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{{Short description|Thematic map visualizing multiple variables}}
[[File:Black Hispanic Bivariate Map.png|thumb|400px|Bivariate choropleth map comparing the Black (blue) and Hispanic (red) populations in the United States, 2010 census; shades of purple show significant proportions of both groups.]]
A '''bivariate map'''
The typical objective of a multivariate map is to visualize any statistical or geographic [[Correlation and dependence|relationship]] between the variables. It has potential to reveal relationships between variables more effectively than a side-by-side comparison of the corresponding univariate maps, but also has the danger of [[Cognitive overload]] when the symbols and patterns are too complex to easily understand.<ref name="slocum2009">T. Slocum, R. McMaster, F. Kessler, H. Howard (2009). Thematic Cartography and Geovisualization, Third Edn. Pearson Prentice Hall: Upper Saddle River, NJ.</ref>{{rp|331}}
==History==
[[File:Minard-carte-viande-1858.png |thumb|
The first multivariate maps appeared in the early [[Industrial era]] (1830-1860), at the same time that [[
[[Charles Joseph Minard]] became a master at creating visualizations that combined multiple variables during the 1850s and 1860s, often mixing [[Choropleth map
Multivariate thematic maps found a resurgence starting in the middle of the 20th Century, coinciding with the [[Quantitative revolution|scientific turn in geography]]. [[George F. Jenks]] introduced the bivariate dot density map in 1953.<ref name="jenks1953" /> The first modern bivariate choropleth maps were published by the U.S. Census Bureau in the 1970s.<ref>{{cite journal |last1=Meyer |first1=Morton A. |last2=Broome |first2=Frederick R. |last3=Schweitzer |first3=Richard H. Jr. |title=Color Statistical Mapping by the U.S. Bureau of the Census |journal=The American Cartographer |date=1975 |volume=2 |issue=2 |pages=
==Methods==
There are a variety of ways in which separate variables can be mapped simultaneously, which generally fall into a few approaches:
[[File:Bivariate.png|thumb|
* A ''multi-layered thematic map'' portrays the variables as separate map layers, using different [[thematic map]] techniques. An example would be showing one variable as a [[choropleth map]], with another variable shown as [[Proportional symbol map|proportional symbols]] on top of the choropleth.
* A ''correlated symbol map'' represents two or more variables in the same thematic map layer, using the same [[visual variable]], designed in such a way as to show the relative combination of the two variables.
** A ''bivariate [[choropleth map]]'' is the most common type of correlated symbol. Contrasting but not complementary colors are generally used, so that their combination is intuitively recognized as "between" the two original colors, such as red+blue=purple.<ref name="trumbo1981" /> They have been found to be more easily used if the map includes a carefully designed legend and an explanation of the technique.<ref name="Olson1981"/> A common legend strategy is a two dimensional matrix, divided into smaller boxes where each box represents a unique relationship of the variables.
** A ''multivariate [[Dot distribution map|dot density map]]'' mixes dots of different colors in each district, typically representing separate subgroups of the overall population.<ref name="jenks1953">{{cite journal |last1=Jenks |first1=George F. |title="Pointillism" as a Cartographic Technique |journal=The Professional Geographer |date=1953 |volume=5 |issue=5 |pages=4–6 |doi=10.1111/j.0033-0124.1953.055_4.x}}</ref>
* A ''multivariate symbol map'' represents two or more variables in the same thematic map layer, using distinct [[visual variable]]s for each variable.<ref name="slocum2009" />{{rp|337}}<ref name="nelson1996"/> For example, a layer of cities might be symbolized with circles of [[Proportional symbol map|proportional size]] representing its total population, and the hue of each circle representing the predominant source type of its electric power, akin to a nominal [[choropleth map]].
** A ''[[cartogram]]'' distorts the size and shape of a set of districts according to a variable, but does not dictate the symbol used to draw each district. Thus it is common to symbolize them as a [[choropleth map]].
** A ''chart map'' represents each geographic feature with a [[Chart|statistical chart]], often a [[pie chart]] or [[bar chart]], which can include a number of variables. Each chart is usually drawn proportionally to a total, making it a multivariate symbol.
** ''[[Chernoff face]]s'' have occasionally been used in maps since the 1970s, generally in an experimental situation.<ref>{{cite journal |last1=Wainer |first1=H. |title=Graphic Experiment in Display of Nine Variables Uses Faces to Show Multiple Properties of States |journal=Newsletter of the Bureau of Social Sciences Research |date=1979 |volume=13 |pages=2–3}}</ref><ref>{{cite journal |last1=Nelson |first1=Elisabeth S. |title=The Face Symbol: Research Issues and Cartographic Potential |journal=Cartographica |date=2007 |volume=42 |issue=1 |page=53}}</ref> This technique constructs a complex point symbol that looks like a face, with various facial features distorted to represent various variables, in an attempt to leverage the innate human experience of interpreting meaning from facial expressions. Experimental results have generally been mixed, and the technique has never gained wide popularity.<ref name="nelson1996">Nelson, E.S., and P. Gilmartin. 1996. ‘‘An Evaluation of Multivariate, Quantitative Point Symbols for Maps.’’ In ''Cartographic Design: Theoretical and Practical Perspectives'', ed. C.H. Wood, and C.P. Keller. Chichester, UK: Wiley. 199–210.</ref>
* A ''[[small multiple]]'' is a series of small maps, arranged in a grid or array, each of which shows a different (but possibly related) variable over the same space.<ref name="Tufte-EI">{{cite book |last1=Tufte |first1=Edward |title=Envisioning Information |date=1990 |publisher=Graphics Press |isbn=978-0961392116 |page=[https://archive.org/details/envisioninginfor0000tuft/page/67 67] |url=https://archive.org/details/envisioninginfor0000tuft/page/67 }}</ref> It has been argued that this is not technically a multivariate map because it is a set of separate maps,<ref name="gistbok" /> but it is included here because it is intended to accomplish the same purpose.
== Advantages and criticisms ==
▲[[File:Bivariate.png|thumb|left|Example of a bivariate thematic map, displaying minority proportion as a choropleth, and family size as a proportional symbol]]
[[File:2016 US Presidential Election Pie Charts.png|thumb|right|300px|A multivariate symbol map of the 2016 U.S. presidential election, using a combination proportional and chart symbol]]
[[File:Dot map black hispanic.png|thumb|left|A bivariate dot density map showing the distribution of the African American (blue) and Latino (red) populations in the contiguous United States in 2010.]]
Multivariate thematic maps can be a very effective tool for discovering intricate geographic patterns in complex data.<ref name="gistbok" /> If executed well, related patterns between variables can be recognized easier in a multivariate map than by comparing separate thematic maps.
The technique works best when the variables happen to have a clear geographic pattern, such as a high degree of [[spatial autocorrelation]], so that there are large regions of similar appearance with gradual changes between them, or a generally strong correlation between the two variables. If there is no clear pattern, the map can become an overwhelming mix of random symbols.
A second problem occurs when the symbols do not harmonize well. In keeping with [[Gestalt psychology]], a multivariate map will work best when map readers can isolate patterns in each variable independently, as well as comparing them to each other. This occurs when the [[map symbol]]s follow the gestalt [[principles of grouping]]. Conversely, it is possible to select thematic symbol strategies that are effective on their own, but do not work together well, such as a proportional point symbol that obscures the choropleth map underneath, or a bivariate choropleth map using base colors that create unintuitive mixed colors.
A third issue arises when a map, or even a single symbol, is overloaded with too many variables that cannot be efficiently interpreted.<ref name="torguson">{{cite book |last1=Dent |first1=Borden D. |last2=Torguson |first2=Jeffrey S. |last3=Hodler |first3=Thomas W. |title=Cartography: Thematic Map Design |date=2009 |publisher=McGraw-Hill |isbn=978-0-07-294382-5 |page=147}}</ref> Chernoff faces have often been criticized for this effect.
Thus, many multivariate maps turn out to be technically impressive, but practically unusable.<ref name="nelson1996"/> This means that the cartographer must be able to critically evaluate whether a multivariate map she has designed is actually effective. It has also been suggested that in some cases, a map might not be the best tool for studying a particular multivariate dataset, and other analytical methods may be more enlightening, such as [[cluster analysis]].<ref name="slocum2009" />{{rp|344}}
==See also==
* [[Domain coloring]]
* [[Four color theorem]]
* [[Multivariate function]]
== References ==
{{reflist|30em}}
*Jeong W. and Gluck M., (2002). [[Multimodal interaction|Multimodal]] bivariate thematic maps with auditory and haptic display. Proceedings of the 2002 International Conference on Auditory Display, Kyoto, Japan, July 2-5.▼
=== Other Literature ===
*Leonowicz, A (2006). Two-variable choropleth maps as a useful tool for visualization of geographical relationship. Geografija (42) pp. 33–37.▼
▲*Jeong W. and Gluck M., (2002). [[Multimodal interaction|Multimodal]] bivariate thematic maps with auditory and haptic display. Proceedings of the 2002 International Conference on Auditory Display, Kyoto, Japan, July
▲*Leonowicz, A (2006). Two-variable choropleth maps as a useful tool for visualization of geographical relationship. Geografija (42) pp.
*Liu L. and Du C., (1999). Environmental System Research Institute (ESRI), online library.
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