# autora.theorist.bms.mcmc

A Markov-Chain Monte-Carlo module.

Module constants

get_ops(): A dictionary of accepted operations: {operation_name: offspring}

operation_name: the operation name, e.g. 'sin' for the sinusoid function

offspring: the number of arguments the function requires.

For instance, get_ops() = {"sin": 1, "**": 2 } means for
sin the function call looks like sin(x1) whereas for
the exponentiation operator **, the function call looks like x1 ** x2


## Node

Object that holds algebraic term. This could be a function, variable, or parameter.

Attributes:

Name Type Description
order int

number of children nodes this term has e.g. cos(x) has one child, whereas add(x,y) has two children

Source code in temp_dir/bms/src/autora/theorist/bms/mcmc.py
  41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 class Node: """ Object that holds algebraic term. This could be a function, variable, or parameter. Attributes: order: number of children nodes this term has e.g. cos(x) has one child, whereas add(x,y) has two children """ def __init__(self, value, parent=None, offspring=[]): """ Initialises the node object. Arguments: parent: parent node - unless this node is the root, this will be whichever node contains the function this node's term is most immediately nested within e.g. f(x) is the parent of g(x) in f(g(x)) offspring: list of child nodes value: the specific term held by this node """ self.parent: Node = parent self.offspring: List[Node] = offspring self.value: str = value self.order: int = len(self.offspring) def pr(self, custom_ops, show_pow=False): """ Converts expression in readable form Returns: String """ if self.offspring == []: return "%s" % self.value elif len(self.offspring) == 2 and self.value not in custom_ops: return "(%s %s %s)" % ( self.offspring[0].pr(custom_ops=custom_ops, show_pow=show_pow), self.value, self.offspring[1].pr(custom_ops=custom_ops, show_pow=show_pow), ) else: if show_pow: return "%s(%s)" % ( self.value, ",".join( [ o.pr(custom_ops=custom_ops, show_pow=show_pow) for o in self.offspring ] ), ) else: if self.value == "pow2": return "(%s ** 2)" % ( self.offspring[0].pr(custom_ops=custom_ops, show_pow=show_pow) ) elif self.value == "pow3": return "(%s ** 3)" % ( self.offspring[0].pr(custom_ops=custom_ops, show_pow=show_pow) ) else: return "%s(%s)" % ( self.value, ",".join( [ o.pr(custom_ops=custom_ops, show_pow=show_pow) for o in self.offspring ] ), ) 

### __init__(value, parent=None, offspring=[])

Initialises the node object.

Parameters:

Name Type Description Default
parent

parent node - unless this node is the root, this will be whichever node contains the function this node's term is most immediately nested within e.g. f(x) is the parent of g(x) in f(g(x))

None
offspring

list of child nodes

[]
value

the specific term held by this node

required
Source code in temp_dir/bms/src/autora/theorist/bms/mcmc.py
 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 def __init__(self, value, parent=None, offspring=[]): """ Initialises the node object. Arguments: parent: parent node - unless this node is the root, this will be whichever node contains the function this node's term is most immediately nested within e.g. f(x) is the parent of g(x) in f(g(x)) offspring: list of child nodes value: the specific term held by this node """ self.parent: Node = parent self.offspring: List[Node] = offspring self.value: str = value self.order: int = len(self.offspring) 

### pr(custom_ops, show_pow=False)

Returns: String

Source code in temp_dir/bms/src/autora/theorist/bms/mcmc.py
  66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 def pr(self, custom_ops, show_pow=False): """ Converts expression in readable form Returns: String """ if self.offspring == []: return "%s" % self.value elif len(self.offspring) == 2 and self.value not in custom_ops: return "(%s %s %s)" % ( self.offspring[0].pr(custom_ops=custom_ops, show_pow=show_pow), self.value, self.offspring[1].pr(custom_ops=custom_ops, show_pow=show_pow), ) else: if show_pow: return "%s(%s)" % ( self.value, ",".join( [ o.pr(custom_ops=custom_ops, show_pow=show_pow) for o in self.offspring ] ), ) else: if self.value == "pow2": return "(%s ** 2)" % ( self.offspring[0].pr(custom_ops=custom_ops, show_pow=show_pow) ) elif self.value == "pow3": return "(%s ** 3)" % ( self.offspring[0].pr(custom_ops=custom_ops, show_pow=show_pow) ) else: return "%s(%s)" % ( self.value, ",".join( [ o.pr(custom_ops=custom_ops, show_pow=show_pow) for o in self.offspring ] ), ) 

## Tree

Object that manages the model equation. It contains the root node, which in turn iteratively holds children nodes. Collectively this represents the model equation tree

Attributes:

Name Type Description
root

the root node of the equation tree

parameters

the settable parameters for this trees model search

op_orders

order of each function within the ops

nops

number of operations of each type

move_types

possible combinations of function nesting

ets

possible elementary equation trees

dist_par

distinct parameters used

nodes

nodes of the tree (operations and leaves)

et_space

space of all possible leaves and elementary trees

rr_space

space of all possible root replacement trees

num_rr

number of possible root replacement trees

x

independent variable data

y

depedent variable data

par_values

The values of the model parameters (one set of values for each dataset)

fit_par

past successful parameter fittings

sse

sum of squared errors (measure of goodness of fit)

bic

bayesian information criterion (measure of goodness of fit)

E

total energy of model

EB

fraction of energy derived from bic score of model

EP

fraction of energy derived from model given prior

representative

representative tree for each canonical formula

Source code in temp_dir/bms/src/autora/theorist/bms/mcmc.py
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1489 1490 1491 1492 1493 1494 1495 1496 class Tree: """ Object that manages the model equation. It contains the root node, which in turn iteratively holds children nodes. Collectively this represents the model equation tree Attributes: root: the root node of the equation tree parameters: the settable parameters for this trees model search op_orders: order of each function within the ops nops: number of operations of each type move_types: possible combinations of function nesting ets: possible elementary equation trees dist_par: distinct parameters used nodes: nodes of the tree (operations and leaves) et_space: space of all possible leaves and elementary trees rr_space: space of all possible root replacement trees num_rr: number of possible root replacement trees x: independent variable data y: depedent variable data par_values: The values of the model parameters (one set of values for each dataset) fit_par: past successful parameter fittings sse: sum of squared errors (measure of goodness of fit) bic: bayesian information criterion (measure of goodness of fit) E: total energy of model EB: fraction of energy derived from bic score of model EP: fraction of energy derived from model given prior representative: representative tree for each canonical formula """ prior, ops = get_priors() def __init__( self, ops=ops, variables=["x"], parameters=["a"], prior_par=prior, x=None, y=None, BT=1.0, PT=1.0, max_size=50, root_value=None, fixed_root=False, custom_ops={}, random_state=None, ): """ Initialises the tree object Args: ops: allowed operations to compose equation variables: dependent variable names parameters: parameters that can be used to better fit the equation to the data prior_par: hyperparameter values over operations within ops x: dependent variables y: independent variables BT: BIC value corresponding to equation PT: prior temperature max_size: maximum size of tree (maximum number of nodes) root_value: algebraic term held at root of equation """ if random_state is not None: seed(random_state) np.random.seed(random_state) # The variables and parameters if custom_ops is None: custom_ops = dict() self.variables = variables self.parameters = [ p if p.startswith("_") and p.endswith("_") else "_%s_" % p for p in parameters ] # The root self.fixed_root = fixed_root if root_value is None: self.root = Node( choice(self.variables + self.parameters), offspring=[], parent=None ) else: self.root = Node(root_value, offspring=[], parent=None) root_order = len(signature(custom_ops[root_value]).parameters) self.root.order = root_order for _ in range(root_order): self.root.offspring.append( Node( choice(self.variables + self.parameters), offspring=[], parent=self.root, ) ) # The possible operations self.ops = ops self.custom_ops = custom_ops # The possible orders of the operations, move types, and move # type probabilities self.op_orders = list(set([0] + [n for n in list(ops.values())])) self.move_types = [p for p in permutations(self.op_orders, 2)] # Elementary trees (including leaves), indexed by order self.ets = dict([(o, []) for o in self.op_orders]) self.ets[0] = [x for x in self.root.offspring] self.ets[self.root.order] = [self.root] # Distinct parameters used self.dist_par = list( set([n.value for n in self.ets[0] if n.value in self.parameters]) ) self.n_dist_par = len(self.dist_par) # Nodes of the tree (operations + leaves) self.nodes = [self.root] # Tree size and other properties of the model self.size = 1 self.max_size = max_size # Space of all possible leaves and elementary trees # (dict. indexed by order) self.et_space = self.build_et_space() # Space of all possible root replacement trees self.rr_space = self.build_rr_space() self.num_rr = len(self.rr_space) # Number of operations of each type self.nops = dict([[o, 0] for o in ops]) if root_value is not None: self.nops[self.root.value] += 1 # The parameters of the prior probability (default: 5 everywhere) if prior_par == {}: self.prior_par = dict([("Nopi_%s" % t, 10.0) for t in self.ops]) else: self.prior_par = prior_par # The datasets if x is None: self.x = {"d0": pd.DataFrame()} self.y = {"d0": pd.Series(dtype=float)} elif isinstance(x, pd.DataFrame): self.x = {"d0": x} self.y = {"d0": y} elif isinstance(x, dict): self.x = x if y is None: self.y = dict([(ds, pd.Series(dtype=float)) for ds in self.x]) else: self.y = y else: raise TypeError("x must be either a dict or a pandas.DataFrame") # The values of the model parameters (one set of values for each dataset) self.par_values = dict( [(ds, deepcopy(dict([(p, 1.0) for p in self.parameters]))) for ds in self.x] ) # BIC and prior temperature self.BT = float(BT) self.PT = float(PT) # For fast fitting, we save past successful fits to this formula self.fit_par = {} # Goodness of fit measures self.sse = self.get_sse() self.bic = self.get_bic() self.E, self.EB, self.EP = self.get_energy() # To control formula degeneracy (i.e. different trees that # correspond to the same canonical formula), we store the # representative tree for each canonical formula self.representative = {} self.representative[self.canonical()] = ( str(self), self.E, deepcopy(self.par_values), ) # Done return # ------------------------------------------------------------------------- def __repr__(self): """ Updates tree's internal representation Returns: root node representation """ return self.root.pr(custom_ops=self.custom_ops) # ------------------------------------------------------------------------- def pr(self, show_pow=True): """ Returns readable representation of tree's root node Returns: root node representation """ return self.root.pr(custom_ops=self.custom_ops, show_pow=show_pow) # ------------------------------------------------------------------------- def canonical(self, verbose=False): """ Provides canonical form of tree's equation so that functionally equivalent trees are made into structurally equivalent trees Return: canonical form of a tree """ try: cansp = sympify(str(self).replace(" ", "")) can = str(cansp) ps = list([str(s) for s in cansp.free_symbols]) positions = [] for p in ps: if p.startswith("_") and p.endswith("_"): positions.append((can.find(p), p)) positions.sort() pcount = 1 for pos, p in positions: can = can.replace(p, "c%d" % pcount) pcount += 1 except SyntaxError: if verbose: print( "WARNING: Could not get canonical form for", str(self), "(using full form!)", file=sys.stderr, ) can = str(self) return can.replace(" ", "") # ------------------------------------------------------------------------- def latex(self): """ translate equation into latex Returns: canonical latex form of equation """ return latex(sympify(self.canonical())) # ------------------------------------------------------------------------- def build_et_space(self): """ Build the space of possible elementary trees, which is a dictionary indexed by the order of the elementary tree Returns: space of elementary trees """ et_space = dict([(o, []) for o in self.op_orders]) et_space[0] = [[x, []] for x in self.variables + self.parameters] for op, noff in list(self.ops.items()): for vs in product(et_space[0], repeat=noff): et_space[noff].append([op, [v[0] for v in vs]]) return et_space # ------------------------------------------------------------------------- def build_rr_space(self): """ Build the space of possible trees for the root replacement move Returns: space of possible root replacements """ rr_space = [] for op, noff in list(self.ops.items()): if noff == 1: rr_space.append([op, []]) else: for vs in product(self.et_space[0], repeat=(noff - 1)): rr_space.append([op, [v[0] for v in vs]]) return rr_space # ------------------------------------------------------------------------- def replace_root(self, rr=None, update_gof=True, verbose=False): """ Replace the root with a "root replacement" rr (if provided; otherwise choose one at random from self.rr_space) Returns: new root (if move was possible) or None (otherwise) """ # If no RR is provided, randomly choose one if rr is None: rr = choice(self.rr_space) # Return None if the replacement is too big if (self.size + self.ops[rr[0]]) > self.max_size: return None # Create the new root and replace existing root newRoot = Node(rr[0], offspring=[], parent=None) newRoot.order = 1 + len(rr[1]) if newRoot.order != self.ops[rr[0]]: raise newRoot.offspring.append(self.root) self.root.parent = newRoot self.root = newRoot self.nops[self.root.value] += 1 self.nodes.append(self.root) self.size += 1 oldRoot = self.root.offspring[0] for leaf in rr[1]: self.root.offspring.append(Node(leaf, offspring=[], parent=self.root)) self.nodes.append(self.root.offspring[-1]) self.ets[0].append(self.root.offspring[-1]) self.size += 1 # Add new root to elementary trees if necessary (that is, iff # the old root was a leaf) if oldRoot.offspring is []: self.ets[self.root.order].append(self.root) # Update list of distinct parameters self.dist_par = list( set([n.value for n in self.ets[0] if n.value in self.parameters]) ) self.n_dist_par = len(self.dist_par) # Update goodness of fit measures, if necessary if update_gof: self.sse = self.get_sse(verbose=verbose) self.bic = self.get_bic(verbose=verbose) self.E = self.get_energy(verbose=verbose) return self.root # ------------------------------------------------------------------------- def is_root_prunable(self): """ Check if the root is "prunable" Returns: boolean of root "prunability" """ if self.size == 1: isPrunable = False elif self.size == 2: isPrunable = True else: isPrunable = True for o in self.root.offspring[1:]: if o.offspring != []: isPrunable = False break return isPrunable # ------------------------------------------------------------------------- def prune_root(self, update_gof=True, verbose=False): """ Cut the root and its rightmost leaves (provided they are, indeed, leaves), leaving the leftmost branch as the new tree. Returns the pruned root with the same format as the replacement roots in self.rr_space (or None if pruning was impossible) Returns: the replacement root """ # Check if the root is "prunable" (and return None if not) if not self.is_root_prunable(): return None # Let's do it! rr = [self.root.value, []] self.nodes.remove(self.root) try: self.ets[len(self.root.offspring)].remove(self.root) except ValueError: pass self.nops[self.root.value] -= 1 self.size -= 1 for o in self.root.offspring[1:]: rr[1].append(o.value) self.nodes.remove(o) self.size -= 1 self.ets[0].remove(o) self.root = self.root.offspring[0] self.root.parent = None # Update list of distinct parameters self.dist_par = list( set([n.value for n in self.ets[0] if n.value in self.parameters]) ) self.n_dist_par = len(self.dist_par) # Update goodness of fit measures, if necessary if update_gof: self.sse = self.get_sse(verbose=verbose) self.bic = self.get_bic(verbose=verbose) self.E = self.get_energy(verbose=verbose) # Done return rr # ------------------------------------------------------------------------- def _add_et(self, node, et_order=None, et=None, update_gof=True, verbose=False): """ Add an elementary tree replacing the node, which must be a leaf Returns: the input node """ if node.offspring != []: raise # If no ET is provided, randomly choose one (of the specified # order if given, or totally at random otherwise) if et is None: if et_order is not None: et = choice(self.et_space[et_order]) else: all_ets = [] for o in [o for o in self.op_orders if o > 0]: all_ets += self.et_space[o] et = choice(all_ets) et_order = len(et[1]) else: et_order = len(et[1]) # Update the node and its offspring node.value = et[0] try: self.nops[node.value] += 1 except KeyError: pass node.offspring = [Node(v, parent=node, offspring=[]) for v in et[1]] self.ets[et_order].append(node) try: self.ets[len(node.parent.offspring)].remove(node.parent) except ValueError: pass except AttributeError: pass # Add the offspring to the list of nodes for n in node.offspring: self.nodes.append(n) # Remove the node from the list of leaves and add its offspring self.ets[0].remove(node) for o in node.offspring: self.ets[0].append(o) self.size += 1 # Update list of distinct parameters self.dist_par = list( set([n.value for n in self.ets[0] if n.value in self.parameters]) ) self.n_dist_par = len(self.dist_par) # Update goodness of fit measures, if necessary if update_gof: self.sse = self.get_sse(verbose=verbose) self.bic = self.get_bic(verbose=verbose) self.E = self.get_energy(verbose=verbose) return node # ------------------------------------------------------------------------- def _del_et(self, node, leaf=None, update_gof=True, verbose=False): """ Remove an elementary tree, replacing it by a leaf Returns: input node """ if self.size == 1: return None if leaf is None: leaf = choice(self.et_space[0])[0] self.nops[node.value] -= 1 node.value = leaf self.ets[len(node.offspring)].remove(node) self.ets[0].append(node) for o in node.offspring: self.ets[0].remove(o) self.nodes.remove(o) self.size -= 1 node.offspring = [] if node.parent is not None: is_parent_et = True for o in node.parent.offspring: if o not in self.ets[0]: is_parent_et = False break if is_parent_et: self.ets[len(node.parent.offspring)].append(node.parent) # Update list of distinct parameters self.dist_par = list( set([n.value for n in self.ets[0] if n.value in self.parameters]) ) self.n_dist_par = len(self.dist_par) # Update goodness of fit measures, if necessary if update_gof: self.sse = self.get_sse(verbose=verbose) self.bic = self.get_bic(verbose=verbose) self.E = self.get_energy(verbose=verbose) return node # ------------------------------------------------------------------------- def et_replace(self, target, new, update_gof=True, verbose=False): """ Replace one elementary tree with another one, both of arbitrary order. target is a Node and new is a tuple [node_value, [list, of, offspring, values]] Returns: target """ oini, ofin = len(target.offspring), len(new[1]) if oini == 0: added = self._add_et(target, et=new, update_gof=False, verbose=verbose) else: if ofin == 0: added = self._del_et( target, leaf=new[0], update_gof=False, verbose=verbose ) else: self._del_et(target, update_gof=False, verbose=verbose) added = self._add_et(target, et=new, update_gof=False, verbose=verbose) # Update goodness of fit measures, if necessary if update_gof: self.sse = self.get_sse(verbose=verbose) self.bic = self.get_bic(verbose=verbose) # Done return added # ------------------------------------------------------------------------- def get_sse(self, fit=True, verbose=False): """ Get the sum of squared errors, fitting the expression represented by the Tree to the existing data, if specified (by default, yes) Returns: sum of square errors (sse) """ # Return 0 if there is no data if list(self.x.values())[0].empty or list(self.y.values())[0].empty: self.sse = 0 return 0 # Convert the Tree into a SymPy expression ex = sympify(str(self)) # Convert the expression to a function that can be used by # curve_fit, i.e. that takes as arguments (x, a0, a1, ..., an) atomd = dict([(a.name, a) for a in ex.atoms() if a.is_Symbol]) variables = [atomd[v] for v in self.variables if v in list(atomd.keys())] parameters = [atomd[p] for p in self.parameters if p in list(atomd.keys())] dic: dict = dict( { "fac": scipy.special.factorial, "sig": scipy.special.expit, "relu": relu, }, **self.custom_ops, ) try: flam = lambdify( variables + parameters, ex, [ "numpy", dic, ], ) except (SyntaxError, KeyError): self.sse = dict([(ds, np.inf) for ds in self.x]) return self.sse if fit: if len(parameters) == 0: # Nothing to fit for ds in self.x: for p in self.parameters: self.par_values[ds][p] = 1.0 elif str(self) in self.fit_par: # Recover previously fit parameters self.par_values = self.fit_par[str(self)] else: # Do the fit for all datasets self.fit_par[str(self)] = {} for ds in self.x: this_x, this_y = self.x[ds], self.y[ds] xmat = [this_x[v.name] for v in variables] def feval(x, *params): args = [xi for xi in x] + [p for p in params] return flam(*args) try: # Fit the parameters res = curve_fit( feval, xmat, this_y, p0=[self.par_values[ds][p.name] for p in parameters], maxfev=10000, ) # Reassign the values of the parameters self.par_values[ds] = dict( [ (parameters[i].name, res[0][i]) for i in range(len(res[0])) ] ) for p in self.parameters: if p not in self.par_values[ds]: self.par_values[ds][p] = 1.0 # Save this fit self.fit_par[str(self)][ds] = deepcopy(self.par_values[ds]) except RuntimeError: # Save this (unsuccessful) fit and print warning self.fit_par[str(self)][ds] = deepcopy(self.par_values[ds]) if verbose: print( "#Cannot_fit:%s # # # # #" % str(self).replace(" ", ""), file=sys.stderr, ) # Sum of squared errors self.sse = {} for ds in self.x: this_x, this_y = self.x[ds], self.y[ds] xmat = [this_x[v.name] for v in variables] ar = [np.array(xi) for xi in xmat] + [ self.par_values[ds][p.name] for p in parameters ] try: se = np.square(this_y - flam(*ar)) if sum(np.isnan(se)) > 0: raise ValueError else: self.sse[ds] = np.sum(se) except ValueError: if verbose: print("> Cannot calculate SSE for %s: inf" % self, file=sys.stderr) self.sse[ds] = np.inf except TypeError: if verbose: print("Complex-valued parameters are invalid") self.sse[ds] = np.inf # Done return self.sse # ------------------------------------------------------------------------- def get_bic(self, reset=True, fit=False, verbose=False): """ Calculate the Bayesian information criterion (BIC) of the current expression, given the data. If reset==False, the value of self.bic will not be updated (by default, it will) Returns: Bayesian information criterion (BIC) """ if list(self.x.values())[0].empty or list(self.y.values())[0].empty: if reset: self.bic = 0 return 0 # Get the sum of squared errors (fitting, if required) sse = self.get_sse(fit=fit, verbose=verbose) # Calculate the BIC parameters = set([p.value for p in self.ets[0] if p.value in self.parameters]) k = 1 + len(parameters) BIC = 0.0 for ds in self.y: if sse[ds] == 0.0: BIC = -np.inf break else: n = len(self.y[ds]) BIC += (k - n) * np.log(n) + n * ( np.log(2.0 * np.pi) + log(sse[ds]) + 1 ) if reset: self.bic = BIC return BIC # ------------------------------------------------------------------------- def get_energy(self, bic=False, reset=False, verbose=False): """ Calculate the "energy" of a given formula, that is, approximate minus log-posterior of the formula given the data (the approximation coming from the use of the BIC instead of the exactly integrated likelihood) Returns: Energy of formula (as E, EB, and EP) """ # Contribution of the data (recalculating BIC if necessary) if bic: EB = self.get_bic(reset=reset, verbose=verbose) / 2.0 else: EB = self.bic / 2.0 # Contribution from the prior EP = 0.0 for op, nop in list(self.nops.items()): try: EP += self.prior_par["Nopi_%s" % op] * nop except KeyError: pass try: EP += self.prior_par["Nopi2_%s" % op] * nop**2 except KeyError: pass # Reset the value, if necessary if reset: self.EB = EB self.EP = EP self.E = EB + EP # Done return EB + EP, EB, EP # ------------------------------------------------------------------------- def update_representative(self, verbose=False): """Check if we've seen this formula before, either in its current form or in another form. *If we haven't seen it, save it and return 1. *If we have seen it and this IS the representative, just return 0. *If we have seen it and the representative has smaller energy, just return -1. *If we have seen it and the representative has higher energy, update the representatitve and return -2. Returns: Integer value (0, 1, or -1) corresponding to: 0: we have seen this canonical form before 1: we haven't seen this canonical form before -1: we have seen this equation's canonical form before but it isn't in that form yet """ # Check for canonical representative canonical = self.canonical(verbose=verbose) try: # We've seen this canonical before! rep, rep_energy, rep_par_values = self.representative[canonical] except TypeError: return -1 # Complex-valued parameters are invalid except KeyError: # Never seen this canonical formula before: # save it and return 1 self.get_bic(reset=True, fit=True, verbose=verbose) new_energy = self.get_energy(bic=False, verbose=verbose) self.representative[canonical] = ( str(self), new_energy, deepcopy(self.par_values), ) return 1 # If we've seen this canonical before, check if the # representative needs to be updated if rep == str(self): # This IS the representative: return 0 return 0 else: return -1 # ------------------------------------------------------------------------- def dE_et(self, target, new, verbose=False): """ Calculate the energy change associated to the replacement of one elementary tree with another, both of arbitrary order. "target" is a Node() and "new" is a tuple [node_value, [list, of, offspring, values]]. Returns: change in energy associated with an elementary tree replacement move """ dEB, dEP = 0.0, 0.0 # Some terms of the acceptance (number of possible move types # from initial and final configurations), as well as checking # if the tree is canonically acceptable. # number of possible move types from initial nif = sum( [ int(len(self.ets[oi]) > 0 and (self.size + of - oi) <= self.max_size) for oi, of in self.move_types ] ) # replace old = [target.value, [o.value for o in target.offspring]] old_bic, old_sse, old_energy = self.bic, deepcopy(self.sse), self.E old_par_values = deepcopy(self.par_values) added = self.et_replace(target, new, update_gof=False, verbose=verbose) # number of possible move types from final nfi = sum( [ int(len(self.ets[oi]) > 0 and (self.size + of - oi) <= self.max_size) for oi, of in self.move_types ] ) # check/update canonical representative rep_res = self.update_representative(verbose=verbose) if rep_res == -1: # this formula is forbidden self.et_replace(added, old, update_gof=False, verbose=verbose) self.bic, self.sse, self.E = old_bic, deepcopy(old_sse), old_energy self.par_values = old_par_values return np.inf, np.inf, np.inf, deepcopy(self.par_values), nif, nfi # leave the whole thing as it was before the back & fore self.et_replace(added, old, update_gof=False, verbose=verbose) self.bic, self.sse, self.E = old_bic, deepcopy(old_sse), old_energy self.par_values = old_par_values # Prior: change due to the numbers of each operation try: dEP -= self.prior_par["Nopi_%s" % target.value] except KeyError: pass try: dEP += self.prior_par["Nopi_%s" % new[0]] except KeyError: pass try: dEP += self.prior_par["Nopi2_%s" % target.value] * ( (self.nops[target.value] - 1) ** 2 - (self.nops[target.value]) ** 2 ) except KeyError: pass try: dEP += self.prior_par["Nopi2_%s" % new[0]] * ( (self.nops[new[0]] + 1) ** 2 - (self.nops[new[0]]) ** 2 ) except KeyError: pass # Data if not list(self.x.values())[0].empty: bicOld = self.bic sseOld = deepcopy(self.sse) par_valuesOld = deepcopy(self.par_values) old = [target.value, [o.value for o in target.offspring]] # replace added = self.et_replace(target, new, update_gof=True, verbose=verbose) bicNew = self.bic par_valuesNew = deepcopy(self.par_values) # leave the whole thing as it was before the back & fore self.et_replace(added, old, update_gof=False, verbose=verbose) self.bic = bicOld self.sse = deepcopy(sseOld) self.par_values = par_valuesOld dEB += (bicNew - bicOld) / 2.0 else: par_valuesNew = deepcopy(self.par_values) # Done try: dEB = float(dEB) dEP = float(dEP) dE = dEB + dEP except (ValueError, TypeError): dEB, dEP, dE = np.inf, np.inf, np.inf return dE, dEB, dEP, par_valuesNew, nif, nfi # ------------------------------------------------------------------------- def dE_lr(self, target, new, verbose=False): """ Calculate the energy change associated to a long-range move (the replacement of the value of a node. "target" is a Node() and "new" is a node_value Returns: energy change associated with a long-range move """ dEB, dEP = 0.0, 0.0 par_valuesNew = deepcopy(self.par_values) if target.value != new: # Check if the new tree is canonically acceptable. old = target.value old_bic, old_sse, old_energy = self.bic, deepcopy(self.sse), self.E old_par_values = deepcopy(self.par_values) target.value = new try: self.nops[old] -= 1 self.nops[new] += 1 except KeyError: pass # check/update canonical representative rep_res = self.update_representative(verbose=verbose) if rep_res == -1: # this formula is forbidden target.value = old try: self.nops[old] += 1 self.nops[new] -= 1 except KeyError: pass self.bic, self.sse, self.E = old_bic, deepcopy(old_sse), old_energy self.par_values = old_par_values return np.inf, np.inf, np.inf, None # leave the whole thing as it was before the back & fore target.value = old try: self.nops[old] += 1 self.nops[new] -= 1 except KeyError: pass self.bic, self.sse, self.E = old_bic, deepcopy(old_sse), old_energy self.par_values = old_par_values # Prior: change due to the numbers of each operation try: dEP -= self.prior_par["Nopi_%s" % target.value] except KeyError: pass try: dEP += self.prior_par["Nopi_%s" % new] except KeyError: pass try: dEP += self.prior_par["Nopi2_%s" % target.value] * ( (self.nops[target.value] - 1) ** 2 - (self.nops[target.value]) ** 2 ) except KeyError: pass try: dEP += self.prior_par["Nopi2_%s" % new] * ( (self.nops[new] + 1) ** 2 - (self.nops[new]) ** 2 ) except KeyError: pass # Data if not list(self.x.values())[0].empty: bicOld = self.bic sseOld = deepcopy(self.sse) par_valuesOld = deepcopy(self.par_values) old = target.value target.value = new bicNew = self.get_bic(reset=True, fit=True, verbose=verbose) par_valuesNew = deepcopy(self.par_values) # leave the whole thing as it was before the back & fore target.value = old self.bic = bicOld self.sse = deepcopy(sseOld) self.par_values = par_valuesOld dEB += (bicNew - bicOld) / 2.0 else: par_valuesNew = deepcopy(self.par_values) # Done try: dEB = float(dEB) dEP = float(dEP) dE = dEB + dEP return dE, dEB, dEP, par_valuesNew except (ValueError, TypeError): return np.inf, np.inf, np.inf, None # ------------------------------------------------------------------------- def dE_rr(self, rr=None, verbose=False): """ Calculate the energy change associated to a root replacement move. If rr==None, then it returns the energy change associated to pruning the root; otherwise, it returns the energy change associated to adding the root replacement "rr" Returns: energy change associated with a root replacement move """ dEB, dEP = 0.0, 0.0 # Root pruning if rr is None: if not self.is_root_prunable(): return np.inf, np.inf, np.inf, self.par_values # Check if the new tree is canonically acceptable. # replace old_bic, old_sse, old_energy = self.bic, deepcopy(self.sse), self.E old_par_values = deepcopy(self.par_values) oldrr = [self.root.value, [o.value for o in self.root.offspring[1:]]] self.prune_root(update_gof=False, verbose=verbose) # check/update canonical representative rep_res = self.update_representative(verbose=verbose) if rep_res == -1: # this formula is forbidden self.replace_root(rr=oldrr, update_gof=False, verbose=verbose) self.bic, self.sse, self.E = old_bic, deepcopy(old_sse), old_energy self.par_values = old_par_values return np.inf, np.inf, np.inf, deepcopy(self.par_values) # leave the whole thing as it was before the back & fore self.replace_root(rr=oldrr, update_gof=False, verbose=verbose) self.bic, self.sse, self.E = old_bic, deepcopy(old_sse), old_energy self.par_values = old_par_values # Prior: change due to the numbers of each operation dEP -= self.prior_par["Nopi_%s" % self.root.value] try: dEP += self.prior_par["Nopi2_%s" % self.root.value] * ( (self.nops[self.root.value] - 1) ** 2 - (self.nops[self.root.value]) ** 2 ) except KeyError: pass # Data correction if not list(self.x.values())[0].empty: bicOld = self.bic sseOld = deepcopy(self.sse) par_valuesOld = deepcopy(self.par_values) oldrr = [self.root.value, [o.value for o in self.root.offspring[1:]]] # replace self.prune_root(update_gof=False, verbose=verbose) bicNew = self.get_bic(reset=True, fit=True, verbose=verbose) par_valuesNew = deepcopy(self.par_values) # leave the whole thing as it was before the back & fore self.replace_root(rr=oldrr, update_gof=False, verbose=verbose) self.bic = bicOld self.sse = deepcopy(sseOld) self.par_values = par_valuesOld dEB += (bicNew - bicOld) / 2.0 else: par_valuesNew = deepcopy(self.par_values) # Done try: dEB = float(dEB) dEP = float(dEP) dE = dEB + dEP except (ValueError, TypeError): dEB, dEP, dE = np.inf, np.inf, np.inf return dE, dEB, dEP, par_valuesNew # Root replacement else: # Check if the new tree is canonically acceptable. # replace old_bic, old_sse, old_energy = self.bic, deepcopy(self.sse), self.E old_par_values = deepcopy(self.par_values) newroot = self.replace_root(rr=rr, update_gof=False, verbose=verbose) if newroot is None: # Root cannot be replaced (due to max_size) return np.inf, np.inf, np.inf, deepcopy(self.par_values) # check/update canonical representative rep_res = self.update_representative(verbose=verbose) if rep_res == -1: # this formula is forbidden self.prune_root(update_gof=False, verbose=verbose) self.bic, self.sse, self.E = old_bic, deepcopy(old_sse), old_energy self.par_values = old_par_values return np.inf, np.inf, np.inf, deepcopy(self.par_values) # leave the whole thing as it was before the back & fore self.prune_root(update_gof=False, verbose=verbose) self.bic, self.sse, self.E = old_bic, deepcopy(old_sse), old_energy self.par_values = old_par_values # Prior: change due to the numbers of each operation dEP += self.prior_par["Nopi_%s" % rr[0]] try: dEP += self.prior_par["Nopi2_%s" % rr[0]] * ( (self.nops[rr[0]] + 1) ** 2 - (self.nops[rr[0]]) ** 2 ) except KeyError: pass # Data if not list(self.x.values())[0].empty: bicOld = self.bic sseOld = deepcopy(self.sse) par_valuesOld = deepcopy(self.par_values) # replace newroot = self.replace_root(rr=rr, update_gof=False, verbose=verbose) if newroot is None: return np.inf, np.inf, np.inf, self.par_values bicNew = self.get_bic(reset=True, fit=True, verbose=verbose) par_valuesNew = deepcopy(self.par_values) # leave the whole thing as it was before the back & fore self.prune_root(update_gof=False, verbose=verbose) self.bic = bicOld self.sse = deepcopy(sseOld) self.par_values = par_valuesOld dEB += (bicNew - bicOld) / 2.0 else: par_valuesNew = deepcopy(self.par_values) # Done try: dEB = float(dEB) dEP = float(dEP) dE = dEB + dEP except (ValueError, TypeError): dEB, dEP, dE = np.inf, np.inf, np.inf return dE, dEB, dEP, par_valuesNew # ------------------------------------------------------------------------- def mcmc_step(self, verbose=False, p_rr=0.05, p_long=0.45): """ Make a single MCMC step Returns: None or expression list """ topDice = random() # Root replacement move if topDice < p_rr: if random() < 0.5: # Try to prune the root dE, dEB, dEP, par_valuesNew = self.dE_rr(rr=None, verbose=verbose) if -dEB / self.BT - dEP / self.PT > 300: paccept = 1 else: paccept = np.exp(-dEB / self.BT - dEP / self.PT) / float( self.num_rr ) dice = random() if dice < paccept: # Accept move self.prune_root(update_gof=False, verbose=verbose) self.par_values = par_valuesNew self.get_bic(reset=True, fit=False, verbose=verbose) self.E += dE self.EB += dEB self.EP += dEP else: # Try to replace the root newrr = choice(self.rr_space) dE, dEB, dEP, par_valuesNew = self.dE_rr(rr=newrr, verbose=verbose) if self.num_rr > 0 and -dEB / self.BT - dEP / self.PT > 0: paccept = 1.0 elif self.num_rr == 0: paccept = 0.0 else: paccept = self.num_rr * np.exp(-dEB / self.BT - dEP / self.PT) dice = random() if dice < paccept: # Accept move self.replace_root(rr=newrr, update_gof=False, verbose=verbose) self.par_values = par_valuesNew self.get_bic(reset=True, fit=False, verbose=verbose) self.E += dE self.EB += dEB self.EP += dEP # Long-range move elif topDice < (p_rr + p_long) and not ( self.fixed_root and len(self.nodes) == 1 ): # Choose a random node in the tree, and a random new operation target = choice(self.nodes) if self.fixed_root: while target is self.root: target = choice(self.nodes) nready = False while not nready: if len(target.offspring) == 0: new = choice(self.variables + self.parameters) nready = True else: new = choice(list(self.ops.keys())) if self.ops[new] == self.ops[target.value]: nready = True dE, dEB, dEP, par_valuesNew = self.dE_lr(target, new, verbose=verbose) try: paccept = np.exp(-dEB / self.BT - dEP / self.PT) except ValueError: _logger.warning("Potentially failing to set paccept properly") if (dEB / self.BT + dEP / self.PT) < 0: paccept = 1.0 # Accept move, if necessary dice = random() if dice < paccept: # update number of operations if target.offspring != []: self.nops[target.value] -= 1 self.nops[new] += 1 # move target.value = new # recalculate distinct parameters self.dist_par = list( set([n.value for n in self.ets[0] if n.value in self.parameters]) ) self.n_dist_par = len(self.dist_par) # update others self.par_values = deepcopy(par_valuesNew) self.get_bic(reset=True, fit=False, verbose=verbose) self.E += dE self.EB += dEB self.EP += dEP # Elementary tree (short-range) move else: target = None while target is None or self.fixed_root and target is self.root: # Choose a feasible move (doable and keeping size<=max_size) while True: oini, ofin = choice(self.move_types) if len(self.ets[oini]) > 0 and ( self.size - oini + ofin <= self.max_size ): break # target and new ETs target = choice(self.ets[oini]) new = choice(self.et_space[ofin]) # omegai and omegaf omegai = len(self.ets[oini]) omegaf = len(self.ets[ofin]) + 1 if ofin == 0: omegaf -= oini if oini == 0 and target.parent in self.ets[ofin]: omegaf -= 1 # size of et_space of each type si = len(self.et_space[oini]) sf = len(self.et_space[ofin]) # Probability of acceptance dE, dEB, dEP, par_valuesNew, nif, nfi = self.dE_et( target, new, verbose=verbose ) try: paccept = ( float(nif) * omegai * sf * np.exp(-dEB / self.BT - dEP / self.PT) ) / (float(nfi) * omegaf * si) except ValueError: if (dEB / self.BT + dEP / self.PT) < -200: paccept = 1.0 # Accept / reject dice = random() if dice < paccept: # Accept move self.et_replace(target, new, verbose=verbose) self.par_values = par_valuesNew self.get_bic(verbose=verbose) self.E += dE self.EB += dEB self.EP += dEP # Done return # ------------------------------------------------------------------------- def mcmc( self, tracefn="trace.dat", progressfn="progress.dat", write_files=True, reset_files=True, burnin=2000, thin=10, samples=10000, verbose=False, progress=True, ): """ Sample the space of formula trees using MCMC, and write the trace and some progress information to files (unless write_files is False) Returns: None or expression list """ self.get_energy(reset=True, verbose=verbose) # Burning if progress: sys.stdout.write("# Burning in\t") sys.stdout.write("[%s]" % (" " * 50)) sys.stdout.flush() sys.stdout.write("\b" * (50 + 1)) for i in range(burnin): self.mcmc_step(verbose=verbose) if progress and (i % (burnin / 50) == 0): sys.stdout.write("=") sys.stdout.flush() # Sample if write_files: if reset_files: tracef = open(tracefn, "w") progressf = open(progressfn, "w") else: tracef = open(tracefn, "a") progressf = open(progressfn, "a") if progress: sys.stdout.write("\n# Sampling\t") sys.stdout.write("[%s]" % (" " * 50)) sys.stdout.flush() sys.stdout.write("\b" * (50 + 1)) for s in range(samples): for i in range(thin): self.mcmc_step(verbose=verbose) if progress and (s % (samples / 50) == 0): sys.stdout.write("=") sys.stdout.flush() if write_files: json.dump( [ s, float(self.bic), float(self.E), str(self.get_energy(verbose=verbose)), str(self), self.par_values, ], tracef, ) tracef.write("\n") tracef.flush() progressf.write("%d %lf %lf\n" % (s, self.E, self.bic)) progressf.flush() # Done if progress: sys.stdout.write("\n") return # ------------------------------------------------------------------------- def predict(self, x): """ Calculate the value of the formula at the given data x. The data x must have the same format as the training data and, in particular, it it must specify to which dataset the example data belongs, if multiple datasets where used for training. Returns: predicted y values """ if isinstance(x, np.ndarray): columns = list() for col in range(x.shape[1]): columns.append("X" + str(col)) x = pd.DataFrame(x, columns=columns) if isinstance(x, pd.DataFrame): this_x = {"d0": x} input_type = "df" elif isinstance(x, dict): this_x = x input_type = "dict" else: raise TypeError("x must be either a dict or a pandas.DataFrame") # Convert the Tree into a SymPy expression ex = sympify(str(self)) # Convert the expression to a function atomd = dict([(a.name, a) for a in ex.atoms() if a.is_Symbol]) variables = [atomd[v] for v in self.variables if v in list(atomd.keys())] parameters = [atomd[p] for p in self.parameters if p in list(atomd.keys())] flam = lambdify( variables + parameters, ex, [ "numpy", dict( { "fac": scipy.special.factorial, "sig": scipy.special.expit, "relu": relu, }, **self.custom_ops, ), ], ) # Loop over datasets predictions = {} for ds in this_x: # Prepare variables and parameters xmat = [this_x[ds][v.name] for v in variables] params = [self.par_values[ds][p.name] for p in parameters] args = [xi for xi in xmat] + [p for p in params] # Predict try: prediction = flam(*args) except SyntaxError: # Do it point by point prediction = [np.nan for i in range(len(this_x[ds]))] predictions[ds] = pd.Series(prediction, index=list(this_x[ds].index)) if input_type == "df": return predictions["d0"] else: return predictions # ------------------------------------------------------------------------- def trace_predict( self, x, burnin=1000, thin=2000, samples=1000, tracefn="trace.dat", progressfn="progress.dat", write_files=False, reset_files=True, verbose=False, progress=True, ): """ Sample the space of formula trees using MCMC, and predict y(x) for each of the sampled formula trees Returns: predicted y values for each of the sampled formula trees """ ypred = {} # Burning if progress: sys.stdout.write("# Burning in\t") sys.stdout.write("[%s]" % (" " * 50)) sys.stdout.flush() sys.stdout.write("\b" * (50 + 1)) for i in range(burnin): self.mcmc_step(verbose=verbose) if progress and (i % (burnin / 50) == 0): sys.stdout.write("=") sys.stdout.flush() # Sample if write_files: if reset_files: tracef = open(tracefn, "w") progressf = open(progressfn, "w") else: tracef = open(tracefn, "a") progressf = open(progressfn, "a") if progress: sys.stdout.write("\n# Sampling\t") sys.stdout.write("[%s]" % (" " * 50)) sys.stdout.flush() sys.stdout.write("\b" * (50 + 1)) for s in range(samples): for kk in range(thin): self.mcmc_step(verbose=verbose) # Make prediction ypred[s] = self.predict(x) # Output if progress and (s % (samples / 50) == 0): sys.stdout.write("=") sys.stdout.flush() if write_files: json.dump( [ s, float(self.bic), float(self.E), float(self.get_energy(verbose=verbose)), str(self), self.par_values, ], tracef, ) tracef.write("\n") tracef.flush() progressf.write("%d %lf %lf\n" % (s, self.E, self.bic)) progressf.flush() # Done if progress: sys.stdout.write("\n") return pd.DataFrame.from_dict(ypred) 

### __init__(ops=ops, variables=['x'], parameters=['a'], prior_par=prior, x=None, y=None, BT=1.0, PT=1.0, max_size=50, root_value=None, fixed_root=False, custom_ops={}, random_state=None)

Initialises the tree object

Parameters:

Name Type Description Default
ops

allowed operations to compose equation

ops
variables

dependent variable names

['x']
parameters

parameters that can be used to better fit the equation to the data

['a']
prior_par

hyperparameter values over operations within ops

prior
x

dependent variables

None
y

independent variables

None
BT

BIC value corresponding to equation

1.0
PT

prior temperature

1.0
max_size

maximum size of tree (maximum number of nodes)

50
root_value

algebraic term held at root of equation

None
Source code in temp_dir/bms/src/autora/theorist/bms/mcmc.py
 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 def __init__( self, ops=ops, variables=["x"], parameters=["a"], prior_par=prior, x=None, y=None, BT=1.0, PT=1.0, max_size=50, root_value=None, fixed_root=False, custom_ops={}, random_state=None, ): """ Initialises the tree object Args: ops: allowed operations to compose equation variables: dependent variable names parameters: parameters that can be used to better fit the equation to the data prior_par: hyperparameter values over operations within ops x: dependent variables y: independent variables BT: BIC value corresponding to equation PT: prior temperature max_size: maximum size of tree (maximum number of nodes) root_value: algebraic term held at root of equation """ if random_state is not None: seed(random_state) np.random.seed(random_state) # The variables and parameters if custom_ops is None: custom_ops = dict() self.variables = variables self.parameters = [ p if p.startswith("_") and p.endswith("_") else "_%s_" % p for p in parameters ] # The root self.fixed_root = fixed_root if root_value is None: self.root = Node( choice(self.variables + self.parameters), offspring=[], parent=None ) else: self.root = Node(root_value, offspring=[], parent=None) root_order = len(signature(custom_ops[root_value]).parameters) self.root.order = root_order for _ in range(root_order): self.root.offspring.append( Node( choice(self.variables + self.parameters), offspring=[], parent=self.root, ) ) # The possible operations self.ops = ops self.custom_ops = custom_ops # The possible orders of the operations, move types, and move # type probabilities self.op_orders = list(set([0] + [n for n in list(ops.values())])) self.move_types = [p for p in permutations(self.op_orders, 2)] # Elementary trees (including leaves), indexed by order self.ets = dict([(o, []) for o in self.op_orders]) self.ets[0] = [x for x in self.root.offspring] self.ets[self.root.order] = [self.root] # Distinct parameters used self.dist_par = list( set([n.value for n in self.ets[0] if n.value in self.parameters]) ) self.n_dist_par = len(self.dist_par) # Nodes of the tree (operations + leaves) self.nodes = [self.root] # Tree size and other properties of the model self.size = 1 self.max_size = max_size # Space of all possible leaves and elementary trees # (dict. indexed by order) self.et_space = self.build_et_space() # Space of all possible root replacement trees self.rr_space = self.build_rr_space() self.num_rr = len(self.rr_space) # Number of operations of each type self.nops = dict([[o, 0] for o in ops]) if root_value is not None: self.nops[self.root.value] += 1 # The parameters of the prior probability (default: 5 everywhere) if prior_par == {}: self.prior_par = dict([("Nopi_%s" % t, 10.0) for t in self.ops]) else: self.prior_par = prior_par # The datasets if x is None: self.x = {"d0": pd.DataFrame()} self.y = {"d0": pd.Series(dtype=float)} elif isinstance(x, pd.DataFrame): self.x = {"d0": x} self.y = {"d0": y} elif isinstance(x, dict): self.x = x if y is None: self.y = dict([(ds, pd.Series(dtype=float)) for ds in self.x]) else: self.y = y else: raise TypeError("x must be either a dict or a pandas.DataFrame") # The values of the model parameters (one set of values for each dataset) self.par_values = dict( [(ds, deepcopy(dict([(p, 1.0) for p in self.parameters]))) for ds in self.x] ) # BIC and prior temperature self.BT = float(BT) self.PT = float(PT) # For fast fitting, we save past successful fits to this formula self.fit_par = {} # Goodness of fit measures self.sse = self.get_sse() self.bic = self.get_bic() self.E, self.EB, self.EP = self.get_energy() # To control formula degeneracy (i.e. different trees that # correspond to the same canonical formula), we store the # representative tree for each canonical formula self.representative = {} self.representative[self.canonical()] = ( str(self), self.E, deepcopy(self.par_values), ) # Done return 

### __repr__()

Returns: root node representation

Source code in temp_dir/bms/src/autora/theorist/bms/mcmc.py
 281 282 283 284 285 286 287 288 def __repr__(self): """ Updates tree's internal representation Returns: root node representation """ return self.root.pr(custom_ops=self.custom_ops) 

### build_et_space()

Build the space of possible elementary trees, which is a dictionary indexed by the order of the elementary tree

Returns: space of elementary trees

Source code in temp_dir/bms/src/autora/theorist/bms/mcmc.py
 342 343 344 345 346 347 348 349 350 351 352 353 354 def build_et_space(self): """ Build the space of possible elementary trees, which is a dictionary indexed by the order of the elementary tree Returns: space of elementary trees """ et_space = dict([(o, []) for o in self.op_orders]) et_space[0] = [[x, []] for x in self.variables + self.parameters] for op, noff in list(self.ops.items()): for vs in product(et_space[0], repeat=noff): et_space[noff].append([op, [v[0] for v in vs]]) return et_space 

### build_rr_space()

Build the space of possible trees for the root replacement move

Returns: space of possible root replacements

Source code in temp_dir/bms/src/autora/theorist/bms/mcmc.py
 357 358 359 360 361 362 363 364 365 366 367 368 369 370 def build_rr_space(self): """ Build the space of possible trees for the root replacement move Returns: space of possible root replacements """ rr_space = [] for op, noff in list(self.ops.items()): if noff == 1: rr_space.append([op, []]) else: for vs in product(self.et_space[0], repeat=(noff - 1)): rr_space.append([op, [v[0] for v in vs]]) return rr_space 

### canonical(verbose=False)

Provides canonical form of tree's equation so that functionally equivalent trees are made into structurally equivalent trees

Return: canonical form of a tree

Source code in temp_dir/bms/src/autora/theorist/bms/mcmc.py
 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 def canonical(self, verbose=False): """ Provides canonical form of tree's equation so that functionally equivalent trees are made into structurally equivalent trees Return: canonical form of a tree """ try: cansp = sympify(str(self).replace(" ", "")) can = str(cansp) ps = list([str(s) for s in cansp.free_symbols]) positions = [] for p in ps: if p.startswith("_") and p.endswith("_"): positions.append((can.find(p), p)) positions.sort() pcount = 1 for pos, p in positions: can = can.replace(p, "c%d" % pcount) pcount += 1 except SyntaxError: if verbose: print( "WARNING: Could not get canonical form for", str(self), "(using full form!)", file=sys.stderr, ) can = str(self) return can.replace(" ", "") 

### dE_et(target, new, verbose=False)

Calculate the energy change associated to the replacement of one elementary tree with another, both of arbitrary order. "target" is a Node() and "new" is a tuple [node_value, [list, of, offspring, values]].

Returns: change in energy associated with an elementary tree replacement move

Source code in temp_dir/bms/src/autora/theorist/bms/mcmc.py
 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896 897 898 899 900 901 902 903 904 905 906 907 908 909 910 911 912 913 914 def dE_et(self, target, new, verbose=False): """ Calculate the energy change associated to the replacement of one elementary tree with another, both of arbitrary order. "target" is a Node() and "new" is a tuple [node_value, [list, of, offspring, values]]. Returns: change in energy associated with an elementary tree replacement move """ dEB, dEP = 0.0, 0.0 # Some terms of the acceptance (number of possible move types # from initial and final configurations), as well as checking # if the tree is canonically acceptable. # number of possible move types from initial nif = sum( [ int(len(self.ets[oi]) > 0 and (self.size + of - oi) <= self.max_size) for oi, of in self.move_types ] ) # replace old = [target.value, [o.value for o in target.offspring]] old_bic, old_sse, old_energy = self.bic, deepcopy(self.sse), self.E old_par_values = deepcopy(self.par_values) added = self.et_replace(target, new, update_gof=False, verbose=verbose) # number of possible move types from final nfi = sum( [ int(len(self.ets[oi]) > 0 and (self.size + of - oi) <= self.max_size) for oi, of in self.move_types ] ) # check/update canonical representative rep_res = self.update_representative(verbose=verbose) if rep_res == -1: # this formula is forbidden self.et_replace(added, old, update_gof=False, verbose=verbose) self.bic, self.sse, self.E = old_bic, deepcopy(old_sse), old_energy self.par_values = old_par_values return np.inf, np.inf, np.inf, deepcopy(self.par_values), nif, nfi # leave the whole thing as it was before the back & fore self.et_replace(added, old, update_gof=False, verbose=verbose) self.bic, self.sse, self.E = old_bic, deepcopy(old_sse), old_energy self.par_values = old_par_values # Prior: change due to the numbers of each operation try: dEP -= self.prior_par["Nopi_%s" % target.value] except KeyError: pass try: dEP += self.prior_par["Nopi_%s" % new[0]] except KeyError: pass try: dEP += self.prior_par["Nopi2_%s" % target.value] * ( (self.nops[target.value] - 1) ** 2 - (self.nops[target.value]) ** 2 ) except KeyError: pass try: dEP += self.prior_par["Nopi2_%s" % new[0]] * ( (self.nops[new[0]] + 1) ** 2 - (self.nops[new[0]]) ** 2 ) except KeyError: pass # Data if not list(self.x.values())[0].empty: bicOld = self.bic sseOld = deepcopy(self.sse) par_valuesOld = deepcopy(self.par_values) old = [target.value, [o.value for o in target.offspring]] # replace added = self.et_replace(target, new, update_gof=True, verbose=verbose) bicNew = self.bic par_valuesNew = deepcopy(self.par_values) # leave the whole thing as it was before the back & fore self.et_replace(added, old, update_gof=False, verbose=verbose) self.bic = bicOld self.sse = deepcopy(sseOld) self.par_values = par_valuesOld dEB += (bicNew - bicOld) / 2.0 else: par_valuesNew = deepcopy(self.par_values) # Done try: dEB = float(dEB) dEP = float(dEP) dE = dEB + dEP except (ValueError, TypeError): dEB, dEP, dE = np.inf, np.inf, np.inf return dE, dEB, dEP, par_valuesNew, nif, nfi 

### dE_lr(target, new, verbose=False)

Calculate the energy change associated to a long-range move (the replacement of the value of a node. "target" is a Node() and "new" is a node_value

Returns: energy change associated with a long-range move

Source code in temp_dir/bms/src/autora/theorist/bms/mcmc.py
  917 918 919 920 921 922 923 924 925 926 927 928 929 930 931 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948 949 950 951 952 953 954 955 956 957 958 959 960 961 962 963 964 965 966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 981 982 983 984 985 986 987 988 989 990 991 992 993 994 995 996 997 998 999 1000 1001 1002 1003 1004 1005 1006 1007 1008 def dE_lr(self, target, new, verbose=False): """ Calculate the energy change associated to a long-range move (the replacement of the value of a node. "target" is a Node() and "new" is a node_value Returns: energy change associated with a long-range move """ dEB, dEP = 0.0, 0.0 par_valuesNew = deepcopy(self.par_values) if target.value != new: # Check if the new tree is canonically acceptable. old = target.value old_bic, old_sse, old_energy = self.bic, deepcopy(self.sse), self.E old_par_values = deepcopy(self.par_values) target.value = new try: self.nops[old] -= 1 self.nops[new] += 1 except KeyError: pass # check/update canonical representative rep_res = self.update_representative(verbose=verbose) if rep_res == -1: # this formula is forbidden target.value = old try: self.nops[old] += 1 self.nops[new] -= 1 except KeyError: pass self.bic, self.sse, self.E = old_bic, deepcopy(old_sse), old_energy self.par_values = old_par_values return np.inf, np.inf, np.inf, None # leave the whole thing as it was before the back & fore target.value = old try: self.nops[old] += 1 self.nops[new] -= 1 except KeyError: pass self.bic, self.sse, self.E = old_bic, deepcopy(old_sse), old_energy self.par_values = old_par_values # Prior: change due to the numbers of each operation try: dEP -= self.prior_par["Nopi_%s" % target.value] except KeyError: pass try: dEP += self.prior_par["Nopi_%s" % new] except KeyError: pass try: dEP += self.prior_par["Nopi2_%s" % target.value] * ( (self.nops[target.value] - 1) ** 2 - (self.nops[target.value]) ** 2 ) except KeyError: pass try: dEP += self.prior_par["Nopi2_%s" % new] * ( (self.nops[new] + 1) ** 2 - (self.nops[new]) ** 2 ) except KeyError: pass # Data if not list(self.x.values())[0].empty: bicOld = self.bic sseOld = deepcopy(self.sse) par_valuesOld = deepcopy(self.par_values) old = target.value target.value = new bicNew = self.get_bic(reset=True, fit=True, verbose=verbose) par_valuesNew = deepcopy(self.par_values) # leave the whole thing as it was before the back & fore target.value = old self.bic = bicOld self.sse = deepcopy(sseOld) self.par_values = par_valuesOld dEB += (bicNew - bicOld) / 2.0 else: par_valuesNew = deepcopy(self.par_values) # Done try: dEB = float(dEB) dEP = float(dEP) dE = dEB + dEP return dE, dEB, dEP, par_valuesNew except (ValueError, TypeError): return np.inf, np.inf, np.inf, None 

### dE_rr(rr=None, verbose=False)

Calculate the energy change associated to a root replacement move. If rr==None, then it returns the energy change associated to pruning the root; otherwise, it returns the energy change associated to adding the root replacement "rr"

Returns: energy change associated with a root replacement move

Source code in temp_dir/bms/src/autora/theorist/bms/mcmc.py
 1011 1012 1013 1014 1015 1016 1017 1018 1019 1020 1021 1022 1023 1024 1025 1026 1027 1028 1029 1030 1031 1032 1033 1034 1035 1036 1037 1038 1039 1040 1041 1042 1043 1044 1045 1046 1047 1048 1049 1050 1051 1052 1053 1054 1055 1056 1057 1058 1059 1060 1061 1062 1063 1064 1065 1066 1067 1068 1069 1070 1071 1072 1073 1074 1075 1076 1077 1078 1079 1080 1081 1082 1083 1084 1085 1086 1087 1088 1089 1090 1091 1092 1093 1094 1095 1096 1097 1098 1099 1100 1101 1102 1103 1104 1105 1106 1107 1108 1109 1110 1111 1112 1113 1114 1115 1116 1117 1118 1119 1120 1121 1122 1123 1124 1125 1126 1127 1128 1129 1130 1131 1132 1133 1134 1135 1136 1137 1138 1139 def dE_rr(self, rr=None, verbose=False): """ Calculate the energy change associated to a root replacement move. If rr==None, then it returns the energy change associated to pruning the root; otherwise, it returns the energy change associated to adding the root replacement "rr" Returns: energy change associated with a root replacement move """ dEB, dEP = 0.0, 0.0 # Root pruning if rr is None: if not self.is_root_prunable(): return np.inf, np.inf, np.inf, self.par_values # Check if the new tree is canonically acceptable. # replace old_bic, old_sse, old_energy = self.bic, deepcopy(self.sse), self.E old_par_values = deepcopy(self.par_values) oldrr = [self.root.value, [o.value for o in self.root.offspring[1:]]] self.prune_root(update_gof=False, verbose=verbose) # check/update canonical representative rep_res = self.update_representative(verbose=verbose) if rep_res == -1: # this formula is forbidden self.replace_root(rr=oldrr, update_gof=False, verbose=verbose) self.bic, self.sse, self.E = old_bic, deepcopy(old_sse), old_energy self.par_values = old_par_values return np.inf, np.inf, np.inf, deepcopy(self.par_values) # leave the whole thing as it was before the back & fore self.replace_root(rr=oldrr, update_gof=False, verbose=verbose) self.bic, self.sse, self.E = old_bic, deepcopy(old_sse), old_energy self.par_values = old_par_values # Prior: change due to the numbers of each operation dEP -= self.prior_par["Nopi_%s" % self.root.value] try: dEP += self.prior_par["Nopi2_%s" % self.root.value] * ( (self.nops[self.root.value] - 1) ** 2 - (self.nops[self.root.value]) ** 2 ) except KeyError: pass # Data correction if not list(self.x.values())[0].empty: bicOld = self.bic sseOld = deepcopy(self.sse) par_valuesOld = deepcopy(self.par_values) oldrr = [self.root.value, [o.value for o in self.root.offspring[1:]]] # replace self.prune_root(update_gof=False, verbose=verbose) bicNew = self.get_bic(reset=True, fit=True, verbose=verbose) par_valuesNew = deepcopy(self.par_values) # leave the whole thing as it was before the back & fore self.replace_root(rr=oldrr, update_gof=False, verbose=verbose) self.bic = bicOld self.sse = deepcopy(sseOld) self.par_values = par_valuesOld dEB += (bicNew - bicOld) / 2.0 else: par_valuesNew = deepcopy(self.par_values) # Done try: dEB = float(dEB) dEP = float(dEP) dE = dEB + dEP except (ValueError, TypeError): dEB, dEP, dE = np.inf, np.inf, np.inf return dE, dEB, dEP, par_valuesNew # Root replacement else: # Check if the new tree is canonically acceptable. # replace old_bic, old_sse, old_energy = self.bic, deepcopy(self.sse), self.E old_par_values = deepcopy(self.par_values) newroot = self.replace_root(rr=rr, update_gof=False, verbose=verbose) if newroot is None: # Root cannot be replaced (due to max_size) return np.inf, np.inf, np.inf, deepcopy(self.par_values) # check/update canonical representative rep_res = self.update_representative(verbose=verbose) if rep_res == -1: # this formula is forbidden self.prune_root(update_gof=False, verbose=verbose) self.bic, self.sse, self.E = old_bic, deepcopy(old_sse), old_energy self.par_values = old_par_values return np.inf, np.inf, np.inf, deepcopy(self.par_values) # leave the whole thing as it was before the back & fore self.prune_root(update_gof=False, verbose=verbose) self.bic, self.sse, self.E = old_bic, deepcopy(old_sse), old_energy self.par_values = old_par_values # Prior: change due to the numbers of each operation dEP += self.prior_par["Nopi_%s" % rr[0]] try: dEP += self.prior_par["Nopi2_%s" % rr[0]] * ( (self.nops[rr[0]] + 1) ** 2 - (self.nops[rr[0]]) ** 2 ) except KeyError: pass # Data if not list(self.x.values())[0].empty: bicOld = self.bic sseOld = deepcopy(self.sse) par_valuesOld = deepcopy(self.par_values) # replace newroot = self.replace_root(rr=rr, update_gof=False, verbose=verbose) if newroot is None: return np.inf, np.inf, np.inf, self.par_values bicNew = self.get_bic(reset=True, fit=True, verbose=verbose) par_valuesNew = deepcopy(self.par_values) # leave the whole thing as it was before the back & fore self.prune_root(update_gof=False, verbose=verbose) self.bic = bicOld self.sse = deepcopy(sseOld) self.par_values = par_valuesOld dEB += (bicNew - bicOld) / 2.0 else: par_valuesNew = deepcopy(self.par_values) # Done try: dEB = float(dEB) dEP = float(dEP) dE = dEB + dEP except (ValueError, TypeError): dEB, dEP, dE = np.inf, np.inf, np.inf return dE, dEB, dEP, par_valuesNew 

### et_replace(target, new, update_gof=True, verbose=False)

Replace one elementary tree with another one, both of arbitrary order. target is a Node and new is a tuple [node_value, [list, of, offspring, values]]

Returns: target

Source code in temp_dir/bms/src/autora/theorist/bms/mcmc.py
 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 def et_replace(self, target, new, update_gof=True, verbose=False): """ Replace one elementary tree with another one, both of arbitrary order. target is a Node and new is a tuple [node_value, [list, of, offspring, values]] Returns: target """ oini, ofin = len(target.offspring), len(new[1]) if oini == 0: added = self._add_et(target, et=new, update_gof=False, verbose=verbose) else: if ofin == 0: added = self._del_et( target, leaf=new[0], update_gof=False, verbose=verbose ) else: self._del_et(target, update_gof=False, verbose=verbose) added = self._add_et(target, et=new, update_gof=False, verbose=verbose) # Update goodness of fit measures, if necessary if update_gof: self.sse = self.get_sse(verbose=verbose) self.bic = self.get_bic(verbose=verbose) # Done return added 

### get_bic(reset=True, fit=False, verbose=False)

Calculate the Bayesian information criterion (BIC) of the current expression, given the data. If reset==False, the value of self.bic will not be updated (by default, it will)

Returns: Bayesian information criterion (BIC)

Source code in temp_dir/bms/src/autora/theorist/bms/mcmc.py
 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 def get_bic(self, reset=True, fit=False, verbose=False): """ Calculate the Bayesian information criterion (BIC) of the current expression, given the data. If reset==False, the value of self.bic will not be updated (by default, it will) Returns: Bayesian information criterion (BIC) """ if list(self.x.values())[0].empty or list(self.y.values())[0].empty: if reset: self.bic = 0 return 0 # Get the sum of squared errors (fitting, if required) sse = self.get_sse(fit=fit, verbose=verbose) # Calculate the BIC parameters = set([p.value for p in self.ets[0] if p.value in self.parameters]) k = 1 + len(parameters) BIC = 0.0 for ds in self.y: if sse[ds] == 0.0: BIC = -np.inf break else: n = len(self.y[ds]) BIC += (k - n) * np.log(n) + n * ( np.log(2.0 * np.pi) + log(sse[ds]) + 1 ) if reset: self.bic = BIC return BIC 

### get_energy(bic=False, reset=False, verbose=False)

Calculate the "energy" of a given formula, that is, approximate minus log-posterior of the formula given the data (the approximation coming from the use of the BIC instead of the exactly integrated likelihood)

Returns: Energy of formula (as E, EB, and EP)

Source code in temp_dir/bms/src/autora/theorist/bms/mcmc.py
 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 def get_energy(self, bic=False, reset=False, verbose=False): """ Calculate the "energy" of a given formula, that is, approximate minus log-posterior of the formula given the data (the approximation coming from the use of the BIC instead of the exactly integrated likelihood) Returns: Energy of formula (as E, EB, and EP) """ # Contribution of the data (recalculating BIC if necessary) if bic: EB = self.get_bic(reset=reset, verbose=verbose) / 2.0 else: EB = self.bic / 2.0 # Contribution from the prior EP = 0.0 for op, nop in list(self.nops.items()): try: EP += self.prior_par["Nopi_%s" % op] * nop except KeyError: pass try: EP += self.prior_par["Nopi2_%s" % op] * nop**2 except KeyError: pass # Reset the value, if necessary if reset: self.EB = EB self.EP = EP self.E = EB + EP # Done return EB + EP, EB, EP 

### get_sse(fit=True, verbose=False)

Get the sum of squared errors, fitting the expression represented by the Tree to the existing data, if specified (by default, yes)

Returns: sum of square errors (sse)

Source code in temp_dir/bms/src/autora/theorist/bms/mcmc.py
 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 def get_sse(self, fit=True, verbose=False): """ Get the sum of squared errors, fitting the expression represented by the Tree to the existing data, if specified (by default, yes) Returns: sum of square errors (sse) """ # Return 0 if there is no data if list(self.x.values())[0].empty or list(self.y.values())[0].empty: self.sse = 0 return 0 # Convert the Tree into a SymPy expression ex = sympify(str(self)) # Convert the expression to a function that can be used by # curve_fit, i.e. that takes as arguments (x, a0, a1, ..., an) atomd = dict([(a.name, a) for a in ex.atoms() if a.is_Symbol]) variables = [atomd[v] for v in self.variables if v in list(atomd.keys())] parameters = [atomd[p] for p in self.parameters if p in list(atomd.keys())] dic: dict = dict( { "fac": scipy.special.factorial, "sig": scipy.special.expit, "relu": relu, }, **self.custom_ops, ) try: flam = lambdify( variables + parameters, ex, [ "numpy", dic, ], ) except (SyntaxError, KeyError): self.sse = dict([(ds, np.inf) for ds in self.x]) return self.sse if fit: if len(parameters) == 0: # Nothing to fit for ds in self.x: for p in self.parameters: self.par_values[ds][p] = 1.0 elif str(self) in self.fit_par: # Recover previously fit parameters self.par_values = self.fit_par[str(self)] else: # Do the fit for all datasets self.fit_par[str(self)] = {} for ds in self.x: this_x, this_y = self.x[ds], self.y[ds] xmat = [this_x[v.name] for v in variables] def feval(x, *params): args = [xi for xi in x] + [p for p in params] return flam(*args) try: # Fit the parameters res = curve_fit( feval, xmat, this_y, p0=[self.par_values[ds][p.name] for p in parameters], maxfev=10000, ) # Reassign the values of the parameters self.par_values[ds] = dict( [ (parameters[i].name, res[0][i]) for i in range(len(res[0])) ] ) for p in self.parameters: if p not in self.par_values[ds]: self.par_values[ds][p] = 1.0 # Save this fit self.fit_par[str(self)][ds] = deepcopy(self.par_values[ds]) except RuntimeError: # Save this (unsuccessful) fit and print warning self.fit_par[str(self)][ds] = deepcopy(self.par_values[ds]) if verbose: print( "#Cannot_fit:%s # # # # #" % str(self).replace(" ", ""), file=sys.stderr, ) # Sum of squared errors self.sse = {} for ds in self.x: this_x, this_y = self.x[ds], self.y[ds] xmat = [this_x[v.name] for v in variables] ar = [np.array(xi) for xi in xmat] + [ self.par_values[ds][p.name] for p in parameters ] try: se = np.square(this_y - flam(*ar)) if sum(np.isnan(se)) > 0: raise ValueError else: self.sse[ds] = np.sum(se) except ValueError: if verbose: print("> Cannot calculate SSE for %s: inf" % self, file=sys.stderr) self.sse[ds] = np.inf except TypeError: if verbose: print("Complex-valued parameters are invalid") self.sse[ds] = np.inf # Done return self.sse 

### is_root_prunable()

Check if the root is "prunable"

Returns: boolean of root "prunability"

Source code in temp_dir/bms/src/autora/theorist/bms/mcmc.py
 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 def is_root_prunable(self): """ Check if the root is "prunable" Returns: boolean of root "prunability" """ if self.size == 1: isPrunable = False elif self.size == 2: isPrunable = True else: isPrunable = True for o in self.root.offspring[1:]: if o.offspring != []: isPrunable = False break return isPrunable 

### latex()

translate equation into latex

Returns: canonical latex form of equation

Source code in temp_dir/bms/src/autora/theorist/bms/mcmc.py
 333 334 335 336 337 338 339 def latex(self): """ translate equation into latex Returns: canonical latex form of equation """ return latex(sympify(self.canonical())) 

### mcmc(tracefn='trace.dat', progressfn='progress.dat', write_files=True, reset_files=True, burnin=2000, thin=10, samples=10000, verbose=False, progress=True)

Sample the space of formula trees using MCMC, and write the trace and some progress information to files (unless write_files is False)

Returns: None or expression list

Source code in temp_dir/bms/src/autora/theorist/bms/mcmc.py
 1285 1286 1287 1288 1289 1290 1291 1292 1293 1294 1295 1296 1297 1298 1299 1300 1301 1302 1303 1304 1305 1306 1307 1308 1309 1310 1311 1312 1313 1314 1315 1316 1317 1318 1319 1320 1321 1322 1323 1324 1325 1326 1327 1328 1329 1330 1331 1332 1333 1334 1335 1336 1337 1338 1339 1340 1341 1342 1343 1344 1345 1346 1347 1348 1349 1350 1351 1352 1353 1354 def mcmc( self, tracefn="trace.dat", progressfn="progress.dat", write_files=True, reset_files=True, burnin=2000, thin=10, samples=10000, verbose=False, progress=True, ): """ Sample the space of formula trees using MCMC, and write the trace and some progress information to files (unless write_files is False) Returns: None or expression list """ self.get_energy(reset=True, verbose=verbose) # Burning if progress: sys.stdout.write("# Burning in\t") sys.stdout.write("[%s]" % (" " * 50)) sys.stdout.flush() sys.stdout.write("\b" * (50 + 1)) for i in range(burnin): self.mcmc_step(verbose=verbose) if progress and (i % (burnin / 50) == 0): sys.stdout.write("=") sys.stdout.flush() # Sample if write_files: if reset_files: tracef = open(tracefn, "w") progressf = open(progressfn, "w") else: tracef = open(tracefn, "a") progressf = open(progressfn, "a") if progress: sys.stdout.write("\n# Sampling\t") sys.stdout.write("[%s]" % (" " * 50)) sys.stdout.flush() sys.stdout.write("\b" * (50 + 1)) for s in range(samples): for i in range(thin): self.mcmc_step(verbose=verbose) if progress and (s % (samples / 50) == 0): sys.stdout.write("=") sys.stdout.flush() if write_files: json.dump( [ s, float(self.bic), float(self.E), str(self.get_energy(verbose=verbose)), str(self), self.par_values, ], tracef, ) tracef.write("\n") tracef.flush() progressf.write("%d %lf %lf\n" % (s, self.E, self.bic)) progressf.flush() # Done if progress: sys.stdout.write("\n") return 

### mcmc_step(verbose=False, p_rr=0.05, p_long=0.45)

Make a single MCMC step

Returns: None or expression list

Source code in temp_dir/bms/src/autora/theorist/bms/mcmc.py
 1142 1143 1144 1145 1146 1147 1148 1149 1150 1151 1152 1153 1154 1155 1156 1157 1158 1159 1160 1161 1162 1163 1164 1165 1166 1167 1168 1169 1170 1171 1172 1173 1174 1175 1176 1177 1178 1179 1180 1181 1182 1183 1184 1185 1186 1187 1188 1189 1190 1191 1192 1193 1194 1195 1196 1197 1198 1199 1200 1201 1202 1203 1204 1205 1206 1207 1208 1209 1210 1211 1212 1213 1214 1215 1216 1217 1218 1219 1220 1221 1222 1223 1224 1225 1226 1227 1228 1229 1230 1231 1232 1233 1234 1235 1236 1237 1238 1239 1240 1241 1242 1243 1244 1245 1246 1247 1248 1249 1250 1251 1252 1253 1254 1255 1256 1257 1258 1259 1260 1261 1262 1263 1264 1265 1266 1267 1268 1269 1270 1271 1272 1273 1274 1275 1276 1277 1278 1279 1280 1281 1282 def mcmc_step(self, verbose=False, p_rr=0.05, p_long=0.45): """ Make a single MCMC step Returns: None or expression list """ topDice = random() # Root replacement move if topDice < p_rr: if random() < 0.5: # Try to prune the root dE, dEB, dEP, par_valuesNew = self.dE_rr(rr=None, verbose=verbose) if -dEB / self.BT - dEP / self.PT > 300: paccept = 1 else: paccept = np.exp(-dEB / self.BT - dEP / self.PT) / float( self.num_rr ) dice = random() if dice < paccept: # Accept move self.prune_root(update_gof=False, verbose=verbose) self.par_values = par_valuesNew self.get_bic(reset=True, fit=False, verbose=verbose) self.E += dE self.EB += dEB self.EP += dEP else: # Try to replace the root newrr = choice(self.rr_space) dE, dEB, dEP, par_valuesNew = self.dE_rr(rr=newrr, verbose=verbose) if self.num_rr > 0 and -dEB / self.BT - dEP / self.PT > 0: paccept = 1.0 elif self.num_rr == 0: paccept = 0.0 else: paccept = self.num_rr * np.exp(-dEB / self.BT - dEP / self.PT) dice = random() if dice < paccept: # Accept move self.replace_root(rr=newrr, update_gof=False, verbose=verbose) self.par_values = par_valuesNew self.get_bic(reset=True, fit=False, verbose=verbose) self.E += dE self.EB += dEB self.EP += dEP # Long-range move elif topDice < (p_rr + p_long) and not ( self.fixed_root and len(self.nodes) == 1 ): # Choose a random node in the tree, and a random new operation target = choice(self.nodes) if self.fixed_root: while target is self.root: target = choice(self.nodes) nready = False while not nready: if len(target.offspring) == 0: new = choice(self.variables + self.parameters) nready = True else: new = choice(list(self.ops.keys())) if self.ops[new] == self.ops[target.value]: nready = True dE, dEB, dEP, par_valuesNew = self.dE_lr(target, new, verbose=verbose) try: paccept = np.exp(-dEB / self.BT - dEP / self.PT) except ValueError: _logger.warning("Potentially failing to set paccept properly") if (dEB / self.BT + dEP / self.PT) < 0: paccept = 1.0 # Accept move, if necessary dice = random() if dice < paccept: # update number of operations if target.offspring != []: self.nops[target.value] -= 1 self.nops[new] += 1 # move target.value = new # recalculate distinct parameters self.dist_par = list( set([n.value for n in self.ets[0] if n.value in self.parameters]) ) self.n_dist_par = len(self.dist_par) # update others self.par_values = deepcopy(par_valuesNew) self.get_bic(reset=True, fit=False, verbose=verbose) self.E += dE self.EB += dEB self.EP += dEP # Elementary tree (short-range) move else: target = None while target is None or self.fixed_root and target is self.root: # Choose a feasible move (doable and keeping size<=max_size) while True: oini, ofin = choice(self.move_types) if len(self.ets[oini]) > 0 and ( self.size - oini + ofin <= self.max_size ): break # target and new ETs target = choice(self.ets[oini]) new = choice(self.et_space[ofin]) # omegai and omegaf omegai = len(self.ets[oini]) omegaf = len(self.ets[ofin]) + 1 if ofin == 0: omegaf -= oini if oini == 0 and target.parent in self.ets[ofin]: omegaf -= 1 # size of et_space of each type si = len(self.et_space[oini]) sf = len(self.et_space[ofin]) # Probability of acceptance dE, dEB, dEP, par_valuesNew, nif, nfi = self.dE_et( target, new, verbose=verbose ) try: paccept = ( float(nif) * omegai * sf * np.exp(-dEB / self.BT - dEP / self.PT) ) / (float(nfi) * omegaf * si) except ValueError: if (dEB / self.BT + dEP / self.PT) < -200: paccept = 1.0 # Accept / reject dice = random() if dice < paccept: # Accept move self.et_replace(target, new, verbose=verbose) self.par_values = par_valuesNew self.get_bic(verbose=verbose) self.E += dE self.EB += dEB self.EP += dEP # Done return 

### pr(show_pow=True)

Returns readable representation of tree's root node

Returns: root node representation

Source code in temp_dir/bms/src/autora/theorist/bms/mcmc.py
 291 292 293 294 295 296 297 298 def pr(self, show_pow=True): """ Returns readable representation of tree's root node Returns: root node representation """ return self.root.pr(custom_ops=self.custom_ops, show_pow=show_pow) 

### predict(x)

Calculate the value of the formula at the given data x. The data x must have the same format as the training data and, in particular, it it must specify to which dataset the example data belongs, if multiple datasets where used for training.

Returns: predicted y values

Source code in temp_dir/bms/src/autora/theorist/bms/mcmc.py
 1357 1358 1359 1360 1361 1362 1363 1364 1365 1366 1367 1368 1369 1370 1371 1372 1373 1374 1375 1376 1377 1378 1379 1380 1381 1382 1383 1384 1385 1386 1387 1388 1389 1390 1391 1392 1393 1394 1395 1396 1397 1398 1399 1400 1401 1402 1403 1404 1405 1406 1407 1408 1409 1410 1411 1412 1413 1414 1415 1416 1417 1418 1419 1420 def predict(self, x): """ Calculate the value of the formula at the given data x. The data x must have the same format as the training data and, in particular, it it must specify to which dataset the example data belongs, if multiple datasets where used for training. Returns: predicted y values """ if isinstance(x, np.ndarray): columns = list() for col in range(x.shape[1]): columns.append("X" + str(col)) x = pd.DataFrame(x, columns=columns) if isinstance(x, pd.DataFrame): this_x = {"d0": x} input_type = "df" elif isinstance(x, dict): this_x = x input_type = "dict" else: raise TypeError("x must be either a dict or a pandas.DataFrame") # Convert the Tree into a SymPy expression ex = sympify(str(self)) # Convert the expression to a function atomd = dict([(a.name, a) for a in ex.atoms() if a.is_Symbol]) variables = [atomd[v] for v in self.variables if v in list(atomd.keys())] parameters = [atomd[p] for p in self.parameters if p in list(atomd.keys())] flam = lambdify( variables + parameters, ex, [ "numpy", dict( { "fac": scipy.special.factorial, "sig": scipy.special.expit, "relu": relu, }, **self.custom_ops, ), ], ) # Loop over datasets predictions = {} for ds in this_x: # Prepare variables and parameters xmat = [this_x[ds][v.name] for v in variables] params = [self.par_values[ds][p.name] for p in parameters] args = [xi for xi in xmat] + [p for p in params] # Predict try: prediction = flam(*args) except SyntaxError: # Do it point by point prediction = [np.nan for i in range(len(this_x[ds]))] predictions[ds] = pd.Series(prediction, index=list(this_x[ds].index)) if input_type == "df": return predictions["d0"] else: return predictions 

### prune_root(update_gof=True, verbose=False)

Cut the root and its rightmost leaves (provided they are, indeed, leaves), leaving the leftmost branch as the new tree. Returns the pruned root with the same format as the replacement roots in self.rr_space (or None if pruning was impossible)

Returns: the replacement root

Source code in temp_dir/bms/src/autora/theorist/bms/mcmc.py
 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 def prune_root(self, update_gof=True, verbose=False): """ Cut the root and its rightmost leaves (provided they are, indeed, leaves), leaving the leftmost branch as the new tree. Returns the pruned root with the same format as the replacement roots in self.rr_space (or None if pruning was impossible) Returns: the replacement root """ # Check if the root is "prunable" (and return None if not) if not self.is_root_prunable(): return None # Let's do it! rr = [self.root.value, []] self.nodes.remove(self.root) try: self.ets[len(self.root.offspring)].remove(self.root) except ValueError: pass self.nops[self.root.value] -= 1 self.size -= 1 for o in self.root.offspring[1:]: rr[1].append(o.value) self.nodes.remove(o) self.size -= 1 self.ets[0].remove(o) self.root = self.root.offspring[0] self.root.parent = None # Update list of distinct parameters self.dist_par = list( set([n.value for n in self.ets[0] if n.value in self.parameters]) ) self.n_dist_par = len(self.dist_par) # Update goodness of fit measures, if necessary if update_gof: self.sse = self.get_sse(verbose=verbose) self.bic = self.get_bic(verbose=verbose) self.E = self.get_energy(verbose=verbose) # Done return rr 

### replace_root(rr=None, update_gof=True, verbose=False)

Replace the root with a "root replacement" rr (if provided; otherwise choose one at random from self.rr_space)

Returns: new root (if move was possible) or None (otherwise)

Source code in temp_dir/bms/src/autora/theorist/bms/mcmc.py
 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 def replace_root(self, rr=None, update_gof=True, verbose=False): """ Replace the root with a "root replacement" rr (if provided; otherwise choose one at random from self.rr_space) Returns: new root (if move was possible) or None (otherwise) """ # If no RR is provided, randomly choose one if rr is None: rr = choice(self.rr_space) # Return None if the replacement is too big if (self.size + self.ops[rr[0]]) > self.max_size: return None # Create the new root and replace existing root newRoot = Node(rr[0], offspring=[], parent=None) newRoot.order = 1 + len(rr[1]) if newRoot.order != self.ops[rr[0]]: raise newRoot.offspring.append(self.root) self.root.parent = newRoot self.root = newRoot self.nops[self.root.value] += 1 self.nodes.append(self.root) self.size += 1 oldRoot = self.root.offspring[0] for leaf in rr[1]: self.root.offspring.append(Node(leaf, offspring=[], parent=self.root)) self.nodes.append(self.root.offspring[-1]) self.ets[0].append(self.root.offspring[-1]) self.size += 1 # Add new root to elementary trees if necessary (that is, iff # the old root was a leaf) if oldRoot.offspring is []: self.ets[self.root.order].append(self.root) # Update list of distinct parameters self.dist_par = list( set([n.value for n in self.ets[0] if n.value in self.parameters]) ) self.n_dist_par = len(self.dist_par) # Update goodness of fit measures, if necessary if update_gof: self.sse = self.get_sse(verbose=verbose) self.bic = self.get_bic(verbose=verbose) self.E = self.get_energy(verbose=verbose) return self.root 

### trace_predict(x, burnin=1000, thin=2000, samples=1000, tracefn='trace.dat', progressfn='progress.dat', write_files=False, reset_files=True, verbose=False, progress=True)

Sample the space of formula trees using MCMC, and predict y(x) for each of the sampled formula trees

Returns: predicted y values for each of the sampled formula trees

Source code in temp_dir/bms/src/autora/theorist/bms/mcmc.py
 1423 1424 1425 1426 1427 1428 1429 1430 1431 1432 1433 1434 1435 1436 1437 1438 1439 1440 1441 1442 1443 1444 1445 1446 1447 1448 1449 1450 1451 1452 1453 1454 1455 1456 1457 1458 1459 1460 1461 1462 1463 1464 1465 1466 1467 1468 1469 1470 1471 1472 1473 1474 1475 1476 1477 1478 1479 1480 1481 1482 1483 1484 1485 1486 1487 1488 1489 1490 1491 1492 1493 1494 1495 1496 def trace_predict( self, x, burnin=1000, thin=2000, samples=1000, tracefn="trace.dat", progressfn="progress.dat", write_files=False, reset_files=True, verbose=False, progress=True, ): """ Sample the space of formula trees using MCMC, and predict y(x) for each of the sampled formula trees Returns: predicted y values for each of the sampled formula trees """ ypred = {} # Burning if progress: sys.stdout.write("# Burning in\t") sys.stdout.write("[%s]" % (" " * 50)) sys.stdout.flush() sys.stdout.write("\b" * (50 + 1)) for i in range(burnin): self.mcmc_step(verbose=verbose) if progress and (i % (burnin / 50) == 0): sys.stdout.write("=") sys.stdout.flush() # Sample if write_files: if reset_files: tracef = open(tracefn, "w") progressf = open(progressfn, "w") else: tracef = open(tracefn, "a") progressf = open(progressfn, "a") if progress: sys.stdout.write("\n# Sampling\t") sys.stdout.write("[%s]" % (" " * 50)) sys.stdout.flush() sys.stdout.write("\b" * (50 + 1)) for s in range(samples): for kk in range(thin): self.mcmc_step(verbose=verbose) # Make prediction ypred[s] = self.predict(x) # Output if progress and (s % (samples / 50) == 0): sys.stdout.write("=") sys.stdout.flush() if write_files: json.dump( [ s, float(self.bic), float(self.E), float(self.get_energy(verbose=verbose)), str(self), self.par_values, ], tracef, ) tracef.write("\n") tracef.flush() progressf.write("%d %lf %lf\n" % (s, self.E, self.bic)) progressf.flush() # Done if progress: sys.stdout.write("\n") return pd.DataFrame.from_dict(ypred) 

### update_representative(verbose=False)

Check if we've seen this formula before, either in its current form or in another form.

*If we haven't seen it, save it and return 1.

*If we have seen it and this IS the representative, just return 0.

*If we have seen it and the representative has smaller energy, just return -1.

*If we have seen it and the representative has higher energy, update the representatitve and return -2.

Integer value (0, 1, or -1) corresponding to:

Name Type Description
0

we have seen this canonical form before

1

we haven't seen this canonical form before

-1: we have seen this equation's canonical form before but it isn't in that form yet

Source code in temp_dir/bms/src/autora/theorist/bms/mcmc.py
 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 def update_representative(self, verbose=False): """Check if we've seen this formula before, either in its current form or in another form. *If we haven't seen it, save it and return 1. *If we have seen it and this IS the representative, just return 0. *If we have seen it and the representative has smaller energy, just return -1. *If we have seen it and the representative has higher energy, update the representatitve and return -2. Returns: Integer value (0, 1, or -1) corresponding to: 0: we have seen this canonical form before 1: we haven't seen this canonical form before -1: we have seen this equation's canonical form before but it isn't in that form yet """ # Check for canonical representative canonical = self.canonical(verbose=verbose) try: # We've seen this canonical before! rep, rep_energy, rep_par_values = self.representative[canonical] except TypeError: return -1 # Complex-valued parameters are invalid except KeyError: # Never seen this canonical formula before: # save it and return 1 self.get_bic(reset=True, fit=True, verbose=verbose) new_energy = self.get_energy(bic=False, verbose=verbose) self.representative[canonical] = ( str(self), new_energy, deepcopy(self.par_values), ) return 1 # If we've seen this canonical before, check if the # representative needs to be updated if rep == str(self): # This IS the representative: return 0 return 0 else: return -1