(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.0' 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[ 1058539, 28476]*) (*NotebookOutlinePosition[ 1059565, 28511]*) (* CellTagsIndexPosition[ 1059473, 28505]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Exploring Data Graphically in Mathematica", "Subtitle", ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ Robert A. Stine Statistics Department The Wharton School University of Pennsylvania Philadelphia, PA 19104-6340 www-stat.wharton.upenn.edu/~stine September 17, 1995 Revised November 4, 2003\ \>", "Subsubtitle", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{ Cell["Introduction", "Section", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ "In 1995, I produced a ", StyleBox["Mathematica", FontSlant->"Italic"], " package that I called ", StyleBox["DataTools.", FontSlant->"Italic"], " This package implemented a variety of statistical functions that were \ then either hard to do or not available in the standard distribution of ", StyleBox["Mathematica", FontSlant->"Italic"], ". Since then, ", StyleBox["Mathematica", FontSlant->"Italic"], " has grown, and the supplied packages have now filled the void that used \ to exist." }], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ "This notebook shows how to do the same things that I once implemented \ \"from scratch\" in ", StyleBox["DataTools", FontSlant->"Italic"], ", but uses features supplied in Version 5 of ", StyleBox["Mathematica", FontSlant->"Italic"], ". The problems (and solutions) are the same, but rather than using \ programs that I developed, I'm going to show how to make the most of \ now-included functions. For example, ", StyleBox["Mathematica", FontSlant->"Italic"], " now has a nice histogram function. It's also got boxplots. I'll use ", StyleBox["Mathematica", FontSlant->"Italic"], "'s version of the first, but I still like my own boxplot. I'll show you \ both and let you choose." }], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ To present these functions in a more realistic (and hopefully \ interesting) context, I introduce them through a small exploratory analysis \ of the daily returns (including dividends) on General Motors common stock \ during 1987-1988. This period includes the big drop in October, 1987. The \ data are in the file gm8788.dat which is available from my Web page. The \ original source for this data is CRSP, the Center for Research in Security \ Prices.\ \>", "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ "The analysis of these data progresses in a fashion intended to introduce \ the functions of originally presented in the ", StyleBox["DataTools", FontSlant->"Italic"], " package, and so might seem a bit contrived. After reading the data, the \ analysis begins with simple descriptive statistics that use binning to \ summarize the data a month at a time. The next two sections use univariate \ and bivariate plots, including histograms, boxplots, and kernel density \ estimates. The last 'technical' section explores how the volatility in these \ data tracks over time using regression. Throughout, the emphasis is upon \ plots that reveal interesting features." }], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ To use the functions that I have written, you'll need to load these \ functions. They're defined in the file DataTools.m.\ \>", "Text"], Cell[BoxData[ \(\(<< \ "\";\)\)], \ "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Reading and Writing Data", "Section", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{ Cell["ReadData", "Subsection"], Cell[TextData[{ "The format of a data file seems a personal choice, given the variety of \ formats out there. My function for reading data simply assumes that the data \ has the following layout. ", StyleBox["ReadData", FontWeight->"Bold"], " expects the data file to possess a very simple format which is often used \ by commercial statistics packages for interchanging data. The first line of \ the file is a quoted string used to document the file; this allows you to \ insert a comment to remind you of the file's contents. The second line of \ the file is a list of variable names separated by one or more spaces. The \ remaining lines of the file are the observation records, with as many values \ per line as there are variable names. For example, a file with two variables \ and 3 observations would appear as follows:" }], "Text"], Cell["\<\ \"A simple file example\" Var1 Var2 2 3 12 na 1 3\ \>", "Text"], Cell[TextData[{ "Missing data can be denoted by whatever symbol you choose, such as the ", StyleBox["na", FontSlant->"Italic"], " symbol used here or any symbolic tag such as the built-in symbol \ Indeterminant. ", StyleBox["ReadData", FontWeight->"Bold"], " prints the documentation string. It then defines the symbols, here Var1 \ and Var2. After running ", StyleBox["ReadData", FontWeight->"Bold"], " on this file, you'll have two new variables, each a list with 3 values. \ ", StyleBox["ReadData", FontWeight->"Bold"], " avoids over-writing a pre-existing symbol, as explained further below." }], "Text"], Cell[TextData[{ "The data file gm8788.dat contains 5 columns of daily information: the day \ month, year, price and return on General Motors common stock for the 506 \ trading days during 1987 and 1988. Each line of the file (after the first \ two descriptive lines) contains five values that define these variables for \ each day. To distinguish the names of variables read from the file from \ other symbols, I used upper-case characters in the name for each (for \ example, DAY for the day of the month). Here are the first few lines of the \ data file gm8788.dat, along with the output of the ", StyleBox["ReadData", FontSlant->"Italic"], " function. (You'll need to tell ", StyleBox["Mathematica", FontSlant->"Italic"], " where to look for the data file on your system.)" }], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ \"Returns on General Motors Stock during 1987-1988\" DAY MONTH YEAR PRICE RETURN 02 1 1987 66.875 0.01325758 05 1 1987 68.5 0.02429906 06 1 1987 68.75 0.003649635 07 1 1987 69.125 0.005454545\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(ReadData["\<~/Documents/Mathematica/DataTools/gm8788.dat\>"]\)], "Input",\ InitializationCell->True, ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ \(ReadData::"doc" \(\(:\)\(\ \)\) "Document string: \!\(\"\\\"Returns on General Motors Stock during \ 1987-1988\\\"\"\)"\)], "Message"], Cell[BoxData[ \(ReadData::"names" \(\(:\)\(\ \)\) "Symbols defined in this file: \!\({\"DAY\", \"MONTH\", \"YEAR\", \ \"PRICE\", \"RETURN\"}\)"\)], "Message"], Cell[BoxData[ InterpretationBox[\("Variables to be read: "\[InvisibleSpace]{DAY, MONTH, YEAR, PRICE, RETURN}\), SequenceForm[ "Variables to be read: ", {DAY, MONTH, YEAR, PRICE, RETURN}], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\(506\[InvisibleSpace]" cases were read."\), SequenceForm[ 506, " cases were read."], Editable->False]], "Print"] }, Open ]], Cell["\<\ After running this function, the named columns are defined. A list \ named by the associated symbol extracted from the second line of the input \ file holds the data for each variable. For example, an abbreviated list of \ the returns is\ \>", "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{ Cell[BoxData[ \(Short[RETURN]\)], "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ TagBox[\({0.01325758`, 0.02429906`, 0.003649635`, \[LeftSkeleton]500\[RightSkeleton], \(-0.001464129`\), 0, \(-0.02052786`\)}\), Short]], "Output"] }, Open ]], Cell["\<\ This output shows the first three daily returns, then indicates 500 \ values are hidden, and then shows the last three daily returns.\ \>", "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ StyleBox["ReadData", FontWeight->"Bold"], " does not overwrite previously defined symbols. If some of the symbols \ which are to denote variables (in this example, the symbols DAY, MONTH, YEAR, \ PRICE and RETURN) are already defined in the active session, a warning \ message is printed. This feature avoids accidentally replacing an important \ piece of data by new data read from the external file. For example, a second \ attempt to read the same file generates a message but does not define any \ variables." }], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{ Cell[BoxData[ \(ReadData["\<~/Desktop/gm8788.dat\>"]\)], "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ RowBox[{\(OpenRead::"noopen"\), \(\(:\)\(\ \)\), "\<\"Cannot open \ \\!\\(\\\"/Users/bob/Desktop/gm8788.dat\\\"\\). \\!\\(\\*ButtonBox[\\\"More\ \[Ellipsis]\\\", ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, \ ButtonData:>\\\"General::noopen\\\"]\\)\"\>"}]], "Message"], Cell[BoxData[ RowBox[{\(ReadData::"notfound"\), \(\(:\)\(\ \)\), "\<\"File \ \\!\\(\\\"~/Desktop/gm8788.dat\\\"\\) not found. \\!\\(\\*ButtonBox[\\\"More\ \[Ellipsis]\\\", ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, \ ButtonData:>\\\"General::notfound\\\"]\\)\"\>"}]], "Message"] }, Open ]], Cell[TextData[{ "To re-read data (if you've made a change to the data, for example) use ", StyleBox["Clear", FontWeight->"Bold"], " to remove the old definitions of the conflicting symbols (as in", StyleBox[" Clear[DAY, MONTH, YEAR, PRICE, RETURN]", FontWeight->"Bold"], ") and repeat the read." }], "Text", ImageRegion->{{0, 1}, {0, 1}}] }, Closed]], Cell[CellGroupData[{ Cell["WriteData", "Subsection"], Cell[TextData[{ "The function ", StyleBox["WriteData", FontWeight->"Bold"], " creates a file with this format by combining several column lists of \ values. With the data defined, we can now turn to some basic plots and \ summary statistics." }], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ \(WriteData[{RETURN}, "\<~/Desktop/test.dat\>", \ "\"]\)], \ "Input"] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["Descriptive Statistics, Sequence Plots, and Binning", "Section", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{ Cell["Descriptive Statistics", "Subsection", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " now includes various descriptive statistics such as the mean and standard \ deviation. With the improvements in the underlying numerical algorithms, \ these run quite fast. Once upon a time (check out the original description of \ this package in Varian) it mattered a great deal how one defined numerical \ calculations in ", StyleBox["Mathematica", FontSlant->"Italic"], ". With the improvements in speed of the host systems and the kernel of ", StyleBox["Mathematica", FontSlant->"Italic"], " itself, these concerns are pretty irrelevant now. That said, I suggest \ using the built-in mean function rather than rolling your own. Its a lot \ faster." }], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ Here are some basic examples. The average and standard deviation of \ the returns over these two years are:\ \>", "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{ Cell[BoxData[ \(ave = Mean[RETURN]\)], "Input", InitializationCell->True, ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ \(0.0009334009565217392`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(sd = StandardDeviation[RETURN]\)], "Input", InitializationCell->True, ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ \(0.02009482277342592`\)], "Output"] }, Open ]], Cell["\<\ Other summaries include the variance, the median, and the \ interquartile range. They've even included the MAD, or median absolute \ deviation about the median. \ \>", "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{ Cell[BoxData[ \(MedianDeviation[RETURN]\)], "Input"], Cell[BoxData[ \(0.0093823355`\)], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Missing Values", "Subsection"], Cell["\<\ These summaries are pretty simple unless you have missing data. For \ example, if we just add a symbol to the end of the returns, the averaging \ function no longer works. Well, not in the sense that you might want for \ analyzing data. You end up with an expression that includes the missing \ symbol. Fine for symbolic algebra (it makes it fun to draw the influence \ curve), but not what you want if the data is simply missing.\ \>", "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{ Cell[BoxData[ \(Mean[Append[RETURN, miss]]\)], "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ \(1\/507\ \((\(\(0.47230088400000003`\)\(\[InvisibleSpace]\)\) + miss)\)\)], "Output"] }, Open ]], Cell["\<\ So, to handle missing data you need to first filter the data to \ remove terms that are not numbers. The function DropNonNumeric from \ Statistics`DataManipulation` handles this conveniently.\ \>", "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{ Cell[BoxData[ \(Mean[DropNonNumeric[Append[RETURN, miss]]]\)], "Input"], Cell[BoxData[ \(0.0009334009565217392`\)], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Care About Speed?", "Subsection"], Cell[TextData[{ "Before moving on, these simple functions offer some insight into the speed \ ", StyleBox["Mathematica", FontSlant->"Italic"], ". The built in functions use the developers considerable knowledge of how \ to make the code run faster. Consider the time that it takes to compute the \ average of 100,000 uniform random numbers." }], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ \(\(x = Table[Random[], {100000}];\)\)], "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ "The obvious way to compute the average is to use the function ", StyleBox["Plus", FontWeight->"Bold"], ". Repeating the calculation 100 times takes about 2.2 seconds. (The \ command ", StyleBox["Plus@@x", FontWeight->"Bold"], " is read as the summation ", StyleBox["x", FontWeight->"Bold"], ".)" }], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{ Cell[BoxData[ \(First[Timing[Table[Plus @@ x, {100}]]]\)], "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ \(21.639999999999997`\ Second\)], "Output"] }, Open ]], Cell["\<\ In comparison, the built-in function is about 50 times faster. You \ can see how they get this to happen by looking at the code that implements \ the various descriptive statistics. It pays to recognize the data is a \ packed array. \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(First[Timing[Table[Mean[x], {100}]]]\)], "Input"], Cell[BoxData[ \(0.4299999999999997`\ Second\)], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Sequence Plots and Dates", "Subsection", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ "A careful analysis of any time series begins with sequence plots. The \ built-in function ", StyleBox["ListPlot", FontWeight->"Bold"], " requires some simple but tedious manipulations which my wrapper ", StyleBox["ScatterPlot", FontWeight->"Bold"], " provides automatically when given a single list as its argument. ", StyleBox["ScatterPlot", FontWeight->"Bold"], " is basically a call to ", StyleBox["ListPlot", FontWeight->"Bold"], ", but Scatterplot adds some touches that are important in data analysis, \ including a transposed argument list and setting the option ", StyleBox["PlotRange", FontWeight->"Bold"], " set to show all of the data. The options of Scatterplot, such as that to \ add a label to the plot, are a superset of those of ", StyleBox["ListPlot", FontWeight->"Bold"], "." }], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{ Cell[BoxData[ \(\(ScatterPlot[PRICE, PlotLabel \[Rule] "\"];\)\)], "Input", ImageRegion->{{0, 1}, {0, 1}}], 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 0.0219236 0.0018859 -0.720159 0.0140983 [ [.21051 .11324 -9 -9 ] [.21051 .11324 9 0 ] [.3991 .11324 -9 -9 ] [.3991 .11324 9 0 ] [.58769 .11324 -9 -9 ] [.58769 .11324 9 0 ] [.77628 .11324 -9 -9 ] [.77628 .11324 9 0 ] [.96488 .11324 -9 -9 ] [.96488 .11324 9 0 ] [.00942 .26672 -12 -4.5 ] [.00942 .26672 0 4.5 ] [.00942 .40771 -12 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Cell["\<\ With a little more effort, we can give the horizontal axis a more \ useful set of labels that identify the time of the big drop in the prices, \ and the large swing in returns (as if you did not know when these happened). \ The first step is to associate dates with the axis labels. Since October 19, \ 1987 (day number 202), is the day the stock market fell substantially, it's \ useful to label this specific date.\ \>", "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{ Cell[BoxData[ \(Position[RETURN, Min[RETURN]]\)], "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ \({{202}}\)], "Output"] }, Open ]], Cell["\<\ Having a function that forms the labels makes the calculations more \ easy to follow. Here is a simple version that builds a date string. The \ result is a (numerical value, date string) pair. \ \>", "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ \(calDate[index_] := With[{m = {"\", "\", "\", "\", "\", \ "\", "\", "\", "\", "\", "\", \ "\"}}, {index, ToString[ DAY\[LeftDoubleBracket]index\[RightDoubleBracket]] <> "\< \>" <> m\[LeftDoubleBracket]MONTH\[LeftDoubleBracket] index\[RightDoubleBracket]\[RightDoubleBracket] <> "\< \>" <> ToString[ YEAR\[LeftDoubleBracket] index\[RightDoubleBracket]]}]\)], "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{ Cell[BoxData[ \(calDate[1]\)], "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell[BoxData[ \({1, "2 Jan 1987"}\)], "Output"] }, Open ]], Cell[TextData[{ "These labels are compatible with the standard ", StyleBox["Ticks", FontWeight->"Bold"], " option used in plots in ", StyleBox["Mathematica", FontSlant->"Italic"], ". It's probably also a good idea with these extended tick marks to move \ the axis beneath the data and make the plot relatively longer. Also, since \ such date labels will crowd on the axis, the ", StyleBox["Tick", FontWeight->"Bold"], " option below provides just three dates. 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