粒子滤波为什么要进行重采样

粒子滤波是一种基于蒙特卡洛方法的滤波算法，它通过在状态空间中产生一组粒子来近似概率分布。在每个时间步骤中，粒子根据其权重被重新采样，以使更多的粒子聚集在高概率区域，从而提高估计的准确性。

如果不进行重采样，那么在滤波过程中，一些粒子的权重会变得非常小，甚至接近于零。这些粒子对于表示概率分布的样本集合来说是没有意义的，因为它们不再代表概率分布的任何部分。另一方面，一些粒子的权重会变得非常大，这些粒子会在下一个时间步骤中占据所有样本的大部分，从而导致估计的方差变大。

通过重采样，可以保留高权重粒子并且减少低权重粒子的数量，从而保持样本的多样性并且提高估计的准确性。

相关问题

粒子滤波四种基本重采样代码

粒子滤波中的重采样是非常重要的一步，可以使得粒子在一定程度上避免漂移。以下是四种基本的重采样代码：

1. Systematic Resampling（系统重采样）

def systematic_resampling(particles, weights):
    N = len(weights)
    positions = (np.arange(N) + np.random.uniform()) / N
    indexes = np.zeros(N, 'i')
    cumulative_sum = np.cumsum(weights)
    i, j = 0, 0
    while i < N and j < N:
        if positions[i] < cumulative_sum[j]:
            indexes[i] = j
            i += 1
        else:
            j += 1
    return particles[indexes]

2. Multinomial Resampling（多项式重采样）

def multinomial_resampling(particles, weights):
    indexes = np.zeros(len(weights), 'i')
    cumulative_sum = np.cumsum(weights)
    cumulative_sum[-1] = 1.  # avoid round-off errors
    for i in range(len(weights)):
        indexes[i] = np.searchsorted(cumulative_sum, np.random.uniform())
    return particles[indexes]

3. Residual Resampling（残余重采样）

def residual_resampling(particles, weights):
    N = len(weights)
    indexes = np.zeros(N, 'i')
    # determine integer parts for each weight
    num_copies = (N * weights).astype(int)
    k = 0
    for i in range(N):
        for _ in range(num_copies[i]):
            indexes[k] = i
            k += 1
    # use multinomial resample on the residual to fill up the rest.
    residual = weights - num_copies     # get fractional part
    residual /= sum(residual)           # normalize
    cumulative_sum = np.cumsum(residual)
    cumulative_sum[-1] = 1.  # avoid round-off errors
    for i in range(k, N):
        indexes[i] = np.searchsorted(cumulative_sum, np.random.uniform())
    return particles[indexes]

4. Stratified Resampling（分层重采样）

def stratified_resampling(particles, weights):
    N = len(weights)
    # make N subdivisions, and chose a random position within each one
    positions = (np.random.uniform(0, 1, N) + range(N)) / N
    indexes = np.zeros(N, 'i')
    cumulative_sum = np.cumsum(weights)
    i, j = 0, 0
    while i < N and j < N:
        if positions[i] < cumulative_sum[j]:
            indexes[i] = j
            i += 1
        else:
            j += 1
    return particles[indexes]

粒子滤波四种基本重采样实例代码

以下是基于Python的四种基本重采样算法的代码实现：

1. 系统重采样

import numpy as np

def system_resampling(particles, weights):
    n_particles = len(weights)
    cum_weights = np.cumsum(weights)
    cum_weights[-1] = 1.0
    indexes = np.zeros(n_particles, dtype=int)
    u0 = np.random.uniform(0, 1/n_particles)
    j = 0
    for i in range(n_particles):
        u = u0 + i/n_particles
        while u > cum_weights[j]:
            j += 1
        indexes[i] = j
    return particles[indexes]

2. 伯努利重采样

import numpy as np

def bernoulli_resampling(particles, weights):
    n_particles = len(weights)
    cum_weights = np.cumsum(weights)
    cum_weights[-1] = 1.0
    indexes = np.zeros(n_particles, dtype=int)
    u = np.random.uniform(0, 1/n_particles)
    j = 0
    for i in range(n_particles):
        u += np.random.uniform(0, 1/n_particles)
        while u > cum_weights[j]:
            j += 1
        indexes[i] = j
    return particles[indexes]

3. 残差重采样

import numpy as np

def residual_resampling(particles, weights):
    n_particles = len(weights)
    indexes = np.zeros(n_particles, dtype=int)
    cum_weights = np.cumsum(weights)
    cum_weights[-1] = 1.0
    step = 1.0/n_particles
    u = np.random.uniform(0, step)
    j = 0
    for i in range(n_particles):
        while u > cum_weights[j]:
            j += 1
        indexes[i] = j
        u += step
    return particles[indexes]

4. 分层重采样

import numpy as np

def stratified_resampling(particles, weights):
    n_particles = len(weights)
    indexes = np.zeros(n_particles, dtype=int)
    cum_weights = np.cumsum(weights)
    cum_weights[-1] = 1.0
    step = 1.0/n_particles
    u = np.random.uniform(0, step)
    j = 0
    for i in range(n_particles):
        while u > cum_weights[j]:
            j += 1
        indexes[i] = j
        u += step
    return particles[indexes]

注意：以上代码中，`particles`表示粒子集合，`weights`表示对应的权重。在系统重采样、伯努利重采样和残差重采样中，`cum_weights`表示累计权重，`indexes`表示选择的粒子下标。在分层重采样中，`step`表示步长。