, "height": 400, "params": [ {"name": "beta", "bind": {"element": viewof beta}}, {"name": "eta", "bind": {"element": viewof eta}} ], "data": { "sequence": {"start": 0, "stop": 1201, "step": 10, "as": "seq_x"} }, "layer": [ { "data":{"values":[{"prob":0.632}]}, "mark":{"type":"rule", "color":"firebrick", "strokeWidth":1}, "encoding":{ "y":{"field":"prob","type":"quantitative"} } }, { "transform": [ { "calculate": "1 - exp(-1 * pow((datum.seq_x / eta), beta))", "as": "seq_y" } ], "mark": {"type": "line", "strokeDash": [10, 5]}, "encoding": { "x": {"field": "seq_x", "type": "quantitative", "title": "Electric Field (kV/mm)"}, "y": {"field": "seq_y", "type": "quantitative", "title": "Cumulative Probability of Failure", "scale":{"domain":[0,1]},"axis":{"format":"p"}} } } ] , "config": { "axis": {"titleFontSize": 20, "labelFontSize": 16, "labelFlush": false, "grid":true} } })36:T9fe,weibull_linear = ({ "$schema": "https://vega.github.io/schema/vega-lite/v5.json", "width": 600, "height": 400, "params": [ {"name": "beta", "bind": {"element": viewof beta}}, {"name": "eta", "bind": {"element": viewof eta}} ], "data": { "sequence": {"start": 0, "stop": 1201, "step": 10, "as": "seq_x"} }, "layer": [ { "data": { "values": [ {"ProbLabel": "99%", "WeibullLinear": 1.527}, {"ProbLabel": "90%", "WeibullLinear": 0.834}, {"ProbLabel": "80%", "WeibullLinear": 0.476}, {"ProbLabel": "70%", "WeibullLinear": 0.186}, {"ProbLabel": "60%", "WeibullLinear": -0.087}, {"ProbLabel": "50%", "WeibullLinear": -0.367}, {"ProbLabel": "40%", "WeibullLinear": -0.672}, {"ProbLabel": "30%", "WeibullLinear": -1.031}, {"ProbLabel": "20%", "WeibullLinear": -1.5}, {"ProbLabel": "10%", "WeibullLinear": -2.25}, {"ProbLabel": "5%", "WeibullLinear": -2.97}, {"ProbLabel": "1%", "WeibullLinear": -4.6} ] }, "layer": [ { "mark": {"type": "rule", "color": "black", "strokeWidth": 0.2, "opacity":1} }, { "mark": {"type": "text","x": -3,"align": "right","fontSize": 16,"clip": false}, "encoding": { "text": {"field": "ProbLabel", "type": "nominal"} } } ], "encoding": {"y": {"field": "WeibullLinear", "type": "quantitative"}} }, { "data":{"values":[{"DBS_prob":0}]}, "mark":{"type":"rule", "color":"firebrick", "strokeWidth":1}, "encoding":{ "y":{"field":"DBS_prob","type":"quantitative"} } }, { "transform": [ {"calculate": "1 - exp(-1 * pow((datum.seq_x / eta), beta))", "as": "seq_y"}, {"calculate": "-log(1 - (datum.seq_y))", "as": "Weibull_inner"}, {"calculate": "log(datum.Weibull_inner)", "as": "Weibull"} ], "mark": {"type": "line", "clip":true, "strokeDash": [10, 5]}, "encoding": { "x": {"field": "seq_x", "type": "quantitative", "title": "Electric Field (kV/mm)","scale":{"type":"log","domain":[40,1200]}}, "y": {"field": "Weibull", "type": "quantitative","scale":{"domain":[-5,2]}, "axis": {"grid": false,"ticks": false,"labels": false,"title": "Cumulative Probability of Failure", "titleFontSize": 20,"titlePadding": 50}} } } ], "config": { "axis": {"titleFontSize": 20, "labelFontSize": 16, "labelFlush": false} } })37:[] 31:{"id":"4bb60c19aa210e87","slug":"overview-of-the-weibull-distribution","title":"The Weibull Distribution","thumbnail":"06ce29264dc320ed147f540054b51edfe60fe4af1ed6abcb9ba7ae1390ab849c","access_level":"public","create_time":"$D2021-06-02T16:58:25.415Z","update_time":"$D2021-06-09T12:48:23.006Z","view_time":null,"first_public_time":"$D2021-06-02T22:17:14.670Z","has_unpublished_edits":null,"fork":false,"stars":0,"starred":false,"owner":{"type":"team","id":"5ca8a3812ea4a326","login":"mdeagen","name":"Michael Deagen","avatar_url":"https://avatars-next.observableusercontent.com/0b8e876e2eac1058faa6c1a2624346f3e72350b8a7e6c9acaec6a9ad920b0476","tier":"starter_2024"},"version":794,"publish_version":794,"role":null,"authored":false,"authors":[{"type":"individual","id":"d3611c0325cb99e9","login":"mdeagen","name":"Michael Deagen","avatar_url":"https://avatars-next.observableusercontent.com/0b8e876e2eac1058faa6c1a2624346f3e72350b8a7e6c9acaec6a9ad920b0476","tier":null}],"members":[{"type":"individual","id":"d3611c0325cb99e9","login":"mdeagen","name":"Michael Deagen","avatar_url":"https://avatars-next.observableusercontent.com/0b8e876e2eac1058faa6c1a2624346f3e72350b8a7e6c9acaec6a9ad920b0476","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":0,"collection":null,"type":"notebook","body":{"title":"The Weibull Distribution","cells":[{"id":0,"value":"# The Weibull Distribution\n\nFound in numerous engineering applications from reliability engineering to manufacturing, the [Weibull distribution](https://en.wikipedia.org/wiki/Weibull_distribution) is also used to describe the probabilistic failure behavior of dielectric materials for electrical insulators. The Weibull probability distribution function (pdf) can be written as,","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":5,"value":"f(x;\\beta,\\eta) =\n\\begin{cases}\n\\frac{\\beta}{\\eta}\\left(\\frac{x}{\\eta}\\right)^{\\beta-1}e^{-(\\frac{x}{\\eta})^{\\beta}} & x\\geq0 ,\\\\\n0 & x<0.\n\\end{cases}","pinned":false,"hidden":false,"mode":"tex","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":473,"value":"Above is the two-parameter Weibull distribution, where ${tex`\\beta`} is the **shape** parameter and ${tex`\\eta`} is the **scale** parameter. Adjust the shape and scale parameters using the sliders below to see how they affect the probability distribution function. When studying dielectric breakdown of insulating materials, ${tex`x`} corresponds to an electric field, and ${tex`f(x)`} is the probability that the material [fails to withstand the electric field](https://en.wikipedia.org/wiki/Electrical_breakdown).","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":33,"value":"viewof beta = Range([0.5,5], {step: 0.1, label: \"shape parameter, β\"})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":35,"value":"viewof eta = Range([50,1000], {step: 10, label: \"scale parameter, η\"})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":479,"value":"vl.render({spec: weibull_pdf})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":12,"value":"You might observe that the distribution loses its peak when the shape parameter is less than 1, or how the distribution \"spreads out\" as the scale parameter is increased. Let's explore further by examining the *cumulative* distribution function,","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":16,"value":"F(x;\\beta,\\eta) = 1 - e^{-(x/\\eta)^\\beta}","pinned":false,"hidden":false,"mode":"tex","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":20,"value":"For dielectric materials, the cumulative distribution is useful for characterizing, or testing, the \"unreliability\" of a material under increasing electric fields. The electric field at which the cumulative probability is equal to ${tex`(1 - \\frac{1}{e})`}, or roughly 63.2%, is defined as the **dielectric breakdown strength ([DBS](https://en.wikipedia.org/wiki/Dielectric_strength))** of the material. In the chart below, the DBS is the *x*-value at which the distribution crosses the red line.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":29,"value":"vl.render({spec: weibull_cdf})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":490,"value":"Inputs.bind(Range([0.5,5], {label: \"shape parameter, β\"}), viewof beta)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":492,"value":"Inputs.bind(Range([50,1000], {label: \"scale parameter, η\"}), viewof eta)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":153,"value":"The sliders in this notebook are all synchronized so that you can see how each plot changes when tuning the shape or scale parameter.\n\nYou may have noticed that changing the **shape** parameter *has no effect on the DBS* and that the **scale** parameter *is equal to the DBS*. Why is that?\n\nFortunately, there are alternative ways to view a Weibull distribution. The above plot shows the distribution along *linear* x- and y-axis scales. What if we use different scales to map data values onto the Euclidean plane? (Creative transformations occur all the time in [map projections](https://observablehq.com/collection/@toja/projections)!) To select the best set of scales for the Weibull distribution, we can start by re-arranging its cumulative distribution function,","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":199,"value":"p = 1 - e^{-(E/\\eta)^\\beta},","pinned":false,"hidden":false,"mode":"tex","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":207,"value":"where ${tex`p`} is probability and ${tex`E`} is electric field, into a slightly different form,","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":759,"value":"1 - p = e^{-(E/\\eta)^\\beta},","pinned":false,"hidden":false,"mode":"tex","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":762,"value":"ln(1 - p) = -(E/\\eta)^\\beta,","pinned":false,"hidden":false,"mode":"tex","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":774,"value":"ln(-ln(1-p)) = \\beta ln(E/\\eta),","pinned":false,"hidden":false,"mode":"tex","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":210,"value":"ln(-ln(1-p)) = \\beta ln(E) - \\beta ln(\\eta).","pinned":false,"hidden":false,"mode":"tex","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":227,"value":"By substituting ${tex`ln(-ln(1-p))`} with _**y**_ and ${tex`ln(E)`} with _**x**_, we get the following,","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":236,"value":"y = \\beta x - b,","pinned":false,"hidden":false,"mode":"tex","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":249,"value":"where ${tex`\\beta ln(\\eta)`}, a constant, has been replaced by _**b**_. We now have a linear equation!\n\nPlotting tools usually offer several axis scale options (*linear*, *logarithmic*, etc.) to choose from. Since ${tex`E`} was shown to linearize with ${tex`ln(E)`}, the x-axis variable should use a *logarithmic* scale. This is provided by Vega-Lite out of the box and specified with `\"scale\": {\"type\": \"log\"}`. But how should we approach that unusual ${tex`ln(-ln(1-p))`} scale transformation for the y-axis? Let's linearize it ourselves!","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":277,"value":"vl.render({spec: weibull_linear})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":328,"value":"Inputs.bind(Range([0.5,5], {label: \"shape parameter, β\"}), viewof beta)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":337,"value":"Inputs.bind(Range([50,1000], {label: \"scale parameter, η\"}), viewof eta)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":345,"value":"$33","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":711,"value":"viewof p = Range([0.01,0.99],{label: \"p\", step: 0.01})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":744,"value":"The current probability is **${p.toLocaleString(undefined,{style: \"percent\"})}**, which corresponds to a *y*-value of about **${weibull_y.toLocaleString(undefined, {maximumFractionDigits: 4})}**.\n\nTo map the cumulative distribution function, each instance of ${tex`p`} was replaced by ${tex`ln(-ln(1-p))`} using the `calculate` transform in Vega-Lite. The y-axis \"grid\" you see is just a set of rule marks for those reference values embedded in the spec.\n\nWith Weibull probability plots, it becomes easier to see how well a dielectric material sample follows a Weibull distribution by checking for linearity, and the slope of the line indicates the reliability of the dielectric failure behavior.\n\nTo see some Vega-Lite charts containing actual dielectric data taken from polymer nanocomposite materials, check out [this chart](https://materialsmine.org/wi/viz/79ea1834fe924be3) or [this one with a DBS calculator](https://materialsmine.org/wi/viz/cb322b69ed19fb59).","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":39,"value":"---\n## Appendix ","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":469,"value":"$34","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":27,"value":"$35","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":274,"value":"$36","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":713,"value":"weibull_y = Math.log(-1 * Math.log(1-p))","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":41,"value":"import {Range} from '@observablehq/inputs'","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W37","stdlib":"1","titling":"auto"}} 32:"$31:body"