import copy
import itertools
import logging
import sympy as sp
import sympy.stats as spstats
import dowhy.utils.cli_helpers as cli
from dowhy.utils.api import parse_state
[docs]class CausalIdentifier:
"""Class that implements different identification methods.
Currently supports backdoor and instrumental variable identification methods. The identification is based on the causal graph provided.
Other specific ways of identification, such as the ID* algorithm, minimal adjustment criteria, etc. will be added in the future.
If you'd like to contribute, please raise an issue or a pull request on Github.
"""
NONPARAMETRIC_ATE="nonparametric-ate"
NONPARAMETRIC_NDE="nonparametric-nde"
NONPARAMETRIC_NIE="nonparametric-nie"
MAX_BACKDOOR_ITERATIONS = 100000
VALID_METHOD_NAMES = {"default", "exhaustive-search"}
def __init__(self, graph, estimand_type,
method_name = "default",
proceed_when_unidentifiable=False):
self._graph = graph
self.estimand_type = estimand_type
self.treatment_name = graph.treatment_name
self.outcome_name = graph.outcome_name
self.method_name = method_name
self._proceed_when_unidentifiable = proceed_when_unidentifiable
self.logger = logging.getLogger(__name__)
[docs] def identify_effect(self):
"""Main method that returns an identified estimand (if one exists).
If estimand_type is non-parametric ATE, then uses backdoor, instrumental variable and frontdoor identification methods, to check if an identified estimand exists, based on the causal graph.
:param self: instance of the CausalEstimator class (or its subclass)
:returns: target estimand, an instance of the IdentifiedEstimand class
"""
if self.estimand_type == CausalIdentifier.NONPARAMETRIC_ATE:
return self.identify_ate_effect()
elif self.estimand_type == CausalIdentifier.NONPARAMETRIC_NDE:
return self.identify_nde_effect()
elif self.estimand_type == CausalIdentifier.NONPARAMETRIC_NIE:
return self.identify_nie_effect()
else:
raise ValueError("Estimand type is not supported. Use either {0}, {1}, or {2}.".format(
CausalIdentifier.NONPARAMETRIC_ATE,
CausalIdentifier.NONPARAMETRIC_NDE,
CausalIdentifier.NONPARAMETRIC_NIE))
[docs] def identify_ate_effect(self):
estimands_dict = {}
mediation_first_stage_confounders = None
mediation_second_stage_confounders = None
### 1. BACKDOOR IDENTIFICATION
# First, checking if there are any valid backdoor adjustment sets
backdoor_sets = self.identify_backdoor(self.treatment_name, self.outcome_name)
estimands_dict, backdoor_variables_dict = self.build_backdoor_estimands_dict(
self.treatment_name,
self.outcome_name,
backdoor_sets,
estimands_dict)
# Setting default "backdoor" identification adjustment set
default_backdoor_id = self.get_default_backdoor_set_id(backdoor_variables_dict)
estimands_dict["backdoor"] = estimands_dict.get(str(default_backdoor_id), None)
backdoor_variables_dict["backdoor"] = backdoor_variables_dict.get(str(default_backdoor_id), None)
### 2. INSTRUMENTAL VARIABLE IDENTIFICATION
# Now checking if there is also a valid iv estimand
instrument_names = self._graph.get_instruments(self.treatment_name,
self.outcome_name)
self.logger.info("Instrumental variables for treatment and outcome:" +
str(instrument_names))
if len(instrument_names) > 0:
iv_estimand_expr = self.construct_iv_estimand(
self.estimand_type,
self._graph.treatment_name,
self._graph.outcome_name,
instrument_names
)
self.logger.debug("Identified expression = " + str(iv_estimand_expr))
estimands_dict["iv"] = iv_estimand_expr
else:
estimands_dict["iv"] = None
### 3. FRONTDOOR IDENTIFICATION
# Now checking if there is a valid frontdoor variable
frontdoor_variables_names = self.identify_frontdoor()
self.logger.info("Frontdoor variables for treatment and outcome:" +
str(frontdoor_variables_names))
if len(frontdoor_variables_names) >0:
frontdoor_estimand_expr = self.construct_frontdoor_estimand(
self.estimand_type,
self._graph.treatment_name,
self._graph.outcome_name,
frontdoor_variables_names
)
self.logger.debug("Identified expression = " + str(frontdoor_estimand_expr))
estimands_dict["frontdoor"] = frontdoor_estimand_expr
mediation_first_stage_confounders = self.identify_mediation_first_stage_confounders(self.treatment_name, frontdoor_variables_names)
mediation_second_stage_confounders = self.identify_mediation_second_stage_confounders(frontdoor_variables_names, self.outcome_name)
else:
estimands_dict["frontdoor"] = None
# Finally returning the estimand object
estimand = IdentifiedEstimand(
self,
treatment_variable=self._graph.treatment_name,
outcome_variable=self._graph.outcome_name,
estimand_type=self.estimand_type,
estimands=estimands_dict,
backdoor_variables=backdoor_variables_dict,
instrumental_variables=instrument_names,
frontdoor_variables=frontdoor_variables_names,
mediation_first_stage_confounders=mediation_first_stage_confounders,
mediation_second_stage_confounders=mediation_second_stage_confounders,
default_backdoor_id = default_backdoor_id
)
return estimand
[docs] def identify_nie_effect(self):
estimands_dict = {}
### 1. FIRST DOING BACKDOOR IDENTIFICATION
# First, checking if there are any valid backdoor adjustment sets
backdoor_sets = self.identify_backdoor(self.treatment_name, self.outcome_name)
estimands_dict, backdoor_variables_dict = self.build_backdoor_estimands_dict(
self.treatment_name,
self.outcome_name,
backdoor_sets,
estimands_dict)
# Setting default "backdoor" identification adjustment set
default_backdoor_id = self.get_default_backdoor_set_id(backdoor_variables_dict)
backdoor_variables_dict["backdoor"] = backdoor_variables_dict.get(str(default_backdoor_id), None)
### 2. SECOND, CHECKING FOR MEDIATORS
# Now checking if there are valid mediator variables
estimands_dict = {} # Need to reinitialize this dictionary to avoid including the backdoor sets
mediation_first_stage_confounders = None
mediation_second_stage_confounders = None
mediators_names = self.identify_mediation()
self.logger.info("Mediators for treatment and outcome:" +
str(mediators_names))
if len(mediators_names) >0:
mediation_estimand_expr = self.construct_mediation_estimand(
self.estimand_type,
self._graph.treatment_name,
self._graph.outcome_name,
mediators_names
)
self.logger.debug("Identified expression = " + str(mediation_estimand_expr))
estimands_dict["mediation"] = mediation_estimand_expr
mediation_first_stage_confounders = self.identify_mediation_first_stage_confounders(self.treatment_name, mediators_names)
mediation_second_stage_confounders = self.identify_mediation_second_stage_confounders(mediators_names, self.outcome_name)
else:
estimands_dict["mediation"] = None
# Finally returning the estimand object
estimand = IdentifiedEstimand(
self,
treatment_variable=self._graph.treatment_name,
outcome_variable=self._graph.outcome_name,
estimand_type=self.estimand_type,
estimands=estimands_dict,
backdoor_variables=backdoor_variables_dict,
instrumental_variables=None,
frontdoor_variables=None,
mediator_variables=mediators_names,
mediation_first_stage_confounders=mediation_first_stage_confounders,
mediation_second_stage_confounders=mediation_second_stage_confounders,
default_backdoor_id = None
)
return estimand
[docs] def identify_nde_effect(self):
estimands_dict = {}
### 1. FIRST DOING BACKDOOR IDENTIFICATION
# First, checking if there are any valid backdoor adjustment sets
backdoor_sets = self.identify_backdoor(self.treatment_name, self.outcome_name)
estimands_dict, backdoor_variables_dict = self.build_backdoor_estimands_dict(
self.treatment_name,
self.outcome_name,
backdoor_sets,
estimands_dict)
# Setting default "backdoor" identification adjustment set
default_backdoor_id = self.get_default_backdoor_set_id(backdoor_variables_dict)
backdoor_variables_dict["backdoor"] = backdoor_variables_dict.get(str(default_backdoor_id), None)
### 2. SECOND, CHECKING FOR MEDIATORS
# Now checking if there are valid mediator variables
estimands_dict = {}
mediation_first_stage_confounders = None
mediation_second_stage_confounders = None
mediators_names = self.identify_mediation()
self.logger.info("Mediators for treatment and outcome:" +
str(mediators_names))
if len(mediators_names) >0:
mediation_estimand_expr = self.construct_mediation_estimand(
self.estimand_type,
self._graph.treatment_name,
self._graph.outcome_name,
mediators_names
)
self.logger.debug("Identified expression = " + str(mediation_estimand_expr))
estimands_dict["mediation"] = mediation_estimand_expr
mediation_first_stage_confounders = self.identify_mediation_first_stage_confounders(self.treatment_name, mediators_names)
mediation_second_stage_confounders = self.identify_mediation_second_stage_confounders(mediators_names, self.outcome_name)
else:
estimands_dict["mediation"] = None
# Finally returning the estimand object
estimand = IdentifiedEstimand(
self,
treatment_variable=self._graph.treatment_name,
outcome_variable=self._graph.outcome_name,
estimand_type=self.estimand_type,
estimands=estimands_dict,
backdoor_variables=backdoor_variables_dict,
instrumental_variables=None,
frontdoor_variables=None,
mediator_variables=mediators_names,
mediation_first_stage_confounders=mediation_first_stage_confounders,
mediation_second_stage_confounders=mediation_second_stage_confounders,
default_backdoor_id = None
)
return estimand
[docs] def identify_backdoor(self, treatment_name, outcome_name):
backdoor_sets = []
backdoor_paths = self._graph.get_backdoor_paths(treatment_name, outcome_name)
# First, checking if empty set is a valid backdoor set
empty_set = set()
check = self._graph.check_valid_backdoor_set(treatment_name, outcome_name, empty_set,
backdoor_paths=backdoor_paths)
if check["is_dseparated"]:
backdoor_sets.append({
'backdoor_set':empty_set,
'num_paths_blocked_by_observed_nodes': check["num_paths_blocked_by_observed_nodes"]})
# Second, checking for all other sets of variables
eligible_variables = self._graph.get_all_nodes() \
- set(treatment_name) \
- set(outcome_name) \
- set(self._graph.get_instruments(treatment_name, outcome_name))
eligible_variables -= self._graph.get_descendants(treatment_name)
num_iterations = 0
found_valid_adjustment_set = False
if self.method_name in CausalIdentifier.VALID_METHOD_NAMES:
for size_candidate_set in range(len(eligible_variables), 0, -1):
for candidate_set in itertools.combinations(eligible_variables, size_candidate_set):
check = self._graph.check_valid_backdoor_set(treatment_name,
outcome_name, candidate_set, backdoor_paths=backdoor_paths)
self.logger.debug("Candidate backdoor set: {0}, is_dseparated: {1}, No. of paths blocked by observed_nodes: {2}".format(candidate_set, check["is_dseparated"], check["num_paths_blocked_by_observed_nodes"]))
if check["is_dseparated"]:
backdoor_sets.append({
'backdoor_set': candidate_set,
'num_paths_blocked_by_observed_nodes': check["num_paths_blocked_by_observed_nodes"]})
if self._graph.all_observed(candidate_set):
found_valid_adjustment_set = True
num_iterations += 1
if self.method_name == "default" and num_iterations > CausalIdentifier.MAX_BACKDOOR_ITERATIONS:
break
if self.method_name == "default" and found_valid_adjustment_set:
break
else:
raise ValueError(f"Identifier method {self.method_name} not supported. Try one of the following: {CausalIdentifier.VALID_METHOD_NAMES}")
#causes_t = self._graph.get_causes(self.treatment_name)
#causes_y = self._graph.get_causes(self.outcome_name, remove_edges={'sources':self.treatment_name, 'targets':self.outcome_name})
#common_causes = list(causes_t.intersection(causes_y))
#self.logger.info("Common causes of treatment and outcome:" + str(common_causes))
observed_backdoor_sets = [ bset for bset in backdoor_sets if self._graph.all_observed(bset["backdoor_set"])]
if len(observed_backdoor_sets)==0:
return backdoor_sets
else:
return observed_backdoor_sets
def get_default_backdoor_set_id(self, backdoor_sets_dict):
# Adding a None estimand if no backdoor set found
if len(backdoor_sets_dict) == 0:
return None
max_set_length = -1
default_key = None
# Default set is the one with the most number of adjustment variables (optimizing for minimum (unknown) bias not for efficiency)
for key, bdoor_set in backdoor_sets_dict.items():
if len(bdoor_set) > max_set_length:
max_set_length = len(bdoor_set)
default_key = key
return default_key
[docs] def build_backdoor_estimands_dict(self, treatment_name, outcome_name,
backdoor_sets, estimands_dict, proceed_when_unidentifiable=None):
"""Build the final dict for backdoor sets by filtering unobserved variables if needed.
"""
backdoor_variables_dict = {}
if proceed_when_unidentifiable is None:
proceed_when_unidentifiable = self._proceed_when_unidentifiable
is_identified = [ self._graph.all_observed(bset["backdoor_set"]) for bset in backdoor_sets ]
if all(is_identified):
self.logger.info("All common causes are observed. Causal effect can be identified.")
backdoor_sets_arr = [list(
bset["backdoor_set"])
for bset in backdoor_sets]
else: # there is unobserved confounding
self.logger.warning("If this is observed data (not from a randomized experiment), there might always be missing confounders. Causal effect cannot be identified perfectly.")
response = False # user response
if proceed_when_unidentifiable:
self.logger.info(
"Continuing by ignoring these unobserved confounders because proceed_when_unidentifiable flag is True."
)
else:
response= cli.query_yes_no(
"WARN: Do you want to continue by ignoring any unobserved confounders? (use proceed_when_unidentifiable=True to disable this prompt)",
default=None
)
if response is False:
self.logger.warn("Identification failed due to unobserved variables.")
backdoor_sets_arr = []
if proceed_when_unidentifiable or response is True:
max_paths_blocked = max( bset['num_paths_blocked_by_observed_nodes'] for bset in backdoor_sets)
backdoor_sets_arr = [list(
self._graph.filter_unobserved_variables(bset["backdoor_set"]))
for bset in backdoor_sets
if bset["num_paths_blocked_by_observed_nodes"]==max_paths_blocked]
for i in range(len(backdoor_sets_arr)):
backdoor_estimand_expr = self.construct_backdoor_estimand(
self.estimand_type, treatment_name,
outcome_name, backdoor_sets_arr[i])
self.logger.debug("Identified expression = " + str(backdoor_estimand_expr))
estimands_dict["backdoor"+str(i+1)] = backdoor_estimand_expr
backdoor_variables_dict["backdoor"+str(i+1)] = backdoor_sets_arr[i]
return estimands_dict, backdoor_variables_dict
[docs] def identify_frontdoor(self):
""" Find a valid frontdoor variable if it exists.
Currently only supports a single variable frontdoor set.
"""
frontdoor_var = None
frontdoor_paths = self._graph.get_all_directed_paths(self.treatment_name, self.outcome_name)
eligible_variables = self._graph.get_descendants(self.treatment_name) \
- set(self.outcome_name)
# For simplicity, assuming a one-variable frontdoor set
for candidate_var in eligible_variables:
is_valid_frontdoor = self._graph.check_valid_frontdoor_set(self.treatment_name,
self.outcome_name, parse_state(candidate_var), frontdoor_paths=frontdoor_paths)
self.logger.debug("Candidate frontdoor set: {0}, is_dseparated: {1}".format(candidate_var, is_valid_frontdoor))
if is_valid_frontdoor:
frontdoor_var = candidate_var
break
return parse_state(frontdoor_var)
[docs] def get_default_backdoor_set_id(self, backdoor_sets_dict):
# Adding a None estimand if no backdoor set found
if len(backdoor_sets_dict) == 0:
return None
max_set_length = -1
default_key = None
# Default set is the one with the most number of adjustment variables (optimizing for minimum (unknown) bias not for efficiency)
for key, bdoor_set in backdoor_sets_dict.items():
if len(bdoor_set) > max_set_length:
max_set_length = len(bdoor_set)
default_key = key
return default_key
[docs] def construct_backdoor_estimand(self, estimand_type, treatment_name,
outcome_name, common_causes):
# TODO: outputs string for now, but ideally should do symbolic
# expressions Mon 19 Feb 2018 04:54:17 PM DST
# TODO Better support for multivariate treatments
expr = None
outcome_name = outcome_name[0]
num_expr_str = outcome_name
if len(common_causes)>0:
num_expr_str += "|" + ",".join(common_causes)
expr = "d(" + num_expr_str + ")/d" + ",".join(treatment_name)
sym_mu = sp.Symbol("mu")
sym_sigma = sp.Symbol("sigma", positive=True)
sym_outcome = spstats.Normal(num_expr_str, sym_mu, sym_sigma)
sym_treatment_symbols = [sp.Symbol(t) for t in treatment_name]
sym_treatment = sp.Array(sym_treatment_symbols)
sym_conditional_outcome = spstats.Expectation(sym_outcome)
sym_effect = sp.Derivative(sym_conditional_outcome, sym_treatment)
sym_assumptions = {
'Unconfoundedness': (
u"If U\N{RIGHTWARDS ARROW}{{{0}}} and U\N{RIGHTWARDS ARROW}{1}"
" then P({1}|{0},{2},U) = P({1}|{0},{2})"
).format(",".join(treatment_name), outcome_name, ",".join(common_causes))
}
estimand = {
'estimand': sym_effect,
'assumptions': sym_assumptions
}
return estimand
[docs] def construct_iv_estimand(self, estimand_type, treatment_name,
outcome_name, instrument_names):
# TODO: support multivariate treatments better.
expr = None
outcome_name = outcome_name[0]
sym_outcome = spstats.Normal(outcome_name, 0, 1)
sym_treatment_symbols = [spstats.Normal(t, 0, 1) for t in treatment_name]
sym_treatment = sp.Array(sym_treatment_symbols)
sym_instrument_symbols = [sp.Symbol(inst) for inst in instrument_names]
sym_instrument = sp.Array(sym_instrument_symbols) # ",".join(instrument_names))
sym_outcome_derivative = sp.Derivative(sym_outcome, sym_instrument)
sym_treatment_derivative = sp.Derivative(sym_treatment, sym_instrument)
sym_effect = spstats.Expectation(sym_outcome_derivative / sym_treatment_derivative)
sym_assumptions = {
"As-if-random": (
"If U\N{RIGHTWARDS ARROW}\N{RIGHTWARDS ARROW}{0} then "
"\N{NOT SIGN}(U \N{RIGHTWARDS ARROW}\N{RIGHTWARDS ARROW}{{{1}}})"
).format(outcome_name, ",".join(instrument_names)),
"Exclusion": (
u"If we remove {{{0}}}\N{RIGHTWARDS ARROW}{{{1}}}, then "
u"\N{NOT SIGN}({{{0}}}\N{RIGHTWARDS ARROW}{2})"
).format(",".join(instrument_names), ",".join(treatment_name),
outcome_name)
}
estimand = {
'estimand': sym_effect,
'assumptions': sym_assumptions
}
return estimand
[docs] def construct_frontdoor_estimand(self, estimand_type, treatment_name,
outcome_name, frontdoor_variables_names):
# TODO: support multivariate treatments better.
expr = None
outcome_name = outcome_name[0]
sym_outcome = spstats.Normal(outcome_name, 0, 1)
sym_treatment_symbols = [spstats.Normal(t, 0, 1) for t in treatment_name]
sym_treatment = sp.Array(sym_treatment_symbols)
sym_frontdoor_symbols = [sp.Symbol(inst) for inst in frontdoor_variables_names]
sym_frontdoor = sp.Array(sym_frontdoor_symbols) # ",".join(instrument_names))
sym_outcome_derivative = sp.Derivative(sym_outcome, sym_frontdoor)
sym_treatment_derivative = sp.Derivative(sym_frontdoor, sym_treatment)
sym_effect = spstats.Expectation(sym_treatment_derivative * sym_outcome_derivative)
sym_assumptions = {
"Full-mediation": (
"{2} intercepts (blocks) all directed paths from {0} to {1}."
).format(",".join(treatment_name), ",".join(outcome_name), ",".join(frontdoor_variables_names)),
"First-stage-unconfoundedness": (
u"If U\N{RIGHTWARDS ARROW}{{{0}}} and U\N{RIGHTWARDS ARROW}{{{1}}}"
" then P({1}|{0},U) = P({1}|{0})"
).format(",".join(treatment_name), ",".join(frontdoor_variables_names)),
"Second-stage-unconfoundedness": (
u"If U\N{RIGHTWARDS ARROW}{{{2}}} and U\N{RIGHTWARDS ARROW}{1}"
" then P({1}|{2}, {0}, U) = P({1}|{2}, {0})"
).format(",".join(treatment_name), outcome_name, ",".join(frontdoor_variables_names))
}
estimand = {
'estimand': sym_effect,
'assumptions': sym_assumptions
}
return estimand
[docs]class IdentifiedEstimand:
"""Class for storing a causal estimand, typically as a result of the identification step.
"""
def __init__(self, identifier, treatment_variable, outcome_variable,
estimand_type=None, estimands=None,
backdoor_variables=None, instrumental_variables=None,
frontdoor_variables=None,
mediator_variables=None,
mediation_first_stage_confounders=None,
mediation_second_stage_confounders=None,
default_backdoor_id=None, identifier_method=None):
self.identifier = identifier
self.treatment_variable = parse_state(treatment_variable)
self.outcome_variable = parse_state(outcome_variable)
self.backdoor_variables = backdoor_variables
self.instrumental_variables = parse_state(instrumental_variables)
self.frontdoor_variables = parse_state(frontdoor_variables)
self.mediator_variables = parse_state(mediator_variables)
self.mediation_first_stage_confounders=mediation_first_stage_confounders
self.mediation_second_stage_confounders=mediation_second_stage_confounders
self.estimand_type = estimand_type
self.estimands = estimands
self.default_backdoor_id = default_backdoor_id
self.identifier_method = identifier_method
[docs] def set_identifier_method(self, identifier_name):
self.identifier_method = identifier_name
[docs] def get_backdoor_variables(self, key=None):
""" Return a list containing the backdoor variables.
If the calling estimator method is a backdoor method, return the
backdoor variables corresponding to its target estimand.
Otherwise, return the backdoor variables for the default backdoor estimand.
"""
if key is None:
if self.identifier_method and self.identifier_method.startswith("backdoor"):
return self.backdoor_variables[self.identifier_method]
elif self.backdoor_variables is not None:
return self.backdoor_variables[self.default_backdoor_id]
else:
return []
else:
return self.backdoor_variables[key]
[docs] def set_backdoor_variables(self, bdoor_variables_arr, key=None):
if key is None:
key = self.identifier_method
self.backdoor_variables[key] = bdoor_variables_arr
[docs] def get_frontdoor_variables(self):
"""Return a list containing the frontdoor variables (if present)
"""
return self.frontdoor_variables
[docs] def get_instrumental_variables(self):
"""Return a list containing the instrumental variables (if present)
"""
return self.instrumental_variables
def __deepcopy__(self, memo):
return IdentifiedEstimand(
self.identifier, # not deep copied
copy.deepcopy(self.treatment_variable),
copy.deepcopy(self.outcome_variable),
estimand_type=copy.deepcopy(self.estimand_type),
estimands=copy.deepcopy(self.estimands),
backdoor_variables=copy.deepcopy(self.backdoor_variables),
instrumental_variables=copy.deepcopy(self.instrumental_variables),
frontdoor_variables=copy.deepcopy(self.frontdoor_variables),
mediator_variables=copy.deepcopy(self.mediator_variables),
default_backdoor_id=copy.deepcopy(self.default_backdoor_id),
identifier_method=copy.deepcopy(self.identifier_method)
)
def __str__(self, only_target_estimand=False, show_all_backdoor_sets=False):
s = "Estimand type: {0}\n".format(self.estimand_type)
i = 1
has_valid_backdoor = sum("backdoor" in key for key in self.estimands.keys())
for k, v in self.estimands.items():
if show_all_backdoor_sets:
# Do not show backdoor key unless it is the only backdoor set.
if k == "backdoor" and has_valid_backdoor > 1:
continue
else:
# Just show the default backdoor set
if k.startswith("backdoor") and k != "backdoor":
continue
if only_target_estimand and k != self.identifier_method:
continue
s += "\n### Estimand : {0}\n".format(i)
s += "Estimand name: {0}".format(k)
if k == self.default_backdoor_id:
s += " (Default)"
s += "\n"
if v is None:
s += "No such variable found!\n"
else:
sp_expr_str = sp.pretty(v["estimand"], use_unicode=True)
s += "Estimand expression:\n{0}\n".format(sp_expr_str)
j = 1
for ass_name, ass_str in v["assumptions"].items():
s += "Estimand assumption {0}, {1}: {2}\n".format(j, ass_name, ass_str)
j += 1
i += 1
return s