Instruction:
Can you generate the documentation for the following Python 3 function
def process_alignment_dict(self):

        # number of sequences in alignment
        nseq = len(self.aln)

        inv_map = defaultdict(list)
        for k,v in self.aln.items():
            for pos, bs in v.items():
                inv_map[pos].append(bs)

        self.nonref_positions = np.sort(list(inv_map.keys()))
        self.inferred_const_sites = []

        ambiguous_char = self.gtr.ambiguous
        nonref_const = []
        nonref_alleles = []
        ambiguous_const = []
        variable_pos = []
        for pos, bs in inv_map.items(): #loop over positions and patterns
            bases = "".join(np.unique(bs))
            if len(bs) == nseq:
                if (len(bases)<=2 and ambiguous_char in bases) or len(bases)==1:
                    # all sequences different from reference, but only one state
                    # (other than ambiguous_char) in column
                    nonref_const.append(pos)
                    nonref_alleles.append(bases.replace(ambiguous_char, ''))
                    if ambiguous_char in bases: #keep track of sites 'made constant'
                        self.inferred_const_sites.append(pos)
                else:
                    # at least two non-reference alleles
                    variable_pos.append(pos)
            else:
                # not every sequence different from reference
                if bases==ambiguous_char:
                    ambiguous_const.append(pos)
                    self.inferred_const_sites.append(pos) #keep track of sites 'made constant'
                else:
                    # at least one non ambiguous non-reference allele not in
                    # every sequence
                    variable_pos.append(pos)

        refMod = np.array(list(self.ref))
        # place constant non reference positions by their respective allele
        refMod[nonref_const] = nonref_alleles
        # mask variable positions
        states = self.gtr.alphabet
        # maybe states = np.unique(refMod)
        refMod[variable_pos] = '.'

        # for each base in the gtr, make constant alignment pattern and
        # assign it to all const positions in the modified reference sequence
        reduced_alignment_const = []
        alignment_patterns_const = {}
        for base in states:
            p = base*nseq
            pos = list(np.where(refMod==base)[0])
            #if the alignment doesn't have a const site of this base, don't add! (ex: no '----' site!)
            if len(pos):
                alignment_patterns_const[p] = [len(reduced_alignment_const), pos]
                reduced_alignment_const.append(list(p))

        return reduced_alignment_const, alignment_patterns_const, variable_pos

Response: process the alignment dictionary for non - variable postitions and non - reference alleles and non - reference positions


Instruction:
Explain what the following Python 3 code does
def prepare_tree(self):
        self.tree.root.branch_length = 0.001
        self.tree.root.mutation_length = self.tree.root.branch_length
        self.tree.root.mutations = []
        self.tree.ladderize()
        self._prepare_nodes()
        self._leaves_lookup = {node.name:node for node in self.tree.get_terminals()}

Response: Prepare the tree for the tree.


Instruction:
How would you implement a function in Python 3 that
sets auxilliary parameters to every node of the tree.

Response:
def _prepare_nodes(self):
        """
        Set auxilliary parameters to every node of the tree.
        """
        self.tree.root.up = None
        self.tree.root.bad_branch=self.tree.root.bad_branch if hasattr(self.tree.root, 'bad_branch') else False
        internal_node_count = 0
        for clade in self.tree.get_nonterminals(order='preorder'): # parents first
            internal_node_count+=1
            if clade.name is None:
                clade.name = "NODE_" + format(self._internal_node_count, '07d')
                self._internal_node_count += 1
            for c in clade.clades:
                if c.is_terminal():
                    c.bad_branch = c.bad_branch if hasattr(c, 'bad_branch') else False
                c.up = clade

        for clade in self.tree.get_nonterminals(order='postorder'): # parents first
            clade.bad_branch = all([c.bad_branch for c in clade])

        self._calc_dist2root()
        self._internal_node_count = max(internal_node_count, self._internal_node_count)


Instruction:
Make a summary of the following Python 3 code
def _calc_dist2root(self):
        self.tree.root.dist2root = 0.0
        for clade in self.tree.get_nonterminals(order='preorder'): # parents first
            for c in clade.clades:
                if not hasattr(c, 'mutation_length'):
                    c.mutation_length=c.branch_length
                c.dist2root = c.up.dist2root + c.mutation_length

Response: Calculates the distance between the root and the root node.


Instruction:
Explain what the following Python 3 code does
def infer_gtr(self, print_raw=False, marginal=False, normalized_rate=True,
                  fixed_pi=None, pc=5.0, **kwargs):
        """
        Calculates a GTR model given the multiple sequence alignment and the tree.
        It performs ancestral sequence inferrence (joint or marginal), followed by
        the branch lengths optimization. Then, the numbers of mutations are counted
        in the optimal tree and related to the time within the mutation happened.
        From these statistics, the relative state transition probabilities are inferred,
        and the transition matrix is computed.

        The result is used to construct the new GTR model of type 'custom'.
        The model is assigned to the TreeAnc and is used in subsequent analysis.

        Parameters
        -----------

         print_raw : bool
            If True, print the inferred GTR model

         marginal : bool
            If True, use marginal sequence reconstruction

         normalized_rate : bool
            If True, sets the mutation rate prefactor to 1.0.

         fixed_pi : np.array
            Provide the equilibrium character concentrations.
            If None is passed, the concentrations will be inferred from the alignment.

         pc: float
            Number of pseudo counts to use in gtr inference

        Returns
        -------

         gtr : GTR
            The inferred GTR model
        """

        # decide which type of the Maximum-likelihood reconstruction use
        # (marginal) or (joint)
        if marginal:
            _ml_anc = self._ml_anc_marginal
        else:
            _ml_anc = self._ml_anc_joint

        self.logger("TreeAnc.infer_gtr: inferring the GTR model from the tree...", 1)
        if (self.tree is None) or (self.aln is None):
            self.logger("TreeAnc.infer_gtr: ERROR, alignment or tree are missing", 0)
            return ttconf.ERROR

        _ml_anc(final=True, **kwargs) # call one of the reconstruction types
        alpha = list(self.gtr.alphabet)
        n=len(alpha)
        # matrix of mutations n_{ij}: i = derived state, j=ancestral state
        nij = np.zeros((n,n))
        Ti = np.zeros(n)

        self.logger("TreeAnc.infer_gtr: counting mutations...", 2)
        for node in self.tree.find_clades():
            if hasattr(node,'mutations'):
                for a,pos, d in node.mutations:
                    i,j = alpha.index(d), alpha.index(a)
                    nij[i,j]+=1
                    Ti[j] += 0.5*self._branch_length_to_gtr(node)
                    Ti[i] -= 0.5*self._branch_length_to_gtr(node)
                for ni,nuc in enumerate(node.cseq):
                    i = alpha.index(nuc)
                    Ti[i] += self._branch_length_to_gtr(node)*self.multiplicity[ni]
        self.logger("TreeAnc.infer_gtr: counting mutations...done", 3)
        if print_raw:
            print('alphabet:',alpha)
            print('n_ij:', nij, nij.sum())
            print('T_i:', Ti, Ti.sum())
        root_state = np.array([np.sum((self.tree.root.cseq==nuc)*self.multiplicity) for nuc in alpha])

        self._gtr = GTR.infer(nij, Ti, root_state, fixed_pi=fixed_pi, pc=pc,
                              alphabet=self.gtr.alphabet, logger=self.logger,
                              prof_map = self.gtr.profile_map)

        if normalized_rate:
            self.logger("TreeAnc.infer_gtr: setting overall rate to 1.0...", 2)
            self._gtr.mu=1.0
        return self._gtr

Response: This function calculates the GTR model from the tree and the tree s alignment and the tree s tree and the tree s pseudo counts.


Instruction:
How would you implement a function in Python 3 that
reconstructs the ancestral sequences for a particular tree and anomaly level.

Response:
def reconstruct_anc(self, method='probabilistic', infer_gtr=False,
                        marginal=False, **kwargs):
        """Reconstruct ancestral sequences

        Parameters
        ----------
        method : str
           Method to use. Supported values are "fitch" and "ml"

        infer_gtr : bool
           Infer a GTR model before reconstructing the sequences

        marginal : bool
           Assign sequences that are most likely after averaging over all other nodes
           instead of the jointly most likely sequences.
        **kwargs
            additional keyword arguments that are passed down to :py:meth:`TreeAnc.infer_gtr` and :py:meth:`TreeAnc._ml_anc`

        Returns
        -------
        N_diff : int
           Number of nucleotides different from the previous
           reconstruction.  If there were no pre-set sequences, returns N*L

        """
        self.logger("TreeAnc.infer_ancestral_sequences with method: %s, %s"%(method, 'marginal' if marginal else 'joint'), 1)
        if (self.tree is None) or (self.aln is None):
            self.logger("TreeAnc.infer_ancestral_sequences: ERROR, alignment or tree are missing", 0)
            return ttconf.ERROR

        if method in ['ml', 'probabilistic']:
            if marginal:
                _ml_anc = self._ml_anc_marginal
            else:
                _ml_anc = self._ml_anc_joint
        else:
            _ml_anc = self._fitch_anc

        if infer_gtr:
            tmp = self.infer_gtr(marginal=marginal, **kwargs)
            if tmp==ttconf.ERROR:
                return tmp
            N_diff = _ml_anc(**kwargs)
        else:
            N_diff = _ml_anc(**kwargs)

        return N_diff


Instruction:
Create a Python 3 function for
getting the mutations on a tree branch.

Response:
def get_mutations(self, node, keep_var_ambigs=False):
        """
        Get the mutations on a tree branch. Take compressed sequences from both sides
        of the branch (attached to the node), compute mutations between them, and
        expand these mutations to the positions in the real sequences.

        Parameters
        ----------
        node : PhyloTree.Clade
           Tree node, which is the child node attached to the branch.

        keep_var_ambigs : boolean
           If true, generates mutations based on the *original* compressed sequence, which
           may include ambiguities. Note sites that only have 1 unambiguous base and ambiguous
           bases ("AAAAANN") are stripped of ambiguous bases *before* compression, so ambiguous
           bases will **not** be preserved.

        Returns
        -------
        muts : list
          List of mutations. Each mutation is represented as tuple of
          :code:`(parent_state, position, child_state)`.
        """

        # if ambiguous site are to be restored and node is terminal,
        # assign original sequence, else reconstructed cseq
        node_seq = node.cseq
        if keep_var_ambigs and hasattr(node, "original_cseq") and node.is_terminal():
            node_seq = node.original_cseq

        muts = []
        diff_pos = np.where(node.up.cseq!=node_seq)[0]
        for p in diff_pos:
            anc = node.up.cseq[p]
            der = node_seq[p]
            # expand to the positions in real sequence
            muts.extend([(anc, pos, der) for pos in self.reduced_to_full_sequence_map[p]])

        #sort by position
        return sorted(muts, key=lambda x:x[1])


Instruction:
Can you generate a brief explanation for the following Python 3 code
def get_branch_mutation_matrix(self, node, full_sequence=False):
        pp,pc = self.marginal_branch_profile(node)

        # calculate pc_i [e^Qt]_ij pp_j for each site
        expQt = self.gtr.expQt(self._branch_length_to_gtr(node))
        if len(expQt.shape)==3: # site specific model
            mut_matrix_stack = np.einsum('ai,aj,ija->aij', pc, pp, expQt)
        else:
            mut_matrix_stack = np.einsum('ai,aj,ij->aij', pc, pp, expQt)

        # normalize this distribution
        normalizer = mut_matrix_stack.sum(axis=2).sum(axis=1)
        mut_matrix_stack = np.einsum('aij,a->aij', mut_matrix_stack, 1.0/normalizer)

        # expand to full sequence if requested
        if full_sequence:
            return mut_matrix_stack[self.full_to_reduced_sequence_map]
        else:
            return mut_matrix_stack

Response: uses results from marginal ancestral inference to return a joint mutation matrix of the sequence states at both ends of the branch.


Instruction:
Can you generate the documentation for the following Python 3 function
def expanded_sequence(self, node, include_additional_constant_sites=False):
        if include_additional_constant_sites:
            L = self.seq_len
        else:
            L = self.seq_len - self.additional_constant_sites

        return node.cseq[self.full_to_reduced_sequence_map[:L]]

Response: Expand a nodes compressed sequence into the real sequence


Instruction:
Make a summary of the following Python 3 code
def dict_sequence(self, node, keep_var_ambigs=False):
        seq = {}

        node_seq = node.cseq
        if keep_var_ambigs and hasattr(node, "original_cseq") and node.is_terminal():
            node_seq = node.original_cseq

        for pos in self.nonref_positions:
            cseqLoc = self.full_to_reduced_sequence_map[pos]
            base = node_seq[cseqLoc]
            if self.ref[pos] != base:
                seq[pos] = base

        return seq

Response: Returns a dict of all the basepairs of the node s variants and their positions.


Instruction:
Can you tell what is the following Python 3 function doing
def _fitch_anc(self, **kwargs):
        # set fitch profiiles to each terminal node

        for l in self.tree.get_terminals():
            l.state = [[k] for k in l.cseq]

        L = len(self.tree.get_terminals()[0].cseq)

        self.logger("TreeAnc._fitch_anc: Walking up the tree, creating the Fitch profiles",2)
        for node in self.tree.get_nonterminals(order='postorder'):
            node.state = [self._fitch_state(node, k) for k in range(L)]

        ambs = [i for i in range(L) if len(self.tree.root.state[i])>1]
        if len(ambs) > 0:
            for amb in ambs:
                self.logger("Ambiguous state of the root sequence "
                                    "in the position %d: %s, "
                                    "choosing %s" % (amb, str(self.tree.root.state[amb]),
                                                     self.tree.root.state[amb][0]), 4)
        self.tree.root.cseq = np.array([k[np.random.randint(len(k)) if len(k)>1 else 0]
                                           for k in self.tree.root.state])

        if self.is_vcf:
            self.tree.root.sequence = self.dict_sequence(self.tree.root)
        else:
            self.tree.root.sequence = self.expanded_sequence(self.tree.root)

        self.logger("TreeAnc._fitch_anc: Walking down the self.tree, generating sequences from the "
                         "Fitch profiles.", 2)
        N_diff = 0
        for node in self.tree.get_nonterminals(order='preorder'):
            if node.up != None: # not root
                sequence =  np.array([node.up.cseq[i]
                        if node.up.cseq[i] in node.state[i]
                        else node.state[i][0] for i in range(L)])
                if hasattr(node, 'sequence'):
                    N_diff += (sequence!=node.cseq).sum()
                else:
                    N_diff += L
                node.cseq = sequence
                if self.is_vcf:
                    node.sequence = self.dict_sequence(node)
                else:
                    node.sequence = self.expanded_sequence(node)
                node.mutations = self.get_mutations(node)

            node.profile = seq2prof(node.cseq, self.gtr.profile_map)
            del node.state # no need to store Fitch states
        self.logger("Done ancestral state reconstruction",3)
        for node in self.tree.get_terminals():
            node.profile = seq2prof(node.original_cseq, self.gtr.profile_map)
        return N_diff

Response: This method reconstructs the ancestral states using Fitch s algorithm.


Instruction:
How would you explain what the following Python 3 function does
def _fitch_state(self, node, pos):
        state = self._fitch_intersect([k.state[pos] for k in node.clades])
        if len(state) == 0:
            state = np.concatenate([k.state[pos] for k in node.clades])
        return state

Response: Returns the Fitch state for a single character of the given node at the given position.


Instruction:
Can you implement a function in Python 3 that
finds the intersection of any number of 1D arrays.

Response:
def _fitch_intersect(self, arrays):
        """
        Find the intersection of any number of 1D arrays.
        Return the sorted, unique values that are in all of the input arrays.
        Adapted from numpy.lib.arraysetops.intersect1d
        """
        def pairwise_intersect(arr1, arr2):
            s2 = set(arr2)
            b3 = [val for val in arr1 if val in s2]
            return b3

        arrays = list(arrays) # allow assignment
        N = len(arrays)
        while N > 1:
            arr1 = arrays.pop()
            arr2 = arrays.pop()
            arr = pairwise_intersect(arr1, arr2)
            arrays.append(arr)
            N = len(arrays)

        return arrays[0]


Instruction:
Given the following Python 3 function, write the documentation
def sequence_LH(self, pos=None, full_sequence=False):
        if not hasattr(self.tree, "total_sequence_LH"):
            self.logger("TreeAnc.sequence_LH: you need to run marginal ancestral inference first!", 1)
            self.infer_ancestral_sequences(marginal=True)
        if pos is not None:
            if full_sequence:
                compressed_pos = self.full_to_reduced_sequence_map[pos]
            else:
                compressed_pos = pos
            return self.tree.sequence_LH[compressed_pos]
        else:
            return self.tree.total_sequence_LH

Response: return the likelihood of the observed sequences given the tree


Instruction:
Write a Python 3 script to
calculate the likelihood of the given realization of the sequences in the tree returns the likelihood of the given realization of the sequences in the tree

Response:
def ancestral_likelihood(self):
        """
        Calculate the likelihood of the given realization of the sequences in
        the tree

        Returns
        -------

         log_lh : float
            The tree likelihood given the sequences
        """
        log_lh = np.zeros(self.multiplicity.shape[0])
        for node in self.tree.find_clades(order='postorder'):

            if node.up is None: #  root node
                # 0-1 profile
                profile = seq2prof(node.cseq, self.gtr.profile_map)
                # get the probabilities to observe each nucleotide
                profile *= self.gtr.Pi
                profile = profile.sum(axis=1)
                log_lh += np.log(profile) # product over all characters
                continue

            t = node.branch_length

            indices = np.array([(np.argmax(self.gtr.alphabet==a),
                        np.argmax(self.gtr.alphabet==b)) for a, b in zip(node.up.cseq, node.cseq)])

            logQt = np.log(self.gtr.expQt(t))
            lh = logQt[indices[:, 1], indices[:, 0]]
            log_lh += lh

        return log_lh


Instruction:
Write a Python 3 function for
setting the branch lengths to either mutation lengths of given branch lengths.

Response:
def _branch_length_to_gtr(self, node):
        """
        Set branch lengths to either mutation lengths of given branch lengths.
        The assigend values are to be used in the following ML analysis.
        """
        if self.use_mutation_length:
            return max(ttconf.MIN_BRANCH_LENGTH*self.one_mutation, node.mutation_length)
        else:
            return max(ttconf.MIN_BRANCH_LENGTH*self.one_mutation, node.branch_length)


Instruction:
Create a Python 3 function to
perform marginal ML reconstruction of the ancestral states. In contrast to joint reconstructions, this needs to access the probabilities rather than only log probabilities and is hence handled by a separate function. Parameters ---------- store_compressed : bool, default True attach a reduced representation of sequence changed to each branch final : bool, default True stop full length by expanding sites with identical alignment patterns sample_from_profile : bool or str assign sequences probabilistically according to the inferred probabilities of ancestral states instead of to their ML value. This parameter can also take the value 'root' in which case probabilistic sampling will happen at the root but at no other node.

Response:
def _ml_anc_marginal(self, store_compressed=False, final=True, sample_from_profile=False,
                         debug=False, **kwargs):
        """
        Perform marginal ML reconstruction of the ancestral states. In contrast to
        joint reconstructions, this needs to access the probabilities rather than only
        log probabilities and is hence handled by a separate function.

        Parameters
        ----------

         store_compressed : bool, default True
            attach a reduced representation of sequence changed to each branch

         final : bool, default True
            stop full length by expanding sites with identical alignment patterns

         sample_from_profile : bool or str
            assign sequences probabilistically according to the inferred probabilities
            of ancestral states instead of to their ML value. This parameter can also
            take the value 'root' in which case probabilistic sampling will happen
            at the root but at no other node.

        """

        tree = self.tree
        # number of nucleotides changed from prev reconstruction
        N_diff = 0

        L = self.multiplicity.shape[0]
        n_states = self.gtr.alphabet.shape[0]
        self.logger("TreeAnc._ml_anc_marginal: type of reconstruction: Marginal", 2)

        self.logger("Attaching sequence profiles to leafs... ", 3)
        #  set the leaves profiles
        for leaf in tree.get_terminals():
            # in any case, set the profile
            leaf.marginal_subtree_LH = seq2prof(leaf.original_cseq, self.gtr.profile_map)
            leaf.marginal_subtree_LH_prefactor = np.zeros(L)

        self.logger("Walking up the tree, computing likelihoods... ", 3)
        # propagate leaves --> root, set the marginal-likelihood messages
        for node in tree.get_nonterminals(order='postorder'): #leaves -> root
            # regardless of what was before, set the profile to ones
            tmp_log_subtree_LH = np.zeros((L,n_states), dtype=float)
            node.marginal_subtree_LH_prefactor = np.zeros(L, dtype=float)
            for ch in node.clades:
                ch.marginal_log_Lx = self.gtr.propagate_profile(ch.marginal_subtree_LH,
                    self._branch_length_to_gtr(ch), return_log=True) # raw prob to transfer prob up
                tmp_log_subtree_LH += ch.marginal_log_Lx
                node.marginal_subtree_LH_prefactor += ch.marginal_subtree_LH_prefactor

            node.marginal_subtree_LH, offset = normalize_profile(tmp_log_subtree_LH, log=True)
            node.marginal_subtree_LH_prefactor += offset # and store log-prefactor

        self.logger("Computing root node sequence and total tree likelihood...",3)
        # Msg to the root from the distant part (equ frequencies)
        if len(self.gtr.Pi.shape)==1:
            tree.root.marginal_outgroup_LH = np.repeat([self.gtr.Pi], tree.root.marginal_subtree_LH.shape[0], axis=0)
        else:
            tree.root.marginal_outgroup_LH = np.copy(self.gtr.Pi.T)

        tree.root.marginal_profile, pre = normalize_profile(tree.root.marginal_outgroup_LH*tree.root.marginal_subtree_LH)
        marginal_LH_prefactor = tree.root.marginal_subtree_LH_prefactor + pre

        # choose sequence characters from this profile.
        # treat root node differently to avoid piling up mutations on the longer branch
        if sample_from_profile=='root':
            root_sample_from_profile = True
            other_sample_from_profile = False
        elif isinstance(sample_from_profile, bool):
            root_sample_from_profile = sample_from_profile
            other_sample_from_profile = sample_from_profile

        seq, prof_vals, idxs = prof2seq(tree.root.marginal_profile,
                                        self.gtr, sample_from_prof=root_sample_from_profile, normalize=False)

        self.tree.sequence_LH = marginal_LH_prefactor
        self.tree.total_sequence_LH = (self.tree.sequence_LH*self.multiplicity).sum()
        self.tree.root.cseq = seq
        gc.collect()

        if final:
            if self.is_vcf:
                self.tree.root.sequence = self.dict_sequence(self.tree.root)
            else:
                self.tree.root.sequence = self.expanded_sequence(self.tree.root)

        self.logger("Walking down the tree, computing maximum likelihood sequences...",3)
        # propagate root -->> leaves, reconstruct the internal node sequences
        # provided the upstream message + the message from the complementary subtree
        for node in tree.find_clades(order='preorder'):
            if node.up is None: # skip if node is root
                continue

            # integrate the information coming from parents with the information
            # of all children my multiplying it to the prev computed profile
            node.marginal_outgroup_LH, pre = normalize_profile(np.log(node.up.marginal_profile) - node.marginal_log_Lx,
                                                               log=True, return_offset=False)
            tmp_msg_from_parent = self.gtr.evolve(node.marginal_outgroup_LH,
                                                 self._branch_length_to_gtr(node), return_log=False)

            node.marginal_profile, pre = normalize_profile(node.marginal_subtree_LH * tmp_msg_from_parent, return_offset=False)

            # choose sequence based maximal marginal LH.
            seq, prof_vals, idxs = prof2seq(node.marginal_profile, self.gtr,
                                            sample_from_prof=other_sample_from_profile, normalize=False)

            if hasattr(node, 'cseq') and node.cseq is not None:
                N_diff += (seq!=node.cseq).sum()
            else:
                N_diff += L

            #assign new sequence
            node.cseq = seq
            if final:
                if self.is_vcf:
                    node.sequence = self.dict_sequence(node)
                else:
                    node.sequence = self.expanded_sequence(node)
                node.mutations = self.get_mutations(node)

        # note that the root doesn't contribute to N_diff (intended, since root sequence is often ambiguous)
        self.logger("TreeAnc._ml_anc_marginal: ...done", 3)
        if store_compressed:
            self._store_compressed_sequence_pairs()

        # do clean-up:
        if not debug:
            for node in self.tree.find_clades():
                try:
                    del node.marginal_log_Lx
                    del node.marginal_subtree_LH_prefactor
                except:
                    pass
        gc.collect()

        return N_diff


Instruction:
How would you explain what the following Python 3 function does
def _ml_anc_joint(self, store_compressed=True, final=True, sample_from_profile=False,
                            debug=False, **kwargs):

        """
        Perform joint ML reconstruction of the ancestral states. In contrast to
        marginal reconstructions, this only needs to compare and multiply LH and
        can hence operate in log space.

        Parameters
        ----------

         store_compressed : bool, default True
            attach a reduced representation of sequence changed to each branch

         final : bool, default True
            stop full length by expanding sites with identical alignment patterns

         sample_from_profile : str
            This parameter can take the value 'root' in which case probabilistic
            sampling will happen at the root. otherwise sequences at ALL nodes are
            set to the value that jointly optimized the likelihood.

        """
        N_diff = 0 # number of sites differ from perv reconstruction
        L = self.multiplicity.shape[0]
        n_states = self.gtr.alphabet.shape[0]

        self.logger("TreeAnc._ml_anc_joint: type of reconstruction: Joint", 2)

        self.logger("TreeAnc._ml_anc_joint: Walking up the tree, computing likelihoods... ", 3)
        # for the internal nodes, scan over all states j of this node, maximize the likelihood
        for node in self.tree.find_clades(order='postorder'):
            if node.up is None:
                node.joint_Cx=None # not needed for root
                continue

            # preallocate storage
            node.joint_Lx = np.zeros((L, n_states))             # likelihood array
            node.joint_Cx = np.zeros((L, n_states), dtype=int)  # max LH indices
            branch_len = self._branch_length_to_gtr(node)
            # transition matrix from parent states to the current node states.
            # denoted as Pij(i), where j - parent state, i - node state
            log_transitions = np.log(np.maximum(ttconf.TINY_NUMBER, self.gtr.expQt(branch_len)))
            if node.is_terminal():
                try:
                    msg_from_children = np.log(np.maximum(seq2prof(node.original_cseq, self.gtr.profile_map), ttconf.TINY_NUMBER))
                except:
                    raise ValueError("sequence assignment to node "+node.name+" failed")
                msg_from_children[np.isnan(msg_from_children) | np.isinf(msg_from_children)] = -ttconf.BIG_NUMBER
            else:
                # Product (sum-Log) over all child subtree likelihoods.
                # this is prod_ch L_x(i)
                msg_from_children = np.sum(np.stack([c.joint_Lx for c in node.clades], axis=0), axis=0)

            # for every possible state of the parent node,
            # get the best state of the current node
            # and compute the likelihood of this state
            for char_i, char in enumerate(self.gtr.alphabet):
                # Pij(i) * L_ch(i) for given parent state j
                msg_to_parent = (log_transitions[:,char_i].T + msg_from_children)
                # For this parent state, choose the best state of the current node:
                node.joint_Cx[:, char_i] = msg_to_parent.argmax(axis=1)
                # compute the likelihood of the best state of the current node
                # given the state of the parent (char_i)
                node.joint_Lx[:, char_i] = msg_to_parent.max(axis=1)

        # root node profile = likelihood of the total tree
        msg_from_children = np.sum(np.stack([c.joint_Lx for c in self.tree.root], axis = 0), axis=0)
        # Pi(i) * Prod_ch Lch(i)
        self.tree.root.joint_Lx = msg_from_children + np.log(self.gtr.Pi).T
        normalized_profile = (self.tree.root.joint_Lx.T - self.tree.root.joint_Lx.max(axis=1)).T

        # choose sequence characters from this profile.
        # treat root node differently to avoid piling up mutations on the longer branch
        if sample_from_profile=='root':
            root_sample_from_profile = True
        elif isinstance(sample_from_profile, bool):
            root_sample_from_profile = sample_from_profile

        seq, anc_lh_vals, idxs = prof2seq(np.exp(normalized_profile),
                                    self.gtr, sample_from_prof = root_sample_from_profile)

        # compute the likelihood of the most probable root sequence
        self.tree.sequence_LH = np.choose(idxs, self.tree.root.joint_Lx.T)
        self.tree.sequence_joint_LH = (self.tree.sequence_LH*self.multiplicity).sum()
        self.tree.root.cseq = seq
        self.tree.root.seq_idx = idxs
        if final:
            if self.is_vcf:
                self.tree.root.sequence = self.dict_sequence(self.tree.root)
            else:
                self.tree.root.sequence = self.expanded_sequence(self.tree.root)

        self.logger("TreeAnc._ml_anc_joint: Walking down the tree, computing maximum likelihood sequences...",3)
        # for each node, resolve the conditioning on the parent node
        for node in self.tree.find_clades(order='preorder'):

            # root node has no mutations, everything else has been alread y set
            if node.up is None:
                node.mutations = []
                continue

            # choose the value of the Cx(i), corresponding to the state of the
            # parent node i. This is the state of the current node
            node.seq_idx = np.choose(node.up.seq_idx, node.joint_Cx.T)
            # reconstruct seq, etc
            tmp_sequence = np.choose(node.seq_idx, self.gtr.alphabet)
            if hasattr(node, 'sequence') and node.cseq is not None:
                N_diff += (tmp_sequence!=node.cseq).sum()
            else:
                N_diff += L

            node.cseq = tmp_sequence
            if final:
                node.mutations = self.get_mutations(node)
                if self.is_vcf:
                    node.sequence = self.dict_sequence(node)
                else:
                    node.sequence = self.expanded_sequence(node)

        self.logger("TreeAnc._ml_anc_joint: ...done", 3)
        if store_compressed:
            self._store_compressed_sequence_pairs()

        # do clean-up
        if not debug:
            for node in self.tree.find_clades(order='preorder'):
                del node.joint_Lx
                del node.joint_Cx
                del node.seq_idx

        return N_diff

Response: Compute joint ML reconstruction of the ancestral states.


Instruction:
Can you tell what is the following Python 3 function doing
def _store_compressed_sequence_pairs(self):
        self.logger("TreeAnc._store_compressed_sequence_pairs...",2)
        for node in self.tree.find_clades():
            if node.up is None:
                continue
            self._store_compressed_sequence_to_node(node)
        self.logger("TreeAnc._store_compressed_sequence_pairs...done",3)

Response: Store the compressed sequence pairs in the tree.


Instruction:
Write a Python 3 script for
performing optimization for the branch lengths of the entire tree. This method only does a single path and needs to be iterated. **Note** this method assumes that each node stores information about its sequence as numpy.array object (node.sequence attribute). Therefore, before calling this method, sequence reconstruction with either of the available models must be performed. Parameters ---------- mode : str Optimize branch length assuming the joint ML sequence assignment of both ends of the branch (:code:`joint`), or trace over all possible sequence assignments on both ends of the branch (:code:`marginal`) (slower, experimental). **kwargs : Keyword arguments Keyword Args ------------ verbose : int Output level store_old : bool If True, the old lengths will be saved in :code:`node._old_dist` attribute. Useful for testing, and special post-processing.

Response:
def optimize_branch_length(self, mode='joint', **kwargs):
        """
        Perform optimization for the branch lengths of the entire tree.
        This method only does a single path and needs to be iterated.

        **Note** this method assumes that each node stores information
        about its sequence as numpy.array object (node.sequence attribute).
        Therefore, before calling this method, sequence reconstruction with
        either of the available models must be performed.

        Parameters
        ----------

         mode : str
            Optimize branch length assuming the joint ML sequence assignment
            of both ends of the branch (:code:`joint`), or trace over all possible sequence
            assignments on both ends of the branch (:code:`marginal`) (slower, experimental).

         **kwargs :
            Keyword arguments

        Keyword Args
        ------------

         verbose : int
            Output level

         store_old : bool
            If True, the old lengths will be saved in :code:`node._old_dist` attribute.
            Useful for testing, and special post-processing.

        """

        self.logger("TreeAnc.optimize_branch_length: running branch length optimization in mode %s..."%mode,1)
        if (self.tree is None) or (self.aln is None):
            self.logger("TreeAnc.optimize_branch_length: ERROR, alignment or tree are missing", 0)
            return ttconf.ERROR

        store_old_dist = False

        if 'store_old' in kwargs:
            store_old_dist = kwargs['store_old']

        if mode=='marginal':
            # a marginal ancestral reconstruction is required for
            # marginal branch length inference
            if not hasattr(self.tree.root, "marginal_profile"):
                self.infer_ancestral_sequences(marginal=True)

        max_bl = 0
        for node in self.tree.find_clades(order='postorder'):
            if node.up is None: continue # this is the root
            if store_old_dist:
                node._old_length = node.branch_length

            if mode=='marginal':
                new_len = self.optimal_marginal_branch_length(node)
            elif mode=='joint':
                new_len = self.optimal_branch_length(node)
            else:
                self.logger("treeanc.optimize_branch_length: unsupported optimization mode",4, warn=True)
                new_len = node.branch_length

            if new_len < 0:
                continue

            self.logger("Optimization results: old_len=%.4e, new_len=%.4e, naive=%.4e"
                   " Updating branch length..."%(node.branch_length, new_len, len(node.mutations)*self.one_mutation), 5)

            node.branch_length = new_len
            node.mutation_length=new_len
            max_bl = max(max_bl, new_len)

        # as branch lengths changed, the params must be fixed
        self.tree.root.up = None
        self.tree.root.dist2root = 0.0
        if max_bl>0.15 and mode=='joint':
            self.logger("TreeAnc.optimize_branch_length: THIS TREE HAS LONG BRANCHES."
                        " \n\t ****TreeTime IS NOT DESIGNED TO OPTIMIZE LONG BRANCHES."
                        " \n\t ****PLEASE OPTIMIZE BRANCHES WITH ANOTHER TOOL AND RERUN WITH"
                        " \n\t ****branch_length_mode='input'", 0, warn=True)
        self._prepare_nodes()
        return ttconf.SUCCESS


Instruction:
How would you explain what the following Python 3 function does
def optimal_branch_length(self, node):
        '''
        Calculate optimal branch length given the sequences of node and parent

        Parameters
        ----------
        node : PhyloTree.Clade
           TreeNode, attached to the branch.

        Returns
        -------
        new_len : float
           Optimal length of the given branch

        '''
        if node.up is None:
            return self.one_mutation

        parent = node.up
        if hasattr(node, 'compressed_sequence'):
            new_len = self.gtr.optimal_t_compressed(node.compressed_sequence['pair'],
                                                    node.compressed_sequence['multiplicity'])
        else:
            new_len = self.gtr.optimal_t(parent.cseq, node.cseq,
                                         pattern_multiplicity=self.multiplicity,
                                         ignore_gaps=self.ignore_gaps)
        return new_len

Response: Calculate the optimal branch length given the sequences of node and parent.


Instruction:
Explain what the following Python 3 code does
def marginal_branch_profile(self, node):
        '''
        calculate the marginal distribution of sequence states on both ends
        of the branch leading to node,

        Parameters
        ----------
        node : PhyloTree.Clade
           TreeNode, attached to the branch.

        Returns
        -------
        pp, pc : Pair of vectors (profile parent, pp) and (profile child, pc)
           that are of shape (L,n) where L is sequence length and n is alphabet size.
           note that this correspond to the compressed sequences.
        '''
        parent = node.up
        if parent is None:
            raise Exception("Branch profiles can't be calculated for the root!")
        if not hasattr(node, 'marginal_outgroup_LH'):
            raise Exception("marginal ancestral inference needs to be performed first!")

        pc = node.marginal_subtree_LH
        pp = node.marginal_outgroup_LH
        return pp, pc

Response: Calculates the marginal distribution of sequence states on both ends of the branch leading to node and returns the parent profile and child profile.


Instruction:
Explain what the following Python 3 code does
def optimal_marginal_branch_length(self, node, tol=1e-10):
        '''
        calculate the marginal distribution of sequence states on both ends
        of the branch leading to node,

        Parameters
        ----------
        node : PhyloTree.Clade
           TreeNode, attached to the branch.

        Returns
        -------
        branch_length : float
           branch length of the branch leading to the node.
           note: this can be unstable on iteration
        '''

        if node.up is None:
            return self.one_mutation
        pp, pc = self.marginal_branch_profile(node)
        return self.gtr.optimal_t_compressed((pp, pc), self.multiplicity, profiles=True, tol=tol)

Response: Calculates the optimal marginal distribution of sequence states on both ends of the branch leading to node.


Instruction:
Given the following Python 3 function, write the documentation
def prune_short_branches(self):
        self.logger("TreeAnc.prune_short_branches: pruning short branches (max prob at zero)...", 1)
        for node in self.tree.find_clades():
            if node.up is None or node.is_terminal():
                continue

            # probability of the two seqs separated by zero time is not zero
            if self.gtr.prob_t(node.up.cseq, node.cseq, 0.0,
                               pattern_multiplicity=self.multiplicity) > 0.1:
                # re-assign the node children directly to its parent
                node.up.clades = [k for k in node.up.clades if k != node] + node.clades
                for clade in node.clades:
                    clade.up = node.up

Response: Removes the short branches from the tree.


Instruction:
Explain what the following Python 3 code does
def optimize_seq_and_branch_len(self,reuse_branch_len=True, prune_short=True,
                                    marginal_sequences=False, branch_length_mode='joint',
                                    max_iter=5, infer_gtr=False, **kwargs):
        """
        Iteratively set branch lengths and reconstruct ancestral sequences until
        the values of either former or latter do not change. The algorithm assumes
        knowing only the topology of the tree, and requires that sequences are assigned
        to all leaves of the tree.

        The first step is to pre-reconstruct ancestral
        states using Fitch reconstruction algorithm or ML using existing branch length
        estimates. Then, optimize branch lengths and re-do reconstruction until
        convergence using ML method.

        Parameters
        -----------

         reuse_branch_len : bool
            If True, rely on the initial branch lengths, and start with the
            maximum-likelihood ancestral sequence inference using existing branch
            lengths. Otherwise, do initial reconstruction of ancestral states with
            Fitch algorithm, which uses only the tree topology.

         prune_short : bool
            If True, the branches with zero optimal length will be pruned from
            the tree, creating polytomies. The polytomies could be further
            processed using :py:meth:`treetime.TreeTime.resolve_polytomies` from the TreeTime class.

         marginal_sequences : bool
            Assign sequences to their marginally most likely value, rather than
            the values that are jointly most likely across all nodes.

         branch_length_mode : str
            'joint', 'marginal', or 'input'. Branch lengths are left unchanged in case
            of 'input'. 'joint' and 'marginal' cause branch length optimization
            while setting sequences to the ML value or tracing over all possible
            internal sequence states.

         max_iter : int
            Maximal number of times sequence and branch length iteration are optimized

         infer_gtr : bool
            Infer a GTR model from the observed substitutions.

        """
        if branch_length_mode=='marginal':
            marginal_sequences = True

        self.logger("TreeAnc.optimize_sequences_and_branch_length: sequences...", 1)
        if reuse_branch_len:
            N_diff = self.reconstruct_anc(method='probabilistic', infer_gtr=infer_gtr,
                                          marginal=marginal_sequences, **kwargs)
            self.optimize_branch_len(verbose=0, store_old=False, mode=branch_length_mode)
        else:
            N_diff = self.reconstruct_anc(method='fitch', infer_gtr=infer_gtr, **kwargs)

            self.optimize_branch_len(verbose=0, store_old=False, marginal=False)

        n = 0
        while n<max_iter:
            n += 1
            if prune_short:
                self.prune_short_branches()
            N_diff = self.reconstruct_anc(method='probabilistic', infer_gtr=False,
                                          marginal=marginal_sequences, **kwargs)

            self.logger("TreeAnc.optimize_sequences_and_branch_length: Iteration %d."
                   " #Nuc changed since prev reconstructions: %d" %(n, N_diff), 2)

            if N_diff < 1:
                break
            self.optimize_branch_len(verbose=0, store_old=False, mode=branch_length_mode)

        self.tree.unconstrained_sequence_LH = (self.tree.sequence_LH*self.multiplicity).sum()
        self._prepare_nodes() # fix dist2root and up-links after reconstruction
        self.logger("TreeAnc.optimize_sequences_and_branch_length: Unconstrained sequence LH:%f" % self.tree.unconstrained_sequence_LH , 2)
        return ttconf.SUCCESS

Response: Optimize the sequences and branch lengths of the tree.


Instruction:
How would you code a function in Python 3 to
get the multiple sequence alignment including reconstructed sequences for the internal nodes.

Response:
def get_reconstructed_alignment(self):
        """
        Get the multiple sequence alignment, including reconstructed sequences for
        the internal nodes.

        Returns
        -------
        new_aln : MultipleSeqAlignment
           Alignment including sequences of all internal nodes

        """
        from Bio.Align import MultipleSeqAlignment
        from Bio.Seq import Seq
        from Bio.SeqRecord import SeqRecord
        self.logger("TreeAnc.get_reconstructed_alignment ...",2)
        if not hasattr(self.tree.root, 'sequence'):
            self.logger("TreeAnc.reconstructed_alignment... reconstruction not yet done",3)
            self.reconstruct_anc('probabilistic')

        new_aln = MultipleSeqAlignment([SeqRecord(id=n.name, seq=Seq("".join(n.sequence)), description="")
                                        for n in self.tree.find_clades()])

        return new_aln


Instruction:
How would you explain what the following Python 3 function does
def get_tree_dict(self, keep_var_ambigs=False):
        if self.is_vcf:
            tree_dict = {}
            tree_dict['reference'] = self.ref
            tree_dict['positions'] = self.nonref_positions

            tree_aln = {}
            for n in self.tree.find_clades():
                if hasattr(n, 'sequence'):
                    if keep_var_ambigs: #regenerate dict to include ambig bases
                        tree_aln[n.name] = self.dict_sequence(n, keep_var_ambigs)
                    else:
                        tree_aln[n.name] = n.sequence

            tree_dict['sequences'] = tree_aln

            if len(self.inferred_const_sites) != 0:
                tree_dict['inferred_const_sites'] = self.inferred_const_sites

            return tree_dict
        else:
            raise TypeError("A dict can only be returned for trees created with VCF-input!")

Response: Returns a nested dict with all the information required to reconstruct sequences for all the nodes in the VCF.


Instruction:
Write a Python 3 function that can
function that returns the product of the transition matrix and the equilibrium frequencies to obtain the rate matrix of the GTR model

Response:
def Q(self):
        """function that return the product of the transition matrix
           and the equilibrium frequencies to obtain the rate matrix
           of the GTR model
        """
        tmp = np.einsum('ia,ij->ija', self.Pi, self.W)
        diag_vals = np.sum(tmp, axis=0)
        for x in range(tmp.shape[-1]):
            np.fill_diagonal(tmp[:,:,x], -diag_vals[:,x])
        return tmp


Instruction:
Given the following Python 3 function, write the documentation
def assign_rates(self, mu=1.0, pi=None, W=None):
        n = len(self.alphabet)
        self.mu = np.copy(mu)

        if pi is not None and pi.shape[0]==n:
            self.seq_len = pi.shape[-1]
            Pi = np.copy(pi)
        else:
            if pi is not None and len(pi)!=n:
                self.logger("length of equilibrium frequency vector does not match alphabet length", 4, warn=True)
                self.logger("Ignoring input equilibrium frequencies", 4, warn=True)
            Pi = np.ones(shape=(n,self.seq_len))

        self.Pi = Pi/np.sum(Pi, axis=0)

        if W is None or W.shape!=(n,n):
            if (W is not None) and W.shape!=(n,n):
                self.logger("Substitution matrix size does not match alphabet size", 4, warn=True)
                self.logger("Ignoring input substitution matrix", 4, warn=True)
            # flow matrix
            W = np.ones((n,n))
        else:
            W=0.5*(np.copy(W)+np.copy(W).T)

        np.fill_diagonal(W,0)
        avg_pi = self.Pi.mean(axis=-1)
        average_rate = W.dot(avg_pi).dot(avg_pi)
        self.W = W/average_rate
        self.mu *=average_rate

        self._eig()

Response: Assign the rates of the current GTR model to the given data.


Instruction:
Can you generate the documentation for the following Python 3 function
def random(cls, L=1, avg_mu=1.0, alphabet='nuc', pi_dirichlet_alpha=1,
               W_dirichlet_alpha=3.0, mu_gamma_alpha=3.0):
        """
        Creates a random GTR model

        Parameters
        ----------

         mu : float
            Substitution rate

         alphabet : str
            Alphabet name (should be standard: 'nuc', 'nuc_gap', 'aa', 'aa_gap')

        """
        from scipy.stats import gamma
        alphabet=alphabets[alphabet]
        gtr = cls(alphabet=alphabet, seq_len=L)
        n = gtr.alphabet.shape[0]

        if pi_dirichlet_alpha:
            pi = 1.0*gamma.rvs(pi_dirichlet_alpha, size=(n,L))
        else:
            pi = np.ones((n,L))

        pi /= pi.sum(axis=0)
        if W_dirichlet_alpha:
            tmp = 1.0*gamma.rvs(W_dirichlet_alpha, size=(n,n))
        else:
            tmp = np.ones((n,n))
        tmp = np.tril(tmp,k=-1)
        W = tmp + tmp.T

        if mu_gamma_alpha:
            mu = gamma.rvs(mu_gamma_alpha, size=(L,))
        else:
            mu = np.ones(L)

        gtr.assign_rates(mu=mu, pi=pi, W=W)
        gtr.mu *= avg_mu/np.mean(gtr.mu)

        return gtr

Response: Creates a random GTR model from a set of random values.


Instruction:
How would you implement a function in Python 3 that
creates a custom GTR model by specifying the matrix explicitly passed as .

Response:
def custom(cls, mu=1.0, pi=None, W=None, **kwargs):
        """
        Create a GTR model by specifying the matrix explicitly

        Parameters
        ----------

         mu : float
            Substitution rate

         W : nxn matrix
            Substitution matrix

         pi : n vector
            Equilibrium frequencies

         **kwargs:
            Key word arguments to be passed

        Keyword Args
        ------------

         alphabet : str
            Specify alphabet when applicable. If the alphabet specification is
            required, but no alphabet is specified, the nucleotide alphabet will be used as
            default.

        """
        gtr = cls(**kwargs)
        gtr.assign_rates(mu=mu, pi=pi, W=W)
        return gtr


Instruction:
Implement a Python 3 function for
inferring a GTR model by specifying the number of transitions and time spent in each character state.

Response:
def infer(cls, sub_ija, T_ia, root_state, pc=0.01,
              gap_limit=0.01, Nit=30, dp=1e-5, **kwargs):
        """
        Infer a GTR model by specifying the number of transitions and time spent in each
        character. The basic equation that is being solved is

        :math:`n_{ij} = pi_i W_{ij} T_j`

        where :math:`n_{ij}` are the transitions, :math:`pi_i` are the equilibrium
        state frequencies, :math:`W_{ij}` is the "substitution attempt matrix",
        while :math:`T_i` is the time on the tree spent in character state
        :math:`i`. To regularize the process, we add pseudocounts and also need
        to account for the fact that the root of the tree is in a particular
        state. the modified equation is

        :math:`n_{ij} + pc = pi_i W_{ij} (T_j+pc+root\_state)`

        Parameters
        ----------

         nija : nxn matrix
            The number of times a change in character state is observed
            between state j and i at position a

         Tia :n vector
            The time spent in each character state at position a

         root_state : n vector
            The number of characters in state i in the sequence
            of the root node.

         pc : float
            Pseudocounts, this determines the lower cutoff on the rate when
            no substitutions are observed

         **kwargs:
            Key word arguments to be passed

        Keyword Args
        ------------

         alphabet : str
            Specify alphabet when applicable. If the alphabet specification
            is required, but no alphabet is specified, the nucleotide alphabet will be used as default.

        """
        from scipy import linalg as LA
        gtr = cls(**kwargs)
        gtr.logger("GTR: model inference ",1)
        q = len(gtr.alphabet)
        L = sub_ija.shape[-1]

        n_iter = 0
        n_ija = np.copy(sub_ija)
        n_ija[range(q),range(q),:] = 0
        n_ij = n_ija.sum(axis=-1)

        m_ia = np.sum(n_ija,axis=1) + root_state + pc
        n_a = n_ija.sum(axis=1).sum(axis=0) + pc

        Lambda = np.sum(root_state,axis=0) + q*pc
        p_ia_old=np.zeros((q,L))
        p_ia = np.ones((q,L))/q
        mu_a = np.ones(L)

        W_ij = np.ones((q,q)) - np.eye(q)

        while (LA.norm(p_ia_old-p_ia)>dp) and n_iter<Nit:
            n_iter += 1
            p_ia_old = np.copy(p_ia)
            S_ij = np.einsum('a,ia,ja',mu_a, p_ia, T_ia)
            W_ij = (n_ij + n_ij.T + pc)/(S_ij + S_ij.T + pc)

            avg_pi = p_ia.mean(axis=-1)
            average_rate = W_ij.dot(avg_pi).dot(avg_pi)
            W_ij = W_ij/average_rate
            mu_a *=average_rate

            p_ia = m_ia/(mu_a*np.dot(W_ij,T_ia)+Lambda)
            p_ia = p_ia/p_ia.sum(axis=0)

            mu_a = n_a/(pc+np.einsum('ia,ij,ja->a', p_ia, W_ij, T_ia))

        if n_iter >= Nit:
            gtr.logger('WARNING: maximum number of iterations has been reached in GTR inference',3, warn=True)
            if LA.norm(p_ia_old-p_ia) > dp:
                gtr.logger('the iterative scheme has not converged',3,warn=True)
        if gtr.gap_index is not None:
            for p in range(p_ia.shape[-1]):
                if p_ia[gtr.gap_index,p]<gap_limit:
                    gtr.logger('The model allows for gaps which are estimated to occur at a low fraction of %1.3e'%p_ia[gtr.gap_index]+
                           '\n\t\tthis can potentially result in artificats.'+
                           '\n\t\tgap fraction will be set to %1.4f'%gap_limit,2,warn=True)
                p_ia[gtr.gap_index,p] = gap_limit
                p_ia[:,p] /= p_ia[:,p].sum()

        gtr.assign_rates(mu=mu_a, W=W_ij, pi=p_ia)
        return gtr


Instruction:
Write a Python 3 function that can
compute the probability to observe seq_ch after time t starting from seq_p.

Response:
def prob_t(self, seq_p, seq_ch, t, pattern_multiplicity = None,
               return_log=False, ignore_gaps=True):
        """
        Compute the probability to observe seq_ch (child sequence) after time t starting from seq_p
        (parent sequence).

        Parameters
        ----------

         seq_p : character array
            Parent sequence

         seq_c : character array
            Child sequence

         t : double
            Time (branch len) separating the profiles.

         pattern_multiplicity : numpy array
            If sequences are reduced by combining identical alignment patterns,
            these multplicities need to be accounted for when counting the number
            of mutations across a branch. If None, all pattern are assumed to
            occur exactly once.

         return_log : bool
            It True, return log-probability.

        Returns
        -------
         prob : np.array
            Resulting probability

        """
        if t<0:
            logP = -ttconf.BIG_NUMBER
        else:
            tmp_eQT = self.expQt(t)
            bad_indices=(tmp_eQT==0)
            logQt = np.log(tmp_eQT + ttconf.TINY_NUMBER*(bad_indices))
            logQt[np.isnan(logQt) | np.isinf(logQt) | bad_indices] = -ttconf.BIG_NUMBER
            seq_indices_c = np.zeros(len(seq_ch), dtype=int)
            seq_indices_p = np.zeros(len(seq_p), dtype=int)
            for ai, a in self.alphabet:
                seq_indices_p[seq_p==a] = ai
                seq_indices_c[seq_ch==a] = ai

            if len(logQt.shape)==2:
                logP = np.sum(logQt[seq_indices_p, seq_indices_c]*pattern_multiplicity)
            else:
                logP = np.sum(logQt[seq_indices_p, seq_indices_c, np.arange(len(seq_c))]*pattern_multiplicity)

        return logP if return_log else np.exp(logP)


Instruction:
Can you generate a brief explanation for the following Python 3 code
def assign_rates(self, mu=1.0, pi=None, W=None):
        n = len(self.alphabet)
        self.mu = mu

        if pi is not None and len(pi)==n:
            Pi = np.array(pi)
        else:
            if pi is not None and len(pi)!=n:
                self.logger("length of equilibrium frequency vector does not match alphabet length", 4, warn=True)
                self.logger("Ignoring input equilibrium frequencies", 4, warn=True)
            Pi = np.ones(shape=(n,))

        self.Pi = Pi/np.sum(Pi)

        if W is None or W.shape!=(n,n):
            if (W is not None) and W.shape!=(n,n):
                self.logger("Substitution matrix size does not match alphabet size", 4, warn=True)
                self.logger("Ignoring input substitution matrix", 4, warn=True)
            # flow matrix
            W = np.ones((n,n))
            np.fill_diagonal(W, 0.0)
            np.fill_diagonal(W, - W.sum(axis=0))
        else:
            W=np.array(W)

        self.W = 0.5*(W+W.T)
        self._check_fix_Q(fixed_mu=True)
        self._eig()

Response: Assign the rates mu pi and W to the GTR model.


Instruction:
Can you generate the documentation for the following Python 3 function
def standard(model, **kwargs):
        from .nuc_models import JC69, K80, F81, HKY85, T92, TN93
        from .aa_models  import JTT92

        if model.lower() in ['jc', 'jc69', 'jukes-cantor', 'jukes-cantor69', 'jukescantor', 'jukescantor69']:
            return JC69(**kwargs)
        elif model.lower() in ['k80', 'kimura80', 'kimura1980']:
            return K80(**kwargs)
        elif model.lower() in ['f81', 'felsenstein81', 'felsenstein1981']:
            return F81(**kwargs)
        elif model.lower() in ['hky', 'hky85', 'hky1985']:
            return HKY85(**kwargs)
        elif model.lower() in ['t92', 'tamura92', 'tamura1992']:
            return T92(**kwargs)
        elif model.lower() in ['tn93', 'tamura_nei_93', 'tamuranei93']:
            return TN93(**kwargs)
        elif model.lower() in ['jtt', 'jtt92']:
            return JTT92(**kwargs)
        else:
            raise KeyError("The GTR model '{}' is not in the list of available models."
                "".format(model))

Response: Create a standard evolution model of the specified model.


Instruction:
How would you implement a function in Python 3 that
creates a random GTR model from a random mu and alphabet

Response:
def random(cls, mu=1.0, alphabet='nuc'):
        """
        Creates a random GTR model

        Parameters
        ----------

         mu : float
            Substitution rate

         alphabet : str
            Alphabet name (should be standard: 'nuc', 'nuc_gap', 'aa', 'aa_gap')

        """

        alphabet=alphabets[alphabet]
        gtr = cls(alphabet)
        n = gtr.alphabet.shape[0]
        pi = 1.0*np.random.randint(0,100,size=(n))
        W = 1.0*np.random.randint(0,100,size=(n,n)) # with gaps

        gtr.assign_rates(mu=mu, pi=pi, W=W)
        return gtr


Instruction:
Write a Python 3 function for
inferring a GTR model by specifying the number of times a change in each character state is observed in each character state Ti and the time spent in each character state Ti.

Response:
def infer(cls, nij, Ti, root_state, fixed_pi=None, pc=5.0, gap_limit=0.01, **kwargs):
        """
        Infer a GTR model by specifying the number of transitions and time spent in each
        character. The basic equation that is being solved is

        :math:`n_{ij} = pi_i W_{ij} T_j`

        where :math:`n_{ij}` are the transitions, :math:`pi_i` are the equilibrium
        state frequencies, :math:`W_{ij}` is the "substitution attempt matrix",
        while :math:`T_i` is the time on the tree spent in character state
        :math:`i`. To regularize the process, we add pseudocounts and also need
        to account for the fact that the root of the tree is in a particular
        state. the modified equation is

        :math:`n_{ij} + pc = pi_i W_{ij} (T_j+pc+root\_state)`

        Parameters
        ----------

         nij : nxn matrix
            The number of times a change in character state is observed
            between state j and i

         Ti :n vector
            The time spent in each character state

         root_state : n vector
            The number of characters in state i in the sequence
            of the root node.

         pc : float
            Pseudocounts, this determines the lower cutoff on the rate when
            no substitutions are observed

         **kwargs:
            Key word arguments to be passed

        Keyword Args
        ------------

         alphabet : str
            Specify alphabet when applicable. If the alphabet specification
            is required, but no alphabet is specified, the nucleotide alphabet will be used as default.

        """
        from scipy import linalg as LA
        gtr = cls(**kwargs)
        gtr.logger("GTR: model inference ",1)
        dp = 1e-5
        Nit = 40
        pc_mat = pc*np.ones_like(nij)
        np.fill_diagonal(pc_mat, 0.0)
        count = 0
        pi_old = np.zeros_like(Ti)
        if fixed_pi is None:
            pi = np.ones_like(Ti)
        else:
            pi = np.copy(fixed_pi)
        pi/=pi.sum()
        W_ij = np.ones_like(nij)
        mu = nij.sum()/Ti.sum()
        # if pi is fixed, this will immediately converge
        while LA.norm(pi_old-pi) > dp and count < Nit:
            gtr.logger(' '.join(map(str, ['GTR inference iteration',count,'change:',LA.norm(pi_old-pi)])), 3)
            count += 1
            pi_old = np.copy(pi)
            W_ij = (nij+nij.T+2*pc_mat)/mu/(np.outer(pi,Ti) + np.outer(Ti,pi)
                                                    + ttconf.TINY_NUMBER + 2*pc_mat)

            np.fill_diagonal(W_ij, 0)
            scale_factor = np.einsum('i,ij,j',pi,W_ij,pi)

            W_ij = W_ij/scale_factor
            if fixed_pi is None:
                pi = (np.sum(nij+pc_mat,axis=1)+root_state)/(ttconf.TINY_NUMBER + mu*np.dot(W_ij,Ti)+root_state.sum()+np.sum(pc_mat, axis=1))
                pi /= pi.sum()
            mu = nij.sum()/(ttconf.TINY_NUMBER + np.sum(pi * (W_ij.dot(Ti))))
        if count >= Nit:
            gtr.logger('WARNING: maximum number of iterations has been reached in GTR inference',3, warn=True)
            if LA.norm(pi_old-pi) > dp:
                gtr.logger('the iterative scheme has not converged',3,warn=True)
            elif np.abs(1-np.max(pi.sum(axis=0))) > dp:
                gtr.logger('the iterative scheme has converged, but proper normalization was not reached',3,warn=True)
        if gtr.gap_index is not None:
            if pi[gtr.gap_index]<gap_limit:
                gtr.logger('The model allows for gaps which are estimated to occur at a low fraction of %1.3e'%pi[gtr.gap_index]+
                       '\n\t\tthis can potentially result in artificats.'+
                       '\n\t\tgap fraction will be set to %1.4f'%gap_limit,2,warn=True)
            pi[gtr.gap_index] = gap_limit
            pi /= pi.sum()

        gtr.assign_rates(mu=mu, W=W_ij, pi=pi)
        return gtr


Instruction:
Can you write a function in Python 3 where it
fixes the main diagonal of the rate matrix and fix the mu if fixed_mu is True.

Response:
def _check_fix_Q(self, fixed_mu=False):
        """
        Check the main diagonal of Q and fix it in case it does not corresond
        the definition of the rate matrix. Should be run every time when creating
        custom GTR model.
        """
        # fix Q
        self.Pi /= self.Pi.sum() # correct the Pi manually
        # NEEDED TO BREAK RATE MATRIX DEGENERACY AND FORCE NP TO RETURN REAL ORTHONORMAL EIGENVECTORS
        self.W += self.break_degen + self.break_degen.T
        # fix W
        np.fill_diagonal(self.W, 0)
        Wdiag = -(self.Q).sum(axis=0)/self.Pi
        np.fill_diagonal(self.W, Wdiag)
        scale_factor = -np.sum(np.diagonal(self.Q)*self.Pi)
        self.W /= scale_factor
        if not fixed_mu:
            self.mu *= scale_factor
        if (self.Q.sum(axis=0) < 1e-10).sum() <  self.alphabet.shape[0]: # fix failed
            print ("Cannot fix the diagonal of the GTR rate matrix. Should be all zero", self.Q.sum(axis=0))
            import ipdb; ipdb.set_trace()
            raise ArithmeticError("Cannot fix the diagonal of the GTR rate matrix.")


Instruction:
Write a Python 3 function that can
perform eigendecomposition of the rate matrix and store the left and right - sequence profiles in self. v and self. v_inv.

Response:
def _eig(self):
        """
        Perform eigendecompositon of the rate matrix and stores the left- and right-
        matrices to convert the sequence profiles to the GTR matrix eigenspace
        and hence to speed-up the computations.
        """
        # eigendecomposition of the rate matrix
        eigvals, eigvecs = np.linalg.eig(self.Q)
        self.v = np.real(eigvecs)
        self.v_inv = np.linalg.inv(self.v)
        self.eigenvals = np.real(eigvals)


Instruction:
Write a Python 3 script for
performing eigendecompositon of the rate matrix and stores the left - and right - matrices and the GTR matrix eigenvals.

Response:
def _eig_sym(self):
        """
        Perform eigendecompositon of the rate matrix and stores the left- and right-
        matrices to convert the sequence profiles to the GTR matrix eigenspace
        and hence to speed-up the computations.
        """
        # eigendecomposition of the rate matrix
        tmpp = np.sqrt(self.Pi)
        symQ = self.W*np.outer(tmpp, tmpp)
        eigvals, eigvecs = np.linalg.eigh(symQ)
        tmp_v = eigvecs.T*tmpp
        one_norm = np.sum(np.abs(tmp_v), axis=1)
        self.v = tmp_v.T/one_norm
        self.v_inv = (eigvecs*one_norm).T/tmpp
        self.eigenvals = eigvals


Instruction:
Write a Python 3 function that can
make a compressed representation of a pair of sequences, only counting the number of times a particular pair of states (e.g. (A,T)) is observed in the aligned sequences of parent and child. Parameters ---------- seq_p: numpy array Parent sequence as numpy array of chars seq_ch: numpy array Child sequence as numpy array of chars pattern_multiplicity : numpy array If sequences are reduced by combining identical alignment patterns, these multplicities need to be accounted for when counting the number of mutations across a branch. If None, all pattern are assumed to occur exactly once. ignore_gap: bool Whether or not to include gapped positions of the alignment in the multiplicity count Returns ------- seq_pair : list :code:`[(0,1), (2,2), (3,4)]` list of parent_child state pairs as indices in the alphabet multiplicity : numpy array Number of times a particular pair is observed

Response:
def compress_sequence_pair(self, seq_p, seq_ch, pattern_multiplicity=None,
                               ignore_gaps=False):
        '''
        Make a compressed representation of a pair of sequences, only counting
        the number of times a particular pair of states (e.g. (A,T)) is observed
        in the aligned sequences of parent and child.

        Parameters
        ----------

         seq_p: numpy array
            Parent sequence as numpy array of chars

         seq_ch: numpy array
            Child sequence as numpy array of chars

         pattern_multiplicity : numpy array
            If sequences are reduced by combining identical alignment patterns,
            these multplicities need to be accounted for when counting the number
            of mutations across a branch. If None, all pattern are assumed to
            occur exactly once.

         ignore_gap: bool
            Whether or not to include gapped positions of the alignment
            in the multiplicity count

        Returns
        -------
          seq_pair : list
            :code:`[(0,1), (2,2), (3,4)]` list of parent_child state pairs
            as indices in the alphabet

          multiplicity : numpy array
            Number of times a particular pair is observed

        '''
        if pattern_multiplicity is None:
            pattern_multiplicity = np.ones_like(seq_p, dtype=float)

        from collections import Counter
        if seq_ch.shape != seq_p.shape:
            raise ValueError("GTR.compress_sequence_pair: Sequence lengths do not match!")

        if len(self.alphabet)<10: # for small alphabet, repeatedly check array for all state pairs
            pair_count = []
            bool_seqs_p = []
            bool_seqs_ch = []
            for seq, bs in [(seq_p,bool_seqs_p), (seq_ch, bool_seqs_ch)]:
                for ni,nuc in enumerate(self.alphabet):
                    bs.append(seq==nuc)

            for n1,nuc1 in enumerate(self.alphabet):
                if (self.gap_index is None) or (not ignore_gaps) or (n1!=self.gap_index):
                    for n2,nuc2 in enumerate(self.alphabet):
                        if (self.gap_index is None) or (not ignore_gaps) or (n2!=self.gap_index):
                            count = ((bool_seqs_p[n1]&bool_seqs_ch[n2])*pattern_multiplicity).sum()
                            if count: pair_count.append(((n1,n2), count))
        else: # enumerate state pairs of the sequence for large alphabets
            num_seqs = []
            for seq in [seq_p, seq_ch]: # for each sequence (parent and child) construct a numerical sequence [0,5,3,1,2,3...]
                tmp = np.ones_like(seq, dtype=int)
                for ni,nuc in enumerate(self.alphabet):
                    tmp[seq==nuc] = ni  # set each position corresponding to a state to the corresponding index
                num_seqs.append(tmp)
            pair_count = defaultdict(int)
            if ignore_gaps:  # if gaps are ignored skip positions where one or the other sequence is gapped
                for i in range(len(seq_p)):
                    if self.gap_index!=num_seqs[0][i] and self.gap_index!=num_seqs[1][i]:
                        pair_count[(num_seqs[0][i],num_seqs[1][i])]+=pattern_multiplicity[i]
            else: # otherwise, just count
                for i in range(len(seq_p)):
                    pair_count[(num_seqs[0][i],num_seqs[1][i])]+=pattern_multiplicity[i]
            pair_count = pair_count.items()

        return (np.array([x[0] for x in pair_count], dtype=int),    # [(child_nuc, parent_nuc),()...]
                np.array([x[1] for x in pair_count], dtype=int))


Instruction:
Can you generate the documentation for the following Python 3 function
def prob_t_compressed(self, seq_pair, multiplicity, t, return_log=False):
        '''
        Calculate the probability of observing a sequence pair at a distance t,
        for compressed sequences

        Parameters
        ----------

          seq_pair : numpy array
            :code:`np.array([(0,1), (2,2), ()..])` as indicies of
            pairs of aligned positions. (e.g. 'A'==0, 'C'==1 etc).
            This only lists all occuring parent-child state pairs, order is irrelevant

          multiplicity : numpy array
            The number of times a parent-child state pair is observed.
            This allows compression of the sequence representation

          t : float
            Length of the branch separating parent and child

          return_log : bool
            Whether or not to exponentiate the result

        '''
        if t<0:
            logP = -ttconf.BIG_NUMBER
        else:
            tmp_eQT = self.expQt(t)
            bad_indices=(tmp_eQT==0)
            logQt = np.log(tmp_eQT + ttconf.TINY_NUMBER*(bad_indices))
            logQt[np.isnan(logQt) | np.isinf(logQt) | bad_indices] = -ttconf.BIG_NUMBER
            logP = np.sum(logQt[seq_pair[:,1], seq_pair[:,0]]*multiplicity)

        return logP if return_log else np.exp(logP)

Response: Calculates the probability of observing a sequence pair at a distance t for compressed sequences.


Instruction:
Create a Python 3 function for
computing the probability to observe seq_ch after time t starting from seq_p.

Response:
def prob_t(self, seq_p, seq_ch, t, pattern_multiplicity = None,
               return_log=False, ignore_gaps=True):
        """
        Compute the probability to observe seq_ch (child sequence) after time t starting from seq_p
        (parent sequence).

        Parameters
        ----------

         seq_p : character array
            Parent sequence

         seq_c : character array
            Child sequence

         t : double
            Time (branch len) separating the profiles.

         pattern_multiplicity : numpy array
            If sequences are reduced by combining identical alignment patterns,
            these multplicities need to be accounted for when counting the number
            of mutations across a branch. If None, all pattern are assumed to
            occur exactly once.

         return_log : bool
            It True, return log-probability.

        Returns
        -------
         prob : np.array
            Resulting probability

        """
        seq_pair, multiplicity = self.compress_sequence_pair(seq_p, seq_ch,
                                        pattern_multiplicity=pattern_multiplicity, ignore_gaps=ignore_gaps)
        return self.prob_t_compressed(seq_pair, multiplicity, t, return_log=return_log)


Instruction:
Here you have a function in Python 3, explain what it does
def optimal_t(self, seq_p, seq_ch, pattern_multiplicity=None, ignore_gaps=False):
        '''
        Find the optimal distance between the two sequences

        Parameters
        ----------

         seq_p : character array
            Parent sequence

         seq_c : character array
            Child sequence

         pattern_multiplicity : numpy array
            If sequences are reduced by combining identical alignment patterns,
            these multplicities need to be accounted for when counting the number
            of mutations across a branch. If None, all pattern are assumed to
            occur exactly once.

         ignore_gaps : bool
            If True, ignore gaps in distance calculations

        '''
        seq_pair, multiplicity = self.compress_sequence_pair(seq_p, seq_ch,
                                                            pattern_multiplicity = pattern_multiplicity,
                                                            ignore_gaps=ignore_gaps)
        return self.optimal_t_compressed(seq_pair, multiplicity)

Response: This function returns the optimal distance between two sequences.


Instruction:
Explain what the following Python 3 code does
def optimal_t_compressed(self, seq_pair, multiplicity, profiles=False, tol=1e-10):

        def _neg_prob(t, seq_pair, multiplicity):
            """
            Probability to observe a child given the the parent state, transition
            matrix, and the time of evolution (branch length).

            Parameters
            ----------

             t : double
                Branch length (time between sequences)

             parent :  numpy.array
                Parent sequence

             child : numpy.array
                Child sequence

             tm :  GTR
                Model of evolution

            Returns
            -------

             prob : double
                Negative probability of the two given sequences
                to be separated by the time t.
            """
            if profiles:
                res = -1.0*self.prob_t_profiles(seq_pair, multiplicity,t**2, return_log=True)
                return res
            else:
                return -1.0*self.prob_t_compressed(seq_pair, multiplicity,t**2, return_log=True)

        try:
            from scipy.optimize import minimize_scalar
            opt = minimize_scalar(_neg_prob,
                    bounds=[-np.sqrt(ttconf.MAX_BRANCH_LENGTH),np.sqrt(ttconf.MAX_BRANCH_LENGTH)],
                    args=(seq_pair, multiplicity), tol=tol)
            new_len = opt["x"]**2
            if 'success' not in opt:
                opt['success'] = True
                self.logger("WARNING: the optimization result does not contain a 'success' flag:"+str(opt),4, warn=True)
        except:
            import scipy
            print('legacy scipy', scipy.__version__)
            from scipy.optimize import fminbound
            new_len = fminbound(_neg_prob,
                      -np.sqrt(ttconf.MAX_BRANCH_LENGTH),np.sqrt(ttconf.MAX_BRANCH_LENGTH),
                       args=(seq_pair, multiplicity))
            new_len = new_len**2
            opt={'success':True}

        if new_len > .9 * ttconf.MAX_BRANCH_LENGTH:
            self.logger("WARNING: GTR.optimal_t_compressed -- The branch length seems to be very long!", 4, warn=True)

        if opt["success"] != True:
            # return hamming distance: number of state pairs where state differs/all pairs
            new_len =  np.sum(multiplicity[seq_pair[:,1]!=seq_pair[:,0]])/np.sum(multiplicity)

        return new_len

Response: This function calculates the optimal distance between two compressed sequences at a given time.


Instruction:
Can you generate the documentation for the following Python 3 function
def prob_t_profiles(self, profile_pair, multiplicity, t,
                        return_log=False, ignore_gaps=True):
        '''
        Calculate the probability of observing a node pair at a distance t

        Parameters
        ----------

          profile_pair: numpy arrays
            Probability distributions of the nucleotides at either
            end of the branch. pp[0] = parent, pp[1] = child

          multiplicity : numpy array
            The number of times an alignment pattern is observed

          t : float
            Length of the branch separating parent and child

          ignore_gaps: bool
            If True, ignore mutations to and from gaps in distance calculations

          return_log : bool
            Whether or not to exponentiate the result

        '''
        if t<0:
            logP = -ttconf.BIG_NUMBER
        else:
            Qt = self.expQt(t)
            if len(Qt.shape)==3:
                res = np.einsum('ai,ija,aj->a', profile_pair[1], Qt, profile_pair[0])
            else:
                res = np.einsum('ai,ij,aj->a', profile_pair[1], Qt, profile_pair[0])
            if ignore_gaps and (self.gap_index is not None): # calculate the probability that neither outgroup/node has a gap
                non_gap_frac = (1-profile_pair[0][:,self.gap_index])*(1-profile_pair[1][:,self.gap_index])
                # weigh log LH by the non-gap probability
                logP = np.sum(multiplicity*np.log(res)*non_gap_frac)
            else:
                logP = np.sum(multiplicity*np.log(res))

        return logP if return_log else np.exp(logP)

Response: Calculates the probability of observing a node pair at a given distance t.


Instruction:
How would you explain what the following Python 3 function does
def propagate_profile(self, profile, t, return_log=False):
        Qt = self.expQt(t)
        res = profile.dot(Qt)

        return np.log(res) if return_log else res

Response: Given a profile compute the probability of the sequence state of the parent at time t0 and given the sequence state of the child at time t1.


Instruction:
Given the following Python 3 function, write the documentation
def evolve(self, profile, t, return_log=False):
        Qt = self.expQt(t).T
        res = profile.dot(Qt)
        return np.log(res) if return_log else res

Response: Compute the probability of the sequence state of the child at time t later given the parent profile.


Instruction:
Implement a Python 3 function for
returning the array of values exp that is less than the given time

Response:
def _exp_lt(self, t):
        """
        Parameters
        ----------

         t : float
            time to propagate

        Returns
        --------

         exp_lt : numpy.array
            Array of values exp(lambda(i) * t),
            where (i) - alphabet index (the eigenvalue number).
        """
        return np.exp(self.mu * t * self.eigenvals)


Instruction:
Can you generate the documentation for the following Python 3 function
def expQt(self, t):
        '''
        Parameters
        ----------

         t : float
            Time to propagate

        Returns
        --------

         expQt : numpy.array
            Matrix exponential of exo(Qt)
        '''
        eLambdaT = np.diag(self._exp_lt(t)) # vector length = a
        Qs = self.v.dot(eLambdaT.dot(self.v_inv))   # This is P(nuc1 | given nuc_2)
        return np.maximum(0,Qs)

Response: Calculates the exponential of the avec of avec with respect to the time t.


Instruction:
How would you explain what the following Python 3 function does
def expQsds(self, s):
        '''
        Returns
        -------
        Qtds :  Returns 2 V_{ij} \lambda_j s e^{\lambda_j s**2 } V^{-1}_{jk}
                This is the derivative of the branch probability with respect to s=\sqrt(t)
        '''
        lambda_eLambdaT = np.diag(2.0*self._exp_lt(s**2)*self.eigenvals*s) # vector length = a
        Qsds = self.v.dot(lambda_eLambdaT.dot(self.v_inv))
        return Qsds

Response: Returns the derivative of the derivative of the branch probability with respect to s


Instruction:
Can you implement a function in Python 3 that
returns the second derivative of the branch probability wrt time

Response:
def expQsdsds(self, s):
        '''
        Returns
        -------
        Qtdtdt :  Returns V_{ij} \lambda_j^2 e^{\lambda_j s**2} V^{-1}_{jk}
                This is the second derivative of the branch probability wrt time
        '''
        t=s**2
        elt = self._exp_lt(t)
        lambda_eLambdaT = np.diag(elt*(4.0*t*self.eigenvals**2 + 2.0*self.eigenvals))
        Qsdsds = self.v.dot(lambda_eLambdaT.dot(self.v_inv))
        return Qsdsds


Instruction:
Can you generate a brief explanation for the following Python 3 code
def sequence_logLH(self,seq, pattern_multiplicity=None):
        if pattern_multiplicity is None:
            pattern_multiplicity = np.ones_like(seq, dtype=float)
        return np.sum([np.sum((seq==state)*pattern_multiplicity*np.log(self.Pi[si]))
                      for si,state in enumerate(self.alphabet)])

Response: Returns the log - likelihood of sampling a sequence from equilibrium frequency.


Instruction:
Make a summary of the following Python 3 code
def plot_vs_years(tt, step = None, ax=None, confidence=None, ticks=True, **kwargs):
    '''
    Converts branch length to years and plots the time tree on a time axis.

    Parameters
    ----------
     tt : TreeTime object
        A TreeTime instance after a time tree is inferred

     step : int
        Width of shaded boxes indicating blocks of years. Will be inferred if not specified.
        To switch off drawing of boxes, set to 0

     ax : matplotlib axes
        Axes to be used to plot, will create new axis if None

     confidence : tuple, float
        Draw confidence intervals. This assumes that marginal time tree inference was run.
        Confidence intervals are either specified as an interval of the posterior distribution
        like (0.05, 0.95) or as the weight of the maximal posterior region , e.g. 0.9

     **kwargs : dict
        Key word arguments that are passed down to Phylo.draw

    '''
    import matplotlib.pyplot as plt
    tt.branch_length_to_years()
    nleafs = tt.tree.count_terminals()

    if ax is None:
        fig = plt.figure(figsize=(12,10))
        ax = plt.subplot(111)
    else:
        fig = None
    # draw tree
    if "label_func" not in kwargs:
        kwargs["label_func"] = lambda x:x.name if (x.is_terminal() and nleafs<30) else ""
    Phylo.draw(tt.tree, axes=ax, **kwargs)

    offset = tt.tree.root.numdate - tt.tree.root.branch_length
    date_range = np.max([n.numdate for n in tt.tree.get_terminals()])-offset

    # estimate year intervals if not explicitly specified
    if step is None or (step>0 and date_range/step>100):
        step = 10**np.floor(np.log10(date_range))
        if date_range/step<2:
            step/=5
        elif date_range/step<5:
            step/=2
        step = max(1.0/12,step)

    # set axis labels
    if step:
        dtick = step
        min_tick = step*(offset//step)
        extra = dtick if dtick<date_range else dtick
        tick_vals = np.arange(min_tick, min_tick+date_range+extra, dtick)
        xticks = tick_vals - offset
    else:
        xticks = ax.get_xticks()
        dtick = xticks[1]-xticks[0]
        shift = offset - dtick*(offset//dtick)
        xticks -= shift
        tick_vals = [x+offset-shift for x in xticks]

    ax.set_xticks(xticks)
    ax.set_xticklabels(map(str, tick_vals))
    ax.set_xlabel('year')
    ax.set_ylabel('')
    ax.set_xlim((0,date_range))

    # put shaded boxes to delineate years
    if step:
        ylim = ax.get_ylim()
        xlim = ax.get_xlim()
        from matplotlib.patches import Rectangle
        for yi,year in enumerate(np.arange(np.floor(tick_vals[0]), tick_vals[-1]+.01, step)):
            pos = year - offset
            r = Rectangle((pos, ylim[1]-5),
                          step, ylim[0]-ylim[1]+10,
                          facecolor=[0.7+0.1*(1+yi%2)] * 3,
                          edgecolor=[1,1,1])
            ax.add_patch(r)
            if year in tick_vals and pos>=xlim[0] and pos<=xlim[1] and ticks:
                label_str = str(step*(year//step)) if step<1 else  str(int(year))
                ax.text(pos,ylim[0]-0.04*(ylim[1]-ylim[0]), label_str,
                        horizontalalignment='center')
        ax.set_axis_off()

    # add confidence intervals to the tree graph -- grey bars
    if confidence:
        tree_layout(tt.tree)
        if not hasattr(tt.tree.root, "marginal_inverse_cdf"):
            print("marginal time tree reconstruction required for confidence intervals")
            return ttconf.ERROR
        elif type(confidence) is float:
            cfunc = tt.get_max_posterior_region
        elif len(confidence)==2:
            cfunc = tt.get_confidence_interval
        else:
            print("confidence needs to be either a float (for max posterior region) or a two numbers specifying lower and upper bounds")
            return ttconf.ERROR

        for n in tt.tree.find_clades():
            pos = cfunc(n, confidence)
            ax.plot(pos-offset, np.ones(len(pos))*n.ypos, lw=3, c=(0.5,0.5,0.5))
    return fig, ax

Response: Plots the time tree vs. years of the time tree.