Kalman filter (Kalman 1960) 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).

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.

Proofs

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.

Proof Kalman 1d with constant measurement uncertainty and no process noise proof

Plot of the convergence.

Proof Kalman 1d with constant measurement uncertainty and no process noise plot

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.

Proof Kalman 1d with constant measurement uncertainty and constant process noise proof

Plot of the convergence.

Proof Kalman 1d with constant measurement uncertainty and constant process noise plot

References

Appendix

Example Implementation

Below is simplistic implementation used to generate plots presented.

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)


r = 1
q = 0
p = 1
m = 1
ps = []
ks = []
xs = []
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()

Vaclav Kosar

Constant 1D Kalman Filter Is Exponential Or Cumulative Average