Parsimony continued

1
Parsimony continued
2
Many issues glossed over

• What if characters disagree?
• How is the tree score determined?
• How can we root the trees?
• How do we find the optimal tree?
• How can we evaluate the robustness of our
  conclusions?

3
Tree score calculation
In
Ltot ? Ln Wn
I1
The tree score is the sum of the minimum number
of weighted steps (Ln) for each character
multiplied by the weight of that character (Wn)
4
How is the minimum number of steps calculated?

• Postorder traversal algorithm
• The tree is arbitrarily rooted
• Each internal node is inspected to see if there
  is an intersection in the possible states of its
  descendant nodes
• If not, length is increased
• It is not necessary to identify all ancestral
  state reconstructions (this requires a preorder
  traversal)

5
Why weight characters?

• If we think some characters are less prone to
  homoplasy, we can upweight them
• Character weights are multiplied by the character
  length

6
We can also weight character state transitions

• Unordered, flat-weighted Fitch parsimony

To state
From state
Step matrix
7
We can also weight character state transitions

• Common examples
• Ordered character states (morphology)

To state
From state
Step matrix
8
We can also weight character state transitions

• Common examples
• Transitions vs. transversions

To state
From state
Step matrix
9
We can also weight character state transitions

• Common examples
• Gains less likely than loss (restriction sites)

To state
From state
Step matrix (Asymmetric)
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The weighting game

• When should you weight characters/character-states
  ?
• If you think that they differ in evidential power
• How much should you modify weights?
• There is no simple formula
• It is probably better to err on the side of less
  extreme weights
• Often sensible to try a range of weights

11
Many issues glossed over

• What if characters disagree?
• How is the tree score determined?
• How can we root the trees?
• How do we find the optimal tree?
• How can we evaluate the robustness of our
  conclusions?

12
Rooting methods for parsimony

• Outgroup method (time reversible models)
• Include one or more taxa that are known from
  prior information to be outside the ingroup the
  group that is the focus of study
• Assume near clock-like behavior
• Midpoint rooting

13
Rooting methods for parsimony

• Outgroup method (time reversible models)
• Include one or more taxa that are known from
  prior information to be outside the ingroup the
  group that is the focus of study
• Assume near clock-like behavior
• Midpoint rooting
• Include characters with asymmetric step-matrices
  (time irreversible models)
• Pick the root that results in the shortest tree

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1-gt 0
L 1

• A 0
• B 0
• C 0
• D 1
• E 1
• F 1

A
B
C
D
F
E
1-gt 0
1-gt 0
L 2
15
Many issues glossed over

• What if characters disagree?
• How is the tree score determined?
• How can we root the trees?
• How do we find the optimal tree?
• How can we evaluate the robustness of our
  conclusions?

16
Rooted trees
Polytomy
Binary/dichotomous/fully-resolved
polyotomous/unresolved
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Unrooted trees
Polytomy
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Number of unrooted fully resolved trees for t taxa
i t

• ? (2i-5)

i 3
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How many places can you add another taxon?

• Two taxa
• Three taxa
• Four Taxa
• Five taxa

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Number of rooted trees?

• The root is just one more taxon same formula but
  t number of taxa 1

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The number of trees gets big

• Number of binary unrooted trees
• 1
• 3
• 15
• 105
• 2,027,025
• 2.2 x 1020
• 2.8 x 1074
• 1 x 101074

• Number of tips
• 3
• 4
• 5
• 6
• 10
• 20
• 50
• 500

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How do you find the optimal tree?

• Exhaustive (lt12 taxa)

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How do you find the optimal tree?

• Exhaustive (lt12 taxa)
• Branch-and-bound (lt18 taxa)

• Obtain the length of a random tree (initial upper
  bound)
• As trees are built determine length
• If length exceeds upper bound then that tree and
  all its descendant trees are ignored

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How do you find the optimal tree?

• Exhaustive (lt12 taxa)
• Branch-and-bound (lt18 taxa)
• Heuristic search (unlimited?)

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Heuristic searches

• Search for optimal trees by finding good trees
  and then rearranging them in the hopes of finding
  an even better tree

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Heuristic search
Suboptimal island of trees
Global optimum
Starting trees
Treespace
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Getting starting trees

• Random tree - not done
• User tree (e.g., a NJ tree)
• Build a tree by adding taxa to the location that
  is optimal
• Can hold more than one tree at each step

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Taxon addition order

• As-is
• In the order of the matrix (not done for
  parsimony)
• Simple taxon addition
• use a distance algorithm to decide order
• Closest taxon addition
• Add the taxon that makes the optimal tree
• Random taxon addition order
• Repeat many times

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Branch swapping

• Nearest-neighbour interchange (NNI)

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Branch swapping

• Subtree pruning and regrafting (SPR)

31
Branch swapping

• Tree-bisection reconnection (TBR)

32
Many issues glossed over

• What if characters disagree?
• How is the tree score determined?
• How can we root the trees?
• How do we find the optimal tree?
• How can we evaluate the robustness of our
  conclusions?

33
Even if the shortest trees is the best estimate
of the true tree - the true tree might not be the
shortest
We should consider suboptimal trees
We should use statistical tests to help us
determine what to actually believe
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Questions we can ask

• Are the data random or do they have signal?
• How much homoplasy is there?
• To what extent are particular elements of the
  trees (clades) supported?
• What alternative results can we reject?

More later.