# Noise Whiteness hypothesis in Kalman filtering

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• Unconscious
In summary, the conversation discusses the role of the white noise hypothesis in the mathematical treatment of the Kalman filter. The speaker is unsure of its importance and asks for clarification on where the mathematical proof might fail if the hypothesis is removed. A link to a thesis on the Kalman filter is provided but does not answer the question. The speaker then explains their doubts based on formulas in an article and asks for further clarification on the role of white noise. The expert summarizer explains that white noise is uncorrelated from sample to sample and is used to improve the accuracy of the filter. The speaker then asks for clarification on the mathematical aspect of this and the expert summarizer confirms that without the white noise hypothesis, the equality in the formula would be
Unconscious
In Kalman filter mathematical treatment I have always read that a foundamental hypothesis is represented by the whiteness of the process noise. I have tried to do again the mathematical steps in the Kalman filter derivation but I can't see where such hypothesis is crucial.
Could you help me showing where the mathematical proof fails if I remove such hypothesis?
Thanks.

Anyway, I try to share with you my doubt more precisely.

https://arxiv.org/pdf/1712.01406.pdf

These two formulas have been obtained without assumptions on the process noise. Then, again without assumptions on the process noise, the author arrives at formula (5), in which appears the probability density ##p(x_k | x_{k-1})##. Taking into account the first equation of the system (1), now he says that if we suppose the gaussianity and the whiteness of the process noise, we can then write such probability density as ##\mathcal{N}(f(x_{k-1}),Q)##.
It is exactly in this point that my doubt arises: it seems to me that I could have written ##p(x_k | x_{k-1})=\mathcal{N}(f(x_{k-1}),Q)## even if I had only assumed gaussianity of the process noise, without whiteness, because ##x_k## is conditioned only by ##x_{k-1}## and not also by ##q_{k-1},q_{k-2},...##.

By definition, white noise is noise that is uncorrelated from sample to sample.

Suppose the actual noise of your sensor was totally correlated - on the first sample, you pick a gaussian, and that's the noise the entire time.

Let's say the first pick is a positive noise.

Then a better algorithm would be able to learn that the noise picked is positive (e.g. it notices that 2/3 of the measurements are higher than its estimate), and could adjust to start making lower estimates given the measurements.

On each step, in some sense you could say ok, I don't know anything about the noise, so since I'm ignorant about it it's a gaussian as far as I can tell, and this kind of works (you can use a Kalman filter in a variety of real world situations that don't fit the theoretical framework and it works pretty well still). But it's not the best estimate you can form if you *are* informed about the noise.This is just a modeling question. If you make 8 samples and all of them return higher measurements than you expect, do you chalk it up to bad luck, bad measuring, or a bad understanding of what the process is? The assumption of the noise being white means the Kalman filter says it's bad luck.

This is clear to me, thank you.
What is not clear to me is the pure math, in particular the step highlighted in my last post.
So, are you saying to me that if the noise is not white, then I can’t correctly write the equality ##p(x_k|x_{k-1})=\mathcal{N}(f(x_{k-1}),Q)##? In other words, this equality would be a logic error?

## 1. What is the Noise Whiteness hypothesis in Kalman filtering?

The Noise Whiteness hypothesis in Kalman filtering is a key assumption that states the measurement noise in a system is uncorrelated and has a constant variance over time. This means that the noise is "white" or random, and does not have any patterns or trends that can be predicted or modeled.

## 2. Why is the Noise Whiteness hypothesis important in Kalman filtering?

The Noise Whiteness hypothesis is crucial in Kalman filtering because it allows for accurate estimation and prediction of a system's state. By assuming that the measurement noise is uncorrelated and has a constant variance, the Kalman filter can effectively remove the noise from the system and provide more accurate and reliable estimates.

## 3. What happens if the Noise Whiteness hypothesis is violated?

If the Noise Whiteness hypothesis is violated, it means that the measurement noise in the system is not truly random or uncorrelated. This can lead to inaccurate estimates and predictions, as the Kalman filter will not be able to effectively remove the noise from the system. In some cases, violating this hypothesis can even cause the Kalman filter to diverge and produce unstable results.

## 4. How can the Noise Whiteness hypothesis be tested in Kalman filtering?

The Noise Whiteness hypothesis can be tested by analyzing the residuals, or the difference between the actual measurements and the estimated values from the Kalman filter. If the residuals show a pattern or trend, it indicates that the measurement noise is not truly random and the Noise Whiteness hypothesis may be violated.

## 5. Are there any limitations to the Noise Whiteness hypothesis in Kalman filtering?

Yes, there are some limitations to the Noise Whiteness hypothesis in Kalman filtering. It assumes that the measurement noise is Gaussian, which may not always be the case in real-world systems. Additionally, if the noise in the system is not truly white, it can lead to inaccurate estimates and predictions. Therefore, it is important to carefully consider the validity of this hypothesis when using Kalman filtering in practical applications.

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