| Geometrical analysis |
Directional analysis
| Typical application | Assumptions | Data needed |
| Displaying and testing for random distribution of directional data | See below | One column of directional data in degrees (0-360) |
Plots a rose diagram (polar histogram) of directions given in a column of degree values (0 to 360). Used for plotting current-oriented specimens, orientations of trackways, orientations of morphological features (e.g. terrace lines), etc.
By default, the 'mathematical' angle convention of anticlockwise from east is chosen. If you use the 'geographical' convention of clockwise from north, tick the box.
You can also choose whether to have the abundances proportional to radius in the rose diagram, or proportional to area (equal area).
The mean angle, together with the R value (Rayleigh's spread), are given. R is further tested against a random distribution using Rayleigh's test for directional data (Davis 1986). Note that this procedure assumes evenly or unimodally distributed data - the test is not appropriate for bidirectional data. Also, the test is not accurate for N>50; it will then report a too high p value.
A four-bin chi-square test is also available, giving the probability that the directions are randomly and evenly distributed.
Point distribution
| Typical application | Assumptions | Data needed |
| Testing for clustering or overdispersion of two-dimensional position values | Elements small compared to their distances, mainly convex domain, N>50. | Two columns of x/y positions |
Point distribution statistics using nearest neighbour analysis (modified from Davis 1986). The area is estimated using the convex hull, which is the smallest convex polygon enclosing the points. This is inappropriate for points in very concave domains. Also, there is no correction for boundary effects, meaning that the statistics are reasonably valid only for large N (N>50).
The probability that the distribution is random (Poisson process, giving an exponential nearest neighbour distribution) is presented, together with the R value. Clustered points give R<1, Poisson patterns give R~1, while overdispersed points give R>1.
Applications of this module include spatial ecology (are in-situ brachiopods clustered) and morphology (are trilobite tubercles overdispersed).
Fourier shape analysis
| Typical application | Assumptions | Data needed |
| Analysis of fossil outline shape | Shape expressible in polar coordinates, sufficient number of digitized points to capture featues. | Two columns of digitized x/y positions around an outline |
Accepts X-Y coordinates digitized around an outline. More than one shape can be simultaneously analyzed by giving more than one pair of columns. Points do not need to be totally evenly spaced. The shape must be expressible as a unique function in polar co-ordinates, that is, any straight line radiating from the centre of the shape must cross the outline only once.
The origin for the polar coordinate system is found by numerical approximation to the centroid. 64 points are then produced at equal angular increments around the outline, through linear interpolation. The centroid is then re-computed, and the radii normalized (size is thus removed from the analysis). The cosine and sine components are given for the first ten harmonics, but note that only N/2 harmonics are 'valid', where N is the number of digitized points. The coefficients can be copied to the main spreadsheet for further analysis (e.g. by PCA).
The 'Shape view' window allows graphical viewing of the Fourier shape approximation(s).
Elliptic Fourier shape analysis
| Typical application | Assumptions | Data needed |
| Analysis of fossil outline shape | Sufficient number of digitized points to capture featues. | Two columns of digitized x/y positions around an outline |
Elliptic Fourier shape analysis is in some respects superior to simple Fourier shape analysis. One advantage is that the algorithm can handle complicated shapes which may not be expressible as a unique function in polar co-ordinates. Elliptic Fourier shapes is now a standard method of outline analysis. The algorithm used in PAST is described in Ferson et al. 1985.
Cosine and sine components of x and y increments along the outline for the first 10 harmonics are given, but only the first N/2 harmonics should be used, where N is the number of digitized points. Size and positional translation are normalized away, and do not enter in the coefficients. The coefficients can be copied to the main spreadsheet for further analysis (e.g. by PCA).
The 'Shape view' window allows graphical viewing of the elliptic Fourier shape approximation(s).