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multivariate observations of a (", StyleBox["p x 1", FontSlant->"Italic"], ") random vector X." }], "Text"], Cell[TextData[{ "At the first stage we have so called training set of ", Cell[BoxData[ \(TraditionalForm\`n\)]], " individuals, say, represented by ", Cell[BoxData[ \(TraditionalForm\`\((p\ x\ 1)\)\)]], " observations ", Cell[BoxData[ \(TraditionalForm\`X\_1, \(\(...\) \(X\_n\)\)\)]], ". In what follows let us identify the individuals ", Cell[BoxData[ \(TraditionalForm\`1, ... , n\)]], " with their respective observations. At this stage, we have also the \ information to which of ", Cell[BoxData[ \(TraditionalForm\`k\)]], " populations (classes, groups) an individual belongs to. The populations \ will be marked ", Cell[BoxData[ \(TraditionalForm\`1, ... , k\)]], ". The information about belonging of an individual ", Cell[BoxData[ \(TraditionalForm\`X\_j\)]], " to a population is expressed by the indicator (tag) ", Cell[BoxData[ \(TraditionalForm\`i\_j\)]], ", ", Cell[BoxData[ \(TraditionalForm\`i\_j\ \[Element] \ {1, ... , k}\)]], ", ", Cell[BoxData[ \(TraditionalForm\`j = 1, ... , n\)]], ". The input data for the first stage are in the form ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox["data", FontSlant->"Italic"], "=", \({{X\_1, i\_1}, \(\(...\) \({X\_n, i\_n}\)\)}\)}], TraditionalForm]]], ", a three-dimensional structure. The purpose of the first stage is to \ build up criteria for classifying other individuals into respective groups in \ the second stage. The methods considered here are either purely nonparametric \ (Mahalanobis), semiparametric (Bayesian), or parametric ones based on the \ assumption of normality of the data involved. In the latter case, linear or \ quadratic DA is under consideration. \n\nIn practice the exact distributions \ are rarely known. In better case we know just an analytical form of the \ distributions involved. In case we know the analytical form of the \ distributions (parametric case, we have to estimate the unknown parameters. \ In this case we deal with the multinormal distribution. The populations may \ differ either in means or both in means and covariance structure. In other \ cases we use the Bayesian or Mahalanobis approach.\n\nAt the second stage, \ classification, we use discriminant scores created at the first stage to \ classify newly incoming individuals with known observations ", Cell[BoxData[ \(TraditionalForm\`X\)]], " (and apparently unknown class) into one of the ", Cell[BoxData[ \(TraditionalForm\`k\)]], " classes. Usually an individual is assigned to population with maximum or \ minimum discriminant score.\n\nFinally, it is of interest to evaluate the \ performance of the discrimination rules. The simplest method is the ", StyleBox["resubstitution method", FontSlant->"Italic"], ". We allocate the individuals from the training set using the \ discrimination rules and look how often the data are misclassified. The \ drawback of this method is that the individual is classified by rules which \ have been created utilizing the information of that individual. More \ sophisticated methods are know as ", StyleBox["cross-validation", FontSlant->"Italic"], ", ", StyleBox["jackknife", FontSlant->"Italic"], ", see [1], [2], e. g.\n\nIn what follows let ", Cell[BoxData[ \(TraditionalForm\`\(L\_i\)(x)\)]], " denote the likelihood or loglikelihood function of the population ", Cell[BoxData[ \(TraditionalForm\`i\)]], ", ", Cell[BoxData[ \(TraditionalForm\`i = 1, ... , k\)]], ". These functions help us to construct so called ", StyleBox["discriminant scores", FontSlant->"Italic"], " which are used in the second stage. 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0.00661739, 0.00588551}}] }, Open ]], Cell[BoxData[""], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Data equipped with tags", "Subsubsection"], Cell["The indicators of the populations are added:", "Text"], Cell[BoxData[ \(\(newdata = Join[Map[Join[#, {1}] &, x1], Map[Join[#, {2}] &, x2], Map[Join[#, {3}] &, x3]];\)\)], "Input", CellLabel->"In[182]:="], Cell[BoxData[ \(newdata\)], "Input", CellLabel->"In[183]:="] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["First stage", "Subsection"], Cell[CellGroupData[{ Cell["Calculations", "Subsubsection"], Cell[TextData[{ "Description of the symbols\n\nn ... the size of the training set\n\ populationmarks ... tags of populations\nk ... number of populations\n\ populations ... the data of the training set partioned into k classes \ together with their tags\npopulationsred ... the same structure as above but \ without tags (reduced data); necessary for estimating unknownparameters\n\ frequencies ... the sizes ", Cell[BoxData[ \(TraditionalForm\`n\_1, \(\(...\) \(n\_k\)\)\)]], " of populations in the training set\npriorrel ... prior probabilities of \ the populations bases on relative frequencies in the training set\n\ covmatrices ... covariance matrices of the populations (unbiased estimates)\n\ pooledcovariance matrix ... the unbiased estimate of the covariance matrix of \ the whole data assuming the equal covariance matrices in the populations" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(\(Needs["\"]\)\(\ \[IndentingNewLine]\)\( (*\ \(data = {{11. , 12, 1}, {33, 332, 3}, {22, 222, 2}, {333, 3332, 3}, {3333, 33332, 3}, {111, 1112, 1}, {222, 2222, 2}, {1111, 11112, 1}, {33333, 333332, 3}};\)*) \)\(\[IndentingNewLine]\) \) (*\ Size\ of\ the\ training\ set\ *) \), "\[IndentingNewLine]", \(n = Length[newdata]\), "\[IndentingNewLine]", \(\(\(populationmarks = Union[Last[Transpose[newdata]]]\)\(\[IndentingNewLine]\) \) (*\ Number\ of\ populations\ *) \), "\[IndentingNewLine]", \(\(\(k = Length[populationmarks]\)\(\[IndentingNewLine]\) \) (*\ Divides\ training\ set\ into\ populations\ *) \), \ "\[IndentingNewLine]", \(\(populations = Map[Cases[newdata, {__, #}] &, populationmarks];\)\[IndentingNewLine] (*\ frequencies\ of\ populations\ *) \), "\[IndentingNewLine]", \(\(\(frequencies = Map[Length, populations]\)\(\[IndentingNewLine]\) \) (*\ prior\ probabilities\ as\ relative\ frequencies\ in\ the\ training\ set\ \ *) \), "\[IndentingNewLine]", \(\(\(priorrel = frequencies/n\)\(\[IndentingNewLine]\) \) (*\ Removes\ the\ marks\ of\ populations\ from\ the\ data\ set\ but\ saving\ \ the\ structure\ of\ populations\ *) \), "\[IndentingNewLine]", \(\(populationsred = Drop[populations, {}, {}, {\(-1\)}] // N;\)\), "\[IndentingNewLine]", \(\(means = Map[Mean, populationsred];\)\), "\[IndentingNewLine]", \(\(covmatrices = Map[CovarianceMatrix, populationsred];\)\), "\[IndentingNewLine]", \(\(\(invcovs = Map[Inverse, covmatrices];\)\(\[IndentingNewLine]\) \) (*\ Creates\ pooled\ covariance\ matrix\ *) \), "\[IndentingNewLine]", \(\(pooledcovariancematrix = Apply[Plus, \((frequencies - 1)\)*covmatrices]/\((n - k)\);\)\), "\[IndentingNewLine]", \(\)}], "Input", CellLabel->"In[184]:="], Cell[BoxData[ \(150\)], "Output", CellLabel->"Out[185]="], Cell[BoxData[ \({1, 2, 3}\)], "Output", CellLabel->"Out[186]="], Cell[BoxData[ \(3\)], "Output", CellLabel->"Out[187]="], Cell[BoxData[ \({40, 50, 60}\)], "Output", CellLabel->"Out[189]="], Cell[BoxData[ \({4\/15, 1\/3, 2\/5}\)], "Output", CellLabel->"Out[190]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(means\)], "Input", CellLabel->"In[130]:="], Cell[BoxData[ \({{5.00258597768326`, 3.4094374398803984`}, {5.890191103717047`, 2.724642812906938`}, {6.498733520519508`, 3.0135413721763076`}}\)], "Output", CellLabel->"Out[130]="] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(Map[Mean, populationsred]\)], "Input", CellLabel->"In[83]:="], Cell[BoxData[ \({{5.00258597768326`, 3.4094374398803984`}, {5.890191103717047`, 2.724642812906938`}, {6.498733520519508`, 3.0135413721763076`}}\)], "Output", CellLabel->"Out[83]="] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(covmatrices\)], "Input", CellLabel->"In[106]:="], Cell[BoxData[ \({{{0.10834199811286666`, 0.09753955989093079`}, {0.09753955989093079`, 0.13022780521916527`}}, {{0.2861506312101881`, 0.05547827330826006`}, {0.05547827330826006`, 0.0877207465546453`}}, {{0.4269300157783464`, 0.05152662143333301`}, {0.05152662143333301`, 0.07410226252351942`}}}\)], "Output", CellLabel->"Out[106]="] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(Map[CovarianceMatrix, populationsred]\)], "Input", CellLabel->"In[84]:="], Cell[BoxData[ \({{{0.10834199811286666`, 0.09753955989093079`}, {0.09753955989093079`, 0.13022780521916527`}}, {{0.2861506312101881`, 0.05547827330826006`}, {0.05547827330826006`, 0.0877207465546453`}}, {{0.4269300157783464`, 0.05152662143333301`}, {0.05152662143333301`, 0.07410226252351942`}}}\)], "Output", CellLabel->"Out[84]="] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["Procedures", "Subsubsection"], Cell["Linear DA", "Text"], Cell[BoxData[ RowBox[{ StyleBox[" ", FontColor->RGBColor[1, 0, 0]], RowBox[{ StyleBox[\(Clear[lDA];\), FontColor->RGBColor[1, 0, 0]], StyleBox["\[IndentingNewLine]", FontColor->RGBColor[1, 0, 0]], RowBox[{ StyleBox[\(lDA[populationsred_, pooledcovariancematrix_, prior_]\), FontColor->GrayLevel[0]], StyleBox[":=", FontColor->GrayLevel[0]], StyleBox["\[IndentingNewLine]", FontColor->GrayLevel[0]], "\t", \(Module[{means, invcov, scores}, \[IndentingNewLine]\t\tmeans = Map[Mean, populationsred]; \[IndentingNewLine]\t\tinvcov = Inverse[pooledcovariancematrix]; \[IndentingNewLine]\t\tscores \ = {means . invcov, \(-0.5\)\ Map[# . invcov . # &, means] + Log[prior]}\[IndentingNewLine]\t\t]\)}]}]}]], "Input", CellLabel->"In[205]:="], Cell["Quadratic DA", "Text"], Cell[BoxData[ RowBox[{ RowBox[{ StyleBox[\(Clear[qDA];\), FontColor->RGBColor[1, 0, 0]], "\[IndentingNewLine]", \(qDA[populationsred_, covmatrices_, prior_] := Module[{means, invcovs, dets, scores}, \[IndentingNewLine]means = Map[Mean, populationsred]; \[IndentingNewLine]\t invcovs = Map[Inverse, covmatrices]; \[IndentingNewLine]\t dets = Map[Det, covmatrices]; \[IndentingNewLine]\t scores = {\(-\(1\/2\)\) Log[dets] + Log[prior], means, invcovs}\[IndentingNewLine]\t\ ]\)}], " "}]], "Input", CellLabel->"In[200]:="], Cell["Mahalanobis DA", "Text"], Cell[BoxData[{ StyleBox[\(Clear[mahalDA];\), FontColor->RGBColor[1, 0, 0]], "\[IndentingNewLine]", \(mahalDA[populationsred_, pooledcovariancematrix_] := Module[{means, invcov, scores}, \[IndentingNewLine]means = Map[Mean, populationsred]; \[IndentingNewLine]\t invcov = Map[Inverse, pooledcovariancematrix]; \[IndentingNewLine]\t scores = {means, invcov}\[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ];\)}], "Input", CellLabel->"In[210]:="], Cell["Maximum Likelihood DA", "Text"], Cell[BoxData[{ StyleBox[\(Clear[mlDA];\), FontColor->RGBColor[1, 0, 0]], "\[IndentingNewLine]", \(mlDA[populationsred_, covmatrices_] := \[IndentingNewLine]Module[{means, invcovs, dets, scores}, \[IndentingNewLine]\t means = Map[Mean, populationsred]; \[IndentingNewLine]\t invcovs = Map[Inverse, covmatrices]; \[IndentingNewLine]\t dets = Map[Det, covmatrices]; \[IndentingNewLine]\t scores = {dets, means, invcovs}\[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ];\)}], "Input", CellLabel->"In[215]:="], Cell["Bayesian DA", "Text"], Cell[BoxData[{ StyleBox[\(Clear[bayesDA];\), FontColor->RGBColor[1, 0, 0]], "\[IndentingNewLine]", RowBox[{\(bayesDA[populationsred_, covmatrices_, prior_] := Module[{means, invcovs, dets, scores}, \[IndentingNewLine]\t means = Map[Mean, populationsred]; \[IndentingNewLine]\t invcovs = Map[Inverse, covmatrices]; \[IndentingNewLine]\t dets = Map[Det, covmatrices]; \[IndentingNewLine]\t scores = {\(-\(1\/2\)\) Log[dets] + Log[prior], means, invcovs}\ ];\), " "}]}], "Input", CellLabel->"In[219]:="], Cell[CellGroupData[{ Cell[BoxData[ \(lDA[populationsred, pooledcovariancematrix, priorrel]\)], "Input", CellLabel->"In[221]:="], Cell[BoxData[ \({{{10.515355595489835`, 29.138618766614115`}, {15.96569491324983`, 18.026440363906293`}, {17.59460699204144`, 19.982297387750968`}}, {\(-77.29688995080554`\), \ \(-72.67691494991685`\), \(-88.19636179620652`\)}}\)], "Output", CellLabel->"Out[221]="] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Construction of the criteriae", "Subsubsection"], Cell["\<\ Possible options: priors\[Rule]uniform, relativeFrequencies (default) method\[Rule] linearDA (default),quadraticDA, MahalanobisDA, MLDA, BayesianDA \ \ \>", "Text"], Cell[BoxData[{ StyleBox[\(Clear[criteriaDA];\), FontColor->RGBColor[1, 0, 0]], "\[IndentingNewLine]", RowBox[{\(Options[criteriaDA] = {method \[Rule] linearDA, priors \[Rule] relativeFrequencies};\), "\[IndentingNewLine]", \( (*\ \[IndentingNewLine]Other\ \(options : \ priors\) \[Rule] uniform, \ method \[Rule] \ quadraticDA, \ MahalanobisDA, MLDA, BayesianDA\ \ \[IndentingNewLine]*) \)}], "\[IndentingNewLine]", \ \(criteriaDA[data_, opts___] := \[IndentingNewLine]Module[{n, k, populationmarks, populations, frequencies, priorrel, covmatrices, pooledcovariancematrix, met, pri, prior (*\(,\)\(lDA\)\(,\)\(qDA\)\(,\)\(mahalDA\)\(,\)\(mlDA\)\(,\)\(\ bayesDA\)*) }, \[IndentingNewLine]n = Length[data]; \ (*\ the\ size\ of\ the\ training\ set\ \ *) \[IndentingNewLine]populationmarks = Union[Last[Transpose[data]]]; \[IndentingNewLine]k = Length[populationmarks]; (*\ No . \ of\ populations\ *) \[IndentingNewLine]populations = Map[Cases[data, {__, #}] &, populationmarks]; \[IndentingNewLine]frequencies = Map[Length, populations]; \[IndentingNewLine]priorrel = frequencies/n; \[IndentingNewLine] (*\ Removes\ the\ marks\ of\ populations\ from\ the\ data\ set\ but\ \ saving\ the\ structure\ of\ populations\ *) \[IndentingNewLine]populationsred \ = Drop[populations, {}, {}, {\(-1\)}] // N; \[IndentingNewLine]covmatrices = Map[CovarianceMatrix, populationsred]; \[IndentingNewLine] (*\ Creates\ pooled\ covariance\ matrix\ \ *) \[IndentingNewLine]pooledcovariancematrix = Apply[Plus, \((frequencies - 1)\)*covmatrices]/\((n - k)\); {met, pri} = \({method, priors} /. {opts}\) /. Options[criteriaDA]; \[IndentingNewLine]If[pri === uniform, prior = 1/k, prior = priorrel]; \[IndentingNewLine]Switch[ met, \[IndentingNewLine]\t\ \ linearDA, \ {met, lDA[populationsred, pooledcovariancematrix, prior]} // Return, \[IndentingNewLine]\t\ quadraticDA, {met, qDA[populationsred, covmatrices, prior]} // Return, \[IndentingNewLine]\t\ MahalanobisDA, {met, mahalDA[populationsred, covmatrices]} // Return, \[IndentingNewLine]\t\ MLDA, {met, mlDA[populationsred, covmatrices]} // Return, \[IndentingNewLine]\t\ BayesianDA, \ {met, bayesDA[populationsred, covmatrices, prior]} // Return]\[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ]\)}], "Input", CellLabel->"In[222]:="], Cell[CellGroupData[{ Cell[BoxData[ \(crlin = criteriaDA[newdata]\)], "Input", CellLabel->"In[225]:="], Cell[BoxData[ \({linearDA, {{{10.515355595489835`, 29.138618766614115`}, {15.96569491324983`, 18.026440363906293`}, {17.59460699204144`, 19.982297387750968`}}, {\(-77.29688995080554`\), \ \(-72.67691494991685`\), \(-88.19636179620652`\)}}}\)], "Output", CellLabel->"Out[225]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(crqua = criteriaDA[newdata, method -> quadraticDA]\)], "Input", CellLabel->"In[226]:="], Cell[BoxData[ \({quadraticDA, {{1.3696183921976133`, 0.8092101324021852`, 0.85425827744058`}, {{5.00258597768326`, 3.4094374398803984`}, {5.890191103717047`, 2.724642812906938`}, {6.498733520519508`, 3.0135413721763076`}}, {{{28.34011947716701`, \ \(-21.226517458442537`\)}, {\(-21.226517458442537`\), 23.577339461003046`}}, {{3.9830505570468544`, \ \(-2.5190479571079427`\)}, {\(-2.5190479571079427`\), 12.992963190652329`}}, {{2.556882658267697`, \ \(-1.7779150095477618`\)}, {\(-1.7779150095477618`\), 14.731128530537653`}}}}}\)], "Output", CellLabel->"Out[226]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(crmahal = criteriaDA[newdata, method -> \ MahalanobisDA]\)], "Input", CellLabel->"In[227]:="], Cell[BoxData[ \({MahalanobisDA, {{{5.00258597768326`, 3.4094374398803984`}, {5.890191103717047`, 2.724642812906938`}, {6.498733520519508`, 3.0135413721763076`}}, {{{28.34011947716701`, \ \(-21.226517458442537`\)}, {\(-21.226517458442537`\), 23.577339461003046`}}, {{3.9830505570468544`, \ \(-2.5190479571079427`\)}, {\(-2.5190479571079427`\), 12.992963190652329`}}, {{2.556882658267697`, \ \(-1.7779150095477618`\)}, {\(-1.7779150095477618`\), 14.731128530537653`}}}}}\)], "Output", CellLabel->"Out[227]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(crML = criteriaDA[newdata, method \[Rule] MLDA]\)], "Input", CellLabel->"In[228]:="], Cell[BoxData[ \({MLDA, {{0.004595174883581096`, 0.022023508187574686`, 0.028981487392043304`}, {{5.00258597768326`, 3.4094374398803984`}, {5.890191103717047`, 2.724642812906938`}, {6.498733520519508`, 3.0135413721763076`}}, {{{28.34011947716701`, \ \(-21.226517458442537`\)}, {\(-21.226517458442537`\), 23.577339461003046`}}, {{3.9830505570468544`, \ \(-2.5190479571079427`\)}, {\(-2.5190479571079427`\), 12.992963190652329`}}, {{2.556882658267697`, \ \(-1.7779150095477618`\)}, {\(-1.7779150095477618`\), 14.731128530537653`}}}}}\)], "Output", CellLabel->"Out[228]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(crbayes\)\(=\)\(criteriaDA[newdata, method \[Rule] BayesianDA]\)\(\ \)\)\)], "Input", CellLabel->"In[229]:="], Cell[BoxData[ \({BayesianDA, {{1.3696183921976133`, 0.8092101324021852`, 0.85425827744058`}, {{5.00258597768326`, 3.4094374398803984`}, {5.890191103717047`, 2.724642812906938`}, {6.498733520519508`, 3.0135413721763076`}}, {{{28.34011947716701`, \ \(-21.226517458442537`\)}, {\(-21.226517458442537`\), 23.577339461003046`}}, {{3.9830505570468544`, \ \(-2.5190479571079427`\)}, {\(-2.5190479571079427`\), 12.992963190652329`}}, {{2.556882658267697`, \ \(-1.7779150095477618`\)}, {\(-1.7779150095477618`\), 14.731128530537653`}}}}}\)], "Output", CellLabel->"Out[229]="] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Second stage", "Subsection"], Cell[CellGroupData[{ Cell["Construction of classifiers", "Subsubsection"], Cell["\<\ Important note: The second argument of the following procedures \ should be criteriaDA\[LeftDoubleBracket]2\[RightDoubleBracket]\ \>", "Text"], Cell["Linear DA", "Text"], Cell[BoxData[{ RowBox[{ StyleBox[\(Clear[critlDA]\), FontColor->RGBColor[1, 0, 0]], ";"}], "\[IndentingNewLine]", \(critlDA[individual_, crit_] := \[IndentingNewLine]\t Module[{scores, ma}, \[IndentingNewLine]\t scores = crit\[LeftDoubleBracket]1\[RightDoubleBracket] . individual + crit\[LeftDoubleBracket]2\[RightDoubleBracket]; \ \[IndentingNewLine]\t ma = Max[ scores]; \[IndentingNewLine]\t\((Position[scores, ma] // Flatten)\)\[LeftDoubleBracket]1\[RightDoubleBracket]\ \[IndentingNewLine]\t\t\ \ \ \ ]\)}], "Input", CellLabel->"In[230]:="], Cell[CellGroupData[{ Cell[BoxData[ \(critlDA[{15, 5}, crlin\[LeftDoubleBracket]2\[RightDoubleBracket]]\)], "Input", CellLabel->"In[232]:="], Cell[BoxData[ \(3\)], "Output", CellLabel->"Out[232]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Map[critlDA[#, crlin\[LeftDoubleBracket]2\[RightDoubleBracket]] &, x1]\)], "Input", CellLabel->"In[233]:="], Cell[BoxData[ \({1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}\)], "Output", CellLabel->"Out[233]="] }, Open ]], Cell["Quadratic DA", "Text"], Cell[BoxData[{ RowBox[{ StyleBox[\(Clear[critqDA]\), FontColor->RGBColor[1, 0, 0]], ";"}], "\[IndentingNewLine]", \(critqDA[individual_, crit_] := \[IndentingNewLine]\t Module[{scores, ma, means, invcovs, au, k}, \[IndentingNewLine]\t means = crit\[LeftDoubleBracket]2\[RightDoubleBracket]; \ \[IndentingNewLine]\tk = Length[means]; \[IndentingNewLine]\t invcovs = crit\[LeftDoubleBracket]3\[RightDoubleBracket]; \[IndentingNewLine]\ \tau = Map[individual - # &, means]; \[IndentingNewLine]\t scores = crit\[LeftDoubleBracket]1\[RightDoubleBracket] - 0.5 Map[ au\[LeftDoubleBracket]#\[RightDoubleBracket] . invcovs\[LeftDoubleBracket]#\[RightDoubleBracket] . au\[LeftDoubleBracket]#\[RightDoubleBracket] &, Range[k]]; \[IndentingNewLine]\t ma = Max[ scores]; \[IndentingNewLine]\t\((Position[scores, ma] // Flatten)\)\[LeftDoubleBracket]1\[RightDoubleBracket]\ \[IndentingNewLine]\t\t\ \ \ \ ]\)}], "Input", CellLabel->"In[234]:="], Cell[CellGroupData[{ Cell[BoxData[ \(critqDA[{5, 3}, crqua\[LeftDoubleBracket]2\[RightDoubleBracket]]\)], "Input", CellLabel->"In[236]:="], Cell[BoxData[ \(1\)], "Output", CellLabel->"Out[236]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Map[critqDA[#, crqua\[LeftDoubleBracket]2\[RightDoubleBracket]] &, x1]\)], "Input", CellLabel->"In[237]:="], Cell[BoxData[ \({3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}\)], "Output", CellLabel->"Out[237]="] }, Open ]], Cell[TextData[StyleBox["Mahalanobis DA", "Text"]], "SmallText"], Cell[BoxData[{ RowBox[{ StyleBox[\(Clear[critmahalDA]\), FontColor->RGBColor[1, 0, 0]], ";"}], "\[IndentingNewLine]", \(critmahalDA[individual_, crit_] := \[IndentingNewLine]\t Module[{scores, ma, means, invcovs, au, k}, \[IndentingNewLine]\t means = crit\[LeftDoubleBracket]2\[RightDoubleBracket]; \ \[IndentingNewLine]\tk = Length[means]; \[IndentingNewLine]\t invcovs = crit\[LeftDoubleBracket]3\[RightDoubleBracket]; \[IndentingNewLine]\ \tau = Map[individual - # &, means]; \[IndentingNewLine]\t scores = \(-0.5\) Map[au\[LeftDoubleBracket]#\[RightDoubleBracket] . invcovs\[LeftDoubleBracket]#\[RightDoubleBracket] . au\[LeftDoubleBracket]#\[RightDoubleBracket] &, Range[k]]; \[IndentingNewLine]\t ma = Max[ scores]; \[IndentingNewLine]\t\((Position[scores, ma] // Flatten)\)\[LeftDoubleBracket]1\[RightDoubleBracket]\ \[IndentingNewLine]\t\t\ \ \ \ ]\)}], "Input", CellLabel->"In[238]:="], Cell[CellGroupData[{ Cell[BoxData[ \(Map[critmahalDA[#, crqua\[LeftDoubleBracket]2\[RightDoubleBracket]] &, x2]\)], "Input", CellLabel->"In[240]:="], Cell[BoxData[ \({2, 2, 3, 3, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 3, 3, 2, 2, 3, 2, 3, 2, 3, 2, 2, 2, 2, 2, 3, 3, 3, 2, 2, 3, 3, 2, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 2, 2, 3}\)], "Output", CellLabel->"Out[240]="] }, Open ]], Cell["Maximum Likelihood DA", "Text"], Cell[BoxData[{ StyleBox[\(Clear[critMLDA];\), FontColor->RGBColor[1, 0, 0]], "\[IndentingNewLine]", \(critMLDA[individual_, crit_] := critqDA[individual, crit]\)}], "Input", CellLabel->"In[244]:="], Cell[CellGroupData[{ Cell[BoxData[ \(Map[critMLDA[#, crqua\[LeftDoubleBracket]2\[RightDoubleBracket]] &, x3]\)], "Input", CellLabel->"In[246]:="], Cell[BoxData[ \({3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2, 3, 3, 1, 2, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 2, 2, 3, 2, 3}\)], "Output", CellLabel->"Out[246]="] }, Open ]], Cell["Bayesian DA", "Text"], Cell[BoxData[{ StyleBox[\(Clear[critbayesDA];\), FontColor->RGBColor[1, 0, 0]], "\[IndentingNewLine]", \(critbayesDA[individual_, crit_] := Module[{scores, ma, means, invcovs, au, k}, \[IndentingNewLine]\t means = crit\[LeftDoubleBracket]2\[RightDoubleBracket]; \ \[IndentingNewLine]\tk = Length[means]; \[IndentingNewLine]\t invcovs = crit\[LeftDoubleBracket]3\[RightDoubleBracket]; \[IndentingNewLine]\ \tau = Map[individual - # &, means]; \[IndentingNewLine]\t scores = crit\[LeftDoubleBracket]1\[RightDoubleBracket] - 0.5 Map[ au\[LeftDoubleBracket]#\[RightDoubleBracket] . invcovs\[LeftDoubleBracket]#\[RightDoubleBracket] . au\[LeftDoubleBracket]#\[RightDoubleBracket] &, Range[k]]; \[IndentingNewLine]\t ma = Max[ scores]; \[IndentingNewLine]\t\((Position[scores, ma] // Flatten)\)\[LeftDoubleBracket]1\[RightDoubleBracket]\ \[IndentingNewLine]\t\t\ \ \ \ ]\)}], "Input", CellLabel->"In[247]:="], Cell[CellGroupData[{ Cell[BoxData[ \(Map[critbayesDA[#, crqua\[LeftDoubleBracket]2\[RightDoubleBracket]] &, x3]\)], "Input", CellLabel->"In[249]:="], Cell[BoxData[ \({3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2, 3, 3, 1, 2, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 2, 2, 3, 2, 3}\)], "Output", CellLabel->"Out[249]="] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Procedure for classification", "Subsubsection"], Cell[TextData[{ "Variable", StyleBox[" individual", FontSlant->"Italic"], " is the vector of observations on an individual. Variable ", StyleBox["criteria", FontSlant->"Italic"], " should be obtained using procedure criteriaDA. " }], "Text"], Cell[BoxData[{ StyleBox[\(Clear[classifyDA];\), FontColor->RGBColor[1, 0, 0]], "\[IndentingNewLine]", RowBox[{ StyleBox[\(classifyDA[individual_, criteria_]\), FontColor->GrayLevel[0]], StyleBox[":=", FontColor->GrayLevel[0]], StyleBox["\[IndentingNewLine]", FontColor->GrayLevel[0]], "\t", \(Module[{met, crit}, \[IndentingNewLine]\t Switch[criteria\[LeftDoubleBracket]1\[RightDoubleBracket], \ \[IndentingNewLine]\tlinearDA, critlDA[individual, criteria\[LeftDoubleBracket]2\[RightDoubleBracket]] // Return, \[IndentingNewLine]\tquadraticDA, critqDA[individual, criteria\[LeftDoubleBracket]2\[RightDoubleBracket]] // Return, \[IndentingNewLine]\tmahalDA, critmahalDA[individual, criteria\[LeftDoubleBracket]2\[RightDoubleBracket]] // Return, \[IndentingNewLine]\tMLDA, critMLDA[individual, criteria\[LeftDoubleBracket]2\[RightDoubleBracket]] // Return, \[IndentingNewLine]\tbayesDA, critbayesDA[individual, criteria\[LeftDoubleBracket]2\[RightDoubleBracket]] // Return\[IndentingNewLine]\t\ \ \ \ \ \ \ \ \ \ \ ]\ \[IndentingNewLine]\t\ \ \ \ \ \ \ \ \ \ \ ]\)}]}], "Input", CellLabel->"In[252]:="], Cell[CellGroupData[{ Cell[BoxData[ \(Map[classifyDA[#, crlin] &, x1]\)], "Input", CellLabel->"In[254]:="], Cell[BoxData[ \({1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}\)], "Output", CellLabel->"Out[254]="] }, Open ]] }, Open ]], Cell[TextData[StyleBox["Misclassification", "Subsection"]], "Subsubsection"], Cell[CellGroupData[{ Cell["Resubstitution method", "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[ \(allocationresults = Table[Map[classifyDA[#, crlin] &, populationsred\[LeftDoubleBracket]i\[RightDoubleBracket]], {i, Length[populationsred]}]\)], "Input", CellLabel->"In[255]:="], Cell[BoxData[ \({{1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {2, 2, 3, 3, 2, 2, 2, 2, 2, 3, 2, 3, 2, 2, 2, 3, 3, 2, 2, 3, 2, 3, 2, 3, 2, 2, 2, 2, 2, 3, 1, 3, 2, 2, 3, 3, 2, 2, 3, 2, 2, 3, 3, 2, 2, 2, 2, 2, 2, 3}, {3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2, 3, 3, 1, 2, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 3, 3, 2, 2, 3, 2, 3, 3, 3, 2, 2, 3, 3, 3, 3, 3, 2, 2, 3, 2, 3}}\)], "Output", CellLabel->"Out[255]="] }, Open ]], Cell["Probabilities of misclassification", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(probabilitiesofmisclassification = 1 - Table[ Count[allocationresults\[LeftDoubleBracket]i\[RightDoubleBracket], i]/Length[ allocationresults\[LeftDoubleBracket] i\[RightDoubleBracket]], {i, Length[allocationresults]}]\)], "Input", CellLabel->"In[256]:="], Cell[BoxData[ \({1\/40, 9\/25, 1\/4}\)], "Output", CellLabel->"Out[256]="] }, Open ]] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Multidimensional Scaling", "Section"], Cell[CellGroupData[{ Cell["Introduction", "Subsection"], Cell[TextData[{ "The purpose of the multidimensional scaling (MS) is in some respect \ reverse to usual statistical methods. Rather than observations we have the \ distances or, better to say, dissimilarities among objects in a real linear \ space of possibly unknown dimension. The purpose of MS is to create a \ configuration of points in a linear space such that the actual distances \ between points are as close as possible to the prescribed ones. It is shown \ that if the distance matrix is Euclidean then such a configuration exists and \ since the proof is in fact the algorithm for constructing the desired \ configuration. The construction uses the spectral decomposition of a \ symmetric matrix. In ", StyleBox["Mathematica", FontSlant->"Italic"], " this is done by use of functions ", StyleBox["Eigenvalue", FontWeight->"Bold"], "s and ", StyleBox["Eigenvectors", FontWeight->"Bold"], "." }], "Text"], Cell[TextData[{ "Notation:", StyleBox[" n", FontSlant->"Italic"], " number of points, ", StyleBox["p", FontSlant->"Italic"], " \n\nmakeconfiguration::usage=\"makeconfiguration[dit_, pdim___] gives a \ configuration of points in a p-dimensional Euclidean space where ", StyleBox["p", FontSlant->"Italic"], " is the number of positive eigenvalues of the respective matrix, or, if \ pdim given, ", StyleBox["p", FontSlant->"Italic"], ":=pdim. The construction uses the distances given in the form of lower \ diagonal list dit (without zeros on the diagonal), i.e., dit = {{", Cell[BoxData[ \(TraditionalForm\`d\_21\)]], "}, {", Cell[BoxData[ \(TraditionalForm\`d\_31\)]], ",", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(d\_32\)\)\)]], "}, {", Cell[BoxData[ \(TraditionalForm\`d\_41\)]], ",", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(d\_42\)\)\)]], ",", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(d\_43\)\)\)]], "}, ... }. The configuration is in the form of ", StyleBox["(nxp)", FontSlant->"Italic"], " matrix with the centre of gravity (the average) 0.\"" }], "Text"], Cell[TextData[{ "newdistancematrix::usage=\"newdistancematrix[pot_] returns the distance \ matrix of the points given in the ", StyleBox["(nxp)", FontSlant->"Italic"], " matrix pot. pot should be of the same form as the output of \ makeconfiguration.\"" }], "Text"], Cell[TextData[{ "newdistancediag::usage=\"newdistancediag[pot] returns the lower diagonal \ of the distance matrix of the points given in the ", StyleBox["(nxp)", FontSlant->"Italic"], " matrix pot. pot should be of the same form as the output of \ makeconfiguration.\"" }], "Text"], Cell[TextData[{ "stress::usage=\"stress[distancediag, distancehatdiag] returns \n\[Psi] \ = ", Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(i = 1\)\%n\(\[Sum]\+\(j = 1\)\%n\((d\_ij\)\ \)\)]], " - ", Cell[BoxData[ \(TraditionalForm\`d\&^\)]], Cell[BoxData[ \(TraditionalForm\`\_ij\)]], Cell[BoxData[ \(TraditionalForm\`)\^2\)]], ", also called raw stress, where ", Cell[BoxData[ \(TraditionalForm\`d\_ij\)]], " are the original distances and ", Cell[BoxData[ \(TraditionalForm\`d\&^\)]], Cell[BoxData[ \(TraditionalForm\`\_ij\)]], " are the \"estimated\" or reproduced distances from the configuration in p \ dimensions. Both of the parameters should be in the form of lower diagonal \ structures {{", Cell[BoxData[ \(TraditionalForm\`d\_21\)]], "}, {", Cell[BoxData[ \(TraditionalForm\`d\_31\)]], ",", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(d\_32\)\)\)]], "}, {", Cell[BoxData[ \(TraditionalForm\`d\_41\)]], ",", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(d\_42\)\)\)]], ",", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(d\_43\)\)\)]], "}, ... } (without zeros) of the same dimension." }], "Text"], Cell[CellGroupData[{ Cell["\<\ Illustration of the method - road distances between Czech cities\ \ \>", "Subsubsection"], Cell["\<\ Praha (Pr), Podebrady (Po), Hradec Kralove (Hr), Jihlava (Ji) The distances {{Po-Pr},{Hr-Pr,Hr-Po},{Ji-Pr,Ji-Po,Ji-Hr}}:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(dit = {{54}, {112, 58}, {123, 102, 110}};\)\[IndentingNewLine] (*\ Number\ of\ objects\ *) \), "\n", \(\(n = Length[dit] + 1;\)\), "\n", \(\(\(lowdiag = Join[{Table[0, {n}]}, Table[Join[dit[\([i]\)], Table[0, {n - Length[dit\[LeftDoubleBracket] i\[RightDoubleBracket]]}]], {i, n - 1}]];\)\(\[IndentingNewLine]\) \) (*\ Creates\ the\ nxn\ matrix\ of\ distances\ *) \), "\[IndentingNewLine]", \ \(Print["\"]\), "\n", \(\((d = lowdiag + Transpose[lowdiag])\) // MatrixForm\), "\n", \( (*\ nxn\ matrix\ of\ ones\ *) \(j = IdentityMatrix[n] - IdentityMatrix[n] + 1;\)\n (*\ centering\ matrix\ *) \), "\[IndentingNewLine]", \(\(h = IdentityMatrix[n] - j/n;\)\), "\n", \(\(a = \(-d\)*d/2. ;\)\), "\n", \(\(b = h . a . h;\)\), "\n", \(Print["\", lambda = Eigenvalues[b] (*\ \(//\)\(\ \)\(Chop\)\ *) ]\), "\n", \(Print["\", p = Count[lambda, x_ /; x > 0]]\), "\n", \(\(charvect = Take[Eigenvectors[b], p];\)\), "\n", \(\(\(points = Take[Sqrt[lambda], p]*charvect\ // \ Chop;\)\(\[IndentingNewLine]\) \) (*\ test : \ Transpose[points] . points\ (*\ should\ equal\ to\ b\ *) \n\ *) \), "\n", \(Print["\", \((pot = Transpose[points])\) // MatrixForm]\), "\n", \(Print["\", \(\(Table[ Apply[Plus, 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Cell[StyleData["Print"], CellMargins->{{66, Inherited}, {Inherited, Inherited}}, CellGroupingRules->"OutputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, DefaultFormatType->DefaultOutputFormatType, "TwoByteSyntaxCharacterAutoReplacement"->True, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, AutoItalicWords->{}, LanguageCategory->None, FormatType->InputForm, CounterIncrements->"Print", StyleMenuListing->None], Cell[StyleData["Print", "Presentation"], CellMargins->{{72, Inherited}, {Inherited, Inherited}}, LineSpacing->{1, 0}, FontSize->18], Cell[StyleData["Print", "SlideShow"], CellMargins->{{100, Inherited}, {Inherited, Inherited}}, LineSpacing->{1, 0}, FontSize->18], Cell[StyleData["Print", "Printout"], CellMargins->{{39, Inherited}, {Inherited, Inherited}}, FontSize->8] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Graphics"], CellMargins->{{4, Inherited}, {Inherited, Inherited}}, CellGroupingRules->"GraphicsGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, DefaultFormatType->DefaultOutputFormatType, LanguageCategory->None, FormatType->InputForm, CounterIncrements->"Graphics", ImageMargins->{{43, Inherited}, {Inherited, 0}}, StyleMenuListing->None, FontFamily->"Courier", FontSize->10], Cell[StyleData["Graphics", "Presentation"], ImageMargins->{{62, Inherited}, {Inherited, 0}}], Cell[StyleData["Graphics", "SlideShow"], ImageMargins->{{100, Inherited}, {Inherited, 0}}], Cell[StyleData["Graphics", "Printout"], ImageMargins->{{30, Inherited}, {Inherited, 0}}, Magnification->0.8] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["CellLabel"], LanguageCategory->None, StyleMenuListing->None, FontFamily->"Helvetica", FontSize->9, FontColor->RGBColor[0.23621, 0.628321, 0.559747]], Cell[StyleData["CellLabel", "Presentation"], FontSize->12], Cell[StyleData["CellLabel", "SlideShow"], FontSize->12], Cell[StyleData["CellLabel", "Printout"], FontFamily->"Courier", FontSize->8, FontSlant->"Italic"] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["FrameLabel"], LanguageCategory->None, StyleMenuListing->None, FontFamily->"Helvetica", FontSize->9], Cell[StyleData["FrameLabel", "Presentation"], FontSize->12], Cell[StyleData["FrameLabel", "SlideShow"], FontSize->12], Cell[StyleData["FrameLabel", "Printout"], FontFamily->"Courier", FontSize->8, FontSlant->"Italic", FontColor->GrayLevel[0]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Presentation Specific", "Section"], Cell[CellGroupData[{ Cell[StyleData["BulletedList"], CellMargins->{{45, 10}, {7, 7}}, CellFrameLabels->{{ Cell[ "\[FilledSmallSquare]", "BulletedList", CellBaseline -> Baseline], Inherited}, {Inherited, Inherited}}, InputAutoReplacements->{"TeX"->StyleBox[ RowBox[ {"T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "LaTeX"->StyleBox[ RowBox[ {"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-0.36, -0.1}, {0, 0}}, BoxBaselineShift -> -0.2], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "mma"->"Mathematica", "Mma"->"Mathematica", "MMA"->"Mathematica", "gridMathematica"->FormBox[ RowBox[ {"grid", AdjustmentBox[ StyleBox[ "Mathematica", FontSlant -> "Italic"], BoxMargins -> {{-0.175, 0}, {0, 0}}]}], TextForm], "webMathematica"->FormBox[ RowBox[ {"web", AdjustmentBox[ StyleBox[ "Mathematica", FontSlant -> "Italic"], BoxMargins -> {{-0.175, 0}, {0, 0}}]}], TextForm], Inherited}, CounterIncrements->"BulletedList", FontFamily->"Helvetica"], Cell[StyleData["BulletedList", "Presentation"], CellMargins->{{56, 50}, {10, 10}}, LineSpacing->{1, 5}, FontSize->18], Cell[StyleData["BulletedList", "SlideShow"], CellMargins->{{85, 50}, {10, 10}}, FontSize->18], Cell[StyleData["BulletedList", "Printout"], CellMargins->{{2, 2}, {6, 6}}, TextJustification->0.5, Hyphenation->True, FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Author"], CellMargins->{{139, 27}, {2, 20}}, FontFamily->"Times", FontSize->24, FontSlant->"Italic"], Cell[StyleData["Author", "Presentation"], CellMargins->{{198, 27}, {2, 25}}, FontSize->32], Cell[StyleData["Author", "SlideShow"], CellMargins->{{198, 27}, {2, 50}}, FontSize->32], Cell[StyleData["Author", "Printout"], CellMargins->{{100, 27}, {2, 20}}, FontSize->14] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Affiliation"], CellMargins->{{141, 27}, {30, 12}}, FontFamily->"Times", FontSize->24, FontSlant->"Italic"], Cell[StyleData["Affiliation", "Presentation"], CellMargins->{{198, 27}, {35, 10}}, FontSize->32], Cell[StyleData["Affiliation", "SlideShow"], CellMargins->{{198, 27}, {100, 10}}, FontSize->32], Cell[StyleData["Affiliation", "Printout"], CellMargins->{{100, 27}, {2, 12}}, FontSize->14] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Header Graphic", "Section"], Cell[CellGroupData[{ Cell[StyleData["ConferenceGraphicCell"], ShowCellBracket->True, CellMargins->{{0, 0}, {0, 0}}, Evaluatable->False, PageBreakBelow->False, CellFrameMargins->False, ImageMargins->{{0, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, Background->RGBColor[0.827451, 0.333333, 0.258824], Magnification->1], Cell[StyleData["ConferenceGraphicCell", "Presentation"], ShowCellBracket->False], Cell[StyleData["ConferenceGraphicCell", "SlideShow"], ShowCellBracket->False], Cell[StyleData["ConferenceGraphicCell", "Printout"], FontSize->8, Magnification->0.75] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["GraphicNoMagnification"], CellMargins->{{60, 10}, {7, 7}}, LineSpacing->{1, 3}, CounterIncrements->"Text", FontFamily->"Helvetica", Magnification->1], Cell[StyleData["GraphicNoMagnification", "Presentation"], CellMargins->{{72, 50}, {10, 10}}, LineSpacing->{1, 5}, FontSize->17], Cell[StyleData["GraphicNoMagnification", "SlideShow"], CellMargins->{{100, 50}, {10, 10}}, FontSize->17], Cell[StyleData["GraphicNoMagnification", "Printout"], CellMargins->{{2, 2}, {6, 6}}, FontSize->10] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Inline Formatting", "Section"], Cell["\<\ These styles are for modifying individual words or letters in a \ cell exclusive of the cell tag.\ \>", "Text"], Cell[StyleData["RM"], StyleMenuListing->None, FontWeight->"Plain", FontSlant->"Plain"], Cell[StyleData["BF"], StyleMenuListing->None, FontWeight->"Bold"], Cell[StyleData["IT"], StyleMenuListing->None, FontSlant->"Italic"], Cell[StyleData["TR"], StyleMenuListing->None, FontFamily->"Times", FontWeight->"Plain", FontSlant->"Plain"], Cell[StyleData["TI"], StyleMenuListing->None, FontFamily->"Times", FontWeight->"Plain", FontSlant->"Italic"], Cell[StyleData["TB"], StyleMenuListing->None, FontFamily->"Times", FontWeight->"Bold", FontSlant->"Plain"], Cell[StyleData["TBI"], StyleMenuListing->None, FontFamily->"Times", FontWeight->"Bold", FontSlant->"Italic"], Cell[StyleData["MR"], "TwoByteSyntaxCharacterAutoReplacement"->True, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, StyleMenuListing->None, FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], Cell[StyleData["MO"], "TwoByteSyntaxCharacterAutoReplacement"->True, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, StyleMenuListing->None, FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Italic"], Cell[StyleData["MB"], "TwoByteSyntaxCharacterAutoReplacement"->True, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, StyleMenuListing->None, FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain"], Cell[StyleData["MBO"], "TwoByteSyntaxCharacterAutoReplacement"->True, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, StyleMenuListing->None, FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Italic"], Cell[StyleData["SR"], StyleMenuListing->None, FontFamily->"Helvetica", FontWeight->"Plain", FontSlant->"Plain"], Cell[StyleData["SO"], StyleMenuListing->None, FontFamily->"Helvetica", FontWeight->"Plain", FontSlant->"Italic"], Cell[StyleData["SB"], StyleMenuListing->None, FontFamily->"Helvetica", FontWeight->"Bold", FontSlant->"Plain"], Cell[StyleData["SBO"], StyleMenuListing->None, FontFamily->"Helvetica", FontWeight->"Bold", FontSlant->"Italic"], Cell[CellGroupData[{ Cell[StyleData["SO10"], StyleMenuListing->None, FontFamily->"Helvetica", FontSize->10, FontWeight->"Plain", FontSlant->"Italic"], Cell[StyleData["SO10", "Printout"], StyleMenuListing->None, FontFamily->"Helvetica", FontSize->7, FontWeight->"Plain", FontSlant->"Italic"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Formulas and Programming", "Section"], Cell[CellGroupData[{ Cell[StyleData["InlineFormula"], CellMargins->{{10, 4}, {0, 8}}, CellHorizontalScrolling->True, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, LanguageCategory->"Formula", ScriptLevel->1, SingleLetterItalics->True, StyleMenuListing->None], Cell[StyleData["InlineFormula", "Presentation"], LineSpacing->{1, 5}, FontSize->16], Cell[StyleData["InlineFormula", "SlideShow"], LineSpacing->{1, 5}, FontSize->16], Cell[StyleData["InlineFormula", "Printout"], CellMargins->{{2, 0}, {6, 6}}, FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["DisplayFormula"], CellMargins->{{60, Inherited}, {Inherited, Inherited}}, CellHorizontalScrolling->True, DefaultFormatType->DefaultInputFormatType, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, LanguageCategory->"Formula", ScriptLevel->0, SingleLetterItalics->True, UnderoverscriptBoxOptions->{LimitsPositioning->True}], Cell[StyleData["DisplayFormula", "Presentation"], CellMargins->{{72, Inherited}, {Inherited, Inherited}}, LineSpacing->{1, 5}, FontSize->16], Cell[StyleData["DisplayFormula", "SlideShow"], CellMargins->{{100, 50}, {Inherited, Inherited}}, LineSpacing->{1, 5}, FontSize->16], Cell[StyleData["DisplayFormula", "Printout"], FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Program"], CellFrame->{{0, 0}, {0.5, 0.5}}, CellMargins->{{60, 4}, {0, 8}}, CellHorizontalScrolling->True, Hyphenation->False, LanguageCategory->"Formula", ScriptLevel->1, FontFamily->"Courier"], Cell[StyleData["Program", "Presentation"], CellMargins->{{72, 50}, {10, 10}}, LineSpacing->{1, 5}, FontSize->16], Cell[StyleData["Program", "SlideShow"], CellMargins->{{100, 50}, {10, 10}}, LineSpacing->{1, 5}, FontSize->16], Cell[StyleData["Program", "Printout"], CellMargins->{{2, 0}, {6, 6}}, FontSize->9] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Hyperlink Styles", "Section"], Cell["\<\ The cells below define styles useful for making hypertext \ ButtonBoxes. The \"Hyperlink\" style is for links within the same Notebook, \ or between Notebooks.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["Hyperlink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0.376471, 0.490196], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookLocate[ #2]}]&), Active->True, ButtonFrame->"None", ButtonNote->ButtonData}], Cell[StyleData["Hyperlink", "Presentation"], FontSize->16], Cell[StyleData["Hyperlink", "SlideShow"]], Cell[StyleData["Hyperlink", "Printout"], FontSize->10, FontVariations->{"Underline"->False}] }, Closed]], Cell["\<\ The following styles are for linking automatically to the on-line \ help system.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["MainBookLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0.269993, 0.308507, 0.6], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "MainBook", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["MainBookLink", "Presentation"], FontSize->16], Cell[StyleData["MainBookLink", "SlideShow"]], Cell[StyleData["MainBookLink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["AddOnsLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontFamily->"Courier", FontColor->RGBColor[0.269993, 0.308507, 0.6], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "AddOns", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["AddOnsLink", "Presentation"], FontSize->16], Cell[StyleData["AddOnsLink", "SlideShow"]], Cell[StyleData["AddOnsLink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["RefGuideLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontFamily->"Courier", FontColor->RGBColor[0.269993, 0.308507, 0.6], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "RefGuide", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["RefGuideLink", "Presentation"], FontSize->16], Cell[StyleData["RefGuideLink", "SlideShow"]], Cell[StyleData["RefGuideLink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["RefGuideLinkText"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0.269993, 0.308507, 0.6], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "RefGuide", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["RefGuideLinkText", "Presentation"], FontSize->16], Cell[StyleData["RefGuideLinkText", "SlideShow"]], Cell[StyleData["RefGuideLinkText", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["GettingStartedLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0.269993, 0.308507, 0.6], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "GettingStarted", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["GettingStartedLink", "Presentation"], FontSize->16], Cell[StyleData["GettingStartedLink", "SlideShow"]], Cell[StyleData["GettingStartedLink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["DemosLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0.269993, 0.308507, 0.6], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "Demos", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["DemosLink", "SlideShow"]], Cell[StyleData["DemosLink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["TourLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0.269993, 0.308507, 0.6], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "Tour", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["TourLink", "SlideShow"]], Cell[StyleData["TourLink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["OtherInformationLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0.269993, 0.308507, 0.6], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "OtherInformation", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["OtherInformationLink", "Presentation"], FontSize->16], Cell[StyleData["OtherInformationLink", "SlideShow"]], Cell[StyleData["OtherInformationLink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["MasterIndexLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0.269993, 0.308507, 0.6], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "MasterIndex", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["MasterIndexLink", "SlideShow"]], Cell[StyleData["MasterIndexLink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Headers and Footers", "Section"], Cell[StyleData["Header"], CellMargins->{{0, 0}, {4, 1}}, DefaultNewInlineCellStyle->"None", LanguageCategory->"NaturalLanguage", StyleMenuListing->None, FontSize->10, FontSlant->"Italic"], Cell[StyleData["Footer"], CellMargins->{{0, 0}, {0, 4}}, DefaultNewInlineCellStyle->"None", LanguageCategory->"NaturalLanguage", StyleMenuListing->None, FontSize->9, FontSlant->"Italic"], Cell[StyleData["PageNumber"], CellMargins->{{0, 0}, {4, 1}}, StyleMenuListing->None, FontFamily->"Times", FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell["Palette Styles", "Section"], Cell["\<\ The cells below define styles that define standard \ ButtonFunctions, for use in palette buttons.\ \>", "Text"], Cell[StyleData["Paste"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, Placeholder]}]&)}], Cell[StyleData["Evaluate"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], FrontEnd`SelectionEvaluate[ FrontEnd`InputNotebook[ ], All]}]&)}], Cell[StyleData["EvaluateCell"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], FrontEnd`SelectionMove[ FrontEnd`InputNotebook[ ], All, Cell, 1], FrontEnd`SelectionEvaluateCreateCell[ FrontEnd`InputNotebook[ ], All]}]&)}], Cell[StyleData["CopyEvaluate"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`SelectionCreateCell[ FrontEnd`InputNotebook[ ], All], FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], FrontEnd`SelectionEvaluate[ FrontEnd`InputNotebook[ ], All]}]&)}], Cell[StyleData["CopyEvaluateCell"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`SelectionCreateCell[ FrontEnd`InputNotebook[ ], All], FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], FrontEnd`SelectionEvaluateCreateCell[ FrontEnd`InputNotebook[ ], All]}]&)}] }, Closed]], Cell[CellGroupData[{ Cell["Placeholder Styles", "Section"], Cell["\<\ The cells below define styles useful for making placeholder \ objects in palette templates.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["Placeholder"], Placeholder->True, StyleMenuListing->None, FontSlant->"Italic", FontColor->RGBColor[0.890623, 0.864698, 0.384756], TagBoxOptions->{Editable->False, Selectable->False, StripWrapperBoxes->False}], Cell[StyleData["Placeholder", "Presentation"]], Cell[StyleData["Placeholder", "SlideShow"]], Cell[StyleData["Placeholder", "Printout"]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["PrimaryPlaceholder"], StyleMenuListing->None, DrawHighlighted->True, FontSlant->"Italic", Background->RGBColor[0.912505, 0.891798, 0.507774], TagBoxOptions->{Editable->False, Selectable->False, StripWrapperBoxes->False}], Cell[StyleData["PrimaryPlaceholder", "Presentation"]], Cell[StyleData["PrimaryPlaceholder", "SlideShow"]], Cell[StyleData["PrimaryPlaceholder", "Printout"]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["FormatType Styles", "Section"], Cell["\<\ The cells below define styles that are mixed in with the styles \ of most cells. If a cell's FormatType matches the name of one of the styles \ defined below, then that style is applied between the cell's style and its \ own options. This is particularly true of Input and Output.\ \>", "Text"], Cell[StyleData["CellExpression"], PageWidth->Infinity, CellMargins->{{6, Inherited}, {Inherited, Inherited}}, ShowCellLabel->False, ShowSpecialCharacters->False, AllowInlineCells->False, Hyphenation->False, AutoItalicWords->{}, StyleMenuListing->None, FontFamily->"Courier", FontSize->12, Background->GrayLevel[1]], Cell[StyleData["InputForm"], InputAutoReplacements->{}, AllowInlineCells->False, Hyphenation->False, StyleMenuListing->None, FontFamily->"Courier"], Cell[StyleData["OutputForm"], PageWidth->Infinity, TextAlignment->Left, LineSpacing->{0.6, 1}, StyleMenuListing->None, FontFamily->"Courier"], Cell[StyleData["StandardForm"], InputAutoReplacements->{ "->"->"\[Rule]", ":>"->"\[RuleDelayed]", "<="->"\[LessEqual]", ">="->"\[GreaterEqual]", "!="->"\[NotEqual]", "=="->"\[Equal]", Inherited}, "TwoByteSyntaxCharacterAutoReplacement"->True, LineSpacing->{1.25, 0}, StyleMenuListing->None, FontFamily->"Courier"], Cell[StyleData["TraditionalForm"], InputAutoReplacements->{ "->"->"\[Rule]", ":>"->"\[RuleDelayed]", "<="->"\[LessEqual]", ">="->"\[GreaterEqual]", "!="->"\[NotEqual]", "=="->"\[Equal]", Inherited}, "TwoByteSyntaxCharacterAutoReplacement"->True, LineSpacing->{1.25, 0}, SingleLetterItalics->True, TraditionalFunctionNotation->True, DelimiterMatching->None, StyleMenuListing->None], Cell["\<\ The style defined below is mixed in to any cell that is in an \ inline cell within another.\ \>", "Text"], Cell[StyleData["InlineCell"], LanguageCategory->"Formula", ScriptLevel->1, StyleMenuListing->None], Cell[StyleData["InlineCellEditing"], StyleMenuListing->None, Background->RGBColor[0.964706, 0.929412, 0.839216]] }, Closed]], Cell[CellGroupData[{ Cell["Automatic Styles", "Section"], Cell["\<\ The cells below define styles that are used to affect the display \ of certain types of objects in typeset expressions. 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