We do not discuss any further this application which is not in the main scope of this paper. We know of only one use of MST for galaxy classification. They find that the majority of the spectral classes are distributed along a well-defined branch going from the earliest to the latest types, with optically bright active galaxies forming an independent branch that intersects the main sequence exactly at the transition between early and late types. This description is already an interpretation of the 23 ASK classes that present a regular distribution of their spectra as already mentioned in Section 6, so that the very linear structure of the MST tree is not surprising.

However, the approach is interesting because this is a rather simple and objective method to obtain relationships between classes. Basically, all galaxies share a common origin which is the gathering of baryonic matter as a self-gravitating object. This baryonic matter was very primitive and has subsequently being enriched and diversified by several generations of stars and many transforming processes like galaxy interactions and mergers. There are thus obvious evolutionary relationships between different kinds of galaxies as immediately understood by Hubble when he discovered galaxies and established his famous tuning fork diagram.

Taking into account the galaxy diversity of morphologies known at that time, he built a phylogenetic tree in which the relationships are due to the evolution of the stellar orbits which, he thought, should flatten with time because of the dynamical friction.

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Even though we now know that this process cannot be accomplished in a time shorter than the known age of our Universe, this tuning fork diagram is still used to represent galaxy diversity. Somewhat strangely enough, phylogenetic analyses of galaxy diversity has not been attempted again for a century.

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This is probably because the data did not allow much progress into this direction. But we now have huge multivariate databases and it seems timely to reconsider this question. We here present only a few techniques, those that have been already used on astrophysical data sets. Before describing some of the most important methods, let us point out that the development of phylogenetic methods has been hindered till the s by very heated discussions on the philosophical merits of the different approaches. It is only in recent years that most of the barriers between the different schools of thoughts could be overcome by a new generation of researchers.

Recently a new picture of phylogenetic methods is emerging. It becomes nowadays increasingly clear that all the different approaches can be discussed within a common framework including distance- and character-based approaches, and that this theoretical framework applies both to phylogenetic trees and networks. There are two main categories of methods: the distance-based and the character-based. For continuous parameters, these states can be obtained through discretization.

For distance-based approaches, Neighbor-Joining is the most popular approach to construct a phylogenetic tree. This method is a bottom-up hierarchical clustering methods.

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It starts from a star tree unresolved tree. The branches of the two objects with the lowest Q i, j are linked together by a new node u on the tree. This node replaces the pair i, j in the subsequent iterations through the distance to any other object k :. Neighbor-Joining minimizes a tree length, according to a criteria that can be viewed as a Balanced Minimum Evolution Gascuel and Steel, For a tree metrics, Neighbor-Joining furnishes a simple algorithm to reconstruct a tree from the distance matrix. There is a large literature on how to best approximate a metrics by a tree metrics see for instance Fakcharoenphol et al.

Neighbor-Joining is justified if the difference between the original distance matrix and the distance matrix describing the X-tree obtained with Neighbor-Joining is not too large. Cladistics has been associated in the 80's to the search of a maximum parsimony tree. Maximum Parsimony is a powerful approach to find tree-like arrangements of objects Figure 5.

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The drawback is that the analysis must consider all possible trees before selecting the most parsimonious one. The computation complexity depends on the number of objects and character states, so that too large samples say more than a few thousands cannot be analyzed. The Maximum Parsimony algorithm can take uncertainties or unknowns into account by evaluating the different possibilities allowed by the range of values and selecting among them the one that provides the smallest score.

In the case of unknown parameters, the most parsimonious diversification scenario provides a prediction for the unknown values. Figure 5.

A example tree obtained with cladistics, represented here as unrooted. When a root is chosen, the tree takes the shape of hierarchical trees. In recent years the definition of cladistics has been extended to the classification of taxa individuals or species defined by characters on a rooted tree. In biological applications, a phylogenetic tree describes the possible evolution of a taxon corresponding to the root. The root may either be a real taxon or be inferred from the descendant taxa. The success of a cladistics analysis much depends on the behavior of the parameters.

In particular, it is sensitive to redundancies, incompatibilities, too much variability reversals , and parallel and convergent evolutions. It is thus a very good tool for investigating whether a given set of parameters can lead to a robust and pertinent diversification scenario. If a set of characters exactly defines a phylogeny, then the phylogeny is called perfect. In practical applications, the available characters seldom define a perfect phylogeny. A supplementary measure of the deviation to a perfect phylogeny is necessary to determine how well a candidate tree fits the characters.

The tree with the minimum score is searched for with some heuristics Felsenstein, The maximum parsimony approach can be directly extended to continuous characters or values.

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To each internal node is associated a real value f u. The score s of a tree equals the sum over all edges of the absolute difference between those values:. Robinson has shown that for a tree defined by continuous characters, a maximum parsimony score is reached for values of the internal nodes belonging to the set of values or states defined on the leaves.

The main method to search for the best tree representation of data beyond Maximum Parsimony include Maximum Likelihood. We note this technique which has never been applied to astrophysics in the context of classification but may be a pertinent approach. The problem here is that an evolutionary model must be used, and naturally the result will depend significantly on it. Maximum Likelihood is used standardly in biology, and it may be possible that astrophysicists could also have well constrained physical models of the evolution of galaxies and their properties.

The phylogenetic tree of Maximum Likelihood is the tree for which the observed data are most probable Williams and Moret, Distance-based approaches are also often quite appropriate for reconstructing a phylogenetic tree from continuous characters. Distance-based approaches are fast and can be used for data exploration and for the selection of the most appropriate variables.

Cladistics when applied to domain outside of biology, like in astrocladistics, refers more generally to the classification of objects by a rooted or an unrooted tree Figure 5. In that case, the tree represents possible relationships between taxa. The search of the best tree described by a set of characters on a set of objects or taxa in phylogeny can be done by several different approaches. The most popular methods are the one using Maximum Parsimony or Maximum Likelihood.

For continuous parameters, the software program TNT Goloboff et al. As an alternative, the data can be discretized through appropriate binning. As mentioned earlier, a new picture of phylogenies is emerging after the understanding that phylogenies on multistate characters reduce through a conceptually simple grouping of the characters into a phylogeny on binary characters. For binary characters, both distance- and character-based approaches are equivalent. This approach opens new perspectives as it furnishes also a bridge between character-based phylogenies and split networks or more precisely outer planar networks.

Outer planar networks permit the simultaneous representation of alternative trees with reticulations, and are thus generalizations of trees Huson and Bryant, In order to understand the connection between outer planar networks and phylogenetic trees, one has to explain succinctly what is called a split on a circular order of the taxa.

A circular order on a phylogenetic tree corresponds to an indexing of the n end nodes according to a circular clockwise or anti-clockwise scanning of the end nodes.

A split on a circular order of the taxa is a partition of the objects into two disjoint sets Figure 6. Figure 6. A circular order for objects A—G, with their pairs of binary states, arranged according to the circular consecutive-ones condition. For multistate characters, a split can be defined after transformation of each multistate character into a binary character. For each pair of states A,B , a subset of states containing the state A is attributed the 1 state and the complementary subset including the subset B is given the binary state 0.

If the transformation can be done on each states and characters for details see Thuillard and Fraix-Burnet, in revision so that each binary character fulfills the circular consecutive-ones condition, then the data can be described exactly by an outer planar network. By definition the circular consecutive-ones condition are fulfilled if for any binary state, the taxa with the 1 state are consecutive on the circular order Figure 6. Splits in an outer planar network Figure 7 furnish neighboring relationships between objects.

Objects sharing a common property, as defined by splits, are consecutive in a circular order. Outer planar networks can be regarded as a generalization of phylogenetic trees. An outer planar network reduces to a phylogenetic tree if for each pair of binary characters, the so-called 4-gamete rule is fulfilled.

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The 4-gamete rule states that for each pair of binary characters there is at least one of the 4 possible gametes [either 1,0 , 0,1 , 1,1 , or 0,0 ] that is missing. Figure 7. An example of an outer planar network showing the eight splits of the eight parameters s1…s8.

For distance-based approach, the circular consecutive-ones conditions have to be replaced by the fulfillment of the Kalmanson inequalities. This distance matrix fulfills the so-called Kalmanson inequalities:. The program SplitTrees4 Huson and Bryant, permits to construct outer planar networks from a distance matrix. In practice, the perfect order is not known or not feasible. The difference between the perfect order and the order one obtains with a given data set is called the contradiction. The minimum contradiction analysis Thuillard, , finds the best order one can get.

It is a powerful tool for ascertaining whether the parameters can lead to a tree-like arrangement of the objects Thuillard and Fraix-Burnet, Using the parameters that fulfill this property, the method then performs an optimization of the order and provides groupings with an assessment of their robustness.

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## Multivariate analysis

We believe that outer planar networks will gain importance in applications outside of biology as they furnish a real alternative to the standard classification methods. Farrah et al. An evolutionary description of these galaxies is proposed from the properties of these groups. Even though their method is not a phylogenetic technique per se since the relationships are constructed after the clustering analysis, this work illustrates the potential need of phylogenetic tools in astrophysics.

The use of phylogenetic approaches in astrophysics has been pioneered and pursued through the denomination of astrocladistics Fraix-Burnet et al. Applications have been successfully performed for galaxies Fraix-Burnet et al. The phylogenetic approaches used on galaxy samples are clearly oriented toward a multivariate and evolutionary classification of galaxies Fraix-Burnet et al.

To this end, several statistical analyses PCA, k-means, cladistics, and minimum contradiction analysis are used to select the set of parameters that yields a robust classification according to several clustering analyses k-means, cladistics, and Minimum Contradiction Analysis.