Source code for SALib.analyze.delta

from typing import Dict
from scipy.stats import norm, gaussian_kde, rankdata

import numpy as np

from . import common_args
from ..util import read_param_file, ResultDict


[docs]def analyze(problem: Dict, X: np.ndarray, Y: np.ndarray, num_resamples: int = 100, conf_level: float = 0.95, print_to_console: bool = False, seed: int = None) -> Dict: """Perform Delta Moment-Independent Analysis on model outputs. Returns a dictionary with keys 'delta', 'delta_conf', 'S1', and 'S1_conf', where each entry is a list of size D (the number of parameters) containing the indices in the same order as the parameter file. Compatible with --------------- * all samplers Parameters ---------- problem : dict The problem definition X: numpy.matrix A NumPy matrix containing the model inputs Y : numpy.array A NumPy array containing the model outputs num_resamples : int The number of resamples when computing confidence intervals (default 10) conf_level : float The confidence interval level (default 0.95) print_to_console : bool Print results directly to console (default False) References ---------- .. [1] Borgonovo, E. (2007). "A new uncertainty importance measure." Reliability Engineering & System Safety, 92(6):771-784, doi:10.1016/j.ress.2006.04.015. .. [2] Plischke, E., E. Borgonovo, and C. L. Smith (2013). "Global sensitivity measures from given data." European Journal of Operational Research, 226(3):536-550, doi:10.1016/j.ejor.2012.11.047. Examples -------- >>> X = latin.sample(problem, 1000) >>> Y = Ishigami.evaluate(X) >>> Si = delta.analyze(problem, X, Y, print_to_console=True) """ if seed: np.random.seed(seed) D = problem['num_vars'] N = Y.size if not 0 < conf_level < 1: raise RuntimeError("Confidence level must be between 0-1.") # equal frequency partition exp = (2.0 / (7.0 + np.tanh((1500.0 - N) / 500.0))) M = int(np.round( min(int(np.ceil(N**exp)), 48) )) m = np.linspace(0, N, M + 1) Ygrid = np.linspace(np.min(Y), np.max(Y), 100) keys = ('delta', 'delta_conf', 'S1', 'S1_conf') S = ResultDict((k, np.zeros(D)) for k in keys) S['names'] = problem['names'] try: for i in range(D): X_i = X[:, i] S['delta'][i], S['delta_conf'][i] = bias_reduced_delta( Y, Ygrid, X_i, m, num_resamples, conf_level) S['S1'][i] = sobol_first(Y, X_i, m) S['S1_conf'][i] = sobol_first_conf( Y, X_i, m, num_resamples, conf_level) except np.linalg.LinAlgError as e: msg = "Singular matrix detected\n" msg += "This may be due to the sample size ({}) being too small\n".format(Y.size) msg += "If this is not the case, check Y values or raise an issue with the\n" msg += "SALib team" raise np.linalg.LinAlgError(msg) if print_to_console: print(S.to_df()) return S
[docs]def calc_delta(Y, Ygrid, X, m): """Plischke et al. (2013) delta index estimator (eqn 26) for d_hat.""" N = len(Y) fy = gaussian_kde(Y, bw_method='silverman')(Ygrid) abs_fy = np.abs(fy) xr = rankdata(X, method='ordinal') d_hat = 0 for j in range(len(m) - 1): ix = np.where((xr > m[j]) & (xr <= m[j + 1]))[0] nm = len(ix) Y_ix = Y[ix] if not np.all(np.equal(Y_ix, Y_ix[0])): fyc = gaussian_kde(Y_ix, bw_method='silverman')(Ygrid) fy_ = np.abs(fy - fyc) else: fy_ = abs_fy d_hat += (nm / (2 * N)) * np.trapz(fy_, Ygrid) return d_hat
[docs]def bias_reduced_delta(Y, Ygrid, X, m, num_resamples, conf_level): """Plischke et al. 2013 bias reduction technique (eqn 30)""" d = np.zeros(num_resamples) d_hat = calc_delta(Y, Ygrid, X, m) N = len(Y) r = np.random.randint(N, size=(num_resamples, N)) for i in range(num_resamples): r_i = r[i, :] d[i] = calc_delta(Y[r_i], Ygrid, X[r_i], m) d = 2 * d_hat - d return (d.mean(), norm.ppf(0.5 + conf_level / 2) * d.std(ddof=1))
[docs]def sobol_first(Y, X, m): xr = rankdata(X, method='ordinal') Vi = 0 N = len(Y) Y_mean = Y.mean() for j in range(len(m) - 1): ix = np.where((xr > m[j]) & (xr <= m[j + 1]))[0] nm = len(ix) Vi += (nm / N) * ((Y[ix].mean() - Y_mean)**2) return Vi / np.var(Y)
[docs]def sobol_first_conf(Y, X, m, num_resamples, conf_level): s = np.zeros(num_resamples) N = len(Y) r = np.random.randint(N, size=(num_resamples, N)) for i in range(num_resamples): r_i = r[i, :] s[i] = sobol_first(Y[r_i], X[r_i], m) return norm.ppf(0.5 + conf_level / 2) * s.std(ddof=1)
[docs]def cli_parse(parser): parser.add_argument('-X', '--model-input-file', type=str, required=True, default=None, help='Model input file') parser.add_argument('-r', '--resamples', type=int, required=False, default=10, help='Number of bootstrap resamples for \ Sobol confidence intervals') return parser
[docs]def cli_action(args): problem = read_param_file(args.paramfile) Y = np.loadtxt(args.model_output_file, delimiter=args.delimiter, usecols=(args.column,)) X = np.loadtxt(args.model_input_file, delimiter=args.delimiter, ndmin=2) if len(X.shape) == 1: X = X.reshape((len(X), 1)) analyze(problem, X, Y, num_resamples=args.resamples, print_to_console=True, seed=args.seed)
if __name__ == "__main__": common_args.run_cli(cli_parse, cli_action)