Some Interesting Plots

By Jin-Ting Zhang

National University of Singapore

All these plots were generated using the statistical software MATLAB.    The beauty of these plots is sometimes out of  people's imagination. That is why I present them here and hope others can enjoy by viewing them.

I.  Cosine functions in 2-dimensional space.

Fig. 1  A cosine function over  a small range.

Cosine functions are useful in mathematics and statistics. A given cosine function has different images over different ranges.  Fig. 1 shows a cosine function in a small interval. In the plot, one can only see a peak. It is a beautiful bump though.

Fig. 2  A cosine function over  a middle range.

However, if we enlarge the range, more peaks will appear.  Fig.2  shows a middle range plot for the same cosine function. It shows  four beautiful peaks.  The feeling for Fig. 2 is quite different from Fig. 1. Can you imagine if we enlarge the range further, what would happen?  You are right, more peaks will appear as in Figs. 3 and 4 below.

Fig. 3  A cosine function over  a large range.

Fig. 4  A cosine function over  a  larger  range.

However, if  we change other parameters for  the cosine function, we can see a different picture like in Fig. 5 below.

Fig. 5  A cosine function with different parameters.

II.  Eigen functions for smoothing splines

Smoothing splines is an important technique for estimating an underlying nonparametric function. The roughness matrix is a key component of the smoothing splines; see Green and Silverman (1994) for the construction of the roughness matrix for a given sequence of knots.  An eigen-decomposition of the roughness matrix will reveal some interesting features.  Here we use a sequence of 100 equispaced knots over the unit interval [0,1] and then do an eigen-decomposition of the associated roughness matrix.   .

Fig. 1  The 100-th eigen function of the roughness matrix, constructed based on a sequence of 100 equispaced knots over [0,1].

Fig. 2  Same caption as in Fig. 1 but now for the 99-th eigen function.

Fig. 3  Same caption as in Fig. 1 but now for the 98-th eigen function.

Fig. 4  Same caption as in Fig. 1 but now for the 95-th eigen function.

Fig. 5  Same caption as in Fig. 1 but now for the 90-th eigen function.

After observing  Figs.1- 5, what do you conclude?  Can we generate this conclusion to all the remaining eigen functions?  Anyway, I wish you do enjoy seeing these nice figures.

III.  SiZerSS  maps for complicated functions

SiZerSS represents SiZer for smoothing splines developed by Marron and Zhang (2004). It is a technique for discovering the underlying features of a complicated function.  Here I would like show us some interesting pictures. Fig.1 shows a SiZerSS analysis for a real data set.  The upper panel shows a family of smooths  generated by spline smoothing. They represents different smooths over a large range of smoothing parameters. The lower panel shows the SiZerSS map which uses color to represent where the first derivative increasing (color 1, dark gray), decreasing (color 4, very light/white), not significant different from 0 (color 3, lighter  gray)  or not  sufficient data available for doing convincing statistical inference (color 2, gray). The full-color version is also available but is not good for being published in a statistical journal.  From Fig. 1, we can see  when the smoothing parameter is too small, there are no sufficient data available for convincing inferences. That why the bottom part of the SiZerSS map is basically in color 2.  Within a range of proper smoothing parameters, we can see that the underlying function may first increase and then decrease as showed in the upper panel.

Fig. 1  A SiZerSS  analysis for  a real data set.

Real data are usually simple. For more complicated SiZerSS maps, we can do simulation. That is, we can generate data from a known function specified by the user. Fig. 2 below shows a SiZerSS map for a simulated function with a small sample size.  For the SiZerSS map for  a large sample size, see Fig. 3.  You can see that when the sample size is increasing, more underlying features can be detected as significant.

Fig. 2  A SiZerSS  analysis for  a simulated  data set with a small sample size.

Fig. 3  A SiZerSS  analysis for  a simulated  data set with a large sample size.

IV.  Corneal Images

A cornea is the surface of the human being's eyeball.  It is of interest since investigation of the feature of corneas may lead to some insight about the eyeball. Below are two corneas presented as 3-dimensional surface.

Fig. 1  Two corneal surfaces.

It is difficult to detect any important features about the corneas except we may see that cornea 1 has some missing area that has no data available .  However, the curvature map for corneas is quite different. We use heavy color such as red to represent high curvature and use light color such as blue to represent low curvature. This allows  us to see where the curvature is high and where the curvature is low. High curvature area usually means the eyeball there has been damaged or deformed and hence means eye disease. Therefore, curvature maps can be used to diagnose the eye disease.  For eaxmple, Fig. 2 (a)  shows a normal corneal curvature map and we can see the curvature is about the same for the whole surface while for Fig.2 (b) some eye disease has being developping. In Fig.2 (c), the corneal curvature map shows that the eyeball has serious disease and some correction or operation may be needed.

(a)Normal           (b)Some Disease     (c)Serious Disease

Fig. 2  Three corneal curvature maps.