""" Utilities for analytic description of parameter-dependent model attributes. This module provides classes and helper functions useful for least-squares regression and general handling of model functions. """ from itertools import chain, combinations import logging import numpy as np import re from scipy import optimize from .utils import is_numeric arg_support_enabled = True logger = logging.getLogger(__name__) def powerset(iterable): """ Return powerset of `iterable` elements. Example: `powerset([1, 2])` -> `[(), (1), (2), (1, 2)]` """ s = list(iterable) return chain.from_iterable(combinations(s, r) for r in range(len(s) + 1)) def gplearn_to_function(function_str: str): """ Convert gplearn-style function string to Python function. Takes a function string like "mul(add(X0, X1), X2)" and returns a Python function implementing the specified behaviour, e.g. "lambda x, y, z: (x + y) * z". Supported functions: add -- x + y sub -- x - y mul -- x * y div -- x / y if |y| > 0.001, otherwise 1 sqrt -- sqrt(|x|) log -- log(|x|) if |x| > 0.001, otherwise 0 inv -- 1 / x if |x| > 0.001, otherwise 0 """ eval_globals = { "add": lambda x, y: x + y, "sub": lambda x, y: x - y, "mul": lambda x, y: x * y, "div": lambda x, y: np.divide(x, y) if np.abs(y) > 0.001 else 1.0, "sqrt": lambda x: np.sqrt(np.abs(x)), "log": lambda x: np.log(np.abs(x)) if np.abs(x) > 0.001 else 0.0, "inv": lambda x: 1.0 / x if np.abs(x) > 0.001 else 0.0, } last_arg_index = 0 for i in range(0, 100): if function_str.find("X{:d}".format(i)) >= 0: last_arg_index = i arg_list = [] for i in range(0, last_arg_index + 1): arg_list.append("X{:d}".format(i)) eval_str = "lambda {}, *whatever: {}".format(",".join(arg_list), function_str) logger.debug(eval_str) return eval(eval_str, eval_globals) class ParamFunction: """ A one-dimensional model function, ready for least squares optimization and similar. Supports validity checks (e.g. if it is undefined for x <= 0) and an error measure. """ def __init__(self, param_function, validation_function, num_vars): """ Create function object suitable for regression analysis. This documentation assumes that 1-dimensional functions (-> single float as model input) are used. However, n-dimensional functions (-> list of float as model input) are also supported. :param param_function: regression function (reg_param, model_param) -> float. reg_param is a list of regression variable values, model_param is the model input value (float). Example: `lambda rp, mp: rp[0] + rp[1] * mp` :param validation_function: function used to check whether param_function is defined for a given model_param. Signature: model_param -> bool Example: `lambda mp: mp > 0` :param num_vars: How many regression variables are used by this function, i.e., the length of param_function's reg_param argument. """ self._param_function = param_function self._validation_function = validation_function self._num_variables = num_vars def is_valid(self, arg: float) -> bool: """ Check whether the regression function is defined for the given argument. :param arg: argument (e.g. model parameter) to check for :returns: True iff the function is defined for `arg` """ return self._validation_function(arg) def eval(self, param: list, arg: float) -> float: """ Evaluate regression function. :param param: regression variable values (list of float) :param arg: model input (float) :returns: regression function output (float) """ return self._param_function(param, arg) def error_function(self, P: list, X: float, y: float) -> float: """ Calculate model error. :param P: regression variables as returned by optimization (list of float) :param X: model input (float) :param y: expected model output / ground truth for model input (float) :returns: Deviation between model output and ground truth (float) """ return self._param_function(P, X) - y class NormalizationFunction: """ Wrapper for parameter normalization functions used in YAML PTA/DFA models. """ def __init__(self, function_str: str): """ Create a new normalization function from `function_str`. :param function_str: Function string. Must use the single argument `param` and return a float. """ self._function_str = function_str self._function = eval("lambda param: " + function_str) def eval(self, param_value: float) -> float: """ Evaluate the normalization function and return its output. :param param_value: Parameter value """ return self._function(param_value) class ModelInfo: def __init__(self): pass class StaticInfo: def __init__(self, data): self.mean = np.mean(data) self.median = np.median(data) self.std = np.std(data) class AnalyticInfo(ModelInfo): def __init__(self, fit_result, function): self.fit_result = fit_result self.function = function class SplitInfo(ModelInfo): def __init__(self, param_index, child): self.param_index = param_index self.child = child class ModelFunction: def __init__(self): pass def is_predictable(self, param_list): raise NotImplementedError def eval(self, param_list, arg_list): raise NotImplementedError class StaticFunction(ModelFunction): def __init__(self, value): self.value = value def is_predictable(self, param_list=None): """ Return whether the model function can be evaluated on the given parameter values. For a StaticFunction, this is always the case (i.e., this function always returns true). """ return True def eval(self, param_list=None, arg_list=None): """ Evaluate model function with specified param/arg values. Far a Staticfunction, this is just the static value """ return self.value class SplitFunction(ModelFunction): def __init__(self, param_index, child): self.param_index = param_index self.child = child def is_predictable(self, param_list): """ Return whether the model function can be evaluated on the given parameter values. The first value corresponds to the lexically first model parameter, etc. All parameters must be set, not just the ones this function depends on. Returns False iff a parameter the function depends on is not numeric (e.g. None). """ param_value = param_list[self.param_index] if param_value not in self.child: return False return self.child[param_value].is_predictable(param_list) def eval(self, param_list, arg_list=list()): param_value = param_list[self.param_index] return self.child[param_value].eval(param_list, arg_list) class AnalyticFunction(ModelFunction): """ A multi-dimensional model function, generated from a string, which can be optimized using regression. The function describes a single model attribute (e.g. TX duration or send(...) energy) and how it is influenced by model parameters such as configured bit rate or packet length. """ def __init__(self, function_str, parameters, num_args, regression_args=None): """ Create a new AnalyticFunction object from a function string. :param function_str: the function. Refer to regression variables using regression_arg(123), to parameters using parameter(name), and to function arguments (if any) using function_arg(123). Example: "regression_arg(0) + regression_arg(1) * parameter(txbytes)" :param parameters: list containing the names of all model parameters, including those not used in function_str, sorted lexically. Sorting is mandatory, as parameter indexes (and not names) are used internally. :param num_args: number of local function arguments, if any. Set to 0 if the model attribute does not belong to a function or if function arguments are not included in the model. :param regression_args: Initial regression variable values, both for function usage and least squares optimization. If unset, defaults to [1, 1, 1, ...] """ self._parameter_names = parameters self._num_args = num_args self.model_function = function_str rawfunction = function_str self._dependson = [False] * (len(parameters) + num_args) self.fit_success = False if type(function_str) == str: num_vars_re = re.compile(r"regression_arg\(([0-9]+)\)") num_vars = max(map(int, num_vars_re.findall(function_str))) + 1 for i in range(len(parameters)): if rawfunction.find("parameter({})".format(parameters[i])) >= 0: self._dependson[i] = True rawfunction = rawfunction.replace( "parameter({})".format(parameters[i]), "model_param[{:d}]".format(i), ) for i in range(0, num_args): if rawfunction.find("function_arg({:d})".format(i)) >= 0: self._dependson[len(parameters) + i] = True rawfunction = rawfunction.replace( "function_arg({:d})".format(i), "model_param[{:d}]".format(len(parameters) + i), ) for i in range(num_vars): rawfunction = rawfunction.replace( "regression_arg({:d})".format(i), "reg_param[{:d}]".format(i) ) self._function_str = rawfunction self._function = eval("lambda reg_param, model_param: " + rawfunction) else: self._function_str = "raise ValueError" self._function = function_str if regression_args: self.model_args = regression_args.copy() self._fit_success = True elif type(function_str) == str: self.model_args = list(np.ones((num_vars))) else: self.model_args = [] def get_fit_data(self, by_param): """ Return training data suitable for scipy.optimize.least_squares. :param by_param: measurement data, partitioned by parameter/arg values. by_param[*] must be a list or 1-D NumPy array containing the ground truth. The parameter values (dict keys) must be numeric for all parameters this function depends on -- otherwise, the corresponding data will be left out. Parameter values must be ordered according to the order of parameter names used in the ParamFunction constructor. Argument values (if any) always come after parameters, in the order of their index in the function signature. :return: (X, Y, num_valid, num_total): X -- 2-D NumPy array of parameter combinations (model input). First dimension is the parameter/argument index, the second dimension contains its values. Example: X[0] contains the first parameter's values. Y -- 1-D NumPy array of training data (desired model output). num_valid -- amount of distinct parameter values suitable for optimization num_total -- total amount of distinct parameter values """ dimension = len(self._parameter_names) + self._num_args X = [[] for i in range(dimension)] Y = [] num_valid = 0 num_total = 0 for key, val in by_param.items(): if len(key) == dimension: valid = True num_total += 1 for i in range(dimension): if self._dependson[i] and not is_numeric(key[i]): valid = False if valid: num_valid += 1 Y.extend(val) for i in range(dimension): if self._dependson[i]: X[i].extend([float(key[i])] * len(val)) else: X[i].extend([np.nan] * len(val)) else: logger.warning( "Invalid parameter key length while gathering fit data. is {}, want {}.".format( len(key), dimension ) ) X = np.array(X) Y = np.array(Y) return X, Y, num_valid, num_total def fit(self, by_param): """ Fit the function on measurements via least squares regression. :param by_param: measurement data, partitioned by parameter/arg values The ground truth is read from by_param[*], which must be a list or 1-D NumPy array. Parameter values must be ordered according to the parameter names in the constructor. If argument values are present, they must come after parameter values in the order of their appearance in the function signature. """ X, Y, num_valid, num_total = self.get_fit_data(by_param) if num_valid > 2: error_function = lambda P, X, y: self._function(P, X) - y try: res = optimize.least_squares( error_function, self.model_args, args=(X, Y), xtol=2e-15 ) except ValueError as err: logger.warning(f"Fit failed: {err} (function: {self.model_function})") return if res.status > 0: self.model_args = res.x self.fit_success = True else: logger.warning( f"Fit failed: {res.message} (function: {self.model_function})" ) else: logger.warning("Insufficient amount of valid parameter keys, cannot fit") def is_predictable(self, param_list): """ Return whether the model function can be evaluated on the given parameter values. The first value corresponds to the lexically first model parameter, etc. All parameters must be set, not just the ones this function depends on. Returns False iff a parameter the function depends on is not numeric (e.g. None). """ for i, param in enumerate(param_list): if self._dependson[i] and not is_numeric(param): return False return True def eval(self, param_list, arg_list=[]): """ Evaluate model function with specified param/arg values. :param param_list: parameter values (list of float). First item corresponds to lexically first parameter, etc. :param arg_list: argument values (list of float), if arguments are used. """ if len(self.model_args) == 0: return self._function(param_list, arg_list) return self._function(self.model_args, param_list) class analytic: """ Utilities for analytic description of parameter-dependent model attributes and regression analysis. provided functions: functions -- retrieve pre-defined set of regression function candidates function_powerset -- combine several per-parameter functions into a single AnalyticFunction """ _num0_8 = np.vectorize(lambda x: 8 - bin(int(x)).count("1")) _num0_16 = np.vectorize(lambda x: 16 - bin(int(x)).count("1")) _num1 = np.vectorize(lambda x: bin(int(x)).count("1")) _safe_log = np.vectorize(lambda x: np.log(np.abs(x)) if np.abs(x) > 0.001 else 1.0) _safe_inv = np.vectorize(lambda x: 1 / x if np.abs(x) > 0.001 else 1.0) _safe_sqrt = np.vectorize(lambda x: np.sqrt(np.abs(x))) _function_map = { "linear": lambda x: x, "logarithmic": np.log, "logarithmic1": lambda x: np.log(x + 1), "exponential": np.exp, "square": lambda x: x ** 2, "inverse": lambda x: 1 / x, "sqrt": lambda x: np.sqrt(np.abs(x)), "num0_8": _num0_8, "num0_16": _num0_16, "num1": _num1, "safe_log": lambda x: np.log(np.abs(x)) if np.abs(x) > 0.001 else 1.0, "safe_inv": lambda x: 1 / x if np.abs(x) > 0.001 else 1.0, "safe_sqrt": lambda x: np.sqrt(np.abs(x)), } @staticmethod def functions(safe_functions_enabled=False): """ Retrieve pre-defined set of regression function candidates. :param safe_functions_enabled: Include "safe" variants of functions with limited argument range, e.g. a safe inverse which returns 1 when dividing by 0. Returns a dict of functions which are typical for energy/timing behaviour of embedded hardware, e.g. linear, exponential or inverse dependency on a configuration setting/runtime variable. Each function is a ParamFunction object. In most cases, two regression variables are expected. """ functions = { "linear": ParamFunction( lambda reg_param, model_param: reg_param[0] + reg_param[1] * model_param, lambda model_param: True, 2, ), "logarithmic": ParamFunction( lambda reg_param, model_param: reg_param[0] + reg_param[1] * np.log(model_param), lambda model_param: model_param > 0, 2, ), "logarithmic1": ParamFunction( lambda reg_param, model_param: reg_param[0] + reg_param[1] * np.log(model_param + 1), lambda model_param: model_param > -1, 2, ), "exponential": ParamFunction( lambda reg_param, model_param: reg_param[0] + reg_param[1] * np.exp(model_param), lambda model_param: model_param <= 64, 2, ), #'polynomial' : lambda reg_param, model_param: reg_param[0] + reg_param[1] * model_param + reg_param[2] * model_param ** 2, "square": ParamFunction( lambda reg_param, model_param: reg_param[0] + reg_param[1] * model_param ** 2, lambda model_param: True, 2, ), "inverse": ParamFunction( lambda reg_param, model_param: reg_param[0] + reg_param[1] / model_param, lambda model_param: model_param != 0, 2, ), "sqrt": ParamFunction( lambda reg_param, model_param: reg_param[0] + reg_param[1] * np.sqrt(model_param), lambda model_param: model_param >= 0, 2, ), "num0_8": ParamFunction( lambda reg_param, model_param: reg_param[0] + reg_param[1] * analytic._num0_8(model_param), lambda model_param: True, 2, ), "num0_16": ParamFunction( lambda reg_param, model_param: reg_param[0] + reg_param[1] * analytic._num0_16(model_param), lambda model_param: True, 2, ), "num1": ParamFunction( lambda reg_param, model_param: reg_param[0] + reg_param[1] * analytic._num1(model_param), lambda model_param: True, 2, ), } if safe_functions_enabled: functions["safe_log"] = ParamFunction( lambda reg_param, model_param: reg_param[0] + reg_param[1] * analytic._safe_log(model_param), lambda model_param: True, 2, ) functions["safe_inv"] = ParamFunction( lambda reg_param, model_param: reg_param[0] + reg_param[1] * analytic._safe_inv(model_param), lambda model_param: True, 2, ) functions["safe_sqrt"] = ParamFunction( lambda reg_param, model_param: reg_param[0] + reg_param[1] * analytic._safe_sqrt(model_param), lambda model_param: True, 2, ) return functions @staticmethod def _fmap(reference_type, reference_name, function_type): """Map arg/parameter name and best-fit function name to function text suitable for AnalyticFunction.""" ref_str = "{}({})".format(reference_type, reference_name) if function_type == "linear": return ref_str if function_type == "logarithmic": return "np.log({})".format(ref_str) if function_type == "logarithmic1": return "np.log({} + 1)".format(ref_str) if function_type == "exponential": return "np.exp({})".format(ref_str) if function_type == "exponential": return "np.exp({})".format(ref_str) if function_type == "square": return "({})**2".format(ref_str) if function_type == "inverse": return "1/({})".format(ref_str) if function_type == "sqrt": return "np.sqrt({})".format(ref_str) return "analytic._{}({})".format(function_type, ref_str) @staticmethod def function_powerset(fit_results, parameter_names, num_args=0): """ Combine per-parameter regression results into a single multi-dimensional function. :param fit_results: results dict. One element per parameter, each containing a dict of the form {'best' : name of function with best fit}. Must not include parameters which do not influence the model attribute. Example: {'txpower' : {'best': 'exponential'}} :param parameter_names: Parameter names, including those left out in fit_results because they do not influence the model attribute. Must be sorted lexically. Example: ['bitrate', 'txpower'] :param num_args: number of local function arguments, if any. Set to 0 if the model attribute does not belong to a function or if function arguments are not included in the model. Returns an AnalyticFunction instantce corresponding to the combined function. """ buf = "0" arg_idx = 0 for combination in powerset(fit_results.items()): buf += " + regression_arg({:d})".format(arg_idx) arg_idx += 1 for function_item in combination: if arg_support_enabled and is_numeric(function_item[0]): buf += " * {}".format( analytic._fmap( "function_arg", function_item[0], function_item[1]["best"] ) ) else: buf += " * {}".format( analytic._fmap( "parameter", function_item[0], function_item[1]["best"] ) ) return AnalyticFunction(buf, parameter_names, num_args)