Spacing measures: heterogeneity in numerical distributions

Numerically-coded data sequences can exhibit a very wide range of distributional characteristics, including near-Gaussian (historically, the most popular working assumption), strongly asymmetric, light- or heavy-tailed, multi-modal, or discrete (e.g., count data).  In addition, numerically coded values can be effectively categorical, either ordered, or unordered.  A specific example that illustrates the range of distributional behavior often seen in a collection of numerical variables is the Boston housing dataframe (Boston) from the MASS package in R.  This dataframe includes 14 numerical variables that characterize 506 suburban housing tracts in the Boston area: 12 of these variables have class “numeric” and the remaining two have class “integer”.  The integer variable chas is in fact a binary flag, taking the value 1 if the tract bounds the Charles river and 0 otherwise, and the integer variable rad is described as “an index of accessibility to radial highways,’’ assuming one of nine values: the integers 1 through 8, and 24.  The other 12 variables assume anywhere between 26 unique values (for the zoning variable zn) to 504 unique values (for the per capita crime rate crim).  The figure below shows nonparametric density estimates for four of these variables: the per-capita crime rate (crim, upper left plot), the percentage of the population designated “lower status” by the researchers who provided the data (lstat, upper right plot), the average number of rooms per dwelling (rm, lower left plot), and the zoning variable (zn, lower right plot).  Comparing the appearances of these density estimates, considerable variability is evident: the distribution of crim is very asymmetric with an extremely heavy right tail, the distribution of lstat is also clearly asymmetric but far less so, while the distribution of rm appears to be almost Gaussian.  Finally, the distribution of zn appears to be tri-modal, mostly concentrated around zero, but with clear secondary peaks at around 20 and 80.

Each of these four plots also includes some additional information about the corresponding variable: three vertical reference lines at the mean (the solid line) and the mean offset by plus or minus three standard deviations (the dotted lines), and the value of the normalized Shannon entropy, listed in the title of each plot.  This normalized entropy value is discussed in detail in Chapter 3 of Exploring Data in Engineering, the Sciences, and Medicine and in two of my previous posts (April 3, 2011 and May 21, 2011), and it forms the basis for the spacing measure described below.  First, however, the reason for including the three vertical reference lines on the density plots is to illustrate that, while popular “Gaussian expectations” for data are approximately met for some numerical variables (the rm variable is a case in point here), often these expectations are violated so much that they are useless.  Specifically, note that under approximately Gaussian working assumptions, most of the observed values for the data sequence should fall between the two dotted reference lines, which should correspond approximately to the smallest and largest values seen in the dataset.  This description is reasonably accurate for the variable rm, and the upper limit appears fairly reasonable for the variable lstat, but the lower limit is substantially negative here, which is not reasonable for this variable since it is defined as a percentage.  These reference lines appear even more divergent from the general shapes of the distributions for the crim and zn data, where again, the lower reference lines are substantially negative, infeasible values for both of these variables.

The reason the reference values defined by these lines are not particularly representative is the extremely heterogeneous nature of the data distributions, particularly for the variables crim – where the distribution exhibits a very long right tail – and zn – where the distribution exhibits multiple modes.  For categorical variables, distributional heterogeneity can be assessed by measures like the normalized Shannon entropy, which varies between 0 and 1, taking the value zero when all levels of the variable are equally represented, and taking the value 1 when only one of several possible values are present.  This measure is easily computed and, while it is intended for use with categorical variables, the procedures used to compute it will return results for numerical variables as well.  These values are shown in the figure captions of each of the above four plots, and it is clear from these results that the Shannon measure does not give a reliable indication of distributional heterogeneity here.  In particular, note that the Shannon measure for the crim variable is zero to three decimal places, suggesting a very homogeneous distribution, while the variables lstat and rm – both arguably less heterogeneous than crim – exhibit slightly larger values of 0.006 and 0.007, respectively.  Further, the variable zn, whose density estimate resembles that of crim more than that of either of the other two variables, exhibits the much larger Shannon entropy value of 0.585.

The basic difficulty here is that all observations of a continuously distributed random variable should be unique.  The normalized Shannon entropy – along with the other heterogeneity measures discussed in Chapter 3 of Exploring Data – effectively treat variables as categorical, returning a value that is computed from the fractions of total observations assigned to each possible value for the variable.  Thus, for an ideal continuously-distributed variable, every possible value appears once and only once, so these fractions should be 1/N for each of the N distinct values observed for the variable.  This means that the normalized Shannon measure – along with all of the alternative measures just noted – should be identically zero for this case, regardless of whether the continuous distribution in question is Gaussian, Cauchy, Pareto, uniform, or anything else.  In fact, the crim variable considered here almost meets this ideal requirement: in 506 observations, crim exhibits 504 unique values, which is why its normalized Shannon entropy value is zero to three significant figures.  In marked contrast, the variable zn exhibits only 26 distinct values, meaning that each of these values occurs, on average, just over 19 times.  However, this average behavior is not representative of the data in this case, since the smallest possible value (0) occurs 372 times, while the largest possible value (100) occurs only once.  It is because of the discrete character of this distribution that the normalized Shannon entropy is much larger here, accurately reflecting the pronounced distributional heterogeneity of this variable.

Taken together, these observations suggest a simple extension of the normalized Shannon entropy that can give us a more adequate characterization of distributional differences for numerical variables.  Specifically, the idea is this: begin by dividing the total range of a numerical variable x into M equal intervals.  Then, count the number of observations that fall into each of these intervals and divide by the total number of observations N to obtain the fraction of observations falling into each group.  By doing this, we have effectively converted the original numerical variable into an M-level categorical variable, to which we can apply heterogeneity measures like the normalized Shannon entropy.  The four plots below illustrate this basic idea for the four Boston housing variables considered above.  Specifically, each plot shows the fraction of observations falling into each of 10 equally spaced intervals, spanning the range from the smallest observed value of the variable to the largest.

As a specific example, consider the results shown in the upper left plot for the variable crim, which varies from a minimum of 0.00632 to a maximum of 89.0.  Almost 87% of the observations fall into the smallest 10% of this range, from 0.00632 to 8.9, while the next two groups account for almost all of the remaining observations.  In fact, none of the other groups (4 through 10) account for more than 1% of the observations, and one of these groups – group 7 – is completely empty.  Computing the normalized Shannon entropy from this ten-level categorical variable yields 0.767, as indicated in the title of the upper left plot.  In contrast, the corresponding plot for the lstat variable, shown in the upper right, is much more uniform, with the first five groups exhibiting roughly the same fractional occupation.  As a consequence, the normalized Shannon entropy for this grouped variable is much smaller than that for the more heterogeneously distributed crim variable: 0.138 versus 0.767.  Because the distribution is more sharply peaked for the rm variable than for lstat, the occupation fractions for the grouped version of this variable (lower left plot) are less homogeneous, and the normalized Shannon entropy is correspondingly larger, at 0.272.  Finally, for the zn variable (lower right plot), the grouped distribution appears similar to that for the crim variable, and the normalized Shannon entropy values are also similar: 0.525 versus 0.767. 

The key point here is that, in contrast to the normalized Shannon entropy applied directly to the numerical variables in the Boston dataframe, grouping these values into 10 equally-spaced intervals and then computing the normalized Shannon entropy gives a number that seems to be more consistent with the distributional differences between these variables that can be seen clearly in their density plots.  Motivation for this numerical measure (i.e., why not just look at the density plots?) comes from the fact that we are sometimes faced with the task of characterizing a new dataset that we have not seen before.  While we can – and should – examine graphical representations of these variables, in cases where we have many such variables, it is desirable to have a few, easily computed numerical measures to use as screening tools, guiding us in deciding which variables to look at first, and which techniques to apply to them.  The spacing measure described here – i.e., the normalized Shannon entropy measure applied to a grouped version of the numerical variable – appears to be a potentially useful measure for this type of preliminary data characterization.  For this reason, I am including it – along with a few other numerical characterizations – in the DataFrameSummary procedure I am implementing as part of the ExploringData package, which I will describe in a later post.  Next time, however, I will explore two obvious extensions of the procedure described here: different choices of the heterogeneity measure, and different choices of the number of grouping levels.  In particular, as I have shown in previous posts on interestingness measures, the normalized Bray, Gini, and Simpson measures all behave somewhat differently than the Shannon measure considered here, raising the question of which one would be most effective in this application.  In addition, the choice of 10 grouping levels considered here was arbitrary, and it is by no means clear that this choice is the best one.  In my next post, I will explore how sensitive the Boston housing results are to changes in these two key design parameters.

Finally, it is worth saying something about how the grouping used here was implemented.  The R code listed below is the function I used to convert a numerical variable x into the grouped variable from which I computed the normalized Shannon entropy.  The three key components of this function are the classIntervals function from the R package classInt (which must be loaded before use; hence, the “library(classInt)” statement at the beginning of the function), and the cut and table functions from base R.  The classIntervals function generates a two-element list with components var, which contains the original observations, and brks, which contains the M+1 boundary values for the M groups to be generated.  Note that the style = “equal” argument is important here, since we want M equal-width groups.  The cut function then takes these results and converts them into an M-level categorical variable, assigning each original data value to the interval into which it falls.  The table function counts the number of times each of the M possible levels occurs for this categorical variable.  Dividing this vector by the sum of all entries then gives the fraction of observations falling into each group.  Plotting the results obtained from this function and reformatting the results slightly yields the four plots shown in the second figure above, and applying the shannon.proc procedure available from the OUP companion website for Exploring Data yields the Shannon entropy values listed in the figure titles.

UniformSpacingFunction <- function(x, nLvls = 10){

  #

  library(classInt)

  #

  xsum = classIntervals(x,n = nLvls, style="equal")

  xcut = cut(xsum$var, breaks = xsum$brks, include.lowest = TRUE)

  xtbl = table(xcut)

  pvec = xtbl/sum(xtbl)

  pvec

}

Implementing the CountSummary Procedure

In my last post, I described and demonstrated the CountSummary procedure to be included in the ExploringData package that I am in the process of developing.  This procedure generates a collection of graphical data summaries for a count data sequence, based on the distplot, Ord_plot, and Ord_estimate functions from the vcd package.  The distplot function generates both the Poissonness plot and the negative-binomialness plot discussed in Chapters 8 and 9 of Exploring Data in Engineering, the Sciences and Medicine.  These plots provide informal graphical assessments of the conformance of a count data sequence with the two most popular distribution models for count data, the Poisson distribution and the negative-binomial distribution.  As promised, this post describes the R code needed to implement the CountSummary procedure, based on these functions from the vcd package.

The key to this implementation lies in the use of the grid package, a set of low-level graphics primitives included in base R.  As I mentioned in my last post, the reason this was necessary - instead of using higher-level graphics packages like lattice or ggplot2 - was that the vcd package is based on grid graphics, making it incompatible with base graphics commands like those used to generate arrays of multiple plots.  The grid package was developed by Paul Murrell, who provides a lot of extremely useful information about both R graphics in general and grid graphics in particular on his home page, including the article “Drawing Diagrams with R,” which provides a nicely focused introduction to grid graphics.  The first example I present here is basically a composite of the first two examples presented in this paper.  Specifically, the code for this example is:

library(grid)

grid.newpage()

pushViewport(viewport(width = 0.8, height = 0.4))

grid.roundrect()

grid.text("This is text in a box")

popViewport()

The first line of this R code loads the grid package and the second tells this package to clear the plot window; failing to do this will cause this particular piece of code to overwrite whatever was there before, which usually isn’t what you want.  The third line creates a viewport, into which the plot will be placed.  In this particular example, we specify a width of 0.8, or 80% of the total plot window width, and a height of 0.4, corresponding to 40% of the total window height.  The next two lines draw a rectangular box with rounded corners and put “This is text in a box” in the center of this box.  The advantage of the grid package is that it provides us with simple graphics primitives to draw this kind of figure, without having to compute exact positions (e.g., in inches) for the different figure components.  Commands like grid.text provide useful defaults (i.e., put the text in the center of the viewport), which can be overridden by specifying positional parameters in a variety of ways (e.g., left- or right-justified, offsets in inches or lines of text, etc.).  The results obtained using these commands are shown in the figure below.

The code for the second example is a simple extension of the first one, essentially consisting of the added initial code required to create the desired two-by-two plot array, followed by four slightly modified copies of the above code.  Specifically, this code is:

grid.newpage()

pushViewport(viewport(layout=grid.layout(nrow=2,ncol=2)))

pushViewport(viewport(layout.pos.row=1,layout.pos.col=1))

grid.roundrect(width = 0.8, height=0.4)

grid.text("Plot 1 goes here")

popViewport()

pushViewport(viewport(layout.pos.row=1,layout.pos.col=2))

grid.roundrect(width = 0.8, height=0.4)

grid.text("Plot 2 goes here")

popViewport()

pushViewport(viewport(layout.pos.row=2,layout.pos.col=1))

grid.roundrect(width = 0.8, height=0.4)

grid.text("Plot 3 goes here")

popViewport()

pushViewport(viewport(layout.pos.row=2,layout.pos.col=2))

grid.roundrect(width = 0.8, height=0.4)

grid.text("Plot 4 goes here")

popViewport()

Here, note that the first “pushViewport” command creates the two-by-two plot array we want, by specifying “layout = grid.layout(nrow=2,ncol=2)”.  As in initializing a data frame in R, we can create an arbitrary two-dimensional array of grid graphics viewports – say m by n – by specifying “layout = grid.layout(nrow=m, ncol=n)”.  Once we have done this, we can use whatever grid commands – or grid-compatible commands, such as those generated by the vcd package – we want, to create the individual elements in our array of plots.  In this example, I have basically repeated the code from the first example to put text into rounded rectangular boxes in each position of the plot array.  The two most important details are, first, the “pushViewport” command at the beginning of each of these individual plot blocks specifies which of the four array elements the following plot will go in, and second, the “popViewport()” command at the end of each block, which tells the grid package that we are finished with this element of the array.  If we leave this command out, the next “pushViewport” command will not move to the desired plot element, but will simply overwrite the previous plot.  Executing this code yields the plot shown below.

The final example replaces the text in the above two-by-two example with the plots I want for the CountSummary procedure.  Before presenting this code, it is important to say something about the structure of the resulting plot and the vcd commands used to generate the different plot elements.  The first plot – in the upper left position of the array shown below – is an Ord plot, generated by the Ord_plot command, which does two things.  The first is to generate the desired plot, but the second is to return estimates of the intercept and slope of one of the two reference lines in the plot.  The first of these lines is fit to the points in the plot via ordinary least squares, while the second – the one whose parameters are returned – is fit via weighted least squares, to down-weight the widely scattered points seen in this plot that correspond to cases with very few observations.  The intent of the Ord plot is to help us decide which of several alternative distributions – including both the Poisson and the negative-binomial – fits our count data sequence better.   This guidance is based on the reference line parameters, and the Ord_estimate function in the vcd package transforms these parameter estimates into distributional recommendations and the distribution parameter values needed by the distplot function in the vcd package to generate either the Poissonness plot or the negative-binomialness plot for the count data sequence.  Although these recommendations are sometimes useful, it is important to emphasize the caution given in the vcd package documentation:

           

“Be careful with the conclusions from Ord_estimate as it implements just some simple heuristics!”

In the CountSummary procedure, I use these results both to generate part of the text summary in the upper right element of the plot array, and to decide which type of plot to display in the lower right element of this array.  Both this plot and the Poissonness reference plot in the lower left element of the display are created using the distplot command in the vcd package.  I include the Poissonness reference plot because the Poisson distribution is the most commonly assumed distribution for count data – analogous in many ways to the Gaussian distribution so often assumed for continuous-valued data – and, by not specifying the single parameter for this distribution, I allow the function to determine it by fitting the data.  In cases where the Ord plot heuristic recommends the Poissonness plot, it also provides this parameter, which I provide to the distplot function for the lower right plot.  Thus, while both the lower right and lower left plots are Poissonness plots in this case, they are generally based on different distribution parameters.  In the particular example shown here – constructed from the “number of times pregnant” variable in the Pima Indians diabetes dataset that I have discussed in several previous posts (available from the UCI Machine Learning Repository) – the Ord plot heuristic recommends the negative binomial distribution.  Comparing the Poissonness and negative-binomialness plots in the bottom row of the above plot array, it does appear that the negative binomial distribution fits the data better.

Finally, before examining the code for the CountSummary procedure, it is worth noting that the vcd package’s implementation of the Ord_plot and Ord_estimate procedures can generate four different distributional recommendations: the Poisson and negative-binomial distributions discussed here, along with the binomial distribution and the much less well-known log-series distribution.  The distplot procedure is flexible enough to generate plots for the first three of these distributions, but not the fourth, so in cases where the Ord plot heuristic recommends this last distribution, the CountSummary procedure displays the recommended distribution and parameter, but displays a warning message that no distribution plot is available for this case in the lower right plot position.

The code for the CountSummary procedure looks like this:

CountSummary <- function(xCount,TitleString){

  #

  #  Initial setup

  #

  library(vcd)

  grid.newpage()

  #

  #  Set up 2x2 array of plots

  #

  pushViewport(viewport(layout=grid.layout(nrow=2,ncol=2)))

  #

  #  Generate the plots:

  #

  #  1 - upper left = Ord plot

  #

  pushViewport(viewport(layout.pos.row=1,layout.pos.col=1))

  OrdLine = Ord_plot(xCount, newpage = FALSE, pop=FALSE, legend=FALSE)

  OrdType = Ord_estimate(OrdLine)

  popViewport()

  #

  #  2 - upper right = text summary

  #

  OrdTypeText = paste("Type = ",OrdType$type,sep=" ")

  if (OrdType$type == "poisson"){

    OrdPar = "Lambda = "

  }

  else if ((OrdType$type == "nbinomial")|(OrdType$type == "nbinomial")){

    OrdPar = "Prob = "

  } 

  else if (OrdType$type == "log-series"){

    OrdPar = "Theta = "

  }

  else{

    OrdPar = "Parameter = "

  }

  OrdEstText = paste(OrdPar,round(OrdType$estimate,digits=3), sep=" ")

  TextSummary = paste("Ord plot heuristic results:",OrdTypeText,OrdEstText,sep="\n")

  pushViewport(viewport(layout.pos.row=1,layout.pos.col=2))

  grid.text(TitleString,y=2/3,gp=gpar(fontface="bold"))

  grid.text(TextSummary,y=1/3)

  popViewport()

  #

  #  3 - lower left = standard Poissonness plot

  #

  pushViewport(viewport(layout.pos.row=2,layout.pos.col=1))

  distplot(xCount, type="poisson",newpage=FALSE, pop=FALSE, legend = FALSE)

  popViewport() 

  #

  #  4 - lower right = plot suggested by Ord results

  #

  pushViewport(viewport(layout.pos.row=2,layout.pos.col=2))

  if (OrdType$type == "poisson"){

    distplot(xCount, type="poisson",lambda=OrdType$estimate, newpage=FALSE, pop=FALSE, legend=FALSE)

  }

  else if (OrdType$type == "nbinomial"){

    prob = OrdType$estimate

    size = 1/prob - 1

    distplot(xCount, type="nbinomial",size=size,newpage=FALSE, pop=FALSE, legend=FALSE)

  }

  else if (OrdType$type == "binomial"){

    distplot(xCount, type="binomial", newpage=FALSE, pop=FALSE, legend=FALSE)

  }

  else{

    Message = paste("No distribution plot","available","for this case",sep="\n")

    grid.text(Message)

  }

  popViewport()

  #

}

This procedure is a function called with two arguments: the sequence of count values, xCounts, and TitleString, a text string that is displayed in the upper right text box in the plot array, along with the recommendations from the Ord plot heuristic.  When called, the function first loads the vcd library to make the Ord_plot, Ord_estimate, and distplot functions available for use, and it executes the grid.newpage() command to clear the display.  (Note that we don’t have to include “library(grid)” here to load the grid package, since loading the vcd package automatically does this.)   As in the previous example, the first “pushViewport” command creates the two-by-two plot array, and this is again followed by four code segments, one to generate each of the four displays in this array.  The first of these segments invokes the Ord_plot and Ord_estimate commands as discussed above, first to generate the upper left plot (a side-effect of the Ord_plot command) and second, to obtain the Ord plot heuristic recommendations, to be used in structuring the rest of the display.  The second segment creates a text display as in the first example considered here, but the structure of this display depends on the Ord plot heuristic results (i.e., the names of the parameters for the four possible recommended distributions are different, and the logic in this code block matches the display text to this distribution).  As noted in the preceding discussion, the third plot (lower left) is the Poissonness plot generated by the distplot function from the vcd package.  In this case, the function is called only specifying ‘type = “poisson”’ without the optional distribution parameter lambda, which is obtained by fitting the data.  The final element of this plot array, in the lower right, is also generated via a call to the distplot function, but here, the results from the Ord plot heuristic are used to specify both the type parameter and any optional or required shape parameters for the distribution.  As with the displayed text, simple if-then-else logic is required here to match the plot generated with the Ord plot heuristic recommendations.

Finally, it is important to note that in all of the calls made to Ord_plot or distplot in the CountSummary procedure, the parameters newpage, pop, and legend, are all specified as FALSE.  Specifying “newpage = FALSE” prevents these vcd plot commands from clearing the display page and erasing everything we have done so far.  Similarly, specifying “pop = FALSE” allows us to continue working in the current plot window until we notify the grid graphics system that we are done with it by issuing our own “popViewport()” command.  Specifying “legend = FALSE” tells Ord_plot and distplot not to write the default informational legend on each plot.  This is important here because, given the relatively small size of the plots generated in this two-by-two array, including the default legends would obscure important details.