FITTING A LOG LINEAR MODEL

You can use log linear models to fit cross classified catagorical data. In fitting log linear models, the logarithm of the expected counts in the cells of the contingency table are modeled as linear combinations of parameters, where there can be parameters for interactions between dimensions in the table. I follow the approach of Bishop, Fienberg, and Holland in Discrete Multivariate Analysis: Theory and Practice, MIT Press, 1975.

Some Terminology

Some terminology is in order. An hierarchical model is a model in which, if a parameter is not in the model, then neither are the higher level interactions associated with the parameter. A complete model is a model in which none of the expected cell counts are equal to zero. An incomplete model has at least one cell where the expected cell count is zero. A saturated model is a model in which all possible parameters are fit. An unsaturated model is a model in which some of the parameters are set to zero. In Bishop, et.al., each dimension of the table is called a factor and only hierarchical models are fit, which can be either complete (chapter 3) or incomplete (chapter 5). The parameters are modeled such that the sum over the indices for a given parameter and factor equals zero (except of course the first parameter, ‘u’).

The Saturated Model

Following the notation of Bishop, et.al., for a three dimensional contingency table, the saturated model would be

log(mijk) = u + u1(i) + u2(j) + u3(k) + u12(ij) + u13(ik) + u23(jk) + u123(ijk),

where ‘mijk‘ is the expected cell count in cell (i,j,k).

Say that we have two levels for each of the three factors. Then, the indices associated with the factors, ‘i’, ‘j’, and ‘k’, can each take on either the value ‘1’ or the value ‘2’. For this example, since there are only two levels for each factor and the parameters sum to zero over each index, specifying a parameter (say u12(11)) for a given set of factors (here, the interaction between factors 1 and 2) means that the other parameters for the given set of factors (in this case, u12(12), u12(21), and u12(22)) are specified too. It follows that, for this example, there are eight unique parameters in the saturated model, one for each factor combination (‘1′,’2′,’3′,’12’,’13’,’23’, and ‘123’) and one for the unindexed parameter ‘u’.

There are eight possible unique sets of values for the three indices ‘i’, ‘j’, and ‘k’ ((1,1,1), (1,1,2), … , (2,2,2)), representing the eight cells in the table. If ‘i’, ‘j’ and ‘k’ are given, so are ‘ij’, ‘ik’, ‘jk’, and ‘ijk’. This means that the eight estimated cell counts are uniquely defined. In the model, there are eight unique parameters, and there are eight cell counts. It follows that the saturated model fits the data exactly – the estimates of the expected cell counts are the cell counts themselves.

An Unsaturated Models

The saturated model is not usually of interest. For unsaturated models, in order to estimate the expected cell counts, Bishop, et.al., model the data in the cells as either independent Poisson variables or as variables from a single multinomial distribution, and find minimal sufficient statistics for the expected cell counts based on either of the two probability distributions. Under the assumptions, the minimal sufficient statistics are the same for either distribution.

An example of an unsaturated model based on the saturated model given above is

log(mijk) = u + u1(i) + u2(j) + u3(k) + u13(ik) + u23(jk) .

Here, the main effects of each of the factors are included (In Bishop, et.al., the main effects are always included. According to Bishop, et.al., such models are called comprehensive models.) The only higher order effects which are still present are the interactions between the first and third factors and between the second and third factors. Note that, since the u12(ij) term is not in the model, the term u123(ijk) cannot be either if the model is to be heirarchical. For an unsaturated model, Bishop, et.al., give an algorithm to find maximum likelihood estimators for the expected cell counts using the minimal sufficient statistics of the given unsaturated model. The maximum likelihood estimators do not depend on which probability distribution is used. (The results that I give below are based on an R function that I wrote to apply the algorithm.)

Evaluating an Unsaturated Model

Bishop, et.al., follow the method of evaluating the fit of an unsaturated model by using the chi squared distribution of either

Χ2 = Σ ( xi – ̂mi )2 / ̂mi

or

G2 = Σ -2 xi log( ̂mi / xi ) ,

where xi is the cell count in cell i and ̂mi is the estimated expected cell count in cell i, and i goes over all of the cells in the array for which ̂mi does not equal zero. Χ2 then is the Pearson chi square statistic and G 2 is minus twice the log likelihood ratio. Χ2 and G2 are goodness of fit statistics. A model fits if the chi square statistic falls in the ‘normal’ range for a chi square of the appropriate degrees of freedom. Bishop, et.al., only look at the right tail of the appropriate chi square distribution, accepting models that fit the data too well. Among the models considered, the most parsimonious model that still fits the data is accepted.

Fitting a Loglinear Model to the Deficit by Political Party Data Set

To demonstrate fitting a log linear model, I will use the data in the Deficit by Political Party Data Set. The deficits (surpluses) as a percentage of the gross domestic product are classed into four classes and the political parties in control are associated with the deficit (surplus) of the following year. The political parties go from 1946 to 2007 and the deficits (surpluses) as a percentage of gross domestic product go from 1947 to 2008. The data array is given below.

On Budget Deficit (-) (Surplus (+)) per GDPby the Controlling Political Parties

of the Presidency, Senate, and HousePolitical Parties Lagged One Year

President Senate House (-6.2,-4] (-4,-2] (-2,0] (0,4.2]
D D D 0 7 10 2
R
R D
R 0 1 4 3
R D D 5 8 5 3
R 1 1 0 0
R D 4 2 2 0
R 1 3 0 0

Since the interaction between the deficit (surplus) is of interest and the senate and house seem to be interrelated, the following models were fit.

1: log(mijk) = u + u1 + u2 + u3 + u4 + u12 + u13 + u14+ u23 + u24 + u34

2: log(mijk) = u + u1 + u2 + u3 + u4 + u12 + u13 + u14 + u34

3: log(mijk) = u + u1 + u2 + u3 + u4 + u12 + u13 + u14

4: log(mijk) = u + u1 + u2 + u3 + u4 + u12 + u14 + u34

5: log(mijk) = u + u1 + u2 + u3 + u4 + u12 + u14

6: log(mijk) = u + u1 + u2 + u3 + u4

The subscript 1 refers to the deficit (surplus) classes and takes on 4 levels, the subscript 2 refers to the party of the president and takes on 2 levels, the subscipt 3 refers to the party controlling the senate and takes on 2 levels, and the subscript 4 refers to the party controlling the house and takes on 2 levels. The G2 values found by fitting the six models, as well as the degrees of freedom for and the p-values from the chi squared tests, are given in the table below.

Model G2 df p-value
1: 9.14 5 0.1036
2: 10.83 7 0.1460
3: 16.79 8 0.0324
4: 12.19 10 0.2725
5: 17.47 11 0.0947
6: 37.66 17 0.0027

From the table, models 1, 2, 4, and 5 all fit the data. Chi squared tests to compare the models
are given in the table below.

Models Difference

in G2

Difference

in df

p-value
1: & 2: 1.69 2 0.4296
2: & 4: 1.36 3 0.7149
4: & 5: 5.28 1 0.0216

There is little difference between models 1 and 2, and 2 and 4, so out of models 1, 2, and 4, the most parsimonius model is model 4. Model 5 is more parsimonious than model 4, however the test of the difference between the models shows a significant difference from the central chi squared distribution with 1 degree of freedom. It appears that something is lost when going from model 4 to model 5. We conclude that model 4 is the best model and that the interaction between the house and senate, the interaction between the deficit level and the party of the president, and the interaction between the deficit level and the party controlling the house are important and that the interaction between the party controlling the senate and the deficit level is not.

Below is a table of the fitted cells, normalized to sum to 100 across rows, for model 4.

Estimated Expected Cell Counts Normalized by RowUsing Model 4 for the On Budget Deficit (-) (Surplus (+)) per GDP

by the Controlling Political Parties

of the Presidency, Senate, and HousePolitical Parties Lagged One Year

Rows Sum to 100 Except for Rounding

President Senate House (-6.2,-4] (-4,-2] (-2,0] (0,4.2]
D D D 00 30 55 15
R
R D
R 00 29 42 29
R D D 33 40 21 07
R 26 42 17 15
R D 33 40 21 07
R 26 42 17 15

Democratic presidents perform much better than Republican presidents. With regard to Republican presidents, there is no difference between Democratic and Republican senates. Republican houses perform somewhat better than Democratic houses under both Democratic and Republican presidencies.

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

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.

Counts for the Sun Placements
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.

moon.faces

asc.faces

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:

faces.com data - proportions by element

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)

faces.com data - Chernoff faces

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:

faces.com data - Multiple Correspondence Analysis

 

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.

Here is the link: https://1drv.ms/b/s!AgjNmjy3BpFDmBBTiEPHc3qPr9ls

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:

const-trop2.2018

 

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.

FOR ENERGY RESEARCHERS:

Sampling Plans – Analysis of Samples – Modeling – Graphing

Below are graphs of altitudes and azimuths of the sun, for differing times of the year, for Stratford, Iowa, which give you an example of the capability of R for use in planning solar installations:

altitude-azimuth

You can download the R function – which plots individual altitude / azimuth graphs, for any place in the world, for any time of year. You can download the file here: angle.plot.

If you call the function ‘angle.plot’, to run the program with default values, you would type ‘angle.plot()’ at the R prompt. To change the defaults, you can manually change the defaults when the function is run; for example, angle.plot(loc=”Hull, MA”,  mo=8,  da=28,  ye=2013,  tz=4,  la=42.3019,  lo=70.9083).

function(loc="Stratford, Iowa", mo=1, da=1, ye=2010, tz=6, 
la=42.26782, lo=93.92097) {

#  The algorithms for this function come from Astrological Algorithms, 2nd ed.,
#  by Jean Meeus, published in 1998, by Willmann-Bell, Inc.  The page 
#  references are to Meeus. 
   
#  check the inputs

      cat("the location entered is ", loc, "\n")
      cat("the date entered is ", mo, "/", da, "/", ye, "\n")
      cat("the time zone is ", tz, " hours from UT", "\n")
      cat("the latitude is ", la, "degrees and the longitude is ", lo, 
          " degrees\n", "\n")
      cat("  if not correct then enter n ")
      tes <- readline()
      if(tes == "n") stop("change the value of the arguments")
	
#
#  define tosid - the solar to sidereal time correction factor (p. 408)
#
      tosid <- 366.2422 / 365.2422	
#
#  create local - the local dynamical times for the plot - in degrees 
#  (see chapter 10 in Meeus)
#
      local <- seq(-150, 150, by = .02)	
#
#  find jd - the julian days for the local dynamical times - in days (p. 61)
#
  if(mo == 1 | mo == 2) { 
    mo <- mo + 12
    ye <- ye - 1
   }
  ga <- floor(ye/100 + 1e-008)
  gb <- 2 - ga + floor(ga/4 + 1e-008)
  jd <- (floor(365.25 * (ye + 4716)) + floor(30.6001 * (mo + 1)) +
       (da +.5) + gb - 1524.5) + local/(15 * 24) + lo/(15 * 24)
     
#
#   find T - the increment from January 1, 2000 at noon TD
 
#            for the local dynamical times - in centuries (p. 163)
#
     T <- (jd - 2451545)/36525
     Tsq <- T^2
#
#   find L0 - the mean longitudes of the sun - in radians (p. 163)
#
           L0 <- (280.46646 + 36000.76983 * T + 0.0003032 * Tsq)/180 * pi
#
#   find M - the mean anomolies of the sun - in radians (p. 163)
#
      M <- (357.52911 + 35999.05029 * T - 0.0001537 * Tsq)/180 * pi
#
#   find C - the sun's equations of the center - in radians (p. 164)
#
      C <- (1.914602 - 0.004817 * T - 1.4e-005 * Tsq) * sin(M)
      C <- C + (0.019993 - 0.000101 * T) * sin(2 * M)
      C <- (C + 0.000289 * sin(3 * M))/180 * pi
#
#   find omega - the correction factors for finding the apparent longitudes
#                of the sun - in radians (p. 164)
#
      omega <- (125.04 - 1934.136 * T)/180 * pi 

#   find lambda - the apparent longitudes of the sun - in radians (p. 164)
#
      lambda <- L0 + C - (0.005689999999999997 + 
                          0.004779999999999999 * sin(omega))/180 * pi
#
#   find epsilon - the apparent obliquities of the ecliptic - in radians 
#   (pp. 144-147)
#
      Lprime <- (218.3165 + 481267.8813*T) / 180 * pi
      deltapsi <- (-17.2*sin(omega) - 1.32*sin(2*L0) - 
                   0.23*sin(2*Lprime) + 0.21*sin(2*omega))/3600
      deltaep <- (9.2*cos(omega) + 0.57*cos(2*L0) + 0.10*cos(2*Lprime) - 
                  0.09*cos(2*omega))/3600
      epsilon <- (23.439291111 - 0.013004167 * T - 1.638889e-07 * Tsq + 
                  5.036111111e-07 * Tsq * T + deltaep)/180 * pi
#   find alpha - the apparent right ascensions of the sun - in radians (p. 165)
#
      alpha <- atan2(cos(epsilon) * sin(lambda), cos(lambda)) 
#
#   find delta - the apparent declinations of the sun - in radians (p. 165)
#
      delta <- asin(sin(epsilon) * sin(lambda))	
#
#   find e - the eccentricities of the earth's orbit - unitless (p. 163)
#
      e <- 0.016708634 - 0.000042037 * T - 0.0000001267 * Tsq 
#
#   find deltaT - the factors to convert to mean times from dynamical times
#                 and vice versa (referenced to solar time) - 
#                 in degrees (p. 78)
 
#
      deltaT <- (65 + 88*T + 25.3*T^2)/3600*15 
#
#   find jd_UT - the julian day for 0h UT on the day of the plot - in days
#   and T_UT - the increment from 1/1/2000 noon UT to jd_UT - in centuries  
#   (p. 87)
# 
      jd_UT <- jd[local==0] - lo/(15 * 24) - .5
      T_UT <- (jd_UT -2451545)/36525
#
#   find THETA0 - the mean sidereal time of the point due south at Greenwich
#                 at 0h UT on the day of the plot - in radians (p. 87)
#
      THETA0 <- 100.46061837 + 36000.770053608*T_UT 
      THETA0 <- (THETA0 + 0.000387933*T_UT^2 - (T_UT^3)/38710000)/180*pi
#
#   find theta - the apparent sidereal times of the points due south at 
#                the location for the local dynamical times - 
#                in radians (p. 88)
#
      theta0 <- THETA0 + (lo + local + 180 + deltaT)*tosid/180*pi
      theta <- (theta0 - lo/180*pi) + (deltapsi*cos(epsilon))/180*pi
#
#   find H - the hour angles of the sun for the local dynamical times - 
#            in radians (referenced to sidereal time) (p. 92)
#
       H <- (theta - alpha)%%(2*pi)
       H[H>pi] <- H[H>pi]-2*pi	
#
#   convert the geographical latitude to radians
#
      phi 
#
#   find the altitudes and azimuths of the sun - in degrees (p. 93)
# 
     azimuth <- atan2(sin(H), (cos(H) * sin(phi) - tan(delta) * 
                      cos(phi)))* 180 / pi
     altitude <- asin(sin(phi) * sin(delta) + 
                      cos(phi) * cos(delta) * cos(H)) * 180 / pi
#
#   convert local - the local dynamical times - to Time - the civil times - 
#   in solar hours (p. 77)
#
      Time <- (local - deltaT + 180 + lo)/15 - tz
#
#   make the plot
#
      par(mar = c(6, 6, 7, 7) + 0.1, font.main=1)
      plot(Time[altitude>=0], altitude[altitude>=0], axes = F, type = "l",
 
      lty = 1, xlab="", ylab = "", 
      xaxs = "r", xaxt = "n", yaxt = "n", yaxs = "r", ylim = c(-10, 90),
      xlim = c(4, 22), col=2)
      axis(1, at = c(4, 6, 8, 10, 12, 14, 16, 18, 20, 22), labels = c("4AM",
           "6AM", "8AM", "10AM", "12PM", "2PM", "4PM", "6PM", "8PM",
           "10PM"), tck = 1)
      axis(2, at = c(-10, 0, 10, 20, 30, 40, 50, 60, 70, 80, 90), labels = c(
           "-10", "0", "10", "20", "30", "40", "50", "60", "70", "80",
           "90"), tck = 1, adj = 1)
      legend(3.4,90, c("Altitude", "Azimuth"), lty=1,
           col = 2:3, bg="white")
      mtext(side = 2, line = 3, "Altitude of the Sun (degrees)")
      par(new = "T")
      plot(Time[altitude>=0], azimuth[altitude>=0], axes = F, type = "l", 
          lty = 1,
          xaxt = "n", ylim = c(-140, 140), xlim = c(4, 22), ylab = "", 
          xlab="", col=3)
      axis(side = 4, at = c(-120, -90, -60, -30, 0, 30, 60, 90, 120), labels
           = c("120 E", "90 E", "60 E", "30 E", "0", "30 W", "60 W", "90 W",
           "120 W"), adj = 0, cex.axis=.7)
      mtext(side = 4, line = 3, "Azimuth of the Sun (degrees)", srt=270)
      mtext(side = 3, line = 2, paste(loc, "\n \n", "Latitude =", 
            round(la, 5), "  Longitude =", round(lo, dig = 5), " "))
      mtext(side = 1, line = 3, "Time")
      box()
#
#   print out some variables
#
      for( i in 1:(length(local)-1) ) {
            if (altitude[i+1] >= 0 & altitude[i] < 0) {kr              
            if (altitude[i+1] >= altitude[i]) {kn 
            if (altitude[i+1] < 0 & altitude[i] >= 0) {ks 
            }
      cat(
      "L0 ",(L0[c(kr,kn,ks)]*180/pi)%%360,"\n",
      "M ",(M[c(kr,kn,ks)]*180/pi)%%360,"\n",
      "C ",(C[c(kr,kn,ks)]*180/pi)%%360,"\n",
      "omega ",(omega[c(kr,kn,ks)]*180/pi)%%360,"\n",
      "lambda ",(lambda[c(kr,kn,ks)]*180/pi)%%360,"\n",
      "epsilon ",(epsilon[c(kr,kn,ks)]*180/pi)%%360,"\n",
      "alpha ",(alpha[c(kr,kn,ks)]*180/pi/15)%%24,"\n",
      "delta ",delta[c(kr,kn,ks)]*180/pi,"\n",
      "e ",e[c(kr,kn,ks)],"\n",
      "deltaT ",(deltaT[c(kr,kn,ks)]*180/pi/15)%%24,"\n",
      "THETA0 ",(THETA0*180/pi/15)%%24,"\n",
      "theta0 ",(theta0[c(kr,kn,ks)]*180/pi/15)%%24,"\n",
      "theta ",(theta[c(kr,kn,ks)]*180/pi/15)%%24,"\n",
      "H ",(H[c(kr,kn,ks)]*180/pi/15),"\n",
      "azimuth ",azimuth[c(kr,kn,ks)],"\n",
      "altitude ",altitude[c(kr,kn,ks)],"\n",
      "Time ", Time[c(kr,kn,ks)],"\n",
      "rise ",kr,"noon ",kn,"set ",ks,"\n")
}

Some Tools for Politicians – Researchers – Pundits

Word clouds tell you what people are saying about candidates:

You can see that tweets containing Hillary or Clinton and Bernie or Sanders have more words than tweets containing just Cruz or just Trump.  Tweets are centered on Indiana, which was just having their primary.  Each word cloud was based on 100 tweets.

Word clouds of tweets containing Hillary, Clinton, Bernie, Sanders, Cruz, and Trump

You can use Box plots to provide visual information about a Data Set:

For example, You might find it is Useful to know How the different Mixes of Political Parties performed with respect to the Budget Deficit, using the Majority Party of the House and Senate and the Political Party of the President.

Box plots of deficit / surplus as a percentage if GDP by party of president, senate, and house

Bar graphs compare categories:

Comparing the different Tools governments use to Raise Money might be of interest to you for setting or evaluating policy on taxes.

iowa-2009 taxes by class

Principal component plots show how numeric data clusters:

You can see from the plot that the parties of the Senate and House cluster together – Republican  Houses and Senates are similar and Democratic Houses and Senates are similar – and the  party of the President clusters with the Budget Deficit. You could find this information very useful.  These plots do not include the Obama years or the years of WWII.

Principal component plots show how numeric data clusters

Multiple correspondence plots show how categorical data clusters:

Multiple correspondence analysis plots of deficit / surplus as a percentage if GDP by party of president, senate, and house

You can see that the House and Senate cluster together for both Republicans and Democrats while Democratic presidents cluster with small deficits than Republican presidents. This might be of interest for your campaign or analysis.

Log linear modeling allows comparisons between classes:

Expected Cell Counts Normalized to Percentages by Row
For the On Budget Deficit (-) (Surplus (+)) per GDP
by the Controlling Political Parties
of the Presidency, Senate, and House
Budget Deficit from 1947 to 2008Political Parties Lagged One Year
Rows Sum to 100 Except for Rounding
President Senate House (-6.2,4] (-4,-2] (-2,0] (0,4.2]
D D D 0% 30% 55% 15%
R
R D
R 0% 29% 42% 29%
R D D 33% 40% 21% 7%
R 26% 42% 17% 15%
R D 33% 40% 21% 7%
R 26% 42% 17% 15%

In the table, each row is the combination of the  Presidential, Senate, and House parties. You find in the rows the log linear modeled percentage of years in which the President, Senate, and House combination falls in each Budget Deficit class. The Party of the Senate does not affect the outcome and the Parties of the President and House do – Republican Houses do a bit better –  Democratic Presidents do much better – which might be of interest to you.

Odds ratios can be used to compare parties:

Iowa 2013 Legislature
Odds Legislator is Male
House Senate
Democrats 3 : 2 10 : 3
Republicans 16 : 2 10 : 2

You could be interested in the odds a legislator is a male, by party.
For both the House and Senate in Iowa, Democrats have lower odds of being male than Republicans. Depending on your political philosophy, this could be bad or good.

Box Plots: A Political Example

BOX PLOTS

Box plots are used to provide information to you about the sizes of a variable in a data set. In the example below, the sizes of the variable – on budget deficit as a percentage of Gross Domestic Product – are compared for differing values of a second variable, a categorical variable describing the controlling political parties of the President, Senate, and House of Representatives. The heavy horizontal line in each box is plotted at the median value for the data group. The top of the box is plotted at the 75th percentile while the bottom of the box is plotted at the 25th percentile. The top and bottom whiskers end at the most extreme values of the points not calculated to be outliers. The circles represent points calculated to be outliers.

The data is from a deficit by political party data set. The deficit by political party data set contains data on the on balance deficits as a percent of the gross domestic product for the years 1940 to 2015, and the political parties controlling the presidency, the senate, and the house for the years 1939 to 2014. The data for the deficit and the gross domestic product are from the Office of Management and Budget and can be found at http://www.whitehouse.gov/omb/budget/Historicals (except the first 8 values for the deficit data, which were from a series published earlier). Before 1978, the budget year ran from July 1st to June 30th. From 1978 onward, the budget year ran from October 1st to September 31st. The tables contain a one quarter correction between the years 1977 and 1978, which I have ignored. The year with which the deficit and gross domestic product are associated is the year at the end of the budget year for which the deficit and gross domestic product are calculated.

For 1940 to 2011, the data for the controlling political parties of the president, senate, and house were taken from the website of Dave Manuel,
http://www.davemanuel.com/history-of-deficits-and-surpluses-in-the-united-states.php
, a good resource on the history of the deficit.  The last three years were taken from the website https://en.wikipedia.org/wiki/United_States_Presidents_and_control_of_Congress

We look at the size of the deficit (-) / surplus (+) using the political affiliations of the President and the controlling parties of the Senate and the House of Representatives. Below are two sets of box plots, one set for the full number of years in the data set and one set for a reduced number of years. Since the budget is created in the year before the budget is in effect, the deficit (-) / surplus (+) percentages are associated with the political parties in power the year before the end of budget year.

Box plots of deficit / surplus as a percentage if GDP by party of president, senate, and house

The first set of box plots are for the years 1940 to 2015 with regard to the budgets and for the years 1939 to 2014 with regard to the party affiliations. We can see in the first set of box plots that in the years of a Democratic President, House, and Senate, there were some very large deficits. These large deficits occurred during World War II. The large deficit under a Republican President and Democratic Senate and House was in 2009, which budget was associated with the last year of the G. W. Bush presidency. The large surplus in the years of a Democratic President and a Republican Senate and House was in the year 1948, when Truman was president.

The second set of box plots are for the years 1947 to 2015 with regard to the budgets and for the years 1946 to 2014 with regard to the party affiliations. With the years of extreme deficits removed, there is a clearer picture of how the differing party combinations perform in more normal years.  The rather large deficits under the Democrats controlling all bodies were under Obama in his first term, as were the four deficits from 2012 to 2015, when Obama was president, the Democrats controlled the Senate, and the Republicans controlled the House.  Obama inherited a dangerous economic situation and got us through the economic turmoil, but in the process, Obama was running quite large deficits.

The data is here:

comb.party.39to14        defsur.40to15
“1939” “DDD”             “1940” “-3.59917”
“1940” “DDD”             “1941” “-4.90272”
“1941” “DDD”             “1942” “-14.78378”
“1942” “DDD”             “1943” “-30.83472”
“1943” “DDD”             “1944” “-23.29589”
“1944” “DDD”             “1945” “-22.00542”
“1945” “DDD”             “1946” “-7.62084”
“1946” “DDD”             “1947” “1.22684”
“1947” “DRR”             “1948” “4.0015243902439”
“1948” “DRR”             “1949” “-0.252890173410405”
“1949” “DDD”             “1950” “-1.68458781362007”
“1950” “DDD”             “1951” “1.31337813072694”
“1951” “DDD”             “1952” “-0.951048951048951”
“1952” “DDD”             “1953” “-2.16993464052288”
“1953” “RRD”             “1954” “-0.722207892700542”
“1954” “RRD”             “1955” “-1.00737100737101”
“1955” “RDD”             “1956” “0.569476082004556”
“1956” “RDD”             “1957” “0.560103403705299”
“1957” “RDD”             “1958” “-0.695762175838077”
“1958” “RDD”             “1959” “-2.39319620253165”
“1959” “RDD”             “1960” “0.0934404784152495”
“1960” “RDD”             “1961” “-0.693937180423667”
“1961” “DDD”             “1962” “-1.00528199011757”
“1962” “DDD”             “1963” “-0.645890521556596”
“1963” “DDD”             “1964” “-0.980540051289787”
“1964” “DDD”             “1965” “-0.225130153369917”
“1965” “DDD”             “1966” “-0.396470136846144”
“1966” “DDD”             “1967” “-1.50322118826056”
“1967” “DDD”             “1968” “-3.08017346825309”
“1968” “DDD”             “1969” “-0.0509009467576097”
“1969” “RDD”             “1970” “-0.829282241921647”
“1970” “RDD”             “1971” “-2.33181452693648”
“1971” “RDD”             “1972” “-2.14022140221402”
“1972” “RDD”             “1973” “-1.12094395280236”
“1973” “RDD”             “1974” “-0.484457004440856”
“1974” “RDD”             “1975” “-3.35899664721222”
“1975” “RDD”             “1976” “-3.87644528849913”
“1976” “RDD”             “1977” “-2.46006704791954”
“1977” “DDD”             “1978” “-2.43174435958213”
“1978” “DDD”             “1979” “-1.5408560311284”
“1979” “DDD”             “1980” “-2.61370137299771”
“1980” “DDD”             “1981” “-2.35470303339281”
“1981” “RRD”             “1982” “-3.63921663297022”
“1982” “RRD”             “1983” “-5.86540905368388”
“1983” “RRD”             “1984” “-4.68781623153208”
“1984” “RRD”             “1985” “-5.18686774072686”
“1985” “RRD”             “1986” “-5.24459337316197”
“1986” “RRD”             “1987” “-3.52161274807085”
“1987” “RDD”             “1988” “-3.73028651238579”
“1988” “RDD”             “1989” “-3.68761220825853”
“1989” “RDD”             “1990” “-4.693470395293”
“1990” “RDD”             “1991” “-5.26014304184874”
“1991” “RDD”             “1992” “-5.29006791303402”
“1992” “RDD”             “1993” “-4.42096278090921”
“1993” “DDD”             “1994” “-3.59554308260858”
“1994” “DDD”             “1995” “-2.9854682596197”
“1995” “DRR”             “1996” “-2.18091573392828”
“1996” “DRR”             “1997” “-1.21652206714447”
“1997” “DRR”             “1998” “-0.333899137892527”
“1998” “DRR”             “1999” “0.0199779191420009”
“1999” “DRR”             “2000” “0.851382511184249”
“2000” “DRR”             “2001” “-0.306684588152888”
“2001” “RDR”             “2002” “-2.91811085879249”
“2002” “RDR”             “2003” “-4.75097949242879”
“2003” “RRR”             “2004” “-4.69864169548169”
“2004” “RRR”             “2005” “-3.82965187098977”
“2005” “RRR”             “2006” “-3.17507873756823”
“2006” “RRR”             “2007” “-2.38918096195603”
“2007” “RDD”             “2008” “-4.3504785661994”
“2008” “RDD”             “2009” “-10.7509053320938”
“2009” “DDD”             “2010” “-9.26715545494476”
“2010” “DDD”             “2011” “-8.88732833957553”
“2011” “DDR”             “2012” “-7.16843865428771”
“2012” “DDR”             “2013” “-4.35807759681418”
“2013” “DDR”             “2014” “-2.99182355166293”
“2014” “DDR”             “2015” “-2.61579248907512”

The code for the plot is here:

function () {
par(mfrow = c(1, 2), oma = c(2, 2, 4, 2) + 0.1)
boxplot(defsur.40to15 ~ comb.party.39to14, cex.axis = 0.8)
title(main = “Percentage of GDP – 1940 to 2015\nPolitical Parties – 1939 to 2014”,
xlab = “President, Senate, House”, ylab = “Percentage”,
cex.main = 1, font.main = 1)
boxplot(defsur.40to15[8:76] ~ comb.party.39to14[8:76], cex.axis = 0.8)
title(main = “Percentage of GDP – 1947 to 2015\nPolitical Parties – 1946 to 2014”,
xlab = “President, Senate, House”, ylab = “Percentage”,
cex.main = 1, font.main = 1)
mtext(“On Budget Deficit as a Percetage of GDP\nGrouped by Political Party Controling the Presidency, Senate, and House\nParty Lagged One Year”,
side = 3, cex = 1, font = 1, outer = T)
}

FITTING A SIMPLE REGRESSION MODEL WITH AUTOCORRELATED ERRORS: AN ASTROLOGICAL EXAMPLE

The assumptions of simple linear regression include the assumption that the errors are independent with constant variance. Fitting a simple regression when the errors are autocorrelated requires techniques from the field of time series. If you are interested in fitting a model to an evenly spaced series where the terms are autocorrelated, I have given below an example of fitting such a model.

The Astronomical/Astrological Hypotheses to be Tested

For some years now, I have been interested in the possible astrological influence of a composite astronomical point, the mean over the shortest arc of the celestial longitudes of the celestial bodies Juno, Ceres, Vesta, and Pallas. I wanted to see if the point affected divorce rates. In astrology, planets rule zodiacal signs. In older systems of astrology, from before the discoveries of the outer planets, each of the five known planets (Mercury, Venus, Mars, Jupiter, and Saturn – we exclude the earth) ruled two signs. As Uranus, Neptune, and Pluto were discovered, the planets were given rulership over one of Saturn’s, Jupiter’s, and Mars’s signs, respectively. In 1977, the celestial body Chiron was discovered. Some astrologers give Chiron the rulership of one of Mercury’s signs, which would leave Venus as the only planet with dual rulership.

The celestial bodies Juno, Ceres, Vesta, and Pallas were discovered in the early 1800’s. Referencing Roman mythology, some astrologers feel that the discoveries of Juno – the wife, Ceres – the mother, Vesta – the sister, and Pallas – the daughter synchronized with the emergence of the struggle for the full participation of women within society. Venus (which, from the above, is for some astrologers the only planet that still rules two signs) is one of the traditional rulers of women in astrology. The two signs that Venus rules are those associated with money and marriage. I had thought that the mean of the celestial longitudes over of the shortest arc connecting Juno, Ceres, Vesta, and Pallas might be the ruler of the sign of marriage, since Juno, Ceres, Vesta, and Pallas refer to female archetypes. Then Venus would just rule the sign of money. The mean over the shortest arc jumps 90 degrees or 180 degrees from time to time. To test the hypothesis, I had thought that the jump point of the mean over the shortest arc might be related to divorce. An ephemeris for the average, which I call Vulcan – as well as a point I call Cent, from 1900 to 2050, can be found here – https://1drv.ms/b/s!AgjNmjy3BpFDmBBTiEPHc3qPr9ls

The Data Sets: Divorce Counts for Iowa and the Jump Points

I created a data set using monthly divorce counts in Iowa from August of 2006 to November of 2009 and information about the jump points of the mean over the shortest arc of the celestial longitudes of Juno, Vesta, Ceres, and Pallas. The divorce counts were adjusted for the number of business days in a month. The adjusted divorce counts are the dependent variable. The independent variables are the usual column of ones to fit a mean and a column which linearly increases as time increases after a jump point and which is set back to zero at each jump point.

The marriage and divorce data set contains data from the Center for Disease Control and Prevention website, from the National Vital Statistics Report page. I took marriage and divorce data for Iowa from the reports from August of 2006 to November of 2009. The divorce data for April of 2008 was 161, which was unusually small. In my analyses, I replaced the number with 703, the average of the numbers on either side.

Also contained in the data set is a variable which measures an astronomical phenomena. When one takes the average of the longitudes of the four major asteroids, Juno, Ceres, Pallas, and Vesta (yes, I know Ceres is now a dwarf planet) along the shortest arc, the average longitude will jump by 90 or 180 degrees from time to time. The third variable in the data set is the time in days from the last day in which a jump occurred. Since the marriage and divorce data is monthly, each data point is assumed to be at the 15th of the month for the calculation of the third variable.

The question of interest is, are the jump points associated the divorce counts in a linear way? Since the months are not of equal length, I have adjusted the divorce counts by dividing the divorce count for a month by the number of state business days in the month and multiplying the result by the average number of state business days in a month over the observed months. The holidays I excluded are New Year’s Day, Martin Luther King Day, Memorial Day, Independence Day, Labor Day, Veteran’s Day, the two days for Thanksgiving, and Christmas Day, as these days were state holidays in Iowa in 2012. The number of state business days that I found for each month are given in the table below.

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
2006 23 20 22 19 20
2007 21 20 22 21 22 21 21 23 19 23 19 20
2008 21 21 21 22 21 21 22 21 21 23 17 22
2009 20 20 22 22 20 22 22 21 21 22 18

Plotting the Data Sets

Since the dependent variable is a time series, one could expect autocorrelation in the error terms which may need to be taken into account. Below, I have plotted the adjusted divorce counts and the jump point variable.

time series plots of divorce counts and the jump point variable

A relationship between the two plots is not immediately evident.

Fitting a Simple Regression Model

I started by fitting a simple linear regression model. The estimated model parameters are given below.

Coefficient Standard Error
Intercept 621.2 20.2
Slope 0.06049 0.05440

Clearly, there is little reason to think that the jump points are related to the adjusted divorce counts given the simple regression model. However, since the data is time series data, there is a possibility that the errors are autocorrelated.

The Box-Jenkins Approach

Following the Box-Jenkins approach to fitting time series, I decided to start by looking at the residuals, from the adjusted divorce count model, as a stationary time series.

I generated two plots; a plot of the correlations between present and lagged variables using the R function acf() and a plot of the partial correlations between present and lagged variables using the R function pacf(). For the autocorrelation and the partial autocorrelation plots the blue lines represent the 95% level.

correlation and partial correlation plots for the divorce count residuals

Following the Box-Jenkins approach, the correlation plot will indicate if the time series is a moving average. If there are spikes in the correlation plot, the spikes indicate the orders of the moving average. An autoregressive time series will have an exponentially decaying correlation plot.

The partial correlation plot indicates the order of the process if the process is autoregressive. The order of the autoregressive process is given by the value of the lag just before the partial correlation goes to essentially zero.

A process with both moving average and autoregressive terms is hard to identify using correlation and partial correlation plots, but the process is indicated by decay that is not exponential, but starts after a few lags (see the Wikipedia page Box-Jenkins).

Looking at the above plots, the autocorrelation plot shows negative spikes at lag 1 and lag 13 and is large in absolute value at lags 3, 12, and 14, but otherwise is consistent with random noise. The partial correlation plot has negative spikes at lags 1, 2, and 13 and is large in absolute value at lag 15, indicating that the process may not be stationary. Neither plot decays.

The Periodogram and Normalized Cumulative Periodogram

Using periodograms is another way you can evaluate autocorrelation. A periodogram can find periodicities in a time series. Below are plots of the periodogram and the normalized cumulative periodogram created using spec.pgram() and cpgram() in R.  In the periodogram, there is a spikes at about the frequencies .35 and .40 that are quite large. The frequencies correspond to lags of 2.9 and 2.5, somewhat close to the lags of 2 and 3 months found in the correlation plot. The normalized cumulative periodogram shows a departure from random noise at a frequency of around .29, or a lag of around 3.4. The blue lines in the normalized cumulative periodogram plot are the 95% confidence limits for random noise.

periodogram and normalized cumulative periodogram for the divorce count residuals

Note also that at the frequency of twenty, corresponding to a lag of two, the periodogram is also high.

Using arima() in R to Compare Models

I used the function arima() in R to fit some autoregressive moving average models with orders up to thirteen, both with and without the astronomical regressor. The best model, using the Aktaike information criteria (see the Wikipedia page Aktaike information criterion), was

Yt = β0 + β1 Xt + θ1 εt-1 + θ3 εt-3 + θ13 εt-13 + εt,

where Yt is the number of divorces in month t, Xt is the number of days since the last shift in the mean of the celestial longitudes of Ceres, Pallas, Vesta, and Juno over the shortest arc, and εt is the difference between the expected number of divorces and the actual number of divorces for month t. The estimated coefficients, with the respective standard errors, are given in the table below.

Model for Adjusted Divorces in Iowa by Month
Coefficient Standard Error
βhat0 624 10.5
βhat1 0.054 0.031
θhat1 -0.682 0.274
θhat3 0.386 0.180
θhat13 -0.598 0.227

The coefficient for the slope of the regression is not at least 1.96 times the size of the standard errors of the coefficient, however, the Aktaike information criteria is the smallest for the models at which I looked. The model without the coefficient is a close contender, with the Aktaike information for the two models being 453.07 and 453.85. Below is a plot of the residuals for the two competing models. To me the residuals look more even for the model given above.

residual plot for two models

The estimates indicate that the intercept is about 624 divorces; that the number of divorces increases by about 5.4 for every one hundred day increase in the time after the midpoint over the shortest arc of the longitudes of Juno, Ceres, Vesta, and Pallas shifts by 90 or 180 degrees; that errors tend to reverse in direction over one lag and thirteen lags, but tend to be in the same direction over three lags.

Some Astrological Observations

The results are interesting from an astrological point of view. Astrological signs are divided into groups by order. The first grouping is by alternating order, with the first sign (Aries) positive and the next sign negative, the third positive, and so on through the circle. The second grouping is in groups of three’s, called the elements – fire, earth, air, and water. The first, fifth, and ninth signs are fire signs, and so on through the circle. The third grouping is in groups of four, called the qualities – cardinal, fixed, and mutable. The first, fourth, seventh, and tenth signs are cardinal signs, and so on through the circle. The months, though I am quite certain that the months were originally concurrent with the zodiac signs, overlap signs. About 2/3 of one sign and 1/3 of the next sign are in each month. The adjusted counts are modeled and used in what follows

Averaging the counts for the February’s, April’s, June’s, August’s, October’s, and December’s, months that are 2/3 positive and 1/3 negative, the result is 616.7 divorces with an estimated standard error of 15.8. Averaging the adjusted counts for the January’s, March’s, May’s, July’s, September’s, and November’s, the months that are 2/3 negative and 1/3 positive, the result is 661.5 divorces with an estimated standard error of 15.8. The difference between the two groupings is -44.9 with an estimated standard error of 25.8. The difference between the counts is significant at the 0.10 level for a two-sided test. Positive months have fewer divorces than negative months in this data set – not what I would expect since the positive signs are associated with action and the negative signs with reaction.

Averaging the adjusted counts for the January’s, April’s, July’s, and October’s, months that are 2/3 cardinal and 1/3 fixed, the result is 627.3 divorces with an estimated standard error of 22.3. Averaging the counts for the February’s, May’s, August’s, and November’s, months that are 2/3 fixed and 1/3 mutable, the result is 611.0 divorces with an estimated standard error of 21.8. Averaging the counts for the March’s, June’s, September’s, and November’s, months that are 2/3 mutable and 1/3 cardinal, the result is 681.2 divorces with an estimated standard error of 22.8.

Looking at the three differences, between the first group and the second group – mainly cardinal versus mainly fixed – the difference is 16.3 with an estimated standard error of 34.4, between the first group and the third group – mainly cardinal versus mainly mutable – there is a difference of -53.9 with an estimated standard error of 35.8, and between the second group and the third group – mainly fixed versus mainly mutable – there is a difference of -70.2 with an estimated standard error of 35.3. The second and third tests are significant at the 0.15 level for two sided tests. The first test is not significant. The test results are in agreement with what one would expect from astrology. Cardinal and mutable signs are associated with change and adjustment, while fixed signs are associated with steadfastness.

The negative direction of the lag of 13, could be explained by the cycles of the Moon. There are approximately 13 Moon cycles per year, so for a given month, the Moon will start in a positive sign one year and in a negative sign the next, over reasonably short time periods. The estimated standard error for the means over the 13 lags is 37.0 except for the mean starting with the first observation, which has an estimated standard error of 30.0. There are 13 means from the 40 observations.

Some Concluding Diagnostic Plots

Below are some diagnostic plots, the plot of the residuals versus the fitted values, the Q-Q plot of the residuals versus normal variates, and a time series plot of the actual values and the fitted values.

residual vs fitted, QQ, and data with fitted plots

The residual versus fitted value plot looks like there is some correlation between the residuals and the fitted values. Testing the correlation gives a p-value of 0.0112, so the model is not ideal. While the model does not pick up most of the extreme data points, the residual and QQ normal plots do not show any great deviations from a model of normal errors with constant variance. The fitted values were found by subtracting the residuals from the actual data points.

The R Code

# First enter monthly divorce counts for Iowa from 8/06 to 11/09

div.ts = ts(c(619, 704, 660, 533, 683, 778, 541, 587, 720, 609, 622, 602, 
        639, 599, 657, 687, 646, 608, 571, 768, 703, 638, 743, 693, 
        624, 694, 660, 522, 639, 613, 446, 863, 524, 564, 716, 635, 
        649, 565, 567, 626), start=c(2006,8), freq=12)

# next enter the break point information: the number of days from the 
# last shift

vulcan.ts = ts(c(63, 94, 124, 155, 185, 216, 247, 275, 306, 336, 367, 397, 
                 428, 459, 489, 520, 550, 581, 612, 641, 672, 702, 733, 
                 763, 19, 50, 80, 4, 34, 65, 96, 124, 155, 185, 216, 246, 
                 277, 308, 3,  34), start=c(2006,8), freq=12)


# there are 1218 days in the 40 months.  the divorce counts are adjusted to
# reflect the number of days tha the county offices are open.

# first create a vector of days of the week, with 1 for Sunday, 
# 2 for Monday, etc.  8/06 started on a Tuesday.

wks = c(3:7, rep(1:7, 173), 1:2)

# next create a daily vector for the months, with 1 for for the first month, 
# 2 for the second month, etc.

mns = c(8:12, rep(1:12,2), 1:11)
mns.d = 1:40
mn =  c(31,28,31,30,31,30,31,31,30,31,30,31)
mns.dys = rep(mns.d,mn[mns])
mns.dys = c(mns.dys[1:553], 19, mns.dys[554:1217]) # add 2008 leap day

# next remove Sundays and Saturdays

mns.dys.r = mns.dys[wks>1 & wks<7]

# next use table() to count the weekdays in each month

mns.dys.s = c(table(mns.dys.r))

# next enter the number of government holidays in each of the 40 months

hol = c(0, 1, 0, 3, 1, 2, 0, 0, 0, 1, 0, 1, 0, 1, 0, 3, 1, 2, 0, 0, 0, 
        1, 0, 1, 0, 1, 0, 3, 1, 2, 0, 0, 0, 1, 0, 1, 0, 1, 0, 3)

# next subtract out the holidays

mns.dys.s = mns.dys.s - hol

# next adjust the divorce counts

div.a.ts = div.ts * mean(mns.dys.s)/mns.dys.s

# next plot the two data vectors

data.d.v = cbind("jump point variable"=vulcan.ts, 
                      "adjusted divorce count"=div.a.ts)
plot(data.d.v, main="Adjusted Divorce Counts in Iowa with the Jump Point Variable
August 2006 to November 2009")

# next fit the simple linear regression

mod.lm = lm(div.a.ts~vulcan.ts)

summary(mod.lm)

# plot the autocorrelations and the partial autocorrelations 
# of the residuals from the model

acf(mod.lm$residuals, main="autocorrelation of the residuals
fitting divorce counts on Ceres, Juno, Vesta, and Pallas breaks")

pacf(mod.lm$residuals, main="partial autocorrelation of the residuals
fitting divorce counts on Ceres, Juno, Vesta, and Pallas breaks")

# plot the periodogram and cumulative periodogram of the residuals from 
# the model

spec.pgram(mod.lm$residuals, main="Residuals from Simple Regression
Raw Periodogram")

cpgram(mod.lm$residuals, main="Residuals from Simple Regression
Cumulative Periodogram")

# here is a function I created to compare models.  the function is set up 
# to only do moving averages.  It takes awhile  to run with or=13.

arima.comp=function(ts=div.a.ts, phi=vulcan.ts, or=3) {
  
  phi <- as.matrix(phi)
  if (nrow(phi)==1) phi <- t(phi)
  ind <- 2*or+ncol(phi)
  fx.lst <- list(0:1)
  #for (i in 2:(2*or)) fx.lst <- c(fx.lst,list(0:1))
  for (i in 2:or) fx.lst <- c(fx.lst,list(0:1))
  
  fx.mat <- as.matrix(expand.grid(fx.lst))
  #fx.phi1 <- matrix(NA, 2^(2*or), 1+ncol(phi))
  fx.phi1 <- matrix(NA, 2^or, 1+ncol(phi))
  
  fx.mat[fx.mat==1] <- NA
  fx.mat1 <- cbind(fx.mat,fx.phi1)
  fx.mat2 <- cbind(fx.mat,fx.phi1[,1])
  
  #res1 <- numeric(2^(2*or))
  #res2 <- numeric(2^(2*or))
  
  res1 <- numeric(2^or)
  res2 <- numeric(2^or)
  
  #for (i in 1:2^(2*or)) {
  for (i in 1:2^or) {
    
    #res1[i] <- arima(ts, order=c(or,0,or), xreg=phi, fixed=fx.mat1[i,])$aic
    #res2[i] <- arima(ts, order=c(or,0,or), fixed=fx.mat2[i,])$aic
    
    res1[i] <- arima(ts, order=c(0,0,or), xreg=phi, fixed=fx.mat1[i,])$aic
    res2[i] <- arima(ts, order=c(0,0,or), fixed=fx.mat2[i,])$aic
    
  }
  
  st1 <- order(res1)
  st1 <- st1[1:min(length(st1),10)]
  print(res1[st1])
  print(fx.mat1[st1,])
  
  st2 <- order(res2)
  st2 <- st2[1:min(length(st2),10)]
  print(res2[st2])
  print(fx.mat2[st2,])
  
}



# the final model was fit using arima().  in fixed, an NA says the 
# parameter is fit and a 0 says no parameter is fit.  in fixed there
# is a value for every value indicated by order and xreg. plus the 
# intercept.  so there are 13 moving average terms, three of which 
# are fit, and there is an intercept and one regressior term, 
# for 15 values in fixed.

mod.arima.r.1 = arima(div.a.ts, order=c(0,0,13), 
                    fixed=c(NA,0,NA,rep(0,9),NA,NA,NA), xreg=vulcan.ts) 

mod.arima.r.1

mod.arima.r.2 = arima(div.a.ts, order=c(0,0,13), 
                      fixed=c(NA,0,NA,rep(0,9),NA,NA))
mod.arima.r.2

# next the residuals from the two models are plottted together

ts.plot(cbind(mod.arima.r.1$residuals, mod.arima.r.2$residuals), col=4:3,
        xlab="Time", ylab="Residuals",
        main="Residuals for Two Moving Average Models")
legend("topright", c("With Regression Coefficient", 
                     "Without Regression Coefficient"), fill=4:3, cex=.7)

# next the estimated covariance matrix for the 40 observations is found,
# using the adjusted counts

cov.1 = matrix(0,40,40)
diag(cov.ua)= sum(c(1,mod.arima.r.1$coef[1:13]^2))

for (i in 1:12) {
  cov.1[cbind(c(1,(i+1)),c(i+1,1))]=
    sum(c(1,mod.arima.r.1$coef[1:(13-i)])*mod.arima.r.1$coef[i:13])
}
cov.1[cbind(c(1,14),c(14,1))] = mod.arima.r.1$coef[13]

for (i in 2:14) {
  cov.1[(i-1)+2:14, i] = cov.1[2:14, 1]
  cov.1[i, (i-1)+2:14] = cov.1[2:14, 1]
}

cov.1 = cov.1*mod.arima.r.1$sigma2

# next the average counts for sets of months along with the standard errors
# t.m is the matrix of multipliers for finding means and standard errors
# the rows of t are positive signs, negative signs, cardinal signs, fixed
# signs, and mutable signs

t.m = matrix(0,5,40)
t.m[1,] = rep(1:0, 20)
t.m[2,] = rep(0:1, 20)
t.m[3,] = c(rep(c(0,0,1),13),0)
t.m[4,] = c(rep(c(1,0,0),13),1)
t.m[5,] = c(rep(c(0,1,0),13),0)

m.m = t.m%*%div.a.ts/apply(t.m,1,sum)
s.m = sqrt(diag(t.m%*%cov.1%*%t(t.m)))/apply(t.m,1,sum)
           
m.m # the five means
s.m # the five standard errors

# next the four tests are done, with the same structure as in the last part
# the tests are positive vs negative, cardinal vs fixed, cardinal vs 
# mutable, and fixed vs mutable

t.t = matrix(0,4,40)
t.t[1,] = rep(c(1/20,-1/20),20)
t.t[2,] = c(rep(c(0,-1/13,1/13), 13), 0)
t.t[3,] = c(rep(c(-1/14,0,1/13), 13),-1/14)
t.t[4,] = c(rep(c(1/14,-1/13,0), 13), 1/14)

m.t = t.t%*%div.a.ts
s.t = sqrt(diag(t.t%*%cov.1%*%t(t.t)))

z.sc = m.t/s.t
p.z.sc = pnorm(z.sc)
m.t # the values of the four test statistics
s.t # the estimated standard errors for the four test statistics

z.sc # the z scores
p.z.sc # the p-values of the z scores

# next, the means and standard errors for the 13 lag means are found

t.14 = c(1,1+seq(13,40,13))
t.13 = c(2,2+seq(13,27,13))

m.13 = matrix(0,13,40)
m.13[1,t.14] = 1/4
for (i in 2:13) m.13[i,t.13+(i-2)] = 1/3

mn.13 = m.13 %*% div.a.ts
sd.13 = sqrt(diag(m.13 %*% cov.1 %*% t(m.13)))

mn.13
sd.13

# next, the diagnostic plots are plotted

par(mfrow=c(1,3))

plot(as.numeric(div.a.ts-mod.arima.r.1$residuals), 
     as.numeric(mod.arima.r.1$residuals),
     main="Plot of Residuals versus Fitted Values
for the Adjusted Divorce Data", xlab="Fitted Value", ylab="Residual", 
     font.main=1)

qqnorm(mod.arima.r.1$residuals, main="QQ Normal Plot for
Residuals from the Divorce Model", font.main=1)

ts.plot(div.a.ts,div.a.ts-mod.arima.r.1$residuals,
        ylab="Counts", xlab="Year",
        main="Adjusted Divorce Counts and Fitted Counts", 
        col=4:3)
legend("bottomleft", c("Adjusted Divorce Counts",
       "Fitted Adjusted Divorce Counts"), col=4:3, lwd=2, cex=.7)

# last, the correlation between the fitted values and the 
# residuals are found

cor.test(div.a.ts-mod.arima.r.1$residuals, mod.arima.r.1$residuals)



The Mechanics of Climate Change: Some Comments on a Sidebar Article in Amstat News

(The article and comments can be found at http://magazine.amstat.org/blog/2010/03/01/climatemar10/. The book Storms of my Grandchildren, by James Hansen, published in 2009, by Bloomsbury USA, is a good overview of the causes of climate change. Dr. Hansen is a climate researcher and much more knowledgeable and experienced than myself in the area of climate research. As a disclaimer, Dr. Hansen’s sister is married to my second cousin, who sent my husband and me the book.)

To the editor of the Amstat News,

I read the article, “The Paper that Convinced Me of the Connection Between Carbon Dioxide and Climate Change”, by Peter Guttorp, in the March 2010 issue of Amstat News, with interest. I have been interested in the greenhouse effect since reading the Whole Earth Catalog in the summer of 1973, going to the sources given in the catalog for information about solar energy and the digestion of waste to produce methane, and doing my senior thesis in physics at Reed College on the potential for using solar energy for space heating in western Oregon. I built two hot water solar absorbers and modeled the heat transfers involved in heating an antifreeze solution pumped through one of the absorbers. The heat transfers involved in a solar absorber are similar to the heat transfers involved in the greenhouse effect of climate change. The Whole Earth Catalog mentions the greenhouse effect, as well as giving many fine references for alternative energy research. Also, I am a student of astrology and, as such, have familiarized myself with the mechanics of the solar system, including the earth’s orbit and rotation. I, also, am a amateur radio operator, so I have practical experience working with electromagnetic waves. And, I am a statistician, having received a masters and PhD from Iowa State in statistics and having done a variety of work using my education over the last 27 years. I feel like I have a unique set of knowledge to comment on Guttorp’s article.

Some Astronomy

To start, what are the mechanics of the earth’s orbit and rotation? For the hard facts involving the astronomy, I used the book, Astronomical Algorithms, 2nd Edition, by Jean Meeus, published by Willmann-Bell, Inc., in 1998. I, also, took some information from Wikipedia and used some information from the physics textbook, Classical Mechanics, by Herbert Goldstein, Ph.D., published by Addison-Wesley, in 1950.

In the early 1600’s, Kepler found, by looking at the detailed and extensive observations of the astronomer, Tycho Brahe, that the orbits of the planets around the sun are essentially ellipses. Elliptical orbits have perihelia and an aphelia, which are of interest in climate change modeling. The perihelion is the point on the orbit where the planet is closest to the sun and the aphelion is the point on the orbit where the planet is farthest from the sun. Kepler also discovered that the orbits sweep out area at a constant rate, which we now know is due to the conservation of angular momentum. We, also, know the actual orbits of the planets are not perfect ellipses. Some terminology is in order. For orbits, deviations from a perfect ellipse are due mainly to the gravitational effect of celestial bodies other than the sun, called perturbations (see Meeus, p. 163). Astronomers use the term, mean, to describe values of variables that, in some sense, are not corrected for perturbations or nutation (to be described below), and, in the case of the sun’s mean longitude, the eccentricity of the earth’s orbit (see Meeus, p. 183). The ‘true’ positions of celestial objects are called geometric positions (see Meeus, p. 412). If one corrects for aberration, which is the correction for the fact that light travels at a finite speed, results are called apparent results.

The earth orbits around the sun in a plane, which is called the ecliptic. Looking down from the north, the earth orbits in a counter-clockwise direction around the sun. The earth also rotates, around the polar axis of the earth (the imaginary line that goes through the south and north poles of the earth). The earth rotates in a counter-clockwise direction, looking down from the north. However, the polar axis is not perpendicular to the ecliptic. The angle between the polar axis and a perpendicular to the ecliptic is called the obliquity of the ecliptic. The earth’s rotation is gradually slowing down, at a non-uniform rate. Since the rotation is slowing down, there is a difference between the time the earth is actually at a given point and the estimated mean time expected. The difference is called the equation of time. The value of the difference, called delta T, is found by observation, so is only known for the past and present, but can be estimated into the future. In Meeus in Chapter 10, the above is discussed and values for delta T are given for the years 1620 to 1998, for every other year. Universal time (UT) is not corrected for delta T and dynamical time (TD) is.

The Precession of the Equinoxes and the Obliquity of the Ecliptic

When a top spins, the top of the top makes slow circles. Similarly, the polar axis of the earth circles around the polar axis of the ecliptic – which can be thought of as an imaginary line through the center of the earth perpendicular to the plane of the ecliptic. A complete circle takes about 26,000 years (Meeus, p. 131). According to Meeus (p. 131) the circling is caused by “the gravitational attraction of the sun and the moon on the earth’s equatorial bulge.” Because of the circling, the polar axis of the earth is tipped from being perfectly perpendicular to the ecliptic by the obliquity of the ecliptic. Right now, the obliquity of the ecliptic is about 23.4 degrees from the vertical. The obliquity of the ecliptic is slowly decreasing over time, with a (almost linear at this time) decrease of about 1.75 degrees over 200 centuries (Meeus, p. 148). There are also periodic oscillations, called nutation, in the circling of the polar axis (see Meeus, p. 143). Nutation is taken into account in calculating geometric placements, but not in mean placements.

Because the polar axis is making a circle over a 26,000 year time period, the places on the ecliptic where the solstices and equinoxes occur precess (move with respect to the fixed stars, which we assume are fixed, though the stars move too, very slowly, see Meeus, p. 150), and the movement is in the opposite direction of the orbit of the earth (the precession is clockwise, looking down from the north). The precession is called the precession of the equinoxes and is about 50 seconds of arc per year (Meeus, p. 131).

During a year, the time it takes for the earth to come back to the point of the spring equinox, or any other point in the seasons, is called a tropical year, and varies with the seasonal point chosen. The mean time, averaged over all seasonal points, is called the mean tropical year and currently takes 365.2422 days, not the 365.2442 days described in Guttorp’s article. The tropical year is mentioned on p. 133 of Meeus and a discussion can be found on Wikipedia, under the entry for tropical year.

The Seasons of the Year

The seasons of the year result because the polar axis is tipped. As the earth orbits around the sun, since the earth always stays tipped in the same direction, at just two points in the orbit the polar axis will be in line with the imaginary line connecting the earth and the sun. When the polar axis is in line with the imaginary line, the solstices occur. For the winter solstice, the polar axis is pointed away from the sun and for the summer solstice, the polar axis is pointed toward the sun. There are, also, two points in the orbit where the polar axis is perpendicular to the earth/sun line. When the polar axes are perpendicular to the earth/sun line, the spring (vernal) and fall (autumnal) equinoxes occur, on which days, day and night have the same length.

A table of the dates and times of equinoxes and solstices for the years 1996 to 2005 can be found on p. 182 of Meeus. Using formulas from Meeus in Chapters 21 and 25 for the mean precession and using a formula found on Wikipedia (under the entry for tropical year) for the length of the mean tropical year, I calculated mean values for the precession and length of the mean tropical year for 1860 and 1980. According to my calculations, in 1860, the mean precession was 50.466 seconds of arc per year and the mean tropical year was 365.242198 days, and in 1980, the mean precession was 50.315 seconds of arc per year and the mean tropical year was 365.242191 days. I calculated the precession of the sun over the year using the mean longitude of the sun. So, by Meeus’s formulas, the mean precession is decreasing in time and the mean tropical year is also decreasing in time.

The Movement of the Perihelion and the Aphelion and the Anomalistic Year

The points of perihelion and aphelion also move, with respect to both the stars and the seasons. The points of perihelion and aphelion move in the direction of the orbit (counter-clockwise, looking down from the north) , currently at the rate of about 47 seconds of arc per century (Meeus, p. 131). According to Meeus, the movement is “due to the gravitational attraction of the planets to the Earth”. As the earth orbits the sun, the time period for the return of the earth to the point of perihelion (or aphelion) is the anomalistic year given in the article by Guttorp. The time period of 365.2596 days given by Guttorp agrees with Meeus, although from Meeus’s formulas the length of the anomalistic year is slowly increasing.

From the above, the solstices and equinoxes move in the opposite direction from the perihelion and aphelion, with respect to the stars. Since the solstices and equinoxes move backward and the perihelion and aphelion move forward, the day on which the perihelion or aphelion occurs is later in the year as time goes by. The perihelion and winter solstice were coincident in 1246 AD, according to Meeus, p. 181. Formulas for the day of mean perihelion and aphelion in a given year can be found in Meeus, on p. 269. Using the formulas in Meeus, the mean times of perihelion and aphelion in 1860 were January 1st at 15.80 hours universal time (UT) and July 2nd at 6.91 hours UT. For 1980, the times were January 3rd at 19.54 hours UT and July 4th at 10.66 hours UT.

On p. 274, Meeus gives the true times of perihelion and aphelion for the years 1991 to 2010. The times are given in dynamical time. Using the table in Chapter 10 of Meeus, I converted the dynamical times of perihelion to universal time for the years 1992, 1994, 1996, and 1998. Then, using the astrological program, Kepler, I calculated the longitudes of the sun at the times of the perihelia. Longitudes are measured in a counter-clockwise direction along the ecliptic from the spring equinox, looking down from the north. On 1/3/1992 at 15h 02m 38s UT, the perihelion was at 282 degrees 29 minutes longitude, on 1/2/1994 at 5h 54m 12s UT the perihelion was at 281 degrees 42 minutes longitude, on 1/4/1996 at 7h 24m 46s UT, the perihelion was at 283 degrees 13 minutes longitude, and on 1/4/1998 at 21h 15m 45s UT the perihelion was at 284 degrees 19 minutes longitude. We can see that the longitudes do not vary linearly with respect to time. There is quite a bit of variation even for the four years given. I am assuming that the times given in Meeus are correct. If the times are correct, the effect of nutation and perturbations can be quite large and swamp the gradual shift of the perihelion and aphelion eastward.

Combining the Cycles

Given the above, the discussion of the two close together cycles, as described by Guttorp, does not make sense. The actual two cycles are, first, the combination of the precession the perihelion/aphelion with the precession of the equinoxes, which cause the seasonal points to shift clockwise (looking down from the north) with regard to the perihelion and aphelion, and, second, the seasons, which result from the tilt of the earth’s polar axis. The precession cycle is very long (about 26,000 years) and the seasonal cycle is just a yearly cycle. (A third celestial mechanical phenomena that might be of interest is the gradual decrease of the obliquity of the ecliptic. A fourth is the changing eccentricity of the orbit of the earth. The earth’s eccentricity is currently decreasing.
See the Wikipedia article on orbital eccentricity.)

Solar Energy Incidence

The heating of the earth mainly depends on energy from the sun. If we are concerned about climate change, we are interested in the heating of the earth, which depends partly on the incidence of energy at the top of the earth’s atmosphere. The incidence of solar radiation at the top of the atmosphere of the earth depends on the distance of the earth from the sun and the solar flux, both of which can vary quite a bit.

First we look at the distance. In Goldstein, Chapter 3 is on the two body central force problem, and Section 3-6 is on Kepler’s results. Goldstein gives the result that the instantaneous area swept out by a planet in an elliptical orbit is equal to [one half] times [the distance between the sun and the planet squared] times [the instantaneous rate of change in the angle swept out by the orbit]. The [distance between the sun and a planet] times [the instantaneous rate of change in the angle] is the speed of the planet in the planet’s orbit, if angles are measured in radians. As stated above, since angular momentum is conserved, the rate at which area is swept out is constant for an elliptical orbit. So [the speed of a planet] times [the distance from the sun to the planet], which is the rate at which area is swept out, would be a constant if the planet’s orbit were perfectly elliptical. At closer distances, the planet moves faster, and at farther distances, the planet moves slower.

We will explore solar energy incident on the earth for a given solar flux. Radiation from the sun spreads out in a cone like fashion from the surface of the sun. The solar energy incident on the earth would be that incident on a flat silhouette of the earth at the distance of the center of the earth. To see how distance affects incident energy, let us define a right triangle, centered at the center of the earth, with one leg going to the center of the sun and the other leg to the surface of the earth. If the earth were a perfect sphere, the radius of the earth would not depend on location, which we will assume. Using the triangle described above, one can see quite easily the radiation incident on the earth varies inversely with the distance from the sun, with the distance raised to the power of two, since the area of a circle is proportional to the square of the radius of the circle. Radiation then should be proportional to the inverse of the distance squared. I do not think that the speed of the earth affects the amount of sunlight incident on the earth.

Insolation at Perihelion and Aphelion

Due to the precession of the equinoxes and the precession of the perihelion and aphelion, the distance from the earth to the sun by time of year is changing. Presently, for the northern hemisphere, the sun is closest to the earth in the winter and farthest in the summer. Meeus, on p. 274, gives a measure of the distance between the earth and the sun at perihelion and aphelion for the years 1991 to 2010. On January 3, 2010, which is the day that the earth went through perihelion this year, the distance between the center of the sun and the center of the earth was .983290 astronomical units. On July 6, 2010, which is the day that the earth went through aphelion this year, the distance between the sun and the earth was 1.016702 astronomical units. So, for a perfectly elliptical orbit and constant solar flux, there would be a 3.4 percent increase in the distance from perihelion to aphelion and, I would think, a 6.5 percent decrease in incident energy. An astronomical unit is the close to the mean distance between the sun and the earth (see Meeus, p. 411) and equals about 150 million kilometers (see Meeus, p. 407).

Changes in Seasonal Lengths

The lengths of the seasons are changing too. On p. 182, Meeus gives the lengths of the seasons, in the northern hemisphere, from -4000 to 6500, in 500 year increments. From
-1500 to 6500, the spring gets shorter. From -4000 to 3500, the summer gets longer. From -1500 to 6500, the autumn gets longer. From -4000 to 3500,the winter gets shorter. At the present time, the transit of the perihelion occurs a few weeks after the winter solstice. So, winter getting shorter in the northern hemisphere, as the day of the fastest speed of the earth moves toward the midpoint of the winter season, makes sense. Following the same logic, spring in the northern hemisphere is also getting shorter as the perihelion approaches the spring equinox and autumn in the northern hemisphere is getting longer as the perihelion moves away from the fall equinox. For the summer in the northern hemisphere, the time of slowest speed is approaching the midpoint of the summer season, so the summers are getting longer.

Celetial Phenomena and the Heating and Cooling of the Earth

What effect might the celestial phenomena have on the heating and cooling of the earth? Two long term major celestial phenomena that might affect weather are the shifting of the perihelion with respect to the seasons and the slowly decreasing size of the obliquity of the ecliptic. The changes in the eccentricity is of much longer duration. The shifting of the perihelion causes both changes in the distance from the earth to the sun by season and the lengths of the seasons.

At this time in geological time, I would think that the effect of the shifting perihelion on the seasons would be to cool winters and springs and to warm summers and falls in the northern hemisphere, since winters and springs are getting shorter and summers and falls are getting longer. In the southern hemisphere, I would expect shifting seasons to warm winters and springs and to cool summers and falls.

I would expect the effect of the shifting perihelion on the distance of the earth from the sun by time of year to warm winters and springs and cool summers and falls in the northern hemisphere, since the earth is getting less distant in the winters and springs and more distant in the summers and falls. For the southern hemisphere, I would expect the effect to cool winters and springs and warm summers and falls.

Looking next at the decrease in the obliquity of the ecliptic, if the angle the polar axis of the earth makes to the ecliptic increases (the obliquity of the ecliptic decreases), I would expect the to see more solar radiation at the higher latitudes in falls and winters and less in springs and summers for either hemisphere. The obliquity of the ecliptic is decreasing, at about 47 seconds of arc per century at the present (Meeus, p. 408).

Combining the two celestial influences, the precession and the change in the obliquity of the ecliptic, in the northern hemisphere changes in the lengths of seasons would make things cooler in the winters and springs and warmer in the falls and summers; changes in the distance from the sun to the earth by season would make things warmer in the winters and springs and cooler in the summers and falls; and changes in the obliquity would warm things in falls and winters and cool things in the springs and summers. So, for the all of the seasons there are both warming and cooling influences.

For the southern hemisphere, changes in the lengths of seasons would make things warmer in the winters and springs and cooler in the summers and falls; changes in the distance from the sun to the earth by season would make things cooler in the winters and springs and warmer in the summers and falls; and changes in the obliquity would warm things in falls and winters and cool things in the springs and summers. Once again, all of the seasons have influences for both cooling and warming.

The Solar Flux

Actual insolation depends on the solar flux as well as the orbital and rotational influences given above. There is a plot of recent levels of solar radiation incident on the earth’s atmosphere, made possible by satellite measurements, at the Wikipedia entry on the solar cycle. From the plot, the total solar radiation incident at the top of earth’s atmosphere is about 1366 watts per meter squared and varies daily and with the sunspot cycle of eleven years. The plot shows that the solar flux increases with increasing sunspots.

Electromagnetic Radiation

We are interested in the causes of climate change, so I will write a little about the greenhouse effect. What is the greenhouse effect? The greenhouse effect is the natural process by which the earth stays warm enough for life as we know life. The effect is a result of the absorption, and emission of electromagnetic waves, also called photons, by the atmosphere and the earth. Electromagnetic waves are oscillating electric and magnetic fields that propagate through space. Electromagnetic waves are characterized by the wavelengths of the waves. Some familiar types of electromagnetic waves are light, with wavelengths from 0.4 to 0.7 microns, the infrared, with wavelengths from 0.7 to 300 microns, microwaves, with wavelengths from 1 millimeter to 1 meter, FM radio, with wavelengths from 2.78 to 3.43 meters, and AM radio, with wavelengths from 175 to 570 meters. As wavelength increases, the energy of an electromagnetic wave decreases. For the greenhouse effect, wavelengths in the visible and infrared are of interest. (The sources for the wavelengths given above are the Wikipedia entries for visible spectrum, infrared, microwave, FM broadcasting, and AM broadcasting. Note that, for electromagnetic radiation, [wavelength in meters] is the same as [three hundred] divided by [frequency in megahertz], which I know from my amateur radio work.)

The mechanism of the greenhouse effect is the absorption and emission of electromagnetic thermal radiation by molecules in the atmosphere. Basically, the atmosphere passes light, which heats the earth, which emits in the infrared, some of which radiation is absorbed by the atmosphere, which emits infrared in all directions including back to the earth, which slows the cooling of the earth. When an electromagnetic wave of the right wavelength hits a molecule, the wave is absorbed and the energy of the molecule increases. However, a wave will be emitted by the molecule at a later time. Between the atmosphere and the ear and within the atmosphere, there is a constant exchange of energies through the absorption and emission of electromagnetic waves. To repeat, thermal radiation is emitted by the earth and some of the energy is absorbed by the atmosphere, and emitted by the atmosphere in all directions. If the greenhouse gases in the atmosphere increase, the process of cooling is slowed, since more thermal radiation is returned to the earth with the increased absorption and emission of waves by the atmosphere.  Also, the heating of the atmosphere pushes the surface of the atmosphere into a cooler region, which means the amount of energy emitted into space from the earth decreases – by Planck’s Law.

Blackbody Radiation

What is described above is a quick description of what is going on physically with the greenhouse effect. On a deeper level, our understanding of quantum mechanics explains the effect. As was discovered by physicists in the last half of the nineteenth century, all materials radiate thermal electromagnetic radiation. With the development of quantum mechanics in the early part of the twentieth century, scientists understood why. For any material, the spectrum and intensity of the thermal radiation from the material depends on the type of material and the temperature of the material. The spectrum is the product of [the spectrum of the radiation from a blackbody at the given temperature] and [the emissivity of the material at the given temperature]. The emissivity of a material, which varies with wavelength and temperature, can take on values from zero to one, inclusive.

The radiation from a blackbody is described by Planck’s law, which was found empirically by Max Planck at the end of the nineteenth century and published in 1901. Planck’s law gives the power of the emissions of a blackbody at a given temperature per unit area per wavelength or frequency per solid angle for any given electromagnetic wavelength or frequency. From quantum mechanics, we know blackbody radiation is the result of electrons jumping between electron shells within atoms as atoms absorb and emit electromagnetic waves. Two formulas for Planck’s law are given in the Wikipedia entry for Planck’s law, one for frequencies and one for wavelengths. At first glance, the two formulas appear different, since we know frequency is just the product of [the speed of light] and [the inverse of the wavelength]. However, the formulas are densities and the difference between the formulas is a result of changing the variable of integration from frequency to wavelength.

The radiation from the sun is essentially blackbody radiation for material at the temperature of the surface of the sun, which is about 5800 degrees Kelvin. Using the formula for the spectral density by wavelength from Wikipedia and the computer program R, I plotted the spectrum for a blackbody at 5800 K. The radiation is most intense between 0.4 microns and 0.7 microns, the range of visible light, with the peak of the energy spectrum being in the wavelengths of green light, around 0.5 microns. Again using R and the formula for the spectral density by wavelength from Wikipedia, I plotted the spectrums for temperatures of -33 C, -8 C, 17 C, and 42 C, typical temperatures for the materials at the surface of the earth. The heights of the peaks of the power decrease fourfold over the four temperatures, and the wavelengths associated with the peaks vary from around ten to thirteen microns, which wavelengths are in the infrared. So, depending on the emissivity of the material at the surface, the thermal emissions from materials at the temperatures at the surface of the earth peak in the infrared.

Absorptivity, Reflectivity, and Transmissivity

For any material, with respect to electromagnetic waves, incident energy is absorbed, reflected, or transmitted. At any given wavelength and temperature, the sum of the absorptivity, reflectivity, and transmissivity is one. At the surface of the earth, electromagnetic waves in the visible and the infrared are either absorbed or reflected. Of the energy that is reflected by the earth, some is reflected back from the atmosphere, some is absorbed by the atmosphere and radiated in all directions, and some passes through the atmosphere back into space. Of the energy that is absorbed, some of the energy is converted to plant growth and some of the energy in converted to heat (or electricity with the use of solar cells). In the visible and the infrared, no energy is transmitted through the earth.

When solar radiation hits the earth, the rays are absorbed or reflected according to the albedo (reflective nature) of the material at the surface of the earth. The albedo (reflectivity) for some common surfaces are given at http://ice-glaces.ec.gc.ca/WsvPageDsp.cfm?ID=10165&Lang=eng. Looking at some common earth surfaces, the albedo for seawater is between 0.05 and 0.10 in the visible. For the surface of the earth, light that is not reflected is absorbed, so the absorptivity of seawater is from 0.90 to 0.95 in the visible. So, seawater absorbs most of the incident radiation from the sun. For soils the albedo is low in the visible and absorptivity high. Snow has a high albedo in the visible, so reflects much of incident sunlight. (In the infrared, snow has a low albedo and absorbs most of the energy incident.)

We are also interested in the emissivity of materials on the surface of the earth, since radiation emitted at the surface of the earth is absorbed by the atmosphere in the greenhouse effect. At http://www.omega.com/literature/transactions/volume1/emissivityb.html is a list of the emissivity of many materials for given temperatures. At everyday temperatures, polished metals have a low emissivity, metal oxides are quite high, water is quite high, as is snow. Soils are around the middle.

Insolation at the Surface of the Earth

Since the absorption of radiation at the surface heats the earth and determines the spectrum of thermal radiation emitted by the earth, we are interested in insolation at the surface of the earth. For a given solar flux, the amount of energy incident from the sun at the surface of the earth depends on the thickness and content of the atmosphere. The thickness of the atmosphere affects the amount of radiation incident, since light can be both absorbed and reflected in the atmosphere. The thickness depends on both elevation and the angle of the incident radiation. As elevation increases, the distance to the top of the atmosphere decreases. As the angle of incidence gets shallower, the amount of atmosphere solar radiation passes through gets larger. The content of the atmosphere, mainly the amount of water in the atmosphere, also has a large effect on what happens to solar radiation in the atmosphere. Water vapor can absorb sunlight. Water droplets can block and scatter sunlight. Other particles in the atmosphere also affect the transmission of light. Particles, say of soot or volcanic ash, can scatter and absorb sunlight. Energy that does not get through or stay in the atmosphere is either reflected by the atmosphere or absorbed in the atmosphere and then emitted into space.

At sea level, perpendicular to the sun’s rays, on a sunny day, the insolation averages about 1000 watts per square meter, according to the Wikipedia entry for insolation. After passing through the atmosphere, the density of the energy incident on the ground depends on the angle of incidence, which varies with sky cover, latitude, season, and time of day. The angle tends to be shallower at more extreme latitudes, the winter season, and away from true noon. Shallower angles mean less energy incident.

Gases in the Atmosphere and the Greenhouse Effect

From the Wikipedia entry for the Earth’s atmosphere, the main components of the atmosphere, for dry air, are nitrogen (about 78 percent), oxygen (about 21 percent), and argon (about 1 percent) by volume. On average, the about 1 percent of the atmosphere is water. Carbon dioxide makes up about 0.04 percent of the atmosphere. J. L. Schnoor, in Environmental Modeling: Fate and Transport of Pollutants in Water, Air, and Soil, published in 1996 by Wiley, in Chapter 11, gives scientific information about climate change and the greenhouse effect and the atmosphere. Water vapor is the main greenhouse gas. Schnoor states, on p. 612, that water vapor and preindustrial carbon dioxide, are responsible for about 98 percent of the heating resulting from the thermal radiation of the earth. We have life on earth because of the heating.

As stated above, the greenhouse effect is the result of the atmospheric absorption of the radiation emitted by the earth. Why does the atmosphere absorb in the infrared? For absorption, the amount of incident electromagnetic energy absorbed by a material depends on the absorptivity of the material. According to my excellent freshman year physics text, Basic Physics, by Kenneth Ford, published in 1968, by Blaisdell, on p. 596, molecular vibration and rotation is responsible for the emission of electromagnetic waves in the infrared, and good emitters are good absorbers. (Note, molecular bonds are the result of atoms sharing electrons in the outer shells of the atoms. The oscillation of the outermost electron in atoms is responsible for emissions in the visible.)

The earth emits in the infrared and, since greenhouse gases are molecules, greenhouse gases in the atmosphere absorb the earth’s radiation. There are greenhouse gases other than carbon dioxide and water, many of which absorb more radiation per molecule. In Schnoor, on p. 608, the concentrations in the atmosphere, as well as other information such as the relative absorptivity, of several greenhouse gases are given.

Some physical processes that both warm and cool the earth are described in Schnoor. Water droplets, in the visible, block energy from the sun, which has a cooling effect. Water vapour does transmits electromagnetic radiation in the visible. In the infrared, water vapor is a greenhouse gas and slows the cooling of the earth. One of the effects of a warmer earth is an increase in the evaporation of water into the atmosphere, which leads to increased clouds, which blocks sunlight, which cools the earth (Schnoor, p. 614). Also, because of sulfate aerosols generated by the burning of coal, clouds are brighter and reflect more sunlight back into space, which cools the earth. But the gaseous products of coal combustion are greenhouse gases and warm the earth (Schnoor, p. 614). Schnoor writes that weather patterns, such as El Nino, are important, too (Schnoor, p. 615).

Comments

So, do we need to worry about climate change? I think so. The loading of the humanly generated greenhouse gases is small, but steady, even though water vapor changes greatly with location and season. In statistics, we are trained to look at trends in data and to separate noise from signal. However, for observational data, we are trained to not assign cause based on correlation. When I took my methods course at Iowa State from Paul Hinz back in 1983, Dr. Hinz gave the example of cancer and smoking. While all the world wanted statisticians to say because there was a correlation between smoking and cancer, smoking causes cancer, we could not say the reason was causal. We had to wait until bio-chemists found a physical cause. Then, the observed correlation supported the physical theory. Dr. Hinz, in support of our hesitancy, gave the example of storks and babies in Copenhagen. There is a strong correlation between the number of storks and the number of babies in Copenhagen. Do storks bring babies? The physicists who model the greenhouse effect have reason to think we have a problem based on known physical theory. As statisticians, we can only say whether the data does or does not support the theory. As to the Guttorp article, I suspect the effect of the precession’s is too small and the noise too large to pull out a change in the climate over the short period of one hundred twenty years, but I am not familiar with the technique used. I do not know how sensitive such a technique would be.

Three Celestial Coordinate Systems

As an aside, I will describe the coordinate systems used in describing celestial phenomena. Three coordinate systems are commonly used to describe the position of any point of interest in the heavens with respect to the earth. The three systems are with respect to the ecliptic, the earth’s equator, and the horizon. In each system, there are two variables. The first variable measures the angle between some reference point on the circle of reference (a point on the ecliptic, equator, or horizon) and the intersection of the shortest line from the celestial point of interest (such as the sun or a star) to the circle of reference (perpendicular). The second variable measures the elevation, either above or below, of the point of interest from the circle of reference.

Ecliptic

The first system is longitude and latitude. Longitude and latitude are measured with respect to the ecliptic, as seen from the earth. Longitude is measured in degrees of arc east of the point on the ecliptic which the sun occupies at the spring equinox and is measured from spring equinox to spring equinox.  The longitude of the sun is the point on the ecliptic behind the sun, looking out at the ecliptic from the earth, since the sun is essentially on the ecliptic. Latitude is the elevation north or south of the ecliptic and is always essentially zero for the sun.

Equator

The second coordinate system is right ascension and declination. Right ascension and declination are measured with respect to the equator. Right ascension is a measure of the angle around the equator from the point on the equator associated with the spring equinox. Right ascension is measured in sidereal time and increases as the point of interest is more east than the point of the spring equinox. The time is called sidereal time since the time is measured against the stars rather than the sun. A sidereal day is slightly shorter than a solar day, there being 366 sidereal days in a year, as opposed to 365 solar days, or 367 sidereal and 366 solar days in a leap year. (Another equivalent measure of angular distance around the equator is the hour angle, which is measured in degrees and is measured from an arbitrary meridian, either east or west. The relative values of the hour angle and the right ascension are the same if the right ascension is multiplied by 15 to convert hours to degrees.) The declination is the angle of the object of interest above or below the equator, as seen from the equator.

Horizon

The third coordinate system is azimuth and altitude. Azimuth and altitude are measured from the horizon. While the starting point and direction of azimuth varies by reference, Wikipedia, in the entry for azimuth, gives azimuth as being the angle along the horizon to the point of interest, measured east from true north. The altitude is the angle of the point of interest, above or below the horizon.

 

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Fitting Sunspot Numbers and CO2 Concentrations to Temperature Anomalies

In this post, I give a simple model for temperature anomaly for data from 1850 to 2013.  The temperature anomaly data is the land and ocean series from Berkeley Earth; the sunspot data is the old monthly averages of daily sunspot numbers from the Royal Observatory in Belgium; and the CO2 data come from two series of the NOAA ESRL, one series – using ice cores – is of yearly CO2 levels from 1850 to 1958 and the other of monthly levels from the Mauna Loa Observatory in Hawaii from March of 1958 to 2013.  I created a time series out of the two CO2 series by adding the seasonal components to each yearly point in the earlier series, where the seasonal components are found using the later series.  I then filtered the combined series with a linear filter 129 months wide.  I also tried a filter width of 12 months, but the results were essentially the same.  I smoothed the CO2 data because the seasonal variation for the CO2 data is local to Hawaii and the other data is global..  The first sunspot observation is associated with the fourth temperature anomaly observation and the fourth CO2 observation, since I have observed that temperature anomaly is lagged three months behind sunspot number, in terms of correlation.

Fit of Temperature Anomaly to SDunspot numbers and CO2 Concentrations 1850 to 2013

Together, the two variables, sunspots and CO2 account for about 70% of the variation in temperature anomaly, where variation is measured by sums of squares. The regression indicates an increase of 0.045 degrees C for every 100 more sunspots and an increase of 0.92 degree C for every 100 ppm increase in the level of CO2.  The fit does agree with the last ten years of the hiatus in temperature growth (Actually it does not.  I thought it did.)  .

While the fit is not perfect to the linear regression assumptions, the fit is not too bad.  The diagnostic plots from R are given below.

Diagnostic Plots

This is a very simple model, and does not account for other greenhouse gases or other component of the atmosphere, or the composition of the different places on the earth. That said, temperature is a measure of energy, they are linearly related.  So using sunspots which, for the data we have, are a good proxy for energy density from the sun, makes sense.  I do not know enough physics  to say if the energy returned to the earth by greenhouse gases is linearly related to greenhouse gas concentration, but the plots indicate probably yes.

Corrected Plots for Filtered Data and Phases for the 1850 to 2015 Data and the 1750 to 2015 Data

When I filtered the sunspot and temperature data, I used a bandwidth of 257 months rather than 129 months.  The corrected plots are below.

cc.ra.ss.ta.plot129 filtered correctly

cc.corrected.phase.centered.ss.ta

Once again, the sunspot data is the mean of the daily sunspot numbers over each month from the Royal Observatory in Belgium and the temperature anomaly data is from Berkeley Earth and is the land and ocean series.

Here are the plots for the new Belgian series and the Berkeley Earth temperature anomaly series for land data, which goes back to 1750.

cc.filt.new.corrected


cc.phase.centered.new.corrected.ss.ta

The contrast between the two sets of series may be because the temperature anomaly data in the first set is land and ocean data, while the temperature data in the second – longer – set is just land data.  The radiation from the ocean is much more uniform than the radiation from  land.  Also, I do not think we are certain that the observed fluctuation in the solar radiation which has been seen to follow the sunspot cycle during the time that we have had satellite measurements – since 1978 I believe – actually followed sunspot numbers closely over the past.