估计卡尔曼滤波器的置信区间

Question

我一直在努力实现一个卡尔曼滤波器，用来在一个二维数据集中寻找异常值。这和我在这里找到的一个很棒的帖子非常相似。接下来的步骤，我想预测置信区间（比如说95%的置信度，表示下限和上限的值），也就是我预测下一个值会落在什么范围内。所以除了下面的那条线，我还想生成另外两条线，表示有95%的置信度下一个值会高于下限或低于上限。

我想我需要使用卡尔曼滤波器每次生成预测时返回的不确定性协方差矩阵（P），但我不太确定这样做是否正确。任何指导或参考资料都将非常感谢！

Python中的卡尔曼二维滤波器

上面帖子中的代码生成了一组随时间变化的测量值，并使用卡尔曼滤波器来平滑结果。

import numpy as np
import matplotlib.pyplot as plt

def kalman_xy(x, P, measurement, R,
              motion = np.matrix('0. 0. 0. 0.').T,
              Q = np.matrix(np.eye(4))):
    """
Parameters:    
x: initial state 4-tuple of location and velocity: (x0, x1, x0_dot, x1_dot)
P: initial uncertainty convariance matrix
measurement: observed position
R: measurement noise 
motion: external motion added to state vector x
Q: motion noise (same shape as P)
"""
return kalman(x, P, measurement, R, motion, Q,
              F = np.matrix('''
                  1. 0. 1. 0.;
                  0. 1. 0. 1.;
                  0. 0. 1. 0.;
                  0. 0. 0. 1.
                  '''),
              H = np.matrix('''
                  1. 0. 0. 0.;
                  0. 1. 0. 0.'''))

def kalman(x, P, measurement, R, motion, Q, F, H):
    '''
    Parameters:
    x: initial state
    P: initial uncertainty convariance matrix
    measurement: observed position (same shape as H*x)
    R: measurement noise (same shape as H)
    motion: external motion added to state vector x
    Q: motion noise (same shape as P)
    F: next state function: x_prime = F*x
    H: measurement function: position = H*x

    Return: the updated and predicted new values for (x, P)

    See also http://en.wikipedia.org/wiki/Kalman_filter

    This version of kalman can be applied to many different situations by
    appropriately defining F and H 
    '''
    # UPDATE x, P based on measurement m    
    # distance between measured and current position-belief
    y = np.matrix(measurement).T - H * x
    S = H * P * H.T + R  # residual convariance
    K = P * H.T * S.I    # Kalman gain
    x = x + K*y
    I = np.matrix(np.eye(F.shape[0])) # identity matrix
    P = (I - K*H)*P

    # PREDICT x, P based on motion
    x = F*x + motion
    P = F*P*F.T + Q

    return x, P

def demo_kalman_xy():
    x = np.matrix('0. 0. 0. 0.').T 
    P = np.matrix(np.eye(4))*1000 # initial uncertainty

    N = 20
    true_x = np.linspace(0.0, 10.0, N)
    true_y = true_x**2
    observed_x = true_x + 0.05*np.random.random(N)*true_x
    observed_y = true_y + 0.05*np.random.random(N)*true_y
    plt.plot(observed_x, observed_y, 'ro')
    result = []
    R = 0.01**2
    for meas in zip(observed_x, observed_y):
        x, P = kalman_xy(x, P, meas, R)
        result.append((x[:2]).tolist())
    kalman_x, kalman_y = zip(*result)
    plt.plot(kalman_x, kalman_y, 'g-')
    plt.show()

demo_kalman_xy()

置信区间预测模型异常值检测数据平滑卡尔曼滤波器统计推断不确定性协方差矩阵二维数据集

估计卡尔曼滤波器的置信区间

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