Algebraic Phylogenetics

cavender_farris_neyman_model(graph::Graph{Directed})

Creates a PhylogeneticModel based on graph whose transition matrices are of type Cavender-Farris-Neyman.

Examples

julia> cavender_farris_neyman_model(graph_from_edges(Directed,[[4,1],[4,2],[4,3]]))
Group-based phylogenetic model on a tree with 3 leaves and 3 edges 
 with distribution at the root [1//2, 1//2]. 
 The transition matrix associated to edge i is of the form 
 [a[i] b[i];
  b[i] a[i]], 
 and the Fourier parameters are [x[i, 1] x[i, 2]].

jukes_cantor_model(graph::Graph{Directed})

Creates a PhylogeneticModel based on graph whose transition matrices are Jukes Cantor matrices.

Examples

julia> jukes_cantor_model(graph_from_edges(Directed,[[4,1],[4,2],[4,3]]))
Group-based phylogenetic model on a tree with 3 leaves and 3 edges 
 with distribution at the root [1//4, 1//4, 1//4, 1//4]. 
 The transition matrix associated to edge i is of the form 
 [a[i] b[i] b[i] b[i];
  b[i] a[i] b[i] b[i];
  b[i] b[i] a[i] b[i];
  b[i] b[i] b[i] a[i]], 
 and the Fourier parameters are [x[i, 1] x[i, 2] x[i, 2] x[i, 2]].

kimura2_model(graph::Graph{Directed})

Creates a PhylogeneticModel based on graph whose transition matrices are Kimura 2-parameter matrices.

Examples

julia> kimura2_model(graph_from_edges(Directed,[[4,1],[4,2],[4,3]]))
Group-based phylogenetic model on a tree with 3 leaves and 3 edges
 with distribution at the root [1//4, 1//4, 1//4, 1//4]. 
 The transition matrix associated to edge i is of the form 
 [a[i] b[i] c[i] b[i];
  b[i] a[i] b[i] c[i];
  c[i] b[i] a[i] b[i];
  b[i] c[i] b[i] a[i]], 
 and the Fourier parameters are [x[i, 1] x[i, 3] x[i, 2] x[i, 2]].

kimura3_model(graph::Graph{Directed})

Creates a PhylogeneticModel based on graph whose transition matrices are Kimura 3-parameter matrices.

Examples

julia> kimura3_model(graph_from_edges(Directed,[[4,1],[4,2],[4,3]]))
Group-based phylogenetic model on a tree with 3 leaves and 3 edges 
 with distribution at the root [1//4, 1//4, 1//4, 1//4]. 
 The transition matrix associated to edge i is of the form 
 [a[i] b[i] c[i] d[i];
  b[i] a[i] d[i] c[i];
  c[i] d[i] a[i] b[i];
  d[i] c[i] b[i] a[i]], 
 and the Fourier parameters are [x[i, 1] x[i, 2] x[i, 3] x[i, 4]].

general_markov_model(graph::Graph{Directed})

Creates a PhylogeneticModel based on graph whose transition matrices are stochastic with no further constraints.

Examples

julia> general_markov_model(graph_from_edges(Directed,[[4,1],[4,2],[4,3]]))
Phylogenetic model on a tree with 3 leaves and 3 edges 
 with distribution at the root [π[1], π[2], π[3], π[4]] 
 and transition matrix associated to edge i of the form 
 [m[i, 1, 1] m[i, 1, 2] m[i, 1, 3] m[i, 1, 4];
  m[i, 2, 1] m[i, 2, 2] m[i, 2, 3] m[i, 2, 4];
  m[i, 3, 1] m[i, 3, 2] m[i, 3, 3] m[i, 3, 4];
  m[i, 4, 1] m[i, 4, 2] m[i, 4, 3] m[i, 4, 4]].

affine_phylogenetic_model!(pm::PhylogeneticModel)

Moves a PhylogeneticModel or GroupBasedPhylogeneticModel from projective into affine space.

Examples

julia> pm = jukes_cantor_model(graph_from_edges(Directed,[[4,1],[4,2],[4,3]]));

julia> affine_phylogenetic_model!(pm)
Group-based phylogenetic model on a tree with 3 leaves and 3 edges
 with distribution at the root [1//4, 1//4, 1//4, 1//4].
 The transition matrix associated to edge i is of the form
 [-3*b[i]+1 b[i] b[i] b[i];
  b[i] -3*b[i]+1 b[i] b[i];
  b[i] b[i] -3*b[i]+1 b[i];
  b[i] b[i] b[i] -3*b[i]+1],
 and the Fourier parameters are [1 x[i, 2] x[i, 2] x[i, 2]].

graph(pm::PhylogeneticModel)

Return the graph of a PhylogeneticModel pm.

Examples

julia> pm = jukes_cantor_model(graph_from_edges(Directed,[[4,1],[4,2],[4,3]]));

julia> graph(pm)
Directed graph with 4 nodes and the following edges:
(4, 1)(4, 2)(4, 3)

transition_matrices(pm::PhylogeneticModel)

Return a dictionary between the edges of the tree specifying the PhylogeneticModel pm and their attached transition matrices.

Examples

julia> pm = jukes_cantor_model(graph_from_edges(Directed,[[4,1],[4,2],[4,3]]));

julia> transition_matrices(pm)
Dict{Edge, MatElem{QQMPolyRingElem}} with 3 entries:
  Edge(4, 1) => [a[1] b[1] b[1] b[1]; b[1] a[1] b[1] b[1]; b[1] b[1] a[1] b[1];…
  Edge(4, 2) => [a[2] b[2] b[2] b[2]; b[2] a[2] b[2] b[2]; b[2] b[2] a[2] b[2];…
  Edge(4, 3) => [a[3] b[3] b[3] b[3]; b[3] a[3] b[3] b[3]; b[3] b[3] a[3] b[3];…

number_states(pm::PhylogeneticModel)

Return the number of states of the PhylogeneticModel pm.

Examples

julia> pm = jukes_cantor_model(graph_from_edges(Directed,[[4,1],[4,2],[4,3]]));

julia> number_states(pm)
4

probability_ring(pm::PhylogeneticModel)

Return the ring of probability coordinates of the PhylogeneticModel pm.

Examples

julia> pm = jukes_cantor_model(graph_from_edges(Directed,[[4,1],[4,2],[4,3]]));

julia> probability_ring(pm)
Multivariate polynomial ring in 6 variables a[1], a[2], a[3], b[1], ..., b[3]
  over rational field

fourier_ring(pm::GroupBasedPhylogeneticModel)

Return the ring of Fourier coordinates of the PhylogeneticModel pm.

Examples

julia> pm = jukes_cantor_model(graph_from_edges(Directed,[[4,1],[4,2],[4,3]]));

julia> fourier_ring(pm)
Multivariate polynomial ring in 6 variables x[1, 1], x[2, 1], x[3, 1], x[1, 2], ..., x[3, 2]
  over rational field

fourier_parameters(pm::GroupBasedPhylogeneticModel)

Return the Fourier parameters of the GroupBasedPhylogeneticModel pm as a vector of eigenvalues of the transition matrices.

Examples

julia> pm = jukes_cantor_model(graph_from_edges(Directed,[[4,1],[4,2],[4,3]]));

julia> fourier_parameters(pm)
Dict{Edge, Vector{QQMPolyRingElem}} with 3 entries:
  Edge(4, 1) => [x[1, 1], x[1, 2], x[1, 2], x[1, 2]]
  Edge(4, 2) => [x[2, 1], x[2, 2], x[2, 2], x[2, 2]]
  Edge(4, 3) => [x[3, 1], x[3, 2], x[3, 2], x[3, 2]]

group_of_model(pm::GroupBasedPhylogeneticModel)

Returns the group the GroupBasedPhylogeneticModel pm is based on.

Examples

julia> pm = jukes_cantor_model(graph_from_edges(Directed,[[4,1],[4,2],[4,3]]));

julia> group_of_model(pm)
4-element Vector{FinGenAbGroupElem}:
 [0, 0]
 [0, 1]
 [1, 0]
 [1, 1]

probability_map(pm::PhylogeneticModel)

Create a parametrization for a PhylogeneticModel of type Dictionary.

Iterate through all possible states of the leaf random variables and calculates their corresponding probabilities using the root distribution and laws of conditional independence. Return a dictionary of polynomials indexed by the states. Use auxiliary function monomial_parametrization(pm::PhylogeneticModel, states::Dict{Int, Int}) and probability_parametrization(pm::PhylogeneticModel, leaves_states::Vector{Int}).

Examples

julia> pm = jukes_cantor_model(graph_from_edges(Directed,[[4,1],[4,2],[4,3]]));

julia> p = probability_map(pm)
Dict{Tuple{Vararg{Int64}}, QQMPolyRingElem} with 64 entries:
  (1, 2, 1) => 1//4*a[1]*a[3]*b[2] + 1//4*a[2]*b[1]*b[3] + 1//2*b[1]*b[2]*b[3]
  (3, 1, 1) => 1//4*a[1]*b[2]*b[3] + 1//4*a[2]*a[3]*b[1] + 1//2*b[1]*b[2]*b[3]
  (4, 4, 2) => 1//4*a[1]*a[2]*b[3] + 1//4*a[3]*b[1]*b[2] + 1//2*b[1]*b[2]*b[3]
  (1, 2, 3) => 1//4*a[1]*b[2]*b[3] + 1//4*a[2]*b[1]*b[3] + 1//4*a[3]*b[1]*b[2] …
  (3, 1, 3) => 1//4*a[1]*a[3]*b[2] + 1//4*a[2]*b[1]*b[3] + 1//2*b[1]*b[2]*b[3]
  (3, 2, 4) => 1//4*a[1]*b[2]*b[3] + 1//4*a[2]*b[1]*b[3] + 1//4*a[3]*b[1]*b[2] …
  (3, 2, 1) => 1//4*a[1]*b[2]*b[3] + 1//4*a[2]*b[1]*b[3] + 1//4*a[3]*b[1]*b[2] …
  (2, 1, 4) => 1//4*a[1]*b[2]*b[3] + 1//4*a[2]*b[1]*b[3] + 1//4*a[3]*b[1]*b[2] …
  (3, 2, 3) => 1//4*a[1]*a[3]*b[2] + 1//4*a[2]*b[1]*b[3] + 1//2*b[1]*b[2]*b[3]
  (2, 1, 1) => 1//4*a[1]*b[2]*b[3] + 1//4*a[2]*a[3]*b[1] + 1//2*b[1]*b[2]*b[3]
  (1, 3, 2) => 1//4*a[1]*b[2]*b[3] + 1//4*a[2]*b[1]*b[3] + 1//4*a[3]*b[1]*b[2] …
  (1, 4, 2) => 1//4*a[1]*b[2]*b[3] + 1//4*a[2]*b[1]*b[3] + 1//4*a[3]*b[1]*b[2] …
  (2, 1, 3) => 1//4*a[1]*b[2]*b[3] + 1//4*a[2]*b[1]*b[3] + 1//4*a[3]*b[1]*b[2] …
  (2, 2, 4) => 1//4*a[1]*a[2]*b[3] + 1//4*a[3]*b[1]*b[2] + 1//2*b[1]*b[2]*b[3]
  (4, 3, 4) => 1//4*a[1]*a[3]*b[2] + 1//4*a[2]*b[1]*b[3] + 1//2*b[1]*b[2]*b[3]
  (2, 2, 1) => 1//4*a[1]*a[2]*b[3] + 1//4*a[3]*b[1]*b[2] + 1//2*b[1]*b[2]*b[3]
  (4, 4, 4) => 1//4*a[1]*a[2]*a[3] + 3//4*b[1]*b[2]*b[3]
  (4, 3, 1) => 1//4*a[1]*b[2]*b[3] + 1//4*a[2]*b[1]*b[3] + 1//4*a[3]*b[1]*b[2] …
  (3, 3, 2) => 1//4*a[1]*a[2]*b[3] + 1//4*a[3]*b[1]*b[2] + 1//2*b[1]*b[2]*b[3]
  ⋮         => ⋮

fourier_map(pm::GroupBasedPhylogeneticModel)

Create a parametrization for a GroupBasedPhylogeneticModel of type Dictionary.

Iterate through all possible states of the leaf random variables and calculates their corresponding probabilities using group actions and laws of conditional independence. Return a dictionary of polynomials indexed by the states. Use auxiliary function monomial_fourier(pm::GroupBasedPhylogeneticModel, leaves_states::Vector{Int}) and fourier_parametrization(pm::GroupBasedPhylogeneticModel, leaves_states::Vector{Int}).

Examples

julia> pm = jukes_cantor_model(graph_from_edges(Directed,[[4,1],[4,2],[4,3]]));

julia> q = fourier_map(pm)
Dict{Tuple{Vararg{Int64}}, QQMPolyRingElem} with 64 entries:
  (1, 2, 1) => 0
  (3, 1, 1) => 0
  (4, 4, 2) => 0
  (1, 2, 3) => 0
  (3, 1, 3) => x[2, 1]*x[1, 2]*x[3, 2]
  (3, 2, 4) => x[1, 2]*x[2, 2]*x[3, 2]
  (3, 2, 1) => 0
  (2, 1, 4) => 0
  (3, 2, 3) => 0
  (2, 1, 1) => 0
  (1, 3, 2) => 0
  (1, 4, 2) => 0
  (2, 1, 3) => 0
  (2, 2, 4) => 0
  (4, 3, 4) => 0
  (2, 2, 1) => x[3, 1]*x[1, 2]*x[2, 2]
  (4, 4, 4) => 0
  (4, 3, 1) => 0
  (3, 3, 2) => 0
  ⋮         => ⋮

compute_equivalent_classes(parametrization::Dict{Tuple{Vararg{Int64}}, QQMPolyRingElem})

Given the parametrization of a PhylogeneticModel, cancel all duplicate entries and return equivalence classes of states which are attached the same probabilities.

Examples

julia> pm = jukes_cantor_model(graph_from_edges(Directed,[[4,1],[4,2],[4,3]]));

julia> p = probability_map(pm);

julia> q = fourier_map(pm);

julia> p_equivclasses = compute_equivalent_classes(p);

julia> p_equivclasses.parametrization
Dict{Tuple{Vararg{Int64}}, QQMPolyRingElem} with 5 entries:
  (1, 2, 1) => 1//4*a[1]*a[3]*b[2] + 1//4*a[2]*b[1]*b[3] + 1//2*b[1]*b[2]*b[3]
  (1, 1, 1) => 1//4*a[1]*a[2]*a[3] + 3//4*b[1]*b[2]*b[3]
  (1, 2, 2) => 1//4*a[1]*b[2]*b[3] + 1//4*a[2]*a[3]*b[1] + 1//2*b[1]*b[2]*b[3]
  (1, 2, 3) => 1//4*a[1]*b[2]*b[3] + 1//4*a[2]*b[1]*b[3] + 1//4*a[3]*b[1]*b[2] …
  (1, 1, 2) => 1//4*a[1]*a[2]*b[3] + 1//4*a[3]*b[1]*b[2] + 1//2*b[1]*b[2]*b[3]

julia> p_equivclasses.classes
Dict{Tuple{Vararg{Int64}}, Vector{Tuple{Vararg{Int64}}}} with 5 entries:
  (1, 2, 1) => [(1, 2, 1), (1, 3, 1), (1, 4, 1), (2, 1, 2), (2, 3, 2), (2, 4, 2…
  (1, 1, 1) => [(1, 1, 1), (2, 2, 2), (3, 3, 3), (4, 4, 4)]
  (1, 2, 2) => [(1, 2, 2), (1, 3, 3), (1, 4, 4), (2, 1, 1), (2, 3, 3), (2, 4, 4…
  (1, 2, 3) => [(1, 2, 3), (1, 2, 4), (1, 3, 2), (1, 3, 4), (1, 4, 2), (1, 4, 3…
  (1, 1, 2) => [(1, 1, 2), (1, 1, 3), (1, 1, 4), (2, 2, 1), (2, 2, 3), (2, 2, 4…

julia> q_equivclasses = compute_equivalent_classes(q);

julia> q_equivclasses.parametrization
Dict{Tuple{Vararg{Int64}}, QQMPolyRingElem} with 5 entries:
  (1, 1, 1) => x[1, 1]*x[2, 1]*x[3, 1]
  (2, 3, 4) => x[1, 2]*x[2, 2]*x[3, 2]
  (2, 2, 1) => x[3, 1]*x[1, 2]*x[2, 2]
  (1, 2, 2) => x[1, 1]*x[2, 2]*x[3, 2]
  (2, 1, 2) => x[2, 1]*x[1, 2]*x[3, 2]

sum_equivalent_classes(equivalent_classes::NamedTuple{(:parametrization, :classes), Tuple{Dict{Tuple{Vararg{Int64}}, QQMPolyRingElem}, Dict{Tuple{Vararg{Int64}}, Vector{Tuple{Vararg{Int64}}}}}})

Take the output of the function compute_equivalent_classes for PhylogeneticModel and multiply by a factor to obtain probabilities as specified on the original small trees database.

Examples

julia> pm = jukes_cantor_model(graph_from_edges(Directed,[[4,1],[4,2],[4,3]]));

julia> p = probability_map(pm);

julia> p_equivclasses = compute_equivalent_classes(p);

julia> sum_equivalent_classes(p_equivclasses)
Dict{Tuple{Int64, Int64, Int64}, QQMPolyRingElem} with 5 entries:
  (1, 2, 1) => 3*a[1]*a[3]*b[2] + 3*a[2]*b[1]*b[3] + 6*b[1]*b[2]*b[3]
  (1, 1, 1) => a[1]*a[2]*a[3] + 3*b[1]*b[2]*b[3]
  (1, 2, 2) => 3*a[1]*b[2]*b[3] + 3*a[2]*a[3]*b[1] + 6*b[1]*b[2]*b[3]
  (1, 2, 3) => 6*a[1]*b[2]*b[3] + 6*a[2]*b[1]*b[3] + 6*a[3]*b[1]*b[2] + 6*b[1]*…
  (1, 1, 2) => 3*a[1]*a[2]*b[3] + 3*a[3]*b[1]*b[2] + 6*b[1]*b[2]*b[3]

specialized_fourier_transform(pm::GroupBasedPhylogeneticModel, p_classes::Dict{Tuple{Vararg{Int64}}, Vector{Tuple{Vararg{Int64}}}}, q_classes::Dict{Tuple{Vararg{Int64}}, Vector{Tuple{Vararg{Int64}}}})

Reparametrize between a model specification in terms of probability and Fourier coordinates. The input of equivalent classes is optional, if they are not entered they will be computed.

Examples

julia> pm = jukes_cantor_model(graph_from_edges(Directed,[[4,1],[4,2],[4,3]]));

julia> p_equivclasses = compute_equivalent_classes(probability_map(pm));

julia> q_equivclasses = compute_equivalent_classes(fourier_map(pm));

julia> specialized_fourier_transform(pm, p_equivclasses.classes, q_equivclasses.classes)
5×5 Matrix{QQMPolyRingElem}:
 1  1      1      1      1
 1  -1//3  -1//3  1      -1//3
 1  -1//3  1      -1//3  -1//3
 1  1      -1//3  -1//3  -1//3
 1  -1//3  -1//3  -1//3  1//3

inverse_specialized_fourier_transform(pm::GroupBasedPhylogeneticModel, p_classes::Dict{Tuple{Vararg{Int64}}, Vector{Tuple{Vararg{Int64}}}}, q_classes::Dict{Tuple{Vararg{Int64}}, Vector{Tuple{Vararg{Int64}}}})

Reparametrize between a model specification in terms of Fourier and probability coordinates.

Examples

julia> pm = jukes_cantor_model(graph_from_edges(Directed,[[4,1],[4,2],[4,3]]));

julia> p_equivclasses = compute_equivalent_classes(probability_map(pm));

julia> q_equivclasses = compute_equivalent_classes(fourier_map(pm));

julia> inverse_specialized_fourier_transform(pm, p_equivclasses.classes, q_equivclasses.classes)
5×5 Matrix{QQMPolyRingElem}:
 1//16  3//16   3//16   3//16   3//8
 3//16  -3//16  -3//16  9//16   -3//8
 3//16  -3//16  9//16   -3//16  -3//8
 3//16  9//16   -3//16  -3//16  -3//8
 3//8   -3//8   -3//8   -3//8   3//4