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{u|si(u)>si(ui)}

si(u)dU (8)

where dU is the elementary surface about u on , and testing the distribution of the {ti} against a

uniform probability density over the segment [0, 1].

This uniformization procedure simply amounts, for each data ui, to identify on the iso-probability

line of the pdf si(u) on which the data ui lie (note that this pdf depends on the location the data

ui come from, which is why ani index is being added to si(u)), and to assign to ti the value of the

probability that ui could have been lying on a higher iso-probability line on . Then, if the GGP

model is compatible with the data set (hence, if the ui are compatible with the si(u)), the {ti} should

be compatible with a uniform distribution on [0, 1].

Whether this is indeed the case can finally be assessed with the help of various standard tests,

(in paper I we used the Kolmogorov-Smirnov (KS) and the 2 tests). Note that this uniformization

6

procedure is very general and does not rely on any properties of the Angular Gaussian distribution. It

can be applied in more general situations.

Taking measurement error into account

A main issue that paper I did not address is that of data measurement errors. The tests we just

described are appropriate only in the event these errors are negligible. Unfortunately this is far from

being the case and we now need to take this into account. In the case of directional data, these

measurement errors are classically described in terms of a Fisher distribution characterized by a so-

called concentration (or precision) parameter K. Within this Fisherian frame-work, stating that the

observed direction s reflects the actual direction u on with an error characterized by a concentration

parameter K, amounts to state that s is the result of a random drawing from a Fisher distribution

centered on u with pdf:

kK(u, s) =K

2(

eK eK)eK(u,s). (9)

In spherical coordinates and when centered on u0 = ( = 0, = 0), this takes the more usual form:

kK(u0, s) =KeK cos

2(

eK eK) . (10)

This means that in order to test a given GGP model against a given data set {ui} with associated

errors characterized by {Ki}, we now need to test whether each ui can be considered the result of a

realization s of the GGP process (distributed over with the pdf si), shifted to ui as a result of an

error with a Fisher distribution kKi(s,u) centered on s and a concentration parameter Ki.

The infinitesimal probability that the GGP process first produces a direction within the elementary

surface dS about s, and that the error next independently shifts this direction to within the elementary

surface dU about ui is:

si(s)dS kKi(s,u)dU. (11)

Then, integrating over all possible intermediate s directions leads to the probability pi(u)dU of

7

finding ui within dU about u, where:

pi(u) =

si(s)kKi(s,u)dS (12)

is the new pdf against which ui should be tested. Thus, taking data errors into account only amounts

to use pi(u) in place of si(u) in Eq.(8). Tests can then again be used, either on a site-by-site basis or,

more interestingly, on a regional and global scale after using the uniformization procedure, which now

becomes:

ti = P{u|pi(u) > pi(ui)} =

{u|pi(u)>pi(ui)}

pi(u)dU (13)

Practical implementation

In practice, testing a given GGP model against a given directional data set thus involves four successive

steps: 1) for each data ui, to compute the error-free pdf si(u) predicted by the GGP model at the site

where ui was collected (the analytical form of which is given by (6)); 2) to compute the error-included

pdf pi(u) through the convolution (12); 3) to produce the uniformized data ti with the help of (13); 4)

to test the uniformized data set {ti} against a uniform distribution over [0, 1].

Unfortunately, no exact analytical solutions of (12) and (13) are known to us. In principle, this

is not too much of a problem, since both (12) and (13) can be computed numerically. In practice

however, the numerical implementation of formula (12) requires quite some computational time. This

drawback can be considered negligible if we simply test a single GGP model against a small database,

or if all data ui coming from the same site share the same data error (in which case pi(u) only needs to

be computed once for each of the typically thirty something sites). But if pi(u) needs to be computed

for each data of the data set (i.e. typically a thousand times), this computational time can become

very long (on the order of a few days on a current PC). For that kind of more realistic situations (and

for the forward testing method we propose here to possibly become of any use for future much more

computer-intensive inverse searches of best models), faster algorithms are clearly desirable. This

prompted us to further look for an approximate, accurate enough, analytic solution to the convolution

(12).

As is described in the appendix, one such approximation can indeed be found. For any given data

8

ui, this consists in approximating pi(u) by the Angular Gaussian distribution defined by (6) and (7),

where m is then the mean field value predicted by the GGP model to be tested at the site associated

with the data ui (i.e., predicted by gi(x) as defined by (4) for the site associated with the data ui),

and is(

Cov(V , V ) + |m|2

K I)1

, where Cov(V , V ) is the covariance matrix associated with gi(x)

and K(= Ki) is the concentration parameter characterizing the Fisherian error associated with ui.

In what follows, whenever we use this approximation of pi(u) to implement the tests, we will refer to

the approximate method, and whenever a direct numerical implementation of the exact convolution

(12) is used, we will refer to the exact method. It turns out that, as we shall see, the approximate

method is most often just as good as the exact method, and about 100 times faster to run.

Models, data set, and statistical tools used in this study

To illustrate the power of the method we propose, several published GGP models have been tested.

We will refer to these as:

CP model, which is the preferred model of [Constable & Parker 1988];

QC model, C1 (preferred) model of [Quidelleur & Courtillot 1996];

CJ model, which is the CJ98 model proposed by [Constable & Johnson 1999];

JC model, which is the CJ98.nz model also proposed by [Constable & Johnson 1999];

TK model, which is the TK03.GAD model recently proposed by [Tauxe & Kent 2004] and further

discussed in [Tauxe 2005];

HK model, which is the final model for the normal polarity of [Hatakeyama & Kono 2002].

Models CP, QC, CJ, JC and TK share many characteristics. They are defined by simple axisym-

metric mean models for which only E(g01) and E(g02) can take non-zero values, and purely diagonal

covariance matrices with Cov(gmn , gmn ) = Cov(h

mn , h

mn ) = (

mn )

2, except in the case of model JC, which

assumes different values for Cov(g12 , g12) = ((g

12))

2 and Cov(h12, h12) = ((h

12))

2. In all but one case,

for n > 3, the mn are further assumed to be independent of m and defined by mn = n where

n = (c/a)n/((n + 1)(2n + 1))1/2. The only exception is model TK which distinguishes mn = n for

(nm) even from mn = n for (nm) odd. Table 1 gives the values of the relevant parameters for

9

each of those five models. Finally, model HK differs from the other models because of a more elaborate

mean field, defined up to degree and order 4. But its covariance matrices are otherwise defined in much

the same way. A full description of this model can be found in Table 2 of [Hatakeyama & Kono 2002].

All those models have been constructed in the hope that they would properly describe the sta-

tistical properties of the paleomagnetic field at times of normal polarity over the past five mil-

lion years. In particular, they have been constructed with the help of databases including a large

number (if not a majority) of directional data corresponding to the current Bruhnes chron (e.g.

[Quidelleur et al 1994, Johnson & Constable 1996]). They are therefore good candidates for an ex-

ample test against a well-controlled data set covering the Bruhnes chron and corresponding to volcanic

directional data acquired at various sites distributed worldwide. Ideally, such a test data set would

have to be built by extracting data from the most recent and permanently updated IAGA paleomag-

netic reference database for paleofield direction available at the National Geophysical Data Center

(http://www.ngdc.noaa.gov/seg/geomag/paleo.shtml), and by relying on a set of fine-tuned criteria

agreeable to the community. This however is not a trivial matter since, as noted by one of the re-

viewers, there currently is no general agreement among investigators on what comprises a satisfactory

database for this kind of study. Such a substantial endeavour, which we plan to carry on at a later

stage, is therefore clearly beyond the scope of the present paper, which only intends to introduce, test,

and illustrate a new methodology. For such a purpose, we felt that a simpler, more readily available

data set (and one that at least already went through some type of selection processes relevant to the

present study), would be appropriate enough.

We therefore decided to use a test data set extracted from the [Quidelleur et al 1994] database,

originally used by [Quidelleur & Courtillot 1996] to build their QC model last updated in January 1998,

and currently available at http://www.ipgp.jussieu.fr/rech/paleomag/var-secu/. We note however,

even before proceeding further, that because of this choice, a close fit of the QC model to our test

data set can be anticipated, so that test results reported here will necessarily not test the QC model

as stringently as the other models. This test data set consists of a total of 990 independent estimates

of the local direction of the paleomagnetic field at 36 sites (Figure 1, Table 2). Each site is located

with the help of its latitude and longitude. At each site, each estimate is based on the direction of the

resultant vector R of n (> 3) volcanic samples (unit vectors) and is given in the form of a declination

10

D and an inclination I together with both n and the norm R = |R| of the resultant vector. Although

R is usually not published as such in the original papers, it can be accurately recomputed from the

published material. This then makes it possible to present all the data in an homogeneous and complete

form in the database. Indeed, and as is well known (e.g. [McElhinny & McFadden 2000]), an estimate

of the local direction of the paleomagnetic field is completely characterized (with is Fisherian error) as

soon as the four parameters D, I, n and R are given.

For each estimate, the corresponding vector ui could then be located on the unit sphere , and an

estimate of the concentration parameter Ki defining the Fisherian error computed [with the help of

K = R (n1)(nR) ; note that this is not the concentration parameter k =(n1)(nR) classically published with

the data, which characterizes the dispersion of the n individual samples about their average direction,

but the concentration parameter characterizing the error on ui]. Starting from ui and Ki, and for each

GGP model to be tested, a uniformized value ti could finally be computed with the help of Eq. (13)

to produce the final uniformized data set {ti}.

To test this final data set against a uniform distribution over [0, 1], we relied on two different

well-known tests: the Kolmogorov-Smirnov test (KS-test) and the Anderson-Darling test (AD-test).

These two tests rely on the fact that, if a given uniformized data set {ti} (with i = 1 to N) is indeed

compatible with a uniform distribution over [0, 1], its empirical cumulative distribution function (cdf)

FN (x) should fluctuate within predictable limits about the theoretical cdf value F (x) = x. The null

hypothesis should then be rejected if the empirical cdf FN (x) either departs too much from x, or

remains to close to it (the latter revealing the presence of suspicious regularities within the {ti}). The

KS-test and AD-test differ in the measure chosen to assess how distant FN (x) is from x over [0, 1]:

the KS-test (as defined by e.g. [Press et al 1996]) uses the maximum value MN of |FN (x) x|

over [0, 1], and is therefore most sensitive to departures of the {ti} from a uniform distribution

towards the middle of the segment [0, 1];

the AD-test (as defined by e.g. [Marsaglia & Marsaglia 2004]) uses the integral quantity IN =

N1

0

(FN (x)x)2

x(1x) dx. Because of the weight [x(1 x)]1

, it is much more sensitive to the behaviour

of {ti} at both extremes of the segment [0, 1].

11

The KS and AD tests are both excellent tests for small samples and can also be used for large

samples. Neither require any additional parameter and both can therefore be considered as completely

objective. Also, they complement each other very well. In practice, for each of the uniformized

data set {ti} we had to test, we therefore computed the values of MN and IN , together with (from

known softwares, e.g. [Press et al 1996] and [Marsaglia & Marsaglia 2004]) the probabilities P (MN )

and P (IN ) for the null hypothesis to have possibly produced such large, or even larger, values for

respectively MN and IN . Whenever P (MN ) and P (IN ) were found to take values very close to 0

(typically 0.05 or less) or to 1 (typically 0.95 or above), the null hypothesis had to be rejected and the

GGP model under consideration had to be considered incompatible with the data-set at this level of

confidence.

Validation and discriminating power of the method

Before actually implementing the testing method we propose, we ran a number of instructive validating

tests. For that purpose, various artificial data sets have been produced in the following way. Starting

from a given GGP model, we first produced random error-free directional data values. Exactly the

same number of data values were produced for each of the 36 sites, as available in the (real) test data

set (thus amounting to a total of 990 independent artificial values). This could easily be done by

just randomly drawing 990 independent sets of Gauss coefficients (gml , hml ) from a multidimensional

gaussian generator with mean values and covariances matrices chosen as specified by the GGP model

under consideration, and computing the predicted values at each site. A random directional error was

next added to each of those 990 error-free directional data values. Because, as shown in the appendix, a

Fisher distribution is extremely close to an Angular Gaussian distribution (for realistic Fisherian errors

such as those we deal with here), this was actually done by introducing a random Angular Gaussian

error (an easier procedure to control than that involved in producing random Fisherian errors). Using

this procedure and starting from various specified GGP models, a number of artificial directional data

sets affected by specified errors could be produced, which we used to validate our method.

We first validated the exact method and checked that for artificial data produced from any GGP

models and perturbed by some known error, this method would always conclude to the compatibility

of the data set with the starting GGP model, provided the data are assumed to be affected by the

12

correct amount of error. This first series of tests not only allowed us to validate the method. It

also allowed us to check our software and the level of numerical accuracy needed to produce accurate

results. It turns out that from a numerical point of view, the most sensitive step of the entire method

is the numerical implementation of the uniformization procedure (i.e. the computation of Eq. (13)).

This indeed requires the numerical computation of pi(u) on a grid over the unit sphere . The most

demanding computations, in terms of grid size, happen to be those corresponding to pi(u) predicting

the largest dispersions of the ui on the unit sphere . In that case, when the grid is not tight enough,

numerical errors happen to mainly affect the computation of the ti lying close to 0 or 1. This in turn

mainly affects the results of the AD-test which produces too large values of IN , and therefore too small

values of P (IN ). As a result, the AD-test tends to more easily produce negative results than it should.

For the purpose of the present paper, we used a grid corresponding to a decent compromise, tight

enough for the KS-test to always lead to an accurate result, but not too tight to make it possible to

run all tests in a reasonable amount of time. This had the drawback that, for the reasons just outlined,

the AD-test produced slightly erroneous results when testing the CJ and JC models. For those two

models, we therefore concluded that only the KS-test could be considered reliable. For all other models

(CP, QC , TK and HK), both the KS-test and the AD-test were otherwise found to be reliable. Those

conclusions have been reached when testing artificial data affected by errors characterised by 95 of

up to 12.5. [Note also that to produce those artificial data sets and run all the tests reported in this

study, all GGP models have been used only up to degree and order 7, since, as we shall later see, higher

degrees contribute negligibly to the final results for all the GGP models we tested].

We next proceeded to compare the results of the approximate method to those of the exact

method, using exactly the same artificial data sets in both cases. Again and as anticipated (see

appendix), significant disagreements between the two methods were found to arise only when pi(u)

predicts large dispersions of the ui on the unit sphere . In practice, the approximate method was

found to lead to accurate enough results for all the models we tested (testing artificial data with all

levels of errors up to 95 = 12.5), except when testing models CJ and JC against data affected by

errors with 95 greater than 7.5. Table 3 summarises the domain of validity of each type of test as

established from all those preliminary tests.

We then went on to test the ability of the method to properly discriminate the source GGP model

13

and the level of error affecting the data. Because, as we shall soon see, the real data turn out to be

compatible only with model QC (from the point of view of the tests described in the present paper),

and since the average 95 for the 990 data in the real database is of 4.7, this test was carried out

on an artificial data set produced from model QC and affected by errors of 95 = 4.7. This artificial

data set was used to test all GGP models, assuming various possible levels of errors (0, 2.5, 5, 7.5,

10, 12.5). An additional test was also run to check that the method would properly conclude that

this data set is compatible with the starting model QC when assuming an error of 95 = 4.7. Since

both the exact and approximate methods provide reliable results for errors of that magnitude, both

methods have been used with the KS-test and AD-test (except for the the CJ and JC models for which

only the KS test was used, for the reasons outlined above). Table 4 reports the results of those tests.

[Note that Table 4 also provides a good illustration of the accuracy of the results of the approximate

method, when compared to the results of the exact method].

This table shows that, as expected, the artificial data set is compatible with the starting model

(QC model) provided the correct level of error affecting the data is being assumed (95 = 4.7). It

also shows that by contrast, making a wrong a priori assumption with respect to the errors affecting

the data can lead to a negative result. A minimum error of 95 = 2.5 and a maximum error of

95 = 7.5 is indeed required for the test with model QC to produce positive results (with no rejection

at a 95% level of confidence by either of the KS or AD tests). Finally, Table 4 shows that the method

also appears to be able to discriminate the various GGP models. The data set produced by QC with

95 = 4.7 cannot be attributed to a CP model, unless (but marginally so) one wrongly assumes

95 = 10. It can no more be attributed to a CJ, JC or HK model, unless one again wrongly assumes

95 = 12.5. And it can hardly be attributed to a TK model, whatever the value assumed for 95.

Altogether those results thus show that the tests we introduced are sensitive to both the choice of the

model to be tested and the level of error assumed to affect the data. In particular they show that if

we have a reasonable knowledge of the error affecting the data, the method applied to the type of data

set we deal with here (i.e. 990 data distributed over the 36 sites of Figure 1) makes it possible to fully

discriminate the models. Indeed Table 4 shows that if we know a priori that the error affecting the

data is somewhere between 2.5 and 7.5, the method can detect that all of the CP, CJ, JC, HK and

TK models are incompatible with the data set produced from QC model with 95 = 4.7, at more than

14

a 95% level of confidence. Those encouraging results show that the method proposed in the present

paper has enough discriminating power to decide which, if any, of the six GGP models can best explain

the test data set we have assembled.

Testing the real data

Having validated the method, and assessed its limits and discriminating power, we then went on to

test the various GGP models against the real test data set.

Global tests

In a first series of tests, we did not rely on the actual errors provided with the data. Rather, for each

model tested, we successively assumed values of 95 = 0, 5, 7.5, 10, and 12.5. This made it

possible to again run both the exact and approximate methods in parallel for further validation of

the approximate method. It also provided us with a quick way of assessing which GGP model could

possibly be considered compatible with the test data set. Table 5 shows the results of those tests.

Interestingly, we first note some common features with the results of the previous test (recall Table

4). Model QC is rejected at more than a 95% confidence level for 95 = 0, 10, and 12.5, but not

for 95 = 5 and 7.5. Again we see that model CP shows a tendency to be compatible with the data

if one assumes 95 = 10, though this possibility is now clearly less likely. Finally we see that only by

assuming 95 = 12.5 can models CJ, JC, HK possibly be consistent with the data, while model TK is

always rejected at a 99% confidence level. Based on these tests, we therefore reach the conclusion that

if we were to ignore the actual level of error affecting the data in the test data set, all models would

be rejected at more than a 95% confidence level except model QC when assuming 95 = 5 or 7.5, or

model CJ and JC when assuming 95 = 12.5.

The key point, now, is that we do have some knowledge of the errors affecting the data in the

test data set. We already noted that on average, this error is of 95 = 4.7. This value is clearly

incompatible with the 95 = 12.5 value needed for models CJ and JC to possibly be compatible with

the test data set. But it seems compatible with the results for model QC.

To finally assess the issue, we eventually ran the test for all six models, no longer assuming the

same a priori error for all the data of the test data set, but the true error associated (i.e. published)

15

with each of the 990 data. Since, as already noted, this requires considerably more computational

time, only the approximate method was used for this comparative assessment [Note that all previous

tests have anyway shown that for errors of the order of 5, the approximate method always leads to

accurate enough results]. Table 6 gives the results of this final test, and Figure 2 shows the cdfs of

the corresponding uniformized data. Those results clearly show that only model QC is not rejected

by the test data set. All other models can be rejected at more than a 99% confidence level, TK being

the second, CP the third, HK the fourth, CJ the fifth and JC the sixth best models in terms of both

the MN and IN measures (note that when both quantities are available in Table 6, MN and IN lead

to the same relative ranking).

As an ultimate check and only for model QC, we finally duplicated this last test using the exact

method. The result of this test (also shown in Table 6) confirms the previous results. Moreover, the

difference found between those two results for model QC clearly appears to be much smaller than

the differences among the results of the approximate method for all models. We therefore feel quite

confident that these results can be trusted.

It is interesting to take advantage of model QC, the only model not rejected by the test data set

so far, to also quickly address the issue of the number of Gauss coefficients that should be taken into

account to carry on the tests. As stated earlier, we used all Gauss coefficients up to degree and order

7. But all models, including QC, assume non zero variances for all degrees up to infinity. However,

the higher the degree n, the weaker the contribution of the coefficients to the local pdfs pi(u), and

thus to the final result. Table 7 illustrates this point and shows that the calculation is indeed already

converged when taking into account all degrees up to 7. Note that this also shows that the test (applied

to the test data set used in the present study), is therefore hardly sensitive to Gauss coefficients of the

GGP models with degrees above seven.

Local tests

It is also important to keep i

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