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#!/usr/bin/env python3
import itertools
import logging
import numpy as np
import os
from collections import OrderedDict
from copy import deepcopy
from multiprocessing import Pool
import dfatool.functions as df
from .utils import remove_index_from_tuple, is_numeric
from .utils import filter_aggregate_by_param, partition_by_param
logger = logging.getLogger(__name__)
def distinct_param_values(param_tuples):
"""
Return the distinct values of each parameter in param_tuples.
E.g. if param_tuples contains the distinct entries (1, 1), (1, 2), (1, 3), (0, 3),
this function returns [[1, 0], [1, 2, 3]].
Note that this function deliberately also consider None
(uninitialized parameter with unknown value) as a distinct value. Benchmarks
and drivers must ensure that a parameter is only None when its value is
not important yet, e.g. a packet length parameter must only be None when
write() or similar has not been called yet. Other parameters should always
be initialized when leaving UNINITIALIZED.
"""
distinct_values = [OrderedDict() for i in range(len(param_tuples[0]))]
for param_tuple in param_tuples:
for i in range(len(param_tuple)):
distinct_values[i][param_tuple[i]] = True
# Convert sets to lists
distinct_values = list(map(lambda x: list(x.keys()), distinct_values))
return distinct_values
def _depends_on_param(corr_param, std_param, std_lut):
# if self.use_corrcoef:
if False:
return corr_param > 0.1
elif std_param == 0:
# In general, std_param_lut < std_by_param. So, if std_by_param == 0, std_param_lut == 0 follows.
# This means that the variation of param does not affect the model quality -> no influence
return False
return std_lut / std_param < 0.5
def _mean_std_by_param(n_by_param, all_param_values, param_index):
"""
Calculate the mean standard deviation for a static model where all parameters but `param_index` are constant.
:param n_by_param: measurements of a specific model attribute partitioned by parameter values.
Example: `{(0, 2): [2], (0, 4): [4], (0, 6): [6]}`
:param all_param_values: distinct values of each parameter.
E.g. for two parameters, the first being None, FOO, or BAR, and the second being 1, 2, 3, or 4, the argument is
`[[None, 'FOO', 'BAR'], [1, 2, 3, 4]]`.
:param param_index: index of variable parameter
:returns: mean stddev
*mean stddev* is the mean standard deviation of all measurements where parameter `param_index` is dynamic and all other parameters are fixed.
E.g., if parameters are a, b, c ∈ {1,2,3} and 'index' corresponds to b, then
this function returns the mean of the standard deviations of (a=1, b=*, c=1),
(a=1, b=*, c=2), and so on.
"""
param_values = list(remove_index_from_tuple(all_param_values, param_index))
partitions = list()
for param_value in itertools.product(*param_values):
param_partition = list()
std_list = list()
for k, v in n_by_param.items():
if (*k[:param_index], *k[param_index + 1 :]) == param_value:
param_partition.extend(v)
if len(param_partition) > 1:
partitions.append(param_partition)
if len(partitions) == 0:
return 0.0
return np.mean([np.std(partition) for partition in partitions])
def _corr_by_param(attribute_data, param_values, param_index):
"""
Return correlation coefficient (`np.corrcoef`) of `attribute_data` <-> `param_values[param_index]`
A correlation coefficient close to 1 indicates that the attribute likely depends on the value of the parameter denoted by `param_index`, if it is nearly 0, it likely does not depend on it.
If any value of `param_index` is not numeric (i.e., can not be parsed as float), this function returns 0.
:param attribute_data: list or 1-D numpy array of measurements
:param param_values: list of parameter values
:param param_index: index of parameter in `by_name[*]['param']`
"""
if _all_params_are_numeric(param_values, param_index):
param_values = np.array(list((map(lambda x: x[param_index], param_values))))
try:
return np.corrcoef(attribute_data, param_values)[0, 1]
except FloatingPointError:
# Typically happens when all parameter values are identical.
# Building a correlation coefficient is pointless in this case
# -> assume no correlation
return 0.0
except ValueError:
logger.error(
"ValueError in _corr_by_param(param_index={})".format(param_index)
)
logger.error(
"while executing np.corrcoef({}, {}))".format(
attribute_data, param_values
)
)
raise
else:
return 0.0
def _compute_param_statistics(
data,
param_names,
param_tuples,
arg_count=None,
use_corrcoef=False,
codependent_params=list(),
):
"""
Compute standard deviation and correlation coefficient on parameterized data partitions.
It is strongly recommended to vary all parameter values evenly.
For instance, given two parameters, providing only the combinations
(1, 1), (5, 1), (7, 1,) (10, 1), (1, 2), (1, 6) will lead to bogus results.
It is better to provide (1, 1), (5, 1), (1, 2), (5, 2), ... (i.e. a cross product of all individual parameter values)
arguments:
data -- measurement data (ground truth). Must be a list or 1-D numpy array.
param_names -- list of parameter names
param_tuples -- list of parameter values corresponding to the order in param_names
arg_count -- dict providing the number of functions args ("local parameters") for each function.
use_corrcoef -- use correlation coefficient instead of stddev heuristic for parameter detection
:returns: a dict with the following content:
std_static -- static parameter-unaware model error: stddev of data
std_param_lut -- static parameter-aware model error: mean stddev of data[*]
std_by_param -- static parameter-aware model error ignoring a single parameter.
dictionary with one key per parameter. The value is the mean stddev
of measurements where all other parameters are fixed and the parameter
in question is variable. E.g. std_by_param['X'] is the mean stddev of
n_by_param[(X=*, Y=..., Z=...)].
std_by_arg -- same, but ignoring a single function argument
Only set if arg_count is non-zero, empty list otherwise.
corr_by_param -- correlation coefficient
corr_by_arg -- same, but ignoring a single function argument
Only set if arg_count is non-zero, empty list otherwise.
depends_on_param -- dict(parameter_name -> Bool). True if /attribute/ behaviour probably depends on /parameter_name/
depends_on_arg -- list(bool). Same, but for function arguments, if any.
"""
ret = dict()
ret["by_param"] = by_param = partition_by_param(data, param_tuples)
ret["use_corrcoef"] = use_corrcoef
ret["_parameter_names"] = param_names
ret["distinct_values_by_param_index"] = distinct_param_values(param_tuples)
ret["distinct_values_by_param_name"] = dict()
for i, param in enumerate(param_names):
ret["distinct_values_by_param_name"][param] = ret[
"distinct_values_by_param_index"
][i]
ret["std_static"] = np.std(data)
# TODO Gewichtung? Parameterkombinationen mit wenig verfügbaren Messdaten werden
# genau so behandelt wie welchemit vielen verfügbaren Messdaten, in
# std_static haben sie dagegen weniger Gewicht
ret["std_param_lut"] = np.mean([np.std(v) for v in by_param.values()])
ret["std_by_param"] = dict()
ret["std_by_arg"] = list()
ret["corr_by_param"] = dict()
ret["corr_by_arg"] = list()
ret["_depends_on_param"] = dict()
ret["_depends_on_arg"] = list()
np.seterr("raise")
for param_idx, param in enumerate(param_names):
if param_idx < len(codependent_params) and codependent_params[param_idx]:
by_param = partition_by_param(
data, param_tuples, ignore_parameters=codependent_params[param_idx]
)
distinct_values = ret["distinct_values_by_param_index"].copy()
for codependent_param_index in codependent_params[param_idx]:
distinct_values[codependent_param_index] = [None]
else:
by_param = ret["by_param"]
distinct_values = ret["distinct_values_by_param_index"]
mean_std = _mean_std_by_param(by_param, distinct_values, param_idx)
ret["std_by_param"][param] = mean_std
ret["corr_by_param"][param] = _corr_by_param(data, param_tuples, param_idx)
ret["_depends_on_param"][param] = _depends_on_param(
ret["corr_by_param"][param],
ret["std_by_param"][param],
ret["std_param_lut"],
)
if arg_count:
for arg_index in range(arg_count):
param_idx = len(param_names) + arg_index
if param_idx < len(codependent_params) and codependent_params[param_idx]:
by_param = partition_by_param(
data, param_tuples, ignore_parameters=codependent_params[param_idx]
)
distinct_values = ret["distinct_values_by_param_index"].copy()
for codependent_param_index in codependent_params[param_idx]:
distinct_values[codependent_param_index] = [None]
else:
by_param = ret["by_param"]
distinct_values = ret["distinct_values_by_param_index"]
mean_std = _mean_std_by_param(by_param, distinct_values, param_idx)
ret["std_by_arg"].append(mean_std)
ret["corr_by_arg"].append(_corr_by_param(data, param_tuples, param_idx))
if False:
ret["_depends_on_arg"].append(ret["corr_by_arg"][arg_index] > 0.1)
elif ret["std_by_arg"][arg_index] == 0:
# In general, std_param_lut < std_by_arg. So, if std_by_arg == 0, std_param_lut == 0 follows.
# This means that the variation of arg does not affect the model quality -> no influence
ret["_depends_on_arg"].append(False)
else:
ret["_depends_on_arg"].append(
ret["std_param_lut"] / ret["std_by_arg"][arg_index] < 0.5
)
return ret
def codependent_param_dict(param_values):
lut = [dict() for i in param_values[0]]
for param_index in range(len(param_values[0])):
uniqs = set(map(lambda param_tuple: param_tuple[param_index], param_values))
for uniq_index, uniq in enumerate(uniqs):
lut[param_index][uniq] = uniq_index
normed_param_values = list()
for param_tuple in param_values:
normed_param_values.append(
tuple(map(lambda ipv: lut[ipv[0]][ipv[1]], enumerate(param_tuple)))
)
normed_param_values = np.array(normed_param_values)
std_by_param = list()
std_by_param_pair = dict()
ret = dict()
for param1_i in range(len(lut)):
std_by_param.append(np.std(normed_param_values[:, param1_i]))
for param2_i in range(param1_i + 1, len(lut)):
stds = list()
for param1_value in range(len(lut[param1_i])):
tt = normed_param_values[:, param1_i] == param1_value
values = normed_param_values[tt, param2_i]
if len(values) <= 1:
stds.append(0.0)
else:
stds.append(np.std(values))
std_by_param_pair[(param1_i, param2_i)] = np.mean(stds)
for param1_i in range(len(lut)):
for param2_i in range(param1_i + 1, len(lut)):
if std_by_param[param1_i] > 0 and std_by_param[param2_i] > 0:
if std_by_param_pair[(param1_i, param2_i)] == 0:
ret[(param1_i, param2_i)] = True
return ret
def _compute_param_statistics_parallel(arg):
return {"key": arg["key"], "dict": _compute_param_statistics(*arg["args"])}
def _all_params_are_numeric(data, param_idx):
"""Check if all `data['param'][*][param_idx]` elements are numeric, as reported by `utils.is_numeric`."""
param_values = list(map(lambda x: x[param_idx], data))
if len(list(filter(is_numeric, param_values))) == len(param_values):
return True
return False
class ParallelParamStats:
def __init__(self):
self.queue = list()
self.map = dict()
def enqueue(self, key, attr):
self.queue.append(
{
"key": key,
"args": [
attr.data,
attr.param_names,
attr.param_values,
attr.arg_count,
False,
attr.codependent_params,
],
}
)
self.map[key] = attr
def compute(self):
"""
Fit functions on previously enqueue data.
Fitting is one in parallel with one process per core.
Results can be accessed using the public ParallelParamFit.results object.
"""
with Pool() as pool:
results = pool.map(_compute_param_statistics_parallel, self.queue)
for result in results:
self.map[result["key"]].by_param = result["dict"].pop("by_param")
self.map[result["key"]].stats = ParamStats(result["dict"])
class ParamStats:
def __init__(self, data):
self.__dict__.update(data)
@classmethod
def compute_for_attr(cls, attr, use_corrcoef=False):
res = _compute_param_statistics(
attr.data,
attr.param_names,
attr.param_values,
arg_count=attr.arg_count,
use_corrcoef=use_corrcoef,
codependent_params=attr.codependent_params,
)
attr.by_param = res.pop("by_param")
attr.stats = cls(res)
def can_be_fitted(self) -> bool:
"""
Return whether a sufficient amount of distinct numeric parameter values is available, allowing a parameter-aware model to be generated.
"""
for param in self._parameter_names:
if (
len(
list(
filter(
lambda n: is_numeric(n),
self.distinct_values_by_param_name[param],
)
)
)
> 2
):
logger.debug(
"can be fitted for param {} on {}".format(
param,
list(
filter(
lambda n: is_numeric(n),
self.distinct_values_by_param_name[param],
)
),
)
)
return True
return False
def _generic_param_independence_ratio(self):
"""
Return the heuristic ratio of parameter independence.
This is not supported if the correlation coefficient is used.
A value close to 1 means no influence, a value close to 0 means high probability of influence.
"""
if self.use_corrcoef:
# not supported
raise ValueError
if self.std_static == 0:
return 0
return self.std_param_lut / self.std_static
def generic_param_dependence_ratio(self):
"""
Return the heuristic ratio of parameter dependence.
This is not supported if the correlation coefficient is used.
A value close to 0 means no influence, a value close to 1 means high probability of influence.
"""
return 1 - self._generic_param_independence_ratio()
def _param_independence_ratio(self, param: str) -> float:
"""
Return the heuristic ratio of parameter independence for param.
A value close to 1 means no influence, a value close to 0 means high probability of influence.
"""
if self.use_corrcoef:
return 1 - np.abs(self.corr_by_param[param])
if self.std_by_param[param] == 0:
# if self.std_param_lut != 0:
# raise RuntimeError(f"wat: std_by_param[{param}]==0, but std_param_lut=={self.std_param_lut} ≠ 0")
# In general, std_param_lut < std_by_param. So, if std_by_param == 0, std_param_lut == 0 follows.
# This means that the variation of param does not affect the model quality -> no influence, return 1
return 1.0
return self.std_param_lut / self.std_by_param[param]
def param_dependence_ratio(self, param: str) -> float:
"""
Return the heuristic ratio of parameter dependence for param.
A value close to 0 means no influence, a value close to 1 means high probability of influence.
:param param: parameter name
:returns: parameter dependence (float between 0 == no influence and 1 == high probability of influence)
"""
return 1 - self._param_independence_ratio(param)
def _arg_independence_ratio(self, arg_index):
if self.use_corrcoef:
return 1 - np.abs(self.corr_by_arg[arg_index])
if self.std_by_arg[arg_index] == 0:
if self.std_param_lut != 0:
raise RuntimeError(
f"wat: std_by_arg[{arg_index}]==0, but std_param_lut=={self.std_param_lut} ≠ 0"
)
# In general, std_param_lut < std_by_arg. So, if std_by_arg == 0, std_param_lut == 0 follows.
# This means that the variation of arg does not affect the model quality -> no influence, return 1
return 1
return self.std_param_lut / self.std_by_arg[arg_index]
def arg_dependence_ratio(self, arg_index: int) -> float:
return 1 - self._arg_independence_ratio(arg_index)
# This heuristic is very similar to the "function is not much better than
# median" checks in get_fitted. So far, doing it here as well is mostly
# a performance and not an algorithm quality decision.
# --df, 2018-04-18
def depends_on_param(self, param):
"""Return whether attribute of state_or_trans depens on param."""
return self._depends_on_param[param]
# See notes on depends_on_param
def depends_on_arg(self, arg_index):
"""Return whether attribute of state_or_trans depens on arg_index."""
return self._depends_on_arg[arg_index]
class ModelAttribute:
"""
A ModelAttribute instance handles a single model attribute, e.g. TX state power or something() function call duration, and corresponding benchmark data.
It provides three models:
- a static model (`mean`, `median`) as lower bound of model accuracy
- a LUT model (`by_param`) as upper bound of model accuracy
- a fitted model (`model_function`, a `ModelFunction` instance)
"""
def __init__(
self,
name,
attr,
data,
param_values,
param_names,
arg_count=0,
codependent_param=dict(),
):
# Data for model generation
self.data = np.array(data)
# Meta data
self.name = name
self.attr = attr
self.param_values = param_values
self.param_names = sorted(param_names)
self.arg_count = arg_count
self.log_param_names = self.param_names + list(
map(lambda i: f"arg{i}", range(arg_count))
)
# Co-dependent parameters. If (param1_index, param2_index) in codependent_param, they are codependent.
# In this case, only one of them must be used for parameter-dependent model attribute detection and modeling
self.codependent_param_pair = codependent_param
self.codependent_params = [list() for x in self.log_param_names]
self.ignore_param = dict()
# Static model used as lower bound of model accuracy
self.mean = np.mean(data)
self.median = np.median(data)
# LUT model used as upper bound of model accuracy
self.by_param = None # set via ParallelParamStats
# Split (decision tree) information
self.split = None
# param model override
self.function_override = None
# The best model we have. May be Static, Split, or Param (and later perhaps Substate)
self.model_function = None
self._check_codependent_param()
def __repr__(self):
mean = np.mean(self.data)
return f"ModelAttribute<{self.name}, {self.attr}, mean={mean}>"
def to_json(self, **kwargs):
ret = {
"paramNames": self.param_names,
"argCount": self.arg_count,
"modelFunction": self.model_function.to_json(**kwargs),
}
return ret
def to_dref(self, unit=None):
ret = {"mean": (self.mean, unit), "median": (self.median, unit)}
return ret
def webconf_function_map(self):
return self.model_function.webconf_function_map()
@staticmethod
def from_json(cls, name, attr, data):
param_names = data["paramNames"]
arg_count = data["argCount"]
self = cls(name, attr, None, None, param_names, arg_count)
self.model_function = df.ModelFunction.from_json(data["modelFunction"])
return self
def _check_codependent_param(self):
for (
(param1_index, param2_index),
is_codependent,
) in self.codependent_param_pair.items():
if not is_codependent:
continue
param1_values = map(lambda pv: pv[param1_index], self.param_values)
param1_numeric_count = sum(map(is_numeric, param1_values))
param2_values = map(lambda pv: pv[param2_index], self.param_values)
param2_numeric_count = sum(map(is_numeric, param2_values))
if param1_numeric_count >= param2_numeric_count:
self.ignore_param[param2_index] = True
self.codependent_params[param1_index].append(param2_index)
logger.info(
f"{self.name} {self.attr}: parameters ({self.log_param_names[param1_index]}, {self.log_param_names[param2_index]}) are codependent. Ignoring {self.log_param_names[param2_index]}"
)
else:
self.ignore_param[param1_index] = True
self.codependent_params[param2_index].append(param1_index)
logger.info(
f"{self.name} {self.attr}: parameters ({self.log_param_names[param1_index]}, {self.log_param_names[param2_index]}) are codependent. Ignoring {self.log_param_names[param1_index]}"
)
def get_static(self, use_mean=False):
if use_mean:
return self.mean
return self.median
def get_lut(self, param, use_mean=False):
if use_mean:
return np.mean(self.by_param[param])
return np.median(self.by_param[param])
def build_dtree(self):
split_param_index = self.get_split_param_index()
if split_param_index is None:
return
distinct_values = self.stats.distinct_values_by_param_index[split_param_index]
tt1 = list(
map(
lambda i: self.param_values[i][split_param_index] == distinct_values[0],
range(len(self.param_values)),
)
)
tt2 = np.invert(tt1)
pv1 = list()
pv2 = list()
for i, param_tuple in enumerate(self.param_values):
if tt1[i]:
pv1.append(param_tuple)
else:
pv2.append(param_tuple)
# print(
# f">>> split {self.name} {self.attr} by param #{split_param_index}"
# )
child1 = ModelAttribute(
self.name,
self.attr,
self.data[tt1],
pv1,
self.param_names,
self.arg_count,
codependent_param_dict(pv1),
)
child2 = ModelAttribute(
self.name,
self.attr,
self.data[tt2],
pv2,
self.param_names,
self.arg_count,
codependent_param_dict(pv2),
)
ParamStats.compute_for_attr(child1)
ParamStats.compute_for_attr(child2)
child1.build_dtree()
child2.build_dtree()
self.split = (
split_param_index,
{distinct_values[0]: child1, distinct_values[1]: child2},
)
# print(
# f"<<< split {self.name} {self.attr} by param #{split_param_index}"
# )
# None -> kein split notwendig
# andernfalls: Parameter-Index, anhand dessen eine Decision Tree-Ebene aufgespannt wird
# (Kinder sind wiederum ModelAttributes, in denen dieser Parameter konstant ist)
def get_split_param_index(self):
if not self.param_names:
return None
std_by_param = list()
for param_index, param_name in enumerate(self.param_names):
distinct_values = self.stats.distinct_values_by_param_index[param_index]
if (
self.stats.depends_on_param(param_name)
and len(distinct_values) == 2
and not param_index in self.ignore_param
):
val1 = list(
map(
lambda i: self.param_values[i][param_index]
== distinct_values[0],
range(len(self.param_values)),
)
)
val2 = np.invert(val1)
val1_std = np.std(self.data[val1])
val2_std = np.std(self.data[val2])
std_by_param.append(np.mean([val1_std, val2_std]))
else:
std_by_param.append(np.inf)
for arg_index in range(self.arg_count):
distinct_values = self.stats.distinct_values_by_param_index[
len(self.param_names) + arg_index
]
if (
self.stats.depends_on_arg(arg_index)
and len(distinct_values) == 2
and not len(self.param_names) + arg_index in self.ignore_param
):
val1 = list(
map(
lambda i: self.param_values[i][
len(self.param_names) + arg_index
]
== distinct_values[0],
range(len(self.param_values)),
)
)
val2 = np.invert(val1)
val1_std = np.std(self.data[val1])
val2_std = np.std(self.data[val2])
std_by_param.append(np.mean([val1_std, val2_std]))
else:
std_by_param.append(np.inf)
split_param_index = np.argmin(std_by_param)
split_std = std_by_param[split_param_index]
if split_std == np.inf:
return None
return split_param_index
def get_data_for_paramfit(self, safe_functions_enabled=False):
if self.split:
return self.get_data_for_paramfit_split(
safe_functions_enabled=safe_functions_enabled
)
else:
return self.get_data_for_paramfit_this(
safe_functions_enabled=safe_functions_enabled
)
def get_data_for_paramfit_split(self, safe_functions_enabled=False):
split_param_index, child_by_param_value = self.split
ret = list()
for param_value, child in child_by_param_value.items():
child_ret = child.get_data_for_paramfit(
safe_functions_enabled=safe_functions_enabled
)
for key, param, args, kwargs in child_ret:
ret.append((key[:2] + (param_value,) + key[2:], param, args, kwargs))
return ret
def _by_param_for_index(self, param_index):
if not self.codependent_params[param_index]:
return self.by_param
new_param_values = list()
for param_tuple in self.param_values:
for i in self.codependent_params[param_index]:
param_tuple[i] = None
new_param_values.append(param_tuple)
return partition_by_param(self.data, new_param_values)
def get_data_for_paramfit_this(self, safe_functions_enabled=False):
ret = list()
for param_index, param_name in enumerate(self.param_names):
if (
self.stats.depends_on_param(param_name)
and not param_index in self.ignore_param
):
by_param = self._by_param_for_index(param_index)
ret.append(
(
(self.name, self.attr),
param_name,
(by_param, param_index, safe_functions_enabled),
dict(),
)
)
if self.arg_count:
for arg_index in range(self.arg_count):
param_index = len(self.param_names) + arg_index
if (
self.stats.depends_on_arg(arg_index)
and not param_index in self.ignore_param
):
by_param = self._by_param_for_index(param_index)
ret.append(
(
(self.name, self.attr),
arg_index,
(by_param, param_index, safe_functions_enabled),
dict(),
)
)
return ret
def set_data_from_paramfit(self, paramfit, prefix=tuple()):
if self.split:
self.set_data_from_paramfit_split(paramfit, prefix)
else:
self.set_data_from_paramfit_this(paramfit, prefix)
def set_data_from_paramfit_split(self, paramfit, prefix):
split_param_index, child_by_param_value = self.split
function_map = {
"split_by": split_param_index,
"child": dict(),
"child_static": dict(),
}
function_child = dict()
info_child = dict()
for param_value, child in child_by_param_value.items():
child.set_data_from_paramfit(paramfit, prefix + (param_value,))
function_child[param_value] = child.model_function
self.model_function = df.SplitFunction(
self.median, split_param_index, function_child
)
def set_data_from_paramfit_this(self, paramfit, prefix):
fit_result = paramfit.get_result((self.name, self.attr) + prefix)
self.model_function = df.StaticFunction(self.median)
if self.function_override is not None:
function_str = self.function_override
x = df.AnalyticFunction(
self.median,
function_str,
self.param_names,
self.arg_count,
fit_by_param=fit_result,
)
x.fit(self.by_param)
if x.fit_success:
self.model_function = x
elif os.getenv("DFATOOL_NO_PARAM"):
pass
elif len(fit_result.keys()):
x = df.analytic.function_powerset(
fit_result, self.param_names, self.arg_count
)
x.value = self.median
x.fit(self.by_param)
if x.fit_success:
self.model_function = x
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