(************** Content-type: application/mathematica ************** Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 694262, 15565]*) (*NotebookOutlinePosition[ 701724, 15784]*) (* CellTagsIndexPosition[ 700020, 15733]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Introduction to Reliability Analysis with Mathematica", FontSize->24], Cell[BoxData[ FormBox[ StyleBox[ SuperscriptBox["", StyleBox["\[RegisteredTrademark]", FontSize->13, FontWeight->"Plain"]], FontSize->24], TraditionalForm]]] }], "Title"], Cell[TextData[{ StyleBox["Dave Collins", FontSize->16, FontWeight->"Bold", FontSlant->"Plain"], StyleBox["\n", FontSlant->"Plain"], StyleBox["dcollins@outbacksoftware.com", FontSize->12, FontSlant->"Plain"], StyleBox["\n\n", FontSlant->"Plain"], StyleBox["Copyright Outback Software, Ltd., 2002. This notebook may be \ freely copied and distributed, provided it is kept in its original form, \ including this notice. For other rights, contact the author. ", FontFamily->"Times New Roman", FontSize->12, FontSlant->"Plain"], StyleBox["Mathematica", FontFamily->"Times New Roman", FontSize->12, FontSlant->"Italic"], StyleBox[" is a registered trademark of Wolfram Research, Inc.", FontFamily->"Times New Roman", FontSize->12, FontSlant->"Plain"] }], "Subsubtitle"], Cell[TextData[{ StyleBox["Acknowledgements", FontFamily->"Arial", FontWeight->"Bold"], "\n", StyleBox[" ", FontSize->8], "\nThis notebook started as a project for a ", "Mathematica", " course taught by ", ButtonBox["Ken Levasseur", ButtonData:>{ URL[ "http://faculty.uml.edu/klevasseur/"], None}, ButtonStyle->"Hyperlink"], " at the University of Massachusetts. Thanks also to ", ButtonBox["Stan Wagon", ButtonData:>{ URL[ "http://www.stanwagon.com/"], None}, ButtonStyle->"Hyperlink"], " for many hints and tips given at ", ButtonBox["Rocky Mountain Mathematica", ButtonData:>{ URL[ "http://math.lfc.edu/Rocky_Mtn_Mathematica/"], None}, ButtonStyle->"Hyperlink"], "." }], "Text", CellTags->"Introduction"], Cell[CellGroupData[{ Cell["Table of Contents", "Section"], Cell[CellGroupData[{ Cell[TextData[StyleBox[ButtonBox["Introduction", ButtonData:>"Introduction", ButtonStyle->"Hyperlink"], FontFamily->"Arial"]], "Subsubsection"], Cell[TextData[{ "\t", ButtonBox["Initialization", ButtonData:>"Initialization", ButtonStyle->"Hyperlink"] }], "Text"], Cell[TextData[{ "\t", ButtonBox["Mathematical models", ButtonData:>"Mathematical models", ButtonStyle->"Hyperlink"] }], "Text"], Cell[TextData[{ "\t", ButtonBox["Relibility modeling", ButtonData:>"Relibility modeling", ButtonStyle->"Hyperlink"] }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox[ButtonBox["Empirical models of reliability", ButtonData:>"Empirical models of reliability", ButtonStyle->"Hyperlink"], FontFamily->"Arial"]], "Subsubsection"], Cell[TextData[{ "\t", ButtonBox["Non-Parametric estimation of f(t), F(t), R(t) and h(t)", ButtonData:>"NonParametricEstimates", ButtonStyle->"Hyperlink"] }], "Text"], Cell[TextData[{ "\t", ButtonBox["Example: The exponential distribution", ButtonData:>"ExampleExponential", ButtonStyle->"Hyperlink"] }], "Text"], Cell[TextData[{ "\t", ButtonBox["Maximum likelihood parameter estimation", ButtonData:>"Maximum likelihood", ButtonStyle->"Hyperlink"] }], "Text"], Cell[TextData[{ "\t", ButtonBox["Example: The Weibull distribution", ButtonData:>"ExampleWeibull", ButtonStyle->"Hyperlink"] }], "Text"], Cell[TextData[{ "\t", ButtonBox["Testing goodness of fit", ButtonData:>"TestingGoodnessOfFir", ButtonStyle->"Hyperlink"] }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox[ButtonBox["Markov models", ButtonData:>"MarkovModels", ButtonStyle->"Hyperlink"], FontFamily->"Arial"]], "Subsubsection"], Cell[TextData[{ "\t", ButtonBox["Reliability modeling examples", ButtonData:>"Markov-Examples", ButtonStyle->"Hyperlink"] }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox[ButtonBox["Simulation or Monte Carlo methods", ButtonData:>"Simulation or Monte Carlo methods", ButtonStyle->"Hyperlink"], FontFamily->"Arial"]], "Subsubsection"], Cell[TextData[{ "\t", ButtonBox["Generating values of random variables", ButtonData:>"GeneratingValuesOfRandomVariables", ButtonStyle->"Hyperlink"] }], "Text"], Cell[TextData[{ "\t", ButtonBox["Reliability modeling example", ButtonData:>"MonteCarlo-Examples", ButtonStyle->"Hyperlink"] }], "Text"] }, Open ]], Cell[TextData[StyleBox[ButtonBox["Summary", ButtonData:>"Summary", ButtonStyle->"Hyperlink"], FontFamily->"Arial"]], "Subsubsection"], Cell[TextData[StyleBox[ButtonBox["Bibliography", ButtonData:>"Bibliography", ButtonStyle->"Hyperlink"], FontFamily->"Arial"]], "Subsubsection"] }, Open ]], Cell[CellGroupData[{ Cell["Introduction", "Section"], Cell[TextData[{ "Reliability analysis is an engineering discipline that applies various \ mathematical techniques to the measurement and prediction of the reliability \ of components and systems. The components under study may be mechanical, \ electronic, software, or other types. Systems could include anything from \ computers to rail transit. Measurements include failure rates, cumulative \ failures, and component lifetimes (time until failure). A variety of \ techniques are employed, drawn mainly from probability, statistics, and the \ theory of stochastic processes. \n\nMathematica is a software application for \ numeric and symbolic computation, developed by ", ButtonBox["Wolfram Research", ButtonData:>{ URL[ "http://www.wolfram.com/"], None}, ButtonStyle->"Hyperlink"], ". This being an introduction, the assumption throughout is that the reader \ knows something about Mathematica, but is not an expert, either in \ Mathematica or reliability analysis.\n\nThe mathematics of reliability \ analysis, even in terms of specific models, has broad applicability. A \ reliability model with failures and repairs to failed units may have the same \ form as a population model in which \"failures\" become deaths, and \"repairs\ \" become births. A probability distribution of survival times may be used in \ reliability analysis to model component lifetimes, in medical research to \ model patient survival in a treatment group, and by actuaries to compute \ insurance premiums. \n\nIn this paper a context is established by discussing \ the general idea of using mathematics to model real-world processes, and \ outlining the specific methods used by reliability engineers. Following the \ introduction, several techniques of reliability analysis are discussed in \ more detail, with particular attention to the suitability of Mathematica as a \ tool for reliability analysis. Examples have been chosen that use \ mathematical models with broad applicability, and that provide good \ illustrations of the use of Mathematica. " }], "Text", CellTags->"Introduction"], Cell[CellGroupData[{ Cell["Initialization", "Subsubsection", CellTags->"Initialization"], Cell[TextData[{ "This section consolidates data input and definition of common functions \ used in this notebook. Most of the cells in this section are Mathematica \ \"initialization cells,\" meaning they will be automatically evaluated \ whenever any computational cell in the notebook is evaluated. If you are \ presented with a dialogue box asking if you want to perform this automatic \ initialization, always answer \"Yes.\" \n\nThe following cell initializes \ data sets used in subsequent sections. ", StyleBox["observedPopulation", FontFamily->"Courier", FontWeight->"Bold"], " is population data (in millions) for the United States from 1770-2000; ", StyleBox["ballBearingFailures", FontFamily->"Courier", FontWeight->"Bold"], " is time to failure (measured in millions of revolutions) for a set of \ tested ball bearings; ", StyleBox["leukemiaRemission", FontFamily->"Courier", FontWeight->"Bold"], " is time to spontaneous remission of leukemia in a group of control \ subjects for a drug trial. " }], "Text"], Cell[BoxData[{ \(\(observedPopulation\ := \[IndentingNewLine]{{1790, 3.9}, {1800, 5.1}, {1810, 6.8}, {1820, 10.0}, {1830, 12.8}, {1840, 17.0}, {1850, 23.1}, {1860, 31.2}, {1870, 38.2}, {1880, 49.4}, {1890, 62.1}, {1900, 74.6}, {1910, 91.6}, {1920, 105.3}, {1930, 122.3}, {1940, 131.0}, {1950, 149.9}, {1960, 178.6}, {1970, 203.3}, {1980, 226.5}, {1990, 248.7}, {2000, 281.4}};\)\), "\[IndentingNewLine]", \(\(ballBearingFailures\ := \[IndentingNewLine]{{1, 17.88}, {2, 28.92}, {3, 33.00}, {4, 41.52}, {5, 42.12}, {6, 45.60}, {7, 48.48}, {8, 51.84}, {9, 51.96}, {10, 54.12}, {11, 55.56}, {12, 67.8}, {13, 68.64}, {14, 68.64}, {15, 68.88}, {16, 84.12}, {17, 93.12}, {18, 98.64}, {19, 105.12}, {20, 105.84}, {21, 127.92}, {22, 128.04}, {23, 173.40}};\)\), "\[IndentingNewLine]", \(\(leukemiaRemission\ := \[IndentingNewLine]{{1, 1}, {2, 1}, {3, 2}, {4, 2}, {5, 3}, {6, 4}, {7, 4}, {8, 5}, {9, 5}, {10, 8}, {11, 8}, {12, 8}, {13, 8}, {14, 11}, {15, 11}, {16, 12}, {17, 12}, {18, 15}, {19, 17}, {20, 22}, {21, 23}};\)\)}], "Input", InitializationCell->True], Cell[TextData[{ "More typically, data sets like the above would be read in from external \ files. For example, suppose there is a text file, USpop.txt, wherein each \ line is a data pair like \"", StyleBox["1790\t3.9", FontFamily->"Courier", FontWeight->"Bold"], "\", representing the U. S. population (in millions) for a given year. Then \ the variable observedPopulation could be created as follows:\n", StyleBox[" ", FontSize->10], "\n", StyleBox["f=OpenRead[\"USpop.txt\"];\n\ observedPopulation=ReadList[\"USpop.txt\",{Number,Number}];\nClose[f];", FontFamily->"Courier New", FontWeight->"Bold"], "\n", StyleBox[" ", FontSize->10], "\nThis code assumes that ", StyleBox["USpop.txt", FontFamily->"Courier", FontWeight->"Bold"], " is in the default Mathematica folder, as given by ", StyleBox["Directory[ ]", FontWeight->"Bold"], ". Since the location of the default directory may vary depending on \ operating system type and installation options, this is error-prone. Thus the \ data is imbedded directly in the notebook, for portability. (This would be \ impractical, of course, for large data sets.) \n\nData can also be imported \ in popular spreadsheet formats using built-in Mathematica capabilities. Using \ ", StyleBox["MathLink", FontSlant->"Italic"], ", Mathematica's method of linking to programs written in other language \ such as C or Java, data could also be read in from external databases such as \ DB2 or Oracle.\n", StyleBox[" ", FontSize->10], "\nLoad needed packages:" }], "Text"], Cell[BoxData[{ \(\(Needs["\"];\)\), "\ \[IndentingNewLine]", \(\(Needs["\"];\)\), "\[IndentingNewLine]", \(\(Needs["\"];\)\), \ "\[IndentingNewLine]", \(\(Needs["\"];\)\), "\[IndentingNewLine]", \ \(\(Needs["\"];\)\)}], "Input", InitializationCell->True], Cell["Turn off unnecessary warning messages:", "Text"], Cell[BoxData[{ \(\(Off[General::"\"];\)\), "\[IndentingNewLine]", \(\(Off[DesignedRegress::"\"];\)\), "\[IndentingNewLine]", \(\(Off[DesignedRegress::"\"];\)\), "\[IndentingNewLine]", \(\(Off[DesignedRegress::"\"];\)\), "\[IndentingNewLine]", \(\(Off[Solve::"\"];\)\)}], "Input", InitializationCell->True], Cell[TextData[{ StyleBox["stepPlot[ ]", FontWeight->"Bold"], " plots a list of points as a step function (like ", StyleBox["Histogram[ ", FontWeight->"Bold"], StyleBox["]", FontWeight->"Bold"], ", but only showing the outline of the top of the bars). It is useful for \ plotting things like discrete probability distributions." }], "Text"], Cell[BoxData[{ \(\(stepPlot[points_?ListQ, opts___?OptionQ] := Module[{otherFuns := PlotWith /. Flatten[{{opts}, Options[stepPlot]}], stepFun, vals, plotLim, p1, p2}, stepFun[x_] := \((\[IndentingNewLine]vals = Select[Sort[ points], \((First[#1] \[LessEqual] x)\) &]; \[IndentingNewLine]If[vals === {}, 0, Last[Last[ vals]]]\[IndentingNewLine])\); \[IndentingNewLine]plotLim \ := Max[First /@ points] + 1; \[IndentingNewLine]p1 := Plot[stepFun[x], {x, 0, plotLim}, PlotRange \[Rule] All, DisplayFunction \[Rule] Identity]; \[IndentingNewLine]If[ otherFuns === {}, Show[p1, DisplayFunction \[Rule] $DisplayFunction], \((p2 := Plot[Evaluate[Map[Apply[#, {x}] &, otherFuns]], {x, 0, plotLim}, PlotRange \[Rule] All, DisplayFunction \[Rule] Identity]; \[IndentingNewLine]Show[ p1, p2 /. Line[l___] \[Rule] {Hue[1], Line[l]}, DisplayFunction \[Rule] $DisplayFunction])\)];];\)\), "\n", \(\(Options[stepPlot] = {PlotWith \[Rule] {}};\)\)}], "Input", InitializationCell->True], Cell[TextData[{ "It takes one option, ", StyleBox["PlotWith\[Rule]{", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`f\_\(\(1\)\(,\)\)\)], FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`f\_2\)], FontWeight->"Bold"], StyleBox[", . . .}", FontWeight->"Bold"], ", where the ", StyleBox["f", FontSlant->"Italic"], "s are other functions to be plotted. The other functions are plotted in a \ different color. This is useful in comparing the step plot to a fitted \ function, for example:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(stepPlot[leukemiaRemission, PlotWith \[Rule] {\ 23/\ \((1 + \ 50 Exp[\(-0.3\) \((#)\)])\) &}];\)\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.04329 0.0147151 0.0255915 [ [.24026 .00222 -3 -9 ] [.24026 .00222 3 0 ] [.45671 .00222 -6 -9 ] [.45671 .00222 6 0 ] [.67316 .00222 -6 -9 ] [.67316 .00222 6 0 ] [.88961 .00222 -6 -9 ] [.88961 .00222 6 0 ] [.01131 .14267 -6 -4.5 ] [.01131 .14267 0 4.5 ] [.01131 .27063 -12 -4.5 ] [.01131 .27063 0 4.5 ] [.01131 .39859 -12 -4.5 ] [.01131 .39859 0 4.5 ] [.01131 .52654 -12 -4.5 ] [.01131 .52654 0 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FontFamily->"Courier New", FontWeight->"Bold"], " is the ", StyleBox["Mathematica", FontSlant->"Italic"], " \"pure function\" corresponding to ", Cell[BoxData[ \(TraditionalForm\`23\/\(1 + 50 e\^\(\(-0.3\) t\)\)\)]], ".)" }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Mathematical models", "Subsubsection"], Cell[TextData[{ "A mathematical model of a process in the real world is a set of equations \ with independent variables taking values based on quantitative aspects of the \ process state at some time, say ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["t", FontSlant->"Plain"], "0"], TraditionalForm]]], ", and dependent variables yielding quantitative information regarding the \ process state at some later time t:\n\n\t", StyleBox["S", FontWeight->"Bold"], "(t) = (", Cell[BoxData[ \(TraditionalForm\`x\_1\)]], "(t), ", Cell[BoxData[ \(TraditionalForm\`x\_2\)]], "(t), . . ., ", Cell[BoxData[ \(TraditionalForm\`x\_n\)]], "(t)) = \n\t\t", StyleBox["f ", FontWeight->"Bold"], "(", StyleBox["S", FontWeight->"Bold"], "(", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["t", FontSlant->"Plain"], "0"], TraditionalForm]]], "), ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["t", FontSlant->"Plain"], "0"], TraditionalForm]]], ", t) = (", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["f", FontSlant->"Plain"], "1"], TraditionalForm]]], "(", Cell[BoxData[ \(TraditionalForm\`x\_1\)]], "(", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["t", FontSlant->"Plain"], "0"], TraditionalForm]]], "), ", Cell[BoxData[ \(TraditionalForm\`x\_2\)]], "(", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["t", FontSlant->"Plain"], "0"], TraditionalForm]]], "), . . ., ", Cell[BoxData[ \(TraditionalForm\`x\_n\)]], "(", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["t", FontSlant->"Plain"], "0"], TraditionalForm]]], ")), . . ., ", Cell[BoxData[ FormBox[ StyleBox[ SubscriptBox[ StyleBox["f", FontSlant->"Plain"], "n"], FontSlant->"Italic"], TraditionalForm]]], "(", Cell[BoxData[ \(TraditionalForm\`x\_1\)]], "(", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["t", FontSlant->"Plain"], "0"], TraditionalForm]]], "), ", Cell[BoxData[ \(TraditionalForm\`x\_2\)]], "(", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["t", FontSlant->"Plain"], "0"], TraditionalForm]]], "), . . ., ", Cell[BoxData[ \(TraditionalForm\`x\_n\)]], "(", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["t", FontSlant->"Plain"], "0"], TraditionalForm]]], ")), ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["t", FontSlant->"Plain"], "0"], TraditionalForm]]], ", t) \n\nMore generally, particularly if the modeling is done on a \ computer, the function ", StyleBox["f", FontWeight->"Bold"], " may be implemented as an algorithmic procedure that is not necessarily \ expressible in closed form. A mathematical model may be descriptive \ (describing or explaining observed data), predictive (predicting data to be \ observed in the future), or both.\n\nAs an example, suppose we are interested \ in explaining or predicting population growth. One model of growth is that, \ in an environment with no constraints, future population may be predicted \ based on a knowledge of the initial population size, and the net growth rate \ (birth rate minus death rate). It is well known that this model implies \ exponential growth, as is shown in the function ", StyleBox["ExponentialGrowthModel", FontWeight->"Bold"], " below. ", StyleBox["P0", FontWeight->"Bold"], " is the initial population size, and ", StyleBox["g", FontWeight->"Bold"], " is the growth rate. In reality, there are always limits to growth, and \ the logistic model takes this into account, adding a parameter ", StyleBox["C", FontWeight->"Bold"], ", the \"carrying capacity\" of a given environment, which is the maximum \ size that a population can achieve in that environment before resource \ limitations reduce the growth rate to zero. The result is a model that first \ grows exponentially, but then levels off and approaches the carrying capacity \ asymptotically. This is shown in the function ", StyleBox["LogisticGrowthModel", FontWeight->"Bold"], " below. \n\nThese two functions are used below to explain population \ growth in the United States over a two-hundred year period. The actual \ population figures, at ten-year intervals from census data, are in the ", StyleBox["ObservedPopulation", FontWeight->"Bold"], " list (in millions, plotted in black). The parameters used in the two \ models were estimated visually to produce a reasonable fit (more rigorous \ techniques of estimating and validating model parameters are described below \ in ", StyleBox["Empirical models of reliability", FontFamily->"Arial", FontWeight->"Bold"], "). The plot shows that ", StyleBox["LogisticGrowthModel", FontWeight->"Bold"], " (plotted in blue) explains the observed data better than ", StyleBox["ExponentialGrowthModel ", FontWeight->"Bold"], "(plotted in red). (More rigorous methods of measuring \"better\" are also \ described below.)\n\nThese models are described in ", ButtonBox["[23]", ButtonData:>"Thompson", ButtonStyle->"Hyperlink"], " and ", ButtonBox["[8]", ButtonData:>"Gotelli", ButtonStyle->"Hyperlink"], ". The 1790-1960 census data is from ", ButtonBox["[9]", ButtonData:>"Census1", ButtonStyle->"Hyperlink"], ", the 1970-1990 data from ", ButtonBox["[8]", ButtonData:>"Census2", ButtonStyle->"Hyperlink"], ", and the 2000 data from ", ButtonBox["[24]", ButtonData:>"Census3", ButtonStyle->"Hyperlink"], "." }], "Text", CellTags->"Mathematical models"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(exponentialGrowthModel[P0_, \ g_, \ t_]\ := \ P0\ Exp[g\ t];\)\ \), "\n", \(\(logisticGrowthModel[P0_, \ C_, \ g_, \ t_]\ := \ C\ /\ \((1 + \((\((C - P0)\)/P0)\) Exp[\(-g\)\ t])\);\)\), "\n", \(\(Show[\[IndentingNewLine]Plot[ exponentialGrowthModel[3.9, 0.0203, t - 1790], \ {t, \ 1790, \ 2020}, AxesOrigin \[Rule] {1790, 0}, AxesLabel \[Rule] {"\", \ "\"}, PlotStyle \[Rule] {Hue[0], Thickness[0.005]}, DisplayFunction \[Rule] Identity], \[IndentingNewLine]Plot[ logisticGrowthModel[3.9, 350, 0.0278, t - 1790], {t, \ 1790, \ 2020}, PlotStyle \[Rule] {Hue[ .6], Thickness[0.005]}, DisplayFunction \[Rule] 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0?ooo`8000001P3oool00`000000oooo0?ooo`0C0?ooo`030000003oool0oooo0?l0ooooIP3oool0 0?l0oooogP3oool00?l0oooogP3oool00?l0oooogP3oool00?l0oooogP3oool00?l0oooogP3oool0 0?l0oooogP3oool00?l0oooogP3oool00?l0oooogP3oool00?l0oooogP3oool00?l0oooogP3oool0 0?l0oooogP3oool00?l0oooogP3oool00?l0oooogP3oool00?l0oooogP3oool00?l0oooogP3oool0 0001\ \>"], ImageRangeCache->{{{0, 476.375}, {294.063, 0}} -> {1751.2, -48.0722, \ 0.62579, 1.83009}}] }, Open ]], Cell["\<\ There are obvious deficiencies in this exercise. Both models assume a \ constant growth rate, whereas the actual growth rate over the period is known \ to have varied. The logistic model, which is a good fit, assumes a constant \ carrying capacity, and suggestes that the limiting population of the U. S. is \ about 350 million. In models of animal populations in a fixed territory, \ availability of food is the main limiting factor, and clearly this has \ changed over the history of the United States. Possibly there are other \ factors, such as crowding, that will indeed limit the population as the model \ suggests, but this remains to be shown. (Failure to recognize the elasticity \ of human resource limitations was the fallacy of the economist Thomas \ Malthus, who predicted the \"crash\" of human population at a level far below \ what has actually been achieved.) The choice of a mathematical model involves a tension between several \ factors:\ \>", "Text"], Cell["\<\ The model must capture essential aspects of the process under study. \ \>", "Text", CellDingbat->"\[FilledSmallCircle]", CellMargins->{{33, Inherited}, {Inherited, Inherited}}, TextAlignment->AlignmentMarker], Cell["\<\ There must be accurate, repeatable measurement procedures for the input and \ output variables, so that the model can be effectively compared to the real \ process.\ \>", "Text", CellDingbat->"\[FilledSmallCircle]", CellMargins->{{33, Inherited}, {Inherited, Inherited}}, TextAlignment->AlignmentMarker], Cell["\<\ The model must be tractable, both mathematically and \ economically\[LongDash]i.e., the cost of modelling to a given level of \ precision must be justified by the value of the result. \ \>", "Text", CellDingbat->"\[FilledSmallCircle]", CellMargins->{{33, Inherited}, {Inherited, Inherited}}, TextAlignment->AlignmentMarker], Cell[TextData[{ "The standard advice for building a tractable model is summarized by \ Friedman (", ButtonBox["[6]", ButtonData:>"Friedman", ButtonStyle->"Hyperlink"], ", p. 27): \"Faced with a complicated problem, assume away any feature that \ is not essential to what you are trying to understand. When you are finished, \ you are left with the simplest problem whose solution will tell you what you \ want to know.\" In a certain sense this advice is circular\[LongDash]lacking \ a complete understanding of a problem, we generally can't determine what is \ essential and what is not. In practice, modeling proceeds iteratively through \ informed guesswork to trial models which are tested against empirical reality \ and used to produce more refined models. In spite of the apparently haphazard \ nature of this process, in the hands of creative scientists it has produced \ remarkable results. The physicist Eugene Wigner famously remarked on \"the \ unreasonable effectiveness of mathematics\" ", ButtonBox["[25]", ButtonData:>"Wigner", ButtonStyle->"Hyperlink"], " for modeling processes in the real world, and even in disciplines less \ precise than physics, we often can assume away a surprising amount of detail \ and still obtain useful answers.\n\nIn the remainder of this paper, the \ mapping of real-world situations to mathematical models is dealt with only in \ a cursory way. The primary focus is on the purely mathematical aspect of the \ models, and how Mathematica can be used to solve them." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Relibility modeling", "Subsubsection"], Cell[TextData[{ "Reliability analysis is a apecific example of mathematical modeling, which \ mathematically has much in common with other disciplines. The exponential \ model above, for example, shows up in modified form (with a negative \ exponent) in reliability analysis: f(t) = \[Lambda]", Cell[BoxData[ \(TraditionalForm\`e\^\(-\[Lambda]t\)\)]], " , where \[Lambda] is an empirically determined parameter, gives the \ probability density of failure at time 0 < t < \[Infinity] for a certain \ class of components (those whose reliability does not change with age). \n\n\ Reliability analysis starts with entities that can ", StyleBox["fail", FontSlant->"Italic"], ". These entities may or may not be repairable. The smallest entity under \ consideration is a ", StyleBox["component", FontSlant->"Italic"], ". Use of this term usually implies we are analyzing a ", StyleBox["system", FontSlant->"Italic"], " composed of components. Where a model deals with an entity as a whole, \ without regard for whether it is a system or component, it is called a ", StyleBox["unit ", FontSlant->"Italic"], "or ", StyleBox["item", FontSlant->"Italic"], ". The ", StyleBox["reliability", FontSlant->"Italic"], " of an item is the probability that it will perform its specified function \ for a specified period of time under specified environmental conditions. For \ example, a light bulb may be specified to produce a certain output in lumens, \ and is considered failed if the output is reduced by more than 10%; the \ manufacturer indicates a 0.99 probability that it will operate for 1,000 \ hours, under specified conditions of ambient temperature and vibration. The \ remainder of this paper assumes that the various conditions are given, and \ emphasizes the mathematical properties of the reliability function (described \ below).\n\nMost of reliability theory is based on the idea that quantities of \ interest are continuous random variables taking time values, with ", StyleBox["PDF", FontSlant->"Italic"], ", or probability density functions, p(t) satisfying p(t) \[GreaterEqual] \ 0, and ", Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{\(\[Integral]\_0\%\[Infinity]\), RowBox[{ RowBox[{ StyleBox["p", FontSlant->"Plain"], "(", StyleBox["t", FontSlant->"Plain"], ")"}], " ", "dt"}]}]}], TraditionalForm]]], " = 1. It is conventional to use uppercase letters for the general random \ variables (for example, T), and the corresponding lowercase letters for \ specific values of the random variable (for example, t). To be really \ precise, we consider a set E of events, with T being a measurable function T: \ E\[Rule]\[DoubleStruckCapitalR], so that T(e \[Element] E) = t; in applied \ work, however, we just think of T as being somehow representative of a \ repeatable process, and t being the value of a particular observation of that \ process. The domain of t is normally (0, \[Infinity] ). The ", StyleBox["CDF", FontSlant->"Italic"], " (cumulative distribution function) corresponding to p(t) is P(t) = \ Probability(T \[LessEqual] t) = ", Cell[BoxData[ FormBox[ RowBox[{ SubsuperscriptBox["\[Integral]", "0", StyleBox["t", FontSlant->"Plain"]], RowBox[{ RowBox[{ StyleBox["p", FontSlant->"Plain"], "(", StyleBox["t", FontSlant->"Plain"], ")"}], " ", "dt"}]}], TraditionalForm]]], ". Either the PDF or CDF characterizes the random variable's probability \ distribution. In applications, the CDF is actually more fundamental than the \ PDF, since P(T \[LessEqual] t) is observable, whereas \"probability density \ at a point\" is not, but rather is derived as ", Cell[BoxData[ FormBox[ FractionBox[ StyleBox["d", FontSlant->"Plain"], "dt"], TraditionalForm]]], "P(t).\n\nThe ", StyleBox["failure density function", FontSlant->"Italic"], ", f(t), is a PDF giving the density of the probability of failure of a \ unit. The corresponding CDF, F(t), gives the probability that failure occurs \ within a certain \"lifetime\": Probability(unit lifetime \[LessEqual] t) = \ F(t) = ", Cell[BoxData[ FormBox[ RowBox[{ SubsuperscriptBox["\[Integral]", "0", StyleBox["t", FontSlant->"Plain"]], RowBox[{ RowBox[{ StyleBox["f", FontSlant->"Plain"], "(", StyleBox["t", FontSlant->"Plain"], ")"}], " ", "dt"}]}], TraditionalForm]]], ". For repairable units, \"lifetime\" is replaced by \"time to failure\". \ The mean lifetime, or ", StyleBox["mean time to failure", FontSlant->"Italic"], " (MTTF), or ", StyleBox["mean time between failures", FontSlant->"Italic"], " (MTBF) for repairable units, is given by ", Cell[BoxData[ FormBox[ RowBox[{\(\[Integral]\_0\%\[Infinity]\), RowBox[{ StyleBox["t", FontSlant->"Plain"], StyleBox[" ", FontSlant->"Plain"], RowBox[{ StyleBox["f", FontSlant->"Plain"], "(", StyleBox["t", FontSlant->"Plain"], ")"}], " ", "dt"}]}], TraditionalForm]]], ". The negative exponential density above is one example of many that \ might be used as a failure density function. In principle, any probability \ distribution that is zero outside the interval (0, \[Infinity] ) might be \ used for modeling failures. In practice, even distributions with a finite \ probability of negative values, such as the normal, are sometimes used, as \ long as the parameters of the distribution are such that the probability of \ a negative value is sufficiently small. \n\nMore frequently used than the \ failure CDF is the ", StyleBox["reliability", FontSlant->"Italic"], " or ", StyleBox["survivor", FontSlant->"Italic"], " function:\n\tR(t) = (probability that the unit lifetime \[GreaterEqual] \ t) = 1 - F(t).\nBecause the total probability is 1,\n\tR(t) = 1 - F(t) = 1 - \ ", Cell[BoxData[ FormBox[ UnderoverscriptBox["\[Integral]", "0", StyleBox["t", FontSlant->"Plain"]], TraditionalForm]]], "f(t)dt = ", Cell[BoxData[ FormBox[ UnderoverscriptBox["\[Integral]", StyleBox["t", FontSlant->"Plain"], StyleBox["\[Infinity]", FontSlant->"Plain"]], TraditionalForm]]], "f(t)dt.\n\nTaking as an example the negative exponential distribution with \ parameter 0.5, this compares f(t) (in blue), F(t) (in green), and R(t) (in \ 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point in time. This is also called the \"instantaneous \ age-specific failure rate\" or simply the \"failure rate\". \n\nThis \ conditional rate is based on the general concept of conditional probability: \ the (unconditional) probability that events A and B both occur is the product \ of the probability that A occurs, by the probability that B occurs, given \ that A has occurred (the conditional probability of B given A). In symbols:\n\ \tP(A\[Intersection]B) = P(A) \[CenterDot] P(B|A). \nThus the conditional \ probability is expressed as\n\n\tP(B|A) = ", Cell[BoxData[ FormBox[ StyleBox[ FractionBox[ RowBox[{ StyleBox["P", FontSize->12, FontSlant->"Plain"], StyleBox["(", FontSize->12, FontSlant->"Italic"], StyleBox[ RowBox[{ StyleBox["A", FontSlant->"Plain"], StyleBox["\[Intersection]", FontSlant->"Italic"], StyleBox["B", FontSlant->"Plain"]}], FontSize->12], StyleBox[")", FontSize->12, FontSlant->"Italic"]}], RowBox[{ StyleBox["P", FontSize->12, FontSlant->"Plain"], StyleBox["(", FontSize->12, FontSlant->"Italic"], StyleBox["A", FontSize->12, FontSlant->"Plain"], StyleBox[")", FontSize->12, FontSlant->"Italic"]}]], FontSize->18], TraditionalForm]]], ". \t\t\t(1)\n\nTo derive the hazard function, consider the \ probability that failure occurs between times t and t + \[CapitalDelta]t:\n\n\ \tP(t \[LessEqual] T \[LessEqual] t + \[CapitalDelta]t) = ", Cell[BoxData[ FormBox[ UnderoverscriptBox["\[Integral]", StyleBox["t", FontSlant->"Plain"], RowBox[{ StyleBox["t", FontSlant->"Plain"], "+", "\[CapitalDelta]t"}]], TraditionalForm]]], "f(\[Tau])d\[Tau] = F( t + \[CapitalDelta]t) ", StyleBox["-", FontSize->16], " F(t) = (1 - F(t)) ", StyleBox["-", FontSize->16], " (1 ", StyleBox["-", FontSize->16], " F( t + \[CapitalDelta]t)) = R( t + \[CapitalDelta]t) ", StyleBox["-", FontSize->16], " R(t).\n\t\nUsing (1), conditioning the probability on the fact that \ failure has not occurred before t, i.e., that the unit is still a survivor at \ time t,\n\n \tP(t \[LessEqual] T \[LessEqual] t + \[CapitalDelta]t | T > t) = \ ", Cell[BoxData[ FormBox[ StyleBox[ FractionBox[ RowBox[{ StyleBox["P", FontSize->12, FontSlant->"Plain"], StyleBox["(", FontSize->12], StyleBox[ RowBox[{ StyleBox["t", FontSlant->"Plain"], "\[LessEqual]", StyleBox["T", FontSlant->"Plain"], "\[LessEqual]", RowBox[{ StyleBox["t", FontSlant->"Plain"], "+", StyleBox[ RowBox[{"\[CapitalDelta]", StyleBox["t", FontSlant->"Plain"]}]]}]}], FontSize->12], StyleBox[")", FontSize->12]}], RowBox[{ StyleBox["P", FontSize->12, FontSlant->"Plain"], StyleBox["(", FontSize->12], StyleBox[ RowBox[{ StyleBox["T", FontSlant->"Plain"], ">", StyleBox["t", FontSlant->"Plain"]}], FontSize->12], StyleBox[")", FontSize->12]}]], FontSize->18], TraditionalForm]]], "= ", Cell[BoxData[ FormBox[ StyleBox[ FractionBox[ RowBox[{ RowBox[{ StyleBox["R", FontSize->12, FontSlant->"Plain"], StyleBox["(", FontSize->12], StyleBox["t", FontSize->12, FontSlant->"Plain"], StyleBox[")", FontSize->12]}], " ", "-", " ", RowBox[{ StyleBox["R", FontSize->12, FontSlant->"Plain"], StyleBox["(", FontSize->12], RowBox[{ StyleBox["t", FontSize->12, FontSlant->"Plain"], StyleBox["+", FontSize->12], StyleBox["\[CapitalDelta]t", FontSize->12]}], StyleBox[")", FontSize->12]}]}], RowBox[{ StyleBox["R", FontSize->12, FontSlant->"Plain"], StyleBox["(", FontSize->12], StyleBox["t", FontSize->12, FontSlant->"Plain"], StyleBox[")", FontSize->12]}]], FontSize->18], TraditionalForm]]] }], "Text"], Cell[TextData[{ "\n Dividing by \[CapitalDelta]t gives the average rate of failure over the \ interval [t, \[CapitalDelta]t]. Finally, the hazard rate (instantaneous rate \ of failure) is defined as\n \n h(t) = ", Cell[BoxData[ \(TraditionalForm\`\(\(lim\)\(\ \)\)\+\(\[CapitalDelta]t \[Rule] \ 0\)\)]], " ", Cell[BoxData[ FormBox[ StyleBox[ FractionBox[ RowBox[{ RowBox[{ StyleBox["R", FontSize->12, FontSlant->"Plain"], StyleBox["(", FontSize->12], StyleBox["t", FontSize->12, FontSlant->"Plain"], StyleBox[")", FontSize->12]}], " ", "-", " ", RowBox[{ StyleBox["R", FontSize->12, FontSlant->"Plain"], StyleBox["(", FontSize->12], RowBox[{ StyleBox["t", FontSize->12, FontSlant->"Plain"], StyleBox["+", FontSize->12], StyleBox["\[CapitalDelta]t", FontSize->12]}], StyleBox[")", FontSize->12]}]}], RowBox[{ RowBox[{ StyleBox["R", FontSize->12, FontSlant->"Plain"], StyleBox["(", FontSize->12], StyleBox["t", FontSize->12, FontSlant->"Plain"], StyleBox[")", FontSize->12]}], "\[CapitalDelta]t"}]], FontSize->18], TraditionalForm]]], "\n\t \n\t = ", Cell[BoxData[ FormBox[ StyleBox[ FractionBox[ RowBox[{ StyleBox[ RowBox[{"-", " ", RowBox[{ StyleBox["R", FontSlant->"Plain"], "'"}]}], FontSize->12], RowBox[{ StyleBox["(", FontSize->12], StyleBox["t", FontSize->12, FontSlant->"Plain"], StyleBox[")", FontSize->12]}]}], RowBox[{ StyleBox["R", FontSize->12, FontSlant->"Plain"], StyleBox["(", FontSize->12], StyleBox["t", FontSize->12, FontSlant->"Plain"], StyleBox[")", FontSize->12]}]], FontSize->18], TraditionalForm]]], " \n\t \n\t = ", Cell[BoxData[ FormBox[ StyleBox[ FractionBox[ RowBox[{ StyleBox["f", FontSize->12, FontSlant->"Plain"], StyleBox["(", FontSize->12], StyleBox["t", FontSize->12, FontSlant->"Plain"], StyleBox[")", FontSize->12]}], RowBox[{ StyleBox["R", FontSize->12, FontSlant->"Plain"], StyleBox["(", FontSize->12], StyleBox["t", FontSize->12, FontSlant->"Plain"], StyleBox[")", FontSize->12]}]], FontSize->18], TraditionalForm]]], " , since R'(t) = ", Cell[BoxData[ FormBox[ FractionBox[ StyleBox["d", FontSize->12, FontSlant->"Plain"], StyleBox["dt", FontSize->12, FontSlant->"Plain"]], TraditionalForm]], FontSize->18], "(1 ", StyleBox["-", FontSize->16], " F(t)) = ", StyleBox["-", FontSize->16], " f(t)\t\t\t(2)" }], "Text", CellTags->"Equation1"], Cell[TextData[{ "Note that the hazard function is not a PDF, since ", Cell[BoxData[ FormBox[ RowBox[{\(\[Integral]\_0\%\[Infinity]\), RowBox[{ RowBox[{ StyleBox["h", FontSlant->"Plain"], "(", StyleBox["t", FontSlant->"Plain"], ")"}], " ", "dt"}]}], TraditionalForm]]], "\[NotEqual] 1; it can be shown that R(t)\[Rule]0 fast enough as t\[Rule]\ \[Infinity] so that ", Cell[BoxData[ FormBox[ RowBox[{\(\[Integral]\_0\%\[Infinity]\), RowBox[{ RowBox[{ StyleBox["h", FontSlant->"Plain"], "(", StyleBox["t", FontSlant->"Plain"], ")"}], " ", "dt"}]}], TraditionalForm]]], "is unbounded" }], "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Empirical models of reliability", "Section"], Cell[TextData[{ "An empirical model is a distribution characterization, e.g., a failure \ CDF, based in whole or in part on measurement data. The distribution in turn \ is used to make predictions about units similar to the one(s) measured, or \ about the measured unit(s) outside the range of measurement. \n\nIt is \ possible to construct a \"pure empirical\" model, i.e., to construct a \ distribution completely from smoothed data. This is rarely done, since a \ known distribution provides more information, and is more convenient to work \ with. Typically, a known distribution type is chosen based on some \ combination of theoretical considerations, formal analysis of the measurement \ data, and visual inspection of the data. Having chosen a distribution type, \ its parameters are estimated based on the data, and a ", StyleBox["goodness of fit", FontSlant->"Italic"], " test is used to assess how closely it models the measurement data. This \ procedure may be iterated with different distibution types until one is found \ that offers a \"sufficiently good\" fit. \n\nEmpirical models may be used \ alone to model simple situations. For example, suppose a manufacturer of ball \ bearings wishes to define a warranty period that will minimize warranty \ replacement costs. This might be taken to mean a period, starting when the \ bearing is new, during which less than 1% of the bearings will fail. Given an \ empirical model, say a failure CDF derived from testing a sample of bearings, \ the reliability function R(t) can be determined. The maximum warranty period \ is then the maximum value of t such that R(t) \[GreaterEqual] 0.99.\n\n\ Empirical models may also be used as components of other models for more \ complex situations. For example, having derived empirical models for the \ failure of individual components comprising a system, a failure rate for the \ system may be derived\[LongDash]in some cases by a closed-form computation, \ but more often using a Monte Carlo simulation." }], "Text", CellTags->"Empirical models of reliability"], Cell[CellGroupData[{ Cell["Non-Parametric estimation of f(t), F(t), R(t) and h(t)", "Subsubsection", CellTags->"NonParametricEstimates"], Cell[TextData[{ "A common starting point is the development of non-parametric estimates of \ the important reliability functions, i.e., estimates that are free of any \ assumption about the true distribution of the data. This is obviously useful \ in constructing a pure empirical model, but also helps to discriminate \ between different theoretical distributions. For example, the exponential \ distribution has h(t) = constant, and is the ", StyleBox["only", FontSlant->"Italic"], " distribution with this property.\n\nRecall that F(t) is the probability \ that failure occurs in time \[LessEqual] t. All the CDFs commonly used in \ reliability theory are continuous, and strictly increasing in the interval \ [0,\[Infinity]). Thus F(t) is invertible, and for 0 \[LessEqual] \[Rho] \ \[LessEqual] 1, ", Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["F", FontSlant->"Plain"], \(-1\)], TraditionalForm]]], "(\[Rho]) = \[Tau] means that \[Tau] is the time such that P(t \[LessEqual] \ \[Tau]) = \[Rho]. \n\nNow, take a partition (0, ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["t", FontSlant->"Plain"], "1"], TraditionalForm]]], ", . . ., ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["t", FontSlant->"Plain"], "i"], TraditionalForm]]], ", . . ., ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["t", FontSlant->"Plain"], "n"], TraditionalForm]]], ", \[Infinity]) of the range of F such that each subinterval ", Cell[BoxData[ \(TraditionalForm\`\[CapitalDelta]\_i\)]], "has the same probability, i.e., F(t + ", Cell[BoxData[ \(TraditionalForm\`\[CapitalDelta]\_i\)]], ") - F(t) is the same for all ", Cell[BoxData[ \(TraditionalForm\`\[CapitalDelta]\_i\)]], ". Then F maps this partition to a partition of 0 \[LessEqual] p \ \[LessEqual] 1 whose subintervals are equal. Conversely, ", Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["F", FontSlant->"Plain"], \(-1\)], TraditionalForm]]], " maps any partition of 0 \[LessEqual] p \[LessEqual] 1 with equal \ subintervals to an equiprobable partition of the range. \n\nThe mapping \ described above is the basis for constructing an empirical CDF from observed \ data points. Suppose we test a sample of ", StyleBox["n", FontSlant->"Italic"], " units until all have failed, giving ", StyleBox["n", FontSlant->"Italic"], " observed failure times 0 \[LessEqual] ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["t", FontSlant->"Plain"], "1"], TraditionalForm]]], "\[LessEqual] . . .\[LessEqual] ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["t", FontSlant->"Plain"], "i"], TraditionalForm]]], "\[LessEqual] . . .\[LessEqual] ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["t", FontSlant->"Plain"], "n"], TraditionalForm]]], ". This can be taken as a mapping from the ", StyleBox["order statistics", FontSlant->"Italic"], " 1, . . ., ", StyleBox["i", FontSlant->"Italic"], ", . . ., ", StyleBox["n", FontSlant->"Italic"], " to t. Assuming that each ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["t", FontSlant->"Plain"], "i"], TraditionalForm]]], " is drawn from an equally-likely subinterval of t, this induces a \ partition of p which we can take to be a normalization of the order \ statistics, 0 \[LessEqual] 1/n \[LessEqual] . . . i/n \[LessEqual] 1. This is \ shown in the figure below:" }], "Text", CellTags->"InverseCDF"], Cell[TextData[{ Cell[GraphicsData["Metafile", "\<\ CF5dJ6E]HGAYHf4PEfU^I6mgLb15CDHPAVmbKF5d0@000TRh0@0006`000000000000002@2001M0@00 00000000003k@P00fB`00215CDH00040^4P20?8=000>0000000000000000000000@0000300100@00 l0000000000000000000003R1020Z@<0AP0001`D0@0@5040AdA9@`4008000`00oNOKX@00003h4`40 0@090003o8T000l0B0000000500002H63`0N0?ooool401@0001GKg9T3P1=JF=bKg=_IW@PEfmbI0D0 000;0P0000050000309<1NT770000?/240070000002l0P0000010P8RDgUcM6E]000000X000040000 003ooooo0@000000<0040000;@4000D000020@40000500002@80000270000?/2c_l00000002@0@00 0004@00BE6U]IGo`0000000901000000A0019DJFeULb1>IGLPDVm]HFh03EGeMaIEmGL10000000`00@0000]0@D0 1@0000T200000PD0000;0P00000=0000P1i02005P0400009`7oo`L0003l0P00 oon_0000100002d12@040000;@4300@0000]0@T02@0001d68@3`01L4[@HI05D0100002d12@040000 ;@4600@0003`0@T0100002d11`070000o0800?ooo`0000@0000]0@T0100002d10`040000;@4900T0 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"Graphics", GeneratedCell->False, CellAutoOverwrite->False, ImageSize->{431, 274.375}, ImageMargins->{{30, 0}, {0, 8}}, ImageRegion->{{0, 1}, {0, 1}}], "\n\t\t\t\t\t\tFigure 1." }], "Text", CellTags->"Figure3"], Cell[TextData[{ "The blue line is the empirical CDF, which has a step at each ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["t", FontSlant->"Plain"], "i"], TraditionalForm]]], "; the step is from (", StyleBox["i", FontSlant->"Italic"], "-1)/", StyleBox["n", FontSlant->"Italic"], " to ", StyleBox["i", FontSlant->"Italic"], "/", StyleBox["n", FontSlant->"Italic"], ". The red line is a continuous CDF fitted to the empirical CDF; ways of \ doing the fitting are described below.\n\nThe partition 0 \[LessEqual] 1/n \ \[LessEqual] . . . i/n \[LessEqual] 1 is biased, since it assumes that P(t < \ ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["t", FontSlant->"Plain"], "1"], TraditionalForm]]], ") = 0, and that ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["t", FontSlant->"Plain"], "n"], TraditionalForm]]], " is the maximum value for t. There are various methods of compensating for \ this bias; the preferred method for asymmetrical distributions (where the \ mean and median are not equal) is ", StyleBox["Benard's formula", FontSlant->"Italic"], ": \n\n\t", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"P", "(", RowBox[{"t", " ", "\[LessEqual]", " ", FormBox[ SubscriptBox[ StyleBox["t", FontSlant->"Plain"], "i"], "TraditionalForm"]}], ")"}], " ", "\[TildeFullEqual]"}], TraditionalForm]]], " ", Cell[BoxData[ FormBox[ OverscriptBox[ StyleBox["P", FontSlant->"Plain"], "^"], TraditionalForm]]], "(t \[LessEqual] ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["t", FontSlant->"Plain"], "i"], TraditionalForm]]], ") = ", Cell[BoxData[ FormBox[ OverscriptBox[ StyleBox["F", FontSlant->"Plain"], "^"], TraditionalForm]]], "(", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["t", FontSlant->"Plain"], "i"], TraditionalForm]]], ") = ", Cell[BoxData[ \(TraditionalForm\`\(i\ - \ 0.3\)\/\(n\ + \ 0.4\)\)], FontSize->16], ".\n\t\nThis is the best approximation for distributions commonly \ encountered in reliability work. (For a comparison of Benard with other \ methods in the Weibull case, see ", ButtonBox["[1]", ButtonData:>"Abernathy", ButtonStyle->"Hyperlink"], ".)\n\nAs an example, take the data set ", StyleBox["leukemiaRemission", FontWeight->"Bold"], ", which contains times (in months) to spontaneous remission of leukemia in \ a control group for a drug study (from ", ButtonBox["[13]", ButtonData:>"Leemis", ButtonStyle->"Hyperlink"], "). Spontaneous remission of leukemia is apparently common; \"failure\" \ here is thus a failure of the carcinoma to continue its uncontrolled growth. \ The form of the dataset is {. . ., {", StyleBox["i", FontSlant->"Italic"], ", ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["t", FontSlant->"Plain"], "i"], TraditionalForm]]], "}, . . .}.\n\nBefore operating on a set of observation data, taking the \ data as a discrete function of ", StyleBox["t", FontSlant->"Italic"], ", it must be single-valued. The ", StyleBox["leukemiaRemission", FontWeight->"Bold"], " dataset is not\[LongDash]some of the remission times occur more than \ once:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(leukemiaRemission\)], "Input"], Cell[BoxData[ \({{1, 1}, {2, 1}, {3, 2}, {4, 2}, {5, 3}, {6, 4}, {7, 4}, {8, 5}, {9, 5}, {10, 8}, {11, 8}, {12, 8}, {13, 8}, {14, 11}, {15, 11}, {16, 12}, {17, 12}, {18, 15}, {19, 17}, {20, 22}, {21, 23}}\)], "Output"] }, Open ]], Cell[TextData[{ "This means that in the kind of step function shown in ", ButtonBox["Figure 1", ButtonData:>"Figure3", ButtonStyle->"Hyperlink"], ", there is a step greater than one at certain values\[LongDash]e.g.,the \ step from t = 5 to t = 8 is from 9 to 13. The data is adjusted for this by \ aggregating the function pairs {10,8}, {11,8}, {12,8}, and {13,8}, leaving \ only {13,8}. In general,the aggregation leaves only the observation with the \ largest ", StyleBox["i", FontSlant->"Italic"], ", for a given t. It also swaps the parameters so each pair is {", StyleBox["t", FontSlant->"Italic"], ", ", StyleBox["i", FontSlant->"Italic"], "}:" }], "Text"], Cell[BoxData[ \(aggregate[ data_?ListQ] := \[IndentingNewLine]Module[{newData = {}, n}, \[IndentingNewLine]n = Length[data]; \[IndentingNewLine]For[ i = 1, i < n, \(i++\), \[IndentingNewLine]While[ i < n\ && \ \(data[\([i]\)]\)[\([2]\)] \[Equal] \(data[\([i + 1]\)]\)[\([2]\)], \(i++\)]; \ \[IndentingNewLine]newData = Append[newData, data[\([i]\)]]; \[IndentingNewLine]If[ i \[Equal] n - 1, newData = Append[newData, data[\([n]\)]]]]; \[IndentingNewLine]Map[{Last[#], First[#]} &, newData]]\)], "Input", InitializationCell->True], Cell[TextData[{ "So the dataset,when aggregated, represents a single-valued function from \ ", StyleBox["t", FontSlant->"Italic"], " to the order statistics 1, 2, . . . ", StyleBox["n", FontSlant->"Italic"], ":" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(aggregate[leukemiaRemission]\)], "Input"], Cell[BoxData[ \({{1, 2}, {2, 4}, {3, 5}, {4, 7}, {5, 9}, {8, 13}, {11, 15}, {12, 17}, {15, 18}, {17, 19}, {22, 20}, {23, 21}}\)], "Output"] }, Open ]], Cell[TextData[{ "The aggregated data is used in the following ", "Mathematica", " function for ", Cell[BoxData[ FormBox[ OverscriptBox[ StyleBox["F", FontSlant->"Plain"], "^"], TraditionalForm]]], "(t), the empirical CDF, which takes as argument the list of failure time \ observations. (Note that ", StyleBox["n", FontSlant->"Italic"], " is still the number of unaggregated observations.)" }], "Text"], Cell[BoxData[{ \(\(empiricalCDF[data_?ListQ, opts___?OptionQ] \ := \[IndentingNewLine]Module[\[IndentingNewLine]{plotQ := Plot /. Flatten[{{opts}, Options[empiricalCDF]}], \[IndentingNewLine]otherFuns := PlotWith /. Flatten[{{opts}, Options[stepPlot]}], \[IndentingNewLine]n, pFormula, distTable}, \[IndentingNewLine]n = Length[data]; \[IndentingNewLine]pFormula = \((# - 0.3)\)/\((n + 0.4)\) &; \[IndentingNewLine]distTable = Map[{First[#], pFormula[Last[#]]} &, aggregate[data]]; \[IndentingNewLine]If[plotQ, stepPlot[distTable, PlotWith \[Rule] otherFuns]; distTable, distTable]];\)\), "\[IndentingNewLine]", \(\(Options[empiricalCDF] = {Plot \[Rule] False, PlotWith \[Rule] {}};\)\)}], "Input", InitializationCell->True], Cell[TextData[{ " The empirical CDF table is always returned. If ", StyleBox["Plot\[Rule]True", FontWeight->"Bold"], ", it also plots the CDF table; ", StyleBox["PlotWith\[Rule]{", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`f\_\(\(1\)\(,\)\)\)], FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`f\_2\)], FontWeight->"Bold"], StyleBox[", . . .}", FontWeight->"Bold"], ", gives a list of other functions to be plotted, for example if a fitted \ function is to be compared with the empirical CDF (see ", ButtonBox["Figure 3", ButtonData:>"Fig3EmpCdf", ButtonStyle->"Hyperlink"], ")." }], "Text"], Cell[TextData[{ "Having constructed the empirical CDF, it can be used directly to generate \ sample failure times; this is illustrated below in the section below on ", StyleBox[ButtonBox["Simulation or Monte Carlo Methods", ButtonData:>"Simulation or Monte Carlo methods", ButtonStyle->"Hyperlink"], FontFamily->"Times New Roman", FontWeight->"Bold"], ". Here we look at deriving other functions from the empirical data\ \[LongDash]particularly the empirical hazard function, ", Cell[BoxData[ FormBox[ OverscriptBox[ StyleBox["h", FontSlant->"Plain"], "^"], TraditionalForm]]], "(t), which is helpful in selecting a distribution type for fitting to the \ empirical data. \n\nTo obtain ", Cell[BoxData[ FormBox[ OverscriptBox[ StyleBox["h", FontSlant->"Plain"], "^"], TraditionalForm]]], "(t), we start with ", Cell[BoxData[ FormBox[ OverscriptBox[ StyleBox["f", FontSlant->"Plain"], "^"], TraditionalForm]]], "(t), the estimated PDF:\n\n\t ", Cell[BoxData[ FormBox[ OverscriptBox[ StyleBox["f", FontSlant->"Plain"], "^"], TraditionalForm]]], "(", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["t", FontSlant->"Plain"], "i"], TraditionalForm]]], ") = ", Cell[BoxData[ FormBox[ FractionBox[ StyleBox["d", FontSlant->"Plain"], StyleBox["dt", FontSlant->"Plain"]], TraditionalForm]]], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ FormBox[ OverscriptBox[ StyleBox["F", FontSlant->"Plain"], "^"], "TraditionalForm"], "(", FormBox[ SubscriptBox[ StyleBox["t", FontSlant->"Plain"], "i"], "TraditionalForm"], ")"}], " ", "\[TildeFullEqual]"}], TraditionalForm]]], " ", Cell[BoxData[ FormBox[ StyleBox[ FractionBox[ RowBox[{ RowBox[{ OverscriptBox[ StyleBox["F", FontSlant->"Plain"], "^"], "(", FormBox[ SubscriptBox[ StyleBox["t", FontSlant->"Plain"], \(i\ + \ 1\)], "TraditionalForm"], ")"}], " ", "-", " ", RowBox[{ OverscriptBox[ StyleBox["F", FontSlant->"Plain"], "^"], "(", FormBox[ SubscriptBox[ StyleBox["t", FontSlant->"Plain"], "i"], "TraditionalForm"], ")"}]}], RowBox[{ SubscriptBox[ StyleBox["t", FontSlant->"Plain"], \(i\ + \ 1\)], " ", "-", " ", SubscriptBox[ StyleBox["t", FontSlant->"Plain"], "i"]}]], FontSize->16], TraditionalForm]]], " = ", Cell[BoxData[ FormBox[ StyleBox[ FractionBox[\(\(i\ + \ 1\ - \ 0.3\)\/\(n\ + \ 0.4\)\ - \ \(i\ \ - \ 0.3\)\/\(n\ + \ 0.4\)\), RowBox[{ SubscriptBox[ StyleBox["t", FontSlant->"Plain"], \(i\ + \ 1\)], " ", "-", " ", SubscriptBox[ StyleBox["t", FontSlant->"Plain"], "i"]}]], FontSize->16], TraditionalForm]]], " = ", Cell[BoxData[ FormBox[ StyleBox[ FractionBox[\(\(\ \)\(1\/\(n\ + \ 0.4\)\)\), RowBox[{ SubscriptBox[ StyleBox["t", FontSlant->"Plain"], \(i\ + \ 1\)], " ", "-", " ", SubscriptBox[ StyleBox["t", FontSlant->"Plain"], "i"]}]], FontSize->16], TraditionalForm]]], " =", StyleBox[" ", FontSize->18], Cell[BoxData[ FormBox[ FractionBox["1", RowBox[{\((n\ + \ 0.4)\), RowBox[{"(", RowBox[{ SubscriptBox[ StyleBox["t", FontSlant->"Plain"], \(i\ + \ 1\)], " ", "-", " ", SubscriptBox[ StyleBox["t", FontSlant->"Plain"], "i"]}], ")"}]}]], TraditionalForm]], FontSize->16], StyleBox[" \n \n", FontSize->16], "The reliability function is", StyleBox[" ", FontSize->16], Cell[BoxData[ FormBox[ OverscriptBox[ StyleBox["R", FontSlant->"Plain"], "^"], TraditionalForm]]], "(", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["t", FontSlant->"Plain"], "i"], TraditionalForm]]], ") = 1 - ", Cell[BoxData[ FormBox[ OverscriptBox[ StyleBox["F", FontSlant->"Plain"], "^"], TraditionalForm]]], "(", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["t", FontSlant->"Plain"], "i"], TraditionalForm]]], "), so from ", ButtonBox["(2)", ButtonData:>"Equation1", ButtonStyle->"Hyperlink"], ",\n\n\t", Cell[BoxData[ FormBox[ OverscriptBox[ StyleBox["h", FontSlant->"Plain"], "^"], TraditionalForm]]], "(", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["t", FontSlant->"Plain"], "i"], TraditionalForm]]], ") = ", Cell[BoxData[ FormBox[ StyleBox[ FractionBox[ RowBox[{ OverscriptBox[ StyleBox["f", FontSlant->"Plain"], "^"], "(", FormBox[ SubscriptBox[ StyleBox["t", FontSlant->"Plain"], "i"], "TraditionalForm"], ")"}], RowBox[{ OverscriptBox[ StyleBox["R", FontSlant->"Plain"], "^"], "(", FormBox[ SubscriptBox[ StyleBox["t", FontSlant->"Plain"], "i"], "TraditionalForm"], ")"}]], FontSize->14], TraditionalForm]]], " = ", Cell[BoxData[ FormBox[ StyleBox[ FractionBox[ RowBox[{ OverscriptBox[ StyleBox["f", FontSlant->"Plain"], "^"], "(", FormBox[ SubscriptBox[ StyleBox["t", FontSlant->"Plain"], "i"], "TraditionalForm"], ")"}], RowBox[{"1", " ", "-", " ", RowBox[{ OverscriptBox[ StyleBox["F", FontSlant->"Plain"], "^"], "(", FormBox[ SubscriptBox[ StyleBox["t", FontSlant->"Plain"], "i"], "TraditionalForm"], ")"}]}]], FontSize->14], TraditionalForm]]], " = ", Cell[BoxData[ FormBox[ StyleBox[ FractionBox[ FractionBox["1", RowBox[{\((n\ + \ 0.4)\), RowBox[{"(", RowBox[{ SubscriptBox[ StyleBox["t", FontSlant->"Plain"], \(i\ + \ 1\)], " ", "-", " ", SubscriptBox[ StyleBox["t", FontSlant->"Plain"], "i"]}], ")"}]}]], \(\(n\ - \ i\ + \ \ 0.7\)\/\(n\ + \ 0.4\)\)], FontSize->14], TraditionalForm]], FontSize->16], " =", StyleBox[" ", FontSize->18], Cell[BoxData[ FormBox[ FractionBox["1", RowBox[{\((n\ - \ i\ + \ 0.7)\), RowBox[{"(", RowBox[{ SubscriptBox[ StyleBox["t", FontSlant->"Plain"], \(i\ + \ 1\)], " ", "-", " ", SubscriptBox[ StyleBox["t", FontSlant->"Plain"], "i"]}], ")"}]}]], TraditionalForm]], FontSize->16], "\n\t\nThis last formula can be applied directly to the observation data to \ derive an empirical ", Cell[BoxData[ FormBox[ OverscriptBox[ StyleBox["h", FontSlant->"Plain"], "^"], TraditionalForm]]], "(t):" }], "Text"], Cell[BoxData[{ \(\(empiricalHazard[data_?ListQ, opts___?OptionQ] \ := \[IndentingNewLine]Module[\[IndentingNewLine]{plotQ := Plot /. 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Open ]], Cell["\t\t\t\tFigure 2.", "Text"], Cell["\<\ The next step is to use this information to determine a distribution for \ fitting the data.\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Example: The exponential distribution", "Subsubsection", CellTags->"ExampleExponential"], Cell[TextData[{ "If the hazard rate is found to be constant, h(t) = ", Cell[BoxData[ FormBox[ StyleBox[ FractionBox[ RowBox[{ RowBox[{"-", " ", RowBox[{ StyleBox["R", FontSlant->"Plain"], "'"}]}], RowBox[{"(", StyleBox["t", FontSlant->"Plain"], ")"}]}], RowBox[{ StyleBox["R", FontSlant->"Plain"], "(", StyleBox["t", FontSlant->"Plain"], ")"}]], FontSize->16], TraditionalForm]]], "= \[Lambda] , then R'(t) = ", StyleBox["-", FontSize->16], " \[Lambda]R(t) , thus R(t) = ", Cell[BoxData[ \(TraditionalForm\`e\^\(-\[Lambda]t\)\)]], ". Given the constraint R(0) = 1, this solution is unique, so the failure \ distribution is exponential. In Mathematica terms:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(DSolve[{\(R'\)[t] \[Equal] \(-\[Lambda]\)\ R[t], R[0] \[Equal] 1}, R[t], t]\)], "Input"], Cell[BoxData[ \({{R[t] \[Rule] \[ExponentialE]\^\(\(-t\)\ \[Lambda]\)}}\)], "Output"] }, Open ]], Cell[TextData[{ "The theoretical basis for the exponential distribution is that the unit \ has no \"memory\" of its prior life; the conditional reliability function, \ given that a unit has survived until time \[Tau], is the same as the \ reliability function starting at time 0:\n\n\tP(T > \[Tau] + t | T > \[Tau]) \ = ", Cell[BoxData[ FormBox[ StyleBox[ FractionBox[ RowBox[{ StyleBox["R", FontSize->12, FontSlant->"Plain"], StyleBox["(", FontSize->12], StyleBox[ RowBox[{"\[Tau]", " ", "+", " ", StyleBox["t", FontSlant->"Plain"]}], FontSize->12], StyleBox[")", FontSize->12]}], RowBox[{ StyleBox["R", FontSize->12, FontSlant->"Plain"], StyleBox["(", FontSize->12], StyleBox["\[Tau]", FontSize->12, FontSlant->"Plain"], StyleBox[")", FontSize->12]}]], FontSize->18], TraditionalForm]]], "=", StyleBox[" ", FontSize->16], Cell[BoxData[ FormBox[ StyleBox[ FractionBox[ StyleBox[ SuperscriptBox["e", RowBox[{"-", RowBox[{"\[Lambda]", "(", RowBox[{"\[Tau]", " ", "+", " ", StyleBox["t", FontSlant->"Plain"]}], ")"}]}]], FontSize->12], StyleBox[\(e\^\(-\[Lambda]\[Tau]\)\), FontSize->12]], FontSize->20], TraditionalForm]], FontSize->24], "= ", Cell[BoxData[ \(TraditionalForm\`e\^\(-\[Lambda]t\)\)], FontSize->14], " = R(t) = P(T > t).\n\t\nEmpirically, this applies to units such as \ electronic components that do not wear out. Does it also apply to the \ leukemia situation? The empirical hazard rate in ", ButtonBox["Figure 2", ButtonData:>"Figure2", ButtonStyle->"Hyperlink"], " does not make this clear. Given that it does not seem to be definitely \ increasing or decreasing, we can look at an average of the first half of the \ hazard table:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(eh = Map[Last[#] &, empiricalHazard[leukemiaRemission]] // Flatten;\)\), "\[IndentingNewLine]", \(Mean[Take[eh, 6]]\)}], "Input"], Cell[BoxData[ \(0.049954490378611244`\)], "Output"] }, Open ]], Cell["\<\ And the last half, with the last observation dropped because it seems like an \ outlier:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Mean[Drop[Take[eh, \(-6\)], \(-1\)]]\)], "Input"], Cell[BoxData[ \(0.09353982052266126`\)], "Output"] }, Open ]], Cell[TextData[{ "These are not very close, but perhaps (in practical work) close enough to \ warrant trying a fit to an exponential distribution. First, since ", Cell[BoxData[ FormBox[ OverscriptBox[ StyleBox["R", FontSlant->"Plain"], "^"], TraditionalForm]]], "(", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["t", FontSlant->"Plain"], "i"], TraditionalForm]]], ") = 1 - ", Cell[BoxData[ FormBox[ OverscriptBox[ StyleBox["F", FontSlant->"Plain"], "^"], TraditionalForm]]], "(", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["t", FontSlant->"Plain"], "i"], TraditionalForm]]], "), " }], "Text"], Cell[BoxData[ \(empiricalReliability[data_?ListQ] := Map[{#[\([1]\)], 1 - #[\([2]\)]} &, empiricalCDF[data, Plot \[Rule] False]]\)], "Input", InitializationCell->True], Cell[TextData[{ "Given R(t) = ", Cell[BoxData[ \(TraditionalForm\`e\^\(-\[Lambda]t\)\)]], ", log(R(t)) = ", StyleBox["-", FontSize->14], " \[Lambda]t is a straight line. Mathematica's linear regression can be \ used to find the best fit to the linearized data (IncludeConstant\[Rule]False \ is used because we know that there is no constant term):" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Regress[ Map[{#[\([1]\)], Log[#[\([2]\)]]} &, empiricalReliability[leukemiaRemission]], {0, t}, t, IncludeConstant \[Rule] False]\)], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"ParameterTable", "\[Rule]", TagBox[GridBox[{ {"\<\"\"\>", "\<\"Estimate\"\>", "\<\"SE\"\>", \ "\<\"TStat\"\>", "\<\"PValue\"\>"}, {"0", "0", "0.`", "Indeterminate", "Indeterminate"}, {"t", \(-0.1252656316299529`\), "0.004933578059708327`", \(-25.39042255213828`\), "2.0594875186331334`*^-10"} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], Function[ BoxForm`e$, TableForm[ BoxForm`e$, TableHeadings -> {{0, t}, {"Estimate", "SE", "TStat", "PValue"}}]]]}], ",", \(RSquared \[Rule] 0.9620740996100836`\), ",", \(AdjustedRSquared \[Rule] 0.9582815095710919`\), ",", \(EstimatedVariance \[Rule] 0.04651410781253081`\), ",", RowBox[{"ANOVATable", "\[Rule]", InterpretationBox[GridBox[{ {"\<\"\"\>", "\<\"DF\"\>", "\<\"SumOfSq\"\>", \ "\<\"MeanSq\"\>", "\<\"FRatio\"\>", "\<\"PValue\"\>"}, {"\<\"Model\"\>", "1", "11.799329200581054`", "11.799329200581054`", "253.67205253375485`", "1.962269824407059`*^-8"}, {"\<\"Error\"\>", "10", "0.46514107812530814`", "0.04651410781253081`", "\<\"\"\>", "\<\"\"\>"}, {"\<\"Total\"\>", "11", "12.264470278706362`", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>"} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], TableForm[ {{1, 11.799329200581054, 11.799329200581054, 253.67205253375485, .19622698244070591*^-7}, { 10, .46514107812530814, .046514107812530812}, {11, 12.264470278706362}}, TableHeadings -> {{"Model", "Error", "Total"}, {"DF", "SumOfSq", "MeanSq", "FRatio", "PValue"}}]]}]}], "}"}]], "Output"] }, Open ]], Cell[TextData[{ "The ", Cell[BoxData[ FormBox[ 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00040?ooo`030000003oool0oooo0100oooo000E0?ooo`030000003oool0oooo0?40oooo203o000? 0?ooo`005@3oool200000?H0oooo00@000000?ooo`3oool0oooo1@3o000:0?ooo`005@3oool00`00 0000oooo0?ooo`3e0?ooo`030000003oool0oooo00H0oooo0`3o00070?ooo`005@3oool00`000000 oooo0?ooo`3e0?ooo`030000003oool0oooo0100oooo000E0?ooo`030000003oool0oooo0?D0oooo 300000070?ooo`005@3oool00`000000oooo0?ooo`3o0?ooo`T0oooo000E0?ooo`030000003oool0 oooo0?l0oooo2@3oool001D0oooo00<000000?ooo`3oool0o`3oool90?ooo`005@3oool00`000000 oooo0?ooo`3o0?ooo`T0oooo0000\ \>"], ImageRangeCache->{{{0, 287}, {176.938, 0}} -> {-1.9823, -0.0760769, \ 0.0926218, 0.00604012}}] }, Open ]], Cell["\t\t\t\t\tFigure 3.", "Text"], Cell["\<\ This can now be used as a model for prediction. For example, suppose we are \ running a trial on a drug that is claimed to induce remission of leukemia. \ How long should the trial run? Given that spontaneous remissions occur, we \ may not want the trial to be so long that most patients undergo remission \ regardless of the drug regimen. Suppose we want a length such that no more \ than 50% of patients will undergo spontaneous remission. This can be \ determined by solving R(t) = 0.5. Using the fitted model yields a value of \ 5.83, so it would be appropriate to run the trial for six months.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[0.5 == Exp[\(-0.118881\)\ t], t]\)], "Input"], Cell[BoxData[ \({{t \[Rule] 5.83059682001283`}}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Moment ratios", "Subsubsection", CellTags->"ExampleExponential"], Cell[TextData[{ "Given a continuous distribution with PDF ", StyleBox["f", FontSlant->"Italic"], ", or a discrete distribution with probabilities ", StyleBox["p", FontSlant->"Italic"], ", the ", StyleBox["r", FontSlant->"Italic"], "th moment about the mean \[Mu] is ", Cell[BoxData[ FormBox[ RowBox[{\(\[Integral]\_0\%\[Infinity]\), RowBox[{\(f(x)\), \(\((x\_1\ - \[Mu]\ )\)\^r\), StyleBox["dx", FontSlant->"Italic"]}]}], TraditionalForm]]], " or ", Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(i = 1\)\%n\( p( x\_i)\) \((x\_i\ - \[Mu]\ )\)\^r\)]], ". The first moment is the mean, the second is the variance, the third is \ the skewness (a measure of the asymmetry of the distribution), and the fourth \ is the kurtosis (a measure of how sharply the distribution peaks). The entire \ set of moments completely characterizes the distribution, and ratios among \ them (particularly the first four) can be used to test an empirical \ distribution against a theoretical model. \n\nFor example, the ", StyleBox["coefficient of variation", FontSlant->"Italic"], " is the standard deviation \[Sigma] (square root of the variance or second \ moment) divided by the mean \[Mu]; for an exponential distribution this ratio \ is always 1. Calculating the ratio using the discrete ", StyleBox["leukemiaRemission", FontWeight->"Bold"], " data gives" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(With[{data = \(Last[#] &\) /@ leukemiaRemission}, \[IndentingNewLine]Mean[data]/ StandardDeviation[data]\ // N]\)], "Input"], Cell[BoxData[ \(1.3399569872781902`\)], "Output"] }, Open ]], Cell[TextData[{ "which suggests that an exponential distribution may not be the right model \ for this data. The coefficient of skewness, defined as the mean of ", Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", StyleBox[\(\(x\_i - \[Mu]\)\/\[Sigma]\), FontSize->16], ")"}], "3"], TraditionalForm]]], ", is always 2 for an exponential distribution, but is less than 1 for the \ ", StyleBox["leukemiaRemission", FontWeight->"Bold"], " data: " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Module[{data = \(Last[#] &\) /@ leukemiaRemission, mean, sigma}, \[IndentingNewLine]mean = Mean[data] // N; \[IndentingNewLine]sigma = StandardDeviation[data] // N; \[IndentingNewLine]\ Mean[\(\((\((# - mean)\)/sigma)\)^3 &\)\ /@ \ data]\[IndentingNewLine]]\)], "Input"], Cell[BoxData[ \(0.7512191180082449`\)], "Output"] }, Open ]], Cell["\<\ which again suggests that an exponential distribution may not be the right \ model for this data, even though the fit is visually good. A better decision \ could be made by obtaining more data, or by fitting to other types of \ distribution and picking the best. \ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Maximum likelihood parameter estimation", "Subsubsection", CellTags->"Maximum likelihood"], Cell[TextData[{ "Given a set of observations, and an n-parameter distribution hypothesized \ to be the source of the observations, the maximum likelihood method attempts \ to find the set of parameter values that maximizes the probability of the \ given observations. This will be illustrated using the (one-parameter) \ exponential distribution.\n\nSuppose ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["t", FontSlant->"Plain"], "1"], TraditionalForm]]], ", ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["t", FontSlant->"Plain"], "2"], TraditionalForm]]], ", . . ., ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["t", FontSlant->"Plain"], "n"], TraditionalForm]]], " are believed to be observations from an exponential distribution with \ parameter \[Lambda]. Then for each ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["t", FontSlant->"Plain"], "i"], TraditionalForm]]], ", the probability density at that value is f(", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["t", FontSlant->"Plain"], "i"], TraditionalForm]]], ") = \[Lambda]", Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["e", FontSlant->"Plain"], RowBox[{"-", SubscriptBox[ StyleBox[ RowBox[{"\[Lambda]", StyleBox["t", FontSlant->"Plain"]}]], "i"]}]], TraditionalForm]], FontSize->14], ". Since the observations are assumed to be independent, joint probability \ is the product of individual probabilities, and the joint density for all the \ t's is the ", StyleBox["likelihood function", FontSlant->"Italic"], "\n\tL(\[Lambda], ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["t", FontSlant->"Plain"], "1"], TraditionalForm]]], ", ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["t", FontSlant->"Plain"], "2"], TraditionalForm]]], ", . . ., ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["t", FontSlant->"Plain"], "n"], TraditionalForm]]], ") = ", Cell[BoxData[ FormBox[ RowBox[{\(\(\(\[Product]\)\(\ \)\)\+\(i\ = \ 1\)\%n\), StyleBox["\[Lambda]", FontSize->12], StyleBox[ FormBox[ SuperscriptBox[ StyleBox["e", FontSlant->"Plain"], RowBox[{"-", SubscriptBox[ StyleBox[ RowBox[{"\[Lambda]", StyleBox["t", FontSlant->"Plain"]}]], "i"]}]], "TraditionalForm"], FontSize->12]}], TraditionalForm]], FontSize->14], " .\nThe value of \[Lambda] which maximizes L for the given set of t's is \ the maximum likelihood (ML) estimate. \n\nSince log is monotomically \ increasing, L has a maximum iff log(L) is maximized. The log likelihood is \ easier to work with for most distributions used in reliability, so we use\n\t\ log(L(\[Lambda], ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["t", FontSlant->"Plain"], "1"], TraditionalForm]]], ", ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["t", FontSlant->"Plain"], "2"], TraditionalForm]]], ", . . ., ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["t", FontSlant->"Plain"], "n"], TraditionalForm]]], ")) = ", Cell[BoxData[ FormBox[ RowBox[{\(\(\(\[Sum]\)\(\ \)\)\+\(i\ = \ 1\)\%n\), RowBox[{"(", " ", RowBox[{\(log\ \[Lambda]\), " ", "-", " ", SubscriptBox[ StyleBox[ RowBox[{"\[Lambda]", StyleBox["t", FontSlant->"Plain"]}]], "i"]}], ")"}]}], TraditionalForm]]], " = n log(\[Lambda]) ", StyleBox["-", FontSize->16], " ", Cell[BoxData[ FormBox[ RowBox[{\(\(\(\[Sum]\)\(\ \)\)\+\(i\ = \ 1\)\%n\), SubscriptBox[ StyleBox[ RowBox[{"\[Lambda]", StyleBox["t", FontSlant->"Plain"]}]], "i"]}], TraditionalForm]]], " .\n\t", Cell[BoxData[ FormBox[ StyleBox[\(\[PartialD]\/\[PartialD]\[Lambda]\), FontSize->14], TraditionalForm]], FontSize->16], "log(L(\[Lambda], ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["t", FontSlant->"Plain"], "1"], TraditionalForm]]], ", ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["t", FontSlant->"Plain"], "2"], TraditionalForm]]], ", . . ., ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["t", FontSlant->"Plain"], "n"], TraditionalForm]]], ")) = ", StyleBox[" ", FontSize->16], Cell[BoxData[ FormBox[ FractionBox[ StyleBox["n", FontSlant->"Plain"], "\[Lambda]"], TraditionalForm]], FontSize->16], StyleBox[" ", FontSize->14], StyleBox["-", FontSize->18, FontWeight->"Bold"], StyleBox[" ", FontSize->14], Cell[BoxData[ FormBox[ RowBox[{\(\(\(\[Sum]\)\(\ \)\)\+\(i\ = \ 1\)\%n\), SubscriptBox[ StyleBox["t", FontSlant->"Plain"], "i"]}], TraditionalForm]], FontSize->14], "\n\t\t\t\t\t = 0 when\n\t\t\t\t \[Lambda] = n / ", Cell[BoxData[ FormBox[ RowBox[{\(\(\(\[Sum]\)\(\ \)\)\+\(i\ = \ 1\)\%n\), SubscriptBox[ StyleBox["t", FontSlant->"Plain"], "i"]}], TraditionalForm]], FontSize->14], ".\nNote that \n\t", Cell[BoxData[ FormBox[ StyleBox[\(\[PartialD]\^2\/\[PartialD]\[Lambda]\^2\), FontSize->14], TraditionalForm]], FontSize->16], "log(L(\[Lambda], ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["t", FontSlant->"Plain"], "1"], TraditionalForm]]], ", ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["t", FontSlant->"Plain"], "2"], TraditionalForm]]], ", . . ., ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["t", FontSlant->"Plain"], "n"], TraditionalForm]]], ")) = ", StyleBox[" ", FontSize->16], StyleBox["-", FontSize->18, FontWeight->"Bold"], Cell[BoxData[ FormBox[ FractionBox[ StyleBox["n", FontSlant->"Plain"], \(\[Lambda]\^2\)], TraditionalForm]], FontSize->14], " < 0\nso this is a maximum for the given fixed ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["t", FontSlant->"Plain"], "1"], TraditionalForm]]], ", ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["t", FontSlant->"Plain"], "2"], TraditionalForm]]], ", . . ., ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["t", FontSlant->"Plain"], "n"], TraditionalForm]]], ".\n\nApplying this to the example above, the maximum likelihood estimate \ of the exponential rate parameter for the leukemia remission data is:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Length[leukemiaRemission]/ Apply[Plus, Map[First[#] &, leukemiaRemission]] // N\)], "Input"], Cell[BoxData[ \(0.09090909090909091`\)], "Output"] }, Open ]], Cell[TextData[{ "This can be compared with the linear regression estimate using the \ Kolmogorov-Smirnov test for goodness of fit (", ButtonBox["described below", ButtonData:>"K-S test", ButtonStyle->"Hyperlink"], "):" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\({ksStatistic[empiricalCDF[leukemiaRemission, Plot \[Rule] False], Exp[\(-0.09090912\)\ \ #] &], \[IndentingNewLine]\ ksStatistic[empiricalCDF[leukemiaRemission, Plot \[Rule] False], Exp[\(-0.125266\)\ \ #] &]}\)\(\ \)\)\)], "Input"], Cell[BoxData[ \({0.8437150582472646`, 0.9112176803878598`}\)], "Output"] }, Open ]], Cell["\<\ The smaller K-S statistic for the maximum likelihood estimate indicates it is \ a better fit. This is in accord with the theory, which says that ML is the \ least biased estimate of distribution parameters. Nevertheless, in practical \ reliability work, linear fitting is used more often than ML. ML for the \ exponential distribution is exceptionally simple; for multiparameter \ distributions (such as the Weibull, described below) it is more difficult, \ and apparently practicioners feel that the somewhat better fit does not \ justify the effort. 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two-parameter distribution that is \ widely used in reliability analysis. The theoretical justification is a \ theorem that says that in the limit, the smallest value in each of a large \ number of samples from any distribution will be Weibull-distributed. This ", StyleBox["weakest link", FontSlant->"Italic"], " property is valid for many types of units, particularly mechanical parts, \ where a failure in the weakest part of the unit propagates and causes the \ entire unit to fail.\n\nThe Weibull equations are:" }], "Text"], Cell[BoxData[{ \(\(weibullPDF[a_, b_, t_] := \(b\/a\) \(\((t\/a)\)\^\(b - 1\)\) Exp[\(-\((t\/a)\)\^\(\(\ \)\(b\)\)\)];\)\), "\[IndentingNewLine]", \(\(weibullCDF[a_, b_, t_] := 1 - Exp[\(-\((t\/a)\)\^\(\(\ \)\(b\)\)\)];\)\ \), \ "\[IndentingNewLine]", \(\(weibullHazard[a_, b_, t_] := \(b\/a\^b\) t\^\(b - 1\);\)\)}], "Input",\ InitializationCell->True], Cell[TextData[{ "(The Weibull, as well as the exponential distribution, is also available \ in the ", "Mathematica", " package ", StyleBox["Statistics`ContinuousDistributions`", FontWeight->"Bold"], ".) \n\nIn these equations, ", StyleBox["a", FontSlant->"Italic"], " is the ", StyleBox["scale parameter", FontSlant->"Italic"], " and ", StyleBox["b", FontSlant->"Italic"], " is the ", StyleBox["shape parameter", FontSlant->"Italic"], ". Note that if ", StyleBox["b", FontSlant->"Italic"], " = 1, the result is an exponential distribution with \[Lambda] = 1/", StyleBox["a", FontSlant->"Italic"], ". 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The idea is that there is a \"burn-in\" period \ with high, but decreasing, hazard rate, as weak components fail and are \ replaced. Then, during the operating life of the system, the hazard rate is \ constant; finally, as components mechanically wear out, the hazard rate \ increases, and presumably the system is retired. 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