Source code for impedance.models.circuits.fitting

import warnings

import numpy as np
from scipy.linalg import inv
from scipy.optimize import curve_fit, basinhopping

from .elements import circuit_elements, get_element_from_name

ints = '0123456789'

[docs]def rmse(a, b): """ A function which calculates the root mean squared error between two vectors. Notes --------- .. math:: RMSE = \\sqrt{\\frac{1}{n}(a-b)^2} """ n = len(a) return np.linalg.norm(a - b) / np.sqrt(n)
[docs]def set_default_bounds(circuit, constants={}): """ This function sets default bounds for optimization. set_default_bounds sets bounds of 0 and np.inf for all parameters, except the CPE and La alphas which have an upper bound of 1. Parameters ----------------- circuit : string String defining the equivalent circuit to be fit constants : dictionary, optional Parameters and their values to hold constant during fitting (e.g. {"RO": 0.1}). Defaults to {} Returns ------------ bounds : 2-tuple of array_like Lower and upper bounds on parameters. """ # extract the elements from the circuit extracted_elements = extract_circuit_elements(circuit) # loop through bounds lower_bounds, upper_bounds = [], [] for elem in extracted_elements: raw_element = get_element_from_name(elem) for i in range(check_and_eval(raw_element).num_params): if elem in constants or elem + f'_{i}' in constants: continue if raw_element in ['CPE', 'La'] and i == 1: upper_bounds.append(1) else: upper_bounds.append(np.inf) lower_bounds.append(0) bounds = ((lower_bounds), (upper_bounds)) return bounds
[docs]def circuit_fit(frequencies, impedances, circuit, initial_guess, constants={}, bounds=None, weight_by_modulus=False, global_opt=False, **kwargs): """ Main function for fitting an equivalent circuit to data. By default, this function uses `scipy.optimize.curve_fit <>`_ to fit the equivalent circuit. This function generally works well for simple circuits. However, the final results may be sensitive to the initial conditions for more complex circuits. In these cases, the `scipy.optimize.basinhopping <>`_ global optimization algorithm can be used to attempt a better fit. Parameters ----------------- frequencies : numpy array Frequencies impedances : numpy array of dtype 'complex128' Impedances circuit : string String defining the equivalent circuit to be fit initial_guess : list of floats Initial guesses for the fit parameters constants : dictionary, optional Parameters and their values to hold constant during fitting (e.g. {"RO": 0.1}). Defaults to {} bounds : 2-tuple of array_like, optional Lower and upper bounds on parameters. Defaults to bounds on all parameters of 0 and np.inf, except the CPE alpha which has an upper bound of 1 weight_by_modulus : bool, optional Uses the modulus of each data (|Z|) as the weighting factor. Standard weighting scheme when experimental variances are unavailable. Only applicable when global_opt = False global_opt : bool, optional If global optimization should be used (uses the basinhopping algorithm). Defaults to False kwargs : Keyword arguments passed to scipy.optimize.curve_fit or scipy.optimize.basinhopping Returns ------------ p_values : list of floats best fit parameters for specified equivalent circuit p_errors : list of floats one standard deviation error estimates for fit parameters Notes --------- Need to do a better job of handling errors in fitting. Currently, an error of -1 is returned. """ f = np.array(frequencies, dtype=float) Z = np.array(impedances, dtype=complex) # set upper and lower bounds on a per-element basis if bounds is None: bounds = set_default_bounds(circuit, constants=constants) if not global_opt: if 'maxfev' not in kwargs: kwargs['maxfev'] = 1e5 if 'ftol' not in kwargs: kwargs['ftol'] = 1e-13 # weighting scheme for fitting if weight_by_modulus: abs_Z = np.abs(Z) kwargs['sigma'] = np.hstack([abs_Z, abs_Z]) popt, pcov = curve_fit(wrapCircuit(circuit, constants), f, np.hstack([Z.real, Z.imag]), p0=initial_guess, bounds=bounds, **kwargs) # Calculate one standard deviation error estimates for fit parameters, # defined as the square root of the diagonal of the covariance matrix. # perror = np.sqrt(np.diag(pcov)) else: if 'seed' not in kwargs: kwargs['seed'] = 0 def opt_function(x): """ Short function for basinhopping to optimize over. We want to minimize the RMSE between the fit and the data. Parameters ---------- x : args Parameters for optimization. Returns ------- function Returns a function (RMSE as a function of parameters). """ return rmse(wrapCircuit(circuit, constants)(f, *x), np.hstack([Z.real, Z.imag])) class BasinhoppingBounds(object): """ Adapted from the basinhopping documetation """ def __init__(self, xmin, xmax): self.xmin = np.array(xmin) self.xmax = np.array(xmax) def __call__(self, **kwargs): x = kwargs['x_new'] tmax = bool(np.all(x <= self.xmax)) tmin = bool(np.all(x >= self.xmin)) return tmax and tmin basinhopping_bounds = BasinhoppingBounds(xmin=bounds[0], xmax=bounds[1]) results = basinhopping(opt_function, x0=initial_guess, accept_test=basinhopping_bounds, **kwargs) popt = results.x # Calculate perror jac = results.lowest_optimization_result['jac'][np.newaxis] try: # jacobian -> covariance # pcov = inv(, jac)) * opt_function(popt) ** 2 # covariance -> perror (one standard deviation # error estimates for fit parameters) perror = np.sqrt(np.diag(pcov)) except (ValueError, np.linalg.LinAlgError): warnings.warn('Failed to compute perror') perror = None return popt, perror
[docs]def wrapCircuit(circuit, constants): """ wraps function so we can pass the circuit string """ def wrappedCircuit(frequencies, *parameters): """ returns a stacked array of real and imaginary impedance components Parameters ---------- circuit : string constants : dict parameters : list of floats frequencies : list of floats Returns ------- array of floats """ x = eval(buildCircuit(circuit, frequencies, *parameters, constants=constants, eval_string='', index=0)[0], circuit_elements) y_real = np.real(x) y_imag = np.imag(x) return np.hstack([y_real, y_imag]) return wrappedCircuit
[docs]def buildCircuit(circuit, frequencies, *parameters, constants=None, eval_string='', index=0): """ recursive function that transforms a circuit, parameters, and frequencies into a string that can be evaluated Parameters ---------- circuit: str frequencies: list/tuple/array of floats parameters: list/tuple/array of floats constants: dict Returns ------- eval_string: str Python expression for calculating the resulting fit index: int Tracks parameter index through recursive calling of the function """ parameters = np.array(parameters).tolist() frequencies = np.array(frequencies).tolist() circuit = circuit.replace(' ', '') def parse_circuit(circuit, parallel=False, series=False): """ Splits a circuit string by either dashes (series) or commas (parallel) outside of any paranthesis. Removes any leading 'p(' or trailing ')' when in parallel mode """ assert parallel != series, \ 'Exactly one of parallel or series must be True' def count_parens(string): return string.count('('), string.count(')') if parallel: special = ',' if circuit.endswith(')') and circuit.startswith('p('): circuit = circuit[2:-1] if series: special = '-' split = circuit.split(special) result = [] skipped = [] for i, sub_str in enumerate(split): if i not in skipped: if '(' not in sub_str and ')' not in sub_str: result.append(sub_str) else: open_parens, closed_parens = count_parens(sub_str) if open_parens == closed_parens: result.append(sub_str) else: uneven = True while i < len(split) - 1 and uneven: sub_str += special + split[i+1] open_parens, closed_parens = count_parens(sub_str) uneven = open_parens != closed_parens i += 1 skipped.append(i) result.append(sub_str) return result parallel = parse_circuit(circuit, parallel=True) series = parse_circuit(circuit, series=True) if series is not None and len(series) > 1: eval_string += "s([" split = series elif parallel is not None and len(parallel) > 1: eval_string += "p([" split = parallel elif series == parallel: # only single element split = series for i, elem in enumerate(split): if ',' in elem or '-' in elem: eval_string, index = buildCircuit(elem, frequencies, *parameters, constants=constants, eval_string=eval_string, index=index) else: param_string = "" raw_elem = get_element_from_name(elem) elem_number = check_and_eval(raw_elem).num_params param_list = [] for j in range(elem_number): if elem_number > 1: current_elem = elem + '_{}'.format(j) else: current_elem = elem if current_elem in constants.keys(): param_list.append(constants[current_elem]) else: param_list.append(parameters[index]) index += 1 param_string += str(param_list) new = raw_elem + '(' + param_string + ',' + str(frequencies) + ')' eval_string += new if i == len(split) - 1: if len(split) > 1: # do not add closing brackets if single element eval_string += '])' else: eval_string += ',' return eval_string, index
[docs]def extract_circuit_elements(circuit): """ Extracts circuit elements from a circuit string. Parameters ---------- circuit : str Circuit string. Returns ------- extracted_elements : list list of extracted elements. """ p_string = [x for x in circuit if x not in 'p(),-'] extracted_elements = [] current_element = [] length = len(p_string) for i, char in enumerate(p_string): if char not in ints: current_element.append(char) else: # min to prevent looking ahead past end of list if p_string[min(i+1, length-1)] not in ints: current_element.append(char) extracted_elements.append(''.join(current_element)) current_element = [] else: current_element.append(char) extracted_elements.append(''.join(current_element)) return extracted_elements
[docs]def calculateCircuitLength(circuit): """ Calculates the number of elements in the circuit. Parameters ---------- circuit : str Circuit string. Returns ------- length : int Length of circuit. """ length = 0 if circuit: extracted_elements = extract_circuit_elements(circuit) for elem in extracted_elements: raw_element = get_element_from_name(elem) num_params = check_and_eval(raw_element).num_params length += num_params return length
[docs]def check_and_eval(element): """ Checks if an element is valid, then evaluates it. Parameters ---------- element : str Circuit element. Raises ------ ValueError Raised if an element is not in the list of allowed elements. Returns ------- Evaluated element. """ allowed_elements = circuit_elements.keys() if element not in allowed_elements: raise ValueError(f'{element} not in ' + f'allowed elements ({allowed_elements})') else: return eval(element, circuit_elements)