Constant 1D Kalman Filter Is Exponential Or Cumulative Average

Kalman filter (Kalman 1960 paper) also known as linear quadratic estimation (LQE) is an iterative algorithm that uses noisy measurements to estimate values and variance of unknown variables. The Kalman filter allows incorporation of known state space behaviour (e.g. momentum of physical particle) and outside-the-model estimated variance of sensor measurement (measurement uncertainty) and unknown factors (process noise).

Kalman filter can be used in to keep a system in a state of control. Read more about application of Kalman filter in PID Controller.

Kalman Filter vs Exponential Average vs Cumulative Average

This blog post proves that Kalman filter in 1D with constant measurement uncertainty and process noise asymptotically behaves as:

• cumulative average in case of zero process noise
• exponential average in case of non zero process noise

The proof relies on Kalman filter asymptotically doesn’t depend on initial state. In general since Kalman filter equations are differentiable, it is reasonable to expect that above could be generalized to nearly-constant uncertainty and process noise.

Constant Measurement Uncertainty, No Process Noise

Below is the proof relies on good choice of initial value of Kalman variance P0 to simplify recursive equation to match cumulative average equation.

Plot of the convergence.

Constant Measurement Uncertainty, Constant Process Noise

Below is the proof relies on setting initial value of Kalman variance P0 such that Pk becomes constant for recursive equation to match exponential moving average equation.

Plot of the convergence.

Example Implementation

Below is simplistic implementation of Kalman filter in one dimension in Python used to generate plots presented above.

import matplotlib.pyplot as plt
import random
from statistics import mean
import pandas as pd


def current_k(p: float) -> float:
    return (p + q) / (p + q + r)


def next_p(p: float, k: float) -> float:
    return (1 - k) * (p + q)


def next_m(m, x, k: float) -> float:
    return m + k * (x - m)


# configuration of the Kalman filter
r = 1
q = 0
p = 1
m = 1

# xs is the measure input with noise
xs = []

# variables of the Kalman filter
# variance estimate
ps = []
# kalman gain
ks = []


# m is the smoothed output
ms = []
cumulative_avg = []
exponential_avg = []
count = 50

for i in range(count):
    xs.append(random.gauss(0, 1))


m = xs[0]

for i in range(count):
    k = current_k(p)
    ks.append(k)
    p = next_p(p, k)
    ps.append(p)
    m = next_m(m, xs[i], k)
    ms.append(m)
    cumulative_avg.append(mean(xs[:i+1]))


exponential_avg = pd.Series(xs).ewm(alpha=ks[-1]).mean()


plt.plot(ks, label='ks')
plt.plot(ps, label='ps')
plt.legend()
plt.show()

plt.plot(ms, label='kalman filter')
plt.plot(cumulative_avg, label='cumulative avg')
plt.plot(exponential_avg, label='exponential moving avg')
plt.legend()
plt.show()