(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of 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[ 291660, 8315]*) (*NotebookOutlinePosition[ 295593, 8443]*) (* CellTagsIndexPosition[ 294989, 8420]*) (*WindowFrame->Normal*) Notebook[{ Cell["H1 Help File for Fitting Functions to Data", "Subtitle", FontSize->16, FontWeight->"Bold"], Cell["How to Use This Help File", "Subsection", Background->RGBColor[0, 1, 1]], Cell[TextData[{ "When evaluating a cell for the first time, a window with the following \ text will be displayed:\n", StyleBox[ "Do you want to automatically evaluate all the initialization cells in the \ notebook 'DataFitH.nb?", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["\n", FontWeight->"Bold"], "Click on ", StyleBox["Yes", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox[" ", FontWeight->"Bold"], " to insure that all the data and functions are available for this help \ file to function properly." }], "Text", CellFrame->True, Background->GrayLevel[0.849989]], Cell[BoxData[ \(<< DataFit`\)], "Input", InitializationCell->True], Cell["Common Features of the Palette Functions", "Subsection"], Cell[TextData[{ StyleBox[ "The palette functions described in this help file are designed to aid in \ fitting functions to given data. All these functions assume that the data in \ ", CellFrame->True, Background->GrayLevel[0.849989]], StyleBox["list", CellFrame->True, FontWeight->"Bold", Background->GrayLevel[0.849989]], StyleBox[ " is given as pairs of input-output values. If the first entries are the \ names of the input and output variables, then the axes will be labeled \ accordingly. All the functions that fit a model to given data can be given \ optional arguments of the form xplot\[ShortRightArrow]{xmin, xmax} and/or \ yplot\[ShortRightArrow]{ymin, ymax} to change the ranges of input and output \ values that are displayed. By default the display window is slightly larger \ than the data. ", CellFrame->True, Background->GrayLevel[0.849989]] }], "Text", CellFrame->True, Background->GrayLevel[0.849989]], Cell["\<\ How to Locate Specific Palette Functions within the Help File\ \>", "Subsection"], Cell[TextData[{ "When you click on any of the buttons below, you will be linked to a \ description of the selected function. A highlighted cell bracket will \ indicate which cell contains the relevant explanation. The buttons are \ grouped vertically according to their usage. The first column contains \ functions that are not geared toward a specific function type, whereas each \ of the other columns are associated with fitting a specific function type. \ For example, when fitting a polynomial function to data, you may want to use \ the palette functions ", StyleBox["FirstUnitDiff", FontWeight->"Bold"], " or ", StyleBox["SecondUnitDiff", FontWeight->"Bold"], " as a numerical check on whether a first or second degree polynomial is \ appropriate. 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fitted polynomial \ function, where "], StyleBox["list", FontWeight->"Bold"], StyleBox[" contains the data and "], StyleBox["n", FontWeight->"Bold"], StyleBox[" indicates the degree of the polynomial to be fitted. "] }], "Text", CellTags->"FittedPoly"], Cell[CellGroupData[{ Cell[BoxData[ \(PolyFitFunc[A, 2]\)], "Input"], Cell[BoxData[ \(\(44.6849999999986202`\[InvisibleSpace]\) - 2.11435714285703379`\ x + 0.0263571428571418664`\ x\^2\)], "Output"] }, Open ]], Cell[BoxData[ StyleBox[ ButtonBox[ StyleBox[\(Back\ to\ top\), "IndentedText", FontWeight->"Plain"], ButtonData:>"top", ButtonStyle->"Hyperlink"], "IndentedText", FontWeight->"Plain"]], "Input", Evaluatable->False, FontSize->9], Cell[TextData[{ StyleBox["\n/.{x\[RightArrow]\[Placeholder]} ", FontWeight->"Bold"], StyleBox[ "is used to evaluate any of the fitted functions for a particular value of \ the independent variable. For example, after fitting a polynomial function to \ the death rate data, we may want to predict the death rate of a 20 year old \ (not given in the table). 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ButtonData:>"top", ButtonStyle->"Hyperlink"], "IndentedText", FontWeight->"Plain"]], "Input", Evaluatable->False, FontSize->9], Cell[TextData[{ StyleBox["\nFirstUnitDiff[list]", FontWeight->"Bold"], " computes the first unit differences (= slope) of the data given in ", StyleBox["list", FontWeight->"Bold"], ". If the resulting unit differences are approximately constant, then a \ linear function is an appropriate type to be fitted. To check whether the \ death rates follow a linear pattern, we use the command below. " }], "Text", PageBreakBelow->False, CellTags->"FirstUnitDiff"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ StyleBox["FirstUnitDiff", FontWeight->"Bold"], "[", "A", "]"}]], "Input"], Cell[BoxData[ \({0.159999999999999964`, 0.4`, 0.6`, 0.919999999999999928`, 1.21999999999999997`}\)], "Output"] }, Open ]], Cell[TextData[{ "The result indicates that we should ", StyleBox["not", FontVariations->{"Underline"->True}], " fit a linear function, since the first unit differences are not \ approximately constant." }], "Text"], Cell[BoxData[ StyleBox[ ButtonBox[ StyleBox[\(Back\ to\ top\), "IndentedText", FontWeight->"Plain"], ButtonData:>"top", ButtonStyle->"Hyperlink"], "IndentedText", FontWeight->"Plain"]], "Input", Evaluatable->False, FontSize->9], Cell[TextData[{ StyleBox["SecondUnitDiff[list]", FontWeight->"Bold"], " computes the second unit differences of the data given in ", StyleBox["list", FontWeight->"Bold"], ". This function can be used to check whether a quadratic function is a \ suitable choice for a fit. To check whether the death rates follow a \ quadratic relationship, we would use" }], "Text", PageBreakBelow->False, CellTags->"SecondUnitDiff"], Cell[CellGroupData[{ Cell[BoxData[ \(SecondUnitDiff[A]\)], "Input"], Cell[BoxData[ \({0.0240000000000000035`, 0.0199999999999999964`, 0.0319999999999999928`, 0.0300000000000000044`}\)], "Output"] }, Open ]], Cell["\<\ Since the resulting differences are approximately constant, a \ quadratic function should fit reasonably well.\ \>", "Text"], Cell[BoxData[ StyleBox[ ButtonBox[ StyleBox[\(Back\ to\ top\), "IndentedText", FontWeight->"Plain"], ButtonData:>"top", ButtonStyle->"Hyperlink"], "IndentedText", FontWeight->"Plain"]], "Input", Evaluatable->False, FontSize->9], Cell[TextData[{ StyleBox["ExpoFitGraph[list]", FontWeight->"Bold"], " fits an exponential function of the form ", Cell[BoxData[ \(TraditionalForm\`a\ b\^x\)]], Cell[BoxData[ \(TraditionalForm\`, \)]], " where ", Cell[BoxData[ \(TraditionalForm\`\(a\ \)\)]], "and ", Cell[BoxData[ \(TraditionalForm\`b\)]], " are constants, and displays the data together with a graph of the fitted \ function. 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If we assume that the horizontal asymptote is at \ level -3, then we have to adjust the output values by adding +3 (i.e., the \ negative of the value of the level). This can be done with the palette \ function ", StyleBox["ShiftOutput.", FontWeight->"Bold"] }], "Text"], Cell[BoxData[ StyleBox[ ButtonBox[ StyleBox[\(Back\ to\ top\), "IndentedText", FontWeight->"Plain"], ButtonData:>"top", ButtonStyle->"Hyperlink"], "IndentedText", FontWeight->"Plain"]], "Input", Evaluatable->False, FontSize->9], Cell[TextData[{ StyleBox["ShiftOutput[list, s]", FontWeight->"Bold"], " adds an amount ", StyleBox["s", FontWeight->"Bold"], " to each output value in ", StyleBox["list", FontWeight->"Bold"], ". This results in a vertical shift in the graph of the data (for s < 0 the \ graph moves down, and for s > 0 the graph moves up). The shape of the graph \ does not change, only the units on the vertical axis. 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Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]], Cell[BoxData[ StyleBox[ ButtonBox[ StyleBox[\(Back\ to\ top\), "IndentedText", FontWeight->"Plain"], ButtonData:>"top", ButtonStyle->"Hyperlink"], "IndentedText", FontWeight->"Plain"]], "Input", Evaluatable->False, FontSize->9], Cell[TextData[{ StyleBox["ExpoFitFunc[list] ", FontWeight->"Bold"], StyleBox[ "gives the functional expression (= equation) of the fitted exponential \ function, where "], StyleBox["list", FontWeight->"Bold"], StyleBox[" contains the data to which the function is being fitted. "] }], "Text", CellTags->"FittedExpo"], Cell[CellGroupData[{ Cell[BoxData[ \(ExpoFitFunc[exposhift]\)], "Input"], Cell[BoxData[ \(3.00616868377670787`\ 2.71210351913911118`\^x\)], "Output"] }, Open ]], Cell[TextData[{ "Note that we have to make the reverse adjustment that we made to the data \ to get an equation of the function that fits the ", StyleBox["original", FontVariations->{"Underline"->True}], " data. Thus, the equation of the function fitting the data given in the \ list ", StyleBox["expo ", FontWeight->"Bold"], "is given by:\n\t\n\t3.00617 * ", Cell[BoxData[ \(TraditionalForm\`2.7121\^x\)]], " + 3" }], "Text"], Cell[BoxData[ StyleBox[ ButtonBox[ StyleBox[\(Back\ to\ top\), "IndentedText", FontWeight->"Plain"], ButtonData:>"top", ButtonStyle->"Hyperlink"], "IndentedText", FontWeight->"Plain"]], "Input", Evaluatable->False, FontSize->9], Cell[TextData[{ StyleBox["\nUnitRatios[list]", FontWeight->"Bold"], " computes the unit ratios of the data given in ", StyleBox["list", FontWeight->"Bold"], ". The results from ", StyleBox["UnitRatios", FontWeight->"Bold"], " should be approximately constant if an exponential fit (with horizontal \ asymptote at 0) is appropriate. If the horizontal asymptote is not at level \ 0, the data needs to be adjusted using ", StyleBox["ShiftOutput ", FontWeight->"Bold"], "prior to using ", StyleBox["UnitRatios", FontWeight->"Bold"], ". Below is an example using the data in the table expo. We have already \ made the appropriate adjustment in the table ", StyleBox["exposhift", FontWeight->"Bold"], "; hence, we can use ", StyleBox["UnitRatios", FontWeight->"Bold"], " on the modified list." }], "Text", PageBreakBelow->False, CellTags->"UnitRatios"], Cell[CellGroupData[{ Cell[BoxData[ \(UnitRatios[exposhift]\)], "Input", PageBreakBelow->False], Cell[BoxData[ \({2.70100271568898797`, 2.71706315662359587`, 2.72250000000000058`, 2.70707070707070718`, 2.74470928937402502`}\)], "Output", PageBreakBelow->False] }, Open ]], Cell[TextData[{ "Note that the ratios are almost constant, so an exponential function (with \ horizontal asymptote at level 0) should fit reasonably well. However, compare \ the result of ", StyleBox["UnitRatios", FontWeight->"Bold"], " on the shifted data with using ", StyleBox["UnitRatios ", FontWeight->"Bold"], "on the original data:" }], "Text", PageBreakBelow->False], Cell[CellGroupData[{ Cell[BoxData[ \(UnitRatios[expo]\)], "Input", PageBreakBelow->False], Cell[BoxData[ \({0.592090594342288767`, \(-1.65254237288135596`\), 5.33333333333333392`, 3.40828402366863869`}\)], "Output", PageBreakBelow->False] }, Open ]], Cell[TextData[{ "These are not at all constant, which would indicate that the data does not \ follow an exponential function! The reason for this result is that ", StyleBox["UnitRatios", FontWeight->"Bold"], " works only for the case of data that has a horizontal asymptote at level \ 0 - thus,", StyleBox[ " it is very important to first shift the data so that it has the proper \ level for the asymptote!!!", FontWeight->"Bold"] }], "Text", PageBreakBelow->False], Cell[BoxData[ StyleBox[ ButtonBox[ StyleBox[\(Back\ to\ top\), "IndentedText", FontWeight->"Plain"], ButtonData:>"top", ButtonStyle->"Hyperlink"], "IndentedText", FontWeight->"Plain"]], "Input", Evaluatable->False, PageBreakBelow->False, FontSize->9], Cell[TextData[{ StyleBox["\nSineFitGraph[list, period]", FontWeight->"Bold"], " fits a function of the form", StyleBox[" a\[CenterDot]sin(bx+c)+ d ", FontSlant->"Italic"], "to the data given in ", StyleBox["list", FontWeight->"Bold"], ", where ", StyleBox["a", FontSlant->"Italic"], ", ", StyleBox["b", FontSlant->"Italic"], ", ", StyleBox["c,", FontSlant->"Italic"], " and ", StyleBox["d", FontSlant->"Italic"], " are constants, and ", StyleBox["period", FontWeight->"Bold"], " is the length of the repeated segment of the function as estimated from \ the graph of the data. We will use the data in table ", StyleBox["B", FontWeight->"Bold"], " to illustrate how to determine the value for the period." }], "Text", CellTags->"SineFit"], Cell[BoxData[ RowBox[{ RowBox[{"B", "=", GridBox[{ {\(-2.0\), \(-2.1\)}, {\(-1.5\), "2"}, {\(-1\), "2.6"}, {\(-.5\), \(-.58\)}, {"0", \(-1.9\)}, {".5", "1.7"}, {"1", "2.2"}, {"1.5", \(-.57\)}, {"2", \(-1.2\)}, {"2.5", "1.7"}, {"3", "2.6"}, {"3.5", \(-.54\)}, {"4", \(-1.8\)} }, RowLines->True]}], ";"}]], "Input", InitializationCell->True, GridBoxOptions->{RowLines->True, ColumnLines->True}], Cell["\<\ The easiest way to determine the period is to use the graph of the \ data together with the table of values. 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For the period we look for repeated values \ at which the function is either increasing or decreasing for both points. For \ example, we can use the points (0.5, 1.7) and (2.5, 1.7). The ", StyleBox["period", FontWeight->"Bold"], " is given by the difference in ", StyleBox["input", FontVariations->{"Underline"->True}], " values: \n\t\t\t\t", StyleBox["period = ", FontWeight->"Bold"], "2.5 - 0.5 = ", StyleBox["2", FontWeight->"Bold"], ". \n\t\t\t\t\nOther possible pairs of points are (-0.5, -0.58) and (1.5, \ -0.57), or (1.5, -0.57) and (3.5, -0.54). Note that the output values do not \ have to be exactly the same, as this happens very infrequently in real data.\n\ \nNow we can use the function ", StyleBox["SineFitGraph ", FontWeight->"Bold"], "to fit a sine curve, resulting in a graph of the fitted function together \ with the data. ", StyleBox[ "Be prepared to wait a little as the sine fit takes somewhat longer than \ the other function types. 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Cell[OutputFormData[ "\<\ 43607.6003017751/(1 + 602.8090464993892/E^(0.08894841625760907*x))\ \>", "\<\ 43607.6 ---------------- 602.809 1 + ------------ 0.0889484 x E\ \>"], "Output", PageBreakBelow->False] }, Open ]], Cell[BoxData[ StyleBox[ ButtonBox[ StyleBox[\(Back\ to\ top\), "IndentedText", FontWeight->"Plain"], ButtonData:>"top", ButtonStyle->"Hyperlink"], "IndentedText", FontWeight->"Plain"]], "Input", Evaluatable->False, FontSize->9], Cell[TextData[{ StyleBox["\nFitComp[list, {funcs}] ", FontWeight->"Bold"], "creates a table containing the input and output values as given in ", StyleBox["list", FontWeight->"Bold"], ", together with the values predicted for the list of function(s) given in", StyleBox[" funcs", FontWeight->"Bold"], " (expressed in terms of ", StyleBox["x", FontWeight->"Bold"], "). In addition, the differences between prediction and data are computed \ for each input value. As an illustration, we will fit both a quadratic and an \ exponential function to the death rate data (given earlier as table ", StyleBox["A", FontWeight->"Bold"], " in the description of ", StyleBox["PlotData", FontWeight->"Bold"], "). 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