# CHERNOFF FACES IN R

Chernoff faces is a technique from multivariate analysis. The Chernoff faces technique was developed by Herman Chernoff and was presented in a paper in 1973 in the Journal of the American Statistical Association. Chernoff faces provide an intuitive way of looking at variation within a data set where the data is in a matrix of rows and columns (for example, a two way contingency tables).  A different face is created from each row in the data set. The differences between the faces are based on the columns of the data set. Each column is associated with one part of a facial expression (the first column is associated with the height of the face, the second with the width of the face, etc.). Chernoff designed the method for up to 18 facial components.  The faces() function in the TeachingDemos package of R uses 15 facial components, which were used here.

### Components of the Faces

From  help page for the function faces(), the fifteen components in R are the height of the face, width of the face, shape of the face, height of the mouth, width of the mouth, curve of the smile, height of the eyes, width of the eyes, height of the hair, width of the hair, styling of the hair, height of the nose, width of the nose, width of the ears, and height of the ears.  If the number of columns in the matrix is less than 15, the function will cycle back through columns until 15 columns are used.  One way around the problem is to put a constant value in the excess columns.  In what is presented here, cycling was done.

### Description of the Data

The Chernoff faces below were generated using the function faces() in the TeachingDemos package of R and are plots of the differences between facial expressions in the The Astrofaces Data Set. The Astrofaces Data Set was created from the pictures at the website, http://www.astrofaces.com, where an astrological group has gathered photographs from over 4700 persons, along with the placement of the Sun, Moon, and Ascendant for each person. In the spring of 2002, I went though the photographs and classed the faces as to whether the face had an open smile, a closed smile, an open mouthed neutral expression, a closed mouth neutral expression, an open frown, or a closed frown. For each person, I recorded the expression and the elements of the Sun, Moon, and Ascendant (the elements in astrology are air, earth, fire, and water). There were 2015 photographs in the data set at the time.

I have used a variety of techniques over the years to try to find a relationship between the elements of the Sun, Moon, and Ascendant and the expressions in the photographs, with little to show for my effort. So far, correspondence analysis has given the best results. (That said, I encourage anyone who thinks that there is nothing to astrology to visit the Astrofaces website and look at groups of photos.)  Few persons had open frowns.  The expressions on the persons at the Astrofaces website are not to be confused with the facial expressions of the Chernoff faces.

The input file for the faces() function in R was a table of normalized counts with astrological elements in the rows and facial expressions in the columns. In the four rows were counts of the number within the element with each of the six facial expressions divided by the total number of persons in the element. The plots are for the Sun, Moon, and Ascendant data, each done separately. In the post for astrological researchers posted earlier on this site, there is a plot of the combined data.

### The Sun Faces

The first column – open smiles – controls height of face, height of eyes and width of nose; the second – closed smiles – width of face, width of eyes and width of ears; the third – open neutrals – shape of face, height of hair, height of ears; the fourth – closed neutrals – height of mouth and width of hair; the fifth – open frowns – width of mouth and styling of hair; and the sixth – closed frown – curve of smile and height of nose.

In the Sun plot, we can see that the air and fire faces have similar sizes and shapes but different expressions, so air and fire are somewhat similar with regard to the first three columns (open smile, closed smile,  and open neutral), but not with regard to the last three columns (closed neutral, open frown, and closed frown).  Water and earth have the same shape- so are similar with respect to open neutrals, but are different sizes – so are different with respect to open and closed smiles.  Other than the open neutrals, the two elements are different.

The raw table of counts for the Sun data is given below.  The zero under open frowns for the water element causes a degeneracy in the water face.

 Open Smile Closed Smile Open Neutral Closed Neutral Open Frown Closed Frown Air 104 31 32 100 3 26 Earth 97 35 40 104 1 28 Fire 108 32 31 101 2 29 Water 107 41 36 93 0 34

### The Moon and Ascendant Faces

The Chernoff faces for the Moon and Ascendant are plotted below.

In the Moon and Ascendant plots, we can follow the same procedure as with the Sun to evaluate the expressions.

### Conclusions

We can see from the plots for the Sun, Moon, and Ascendant that the faces tell us something intuitive about the differences between the four astrological elements with regard to the six facial expression classes in the Astrofaces dataset. Looking between plots, we can also see similarities across the Sun, Moon, and Ascendant for differing elements.

# FOR ASTROLOGICAL RESEARCHERS:

### Analysis of Count Data – Clustering – Hypotheses Testing – Plotting

#### In this Plot You can see how Plots can Convey Information about the Relationship between Placements:

The data for this plot comes from www.astrofaces.com. Several years ago, I went through the photos at the Astrofaces website and classed the photos by expression and by the element of the Sun, Moon, and Ascendant. The plot is of the classed data. From the plot, Fire has fewer Closed Smiles and Air has fewer Closed Frowns, otherwise, there is not much difference between the expressions.

#### Chernoff Faces show You Differences between Rows of data across Columns within a matrix by using Facial Expressions: (not to be confused with the expressions in the data set)

This plot is based on the data on expressions from www.astrofaces.com. (One reason I included this plot is because the expressions on the faces seem to have an astrological take – pure coincidence.)  You can get Chernoff faces for your data at Vanward Statistics.

#### I can do Multiple Correspondence Plots for you to show Clustering for Categorical Variables:

For the Facial Expression data, the plot shows that different element / point combinations are associated with different facial expressions.

#### I can Model and Test Hypotheses for Astrological Questions for You, Using the Swiss Ephemeris:

In an article in Today’s Astrologer, by Cowell Jeffery, Volume 73, Number 13, Page 23, Mr. Jeffery hypothesized that if an aspect between two planets is present at birth, the same planets are likely to be in aspect at death. Mr. Jeffery used the Elizabethans Henry VIII, Queen Elizabeth I, Robert Devereux, William Cecil, and Mary Stuart, for an example. I modeled the problem for the major aspects, using an orb of 8 degrees for all of the planets and the Sun and an orb of 12 degrees for the Moon, where the planets are the eight usual planets (including Pluto, excluding Chiron).

I estimated the expected number of aspects for each point combination (there are 42 of them – I excluded aspects between Uranus, Neptune, and Pluto) – if the births and deaths occured randomly over the time period from the last decade of the 15th century to the end of the 16th century. I, also, found estimates of the variances and covariances between the point combinations. Different planetary combinations have different expected numbers of aspects and different variances and covariances. I, then, estimated the number of matches to expect, and the variance of the estimate of the number of matches to expect, if the count at birth is independent of the count at death. Using the results, I was able to test if the five Elizabethans had unusually high numbers of matches. The difference was positive but not significantly different from zero. One would need a very strong effect for the p-value of a test to be very small given a sample of five. The number seen for the Elizabethans is larger than the estimated expected count, which is encouraging, but too small to be considered significantly different from zero.

I did not tried to model seasonal, locational, or time of day patterns for the Elizabethans, which would affect the results. This is an area of ongoing research for me. Different parts of the world (or US for the US) have differing distributions of births  (on which planetary placements depend) over the year, so currently I am only looking at data in localized places (in Iowa). Also, I now used the distributions of births in a data set and the seasonal patterns of births from data given to me by the State of Iowa to estimate expected counts and the covariances of the counts. I use an empirical density function generated from the birth observations to do the estimations. Since the data I am using does not have times of birth, I have not tried to account for time of day.

#### Links to a Vulcan Cent Ephemeris:

I have been experimenting with points I call Vulcan and Cent. The longitude of Vulcan is the average over the shortest arc of the longitudes of Ceres, Vesta, Pallas, and Juno. The declination of the point is found using the longitude and the average of the latitudes of the four. Cent is the closer midpoint of the Centaurs Chiron and Pholus – the only two civil Centaurs in Greek mythology. An ephemeris from 1900 to 2050 for the two points can be found in two files, one in rich text format and the other in pdf format. The original files I put up contained an error. The files now should be correct. I did not realize ‘abs’ was an integer function in C in computing the original function. I used the Swiss Ephemeris for the placements in longitude and latitude of Ceres, Vesta, Pallas, Juno, Chiron, and Pholus.

Earlier in this blog, there is a description of doing regression analysis with autocorrelated errors. The point Vulcan jumps 90 or 180 degrees from time to time and I look at divorce numbers (in Iowa) as related to Vulcan jumps.

#### For the Basics of Statistical Theory:

Gator Talk Slides
Here I have provided you with a introduction to some statistical results. Click on the link  above to see the power point slides that I used for a talk to Alphee Lavoie’s AstroInvestigators. The slides cover the Normal and Chi Square distributions, plus the Central Limit Theorem, which are applied to some Astrological Data.

#### Sometimes, You might Find that just Plotting Astronomical Information is Interesting:

Using sidereal longitudinal limits for the constellations of the zodiac, I plotted
the placements of the zero degree points of the tropical signs within the constellations.