Minimalist's Kalman Filter Derivation, Part I
Break Yang August 03, 2020 #robotics #control #statisticsMotivation
State estimation has many applications in general robotics, for example autonomous driving localization and environment prediction. Kalman filter is a classical yet powerful algorithm that tackles such problem beautifully. Although there are already many articles, textbooks and papers on how to derive the algorithm, I found most of them too heavy on the theoretical side and might be hard for a firsttime learner who comes from an engineering background to follow. Therefore, I would shamelessly attempt to fill this hole with a series of posts.
This is going to be the first post of the series that only focuses on the 1 dimensional case. Future posts will talk about the multivariate version of Kalman filter.
Spoiler: there will be a lot of math equations. But rest assured, nothing will exceed the level of basical calculus arithmetics.
Single Step Prediction
Let's say we have a one dimensional linear Markovian System, whose transition function is know. This means

The state of the system can be represented as a single scalar. Let's denote it as $x$.

The transition function is a linear function, where the next state only depends on the current state. Therefore we can write the transition function as
$$ x_{t+1} = a \cdot x_t $$
Suppose you have an estimation of $x_t$ in the form of a Gaussian distribution
$$ x_t \sim N(\hat{x}_t, \sigma_t^2) $$
Based on that, what is your besteffort guess about the next state $x_{t+1}$? First of all, the reason that we can make such a prediction of the next state is because the transition function actually reveals the relationship between $x_t$ and $x_{t+1}$, which happens to be linear in this case. I know that intuitively, you would guess the answer immediately:
$$ x_{t+1} \sim N(a\hat{x}_t, a^2\sigma_t^2) $$
And that is the correct answer. But how do you prove that? Or more generally, if we have a random variable $X \sim N(\mu, \sigma^2)$, can we prove that another random variable that satisfies $Y = aX$ actually follows the distribution $Y \sim N(a\mu, a^2\sigma^2)$?
Let
$$ \phi(x) = \frac{1}{\sqrt{2\pi}} \cdot e^{\frac{1}{2}x^2} $$
be the pdf (probability density function) of a standard gaussian distribution $N(0, 1)$. It is easy to derive that the pdf of a general gaussian distribution $N(\mu, \sigma^2)$ that $X$ follows would be
$$ f_X(x) = \frac{1}{\sqrt{2\pi}\sigma} \cdot e^{\frac{1}{2\sigma^2}(x\mu)^2} = \frac{1}{\sigma}\phi \left( \frac{x  \mu}{\sigma} \right) $$
Using the trick called differential form, the probability of $X$ taking a specific value $x$ is^{1}
$$ \mathbb{P} [X = x] = f_X(x) \mathrm{d}x = \frac{1}{\sigma}\phi \left( \frac{x  \mu}{\sigma} \right) \mathrm{d} x $$
Okay, so what does pdf of $Y$ (i.e. $\mathbb{P}(Y = y)$) look like? It turns out that we can easily derive that with a bit transformation:
$$ \begin{darray}{rcl} \mathbb{P} \left[ Y = y \right] &=& \mathbb{P} \left[ X = \frac{y}{a} \right] \\\ &=& f_X \left(\frac{y}{a}\right) \mathrm{d}x \\\ &=& f_X \left(\frac{y}{a}\right) \frac{\mathrm{d}y}{a} \\\ &=& \frac{1}{\sigma}\phi \left( \frac{\frac{y}{a}  \mu}{\sigma} \right) \frac{\mathrm{d}y}{a} \\\ &=& \frac{1}{a\sigma}\phi \left( \frac{y  a\mu}{a\sigma} \right) \mathrm{d}y \end{darray} $$
This basically shows that $Y$'s pdf is nothing but the pdf of $N(a\mu, a^2\sigma^2)$, hence conclude the proof.
An intuitve perspective here that helps understanding that is $f_X(x) \mathrm{d}x$ is actually the area, whose fundamental unit is probability!
The above proved conclusion enables us to solve the prediction problem at the very beginning of this section,
$$ \textrm{based on } x_{t+1} = a \cdot x_t \textrm{ and } x_t \sim N(\hat{x}_t, \sigma_t^2) $$
$$ \textrm{we can predict } x_{t+1} \sim N(a\hat{x}_t, a^2\sigma_t^2) $$
equivalently, this means we can obtain estimation of $x_{t+1}$ (without observing it) as
$$ \begin{cases} \hat{x}_{t+1} &=& a\hat{x}_t \\ \sigma_{t+1} &=& a^2 \sigma_t^2 \end{cases} $$
Uncertainty in Transition Function
Now it is time to introduce one variation on top of the simplest case we discussed in the previous section. In reality the transition is usually not perfect, which means that there is an error associated with it. Mathematically, it means
$$ x_{t+1} = a \cdot x_t + e_t $$
As usual for simplicity we assume the error is a zeromean random variable follows a Gaussian distribution, i.e.
$$ e_t \sim N \left(0, \sigma_{e_t}^2 \right) $$
How should we revise our prediction under such condition? Remember we are still solving the following question  if we already have an estimation of $x_t$ as
$$ x_t \sim N(\hat{x}_t, \sigma_t^2) $$
what is a good estimation of $x_{t+1}$, given that we know (although not precisely in this case) the transition function?
Here we are going to introduce another useful idea  the generative model. The generative model basically describes the procedure to get a sample value of a random variable. In this particular case, the generative model of $x_{t+1}$ consits of:
 Sample $a \cdot x_t$ out of the distribution $N(a\hat{x}_t, a^2\sigma_t^2)$ (Note that this is the conclusion from the previous section)
 Sample $e_t$ out of the distribution $N(0, \sigma_{e_t}^2)$
 Construct $x_{t+1}$ by adding the two sampled values up
So you can see that the generative model is basically an interpretation of the problem formulation, provides no new knowledge at all. However, with such interpretation it is clear to see that $x_{t+1}$ as a random variable is basically the sum of two independently distributed gaussian random variables!
I will cheat here by referring to the generating function based proof from wikipedia. As you have probably guessed, the conclusion is that
$$ \textrm{if i.i.d } \begin{cases} X &\sim N(\mu_X, \sigma_X^2) \\ Y &\sim N(\mu_Y, \sigma_Y^2) \end{cases} \textrm{ then } Z = X + Y \sim N(\mu_X + \mu_Y, \sigma_X^2 + \sigma_Y^2) $$
By plugging in our generative model, we obtain
$$ x_{t+1} \sim N(a\hat{x}_t, a^2\sigma_t^2 + \sigma_{e_t}^2) $$
which means such good prediction would be
$$ \begin{cases} \hat{x}_{t+1} &=& a\hat{x}_t \\ \sigma_{t+1} &=& a^2 \sigma_t^2 + \sigma_{e_t}^2 \end{cases} $$
Yep, just add the variance of the error to the estimation of variance. Pretty simple, right?
Let There Be Observations
So we know how to predict $x_{t+1}$ given $x_t$, which is great. This means that if we happen to know the initial state of the system, $x_0$, we can start to predict $x_1$, and then $x_2$, ..., till any $x_t$, which will be
$$ \begin{cases} \hat{x}_t &=& a^t\hat{x}_t \\ \sigma_t &=& a \cdot (a \cdot ( a \cdots) + \sigma_{e_{t2}}^2) + \sigma_{e_{t1}}^2 \end{cases} $$
As we can see, there is one fatal problem in the above prediction. As $t$ grows, our estimation will be less precise because the variance is going to grow very quickly. This is because ecah time we make prediction for one more step, the variance of error will be added to it. To see it more clearly, when $a=1$, we will have
$$ \sigma_t = \sigma_{e_0}^2 + \sigma_{e_1}^2 + \cdots + \sigma_{e_{t1}}^2 $$
It is easy to understand this error accumulation intuitively. As we make more predictions, we are not getting new information about the system. Think about it  if you only know what a cat looks like when it is 1week old, how can you precisely predict how it looks like when he is 3 years old? If you only know your weight before COVID19 keeping us at home, how do you precisely estimate your current weight? The key here is that you need constant feedback to guide your estimation when it gets
Okay let's take one more step in the problem formulation, so that it will be more realistic. Now we are allowed to take measurement of the system via an observation function. In the weight example, this translates to you are allowed to weigh yourself with an electric scale every now and then. It seems that with the method to take measurement, we do not need to estimate the state anymore, we can simply observe the readings and get the precise value! Except for that in reality the measurement is usually not always accurate. Therefore, the observation function has an associated error as well. Assuming a linear observation function, we will have the readings mathematically as:
$$ y_t = h_t \cdot x_t + r_t, \textrm{ where } r_t \sim N \left(0, \sigma_{r_t} \right) $$
Note that when you take measurement at time $t + 1$, you can directly observe $y_{t+1}$ (though $y_{t+1}$ is not $x_{t+1}$, and the latter is what we want to estimate). The question now becomes: how to make a good estimation about $x_{t+1}$, given
 A good estiamtion of the previous state $x_t$, and
 The current measurement reading $y_t$
Let's first find the generative model interpretation of this. We can see that $y_{t+1}$ is generated in the following 3 steps:
 Sample $x_{t+1} = a \cdot x_t + e_t$ out of the distribution $N(a\hat{x}_t, a^2\sigma_t^2 + \sigma_{e_t}^2)$ (Note that this is the conclusion from the previous section)
 Sample $r_{t+1}$ out of the distribution $N(0, \sigma_{r_{t+1}}^2)$
 Directly compute $y_{t+1} = h_{t+1} \cdot x_{t+1} + r_{t+1}$
Let's stop for a while to take a closer look at the above generative model. The distribution $N(a\hat{x}_t, a^2\sigma_t^2 + \sigma_{e_t}^2)$ comes from the conclusion of the previous section, which represents the best^{2} estimation of $x_{t+1}$ we can get based pure prediction. The mean and variance of this distribution will be used quite a lot in the derivation below, so it is good to give it some name. Let
$$ \begin{cases} x'_{t+1} &=& a\hat{x}_t \\ \sigma'^2_{t+1} &=& a^2 \sigma_t^2 + \sigma_{e_t}^2 \end{cases} $$
Note that both $x'_{t+1}$ and $sigma'_{t+1}$ are determinsitic values, i.e. neither of them is random variable.
I am being informal here as we haven't formally defined what the best estimation means. We will likely get to this topic in the future, so bear with me for now.
Although the above generative model is about generating $y_{t+1}$, it is $x_{t+1}$ that we actually want to estimate. We can do this by deriving the pdf of $x_{t+1}$. The following derivation will likely seem very tedious, but I will try to be clear on each step and trust me, this will be the last challenge in this post!
Given that we have observed $y_{t+1} = y$, what is the probability of $x_{t+1} = x$? Such probability can be written as
$$ \forall x, \mathbb{P} [ x_{t+1} = x \mid y_{t+1} = y] = f_{x_{t+1}}(x) \mathrm{d}x $$
where $f_{x_{t+1}}(x)$ is the unknown (yet) pdf of $x_{t+1}$ that we want to derive. Also note that there is $\forall x$ in the statement, which is very important. It means that the equation holds for every single $x$.
By applying Bayes's law, the left hand side can also be transformed as
$$ \begin{darray}{rcl} \forall x, \mathbb{P} [ x_{t+1} = x \mid y_{t+1} = y] &=& \frac{\mathbb{P}[y_{t+1}=y \mid x_{t+1} = x] \mathbb{P}[x_{t+1} = x]}{\mathbb{P}[y_{t+1} = y]} \\ &=& \frac{\mathbb{P}[r_{t+1} = y  h_{t+1}x] \mathbb{P}[x_{t+1} = x]}{\mathbb{P}[y_{t+1} = y]} \end{darray} $$
So there are 3 items on the right hand side. Let's crack them one by one.
The simplest one here is $\mathbb{P}[y_{t+1} = y]$. Since it does not depend on $x$, this can be just written as
$$ \mathbb{P}[y_{t+1} = y] = \mathrm{Const} \cdot dy $$
Next comes $\mathbb{P}[x_{t+1} = x]$, without conditioning on the value of $y_{t+1}$. This is the pure prediction we discussed above, which $ \sim N(x'_{t+1}, \sigma'^2_{t+1})$. Therefore it is simply
$$ \mathbb{P}[x_{t+1} = x] = \mathrm{Const} \cdot \exp\left(\frac{(xx'_{t+1})^2}{2\sigma'^2_{t+1}}\right) \mathrm{d} x $$
The last one $\mathbb{P}[r_{t+1} = y  h_{t+1}x]$ is about $r_{t+1}$, which happens to be following a Gaussian distirbution as well (even better, the mean is zero)! This means
$$ \begin{darray}{rcl} \mathbb{P}[r_{t+1} = y  h_{t+1}x] &=& \mathrm{Const} \cdot \exp \left( \frac{(y  h_{t+1}x)^2}{2\sigma^2_{r_{t+1}}} \right) \mathrm{d}r \\ &=& \mathrm{Const} \cdot \exp \left( \frac{(y  h_{t+1}x)^2}{2\sigma^2_{r_{t+1}}} \right) (\mathrm{d}y  h_{t+1}\mathrm{d}x) \end{darray} $$
Note $\mathrm{d}r$ can be written as the form above because of differential form arithmetics. It is good to understand the rules behind them, but when you get familiar with the rules, they are just no more strange than the rules you use to take derivatives.
Therefore, take the above 3 expanded components and plug them back, and keep in mind that by differential form rule $\mathrm{d}x \wedge \mathrm{d}x = 0$, we have
$$ \begin{darray}{rcl} \forall x, && f_{x_{t+1}}(x) \mathrm{d}x \\ &=& \mathbb{P} [ x_{t+1} = x \mid y_{t+1} = y] \\ &=& \frac{\mathrm{Const} \cdot \exp \left( \frac{(xx'_{t+1})^2}{2\sigma'^2_{t+1}} \frac{(y  h_{t+1}x)^2}{2\sigma^2_{r_{t+1}}} \right) (\mathrm{d}x \wedge \mathrm{d} y  h_{t+1} \mathrm{d}x \wedge \mathrm{d}x)} {\mathrm{Const} \cdot dy} \\ &=& \mathrm{Const} \cdot \exp \left( \frac{(xx'_{t+1})^2}{2\sigma'^2_{t+1}} \frac{(y  h_{t+1}x)^2}{2\sigma^2_{r_{t+1}}} \right) \mathrm{d} x \end{darray} $$
Let's then take a closer look at the terms inside $\exp()$
$$ \begin{darray}{rcl} &&\frac{(xx'_{t+1})^2}{2\sigma'^2_{t+1}} \frac{(y  h_{t+1}x)^2}{2\sigma^2_{r_{t+1}}} \\ &=& \frac{(\sigma^2_{r_{t+1}} + h^2_{t+1}\sigma'^2_{t+1})x^2  2(\sigma^2_{r_{t+1}} x'_{t+1} + \sigma'^2_{t+1}h_{t+1}y)x + \mathrm{Const}} {2\sigma'^2_{t+1}\sigma^2_{r_{t+1}}} \\ &=& \frac{1}{2} \frac{x^2  2\frac{\sigma^2_{r_{t+1}} x'_{t+1} + \sigma'^2_{t+1}h_{t+1}y}{\sigma^2_{r_{t+1}} + h^2_{t+1}\sigma'^2_{t+1}}x} {\frac{\sigma'^2_{t+1}\sigma^2_{r_{t+1}}}{\sigma^2_{r_{t+1}} + h^2_{t+1}\sigma'^2_{t+1}}} + \mathrm{Const} \\ &=&  \frac{1}{2}\frac{\left(x  \frac{\sigma^2_{r_{t+1}} x'_{t+1} + \sigma'^2_{t+1}h_{t+1}y}{\sigma^2_{r_{t+1}} + h^2_{t+1}\sigma'^2_{t+1}} \right)^2} {\frac{\sigma'^2_{t+1}\sigma^2_{r_{t+1}}}{\sigma^2_{r_{t+1}} + h^2_{t+1}\sigma'^2_{t+1}}} + \mathrm{Const} \end{darray} $$
Plug this back in the above equation we have
$$ \begin{darray}{rcl} \forall x, f_{x_{t+1}}(x) \mathrm{d}x &=& \mathbb{P} [ x_{t+1} = x \mid y_{t+1} = y] \\ &=& \mathrm{Const} \cdot \exp \left(  \frac{1}{2}\frac{\left(x  \frac{\sigma^2_{r_{t+1}} x'_{t+1} + \sigma'^2_{t+1}h_{t+1}y}{\sigma^2_{r_{t+1}} + h^2_{t+1}\sigma'^2_{t+1}} \right)^2} {\frac{\sigma'^2_{t+1}\sigma^2_{r_{t+1}}}{\sigma^2_{r_{t+1}} + h^2_{t+1}\sigma'^2_{t+1}}} \right) \mathrm{d} x \end{darray} $$
Let's remove $\mathrm{d}x$ from both side, and we have
$$ \forall x, f_{x_{t+1}}(x) = = \mathrm{Const} \cdot \exp \left(  \frac{1}{2} \frac{\left(x  \frac{\sigma^2_{r_{t+1}} x'_{t+1} + \sigma'^2_{t+1}h_{t+1}y_{t+1}}{\sigma^2_{r_{t+1}} + h^2_{t+1}\sigma'^2_{t+1}} \right)^2} {\frac{\sigma'^2_{t+1}\sigma^2_{r_{t+1}}}{\sigma^2_{r_{t+1}} + h^2_{t+1}\sigma'^2_{t+1}}} \right) $$
Note that since $y$ is basically the value of $y_{t+1}$, it is replaced with $y_{t+1}$.
This means that $x_{t+1}$ follows Gaussian distirbution! We can even directly tell what is the mean and what is the variance of the estimation from the above formula, i.e.
$$ \begin{cases} \hat{x}_{t+1} &=& \frac{\sigma^2_{r_{t+1}} x'_{t+1} + \sigma'^2_{t+1}h_{t+1}y_{t+1}}{\sigma^2_{r_{t+1}} + h^2_{t+1}\sigma'^2_{t+1}} \\ \sigma^2_{t+1} &=& \frac{\sigma'^2_{t+1}\sigma^2_{r_{t+1}}}{\sigma^2_{r_{t+1}} + h^2_{t+1}\sigma'^2_{t+1}} \end{cases} $$
Making Sense of the Result
The above answer still looks very complicated, and let me try to interpret it in a more intuitive way in this section.
Let's start with the mean
$$ \begin{darray}{rcl} \hat{x}_{t+1} &=& \frac{\sigma^2_{r_{t+1}} x'_{t+1} + \sigma'^2_{t+1}h_{t+1}y_{t+1}}{\sigma^2_{r_{t+1}} + h^2_{t+1}\sigma'^2_{t+1}} \\ &=& \frac{\sigma^2_{r_{t+1}}}{\sigma^2_{r_{t+1}} + h^2_{t+1}\sigma'^2_{t+1}} \cdot x'_{t+1} + \frac{\sigma'^2_{t+1}h_{t+1}}{\sigma^2_{r_{t+1}} + h^2_{t+1}\sigma'^2_{t+1}} \cdot y_{t+1} \\ &=& \frac{\sigma^2_{r_{t+1}}}{\sigma^2_{r_{t+1}} + h^2_{t+1}\sigma'^2_{t+1}} \cdot x'_{t+1} + \frac{\sigma'^2_{t+1}h^2_{t+1}}{\sigma^2_{r_{t+1}} + h^2_{t+1}\sigma'^2_{t+1}} \cdot \frac{y_{t+1}}{h_{t+1}} \\ &=& K \cdot x'_{t+1} + (1  K) \cdot \frac{y_{t+1}}{h_{t+1}} \end{darray} $$
Note that in the above formula, we let
$$ K = \frac{\sigma^2_{r_{t+1}}}{\sigma^2_{r_{t+1}} + h^2_{t+1}\sigma'^2_{t+1}} $$
$K$ is clearly a number between $0$ and $1$. This means that the mean of estimation of $x_{t+1}$ is actually a weighted combination of $x'_{t+1}$ and $y_{t+1} / h_{t+1}$. It is worth noting that
 $x'_{t+1}$ is the best guess you can have based on pure prediction
 $y_{t+1} / h_{t+1}$ is the best guess you can have based on pure observation
So this is basically about trusting both of those two evidences with a grain of salt. And how much you trust each of them depends on the variance of each guess. The bigger variance, the less trustworthy. Very reasonable, right?
What about the estimated variance of $x_{t+1}$? As we have defined $K$, it can be written as
$$ \sigma^2_{t+1} = K \cdot \sigma'^2_{t+1} $$
This is intuitively just updating the pure prediction based variance estimation as we have observation now. Note that becauese $K < 1$, the final estimated variance is going to be smaller than the actual pure prediction based estimated variance!
So at this moment we can now summarize the procedure of Kalman Filter update, i.e. how to obtain $t+1$step estimation based on $t$step estimation and new observation.
Step I: Compute the pure prediction based estimation.
$$ \begin{cases} x'_{t+1} &=& a\hat{x}_t \\ \sigma'^2_{t+1} &=& a^2 \sigma_t^2 + \sigma_{e_t}^2 \end{cases} $$
Step II: Compute the combination weight $K$, which is often called the Kalman Gain.
$$ K = \frac{\sigma^2_{r_{t+1}}}{\sigma^2_{r_{t+1}} + h^2_{t+1}\sigma'^2_{t+1}} $$
Step III: Use Kalman Gain $K$ and observation $y_{t+1}$ to update the pure prediction based estimation.
$$ \begin{cases} \hat{x}_{t+1} &=& K \cdot x'_{t+1} + (1  K) \cdot \frac{y_{t+1}}{h_{t+1}} \\ \sigma^2_{t+1} &=& K \cdot \sigma'^2_{t+1} \end{cases} $$
Summarry
This post demonstrated the derivation of 1D Kalman Filter, and also slightly touched the intuitive interpretation of it. Also, I think many of the techniques used here such as generative model and differential forms can find their applications in many other situations.
However, in reality, 1D Kalman Filter is rarely useful enough. This post should have prepared you for the next journey  multivariate Kalman Filter. Stay tuned!