Kalman filter Puns

I am developing a localisation model with EKF. I made the prediction to go with odometry from encoders and correction step is done with a sensor of indoor positioning system.

I have calculated the covariance matrix for prediction and measurement step in different scenarios (vehicle going straight, vehicle turning left, vehicle turning right, constant velocity). But, I don't know how to input these covariance matrices calculated for different scenarios. I want to know how in the world of Autonomous vehicle this is done... I can also provide more information regarding the work ....

Hey all, I have been told that for obtaining conditional forecasts when working with VAR models one often uses the Kalman filter to do so. Could someone maybe give me the intuition on why the Kalman filter is useful in this setting? I find it hard to wrap my head around this.

This seems like a stupid question, but I am struggling with this. I have a kalman filter implemented with a simple state variable vector, [x,y,u,v]. I only have observations of position but am using velocity as an output to control the system. My problem is that the velocity calculated in the state variable vector does not correspond to the derivative of position in time. I thought those were dynamically tied but apparently not? Furthermore, the position is a good estimate of my state, but the error between the derivative of position and the predicted velocity is creating positional error in my system which is unacceptable. Here is a an example plot of the data that I'm dealing with and a simplified python script that is my kalman filter.

https://i.imgur.com/5e3c4ut.png

https://pastebin.com/ji0nxJc8

I have built a flight controller for a quadcopter and for attitude estimation I have used a linear kalman filter.

I have noticed some strange behaviour, especially when large sudden attitude changes are made and I'm thinking maybe I need to use an EKF to avoid this behaviour.

What are the downsides of using a linear kalman filter on a quadcopter as opposed to an EKF?

I'm trying to create a simple Kalman Filter tracker using the openCV library. I based on tutorials for similiar goals in: http://opencvexamples.blogspot.com/2014/01/kalman-filter-implementation-tracking.html https://machinelearningspace.com/object-tracking-python/

However, I'm getting an issue with matrix sizes. I'm confident that I'm setting them right and I've been around this for hours without being able to fix it. Any suggestions?

cv::KalmanFilter KF(STATE_SIZE, MEAS_SIZE, CONTR_SIZE, CV_32F);

// Initial State Matrix
// [p_x]
// [p_y]
// [p_z]
// [v_x]
// [v_y]
// [v_z]
KF.statePre = cv::Mat::zeros(cv::Size(1, STATE_SIZE), CV_32FC1);
KF.statePre.at<float>(0) = odometry.x;
KF.statePre.at<float>(1) = odometry.y;
KF.statePre.at<float>(2) = odometry.z;

cv::setIdentity(KF.measurementMatrix);
cv::setIdentity(KF.processNoiseCov, cv::Scalar::all(1e-4));
cv::setIdentity(KF.measurementNoiseCov, cv::Scalar::all(10));
cv::setIdentity(KF.errorCovPost, cv::Scalar::all(.1));

// State Transition Matrix A
// [ 1  0   0   dT  0   0 ]
// [ 0  1   0   0   dT  0 ]
// [ 0  0   1   0   0   dT]
// [ 0  0   0   1   0   0 ]
// [ 0  0   0   0   1   0 ]
// [ 0  0   0   0   0   1 ]
KF.transitionMatrix = cv::Mat::zeros(cv::Size(STATE_SIZE, STATE_SIZE), CV_32FC1);
KF.transitionMatrix.at<float>(0) = 1.0;
KF.transitionMatrix.at<float>(3) = dT;
KF.transitionMatrix.at<float>(7) = 1.0;
KF.transitionMatrix.at<float>(10) = dT;
KF.transitionMatrix.at<float>(14) = 1.0;
KF.transitionMatrix.at<float>(17) = dT;
KF.transitionMatrix.at<float>(21) = 1.0;
KF.transitionMatrix.at<float>(28) = 1.0;
KF.transitionMatrix.at<float>(35) = 1.0;

// Control Input Matrix B
// [ dT^2/2   0        0   ]
// [    0   dT^2/2     0   ]
// [    0     0     dT^2/2 ]
// [   dT     0        0   ]
// [    0     dT       0   ]
// [    0     0        dT  ]
KF.controlMatrix = cv::Mat::zeros(cv::Size(STATE_SIZE, MEAS_SIZE), CV_32FC1);
KF.controlMatrix.at<float>(0) = pow(dT,2)/2;
KF.controlMatrix.at<float>(4) = pow(dT,2)/2;
KF.controlMatrix.at<float>(8) = pow(dT,2)/2;
KF.controlMatrix.at<fl

Hi,

I'm interviewing for a "robotics systems engineer" role. It's a small new team at a very large tech company that is starting to build robots. It's a pretty broad role, but luckily I have a lot of relevant experience after being the only electrical engineer at a few robotics startups over the last 4-5 years.

One thing I don't have any experience with is camera calibration, and also kalman filters. These are things the hiring manager specifically mentioned during the first phase of interviewing.

Can you recommend any good resources for learning how to do these? The role involves evaluating new sensors for their systems, bringing them up, troubleshooting, calibrating, and characterizing.

I'm guessing I'll need more of a practical understanding than a full in-depth theoretical understanding of these topics, but having a good fundamental background would be good too in case this comes up during the interview.

Are there any industry standard software tools that I should definitely know about?

Thanks!

Hello, recently I've had a course during my degree which focuses on Kalman filters for quite some time
I was interested to know if someone has some resources (possibly brief) about how to apply them for market price forecasting

Also interested to hear your personal opinions about

Preamble: For research purposes I built out a kalman filter stat arb. model inspired by Ernie Chans kalman filter mean reversion model. I then backtested it on a long-short bitcoin etherium portfolio. For a more in-depth breakdown of the strategy and concepts see: Chan, E., 2013. Algorithmic trading: winning strategies and their rationale (Vol. 625). John Wiley & Sons.

The model

The model uses a kalman filter regression to calculate a hedge ratio between bitcoin (BTC) and etherium (ETH). It then monitors the value of the hedge portfolio looking for moments of diversion to enter long or short positions. The test data was compiled BTC and ETH data in 4H time intervals spanning 1035 days.

The Backtest

a step by step procedure below:

1. Use kalman filter regression (as seen in EC's book) to calculate the hedge ratio between BTC and ETH

2. Calculate a spread as: S = BTC - (Hedge Ratio * ETH)

3. Calculate Z score of the Spread (S) using a rolling mean and std. (can use half life from kalman calcs or a set lookback period eg. 10)

4. Define long entry as -2, short entry as 2 and trade exit as 0

5. enter a long position when Z score <= -2,exit trade when Z score >= 0

6. enter a short Z score >= 2,exit trade when Z score <= 0

Figures and results

fig 1. Sample of Kalman spread Z score with trade entry

https://preview.redd.it/gq2aqemc5i871.png?width=1615&format=png&auto=webp&s=7c64c35f4b635c3ede36aea70b3a2a5c299eb5ca

fig 2. Sample of cumulative portfolio return with trade entry

https://preview.redd.it/80gl1gpm5i871.png?width=1627&format=png&auto=webp&s=36b6cc63c21a9f699a22de665f06fbef1c2de8f4

fig 3. Total Cumulative Return (1035 days of test data)

https://preview.redd.it/tqnc33406i871.png?width=1527&format=png&auto=webp&s=6a853f545c959b1681567c105a2921d887870905

fig 4. Results Summary

https://preview.redd.it/kfx8etv56i871.png?width=750&format=png&auto=webp&s=04082e874ca1dbf7a48b956019a54d687a79951f

Discussion

• It was cool to see an alpha directly from a book applied to a different asset class still continue to work
• The Z score is calculated as (observed_spread - spread_rolling_mean) / (spread_std)
• Long-short entries were very wide meaning the strategy was low touch (27.05% time in market).would work well paired with other low touch strategies
• No apparent long short bias with strong returns and performance metrics
• Live trading resu

Hey everyone, So I’m trying to use kalman filters in my pairs trading strategy to compute a dynamic ratio for two cointegrated assets. However, I’m unable to find a good resource that helps me achieve this in code. Many articles I found leave out many code bits necessary for a final solution. Do any of you know a good resource for a pairs trading w/ kalman filters implementation? Thanks in advance.

I've seen a few references to using the Kalman Filter as part of a formulaic trading strategy and I'm trying to understand it to see how it might be used for my own strategies. I've read a few different things to try to get a grasp of the technique (I have a broad understanding of common machine learning techniques but limited formal math background). This was the last one I read and it does a better job in my opinion of explaining the different covariance matrices in the context of the prediction and update steps as well as putting a lot of the explanation in plain terms.

https://www.reddit.com/r/MachineLearning/comments/8z19gw/d_how_a_kalman_filter_works_in_pictures/

Note: I'm posting this in r/MachineLearning since the above post was also in this subreddit.

However, what seems to be kind of glossed over in everything I've seen are the noise elements of the Kalman equations. For instance, it might be assumed that a sensor has a predetermined level accuracy (Rk) that is fixed for all updates. The explanation in the above discusses the additional noise parameter (Qk) also assumes that acceleration is known.

Additionally, I've seen indications that Kalman Gain should always be decreasing, implying that the estimate is always getting more accurate, however, this also seems to imply that something is fixed. For example, if the "velocity" (i.e., the period-over-period change in price, the value of an indicator, etc.) is constantly changing, placing greater and greater reliance on the estimate will lead to predictions that drift away from the true/observed value.

So I'm trying to bridge all of this back to the application to trading where:

1. I won't know the acceleration or it won't be relevant.
2. Velocity (period-over-period change in price) should generally be expected to oscillate around zero but won't be fixed and can change in either direction at any given time.
3. I will have to estimate both the measurement and estimate noise, Rk and Qk, respectively, and at least one of these should probably be changing over time.

I think #1 and #2 shouldn't matter if #3 is worked out, but I really have no idea how to update Rk and Qk beyond maybe keeping a rolling estimate of volatility, but I'm not sure if that's even in the ballpark.

I'm wondering if there are reasonable ways to approach this or if some of it is not necessary and I'm overcomplicating

Hello! I have been looking for literature of LQG (LQR + Kalman filter) where the state space changes slowly (i.e. the ss comes from a parametric model whose parameters change slowly wrt the system dynamics).

Unfortunately, I couldn't find much information on adaptive control in the LQG framework (i.e. adaptive Kalman filter). Am I looking in the wrong direction? Is there some technique similar to what I a looking for?

I wonder whether what I am looking for is an "adaptive Kalman filter" where the process covariance matrice is updated to account for the change in underlying parameters, or if there is any other estimation/control method that could be applied.

What does r/algotrading think about them?

So the hidden Markov chain is not bound to discrete states, but can take arbitrary real values (possibly subject to constraints of some kind). This makes sense to me, but I could not find a definite source for this, so a quick confirmation would be helpful. Thanks!

I thought this might be of interest to this sub.

I made an introductory-level video about the inner-workings of Kalman Filters. I am new to making this kind of content, so I am looking for all kinds of feedback.

https://www.youtube.com/watch?v=IFeCIbljreY

The general consensus seems to be that a kalman filter is the best option for filtering the data from an IMU. From my understanding the equations have to be suited for the dynamics of the system so the rocket. How ever most examples/code aren't for rockets. Any good resources/code on this issue?

I know this may be very basic for a lot of all you ML experts out there, but I just spent an hour learning and implementing a simple Kalman filter here: https://github.com/enochkan/kalmanpy and it worked :D pretty darn proud of myself 🙌✨

Please check it out and would appreciate any feedbacks.. my implementation is far from perfect and I plan on implementing more as my knowledge grows.

So I've got a barometer measuring altitude directly, an accelerometer measuring acceleration and a GPS measuring altitude as well.

I've made a Kalman Filter in C++ where the F term looks something like this:

F = {
1, T, 0.5 * T ^2
0, 1, T,
0, 0, 1};

I'm doing the update step for each sensor as it's measurements come in.

So for updating with the barometer I have an H matrix like this:

Matrix<1, 3> H_Baro = {1, 0, 0};

The GPS H matrix looks like the above as well.

As for the accelerometer it looks like:

Matrix<1, 3> H_Accel = {0, 0, 1};

So I have separate update steps rather than bundling the barometer and accelerometer together into one update step. Is this the right way to do this?

Here's an example of one of the predict and update steps:

void predict()
{
currentTime = micros();
delT = (currentTime - prevTime) / 1000000.0f;
data.loopTime = delT;
prevTime = currentTime;
F = {
1, delT, 0.5 * delT * delT,
0, 1, delT,
0, 0, 1};
X = F * X;
P = F * P * ~F + Q;
}

void updateBaro(float altitude)
{
predict();
Z_Baro = {altitude};
K_Baro = P * ~H_Baro * (H_Baro * P * ~H_Baro + R_Baro).Inverse();
X = X + K_Baro * (Z_Baro - H_Baro * X);
P = (I - K_Baro * H_Baro) * P * (~(I - K_Baro * H_Baro)) + K_Baro * R_Baro * ~K_Baro;
}

I've noticed when I make the R for the barometer really high, the filter doesn't seem to ignore the barometer measurements much. If I say, blow air at the barometer and the R is very high, it still affects my Kalman position as if it's trusting the barometer. Any advice?

I am trying to build a system that can measure/estimate the angular acceleration of robotics joints. I'm expanding on the idea presented in this paper which attached an IMU to a single joint and somehow used an EKF to estimate the acceleration and velocity. I built a system that measures the velocity of all joints and just one of them, but after I was done, it seems impossible to me to calculate the joint joint acceleration from the cartesian acceleration.

I mean.. this is essentially inverse kinematics for a kinematically redundant robot, afaik this is not a solvable problem. On the other hand, the recursive method described in the paper (equation 3) is quite interesting because it does in fact describe a valid relation between the cartesian acceleration and the angular acceleration. But I was not able to solve this equation for the angular acceleration because the angular acceleration is inside a cross product with a vector.. another seemingly unsolvable problem.

I just cannot wrap my mind about how this researcher did what they did with the EKF. How does it just take in the linear acceleration and spit out the angular acceleration. Can anyone please point me in the right direction?

Hi,

After studying the theory of Kalman filter for a while, today I implemented it in Python. Now I'd like to actually apply it to some data, so I decided to take a really simple, simulated use case. But I have some difficulties developing an intuition for "ground truth" and my motion model.

I assume a robot is moving along a line in the x-y-plane with constant velocity v, which I use as a control value (control value u = v). So my ground truth are 100 equidistant x-y coordinates along that line (1 xy-pair for one of 100 time steps).

I simulated my measured data by choosing x and y measurement variances and just adding 0-mean Gaussians with these variances to the ground truth data. From these variances I also infer the measurement variance matrix R.

I initialized some random initial position estimate and chose a high variance P, since it's basically a guess. For Q I was just playing around with some values and chose Q=[[0.2, 0],[0, 0.2]].

The problem I am having is, if I know the velocity v, then my motion model (s_2 = s_1+v*t) will describe my ground truth perfectly if I know the initial position. So basically the entire uncertainty of my motion model only stems from not knowing the initial position and if I knew it, I woudln't even need Kalman filter anymore. That's obviously not the point of Kalman filter; I would like to take a noisy measurement and a not perfect model estimate and obtain a value that's more accurate than both (even if I perfectly knew my initial position).

Maybe part of my problem here is the simplicity of the problem I chose? If I would look at a more complex real life problem, then my motion model would actually just be an approximation.

In my simulated example, does Q simulate this uncertainty of not having a perfect motion model? Or how can I model the uncertainty of the motion model in this example? In this case, is it normal that the entire uncertainty of the motion model only stems from not knowing the initial position or am I doing something wrong?

Thanks in advance, any help is appreciated :)

This one got removed: https://www.reddit.com/r/MachineLearning/comments/pd5c65/d_kalman_filters_classical_models_vs_ml_dnn/

So starting one here.

Hello! I am a data engineer currently studying systems engineering part-time. I am really fascinated both by the recent advances in ML technology, but also by the impressive mathematical models used up until now.

Last semester was an intro to the Kalman Filter (system identification methods). What I still haven't understood though, is when to use which tool and for which scenarios. Going through this paper:

https://medium.com/analytics-vidhya/kalman-filter-vs-deep-learning-for-position-estimation-27658e0cf568

It almost seems as if DNNs would be the natural modern choice for time-series predictions. A part of me though is so fascinated by the Kalman filters, that I want to believe there are scenarios where a Kalman filter or similar model would be a better choice.

What would these scenarios be if so?

I need to learn (and apply) the Kalman filter from scratch within 2 months, and I have little knowledge about stats outside of a stats core i barely passed in university. However, I’ve begun reading extensively about it and found it really interesting, and I think I have a basic understanding. So questions:

1. I understand the algorithm as an iterative process to update a multivariate distribution with every new set of time series data to better estimate the true distribution and be able to estimate what the next data might be, then use said data to see how far off the estimate is to again update the distribution. Is my (very basic) view at least on the right path?

2. From what I’ve seen, the math seems to be mostly basic matrix operations and not much outside of (multivariate) normal distribution and covariance, which I have a rough grasp of. Is there anymore pre-requisite maths I need to start applying the process to data? And/Or

3. What kind of maths do I need, to be able to apply it in python/R (both of which are also completely new to me)?

I recognise this is a bit of a lazy question, but would really appreciate your help on this. Would also love any good books/links that you think is a good primer for someone like myself. Thanks a lot!

Sorry if this isn't the best place for this kind of discussion.

I'm currently going through a udacity course and we're covering Kalman filters. They've handed us the formulas which are easy to code, and a conceptual idea of what they're doing, but I'd like to know more, namely why the formulas work the way they work.

Concretely, ;-) We have the following equations. A formula for gaussians.

[;p(x) = \frac{1}{\sigma\sqrt{2\pi}}e^-\frac{(x-\mu)^2}{2\sigma^2};]

[;\sigma^2;] = variance (why squared?)

[;\mu;] = mean

1. The lesson says "Note the term before the exponential serves as a dampening factor" does that mean that the height of the gaussian diminishes as [;\sigma;] grows?

2. How does the power term affect the output of the function? Why is it negative? Now that I'm reading this a little more closely it's starting to make a little sense.

3. Why is the power term negative? [;(x-\mu)^2;] is the absolute value (squared) of the distance of x from the mean, ah divided by the total distance of the variance? why squared and why multiplied by 2?

Next question, to calculate the new mean,

[;\mu^\prime = \frac{r^2\mu + \sigma^2v}{r^2+\sigma^2};] where

[;\mu^\prime;] = resulting new mean

[;\mu;] = mean of the prior gaussian

[;\sigma^2;] variance of the prior gaussian

[;r^2;] = new measurement variance (again why squared)

[;v;] = new measurement mean

The explanation is

"The answer is - the uncertainty of the prior is multiplied by the mean of the measurement, to give it more weight, and similarly the uncertainty of the measurement is multiplied with the mean of the prior. "

My questions are

1. Are the terms all added because basically the result is the average of all of the ... weighted means over the total variance?

2. if a new measurement is more accurate (reduced variance) wouldn't that result in a smaller value, and the old mean with the new variance would be biased less than the new mean with the old variance? If the new variance is better (smaller value), shouldn't it pull the old mean more towards the resulting mean?

lastly, to calculate the new variance

[;\sigma^{2^\prime} = \frac{1}{\frac{1}{r^2} + \frac{1}{\sigma^2}};]

As the variances get larger, the old probability distributions are added together? that makes sense mentally, but in actuality as the variances grow larger, oh.... the resulting variance also grows.... right? Its basically the average of the, what exactly, probability distributions?

Ok!

The title explains it all. I mean why do we have two different explanations for the same thing ?

Hello!

I implemented an EKF on Victoria Park Dataset (Odometry + GPS + Lidar) I read an article that said that GPS coordinates can be considered as ground truth. Is it logical to calculate the euclidean distance between my estimates with the 3 sensors and the gps coordinates?

It seems wierd to do so but I didn't find an article that could explain to me how to test my algorithm.

Thank you!

I am currently doing my PhD specializing in cooperative localization and state estimation. I was wondering if anyone knew any specific job roles or positions that would mostly deal with state estimation. GNC engineers come to mind but wondering if there are other job titles Im missing. I’d like to be certain about what to search for when I’m looking g for positions.

Hello there,

I wrote a little script that gets stock data in a minute timeframe.

I then just use the close prices, calculate the hedge ratios using a kalman filter and test the spread of a pair on stationarity.

If I identify that the spread is stationary and is too high right now and I would enter a trade: Should I then rebalance the trade every minute with the new hedge ratio or shouldn't I?

Thanks in advance!

In many cases I’ve seen that the number of measurements for the h matrix is the same as the number of states. So the H matrix for the measurement model ends up being square. Are there any examples of there being less measurements than states? Is that even possible?

I am trying to build a system that can measure/estimate the angular acceleration of robotics joints. I'm expanding on the idea presented in this paper which attached an IMU to a single joint and somehow used an EKF to estimate the acceleration and velocity. I built a system that measures the velocity of all joints and just one of them, but after I was done, it seems impossible to me to calculate the joint joint acceleration from the cartesian acceleration.

I mean.. this is essentially inverse kinematics for a kinematically redundant robot, afaik this is not a solvable problem. On the other hand, the recursive method described in the paper (equation 3) is quite interesting because it does in fact describe a valid relation between the cartesian acceleration and the angular acceleration. But I was not able to solve this equation for the angular acceleration because the angular acceleration is inside a cross product with a vector.. another seemingly unsolvable problem.

I just cannot wrap my mind about how this researcher did what they did with the EKF. How does it just take in the linear acceleration and spit out the angular acceleration. Can anyone please point me in the right direction?

I am trying to build a system that can measure/estimate the angular acceleration of robotics joints. I'm expanding on the idea presented in this paper which attached an IMU to a single joint and somehow used an EKF to estimate the acceleration and velocity. I built a system that measures the acceleration of all joints and not just one of them, but after I was done, it seems impossible to me to calculate the joint joint acceleration from the linear cartesian acceleration.

I mean.. this is essentially inverse kinematics for a kinematically redundant robot, afaik this is not a solvable problem. On the other hand, the recursive method described in the paper (equation 3) is quite interesting because it does in fact describe a valid relation between the cartesian acceleration and the angular acceleration. But I was not able to solve this equation for the angular acceleration because the angular acceleration is inside a cross product with a vector.. another seemingly unsolvable problem.

I just cannot wrap my mind about how this researcher did what they did with the EKF. How does it just take in the linear acceleration and spit out the angular acceleration. Can anyone please point me in the right direction?

I am trying to build a system that can measure/estimate the angular acceleration of robotics joints. I'm expanding on the idea presented in this paper which attached an IMU to a single joint and somehow used an EKF to estimate the acceleration and velocity. I built a system that measures the velocity of all joints and just one of them, but after I was done, it seems impossible to me to calculate the joint joint acceleration from the cartesian acceleration.

I mean.. this is essentially inverse kinematics for a kinematically redundant robot, afaik this is not a solvable problem. On the other hand, the recursive method described in the paper (equation 3) is quite interesting because it does in fact describe a valid relation between the cartesian acceleration and the angular acceleration. But I was not able to solve this equation for the angular acceleration because the angular acceleration is inside a cross product with a vector.. another seemingly unsolvable problem.

I just cannot wrap my mind about how this researcher did what they did with the EKF. How does it just take in the linear acceleration and spit out the angular acceleration. Can anyone please point me in the right direction?