1/8/2011

Aspect Ratio and Banking to 45^o

Cleveland has shown that we can enhance XY chart visualization by careful selection of the data rectangle (area inside axes) height to width (h/w) aspect ratio.

 "The aspect ratio is vital because it has a large impact on our ability to judge rate of change. A number of studies in visual perception have shown that our ability to judge the relative slopes of line segments on a graph is maximized when the absolute values of the orientations of the segments are centered on 45 degrees." Cleveland

Naomi Robbins, in her book  Creating More Effective Graphs, points out that

"... the human eye can judge a 45^o angle with considerable accuracy. Small angle such as 5^o and 10^o are much more difficult to judge.  Designers can take advantage of this by changing the aspect ratio of a chart so that the orientations of line segments are centered at 45^o, [this is] called banking to 45^o ".

This page presents Excel workbook examples of Cleveland's aspect ratio and banking to 45^o techniques.

 Sunspot Trend Example

In The Elements of Graphing Data (page 4), Cleveland uses the 1750-1924 annual sunspot trend to demonstrate the importance of aspect ratio, showing the same data with plot aspect ratios (h/w) of 1.0 and 0.055.  Cleveland's charts, reproduced in Excel, are shown on the right.

Sunspots demonstrate a clear cyclical pattern, with cycles averaging 11 years. The lower plot h/w aspect ratio (0.055)  shows that the  cycle rise is much faster than the decline, a pattern not apparent in the upper plot.

In Beautiful Evidence, Tufte (page 60), builds on Cleveland's work by asking the question "How should a sparkline aspect ratio be chosen?". Part of Tufte's  answer is reproduced bellow....

".. a graphic's width/ height [w/h] ratio makes a big difference in displaying data." ...
In general , statistical graphics should be moderately greater in length than in height. And, as William Cleveland discovered, for judging slopes and velocities up and down hills in time-series, best is an aspect ratio that yields hill-slopes averaging 45^o, over all the cycles in the time-series. That is, variations in slopes are best detected when the slopes are around 45^o, uphill or downhill. ... the aspect ratio should be such that the time-series graphics tend toward a lumpy  profile rather than a spiky profile .. or a flat profile." Tufte, Beautiful Evidence

CO₂ Trend Example

In an excellent article, Heer and Agrawala of University of California, Berkeley,  further demonstrate the importance of the aspect ratio to viewer's perception. Basing their work on Cleveland's earlier work, they present two trend plots for average monthly CO₂ measurements at the Mauna Loa observatory in Hawaii. The plots show 1959 - 1990 CO2 trends, one with an h/w aspect of 0.85 and the second with an aspect ratio of 0.127. Their plots have been reproduced in Excel , as shown on the right.

Review of these two plots of the same CO₂ data provide important lessons:

1. There is a noticeable inflection point in the 0.85 h/w aspect ratio plot that is not as apparent in the 0.127 aspect ratio plot.

2. The CO₂ data follows a seasonal cycle, with a gradual rise and sharper decline, the opposite pattern to what we saw with the sunspot plots.

3. The 0.127 aspect ratio chart shows the different rise and fall rates of the seasonal cycle better than the 0.85 ratio chart.

Both charts are based on Cleveland's banking to 45^o  technique. The upper chart, with the 0.85 h/w aspect ratio, was developed by fitting a smooth regression curve and using the banking to 45^o of that curve while the 0.127 curve uses the entire data set to establish the banking to 45^o aspect ratio.

Using Cleveland's Techniques in Excel

Advanced graphical analysis packages like R, S Plus, SAS  include data rectangle aspect ratio control and banking to 45^o capabilities. How can Excel Users use Cleveland's data rectangle aspect ratio and banking to 45^o techniques?

To use Cleveland's techniques in Excel, we need to be able to do two things:

1. Control Excel Chart Data Rectangle dimensions

2. Calculate banking to 45^o aspect ratio

The next two section describe how to implement Cleveland's technique in Excel. A downloadable copy of my banking_to_45 workbook is available here.

Controlling Excel Chart Dimensions

 Excel's chart object includes three areas:

1. Overall chart area (Title, horizontal & vertical offsets, Plot Area)
2. Plot area (Axes marks, labels, area inside axes)
3. Data rectangle (actual data area inside axes)

To control the h/w aspect ratio, we must be able to accurately specify the width and height of the data rectangle. How can we directly control the dimensions of the data rectangle?

Excel lets Users manually size the chart area and plot area, Excel does not let the User directly control the data rectangle aspect ratio The chart area includes the plot area and titles and offsets, controlling chart area dimensions does not directly control the data rectangle dimensions. The plot area includes the axis tick marks and labels as well as the data rectangle, so the data rectangle  aspect ratio is affected by both the plot aspect ratio and axis formatting. Differences in X and Y axis formatting will affect the aspect ratio,.

The data rectangle is the actual graph area inside the axis lines. By turning off all X and Y axis major and minor tick formatting, labels and axis titles, we can make the the plot area and data rectangle the same. We can then use dummy axis series to provide X and Y axis labels and text boxes to provide axis titles.

To provide data rectangle dimensions based on target h/w aspect ratio, I have developed a multi step procedure to control the data rectangle aspect ratio:

1. Turn off X and Y axis tick marks and labels and axis titles so that the data rectangle and plot area are equal.
2. Create X and Y dummy axis series
3. Run VBA procedure that gets user provided target data rectangle width, Aspect Ratio and offsets, sets data rectangle dimensions,  establishes plot area and chart area dimensions based on User input.

The VBA procedure lets me specify the data rectangle width and desired h/w aspect ratio. The procedure then sizes the data rectangle - plot area and chart area so that the plot area aspect ratio meet the required value.

Banking to 45^o Calculations

Heer and Agrawala explain the 12 basic methods for calculating the banking to 45^o aspect ratio. I elected to use the average line segment = 45^o   method. I used Excel's Solver to calculate the aspect ratio that yields  an average orientation of 45^o.

Here's the link to the Bank_to_45 Workbook