Source code for quantecon.markov.estimate

import numpy as np
from numba import njit
from .core import MarkovChain
from .._gridtools import cartesian_nearest_index, cartesian

[docs]def estimate_mc(X): r""" Estimate the Markov chain associated with a time series :math:`X = (X_0, \ldots, X_{T-1})` assuming that the state space is the finite set :math:`\{X_0, \ldots, X_{T-1}\}` (duplicates removed). The estimation is by maximum likelihood. The estimated transition probabilities are given by the matrix :math:`P` such that :math:`P[i, j] = N_{ij} / N_i`, where :math:`N_{ij} = \sum_{t=0}^{T-1} 1_{\{X_t=s_i, X_{t+1}=s_j\}}`, the number of transitions from state :math:`s_i` to state :math:`s_j`, while :math:`N_i` is the total number of visits to :math:`s_i`. The result is returned as a `MarkovChain` instance. Parameters ---------- X : array_like A time series of state values, from which the transition matrix will be estimated, where `X[t]` contains the t-th observation. Returns ------- mc : MarkovChain A MarkovChain instance where `mc.P` is a stochastic matrix estimated from the data `X` and `mc.state_values` is an array of values that appear in `X` (sorted in ascending order). """ X = np.asarray(X) axis = 0 if X.ndim > 1 else None state_values, indices = np.unique(X, return_inverse=True, axis=axis) n = len(state_values) P = np.zeros((n, n)) # dtype=float to modify in place upon normalization P = _count_transition_frequencies(indices, P) P /= P.sum(1)[:, np.newaxis] mc = MarkovChain(P, state_values=state_values) return mc
@njit(cache=True) def _count_transition_frequencies(index_series, trans_counter): T = len(index_series) i = index_series[0] for t in range(1, T): j = index_series[t] trans_counter[i, j] += 1 i = j return trans_counter
[docs]def fit_discrete_mc(X, grids, order='C'): r""" Function that takes an arbitrary time series :math: `(X_t)_{t=0}^{T-1}` in :math: `\mathbb R^n` plus a set of grid points in each dimension and converts it to a MarkovChain by first applying discretization onto the grid and then estimation of the Markov chain. Parameters ---------- X: array_like(ndim=2) Time-series such that the t-th row is :math:`x_t`. It should be of the shape T x n, where n is the number of dimensions. grids: array_like(array_like(ndim=1)) Array of `n` sorted arrays. Set of grid points in each dimension Examples -------- >>> grids = (np.arange(3), np.arange(2)) >>> X = [(-0.1, 1.2), (2, 0), (0.6, 0.4), (1.0, 0.1)] >>> mc = fit_discrete_mc(X, grids) >>> mc.state_values array([[0, 1], [1, 0], [2, 0]]) >>> mc.P array([[0., 0., 1.], [0., 1., 0.], [0., 1., 0.]]) Returns ------- mc: MarkovChain An instance of the MarkovChain class constructed after discretization onto the grid. """ X_indices = cartesian_nearest_index(X, grids, order=order) mc = estimate_mc(X_indices) # Assign the visited states in the cartesian product as the state values prod = cartesian(grids, order=order) mc.state_values = prod[mc.state_values] return mc