Taxonomy & Phylogeny Introduction Classification Phylogeny Cladograms Quiz.
Most Likely Rates given Phylogeny L[ |001] = 0 x (P[ A ] + P[ B ]) + 1 x (P[ C ] + P[ D ])
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Transcript of Most Likely Rates given Phylogeny L[ |001] = 0 x (P[ A ] + P[ B ]) + 1 x (P[ C ] + P[ D ])
![Page 1: Most Likely Rates given Phylogeny L[ |001] = 0 x (P[ A ] + P[ B ]) + 1 x (P[ C ] + P[ D ])](https://reader036.fdocuments.us/reader036/viewer/2022062401/5a4d1b617f8b9ab0599adcf5/html5/thumbnails/1.jpg)
Most Likely Rates given Phylogeny
• L[|001] = 0 x (P[A] + P[B])
+ 1 x (P[C] + P[D])
0
0
1
0
0
1
0
0
1
01
00
= +00
0
1
0
11
+ +10
1
A τB
τC τD
![Page 2: Most Likely Rates given Phylogeny L[ |001] = 0 x (P[ A ] + P[ B ]) + 1 x (P[ C ] + P[ D ])](https://reader036.fdocuments.us/reader036/viewer/2022062401/5a4d1b617f8b9ab0599adcf5/html5/thumbnails/2.jpg)
Most Likely Rates given Phylogeny
• If 0 = 1:L[|001] P[∝ 001|,A] + P[001|,B]
+ P[001|,C] + P[001|,D]
0
0
1
0
0
1
0
0
1
01
00
= +00
0
1
0
11
+ +10
1
A τB
τC τD
![Page 3: Most Likely Rates given Phylogeny L[ |001] = 0 x (P[ A ] + P[ B ]) + 1 x (P[ C ] + P[ D ])](https://reader036.fdocuments.us/reader036/viewer/2022062401/5a4d1b617f8b9ab0599adcf5/html5/thumbnails/3.jpg)
Most Likely Rates given Phylogeny
P[001|,A] = (1-)3 x 1
P[001|,B] = (1-)2 x 2
P[001|,C] = (1-)1 x 3
P[001|,D] = (1-)2 x 2
0
0
1
0
0
1
0
0
1
01
00
= +00
0
1
0
11
+ +10
1
A τB
τC τD
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Most Likely Rates given Phylogeny
• If 0 = 1:L[|001] ∝ -2
0
0
1
0
0
1
0
0
1
01
00
= +00
0
1
0
11
+ +10
1
A τB
τC τD
![Page 5: Most Likely Rates given Phylogeny L[ |001] = 0 x (P[ A ] + P[ B ]) + 1 x (P[ C ] + P[ D ])](https://reader036.fdocuments.us/reader036/viewer/2022062401/5a4d1b617f8b9ab0599adcf5/html5/thumbnails/5.jpg)
Most Likely Rates given Phylogeny
• If 0 = 1:L[|001] ∝ -2
dL[|001] ∝ -2 ∴ L[|001] maximized at 2=
0
0
1
0
0
1
0
0
1
01
00
= +00
0
1
0
11
+ +10
1
A τB
τC τD
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Allowing uncertain ancestry favors high rates
P[001|,] = - 2
i.e., probability maximized when character changes half of the time!
0.00
0.05
0.10
0.15
0.20
0.25
000 025 050 075 00
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Matrices have obvious structure (25 steps over 16 taxa & 20 characters)…..
• 00000000000000000000• 00000110000000010100• 00001111000000010100• 00001101010000010100• 00101101010000010100• 01101101111000010100• 11101101111000001100• 11101101111010000100• 11101101111000001100• 11101101111010000110• 11101101111110000110• 11101101111111100110• 11101101111111100111• 11111001111000001100• 11111001111000001100• 11111001110000011100
![Page 8: Most Likely Rates given Phylogeny L[ |001] = 0 x (P[ A ] + P[ B ]) + 1 x (P[ C ] + P[ D ])](https://reader036.fdocuments.us/reader036/viewer/2022062401/5a4d1b617f8b9ab0599adcf5/html5/thumbnails/8.jpg)
….. or lack thereof (50 steps over 16 taxa & 20 characters)
• 00000000000000000000• 00010000000000001100• 00010001000101011100• 00000000000001101000• 00111011000101011100• 00011110101001101001• 01011000000101010000• 01011000100101011000• 01111001000111011000• 01111001000001010000• 01001010111001101111• 01001010111001101111• 10001010110000101111• 10011010110001101111• 11011011110001101101• 10001010110000101111
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Evaluating Matrix Structure:Do the distributions of character states tell us
anything prior to phylogenetic analysis?
Taxon Char A Char B Char CAlpha 0 0 0Beta 1 0 0Gamma 0 0 1Delta 1 0 1Epsilon 1 1 1Frack 1 1 0Kappa 1 1 0
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Character Compatibility
• Compatible character pairs: two characters with state combinations that do not necessarily imply homoplasy
• Incompatible character pairs: two
characters with state combinations that do necessarily imply homoplasy.
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Evaluating Matrix Structure:
Taxon Char A Char B Char CAlpha 0 0 0Beta 1 0 0Gamma 0 0 1Delta 1 0 1Epsilon 1 1 1Frack 1 1 0Kappa 1 1 0
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Compatibility among binary characters
Compatible: no homoplasyon some trees
A B0 01 01 1
Incompatible: homoplasyon all trees
A B0 01 01 10 1
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Graphical depiction of (in)compatibility:closing the circuit is “bad”
00
11
0110
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Compatibility among unordered multistates: break characters down into binaries
D E D E D E D E0 0 0 0 0 01 0 = 1 0 1 01 1 1 1 1 12 0 2 0 2 02 2 2 2 2 0
Compatible: all “pairs” must be compatible AND each pair must have unique combinations.
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Compatibility among unordered multistates: only one “pair” need be incompatible
D F D F D F D F0 0 0 0 0 00 2 0 2 0 21 0 = 1 0 1 01 1 1 1 1 12 0 2 0 2 02 2 2 2 2 0
Incompatible: states 0 and 2 show all combinations.
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Compatibility among unordered multistates: sometimes incompatibility is subtle.
D G D G D G D G0 0 0 0 0 00 1 0 1 0 11 0 = 1 0 1 01 2 1 2 1 22 1 2 1 2 12 2 2 2 2 2
Incompatible: all pairs compatible, but you cannot have the third pair and the first two without homoplasy.
![Page 17: Most Likely Rates given Phylogeny L[ |001] = 0 x (P[ A ] + P[ B ]) + 1 x (P[ C ] + P[ D ])](https://reader036.fdocuments.us/reader036/viewer/2022062401/5a4d1b617f8b9ab0599adcf5/html5/thumbnails/17.jpg)
Graphical depiction of (in)compatibility:circuit completed
00
0110
12 21
22
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Compatibility among Ordered multistates: Order of states must not be broken
H I H J0 0 0 01 0 1 01 1 1 12 1 2 02 2 2 2
HI show no gap in distributions (compatible);
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Compatibility among Ordered multistates: Order of states must not be broken
H I H J0 0 0 01 0 1 01 1 1 12 1 2 02 2 2 2
HJ show a gap in distributions (incompatible).
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Inferring Phylogeny with Compatibility: Clique Analysis
• Clique: a group of characters that all are compatible;
• Take the largest clique and infer phylogeny from those characters– This can produce a general (usually polytomous) tree
with no homoplasy;• Within each section of the phylogeny, find the
largest remaining clique;– Use this to clarify relations among those taxa;– Rinse & repeat….
• Note: “That was them, not us…..” (G. Estabrook, a.k.a., “Mr. Compatibilty”
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Testing Matrix Structure with Compatibility: Permutation Tests
• Calculate number of compatible pairs in matrix. • Permute matrix, scrambling states within each
character;– Each character retains same number of taxa with each
state;– Estimates P[observed compatibility] given such high
rates of change that there is no inheritance. • Calculate compatibility of matrix & characters;
– If observed compatibility is within the range of permuted matrix, then the data likely are cr@p;
– If an individual character’s compatibility is with the range of a permuted character, then it is likely useless.
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Testing Matrix Structure with Compatibility: Simple Inverse Models
• Calculate number of compatible pairs in matrix. • Evolve a tree and matrix of the same dimensions
as the original data;– Use same number of states as seen for each character;– Estimates P[observed compatibility] given particular
frequencies of change. • Calculate compatibility of matrix & characters;
– Tally P[compatibility | overall changes] for matrix or matrix partitions;
– Tally P[compatibility | # changes, # derived taxa] for each character.
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Probability of CompatibilityGiven Total Number of Steps
• As frequencies of change (and thus homoplasy) increase, the expected matrix compatibility drops.
0.00
0.05
0.10
0.15
0.20
0.25
2000 2500 3000 3500 4000 4500
Compatible Pairs
250230210190170150
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One can test partitions for significant differences in frequencies of change
• “Slug” characters are significantly less homoplastic than are shell characters among the Rapaninae.
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One can test partitions for significant differences in frequencies of change
• This is much less true for species in the Nassariidae.
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Effect of Change on Compatibility
• Infrequently changing characters typically have high compatibilities
• frequently changing ones have low compatibilities.
Simulation of 32 taxa with 100 binary characters and 200 total changes
0
20
40
60
80
100
1 2 3 4 5 6Changes
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Extreme distributions of taxa with derived states:End cases automatically compatible
0 0 00 0 10 0 10 0 10 0 10 0 10 0 10 0 10 1 10 1 10 1 10 1 10 1 10 1 10 1 11 1 1
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Effect of Derived Taxa on Compatibility
• Characters with 1 or 31 taxa with 0 or 1 (= autapomorphic) are automatically compatible
• Characters with 16 0’s and 1’s have lowest compatibility.
Simulation of 32 taxa with 100 binary characters and 200 total changes
0
20
40
60
80
100
0 2 4 6 8 10 12 14 16Derived Taxa
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Given X steps, compatibility is correlated with the number of
derived taxa. 0
20
40
60
80
100
1 2 3 4 5 6Changes
0
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40
60
80
100
0 2 4 6 8 10 12 14 16Derived Taxa
0
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80
100
0 2 4 6 8 10 12 14 16Derived Taxa
0
20
40
60
80
100
0 2 4 6 8 10 12 14 16Derived Taxa
1 Change
2 Changes3 Changes
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Effect of Correlated Character Changeon Compatibility
• Simulated case with two suites of characters in which change in one induces a 75% chance of change in others.
• Elevates compatibility because distributions are so similar.
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Testing for Independent Character Change: Mutual Compatibility
• Mutual compatibility: common compatibilities between two characters.– If character i and j both are compatible with
character k, then it is a mutual compatibility.• Character suites that exhibit correlated
change should share more mutual compatibilities than independently changing characters.– Do characters i & j have more mutual
compatibilities than expected given compatibility and independent change?
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Testing for Independent Character Change: Mutual Compatibility
• Multivariate structure among mutual compatibilities clusters correlated suites.– Similarity between each character pair based on proportion
of other characters with which both are compatible.
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Testing for Independent Character Change: Mutual Compatibility
• Mutual compatibility: common compatibilities between two characters.– If character i and j both are compatible with
character k, then it is a mutual compatibility.• Character suites that exhibit correlated
change should share more mutual compatibilities than independently changing characters.– Do characters i & j have more mutual
compatibilities than expected given compatibility and independent change?
![Page 34: Most Likely Rates given Phylogeny L[ |001] = 0 x (P[ A ] + P[ B ]) + 1 x (P[ C ] + P[ D ])](https://reader036.fdocuments.us/reader036/viewer/2022062401/5a4d1b617f8b9ab0599adcf5/html5/thumbnails/34.jpg)
Stratigraphic Compatibility
• For individual characters: no states with gaps in sampled record
• For character pairs: compatible pair in which the appearance of character pairs is also consistent with phylogeny.
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Compatible Pair Compatible with Stratigraphy
Ch 1 Ch2 FA LA 0 0 1 8 1 0 2 4 1 1 2 6 2 0 6 8 2 2 7 7
No necessary homoplasy, nor any necessary stratigraphic gaps between morphotypes.
22
11
20
10
VIIVIVIVIIIIII
VIII00
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Ch 1 Ch2 FA LA 0 0 5 8 0 1 3 8 1 0 2 6 1 2 1 4
Morphotypes 00 and 01 appear out of order given character necessary to avoid homoplasy.
Compatible Pair Incompatible with Stratigraphy
VIIVIVIVIIIIII
VIII00 01
10
12
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Ch 1 Ch2 FA LA 0 0 1 8 1 1 2 7 2 1 2 4 2 2 6 8
Morphotypes appear in the right order, but a gap exists between morphotypes 21 & 22.
Compatible Pair Incompatible with Stratigraphy
VIIVIVIVIIIIII
VIII00
11
22
21
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Ch 1 Ch2 FA LA 0 0 1 8 1 1 2 7 1 1 3 6
Morphotypes appear in the right order, but a gap exists within morphotypes 01.
Compatible Pair Incompatible with Stratigraphy
VIIVIVIVIIIIII
VIII00 01
11