A preliminary analysis on SSL trends data:

Tomo Eguchi

I conducted PCA’s on Steller Sea Lion trends data.  Here are some preliminary results.

 

  1. Variances of each variable (80’s slope, 90’s slope, 91 intercept, latitude, and longitude)

 

0.011   0.003   0.722   6.275   108.154          

 

  1. Because variances are very different (even without latitude and longitude), I use the correlation matrix in the following PCA’s.  For the following analyses, I used locations as samples.

 

(a)    PCA on 80’s slope, 90’s slope, and 91 intercept:

 

Variance explained by each PC:            0.452   0.354   0.193  

Cumulative variance:                             0.452   0.807   1.000  

Eigenvectors:

                                    1          0.513   0.735   -0.443 

                                    2          -0.668  0.017   -0.744 

                                    3          0.539   -0.678  -0.500 

 

The plot PC 1 vs. PC 2 looks like this:

 

 

Looks like there are a few groups of points if I look at from a specific angle: (22, 4, 25), (20, 13), (29, 1, 8, 5, 24, 2, 12), and the rest.  What do you think? 

 

 

(b)   Next, I added the longitude and latitude as variables and ran a PCA:

 

Variance explained by each PC:            0.481   0.224   0.206   0.083   0.006  

Cumulative variance:                             0.481   0.705   0.911   0.994   1.000  

Eigenvectors:

                                    1          0.319   0.212   -0.365  -0.615  0.584  

                                    2          -0.638  -0.656  -0.270  -0.145  0.265  

                                    3          0.416   -0.561  0.663   -0.203  0.177  

                                    4          0.541   -0.458  -0.591  0.129   -0.363 

                                    5          -0.161  0.026   0.073   -0.737  -0.652 

 

And the plot of PC 1 vs. PC 2:

 

 

Of course, the latitude and longitude play a big role in this analysis.  The following is a map of the rookeries:

 

 

 

According to Rencher (1995; p. 433), “the component from a given correlation matrix are not unique to that correlation matrix.  … Thus the statement that the first component from a correlation matrix accounts for, say, 90% of the variance is not very meaningful.  In general, for p > 2 (# variables), the components from a correlation matrix depend only on the ratios (relative values) of the correlations, not on their actual values, and components of a given correlation matrix will serve for other matrices.”