Visual Clustering and Classification: The Oronsay Particle Size Data Set Revisted

- Technical Report -

By

Adalbert F. X. Wilhelm

Edward J. Wegman

and

Jürgen Symanzik

A (Unix) compressed postscript version (2.7MB - extends to 65.4MB) and a pdf version (648KB) of the text are available. All the figures from the text are available as GIF files below.

Legends for Figures

• Figure 1: An excerpt from a map of the Oronsay island, Inner Hebrides. It shows the locations of the two archeological sites labeled Caisteal nan Gillean I and Cnoc Coig and the four transects TG, CC, ECC, and TUS where ``modern'' sand samples have been collected. Reprinted with permission from Fieller et al. (1987).
  
  
• Figure 2(a) and Figure 2(b): (a) Scatterplot of summed up weights (vertical) versus group (horizontal). While most weights scatter around 60g, values around 70g have been observed for most samples from groups 1 and 22. Group 5 has the smallest sum (51.8g) marked with a ``filled circle'' but also the largest sum (82.4g) marked with a ``filled box''. (b) A parallel coordinate plot of group 5 reveals that the ``filled circle'' sample has an unusual small value for variable 9, i. e., particle size [0.125, 0.18) mm, but it also has higher values for variables 1 and 7. The ``filled box'' sample has a shape that matches the shape of the remaining samples of group 5.
  
  
• Figure 3(a) and Figure 3(b): (a) A dotplot of group for the 149 samples from known sampling locations. Within each of the groups 6 to 20 two training samples have been selected. The ``+'' symbols for all observations in groups 18 to 20 have been obtained through linked brushing in (b) a projection of the grand tour. Here, two clusters are clearly distinguishable that separate Caisteal nan Gillean samples (the ``+'' samples) from Cnoc Coig samples.
  
  
• Figure 4(a), Figure 4(b), Figure 4(c), and Figure 4(d): (a) When considering only the Cnoc Coig samples (groups 6 to 17), a dotplot reveals that the cluster brushed with a ``+'' symbol in (b) a projection of the grand tour contains mid beach (group 15) and upper beach (group 17) samples. (c) A dotplot of the final clustering of the Cnoc Coig samples shows the correct classification of dune samples (groups 8, 12 to 14) and the misclassification of two entire groups of upper beach samples (groups 7 and 11) and 1 mid beach sample (from group 10) based on (d) a projection of the grand tour where a homogeneous group of points has been marked with an ``×'' symbol, assuming these are all dune samples. This projection also shows how the ``+'' cluster brushed in (b) separates into two subclusters.
  
  
• Figure 5(a), Figure 5(b), Figure 5(c), Figure 5(d), and Figure 5(e): (a) When considering only the Caisteal nan Gillean samples (groups 18 to 20), a dotplot reveals that the cluster brushed with a ``filled circle'' symbol in (b) a projection of the grand tour contains all lower beach samples (group 18). (c) A dotplot showing how the dune samples (group 20) brushed with a ``open circle'' symbol can be separated from upper beach samples (group 19) based on (d) another projection of the grand tour. The ``filled box'' sample is not identified as a dune sample even though it originates from group 20. (e) Even though this projection of the grand tour shows an instance of a separation among the three groups, it cannot be considered as useful for clustering in an interactive environment since the ``+'' and ``open circle'' symbols moved in the same direction and the ``filled box'' moved outside the area of ``open circle'' symbols almost immediately.
  
  
• Figure 6(a) and Figure 6(b): (a) When considering only groups 6 to 17 for the reevaluation of Figure 4c in Fieller et al. (1984) a dotplot shows the symbols used to mark the 12 known groups at the Cnoc Coig site. (b) A projection showing a local optimum based on the projection-pursuit-guided grand tour and a manually drawn dividing line shows a separation between beach samples (left of the line) and dune samples (right of the line). However, upper beach locations from the CC transect (group 7) and from the TG transect (group 11) also fall right of the line.
  
  
• Figure 7(a), Figure 7(b), and Figure 7(c): (a) A dotplot shows the symbols used to mark the 5 groups from the Caisteal nan Gillean site used for the reevaluation of Figure 4d in Fieller et al. (1984). Projection (b) shows a circular arrangement and projection (c) shows a linear arrangement, each obtained as a local optimum based on the projection-pursuit-guided grand tour. These and many similar projections show more differences than similarities between archaeological samples (groups 5 and 21) and modern samples (groups 18 to 20).
  
  
• Figure 8(a), Figure 8(b), and Figure 8(c): (a) A projection that separates between sites (the big ``+'' and ``×'' symbols are Cnoc Coig samples, the small ``+'' and ``×'' symbols are Caisteal nan Gillean samples) and sands within sites (the ``+'' symbols are beach samples and the ``×'' symbols are ``dune-like'' samples). The small ``.'' symbols represent the archaeological samples. Separation lines have been added manually. (b) A dotplot shows the symbols used for all 226 samples. (c) When adding the symbols used in (b) to the projection in (a), we see that archaeological Caisteal nan Gillean samples fall close to modern Caisteal nan Gillean samples (beach and dune). Archaeological Cnoc Coig samples are clearly distinguishable from modern Cnoc Coig beach. Sands above CC Midden (group 1) and Sands below CC Midden (groups 2 and 3) are close to modern Cnoc Coig dunes. CC Shell Midden (group 22) and CC Soil Pit (group 4) have some overlap with the other archaeological Cnoc Coig samples but they have very little in common with modern Cnoc Coig dunes.
  
  
• Figure 9: Original parallel coordinate plot of Oronsay Cnoc Coig and Caisteal nan Gillean data. The Cnoc Coig data is in black, the Caisteal nan Gillean data in gray (red). Data from the two sources strongly separate with the Cnoc Coig sand generally being much finer than the sand from the Caisteal nan Gillean site.
  
  
• Figure 10: Parallel coordinate plot of Cnoc Coig data after completing the BRUSH-TOUR strategy. The data are divided into six clusters with red and magenta being ``dune-like'' sand. In the present image, all points are rendered in grayscale. The reader is referred to the webpage for full color illustrations.}
  
  
• Figure 11: Sequence of decompositions for Cnoc Coig sand data. Red and magenta are basically the ``dune-like'' sands, other colors represent the ``beach-like'' sands. The strongest splits tend to occur early in the BRUSH-TOUR strategy. Thus, the ``dune-beach'' split was the most evident.
  
  
• Figure 12: Parallel coordinate display of Cnoc Coig known (training) data and Cnoc Coig groups 2 and 3 after partial grand tour. The group 2 and 3 data are given in white. The group 2 and 3 data generally follow the red-magenta ``dune-like'' sand data. However the group 2 and 3 data clearly depart significantly in certain dimensions, notably along the .50-.71 mm axis in this illustration.
  
  
• Figure 13: Scatterplot matrix of Cnoc Coig known (training) data against Cnoc Coig group 4 (unknown) data. The group 4 data are shown in white while the known Cnoc Coig data are shown in colors. The scatterplot diagram shown in the upper right-hand side of this illustration shows that the group 4 data are essentially orthogonal to all of the training data. Group 4 is definitely a different cluster, although if forced to characterize group 4 data, they would be closest to the ``dune-like'' sand data.
  
  
• Figure 14: Simplified parallel coordinate display of all Cnoc Coig data after partial grand tour. ``Dune-like'' sands are shown in red, ``beach-like'' sands are shown in green, and ``unknown'' sands shown in white. The unknown class is distinct from both ``dune-like'' sand and ``beach-like'' sand. This is particularly clear in the .25-.355 axis.
  
  
• Figure 15: Simplified scatterplot matrix of all Cnoc Coig data after partial grand tour. The coloring is as in Figure 14. A density plot of the circled scatterplot is shown in the upper right. In the density plot, the tallest bump/mode corresponds to the red ``dune-like'' sand, the two smaller bump/modes on the right correspond to the green ``beach-like'' sand, and the smaller bump/mode on the left corresponds to the white ``unknown'' sand. The ``unknown'' sand is most like the ``dune-like'' sand, but still rather distinct.
  
  
• Figure 16: Dotplots of all variables for 149 known samples. Bright colors represent many points and dark colors only a few points. Measurements on the extreme particle sizes >2.0mm, 1.4 - 2.0mm, 1.0 - 1.4mm, .71 - 1.4mm, .063 - .09mm, and <.06mm are highly quantized. Large gaps in the data are apparent for variables .355 - .50mm and .25 - .355mm.
  
  
• Figure 17: Boxplots for all 149 training samples. Variables >2.0mm, 1.4 - 2.0mm, 1.0 - 1.4mm, \ldots , .063 - .09mm, and <.06mm are displayed from left to right. Only two variables, .18 - .25mm and .09 - .125mm, show no outliers. These two distributions are also only slightly skewed. Data on all the other variables is skewed to the right, but data on particle size .125 - .18mm is skewed to the left.
  
  
• Figure 18: No classification in dotplots between dune and beach for the 149 known samples for both locations when selecting beach (top row) and dune (bottom row). Due to overplotting it appears that the same points have been selected more than once. In reality, a visible point sometimes represents a larger number of samples and it is already highlighted if just one of these samples is selected.
  
  
• Figure 19: Clear classification between sites Cnoc Coig (top row) and Caisteal nan Gillean (bottom row) when selecting clusters of variable `[0.25, 0.355) mm'. Also the variables `[0.355, 0.5) mm', `[0.125, 0.18) mm', and `[0.09, 0.125) mm' allow a clear separation between Cnoc Coig and Caisteal nan Gillean.
  
  
• Figure 20: We apply the classification rule of training data to the entire data set of 226 samples. (a) Selecting the right-hand cluster in variable .25 - .355mm highlights all training samples at Caisteal nan Gillean, one sample of test group 5 and also all but one of group 21. (b) However, 16 points fall between the previously established clusters. Those points are classified by using the classification rule based on the training data for variable .355 - .50mm. (c) The resulting classification is correct for groups 18 to 21, but misses two samples in group 5 and misclassifies five samples in group 4.
  
  
• Figure 21: Final classification tree for separating all 226 sand samples by site.
  
  
• Figure 22: (a) Outliers for particle size `[1.4, 2.0)mm' fall mainly in groups 15 and 18. (b) Enlarging the selected group to all samples with values greater than 0.1g for variable 1.4 - 2.0mm highlights all but one sample of group 6, all samples of groups 15 and 18, and one sample of group 10 (misclassified).
  
  
• Figure 23: The dune samples at Cnoc Coig split into two groups for variables .18 - .25mm and .125 - .18mm. Group 12 (marked red, i.e. bigger light dots) is different from groups 8, 13, and 14 (all marked blue, i.e. bigger dark dots). From the position of the two groups it can be concluded that dune sands of group 12 are much finer since these samples have heavier weights from the finer sieves and lighter weights from the coarser sieves.
  
  
• Figure 24: The distribution for particle size `[0.09, 0.125)mm' seems to be a mixture of four individual distributions: one for the Caisteal nan Gillean samples, one for groups 7, 8, 11, 13, and 14 (in a), one for groups 6, 10, 12, and 17 (in b), and one for groups 9, 15, and 16 (in c).
  
  
• Figure 25: Individual separation of group 12 by sequentially selecting subclusters (the dashed areas) in the respective highlighting of particle sizes `[0.09, 0.125)mm', `[0.25, 0.355)mm', and `[0.335, 0.50)mm'. No further clustering can be found in the dotplots of the other variables .125 - .18mm and .18 - .25mm (right under bar chart for group).
  
  
• Figure 26: Individual separation of groups 9, 15 and 16 by sequentially selecting subclusters in the respective highlighting of particle sizes `[0.09, 0.125)mm', `[0.125, 0.18)mm', and `[0.335, 0.50)mm'.
  
  
• Figure 27: Histograms of the root variable `particle sizes [0.09, 0.125)mm' for modern samples and entire data set.
  
  
• Figure 28: Classification of 30 known sand samples at Caisteal nan Gillean.
  
  
• Figure 29: Classification tree based on 30 known samples for separating sand types at Caisteal nan Gillean.
  
  
• Figure 30: Classification of the test data at Caisteal nan Gillean. Groups 4 and 5 build a separate cluster that would be classified as neither dune nor beach. Half of the samples in group 21 fall into the beach and the dune clusters.