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A Tour of Trellis Graphics

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1. Le wind temperature o 100 250 0 100 250 0 100 250 o 100 250 0 100 250 Ligii Ce et oa a z P 0 100 250 byiiiiit Cube Root Ozone cube root ppb La 2 ARREA rrr T irr rir TtT o 100 0 100 250 0 100 250 0 100 250 o 250 0 100 250 Radiation langleys Figure 15 Loess fit to the ozone data The key appears in color on a color device The leftmost 6 columns show temperature for the lowest 3 levels of wind speed the rightmost 6 columns are for the highest wind speeds Now that we have used the panel and strip functions it should come as little surprise that there is another function accepted by high level Trellis functions A page function is called at the end of each page of output and can be used to put on page numbers or other identifying information For example xyplot page function x if x gt 1 mtext paste page x side 1 line par oma 1 1 outer TRUE adj 1 would number each page but the first of a multi page display placing the page number at the bot tom right of the page 2 7 Axes Alignment of axes is one characteristic that makes the layout of Trellis displays so power ful Unlike arbitrary sized and positioned windows such as those produced by many software packages Trellis displays ensure that axes are aligned and can be readily compared to one another In support of this capability Trelli
2. conventionally the dependent variable is plotted on the vertical axis the variable for the horizontal axis is given after the operator and given or conditioning variables are mentioned last The data ethanol tells xyplot to look first in the data frame ethanol for the objects NOx E and C in the formula A data frame contains a set of related vectors and can be operated upon as if it were a matrix however data frames can hold data of various types including 6 character and numeric vectors factors and shingles Data mentioned in a formula can come from anywhere on the S search list However data frames are often a convenient way to keep related vectors together Suppose that we would like to control the layout of the five panels in Figure 2 We can do that with the layout argument xyplot NOx E C data ethanol layout c 3 2 1 Figure 3 This produces a layout with 3 columns and 2 rows on 1 page Notice that layout specifies columns and then rows unlike matrix row column notation We do this because we are dealing with graphs and the convention with graphs is to have an origin in the lower left corner We start there and proceed left to right bottom to top page to page The number of panels to be produced in a Trellis display is determined by the number of levels in the given variables However if the layout argument allows less room than required for all of the panels only the panels that will fit
3. Bass 1 feersrceenecnenetnece e Bass 2 jeacee creeper oc He 60 65 70 75 Height inches Figure 9 A box and whisker plot showing heights of members of a choral group arranged ac cording to the part they sing given variable is a character vector it is automatically turned into a factor with the sorted unique values of the vector as the factor s levels if it is numeric it is turned into a shingle with zero width intervals at the unique values of the numeric variable in this case it may be better to create a shingle yourself perhaps by using the equal count function so that you can control the inter vals chosen Certain Trellis functions take slightly different kinds of formulas Univariate functions that produce a whole panel from one set of data omit the y variable and have a formula like x gl g2 A rationale for this is that for example a histogram plots the data values along the x axis and internally computes the y values that determine the heights of the bars So Figure 10 is produced from histogram height voice part data singer xlab Height inches Figure 10 In Figure 10 each value of the given variable has produced a separate panel This is a general principle the x and y variables to the left of the vertical bar in a formula make up each panel and the given variables cause multiple panels to be produced The function qq takes a simple formula y x gl g
4. wind perathperathperathpelathperathperall_ 5 0 4 5 4 0 3 5 3 0 2 5 2 0 wind wind jaa pam en wind perathperathperathpelathperathperatl_ TETTETET TTT TTT TTT TTT TTT TT TTT TT TTT TT 0 250 0 250 0 250 Radiation langleys Figure 14 Loess fit to the ozone data with aspect ratio chosen by banking computations axis of a graph generally increases from bottom to top and if given variables have their levels in increasing order the standard Trellis panel order will make the given variables increase from bot tom to top That said there are occasions when a top to bottom order might be convenient This is called table order left to right top to bottom and is specified by using as table TRUE as an argument to a high level Trellis function For an example see the function example calendar Notice that the plot of Figure 17 contains a key that allows the reader to see how the shingle colors correspond to the given variables Also for those of you who are looking carefully the rectangle parameters given to key are done in reverse order since key draws things top to bottom while strip labels are constructed bottom to top The in line strip function was used to sup press the strip names Also the between argument allows us to insert space in units of charac ters between panels in the x or y direction we used it to distinguish between the two major lev els of wind speed
5. NOxplot2 lt update NOxplot pch X change plotting character NOxplot2 display it You can also save the result of update and update that 2 6 Layout The formula given to a Trellis display determines the order in which subsets of the data are produced The first packet of Trellis data corresponds to the first level of each of the given vari ables or the first interval of a shingle The second packet is at the second level of the first given variable and the first level of each of the other given variables This goes on until the last level of the first given variable Next the second level of the second given variable is reached and all other variables go to their first level All of the data packets are produced in this way This deter mines the packet order If there are N1 N2 and N3 levels for 3 given variables then there will be a total of N1 N2 N3 packets How does this relate to the layout of panels on the page The page is divided into columns and rows as specified by the layout argument The panel order is defined so that the first panel begins at the lower left corner and successive panels fill the bottom row Next panels fill the second row from left to right The total number of panels in the panel order is determined by the layout specification Remember that Trellis displays are filled as graphs from the origin in the lower left not top down as in a table That is also why the number of horizontal panels columns pr
6. No 462 Svansota 0 Trebi Wisconsin No 38 No 457 Glabron Peatland Velvet No 475 Manchuria No 462 Svansota Barley Yield bushels acre Figure 7 A dotplot shows how the yield of 10 varieties of barley varied over 6 sites and 2 years Notice that the 1931 yields were generally higher than 1932 except at the Morris site a likely data transcription error sif supply panel functions for doing many kinds of Trellis displays but you always have the option to write a custom panel function to draw just the right thing in each panel 2 1 Display Functions The Trellis graphics software comes with many display functions that produce various types of complete graphs One way of classifying the display functions is according to the dimension ality of the incoming data Here s a list Data Type Univariate Bivariate Trivariate Hypervariate 3 D Displays Table 1 Trellis Display Functions Function barchart bwplot densityplot dotplot histogram piechart qqmath stripplot qq timeplot xyplot contourplot levelplot splom parallel wireframe cloud 2 2 Customization for Devices Type of Plot bar plots box and whisker plots kernel density plot dot plots histograms pie charts yuck quantile plots against mathematical distribution 1 dimensional scatter plot qq plot for comparing two distributions time series plot scatter plot contour plot level plots
7. nate from the bottom to the top of the page they would all be on the bottom side Each panel s y axis would be logarithmic base 2 would have approximately 17 tick labels although no tick marks and the y axes would all have the same range the sliced specification On both axes the character size would be 75 of standard for that device Of course most of the time you will not need to specify so much about axes but the point is that the scales argument gives you lots of control In fact it also lets you suppress the draw ing of one or both axes scales list draw FALSE suppresses both the x and y axes This might be appropriate if for example the plot were a map or other recognizable entity that did not require a numeric scale One of the most common uses of the scales argument is to specify axis relationships By default the scale argument takes the value relation same which means that the hori zontal axes will be identical on all panels and the vertical axes will be identical on all panels the horizontal and vertical axes are not necessarily the same as one another though The value relation sliced means that the axes are set up to have the same units per inch as if each axis was a slice from one consistent larger axis This means that the maximum minus mini mum value for each axis is identical Finally the value relation free can be used to allow each panel to have freedom in constructing an axis that jus
8. regions Figure 25 Sample output from the show settings function This shows the various cus tomization settings done for a PostScript printer 3 4 Finding the Data In many of the examples so far we have used the data argument to present a data frame to be searched for objects named in a Trellis formula There are a few subtleties of this that are addressed here These same issues affect the functions described in the book Statistical Models in S When the data argument is given the data frame or list specified by that argument is the first place searched for finding objects named in the formula or in the subset or groups arguments This search of the data argument replaces the normal search of the caller s frame In any case if an object is not found in the data argument or in the caller s frame the search will continue through frames 1 0 and the databases explicitly on the search list Why does this matter It matters when you write functions that call Trellis functions For example myfunl lt function histogram height voice part data singer would work and myfun2 lt function histogram sqrt height voice part data singer would also work but myfun3 lt function sqrtht lt sqrt singer height histogram sqrtht voice part data singer 39 would fail with the error message gt myfun3 Error in myfun3 Object sqrtht not found This is because the
9. When we first produced Figure 23 we did not include the aspect 2 5 argument Because we used a prepanel function that returned dx and dy values the banking computation was done automatically Unfortunately the resulting aspect ratio was far too big To see how it looked try the expression without using the aspect argument The problem is that many of the panels have basically horizontal fits and only an extreme aspect ratio will make these panel fits non hor izontal This shows the occasional need for user adjustments to the automatically chosen values 3 2 The subscripts Argument Suppose that we would like to produce the plot of Figure 3 but instead of plotting points we would like to plot the observation number for each point Since a panel function only gets the x and y values that belong on a panel how are we to accomplish this We solve this problem in a very general way by allowing the panel function to request one extra argument named sub scripts The subscripts argument is a numeric vector that tells which observation in the original data is associated with the x and y values Thus we can neatly solve our problem by using xyplot NOx E C data ethanol Figure 24 panel function x y subscripts text x y subscripts This works because xyplot sees that the panel function expects an argument named sub scripts and arranges to pass the appropriate information to the panel function We claimed that the subscrip
10. by reading a data file and making an intermediate data frame named barley2 The data file barley data looked like this 33 yield variety year site 27 00000 Manchuria 1931 University Farm 48 86667 Manchuria 1931 Waseca 27 43334 Manchuria 1931 Morris 39 93333 Manchuria 1931 Crookston 32 96667 Manchuria 1931 Grand Rapids 28 96667 Manchuria 1931 Duluth 43 06666 Glabron 1931 University Farm The columns were separated by tab characters important because sites like University Farm have embedded blanks so we could for example use read table to create the barley2 data frame barley2 lt read table barley data sep t header TRUE When such a table is made the default is to alphabetize the various factors So the expression dotplot variety yield year site Figure 22 data barley2 xlab Barley Yield bushels acre produces the difficult to understand Figure 22 The problem is that this ordering obscures the underlying pattern the main effects ordering that makes the Morris year anomaly stand out in Figure 7 is not present here Notice too that read table has produced a numeric vector for year and hence it is plotted as a shingle rather than a factor In order to produce Figure 7 we changed the factors barley lt barley2 barley variety lt reorder factor barley2S variety barley2 yield median barley year lt ordered barley year c 1932 1931 make it a factor barley site lt reord
11. characters line types and colors Gray scales and different s e line styles and symbol types are used in the black and white version to distinguish areas lines and points If color is available the Trellis color scheme is used It is designed to provide good discrimination amongst a fairly large group of colors This is necessary because Trellis displays are often concerned with multivariate situations The colors are also chosen to be relatively robust across various graphics devices screens and hardcopy they are simple mixtures of the cyan magenta and yellow printing primaries The background and basic annotations are done monochromatically to contrast with the color assigned to data based graphics Black text is used because it is available on all devices and stands out well on white gray and color For hardcopy devices e g those that support PostScript the background color will be the white of the paper on a computer screen the background color is light gray because white is too harsh and tends to hide light colors The colors used are those that empirically offer good dis crimination and are not difficult for the device In particular text is drawn in black and the colors cyan magenta green orange blue yellow and red are employed for various groups The func tion show settings gives a graphical display of the Trellis color and other parameter settings See section 3 3 for an example plot and more details about devic
12. each packet a scatterplot matrix is plotted The axes produced by splom are cleverly constructed to use minimal space there is an argument pscales that can control these axes 5 A GRAB BAG This section contains descriptions of several Trellis functions that don t quite fit in with the general formula and data framework that has been introduced earlier However don t get the idea that they are not important just because they are hard to place Each of these functions does something non trivial The function rfs produces a plot that shows the residual and fitted values spreads as described in Cleveland 1993 Itis used to assess the magnitude of the fit to the magnitude of the residuals You initially fit a model then give the model object to rfs It produces a pair of plots with identical y axis scaling in units per inch By comparing the sizes of the y variation two plots you can determine the importance of the fit The tmd function computes a Tukey mean and difference plot Itis unusual since it takes as an argument the output of one of the other plotting functions and it produces from that another similar object of class trellis However for each plot it transforms the data so that the x axis holds the mean of the original x and y variables and the y axis gives the difference The purpose of this is to allow more effective comparisons Instead of comparing how well data fits the y x line you can look at how well the tmd modif
13. on continuous or discrete variables in the data thus providing a coordinated series of views of high dimensional data This document leads you through Trellis graphics it shows the functions in the Trellis library it describes the common arguments that the functions share and shows how Trellis dis plays are customized for various graphical devices Other information is available about Trellis including a user s manual and a journal article with data analysis examples To find these and more refer to the Trellis web page http netlib att com projects trellis 1 1 New Capabilities for S and S PLUS Graphics Graphics have always been a strong feature of S and S PLUS the commercial version of S distributed by MathSoft Its graphics provide device independence high level plotting functions that produce an entire display low level functions to augment existing displays or build new ones and a collection of graphical parameters that provide a wide range of control over the details of plotting Graphical parameters in S provide the ability to produce several plots on a single page However producing a coordinated set of plots on a page with control over aspect ratios and axes has always taken more knowledge of the graphical functions than even a proficient user is likely to possess In addition graphics devices may vary in their capabilities thus requiring adaptations in order to produce the best plot on each device The Trellis lib
14. panels tall by specifying aspect 2 in the call The reason for this will be made clear later NOx 8 10 12 14 16 18 10 12 14 16 18 fi a EE E OO oo o0 00 EE D e omo 000 00 O o 8 10 12 1 T T 4 16 8 10 12 C 14 16 18 8 T T T T T 10 12 14 16 18 Figure 5 Scatterplots show NOx emissions as a function of compression ratio for various values of equivalence ratio Suppose that for some reason we wanted to recreate Figure 2 but to show it only for com pression ratios greater than 8 One way to do this would be ii lt ethanol C gt 8 xyplot NOx ii E iil another possibility would be xyplot NOx E C data C iil data ethanol ethanol ethanol Cs8 but an easier version can be accomplished using the subset argument xyplot NOx E C data ethanol subset C gt 8 Figure 6 In complicated expressions subset can save you from lots of subscripting and also allows you to refer to components of a data frame by name In addition using subset causes an automatic operation on all factors used in the formula to drop any levels that have no data associated with them The next section describes how factors are handled in Trellis displays 1 4 Conditioning on Factors Barley Data Let s leave the engine example and go on to something else this time involving more than
15. presence of the data argument prevents histogram from looking into myfun3 s frame to find sqrtht Here are several ways to fix the problem a put the transformed object back into the data frame myfun3a lt function singer sqrtht lt sqrt singer height histogram sqrtht voice part data singer b don t use the data arugment myfun3b lt function sqrtht lt sqrt singer height histogram sqrtht singer voice part c explicitly put the data frame on the search list myfun3c lt function attach singer sqrtht lt sqrt height histogram sqrtht voice part 4 HIGHER DIMENSIONS This section describes plots for data of three or more dimensions 4 1 3 D Plotting Perspective Trellis displays are carried out by the function cloud which produces a 3 dimensional point cloud and the function wireframe which draws a 3 D wireframe surface Unfortunately static cloud displays are typically difficult to understand because even though points in 3 space are displayed in perspective there is too little structure for the viewer s eye to put together an integrated view when a static panel is drawn The going is a bit rough here so we will give an extended example The non Trellis func tions mentioned in this section are described in the book Statistical Models in S Suppose we are interested in learning about the environmental ozone data where ozone concentrations are to be rel
16. that Trellis displays are controlled This document gives an entry point to use of the software and explains common features It does not have the space to explain except in the most cursory way the meaning or use of the graphical techniques To find out much more about how to use Trellis displays to 4 understand data read Visualizing Data by William S Cleveland 1993 Although Trellis graphics functions are capable of producing multiple panel displays they are also excellent at doing basic single panel graphs For this example we will use data from an experiment involving 88 trials of an engine running an ethanol mixture contained in the data frame ethanol There are three variables emissions of oxides of nitrogen NOx equivalence ratio a measure of the richness of the fuel air mixture E and five values of compression ratio C We can use the scatterplot function xyplot on the ethanol data to produce a simple scat terplot xyplot NOx E data ethanol Figure 1 4 ra r oS a o Mi e Pig o o 90 g S Oo O o fe 3 5 L O 0 oo O O Z i o 275 n x 8 O o o o oF o fo o 2o o o i 8 i fo fe m o o doa B i 8 o 9 09 g O T T T T 0 6 0 8 1 0 1 2 E Figure 1 A simple scatterplot of the engine data showing Nox emissions as a function of equiv alence ratio The first argument to xyplot and to most Trellis functions is a formula and the second tells where the data in the form
17. 2 where the numeric response x is split into exactly two groups by the y variable The resulting panels each show a quantile quantile plot of the data from one group against the data from the other group The function qqmath produces quantile plots Its formula is x gl g2 It also takes an argument distribution that specifies a quantile function a function of a vector of probabilities that produces a set of quantiles The quantile function is often a standard 16 Soprano 2 Soprano EE essi E Teror Bor 40 4 H Percent of Total 304 204 T 60 65 70 75 60 65 70 75 Height inches Figure 10 Histograms of singer heights by voice part reference distribution qnorm to compare the y data to the normal distribution or qunif to compare to a uniform distribution As an example we can see how the heights of singers within voice part compare to a normal distribution qqmath height voice part distribution qnorm data singer Figure 11 The plot of Figure 11 could be improved perhaps by adding a reference line and grid to make it easier to assess the quality of the fit and by adding a nicer x label The plots might also look bet ter if they were made square later we can see if there is an even better aspect ratio This is accomplished in Figure 12 produced by qqmath height voice part distribution gnorm Figure 12 data singer aspect 1 prep
18. A Tour of Trellis Graphics Richard A Becker William S Cleveland Ming Jen Shyu Bell Laboratories Murray Hill New Jersey 07974 Stephen P Kaluzny Statistical Sciences Division MathSoft 1700 Westlake Avenue North Seattle Washington 98109 April 15 1996 TABLE OF CONTENTS 1 INTRODUCTION 1 1 New Capabilities for S and S PLUS Graphics 1 2 A Simple Scatterplot Ethanol Data 1 3 Conditioning on Numeric Variables Ethanol Data 1 4 Conditioning on Factors Barley Data 1 5 Other Examples 2 HOW TO USE TRELLIS SOFTWARE 2 1 Display Functions 2 2 Customization for Devices 2 3 Panel Functions 2 4 Formulas 2 5 Trellis Objects 2 6 Layout 2 7 Axes 2 8 Aspect Ratio 2 9 Data Structures 2 10 Labeling Titles Strip Labels Keys 3 ADVANCED CONCEPTS 3 1 Prepanel Functions 3 2 The subscripts Argument 3 3 Device Settings 3 4 Finding the Data 4 HIGHER DIMENSIONS 4 1 3 D Plotting 4 2 Contour Plots 4 3 More Than Three Variables 5 A GRAB BAG mee 1 INTRODUCTION Trellis displays are plots which contain one or more panels arranged in a regular grid like structure a trellis Each panel graphs a subset of the data All panels in a Trellis display contain the same type of graph but these graphs are general enough to encompass a wide variety of 2 D and 3 D displays histogram scatter plot dot plot contour plot wireframe 3 D point cloud and more The data subsets are chosen in a regular manner conditioning
19. R NS A MAANINKA pot Ozone pot Ozone NRE SSS OOA SS root ppb SSSR root ppb SESS SS SSS SSE ss ARSS E coe SESS San Stes 3998500 ASIENS SRIAS 99S Temperature Wind Speed mph Temperature F Wind Speed mph l radiation radiation SS SSS SRO pot Ozo Sere SSS SONS STS ILO ASEAS jo root ppb Sh Xx P Sees eee EEN O sssaaa Se SSS bot Ozoni root ppb Ax SSMS EES A tet SSeS KX SSES SS NS SSS ESRI SISSON eSa SSS Se ne set se meee xm eS care RX so se x es X R s se S R w X n w n i me i x ne i i Temperature Wind Speed mph Temperature Wind Speed mph Figure 26 A Trellis wireframe display showing the loess fit of ozone data to wind speed and temperature for four given levels of solar radiation Notice the changes in height relative to the surrounding box especially in the far corner and at intersections with axes 4 2 Contour Plots Wireframe and contour displays share many characteristics the input data is often identical only the way it is displayed changes Once we have our data structures straight a contour plot corresponding to the previous wireframe is simple to produce wind temperature radiation Figure 27 contourplot fit data grid xlab Wind Speed mph ylab Temperature F main Cube Root Ozone
20. and radiation Now to produce the wireframe plots we execute wireframe fit wind temperature radiation Figure 26 data grid xlab Wind Speed mph ylab Temperature F zlab Cube Root Ozone n cube root ppb The wireframe function also allows us to specify that color draping should be applied to the wireframe this means that each patch of the wireframe can be drawn in a solid color controlled by the z value or by any other value given as the drape argument which must be the same length as z For more information on color draping see the online documentation for wire frame There are a few arguments that are unique to the 3 D functions These specify parameters for the viewing transformation and provide for axis control The default viewing perspective was shown in the wireframe example Suppose however that you wish to look at the surface from other vantage points You can do this with the argument screen a list with named components each describing a rotation The name of the element tells which axis the rotation should be around the value gives the degrees of rotation about the axis For example the default for screen is screen list z 40 x 60 which means that we rotate by 40 about the z axis and then by 60 about the x axis To under stand these rotations we need to understand the 3 D axis orientations the x axis extends horizon tally across the screen the positive y axis goes back into t
21. anel prepanel qqmathline panel function x y panel grid panel qqmathline y distribution qnorm panel qqmath x y xlab Unit Normal Quantile The prepanel function will be described later in the ADVANCED CONCEPTS section Suppose we thought that the square roots of the singer heights rather than the raw values were more appropriate to compare to a normal distribution We could have used the sqrt func tion inside the formula qqmath sqrt height voice part distribution qnorm data singer Functions and other S expressions can be used inside the formula However one disadvantage of using complicated expressions inside the formula is that the default labels are also more compli cated and can therefore be more difficult to read This is especially true for strip labels An alter native especially if you are using equal count to generate shingles for given variables is to 217 2 al 0 1 2 2 A 0 1 2 E i i AoT za Sopranoz Soprano 4 H75 o 4 o00 O o o H 70 o o co o o coo o am aD ammo 4 am aD com comm H 65 comp oD o o oo aD on o aD am omm ca ood o0 5 4 o O 00 o H 60 D E Bass 2 Bass 1 TM Tenor 2 efor 1 o o o 754 ooo O don o H faras a o aD o o am 00 o am a000 co 70 4 am amm fa H o aD omm o oam 0o00 an o0 o 0a o aco 65 4 o H p o 60 4 H qnorm Figure 11 Singer heights within voice part compared to t
22. anel function panel xyplot panel xyplot lt function x y plot symbol lt trellis par get plot symbol points x y cex plot symbol cex pch plot symbol pch font plot symbol font col plot symbolS col The object plot symbol is a list with components pch cex font and col to specify the plot ting symbol its size font and color When the S device is set up by an explicit or implicit call to trellis device the Device object is given an attribute that stores a set of Trellis parameter lists plot symbol is one such list The function trellis par get gets the named list As illustrated above it is often convenient to save the list and then extract components from it in var ious graphics routines inside the panel function To see how customization is set up for on any particular device start the device and then execute the function show settings Figure 25 shows an example 38 oe Be ae W p l J o 0o 0 p o l O O eee o o o o o 9 l M o o o o 0 o o l O O 0 0 oo 0 u l O O o o o o o o o l O O o o o o o o o l O O superpose symbol superpose line strip background strip shingle World Hello dot symbol line box dot rectangle umbrella addline text reference line plot shingle bar fill histogram bar fill barchart bar fill coin 1 100 piechart fill
23. are produced For example using layout c 2 2 1 with this example would produce just one page with 4 panels c I c 4 o 2 Oo L 4 o e a o o 4 H3 O o o 4 o o o F2 o Qi o o o o 4 i o eo Pin o ojo a i o ac 4 L O o o ge o o o o 34 fel H e o Oo o o o o o 24 Fa fi 4 o o o e e 14 o o L oo o oo o o o Figure 3 The engine data with layout c 3 2 1 We have conditioned on compression ratio a variable that has only 5 levels gt sort unique ethanol cC 1 7 5 90 T20 15 0 180 Suppose we want to see how NOx concentrations depend on C for various values of E It won t work to condition on E since there are 83 unique values of E for the 88 observations However we can do something similar by conditioning on intervals of E The function equal count con structs a data structure called a shingle from our data with a specified number of intervals and a specified amount of overlap from one interval to another For example EE lt equal count ethanolSE number 9 overlap 1 4 constructs a shingle with 9 intervals spanning the range of E each containing approximately the Te same number of observations and each having about 25 of its points in common with the each of its two adjacent intervals gt EE Data 1 13 25 37 49 61 73 85 OGOGOGoOoG PHPO 907 138 oL75 868 765 749 694 584 aa
24. ated to wind speed temperature and solar radiation One way to learn about the general pattern is to fit a smooth surface to the data and we have previously determined that the cube root of ozone concentration is an appropriate response variable We can use the loess function compute the smooth fit attach environmental ozo m lt loess ozone 1 3 wind temperature radiation parametric c radiation wind span 1 degree 2 Notice how the formula given to loess looks just like the formulas given to Trellis functions To produce a wireframe or surface plot we need to evaluate the surface at a regular grid of val ues We will evaluate the grid for 50 values of wind speed ranging from the minimum to maxi mum observed speeds 50 values of temperature and 4 levels of radiation We first form vectors 40 of the wind speed temperature and radiation marginal values and then make a grid of all 50x50x4 combinations of those w marginal lt seq min wind max wind length 50 t marginal lt seq min temperature max temperature length 50 r marginal lt seq min radiation max radiation length 4 grid lt expand grid wind w marginal temperature t marginal radiation r marginal The grid object is just what the predict function needs to produce the fitted values fit lt predict ozo m grid The fit object is a vector that matches the vectors in the grid data frame wind temperature
25. bel arguments can be given as a list so that the label comes along with associated graphical parameters e g xlab list Equivalence Ratio cex 1 5 col 2 Strip labels are an important part of a Trellis display By looking at the panel and its 30 associated strip labels you can see the given variables and which levels are used for the particular panel By default strip labels for shingles give the name of the shingle variable and show by shading the fraction of the entire data range taken up by the current shingle interval Strip labels for factors give the label corresponding to the factor level and are shaded to show the order of this level within the factor By now you probably can guess that there are many ways to customize strip labels At the most extreme level you can pass an entire strip label drawing routine as the strip argument to high level Trellis plots the default strip label routine is called strip default However most of the time there is no need to do this much The argument par strip text allows you to pass in a list of graphical parameters that controls the rendering of the text strings in the strip labels The most common use of this is to control the size of the text in the strip labels the size of the strip label box changes to accommodate the characters You can also use par strip text to control the color and font of the text par strip text list cex 75 By default the text size varies with the layou
26. ctor with 3 levels giving the names of the original data vectors So we could use bwplot which data data make groups lottery payoff lottery2 payoff lottery3 payoff to produce box and whisker plots or histogram data which data make groups lottery payoff lottery2 payoff lottery3 payoff to produce histograms of the three sets of data Just as make groups converts vectors to data frames for use with Trellis functions there are also functions to facilitate graphics with arrays and time series The functions as data frame array and as data frame ts convert arrays or time series into data frames sorry about the long names for those of you who don t like to type Consider the object iris a 3 way array with a dim vector like this gt dim iris 1 50 4 3 We can turn iris into a data frame in preparation for plotting by using iris df lt as data frame array iris col dims 2 The resulting data frame has what used to be its second dimension turned into 4 columns so that it looks like this gt iris d 1 5 Sepal L Sepal W Petal L Petal W flower species 1 DFT 3 45 1 4 0 2 1 Setosa 2 4 9 3 0 1 4 0 2 2 Setosa 3 4 7 362 1 3 0 2 3 Setosa 4 4 6 Fst 22 5 0 2 4 Setosa 28 5 5 0 3 26 1 4 0 2 5 Setosa So given a data frame like this we can now use it with Trellis functions Let s try a parallel coordinate plot The function parallel takes a formula in the form x gl g2 where x is a mat
27. cube root ppb If you compare the contour and wireframe figures you can probably see that the contour plot is preferable for quantitative information but the wireframe gives a good gestalt of the surface 42 Cube Root Ozone cube root ppb 5 10 15 20 l radiation radiation N N radiation radiation Temperature F P a 90 U 70 7 M 60 z Wind Speed mph Figure 27 A Trellis contour display showing the loess fit of ozone data to wind speed and tem perature for four given levels of solar radiation Compare to the wireframe display in Figure 24 The function 1evelplot produces a color level plot where color or gray levels are used to encode the value of a third variable It is closely related to a contour plot the boundaries between different regions of a levelplot are contours 4 3 More Than Three Variables Two functions are designed to work with more than three variables on each panel of a Trel lis display splom and parallel You saw the result of parallel in Figure 18 The splom function produces a plot in which each column in an input matrix x is plotted against each other column Figure 20 was an example of this What might not be obvious is that the formula may also contain given variables splom x gl g2 43 Here x is a matrix and the given vectors cause the packets to contain sub matrices consisting of various rows of x For
28. d dy The xlim and ylim values are vectors that give the minimum and maximum value on each axis The dx and dy vectors describe the run and rise of line segments that should be banked to 45 Certain types of Trellis displays occur often enough that prepanel functions are already writ ten for them They are prepanel 1lmline prepanel qqmathline and prepanel loess If you want a least squares line fit via the 1m function to the x y data in each panel prepanel 1mline will carry out the computations so that the panels will have enough room to show the fitted line over the range of the x data and so that the lines on the panels will be banked as closely as possible to 45 The prepanel qqmathline function does a similar thing but the line is fit to a quantile plot and goes through the 25th and 75th percentiles It is used in conjunc tion with qqmath and the panel qqmathline functions 35 Perhaps the most commonly used prepanel function is prepanel loess It does a loess smoothing on the data in each panel and banks the segments of the smooths to 45 Let s give an example of this applied to the engine data EE lt equal count ethanolSE number 9 overlap 0 25 Figure 23 xyplot NOx C EE data ethanol prepanel function x y prepanel loess x y span 1 panel function x y panel grid h 2 panel xyplot x y panel loess x y span 1 aspect 2 5 xlab Compress
29. d the plots to hold both key and titles and they can also produce a legend on each page of a multi page display One of the most common uses for a key is when you are using panel superpose to dis tinguish between various groups of points on scatterplot panels For example we can display a scatterplot matrix of the Anderson Iris data using colors to encode the three different species new iris lt iris df 1 4 for i in 1 4 new iris i lt jitter new iris i iris variety lt iris df 6 superpose symbol lt trellis par get superpose symbol splom new iris panel panel superpose groups iris variety key list space top columns 3 transparent TRUE text list levels iris variety points Rows superpose symbol 1 3 The result is shown in Figure 20 The position of the key is controlled here by the space com ponent of the key argument Jf Setosa o Versicolor o Virginica o pee 15 20 25 OL 2 0 o a Has 15 4 Petal W 1 0 4 Petal L 4 amp ad Sepal W 3 0 3 0 4 Figure 20 Scatterplot matrix of the jittered iris data with symbols coding the three species Out put from older versions of the Trellis library may not show the cute axes in the diagonal panels Precise control of key positioning could also have been done by using the x y and cen ter components to the key argument The coordinate system for x and y is a unit square sur rou
30. e panel function is passed arguments x and y the coordinates for the points on the current panel If no points belong on the panel the panel function is not called The panel function is passed in as an argument to a high level Trellis function perhaps as an object panel xyplot as a character string panel xyplot or constructed on the fly in the argument list Eunction x y Any arguments given to the top level Trellis function but not recognized by it are passed unchanged down to the panel function This can be a useful way of communicating extra infor mation to the panel function and is often used in conjunction with simple graphical parameters like col or cex Because Trellis displays may take up many pages the paradigm of producing a plot and then adding to it will not work This was often the way things were done prior to Trellis graph ics However the panel function can do something similar This is relatively easy because panel functions have a nice synergy with the ability to define functions For example suppose you want panels with points and a smooth line You can do this easily by combining two existing panel functions into one Se xyplot NOx E C data ethanol Figure 8 panel function x y panel xyplot x y plot points panel loess x y add smooth line NOx Figure 8 A Trellis display of engine emissions data The smooth line was put
31. e settings 2 3 Panel Functions Panel functions lie at the heart of Trellis displays When any Trellis display function is exe cuted common code parses the formula gets the data and returns a Trellis object If the object is printed the print method print trellis sets up the coordinate system constructs appropri ate axes and then produces the panels one by one When the panel function is called it is pro vided with two arguments x and y giving the horizontal and vertical coordinates of what should be plotted The data passed to the panel function consists of subsets of the independent and dependent variables the subsets determined by appropriate values of the given variables There is a default panel function for each of the high level Trellis functions the name is constructed by gluing panel onto the name of the Trellis function Thus panel xyplot corresponds to the xyplot function Many of these functions are simple basically a call to a plotting routine like points or lines with a few graphical parameters specified to mesh with the Trellis graphics customization rules see section 3 3 Device Settings When a panel function is called the Trellis code has already set up a coordinate system based upon data values and arguments xlim ylim and scales It has also extracted from the entire collection of data the data to be displayed upon the panel the extraction based on the levels of the given variables Th
32. e ts hstart Figure 19 type h xlab Year ylab Housing Starts by Month 1966 1968 1970 1972 1974 1966 1968 1970 1972 1974 L i 1 L L L 1 i L L 1 1 L L sp E Oat Nov ES Dec 200 a 150 J H oa _ May n J Aug E 4 200 a N 5 M 150 on D gl j 100 n x 4 H 50 Jan Feb Mar Apr 200 P 150 F i BON T T T T T T T T T T d 1966 1968 1970 1972 1974 1966 1968 1970 1972 1974 Year Figure 19 Each panel shows the housing starts for a particular month during the years from 1966 to 1974 Notice that Figure 19 would change if time in the formula were replaced with trunc time the vertical lines for each month would line up exactly at the year marks 2 10 Labeling Titles Strip Labels Keys Labeling is an important part of Trellis displays The Trellis functions take arguments xlab and ylab to control the x and y axis labels By default these labels are made up from the names of the variables or the expressions plotted there Of course you can often make the labels more meaningful by giving explicit values for example xlab Equivalence Ratio ylab NOx micrograms J would do a nicer job of labeling Figure 1 Also the arguments main and sub may be given to any high level Trellis function to put a main title at the top and a subtitle at the bottom on each page Any of these la
33. ear space Right columns 2 Trebi Wisconsin No 38 f No 457 Glabron f Peatland f Velvet f No 475 Manchuria No 462 Svansota Crookston E ee 2 0 Trebi Wisconsin No 38 No 457 f Glabron f Peatland f Velvet t No 475 f Manchuria No 462 f Svansota Trebi Wisconsin No 38 f No 457 Glabron f Peatland Velvet f No 475 Manchuria f No 462 Svansota o 1932 o 1931 Trebi Wisconsin No 38 No 457 Glabron Peatland Velvet No 475 Manchuria No 462 f Svansota p Trebi Wisconsin No 38 No 457 f Glabron f Peatland Velvet No 475 Manchuria No 462 Svansota p Trebi Wisconsin No 38 No 457 Glabron Peatland f Velvet f No 475 Manchuria No 462 Svansota Barley Yield bushels acre Figure 21 Barley Data comparison of 1931 and 1932 on a single panel This figure makes the anomaly in Morris stand out boldly it is the only panel where the 1932 symbols are furthest to the right Notice how the panel function draws the horizontal dotted lines and then uses panel superpose to draw the symbols It is important that you realize that the row column and strip labels are ordered according to how the data was initially read For example the data frame barley that we used earlier was created
34. ecedes the number of vertical panels rows in the layout specification When it comes time to display a Trellis graph the packets and panels are associated with one another the first packet in the packet order goes into the first panel in panel order and so on The important concept is that the packets are produced in packet order and the panels in panel order are filled by those packets If you do not specify the Layout argument yourself it has defaults that depend on how many given variables are in the formula For one given variable the default layout is chosen by a layout optimization algorithm It chooses the number of rows and columns to maximize the page area devoted to the panels taking into account the aspect ratio of the panels the size of the device and the size of the strip labels For two or more given variables the default number of columns is determined by the number of levels of the first given variable the rows by the second given vari able and the rest of the given variables vary across pages There is also a way to have the layout optimization algorithm assist you with 2 or more given variables If you specify a layout argument of the form layout c 0 n the optimization is carried out for n plots per page Occasionally the layout of a Trellis display may be problematic For example suppose you have two factors one with 14 levels and another with 10 and would like to have 3 columns and 5 rows on each page f
35. ed and manipulated We chose this strategy to avoid having complicated labeling arguments for Trellis functions Thus to change the labels you change the factor levels month observed lt month name full names levels month observed lt month abb abbreviated names This is a powerful notion and will come up in later examples Two other functions are useful for ordering the levels of a factor The first is ordered ordered grades lt c Poor Fair Good Excellent The second function useful for modifying factors is reorder factor It allows you to rear range the levels of a factor based upon a computed value For example you could reorder a hypothetical factor state from its original alphabetical order to an order based upon the median income of the observations in that state state lt reorder factor state income median The function make groups is often useful with Trellis displays It constructs a data frame from several vectors and the data frame can be passed in to a Trellis function Suppose that we have several vectors and want to see box and whisker plots or histograms of each For example we want to compare payoffs of the New Jersey Pick It lottery from three time periods make groups lottery payoff lottery2 payoff lottery3 payoff creates a data frame with two components data and which The data component is simply the combined numbers from all the make groups arguments The which component is a fa
36. er factor barley2 site barley2 yield median For example the levels of the variety factor were rearranged by reorder factor by increas ing order of barley yield 3 ADVANCED CONCEPTS 3 1 Prepanel Functions One of the strengths of a Trellis display is the independence between a the common code that sets up the coordinate systems labels aspect ratios and b the panel function that takes care of the drawing for each individual panel Generally this decoupling allows you to customize the drawing in the panel without having to work with or understand the common code Unfortu nately this is not always the case The only thing that the common code can do in order to figure out how to set up coordinate systems is to look at the variables in the formula to determine what is going to be plotted on the horizontal and vertical axes Suppose though that your panel function draws more on the panel than just the x and y data For example suppose it puts on a fitted line or curve How then can the Trellis software allocate enough space on the panel for the extra things plotted by the panel function when the common code has no idea what extra plotting is going to take place Another problem involves aspect ratio calculations The Trellis code can bank to 45 based upon the x and y data but sup pose that the important thing for the banking computations is a fitted line not the raw data How can that be taken into account so that an appropria
37. eral Trellis displays consist of one or more panels arranged in a regular grid like structure of columns rows and pages Simple displays are usually easy to create multi panel displays take little more effort A wide range of graphs can be drawn inside each panel although all panels in a particular Trellis display must be alike Each panel displays a subset of the data determined by the values of the given variables The Trellis software is structured so that there is one piece of software the Trellis print method that takes Trellis objects and produces all types of displays It advances from panel to panel sets up axes computes appropriate aspect ratios and generates overall labels The print method calls a panel function once per panel to draw the graph inside the panel This decoupling of the overall setup and panel drawing provides much of the power to the Trellis software We 10 Trebi Wisconsin No 38 No 457 Glabron Peatland Velvet No 475 Manchuria No 462 Svansota Trebi Wisconsin No 38 No 457 Glabron Peatland Velvet No 475 Manchuria No 462 Svansota Trebi Wisconsin No 38 No 457 Glabron Peatland Velvet No 475 Manchuria No 462 Svansota Trebi Wisconsin No 38 No 457 Glabron Peatland Velvet No 475 Manchuria No 462 Svansota Trebi Wisconsin No 38 No 457 Glabron Peatland Velvet No 475 Manchuria
38. excellent plots on all devices without requiring any changes on the part of the user to make it happen If your device has color capabilities Trellis functions will use them they may also use various fonts character sizes line styles area fills etc However when you write your own panel functions you may know for certain that you are producing the plot for a particular device and you know just what values the parameters should have If so just write the panel function with exactly the parameters you want ee 0 6 0 8 1 0 1 2 T L I L L T L L 19 L 4 55 54 74 41 4 L 3 34 7 J 73 L 2 52 8 88 43 B 42 35 14 p 72 43 6 2866 x ae 3387 ape 2 C I T ec 4 L 40 67 1 3 50 63 20 6 L 4 71 51 P v a i 17 bo o l 57 47 14 15 29 69 a A 12 b 83 845 P07 85 Ag T T T T T T T T T 0 6 0 8 1 0 1 2 0 6 0 8 1 0 1 2 E Figure 24 The ethanol data with each observation identified by observation number On the other hand you may want to write your Trellis plot calls that adapt to device charac teristics too In this case an explanation of how Trellis functions deal with graphical parameters may help you to understand how device specific customization works Basically all Trellis functions are careful to specify graphical parameters in a symbolic way by referring to objects that give the actual values of the parameters For example let s look at a simplified version of the most commonly used p
39. ft half and a right half wlevels lt seq 4 16 length 6 c 1 4 2 5 3 6 Figure 15 strip shingle lt trellis par get strip background xyplot env fit radiation temperature shingle wind wlevels data twr grid layout c 12 3 between list x c 0 0 0 0 0 1 strip function strip default strip names FALSE key list text list c wind temperature rectangle Rows strip shingle 2 1 space Top ti panel function x y panel grid h 2 panel xyplot x y type 1 aspect xy xlab Radiation langleys ylab Cube Root Ozone cube root ppb You may find the panel order in Trellis displays somewhat peculiar particularly the part about the rows of panels being filled from bottom to top The reason for this is that the vertical 3 2D 0 250 0 250 0 250 REESE REE Ree ee eee ee ee ee ee wind wind wind wind win wind fperathberathperathpelathperathperatl_ T wind wind sra ung a perathperathperathpeathper 5 0 4 45 4 0 4 35 4 3 0 4 qs Q 20741 wind wind wn win wind S _fperathperathp rathpe athperdthperaij 26 4 4 5 ab 4 4 0 2 4 3 5 4 3 0 Q 4 L 25 oO ar ge iif 20 Cc wind wind wing vin wind wind O 4 _fperathBerathperathpelathperathperall 6 asd oN L 4 L O 354 L O 3 0 4 H am 25 4 o 24 Q O ind Wind Wind wind Wind
40. he screen and the z axis is vertical Another important viewing parameter is named perspective This is a logical flag that tells whether the projection should be orthogonal perspective FALSE or perspective If a perspective projection is desired the parameter distance controls the apparent distance from the eye to the object A value of 0 implies an infinite viewing distance an orthogonal view while 1 has the eye point right up to the data the default is 0 2 There are two ways that the axes can be drawn in 3 D Trellis plots either with 3 D axis lines with ticks and tick labels or by simple arrows that run parallel to the axes and point in the direction of increasing values Arrows are the default conventional axes by turning off the arrow axes via the scale list arrows FALSE argument In either case the axes or arrows are labeled by variable names either taken from the formula or passed in through the arguments xlab ylab and zlab Another quick comment related to 3 D plotting the notion of a simple panel function breaks down here Unfortunately the panel function does projections axis drawing etc and thus it is huge and unwieldy Don t expect to produce your own 3 D panel function without studying and modifying a copy of an existing one 4 radiation radiation M K w ORK 4 XA S N ay So AIO LKS 3 ESR ONO ei SSES AAO SSS ARH SIO N SS
41. he x y position of an nx by ny rectangular layout The more argument tells whether more plotting is to be done on the same page As some of our examples have shown the description of some Trellis displays through arguments to the high level Trellis function may take several lines Suppose you are producing Trellis displays and are trying to adjust things to come up with just the right look One way to do 19 2 1 0 1 2 2 1 0 1 2 1 l Soprano 2 i Soprano D D ta T T T T T T T T T T T T T 2 1 0 1 2 2 1 0 1 2 Unit Normal Quantile 15 4 T He 104 L T fo 2 0 A 5 0 le T T T T 60 65 70 75 height Figure 13 Two different Trellis displays on a single page positioned by print trellis this is to store the expression in a file use your favorite editor to change the file and then use source to execute the expression in the file One thing to note about this plan automatic print ing of results is not done within a source file so you should surround your expression with print for example put the expression print xyplot NOx E C data ethanol in file foo and execute source foo An alternative is to create a Trellis object and then modify and redisplay it with the update function NOxplot lt xyplot NOx E C data ethanol NOxplot display it 20 update NOxplot layout c 3 2 display with new layout
42. heoretical quantiles from a normal dis tribution create a new object with a nice looking name the variable s name is used in labeling shingles prior to calling the S function The contourplot and wireframe functions also take a different sort of formula Since these plots are constructed from 3 D data the formula is z x y gl 92 Here x y and z are numeric vectors and x and y are evaluated on a regular grid For example the z values may represent a surface evaluated at 60 points made up from 6 unique x values and 10 unique y values If given variables are present there should be a regular surface of data for each unique combination of values of the given variables although the x and y values need not be the same for each surface For an example see section 4 1 3 D PLOTTING 2 5 Trellis Objects All of the high level Trellis functions those listed in Table 1 return an object of class trellis as their value These objects are ordinarily plotted straight away because the print method for this class print trellis actually plots the objects However the fact that Trellis functions return objects means that those objects can be stored and replotted later perhaps when a different device is active For example in S NOxplot trellis NOxplot trellis NOxplot lt xyplot NOx E C data device iris4d now plot it device postscript ethanol change devices plot it again for ne
43. ied data follows a horizontal line Finally the function oneway is provided to do a one way analysis of variance Suppose we believe that the heights of singers can be modeled by a typical value for each voice part with vari ation about that typical value We can fit this model with oneway and assess the quality of the fit with rfs attach singer Figure 28 singer model lt oneway height voice part spread 1 rfs singer model aspect 1 ylab Height inches The results are shown in Figure 28 0 0 0 2 0 4 0 6 0 8 1 0 Residuals Height inches f value Figure 28 Residuals and fitted values from a one way fit to the singer data REFERENCES 1 Becker Richard A Chambers John M and Wilks Allan R The New S Language Chap man amp Hall 1988 2 Cleveland William S Visualizing Data Hobart Press Summit NJ 1993 3 Chambers John M and Hastie Trevor J eds Statistical Models in S Chapman amp Hall 1992 4 Becker Richard A Cleveland William S and Shyu Ming Jen The Visual Design and Control of Trellis Display J Computational and Graphical Statistics 1995 to appear 5 MathSoft S PLUS Trellis Graphics User s Manual Version 3 3 Seattle MathSoft Inc
44. ing the default aspect fi11 Now we have a plot where the lines are too flat to compare easily Figure 17 0 50 100 200 300 O 50 100 200 300 O 50 100 200 300 rt Li oss op os Lo i pop a EN C T T ER pop T S SOS ES CRS ES SE E E wind wind wind wind wind wind temperature temperature temperature temperature temperature temperature wind I wind I wind I wind I wind wind I temperature _ temperature tefperature temperat re temperature temperature wind wind wind wind wind temperature temperature temperature temperate temperature wind temperature wind wind wind wind T wind wind temperature temperature temperature temperat re temperature temperature E wind I wind wind I wind temperature temperature temperature wind temperature temperature temperature Cube Root Ozone cube root ppb 5 0 wind wind temperature temperature wind wind wind temperature temperature temperature wind temperature T mr T T 0 50 100 200 300 0 50 100 200 300 Radiation langleys Figure 17 Loess fit to the ozone data no aspect ratio control 2 9 Data Structures There are two data structures that are of particular importance to Trellis displays These are factors and shingles Factors were described in detail in Chambers amp Hastie 1992 They are used to hold categorical value
45. ion Ratio ylab NOx micrograms J 8 10 12 14 16 18 8 10 12 14 16 18 1 CE E 1 ot L et iz ef eet 1 i et L i J La q 3 wn E g D fa 2 E x O Z T T 8 10 12 14 16 18 8 10 12 14 16 18 8 10 12 14 16 18 Compression Ratio Figure 23 Engine data showing how NOx emissions depend on compression ratio for various intervals of equivalence ratio Studying Figure 23 and the xyplot expression that produced it gives a concrete example of many of the issues that we have been discussing For example notice how the span argument to loess is passed to both pane1 loess and prepanel loess As we mentioned earlier the formulas given to Trellis functions are allowed to contain expressions That means we could have made the call to equal count as part of the formula However the problem with this is that there would then be no nice character string to use for the strip labels and we have found that names on strip labels are particularly important for shingles Otherwise the information about what variable is represented by the shingle needs to be placed in the caption or somewhere else in the figure As you can see calls to Trellis functions may get a bit large even though no particular part 36 is all that difficult The way we structured the call with a separate argument on each line and multi line indented panel functions is designed to make it easier to read and understand You may find this layout technique useful too
46. nctions are intended as the place for users to customize their plots so they are generally short and easy to understand and modify However all of the Trellis functions are accessible and can often be understood with a little bit of work In any case the point is that you should not be deterred if the Trellis software doesn t do exactly what you want you can change it yourself 2 4 Formulas The formula argument for Trellis functions often comes in the form y x gl g2 Here x y and the g variables are S objects or S function calls The y variable describes the vari able plotted on the vertical axis In many cases both the x and y variables are numeric as in the examples of the xyplot function that produced Figure 1 However for the univariate display functions such as bwplot the y variable is treated as a factor and is made into one if it isn t a factor already For example Figure 9 is produced from bwplot voice part height data singer xlab Height inches Figure 9 Notice that in producing Figure 9 we have not used any given variables hence the Trellis display consists of only one panel When given variables are used in a formula they are handled as factors or shingles If a 15 Soprano 1 i P EEOAE Soprano 2 a m Alto 1 Penvesstomeniviecni en Alto 2 a e Tenor 1 bassa Tenor 2 A niche e maaa A
47. nding the entire set of panels If x and y are specified and space is not specified no additional space is left for the legend This may be useful if the legend actually is superimposed on the panels presumably in some unused space Another thing to notice about the example is the use of the superpose symbol cus tomization list The panel superpose function uses superpose symbol to encode the groups so it is important to use the same parameters to build the key Function trellis par get returns a list giving graphical parameters used to plot superposed symbols see the online documentation Rows chops it down to the first three values for each component of the list and finally those parameters are used by key to encode a column of points symbols The panel superpose function can help us improve the Barley data plot of Figure 7 If we show the 1931 and 1932 values superimposed on a single panel it will make the comparisons easier We can use superpose symbol lt trellis par get superpose symbol dot line lt trellis par get dot line dotplot variety yield site Figure 21 data barley groups year panel function x y subscripts abline h y lwd dot lineSlwd lty dot lineSlty col dot lines col panel superpose x y subscripts 235 E aspect 0 5 layout c 1 6 xlab Barley Yield bushels acre key list points Rows superpose symbol 1 2 text list levels barleySy
48. on temperature wind Figure 14 data twr grid panel function x y panel grid h 2 panel xyplot x y type 1 aspect xy xlab Radiation langleys ylab Cube Root Ozone cube root ppb The plot has an aspect ratio chosen by the slope of the lines in the panels and these banking com putations described in section 2 7 Axes make it tall and thin Suppose we needed to show this data on a landscape oriented piece of paper or on a display screen that was wider than high How could we do this preserve aspect ratios and still make good use of the display area Rather than a display with 6 rows and 6 columns we display the panels in a layout with 12 columns and 3 rows something that will fit the page quite well The trick to doing this is to reorder the levels of the second given variable wind Why does this work Think of how the layout argument interacts with the levels of the given variable in other words think of the interaction between panel order and packet order Trellis cycles through levels of the first given variable temperature for the first level of the second given variable wind This uses 6 panels starting from the lower left of the display and going horizontally There are still 6 panels left in the bottom row and those are taken up by 6 the temperature values and the second level of wind We have reordered the levels of wind so that the resulting plot appears to be organized into a le
49. on by the in line panel function The previous example used two panel functions that come with the Trellis software panel xyplot and panel loess As described earlier panel xyplot is the default panel function for xyplot The panel loess function is one of a group of functions that are useful in constructing panel functions This group includes panel abline panel fill panel grid panel 1lmline panel loess and panel qqmathline The panel abline function plots one or more lines on the panel The lines can be 14 horizontal vertical or the result of a linear fit expressed in the form y a b x The func tion uses Trellis line style conventions and will not generate warning messages if any of the lines fail to intersect the plot The function panel fil1 is used to fill in the entire panel with a suitable color or gray level Aside from making pretty colored panels it also allows further plotting on the panel in the background color col 0 For example if you are on a device with a white background you can fill each panel with gray and later use panel grid col 0 to draw a reference grid on top in white The panel grid function is designed to put a reference grid on each panel to enhance the visual comparisons from one panel to another It should generally be the first thing drawn on a panel after panel fil1 though so that the reference grid will not obscure any important information Arguments control how many grid lines a
50. one conditioning variable and a dotplot rather than a scatterplot The data frame barley describes the yield in bushels per acre of 10 varieties of barley harvested at 6 sites in 2 different years The expression dotplot variety data xlab barley Barley Yield yield year site bushels acre Figure 7 NOx Figure 6 Engine emissions data for compression ratios larger than 8 produces Figure 7 See Cleveland 1993 for a thorough analysis of the barley data 1 5 Other Examples We have reached the end of this introductory section and you should now have some famil iarity with Trellis displays and how they are produced However there is no way in a document as short as this that we can do more than hint at the variety of displays you can produce with Trel lis expressions One way for you to get more experience is to explore the functions whose names start with example in the Trellis library Execute help example bwplot to get a full list For example you can execute example bwplot to produce a Trellis box and whisker display boxplot and execute example bwplot to print the function so you can understand how the display is created 2 HOW TO USE TRELLIS SOFTWARE The previous section showed a few examples of Trellis displays and how they are specified and controlled in S The purpose of this section is to give a broader look at the Trellis graphics software In gen
51. oooa 00 Intervals min 535 655 733 808 892 990 042 115 s175 EHHO QOO GGOGOG max 686 161 811 899 002 045 125 189 232 PRRRROOOO 761 601 568 762 878 892 816 2562 OrRrROrFROAOOFR count 13 13 12 13 13 13 12 13 13 108 696 2977 144 811 002 037 2535 OrROODOrFAOAOR 016 686 767 045 676 812 181 655 189 072 006 797 045 230 899 orror rrr roor rorrk Overlap between adjacent intervals 1 43334334 001 074 893 e115 968 804 227 roor rror 231 934 152 070 846 813 180 aroraa z123 808 693 219 684 002 795 O O OOHH 042 071 232 637 729 696 990 We can see the intervals graphically by using the plot function on a shingle plot EE Figure 4 A plot of the shingle with 9 equal count intervals constructed from the engine data Interval xlab Range of E ylab Interval Figure 4 a F si i E L A T L F L F a F E F T T T T 0 6 0 8 1 0 1 2 Range of E PROORRR 215 009 036 2733 s911 199 201 OrRFOOrRFRO 930 142 125 715 808 030 629 ooror rrr 152 229 081 872 168 602 608 Now that we have a set of intervals we can use them to produce conditional scatterplots of NOx vs C for the various ranges of E xyplot NOx C EE data ethanol aspect 2 Figure 5 We made the
52. or 10 pages If you use layout c 3 5 the result will not be satisfactory because there will be 15 panels on a page so the arrangement of factor levels will not be consis tent from page to page Things would be better if you could have 14 panels on the page but 7 by 2 or 2 by 7 layouts cause the panels to be too tall or fat You can solve the problem by using the 3 by 5 layout along with the skip argument to skip the center panel or perhaps the upper right panel when panels are traversed in panel order The skip argument is a logical vector that tells for each panel on a page whether it should be skipped The skip vector is replicated as necessary and can even be longer than the number of panels on a page so that complicated multi page skip patterns can be expressed For example the combination of layout c 3 5 skip c rep F 14 T eH ee would skip the upper right hand panel and layout c 3 5 skip c rep F 7 T rep F 7 would skip the middle panel Another similar situation might involve two given factors A and B with 5 and 2 levels respectively As an exercise consider how you could use skip to get 3 columns and 4 rows on a page filled out reasonably Sometimes the interaction of the page size and the number of factor levels can make a nice layout difficult Consider a plot of a fit to the ozone data where ozone is plotted as a function of radiation for 6 levels each of temperature and wind speed xyplot env fit radiati
53. perature 22 24 L 8 20 fi wind wind wind temperature temperature temperature wind wind temperature temperature wind tenjperature 26 30 34 wind wind wind wind wind temperature temperature teniperature temperature temperature temperature 42 36 6 3 0 34 38 wind wind I wnd wind L wnd wind temperature temperature tenjperature temperature temperature temperature Cube Root Ozone cube root ppb 34 38 42 40 f 3 0 wind temperature wind wind wind wind temperature temperature temperature temperature temperature 35 40 45 50 10 35 36 40 44 34 38 42 46 mere rer en ae eG a i att 0 50 150 250 0 50 150 250 0 50 150 250 Radiation langleys Figure 16 Loess fit to the ozone data no aspect ratio control vertical scales allowed to vary from panel to panel 2 8 Aspect Ratio One of the important capabilities of Trellis displays is the ability to control the panel aspect ratio in order to produce more understandable plots The aspect ratio is the physical height of a panel divided by its physical width Physical measurements are in inches or centimeters not in data units The aspect ratio can be specified numerically or it can be computed by banking cal culations In particular Cleveland has experimental evidence that angles near 45 a
54. rary is designed to remedy this situation Besides providing a straightforward way to produce multiple panels on a single page it also sets up a unifying framework for doing this Trellis displays extend S graphics to handle multivariate data situations by using a powerful and general technique conditioning In addition the Trellis software does an excellent job with single panel displays making it a suitable vehicle for doing most high level graphics in S While improving user control of graphics the Trellis software also makes graphics func tions behave just like any other S functions The result of executing a Trellis expression is a Trel lis object Unless it is assigned a name or used in a further computation the Trellis object is dis played The Trellis library is now distributed as a standard part of S PLUS Hold onto your hats jargon to follow S PLUS prior to version 3 3 does not come with the Trellis library In the PC environment S PLUS for Windows Version 3 3 and presumably anything later comes with Trellis Version 2 0 as described in this document Under the Unix operating system S PLUS Version 3 3 contains a slightly older version of Trellis The next release due in 1996 is sched uled to contain Trellis Version 2 0 1 2 A Simple Scatterplot Ethanol Data Perhaps the easiest way to introduce Trellis displays is by examples They illustrate the variety of Trellis displays that can be produced and also introduce the way
55. re the easiest for the viewer to discriminate Banking computes an appropriate aspect ratio to make the impor tant characteristics of the display appear close to 45 The aspect argument controls the aspect ratio of Trellis displays If a numeric value is given that aspect ratio is used for all panels For example aspect 1 says that each panel should be square Another common use for numeric values is to ensure a physical relationship between the x and y coordinates aspect diff range y diff range x would set up an aspect ratio that just matched the ratio of the y and x coordinates This is what is required for making circles appear circular for preserving physical shapes etc Finally using aspect xy performs the banking to 45 computations on the x and y data for all panels The sorted x values and corresponding y values are converted to fractions of the range are differ enced scaled by their length and all thrown together into one computation that finds the aspect ratio that brings the segments closest to 45 Because control over the aspect ratio is best done with at least a hint from the user the default is to use aspect fi11 which makes the aspect ratio appropriate so that the collection of panels fills the display area 26 There is a lot of wasted space in Figure 14 because of the chosen aspect ratio Suppose we had not tried to control the aspect ratio but instead allowed the panels to fill the space us
56. re used panel grid h 1 v 4 draws a horizontal grid aligned with the vertical axis ticks and places 4 vertical reference lines on the panel Although conventional plots often have grids aligned with axes there is no need for this In many cases the grid is not there to help the viewer read off numeric values it provides reference lines against which patterns can be compared Functions panel 1mline and panel qqmathline are designed to put fitted lines on the panel The former draws a line fit with least squares the latter does a robust fit to a theoretical quantile plot As illustrated in Figure 7 the panel loess function adds a smooth curve using the non parametric loess procedure various arguments to loess can be supplied The panel superpose function is often used as a replacement for panel xyplot when each panel is intended to contain multiple lines or points based upon the values of a grouping variable The high level function is given an argument groups that specifies the grouping variable as a factor or something that will be coerced to a factor This groups argument will be passed down to panel superpose to control the plotting symbols and colors used to display the specified groups Examples using panel superpose appear later in Figures 20 and 21 One important thing to remember about the Trellis software everything is written in the S S PLUS language This means that you can read it and modify it The panel fu
57. rix The expression parallel iris df 1 4 iris df 6 layout c 3 1 main Iris Data produces Figure 18 Iris Data in L 1 L L Setosa Virginica A Petal W I WNN M Petal L Sepal W Sepal L T T T Min Max Min Max Figure 18 Parallel coordinate plot showing three groups of the iris data The function as data frame ts takes one or more time series as arguments and pro duces a data frame with components named series which time and cycle The series component is just the data from all of the time series combined into one long vector The time component gives the time associated with each of the points measured in the same units as the original series e g years and cycle gives the periodic component of the time e g 1 Jan 2 Feb Finally the which component is a factor that tells which of the time series the mea surement came from In this case there was only one series hstart but in general as data frame ts can take many arguments For example gt as data frame ts hstart series which time cycle 1 81 9 hstart 1966 000 Jan 2 0 hstart 1966 083 Feb 3 122 4 hstart 1966 167 Mar 4 0 5 9 hstart 1966 250 Apr hstart 1966 333 May 29 Using as data frame ts we can produce a plot that shows housing starts from 1966 to 1974 broken down by their monthly levels timeplot series time cycle data as data fram
58. s for example colors red green blue states Alaska Alabama etc Factors are created by the factor function The other data structure a shingle is a vec tor of numeric data and a set of intervals Shingles get their name because like shingles on a roof the intervals can overlap one another For each interval some subset of the numeric values fall into that interval Two functions are available for dealing with shingles The first is shingle which creates a shingle by specifying the data and intervals By default the intervals are zero width at the unique values of the data This sort of shingle is almost like a factor although the precise numeric value of each interval is preserved Consider for example the engine data and the val ues of compression ratio Treated as a factor there would be 5 values Treated as a shingle there would also be 5 values but the numerical spacing of the compression ratios would be preserved The second function for creating shingles is equal count where a numeric vector is divided up as equally as possible into a specified number of overlapping intervals the amount of overlap between adjacent intervals can also be specified Why are we spending time discussing these data structures One reason is that shingles are sD Jee new to Trellis displays The other is that the ordered level names of factors are used to label Trel lis plots thus it is important to describe how these labels are creat
59. s functions take several arguments that give control over the axes Most basic are the xlim and ylim arguments that allow specification of lower and upper limits for axes For example specifying xlim c 0 100 ensures that the x axis will accommodate values between 0 and 100 Although xlim and ylim are convenient they are only a quick way of specifying one detail of the scales on a Trellis plot Full axis control is available through the scales 24 argument It not only gives precise control over tick marks labels etc but it also controls how the horizontal and vertical axes relate from panel to panel In general scales is a list with components named x and y Each of these components in turn is a list with components in name value form giving information about the x or y axis If the scales list contains other components they are taken to apply to both the x and y axes What kinds of things can you specify in the scales list Here s an example scales list xs Irse corl 2 tekscS aise ols 3 970 aE label c good better best alternating FALSE y list sliced nticks 17 tick FALSE log 2 cex 75 This would place 3 tick marks on the x axis at coordinates 7 9 and 11 labeled with the words good better and best would make the tick marks half their normal length and would draw the axis line tick marks and tick labels in color 2 In addition the x axis labels would not alter
60. scatterplot matrix parallel coordinate plots function as a wire frame 3 dimensional point cloud Trellis displays are adapted to different graphical devices by the trellis device func tion When you are using Trellis graphics on your personal computer or workstation you nor mally use trellis device implicitly when the first Trellis object is printed trellis device is called automatically to produce a window in which to display the graph If you are running Windows the win graph function is invoked and if you are on a workstation running the Unix operating system the motif device function is executed However there may be times when you will want to run trellis device explicitly For example executing trellis device postscript file output ps color TRUE will direct any further Trellis graphics to the postscript device function leaving the file output ps filled with PostScript language commands This file can then be sent to a printer or included in documents The color TRUE argument says that your output should be done in color For a black and white printer use color FALSE Another reason why you may want to execute trellis device explicitly is that you can use it in S Plus to display multiple windows on one screen or you can set up an on screen device in addition to a hardcopy device Not only does trellis device set up the device but it also initializes a list that controls graph characteristics such as plotting
61. t the more rows or columns on the display the smaller the default strip label text size If you want more control you may want to construct your own strip label function or at least change the style given in the default strip function It helps to know that strip default takes two arguments you may want to change strip names and style The strip names argument to strip default is a logical vector of length 2 for factors and shingles respectively that controls whether or not the variable names are included on strip labels By default names are present for shingles but not for factors Similarly there is a style argument to strip default So for example strip function strip default strip names TRUE would put the variable name on both factors and shingles in the strip labels and strip function strip default style 4 would use style 4 for strip labels style 4 is described in the online documentation for strip default In some circumstances you may want to suppress the strip labels altogether In this case you can use strip FALSE Trellis graphics provides a key function that allows great flexibility in labeling plots The key argument to Trellis functions is a list that is passed down almost unchanged to the key function Why then is it an argument to Trellis functions Why not just call key directly The reason is that the key argument is processed by Trellis functions so that they can leave sufficient space aroun
62. t accommodates the data to be plotted there Because it is used so often the relat ion component is the only one of the scale compo nents that need not be named explicitly For both sliced and free scaling axes are drawn for each panel taking up extra space on the display Suppose we had wanted to modify Figure 14 by allowing the y axes to vary from plot to plot If we include scales list y free and remove aspect xy from the call we generate Figure 16 which now makes most compar isons extremely difficult since the vertical axes are now scaled differently When a Trellis display is finished the coordinate system is left in a state that in general has no relation to the scales that were plotted on the page The margins and outer margins are returned to their previous values and the multiple figure parameters are set for one plot per page Because these parameters are reset it is seldom reasonable to add to Trellis displays after they are produced If you want to augment a Trellis display do so by means of the panel or page func tions 05s 0 50 150 250 0 50 150 250 0 50 150 250 Lis y Liss Lisi pau tii it Lisia Lissy wind wind wind wind wind wind temperature temperature temperature temperature _ temperature teniperature aa a S 19 24 23 25 wind wind temperature temperature wind temperature wind temperature wind wind teriperature tem
63. te aspect ratio can be computed Remember by the time the panel function is called the coordinate system and aspect ratio are already locked in These problems are solved by the argument prepane1 which provides a prepanel function The job of the prepanel function is to take the x and y data for a panel and return a list containing 34 Wisconsin No 38 Velvet Trebi Svansota Peatland No 475 No 462 No 457 Manchuria Glabron University Fam ME University Farm MT I year ear Wisconsin No 38 hmm 2 2 2 2 22 20 22 se cee Velvet Trebi Svansota Peatland No 475 No 462 No 457 Manchuria Glabron Wisconsin No 38 Velvet Trebi Svansota Peatland No 475 No 462 No 457 Manchuria Glabron i vear yasr Wisconsin No 38 oe Velvet Trebi Svansota Peatland No 475 No 462 No 457 Manchuria Glabron TTT a_i I Year yaar Wisconsin No 38 oe ee BrE Velvet Trebi Svansota Peatland No 475 No 462 No 457 Manchuria Glabron Crookston i Crookston year year Wisconsin No 38 Velvet Trebi Svansota Peatland No 475 No 462 No 457 Manchuria Glabron 50 60 Barley Yield bushels acre Figure 22 The barley data with factors in alphabetical order Compare to Figure 7 one or more of the following components xlim ylim dx an
64. ts argument was a very general solution to this and similar problems Why is that so Basically once a panel function knows which of the original observa tions correspond to the x and y values that information can be used to deal with any other data that is parallel to the original data In particular if x and y come from a data frame any other variables in the data frame can be subscripted by the subscripts vector To facilitate this even more another argument groups can be given to all high level Trellis functions This argument will be passed down to the panel function along with the subscripts argument Inside the panel function groups subscripts is a vector parallel to x and y that can be used for attribute colors identifiers etc to the observations The panel function panel superpose used earlier in several examples Figures 18 and 20 uses exactly this mechanism to do its work When a high level Trellis function can see that its panel function has a subscripts argu ment it automatically arranges to call the panel function with arguments named x y and sub scripts The subscripts mechanism interacts properly with the subset argument sub scripts are done relative to the original data 3 3 Device Settings Before we get deeply into the concepts of device customization a few words of reassurance Don t worry you normally do not need to write code like this yourself Trellis graphics makes use of customization to produce
65. ula can be found Both of these kinds of arguments were introduced in the book Statistical Models in S by Chambers amp Hastie 1992 We have used this same para digm for Trellis graphics 1 3 Conditioning on Numeric Variables Ethanol Data A simple modification of the previous call to xyplot produces a multi panel display xyplot NOx E C data ethanol Figure 2 This produces Figure 2 which shows NOx emissions plotted against equivalence ratio each panel showing data for one of the five values of compression ratio The Trellis display consists of 5 panels each showing NOx on the vertical axis and E on the C 4 o la ie o 3 4 5 fe 2 a o H o fs 14 or fe fe fe o 0 6 0 8 1 0 1 2 2 l i l i c C I 4 a L 4 fo o fe js a o 4 r L 3 fo x 2 i ao Lo ie o fe m fe fe fe H1 o o o fe o o Cc I Cc 44 L le e o o Oo oo fe z 0 a o fe 24 K L fe fe fo O 14 o 0 a H 00 O oo Oo T T T T T T T 0 6 0 8 1 0 1 2 Figure 2 A Trellis display of engine emissions data horizontal axis The value of C is shown by the strip label at the top of each panel in this case C takes on 5 discrete values as shown by the darkly shaded region of the strip label atop each panel The formula NOx E c is read aloud as NOx is plotted against E given C Note that the variable that goes on the vertical axis is mentioned first in the formula
66. w device save result graphics on the workstation no plotting 18 i f f f i 1 f i 1 f 1 1 f a Alto 1 ea Soprano 2 Soprano 1 height Unit Normal Quantile Figure 12 An improved version of the singer heights data with a comparison to the normal dis tribution S Plus users have slightly different device functions and can have several functions active at once Refer to the Trellis Graphics User s Manual MathSoft 1995 for more details It is sometimes useful to call print trellis directly because it has several arguments that can be used to position Trellis plots on a page Suppose for example that you have two Trellis objects obj1 and obj2 and would like to position them one above the other on a single page of output If the two objects were identical in type both scatterplots for example it might have been easier to use xyplot with a made up given variable to do this However suppose obj1 is a multi panel Trellis display and obj2 is a single display of a different type Let s be more concrete Let obj1 lt qqmath sqrt height voice part distribution qnorm data singer obj2 lt histogram height data singer and now combine them using print trellis print trellis obj1 split c 1 2 1 2 more TRUE Figure 13 print trellis obj2 split c 1 1 1 2 more FALSE The split argument to print trellis is a vector of length 4 c x y nx ny saying to use t

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