Effects of Rooting on Phylogenic Algorithms Margareta Ackerman Joint work with David Loker and Dan Brown

Hierarchical Clustering & Phylogency

Ph ylogeny is an application of Hierarchical Clustering. They are closely related! Phylogeny meets Hierarchical Clustering Unfortunately, there is a disconnect between these fields.

A step towards bridging the gap: We bring techniques from cluster analysis to study Phylogenetic algorithms. We apply a recent framework for clustering algorithm selection to Phylogeny [(Ackerman, Ben-David, and Loker, ‘10), (Ackerman, Ben-David, and Loker, ‘10), (Ackerman & Ben-David, IJCAI ‘11), (Zedah and Ben- David, ‘09)] Bridging the Gap

Given the same input, different Phylogenetic algorithms can produce radically different results. 5 How should a user decide which algorithm to use? Selecting Phylogenetic Algorithms

This framework lets a user utilize prior knowledge to select an algorithm Identify properties that distinguish between different input-output behaviour of clustering paradigms The properties should be: 1) Intuitive and “user-friendly” 2) Useful for distinguishing clustering algorithms 6 Framework for Selecting Phylogenetic Algorithms

Rooting Phylogenetic Trees Formal Framework Properties of Hierarchical Algorithms Analysis of Linkage-Based Algorithms Analysis of Neighbor Joining Conclusions and Future Direction Outline

A common solution: Introduce distant taxa (or, elements) and root where the distant taxa connect with the ingroup. How to Root Phylogenetic Trees? E

The addition of an outgroup can CHANGE the topology of the ingroup. When Rooting Changes the Ingroup After adding outgroup E

Empirical studies demonstrate that when using some algorithms, ingroup topology can be disrupted when an outgroup is added [(Holland et. al., ‘03), (Shavit et. al., ‘07), (Lin et. al, ‘02), (Slack et. al., ‘03) ] We perform a theoretical analysis of this phenomenon, proving that some algorithms are immune to this problem, while others are highly volatile. This Happens in Practice!

Independently of our work, it was shown that when using BME, the ingroup topology can change arbitrarily when an outlier is added (Cueto and Matsen, 2010) Previous Work

Linkage-based algorithms (including UPGMA) do not change ingroup when the outgroup is sufficiently far away Using Neighbor Joining, ingroup topology is effected by outgroups even if the outgroup is arbitrarily far away Our Contributions

Rooting Phylogenetic Trees Formal Framework Properties of Hierarchical Algorithms Analysis of Linkage-Based Algorithms Analysis of Neighbor Joining Conclusions and Future Direction Outline

C_iD C_i C_i is a cluster in a dendrogram D if there exists a node in the dendrogram so that C_i is the set of its leaf descendents. 14 Formal Setup

C = {C 1, …, C k } D C = {C 1, …, C k } is a clustering in a dendrogram D if –C i D1≤ i ≤ k –C i is a cluster in D for all 1≤ i ≤ k, and –Clusters are disjoint 15 Formal Setup

A A Hierarchical Clustering Algorithm A maps X d (X,d) Input: A data set X with a distance function d, denoted (X,d) to X Output: A dendrogram of X Y ⊆ X Z ⊆ X The distance between Y ⊆ X and Z ⊆ X is the length of the minimum edge between them d(Y,Z) = min y in Y, z in Z d(y,z) 16 Formal Setup

Rooting Phylogenetic Trees Formal Framework Properties of Hierarchical Algorithms Analysis of Linkage-Based Algorithms Analysis of Neighbor Joining Conclusions and Future Direction Outline

(X u O, d) A Given a data set (X u O, d) and algorithm A, XO X is unaffected by O A(X, d)A(X u O, d) if A(X, d) is a sub-dendrogram of A(X u O, d). XO Otherwise, X is affected by O. A(X,d)A(O,d) A(X u O,d) Unaffected by an Outgroup

Ingroup A (X, d) (O, d’)(X,d) (O,d’) X O Algorithm A is outgroup-independent if for any data sets (X, d) and (O, d’), if (X,d) and (O,d’) are sufficiently far apart then X is unaffected by O. Outgroup Outgroup Independence

A (X, d) (O, d’)(X,d) (O,d’) X O Algorithm A is outgroup-independent if for any data sets (X, d) and (O, d’), if (X,d) and (O,d’) are sufficiently far apart then X is unaffected by O. A(X,d) A(O,d’) A(X u O,d*) d* (X,d) (O,d’) d* puts (X,d) and (O,d’) sufficiently far apart Outgroup Independence

A (X,d) c (O,d’) X OcX O An algorithm A is outgroup volatile if for any data set (X,d) and any constant c, there exist (O,d’) with distance between X and O at least c, such that X is affected by O. OA If O is a singleton, then A is outlier volatile. Outgroup Volatility

Rooting Phylogenetic Trees Formal Framework Properties of Hierarchical Algorithms Analysis of Linkage-Based Algorithms Analysis of Neighbor Joining Conclusions and Future Direction Outline

Theorem : Any hierarchical algorithm A that is 2-rich, outer-consistent, and local, is outgroup independent. We use the following general result to show that Linkage-Based algorithms are outgroup-independent.

If we select a cluster from the dendrogram, and run the algorithm the data underlying this cluster, we obtain a result that is consistent with the original dendrogram. D = A(X,d) D’ = A(X’,d) X’={x 1, …, x 4 } 24 Locality

A(X,d) C C(X,d) C on dataset (X,d) C(X,d’) C on dataset (X,d’) Outer-consistent change 25 If A is outer-consistent, then A(X,d’) will also include the clustering C. Outer Consistency

(X, d) Given any pair of data sets (X, d) and (X’, d’)d* X u X’X X’A(X u X’, d*) (X’, d’), there exists d* over X u X’, so that X and X’ are the children of the root in A(X u X’, d*). 2-Richness (X,d) (X, d’) (X, d*) X A(X O,d*) A(X u O,d*) X’

Proof: We want to show that given any if the data sets are placed sufficiently far apart, then A(X,d) is a sub-dendrogram of A(X u O, d*). Theorem : Any hierarchical algorithm A that is 2-rich, outer-consistent, and local, is outgroup independent. (X,d) (O, d’) (X O,d’’) (X u O,d’’) A(X,d) A(X O,d*) A(X u O,d*)

Proof: First, apply 2-richness. Given X O there exists d’’ over X u O, X O,d’’). so that X and O are children of A(X u O,d’’). Theorem : Any hierarchical algorithm A that is 2-rich, outer-consistent, and local, is outgroup independent. (X,d) (O, d’) (X O,d’’) (X u O,d’’) X A(X O,d’’) A(X u O,d’’) O c

Proof: c Let d* be any distance function extending d and d’ where the min distance between X and O is at least c. X O,d*). Then by outer-consistency, X and O are children of the root of A(X u O,d*). Theorem : Any hierarchical algorithm A that is 2-rich, outer-consistent, and local, is outgroup independent. (X O,d’’) (X u O,d’’) X A(X O,d*) A(X u O,d*) O c (X O,d*) (X u O,d*)

Proof: A(X O,d*). Finally, by locality, A(X,d) is a sub-dendrogram of A(X u O,d*). Therefore, whenever (X,d) and (O,d’) are sufficiently far apart, X is unaffected by O. Theorem : Any hierarchical algorithm A that is 2-rich, outer-consistent, and local, is outgroup independent. X A(X O,d*) A(X u O,d*) O A(X,d)

XCreate a leaf node for every element of X Insert image 31 Linkage Based Algorithm

XCreate a leaf node for every element of X Repeat the following until a single tree remains: –Consider clusters represented by the remaining root nodes. 32 Linkage Based Algorithm

XCreate a leaf node for every elements of X Repeat the following until a single tree remains: –Consider clusters represented by the remaining root nodes. Merge the closest pair of clusters by assigning them a common parent node. 33 ? Linkage Based Algorithm

The choice of linkage function distinguishes between different linkage-based algorithms. Examples of common linkage-functions –UPGMA : average between-cluster distance –Single-linkage : shortest between-cluster distance –Complete-linkage : maximum between-cluster distance X1X1X1X1 X2X2X2X2 34 Examples of Linkage Based Algorithms

Proof: We can show that all linkage-based algorithms are 2-outer-rich, outer- consistent, and local. Result follows by previous Theorem. Theorem : All Linkage-Based algorithms are outgroup independent.

Rooting Phylogenetic Trees Formal Framework Properties of Hierarchical Algorithms Analysis of Linkage-Based Algorithms Analysis of Neighbor Joining Conclusions and Future Direction Outline

Most widely-used distance-based method for phylogenetic reconstruction Works well in practice If there is a tree that fits the distance matrix (additive), it will find it Neighbour Joining

This remains the case when distances of the ingroup are additive. Theorem : Neighbor joining is outlier volatile.

Theorem : (X,d) O d ∗ X ∪ O dd ∗ (X,O) Given any data set (X,d), there exists a set of outliers O and a distance function d ∗ over X ∪ O extending d, where d ∗ (X,O) can be arbitrarily large, such that NJ(X ∪ O, d ∗ )|X NJ(X ∪ O, d ∗ )|X is an arbitrary dendrogram. Outgroups can lead to arbitrary dendrograms A(X,d) A(X O,d*)|X A(X u O,d*)|X

Rooting Phylogenetic Trees Formal Framework Properties of Hierarchical Algorithms Analysis of Linkage-Based Algorithms Analysis of Neighbor Joining Conclusions and Future Direction Outline

Present a formal framework for the analysis of the effects of outgroups on the ingroup topology for computationally efficiently hierarchical algorithms Prove that all Linkage-Based algorithms, which include UPGMA, are outgroup independent Prove that NJ is outgroup volatile This only addresses rooting - We do not claim that UPGMA is in general better than NJ. Conclusions

How to choose outgroups for rooting NJ? Perform a similar analysis of Likelihood methods Future Work