Teaching:TUW - UE InfoVis WS 2008/09 - Gruppe 02 - Aufgabe 1 - Scatterplot: Difference between revisions

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A scatterplot (also called a ''scatter chart'', ''scatter diagram'' or ''scatter graph'' [Wikipedia1, 2008]) is a diagram in which the values of two ''metric'' variables are applied to the horizontal and vertical axes of a cartesian coordinate system. The resulting point in the graph represents one record from a data set. The distribution pattern of points from multiple records reveals, among other qualities, the correlation between the selected variables in the data set. The scatterplot is not to be confused with the ''correlation plot'' [NIST, 2008] which treats already adopted correlation coefficients in different data groups, while the term ''correlation diagram'' does not seem to be bound.
A scatterplot allows effective visualization of the relation between variables in multidimensional data.
 
 
Also called a ''scatter chart'', ''scatter diagram'' or ''scatter graph'' [Wikipedia1, 2008], it is in its most basic form a diagram in which the values of two ''metric'' variables are applied to the horizontal and vertical axes of a cartesian coordinate system. The resulting point in the graph represents one record from a data set. The distribution pattern of points from multiple records reveals, among other qualities, the correlation between the selected variables in the data set. The scatterplot is not to be confused with the ''correlation plot'' [NIST, 2008] which treats already adopted correlation coefficients in different data groups, while the term ''correlation diagram'' does not seem to be bound.


===Revealed Information===
===Revealed Information===
Line 5: Line 8:
Perfect linear correlation results in all samples lying on the regression line with positive or negative incline dependent on the sign of the correlation coefficient [MSTE, 1997]. Note, that the nonzero incline of the line is insignificant in this kind of diagram [Wikipedia 3, 2008] since it is dependent on axis scales.
Perfect linear correlation results in all samples lying on the regression line with positive or negative incline dependent on the sign of the correlation coefficient [MSTE, 1997]. Note, that the nonzero incline of the line is insignificant in this kind of diagram [Wikipedia 3, 2008] since it is dependent on axis scales.


An example of perfect correlation can be seen in Figure1 together with other patterns: strong  
An example of perfect correlation can be seen in the upper left of Figure 1 together with other patterns: strong positive (upper right), weak negative (lower left) and one example of variables without significant correlation.
positive, weak negative and one example of variables without significant correlation.


[[Image:SomeScatterplots.jpg|right|200px|thumb|Figure1: Some scatterplots.]]
[[Image:SomeScatterplots.jpg|right|200px|thumb|Figure 1: Some scatterplots.]]


Figure2 features a regression line to further increase expressiveness. The regression function is not necessarily chosen linear as in this example. Any kind of curve may fit a plot (quadratic, splines, ...). Generally, the curve with the smallest sum of squared distances to the points is sought after, [NetMBA, 2008]. For an introduction on linear regression, see [Wikipedia 4, 2008; Wikipedia 5, 2008].
Figure 2 features a regression line to further increase expressiveness. It is the line that passes through the plot as close to the points as possible. The regression function is not necessarily chosen linear as in this example. Any kind of curve may fit a plot (quadratic curves, splines, ...) and must be chosen according to subject matter. Generally, the curve with the smallest sum of squared distances to the plotted points is sought after (''least squares fitting''), [NetMBA, 2008]. For an introduction on linear regression, see [Wikipedia 4, 2008; Wikipedia 5, 2008].


[[Image:WeakNegativeCorrelationLine.jpg|right|200px|thumb|Figure2: Regression line.]]
[[Image:WeakNegativeCorrelationLine.jpg|right|200px|thumb|Figure 2: Regression line.]]


Further properties of data sets that are easily discovered are the presence of clusters and outliers(Figure3).
Further properties of data sets that are easily discovered are the presence of clusters and outliers as denoted in Figure 3.


[[Image:ClustersOutlyers.jpg|right|200px|thumb|Figure3: Clusters, outliers.]]
[[Image:ClustersOutlyers.jpg|right|200px|thumb|Figure 3: Clusters, outliers.]]




===Scatterplots of Higher Dimensions===
===Scatterplots of Higher Dimensions===


Scatterplots are not restricted to records with only two variables. Higher dimensional data can be displayed by adding the third axis to the plot or by assigning point properties (color, size, shape).
Scatterplots are not restricted to records with only two variables. Higher dimensional data can be displayed by adding the third axis to the plot or by assigning point properties like color, size or shape. Figure 4 shows a plot done with Matlab's command "plot4D", [Lewis, 2008].


TODO: add figure with colored 3D plot,
[[Image:Plot4D.png|right|200px|thumb|Figure 4: 4D plot.]]


For an example of a threedimensional scatterplot refer to [Wikipedia 1, 2008]. A way of plotting multidimensional data without the use of the third axis can be found in [Demsar, 2008].
For another example of a threedimensional scatterplot refer to [Wikipedia 1, 2008]. A way of plotting multidimensional data without the use of the third axis can be found in [Demsar, 2008].


===Treating Discrete Data===
===Treating Discrete Data===
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For continuously distributed data, scatterplots do well in visualizing density. The problem with discrete data is the possibility of more than one record sharing one point in the diagram (''overplotting''). One solution is to alter the point representation according to density, as is achieved by ''sun flower plots'' in which each point symbol gains radial segments as a consequence, [Wikipedia 2, 2008]. Examples can be found here: [Friendly, 2006; addictedtor, 2005].
For continuously distributed data, scatterplots do well in visualizing density. The problem with discrete data is the possibility of more than one record sharing one point in the diagram (''overplotting''). One solution is to alter the point representation according to density, as is achieved by ''sun flower plots'' in which each point symbol gains radial segments as a consequence, [Wikipedia 2, 2008]. Examples can be found here: [Friendly, 2006; addictedtor, 2005].


===Example of Application===
One prominent example of a scatterplot of two variables is the Hertzsprung-Russell diagramm (HRD for short) in astronomy (Figure 5). It plots absolute magnitude (visual brightness) of stars against their effective temperature and shows a rich set of features such as clusters and a central filament (the ''main sequence''), [Britannica, 2008].
[[Image:hr_diagram_local.png|right|200px|thumb|Figure 5: The HRD of some nearby stars.]]


=References=
=References=
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*[Wikipedia 4, 2008] Linear Regression, Wikipedia. Retrieved at 2 November, 2008. [http://en.wikipedia.org/wiki/Linear_regression http://en.wikipedia.org/wiki/Linear_regression]
*[Wikipedia 4, 2008] Linear Regression, Wikipedia. Retrieved at 2 November, 2008. [http://en.wikipedia.org/wiki/Linear_regression http://en.wikipedia.org/wiki/Linear_regression]
*[Wikipedia 5, 2008] Regression Analysis, Wikipedia. Retrieved at 2 November, 2008. [http://en.wikipedia.org/wiki/Regression_analysis http://en.wikipedia.org/wiki/Regression_analysis]
*[Wikipedia 5, 2008] Regression Analysis, Wikipedia. Retrieved at 2 November, 2008. [http://en.wikipedia.org/wiki/Regression_analysis http://en.wikipedia.org/wiki/Regression_analysis]
*[Lewis, 2008] Antony Lewis, CosmoMC readme. Retrieved at 16 November, 2008. [http://cosmologist.info/cosmomc/readme.html http://cosmologist.info/cosmomc/readme.html]
*[Britannica, 2008] Hertzsprung–Russell diagram. In Encyclopædia Britannica. Retrieved November 16, 2008, from Encyclopædia Britannica Online: [http://www.britannica.com/EBchecked/topic/263951/Hertzsprung-Russell-diagram http://www.britannica.com/EBchecked/topic/263951/Hertzsprung-Russell-diagram]
=External Links=
=External Links=


*Java Applet: [http://matti.usu.edu/nlvm/nav/frames_asid_144_g_4_t_5.html http://matti.usu.edu/nlvm/nav/frames_asid_144_g_4_t_5.html]
*Java Applet: [http://matti.usu.edu/nlvm/nav/frames_asid_144_g_4_t_5.html http://matti.usu.edu/nlvm/nav/frames_asid_144_g_4_t_5.html]

Revision as of 22:25, 16 November 2008

A scatterplot allows effective visualization of the relation between variables in multidimensional data.


Also called a scatter chart, scatter diagram or scatter graph [Wikipedia1, 2008], it is in its most basic form a diagram in which the values of two metric variables are applied to the horizontal and vertical axes of a cartesian coordinate system. The resulting point in the graph represents one record from a data set. The distribution pattern of points from multiple records reveals, among other qualities, the correlation between the selected variables in the data set. The scatterplot is not to be confused with the correlation plot [NIST, 2008] which treats already adopted correlation coefficients in different data groups, while the term correlation diagram does not seem to be bound.

Revealed Information

Perfect linear correlation results in all samples lying on the regression line with positive or negative incline dependent on the sign of the correlation coefficient [MSTE, 1997]. Note, that the nonzero incline of the line is insignificant in this kind of diagram [Wikipedia 3, 2008] since it is dependent on axis scales.

An example of perfect correlation can be seen in the upper left of Figure 1 together with other patterns: strong positive (upper right), weak negative (lower left) and one example of variables without significant correlation.

Figure 1: Some scatterplots.

Figure 2 features a regression line to further increase expressiveness. It is the line that passes through the plot as close to the points as possible. The regression function is not necessarily chosen linear as in this example. Any kind of curve may fit a plot (quadratic curves, splines, ...) and must be chosen according to subject matter. Generally, the curve with the smallest sum of squared distances to the plotted points is sought after (least squares fitting), [NetMBA, 2008]. For an introduction on linear regression, see [Wikipedia 4, 2008; Wikipedia 5, 2008].

Figure 2: Regression line.

Further properties of data sets that are easily discovered are the presence of clusters and outliers as denoted in Figure 3.

Figure 3: Clusters, outliers.


Scatterplots of Higher Dimensions

Scatterplots are not restricted to records with only two variables. Higher dimensional data can be displayed by adding the third axis to the plot or by assigning point properties like color, size or shape. Figure 4 shows a plot done with Matlab's command "plot4D", [Lewis, 2008].

Figure 4: 4D plot.

For another example of a threedimensional scatterplot refer to [Wikipedia 1, 2008]. A way of plotting multidimensional data without the use of the third axis can be found in [Demsar, 2008].

Treating Discrete Data

For continuously distributed data, scatterplots do well in visualizing density. The problem with discrete data is the possibility of more than one record sharing one point in the diagram (overplotting). One solution is to alter the point representation according to density, as is achieved by sun flower plots in which each point symbol gains radial segments as a consequence, [Wikipedia 2, 2008]. Examples can be found here: [Friendly, 2006; addictedtor, 2005].


Example of Application

One prominent example of a scatterplot of two variables is the Hertzsprung-Russell diagramm (HRD for short) in astronomy (Figure 5). It plots absolute magnitude (visual brightness) of stars against their effective temperature and shows a rich set of features such as clusters and a central filament (the main sequence), [Britannica, 2008].

Figure 5: The HRD of some nearby stars.

References

External Links

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