# COPLOT MULTIVARIATE PDF

Many critical questions in medicine require the analysis of complex multivariate data, often from large data sets describing numerous variables. By addressing these issues, CoPlot facilitates rich interpretation of multivariate data. We present an example using CoPlot on a recently. Purpose: To describe CoPlot, a publicly available, novel tool for visualizing multivariate data. Methods: CoPlot simultaneously evaluates associations between.

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Multivariate descriptive displays or plots are designed to reveal the relationship among several variables simulataneously. As was the case when examining relationships among pairs of variables, there are several basic characteristics of the relationship among sets of variables that are of interest. The easiest way to get the data for the multivariate plotting examples is to download a copy of the workspace geog Otherwise, all of the individual data sets are available muotivariate download from the GeogR data page.

To get the workspace, right-click on this link [geog RData] and save it to your working folder. Then read it in to R:. The scatter diagram or scatter plot is the workhorse bivariate plot, and can be enhanced to illustrate relationships among three or four variables.

However, a simple plot of Insolation and O18 and correlation suggests otherwise:. The cloud of points at first glace is quite amorphous, mhltivariate the correlation coefficient is also quite low:. Plotting O18 as a function of Ageand color coding the symbols by Insol levels, reveals the nature of the control of ice volume by insolation:.

Information from multivariat variables at a time can also be displayed. In multiariate example for the Summit Cr. Although these are factors, numerical variables could also be plotted.

Note the use of two applications of the legend function: The bubble plot displays the values of three variables at a time using graduated symbols usually circleswhere the value of one variable determines the relative position of the symbol along the X-axis and the value of a second variable determines the relative position of the symbol along the Y-axis, and the value of the third variable is used to determine the size of the symbol.

There are a number of basic enhancements of the basic 3-D scatter plot, such as the addition of drop lines, lines connecting points, symbol modification and so on. This plot makes use of the lattice package. Notice that you can still see the outline of the state, because elevation is a fairly well behaved variable.

The first part of the code, like in making maps, does some setup like determining the number of colors to plot and getting their definitions. The second block produces the plot. The z-variable, in this case, annual precipitation, is plotted as a dot, and for interpretability a drop line is plotted below the dot. This simple addition facilitates finding the location of each point where it hits the x-y, or latitude-longitude planeas well as the value of annual precipitation.

The map function generates the outlines of a map of Oregon counties, and stores them in or. The rgl package by D.

Alder can be used to plot points and surfaces and lines in a 3-D space. Multivaritae down the left button while dragging rotates the balls, while holding down the right changes the perspective. Often, the issue might arise of how a particular relationship between variables might differ among groups.

Information of that nature can be gained using conditioning plots or coplots.

## Multivariate displays – Coplots

Such plots are part of a general scheme of visual data analysis, known as Trellis Graphics that has been created by the developers of the S language. Trellis Graphics are implemented in R using the package Lattice. Conditioning scatter plots involves creating a multipanel display, where each panel contains a subset of the data.

This subset can be either a those observations that fall in a particular group, or b they may represent a the values that fall within a particular range of the values of a variable.

This coplot contains scatter diagrams for Yes as a function of the log 10 of Population, conditioned by country i.

In other words, coplot selects the observations of Yes and log Pop for a particular panel i. Countrysends these to the panel function, which passes them on relabeled as x and yand plots the points, and then panel.

The general idea is to compare the panels countries seeing where in the panel the points lie and what the relationship looks like.

### CoPlot: a tool for visualizing multivariate data in medicine.

The general relationship between population and percent multivariqte Yes votes is apparent, as well as colpot differences, like the generally greater proportion of Yes votes in Finland. Most of the time, the conditioning variables are continuous numeric variables.

We know the arrangement of the reaches, and so the resulting plot should be no surprise. The idea here is to chop longitude into eight bands from west to east using the equal. The third argument here, 0. Then the lattice plot is made using the xyplot function, which makes a separate scatter plot for each longitude band, showing the relationship between annual precipitation and elevation.

Notice that in each panel, a straight regression line more about regression later and a smooth lowess curve have been added to help summarize the relationships.

The panels are arranged in longitudinal order from low west to high east, remember that in the western hemisphere, longitudes are negative. The plots are certainly interesting. The general idea is that precipitation should increase with increasing elevation, but at least for the western part of the state the reverse seems to be true! What is going on here is that proximity to the Pacific is a much more important control than elevation, and low elevation coastal and inland stations are quite wet.

The following plots explore the seasonality of precipitation in the Yellowstone region. The first block of code below sets things up, and the stars function does the plotting. Here the stars wind up looking more like fans. The legend indicates that stations with fans that open out to the right are stations with winter precipitation maxima like in the southwestern portion of the region while those that open toward the left have summer precipitation maxima like in the southeastern portion of the region.

The next examples show a couple of conditioning plots coplotsthat illustrate the relationship between January and July precipitation, as varies is conditioned on with elevation. The first block of code does some set up. The plot shows that the relationship between January and July precipitation indeed varies with elevation. At low elevations, there is proportionally lower January precipitation for the same July values lower two panels on the lattice plotbut at higher elevations, there is proportionally more top two panels.

This relationship points to some orographic i.

### RobCoP: A Matlab Package for Robust CoPlot Analysis

The next plot shows the variation of the relationship between January and July precipitation as it varies spatially. Notice that the steepest curve lies in the panel representing the southwestern part of the region low latitude and low longitude, i. Notice that at low elevations, most of the stations are behaving similarly, and showing a distinct summer precipitation maximum and only one station seems to show a winter maximum.

At high elevations, there is more variability but a general tendency for winter precipitation to dominate. Lattice plots can extend many of the basic univariate and bivariate plots. The spplot function in the sp package is a Lattice-plot type multivaritae, and can be thought of as either extending the capabilities of Lattice plots to maps, or extending the ability of R to produce multi-panel maps.

The following example uses a data set of locations and elevations Oregon cirque basins upland basins eroded by glaciersand whether or not they are currently early 21st century glaciated. Whether a cirque is occupied by a glacier or not is basically determined by the trade-off between snow accumulation and hence winter precipitation and summer ablation or melting, and hence summer temperature.

In the code below, the two as. The two variables are obviously redundant the elements would sum to 1 for each observationbut it makes the illustration of the method more transparent. The top panel shows unglaciated cirques in pink and glaciated ones in turquoise, while the bottom panel shows the reverse, glaciated cirques in pink, multivaroate in turquoise. Note the aspect argument — this scales the horizontal and vertical axes of the plot in a way that makes the map look projected.

This way of mapping the cirques could also have been done by plotting a multuvariate shape file, and then putting points on top, e.

Finally, here are some multi- and single-panel plots of climate-station data, the interpretation of which is straightforward. Then read it in to R: However, a simple plot of Insolation and O18 and correlation suggests otherwise: