offers an application of incidence geometry to historical biogeography by defining collection localities as points, tracks as lines and generalized tracks as planes.

MultiModel Selection and The Croizat Method asInformation Theoretic Panbiogeography

Looks like there really is a non-partisan == non-"political" solution. It was hard to understand how to realize Croizat's sour and forlorn attitude relative to those that claimed him on 'the' "lunatic fringe" or asserted that without an adequate and reliable null test that his claimed output of his "methodology" - the tracks etc themselves were not really statistical but rather appeared as subjetive artifacts of his a priori approach. The rise of the information theoretic approach ( in wildlife management situations) that differentiates itself from Bayesain techniques seems to provide a resolution to the difference of opinion between say Patterson and Croizat etc. Recent panbiogeographic consideration of congruence brought on by the Martitrack algorithm may be addressed by multiple model selection of generalized tracks.

Thus I will conclude that Croizat's contributions have been eschewed not for simple sociological differences (continental vs anglophonic philosophy for instance("Croizat’s major works represented thedevelopment of a long line of evolutionary thought originating in continental Europe."Carmine Colacino and John Grehan)) but rather because the difference between frequentists and Bayesians do not present the full gamit ("versimultude"(of Kant)) of statistics applied to evolution nor that it was due to Croizat's use of paraphyla etc. It is not possible to logically separate empirically an an original event from full model truth even thought these are different.

The difference in the view of say Patterson and Croizat might be thought of as one where Croizat was referring to f with his own thoughts on the parameters while Patterson realized that science really uses g (given the data). That conversation should have been about if Croizat's records and data used enable him to express a model panbiogeogrpahy g close to f and to say what quantitative statistical support there was for that model of tracks and nodes etc etc.

The development of an anti-node notion means that this functionality could be rigoursly sought and here I try to show to set up generalized track models a priori of various topological Croizat method ontologies that permit model averaging and thus avoid the need to perfectly specify a definition of congruence but yet still provide for changing knoweledge of/in generalized tracks and the Crtoizat method as new data sets are added to the synthesis of space, time and form. Further the information theoretic statisticas are enriched as the the panbiogeographic progression of acceptable models informs larger and larger sets of phyolgenetic tress for use systematically (given a common historical geology) and thus the simple difference between TRUTH and model reality is actually more diverse than is currently be practice with Mark-Recapture Analyses.

Here we show how panbiogeography may develop from its meger
beginnings into a full blown historical biogeography that evolves our
understanding of space time and form by casting it as a form information
theoretic statistics of multiple model inference set creation and recreation of my petrsonal model evolution

Eurycea--> Salamanders--> & Fish--> metacommunities with others

This progression is used to addresses both the information - theorecitc concepetsw relvant but also how other panbiogeogrpher's work might be merged into a whole and tested against further molecular and systematic multiple modeling.

Multimodel Inference in Panbiogeography

There is a suggestion to extend the program of AIC type model
construction beyond mark and recapture into a “void” around phylogeography.

Information theoretic methods are finding
increasing use where designed experiments are not possible. (Model selection in
ecology and evolution JJohnson K Omland Trends in Ecology and Evolution Vol.19
No.2 Feb 2004)

Millington and Perry have presented the general case for MMI
in bioeography. There is no doubt that this form of inference
panbiogeographically is better done with AIC rather than Bayesian wise since we do
not know about the origin of life location and the discipline is going to begin
like it or not from Earth’s data first, no matter what. The truth that Croizat enunciated was difficult for others to realize was new. Now we can understand other non-panbiogeographic biogeographic approaches as some kind of model with a loss of this information from the Croizat Method. The true model however is never going to be in the set so constructing bayesian priors will always be callable into question.

Millington and Perry (2011)
Multi-model inference in biogeography. Geography Compass 5(7) 448–463
10.1111/j.1749-8198.2011.00433.x

Can the Panbiogeographic generalized track provide
geographic closure in mark-capture syntheses? The differnces of opinions over generalized track construction can simply be cognized as different models once one relates the individual tracks to parameterization of the general model track. This way one can avoid my pure math approach ( of axiomatics based on using some kind of logic ( why I called it "terminological" panbiogeography like that below)

that Panbiogeography is a "progressive research program". So many people tried to argue that it can not be one but none of those criticisms take the perspective presented by Anderson et al. It only remains to establish a "definitive" panbiogeographic data set. If I can get my Urodelan application to work within Head's primate structure such records might b e able to become such. "Terms colinear" become expressed as linear combinations of component parameters and do not depend to specific use of some of Bertrand Russell's concepts(Strecke) that I used.

+++++++++++++++++++++++++++++++++++++++

Here we show how to
build those multiple generalized track models with this new methodology by using different
number of individual track, node and mass parameters.

We develop an AIC criterion for
model selection and demonstrate the new methodology to predicting distributions
not in the original data set as well as projecting sets of compatible taxonomic trees for use in taxonomy.

Constructiing a generalized track with more parameters results in a "better fit"(more parameters per area) to the biogeographic record data. Here one might model the apparent geographic trend below with 4 or 13 parameters (Simply select a covering size and place them over locations where all individual tracks are represented and distribute them to maximize the total number of collection records within all of the covergings.

Weston wrote, "Croizat regarded generalised tracks as having a statistical basis, their degree of justification being directly related to the number of individual tracks consistent with them."

Here show a use of the Akaike Information Criterion to develop "degrees of justification" for a particular generalised track model as well via model averaging for cross model average of the track parts (which includes individual tracks but may also have nodes, anti-nodes, masses and different algebraic cosets associated with a particular baseline directed).

Model selection leads
a generalized track statements and individual parameter model averaging can
provide statistical means to predict missing data in other phyla not used to
create the generalized track. This is really exciting!!

In this sense tracks are eminently statisticaland meaningful.(Some of the discrensipciy about Croizat
tracks being statistical or not stem from the differenece of the null vs
multiple model significna e in statstiics). Thus to be logically valid the "explicit statistical basis" of Weston need not have a model wich is "the truth" as in some Bayesian applications which might be tried here in expanding from Pag's use of minimal spanning trees for constructing individual tracks. The statistics will be developed as different probabilites are assoicated with different MST indiviudal tracks, nodes masses and coset basline parts sum to one both for the historical biogeography as well as for the total taxonomic or clade node information derivable.

A history of Panbiogeographic notions of the node are
related to this approach. Nicely, Heads (Molecular Panbiogeography of the Tropics (page 2) wrote "In contrast, geologists (Chamberlin, 1890, repreinted 1965) and now molecular biologists (Hickerson et.al., 2010) cite the method of "multiple working hypotheses," to explain a given phenomenon. Accepting a single interpretation as definitive can be counter productive" and Anderson (Model Based Inference in the Life Sciences (page1) opens with Chamberlin and "science philosophy" based on multiple working hypotheses.

Robin Craw introduced the notionof null hypothesis testing of tracks with
Clique Compatibility.

solution to the congruence problem is presented and a route to combination with phlogenetic
systematic multi-model truthing is hinted at. We will be able to use the mutliple hypotheses of molecular panbiogeography and molecular biology in terms of track parts thus fusing the nodes of panbiogeography and cladistics as Nelson realized was happening panbiogeographically.

Through the creation of multiple models it becomes possible
to lessen the apparent subjectivity in the intial description of the
parameters. I will account this in the "evolution " of models which proceeds wholly within panbiogeography and does not depend on systematics at all.

This may appear counter
intuitive since the multiple model can only approximate truth from the models
it contains. The missing true model however must be phylogentically informative overall. That model just is never the one the panbiogeographer works with.

While the true model need
not be in the set evolution as a fact specifies that there is some true model-
How well the forces implied by the multiple models fits the substance of the
truth however canbe bettered by
improved inference given the initial set.

For intance futher tests of Head's molecular panbiogeography of the tropics will result in global systematic changes that may or may not be able to evolve as he proceeds to do the same for Australia.

So if the intial set is well ordered beyond and organon (threshold) it
may work recursively and objectgively to attain the true substance that gave
rise to the the various sets of forces derivable.A very poor intitally subjective set however
will never be able to do this.

It is
suggested that with respect to salamander speciation in Eastern USA that the
initial set example provided is suffienenht to assomptoicically approach the
true salamanders evolution over increased research time.

As better and better
approaches to reality are constructed (through mutimodel inference to
systematic tree alternatives after parameter testing relative to geology) the
degree of apparent subjectivity can thus
be lessened.

Thus we address the question raised , “Is the union of two
individual tracks to be a generalized track”?

We show that other parameters , nodes, masses and baselines
shape the fit of the union of indivudal tracks when the criteria is either
historical contingency or metacommunity data output creation under the
constraint of geology and systematic.

The GT is both a model approximating a biogeographic realty
and a representation of truth.Union of
individual tracks can proceed first or creation of the GT can be first
depending on if the model inference is within the multiple model
panbiogeograpic model set or if it is to be part of truth (then GT is first)

What guides parameter alteration by the user is the total
baseline set fit to all other systematic trees for the same taxa as well as
better and better biogeographic generalized tracks for larger and larger
spatial taxa and parameter numbers.Admittedly the project remains subjective to a large extent since
mutimodel creation starts with givens that may or may not be true but the
ability to make predictions and get better and better fits overtime indicates
that panbiogeography is indeed progressive and may even eventually come to be
definitive for larger timing suggestions of evolution as a whole.

The multiple model approach allows panbiogeography to
develop repeatable, reliable and quantitative criteria to capture congruence on
species distributions. Maximum likelyhood enables a definition of congruence.We use variance to measure deviance from
similiarty through nodes and anti nodes and masses to influence bias.Model averageing gives rise to diffenet
baseline derivations. Whileparameter
averaging enables fits to other multimodel systematic trees.

The parameters used for GT creation are
it(1),(2),(3)…n(1),(2),(3), antin(1),(2),(3)…m(1),(2),(3)…

Baselines are reserved for the truth model.

Confusions over Croizat’s use of deductive and inductive are
c leared up and the use of null hypothesis as used by Craw with clique
compatibilityis advanced.The null can not becreated because we can not
be certain without all taxa data on the parameters as they work algebraically
through thye southern hemisphere.

Model selection vsModel averaging – panbiogeography melds them together!!

The models are different sets of claimed tracks, nodes,
masses and baselines which series present the parameters and these can be
selected to fit the collection locality data as closely as possible. Model
selection is motivated by panbiogeography but incremental averaging provides
evidence of the need for a futher model closer the truth.Thus in panbiogeography there is a circuit of
model selection to model averaging to better candidate models to better averaging
as the increasing amount of useable data is incoportated.This is not the case in other uses where the
hypothesis are not embedded in larger hypotheses.This will be the case in panbiogeography
until the all fo the temproatily is investgigagte sthe spaces compassed.

“The first
assumes that models can be used to assess the importance of a variable in the
context of a given process/system by evaluating whether its parameter estimate
is different from some specified null value (often zero). This approach
emphasises uncertainty in model parameters but ignores uncertainty in the
selection of the model itself (e.g., which variables should be selected).”

So the importance of the Ozark node may be important in two
different models some in which it is connected the Ouichita node and others
where it does notwhen either of these
larger models has some use of the Ozark node in larger models (where the mass
is to the west or the east for instance) in important nonetheless. Asthe large3r model uncertainty is resolved
refinement of the node differences into track width permits alteration of the
structure through a progressive better fit and predictability.

’” The
second approach, which has been the subject of much recent discussion in many
sub-fields of biogeography and ecology (e.g., Stephens et al., 2007a, Grueber
et al., 2011), holds that uncertainty in model selection must not be ignored
and is as important as, if not more than, uncertainty in parameter estimates.”

The uncertainity in the choice of one model or the other is
due to lack of clear anti-node data (where the localities are not) but this
becomes findable as the models which receive their support wholly
biogeographically are used to predict other missing data in other phyla which
are then inturn used to recreate the intital model with refined parameters.

The reason that one needs multiple working hypotheses in
panbiogeography is that there can be differentiation both within and between
genera and species.This lies at the
heart of the issue withDarwin’s use of
biogeography and its criticism by Croizat as used by Nelson and others over the
center of dispersal.Without mutilple
hypothesis (essentially one process from the origin) the nested and test of
cross taxa within and without differences would not be possible.If there is actaully a binary division in
cause (which can be now hypothesized given DNA and molecular systematic
molecular evolution) then there may be reason to start with multiple rather
than a single origin hypothesis.This
does complicate the issue on the origin of life but that is so far off it may
not be a hindrance to doing some good work in the mean time.

The general panbiogeography MMI divides the total
probability into the three angles a,b,c where the idea of seen of not seen is
through the third angle c.Each
parameter has a decided proportion of each angle totals probability.

T1(a,b,c)+T2(a,b,c)…+N1(a,b,c)….=1.As the model fits and data increases continue
there will be shift in the proportion of the c amount to One.As one moves off Earth and to the origin
question this proportion will be mostly one unless there is strong mixing of
life both on and off Earth..

given {pan(t1,t2,t3,...n1,n2,n3,....m1,m2,m3...)} what is
the alpha, beta, gamma likelihood?

1 track has a lower likelihood than 3 tracks and two
nodes.Try to find the maximum
likelihood for all combinations of track,nodes,and masses.

What is extremely interesting is that one can actually do
the Chamberlin idea but within the Croizat one, namely one can do biogeography
first and foremost and THEN taxonomy.One creates multiple working Panbiogeographic hypotheses (BEFORE LOOKING
AT PHYLOGENETIC DATA) and then uses these to work out evolutionary
relationships systematically.The
inferences from biogeography are used to support different Panbiogeographic
concepts for incorporation later in systematics.Thus one need not know the phyla relations
but only the models.Some differences in
Panbiogeographic models can support the same phylogenetic trees but some of the
models imply different trees.

The complex relationship between biogeography and systematic
really requires that one use MMI.

Computing Panbiogeographic Model Lilkely hoods

The 3 track 2 node
model is more likely than the one track model clearly.It fits the data better. It has more
parameters. Here there are no parameters
in common.

These do

So model averaging may focus down onto one parameter say a
node which shows that it can be used to find anti-nodes in other parts of the
range which causes one to eliminate other nodes and use a single mass and thus
fit a larger systematic tree.

With AIC rather than Bayesian one only needs know that a
true evolution exists not that the evolutionary sequence is actually one of the
models.This is a way around the
creation evolution wrongful controversty.One could still say there is no evolution since we don not yet havethe
origin worked into the model structyure but as the model predcitivitiy in
creases it will be hard to find a non evolution truth for the same.

As more and more phyla are incorporated and more and more c
values occur the AIC number will go up for panbiogeographyasa
whole.

One does not want to use Bayesian methods in panbiogeography
because one wants to keep the relation of biogeogrpqhy and systematic
independent. Integration across all variables and models would not permit this
and can only be used by evolutionary biology if and when it has all of the
Earth’s distributions well understood.There may be time for this but it is notgoing to beprior to the collection of enough data to
bind the species and gene trees together.And that will depend on whether there is indeed a binary underlying
theory or not to the codes expressivity evovabilities.

“So how do we put all of these criteria together to select
the ‘best’ model? Unfortunately, this is where it get a bit more
difficult. As of yet I have not encountered a metric that combines fit,
predictivity, and parsimony into one optimisable statistic. Some of the
metrics described above already attempt to combine two of the above
criteria. The information criteria metrics such as AIC, BIC, and DIC
already consider fit and parsimony jointly. Stone
(1977) shows that model selection by AIC and leave-one-out cross validation
are asymptotically equivalent so this suggests that there is a link between
predictivity and the joint consideration of fit and parsimony.”

Predictivity comes when anti-nodes can be predicted by the
model (places where data is not expected ) as well as where it is expected in
other taxa but missing.Do Desmogs have
other Ozark nodes? Is fish distribution show that salamanders are in Norht
Carolina not connected so far? Is the crawfish anti node part of the Eurycea
data set?

Likelihoods
for mixed continuous–discrete distributions

The above can be extended in a
simple way to allow consideration of distributions which contain both discrete
and continuous components. Suppose that the distribution consists of a number
of discrete probability masses p_{k}(θ) (masses)and a density f(x | θ), (track,node anti-nodes) where the sum of
all the p's added to the integral of f is always one. Assuming
that it is possible to distinguish an observation corresponding to one of the
discrete probability masses from one which corresponds to the density
component, the likelihood function for an observation from the continuous
component can be dealt with as above by setting the interval length short
enough to exclude any of the discrete masses. For an observation from the
discrete component, the probability can either be written down directly or
treated within the above context by saying that the probability of getting an
observation in an interval that does contain a discrete component (of being in
interval j(which is a geographic distance) which
contains discrete component k) is approximately

where can be any point in
interval j. Then, on considering the lengths of the intervals to
decrease to zero, the likelihood function for an observation from the discrete
component is

where k is the index of the
discrete probability mass corresponding to observation x.

The fact that the likelihood
function can be defined in a way that includes contributions that are not
commensurate (the density and the probability mass) arises from the way in
which the likelihood function is defined up to a constant of proportionality,
where this "constant" can change with the observation x,
but not with the parameter θ.

So one could develop a frequentist
orclassical approach.In the classical approach the Croizat method
is presumed to rely on a natural symmetry from track to node to mass to baseline
per dendogram derived.On the
frequentist approach the anti-node can alter subtly thetruth of any symmetry by interacting between
the discrete and continuous differences (masses vs tracknodes
perbaseline).Denial of the existence of
any thing one can call the Croizat method is simply a denial that such a
symmetry can be cognized or was found.

Croizat’s panbiogeography is an
advance over Darwin’s understanding of geographic distributions relative to
evolution because Darwin was not able to separate discrete from continuous
contrtibutions in the modification by descent from that dyanically formed
through natural selection.Fisher worked
out one half of this problem by focusing on discrete (Mendelian) genetic part
but did not include (in principle) any discrete phenotype.Wright’s approach although no explicit about
non continuous phenoytpes relative to the mechanics of antural selection was
able via versimultude (in the inverse probabilithy possibility) to allow for
the possibility in unobserved variables that affect the interaction of mutation
and immigration with selection.

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Panbiogeography depicts distribution
patterns by reducing apparent localization complexity. Mapping many
species distributions on top of each other results in visually dense
images that are really-pretty-useless for the human observer to
dissect in meaninful ways.

One
way to analyze such information is
to automate algorthims and have an computer sort all of the edges
and nodes in some predefined way but another way is to deduce a
generalized track from such
data and thus induce a means to search and encounter patterns simply
by visual inspection later verified statistically. The amount of
published panbiogeographic material more or less requires that the
stronger deductive approach be developed.

As the process evolves computer searches and human recoveries merge as an horizon of past knowlege grows.

The
tension between these two directions/ways was brought out recently by
Ferrari Barao and Simoes (Quantitative panbiogeography: was the
congruence problem solved? 2013) in which they noted that parameter
setup was too subjective and congruence too partiuclarlly objective.
Below I set up a circular pratice that can improve congruence
increasingly and make the parameter setup more objective (statistical)
(with increasing knowledge). I offer this as a possible solution in the
deveolopment of quantitative panbigeoprahy by going beyond the cell or
predefined area as begun with clique analysis and introduce definite use
cases for track width, node shape, mass position, and baseline
direction. A point set state space is used instead.

Millington
and Peryy 2011 reviewed Multi-Model Inference in Biogeography but did
not mention panbiogeography. Here I show how to develop that means
towards the aforementioned goal of axiomatic panbiogeography.
Multimodel panbiogeography enables one to use both subjective guesses
and objective model averaging as the process of getting better patterns
matched over time. Thus there is a role for both the objective and
subjective components and the inductive and deductive modes
of investigation . It is then up to debate as to which on- going
alogrthmic creation flows andassociated logics are considered the
consensus. I make no apology for the increased complexity of the total
working protocol as compared to say that used in vicariance biogeography
or comparative biogeography. As long as blocks are in place we can not
advance historical biogeography into evolutionary theory proper any
further now than before Croizat's time.

One
uses exploratory analyses to hone the biogeoghraphic models that then
are used apriori in comparison with taxonomic nodal models to find an
objective atlas of increasing extents.This differs from vicarance
biogeography in that there is a synthesis of space rather than an
analysis of it. Time however is expected to be the same however. Thus
geographic data is used for hypothesis generaion and taxonomic data for
hypothesis testing. The cladistic node is thus made to format no merely
the panbiogeographic node but other dimensions of fundamental
Croizatian concepts. It repairs a rather dogmatic thininking that had
linked algebraic structure with genetic differentiation (Robinson on
Ethington) and noted by ABC in their anaysis of the hierarichacal
problem withg phlyogentic tree reconstruction. Vicariance
comprehesively applied can divided where only combinations were
otherwise designed. In this way multimodel panbiogeography is also an
advance beyond comparative biogeography.

Axiomatic panbiogeography provides an epistemology from which mult-model formats can be cognized (generalized linear models
as generalized tracks). With this multi-model perspective it is easy
to see how Croizat's panbiogeography was not as warmly recieved as it
could have been. The truth is really not in the model. This will be
apparent as I compare the Bayesian and Multinomial/Maximum likelihood
representations of the multimodel technique.

There
will no doubt be crticism of the ultimatie statstical distributions
behind the models (in the use of point set state spaces) but as the
process continues to integrate more and more taxonomic data (both
molecular and traditional) it will be hard to argue from pure
apriori ness any further than the inshgihts that are to be gainsaid. And
since model averaging will enable a increased precision through
parameter equalization under increased data incorporation the
subjectivity of intial choices/conditions will increasingly appear less.
Bias can decrease.with a better understanding of the variances
involved. Vicarance can be replaced with directed orthogenesis per form.

I
start with a multi-model approach to integrating salamander
distributions across the Mississppi in the central US, expand this to
include fish of similar biogeography (showing how historical
biogeography and taxonomy can dovetail), find plants that fit the same
model set, and then produce statements about metacommunity dispersal
movements, provide geological covariates of further congruent intricacy
and suggest needs for conservation through the same means of landscape
effects on possible speciation all the while testing and generting
increasingly robust potential challenges to existing taxonomic relations
amongst the polyphyletic groups included in the study. Through null
testing of the multinomial metacommunity conclusions we find a role for
both the old and new statitical paradigams in the ever complex field of
historical biogeography panbiogeographicalized to the levels/
disciplines of ecology ,evolution and classification. The use of
node-antinode density will improve this direction against bias but it is
dependent on the data and realizes sampling inclusion directly in the
process. Also where evolution is to go (with human induced cliamte
change say) can be predicted from the movements of the metacommunites.
This is a place for the study of dispersal.

In
attempting to design and develop a multi-model paradigm for panbiogeography it
appears that the situation where the betas are all orthogonal may be a goal of
construction wherein “Croizat concepts” (track width, node, mass, baseline)
thus remain optimized to maximal orthogonality.
This would permit an objective criterion on which generalized tracks
could be compounded from individual ones.
Interestingly, with this format orthogenesis might be interpretable as
when vicariant speciation follows spatially the given concept orthogonalites
and yet the lineage itself may present stochastic variations nonetheless.

What this
formation envisions is that for instance track1beta , track 2beta, node1beta,
baseline1beta are in a limit such that the different concept predictors do not
affect the other predictor betas (through the residual variance). The error in the technique will simply be due
to the concepts themselves not being able to capture the speciation process
itself (more non spatial information(particular genetic info) is involved,
other data from other lineages need be included, not enough data on the
lineages under consideration are included.

Once a
maximally orthogonal set of concepts is developed for a given data set then
this spatial path can be used as probe
for search encounter algorthims as if it was the generalized track so as to
incorporated spatial covariates which might be used to improve the entire
horizon of application.

If this
analysis is correct then one might think that model averaging could be used as
a means to find this maximally orthogonal concept organization when it does not appear
simply by inspection of the data graphed. Model averaging thus enables one to
find rather automatically and nonsubjectively parameters of track width,
nodeshape, mass amount and baseline direction (to sustain orthogonality
necessarily) per generalized track(“good”
model averaged predictions).
These parameters can then be used to write a fully statistical process
in which to evaluate the relation of biogeographic distributions to geographic
spatial divisions on which panbiogeography and also evolution interse depends.

One possible
use of these models may be for dissecting space usage of metacommunities (bound
by cross sections of the conceptual orthoganlities) impacted by human caused
distruption and or climate change. So
panbiogeography can be used as a means to asses how whole communites (rather
than individual species)may respond to global environmental effects.

The notion
of track congruence can be defined through the use of this method of synthesis.

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