Chris' Blog.

My occasional thoughts on iOS development, developers careers, trying to make an income from the App Store, and updates on life in general.

The Kalman Filter with Velocity

Hi all, here’s how to implement a position-and-velocity Kalman filter from scratch, in C++ish pseudocode, from the perspective of a developer who isn’t a professional mathematician and doesn’t understand the kind of symbology used in many Kalman articles. I wrote another article about a simpler Kalman filter as a primer on the subject, if you read it first it will hopefully help make this article easier to understand.

Firstly we need to implement some matrix maths stuff as a prerequisite. It might help to skim-read Wikipedia on the topic first.

// A 2-row column vector.
struct Vector {
    double a // aka Position or Price
    double b // aka Velocity
// A 2x2 matrix.
struct Matrix {
    double tl // Top Left, eg row 1 column 1
    double tr // Top right, r1c2
    double bl // Bottom left, r2c1
    double br // Bottom right, r2c2
// vectorA + vectorB.
+(Vector lhs, Vector rhs) -> Vector {
    return Vector {
        a: lhs.a + rhs.a,
        b: lhs.b + rhs.b
// vectorA - vectorB.
-(Vector lhs, Vector rhs) -> Vector {
    return Vector(
        a: lhs.a - rhs.a,
        b: lhs.b - rhs.b
// matrixA + matrixB.
+(Matrix lhs, Matrix rhs) -> Matrix {
    return Matrix(
        tl: +, tr: +,
        bl: +, br: +
// matrixA - matrixB.
-(Matrix l, Matrix r) -> Matrix {
    return Matrix(
        tl: -, tr: -,
        bl: -, br: -
// matrixA * matrixB.
*(Matrix l, Matrix r) -> Matrix {
    return Matrix(
        tl: * + *,
        tr: * + *,
        bl: * + *,
        br: * + *
// vectorOutput = matrixA * vectorB.
*(Matrix m, Vector v) -> Vector {
    return Vector(
        a: * v.a + * v.b,
        b: * v.a + * v.b
// Invert a matrix (the matrix version of 1 / m).
invert(Matrix m) -> Matrix {
    return Matrix(
        tl:, tr:,
        bl:, br:
// Transpose a matrix (mirror it diagonally).
transpose(Matrix m) -> Matrix {
    return Matrix(
        tl:, tr:,
        bl:, br:

Next, you’ll need the ‘external noise covariance matrix’. This is usually referred to as Q in Kalman literature:

const double Q_POSITION_VARIANCE = 0.1
const double Q_VELOCITY_VARIANCE = 0.1
const Matrix Q_EXTERNAL_NOISE = {

Next, you’ll need the ‘measurement noise covariance matrix’, aka R in Kalman articles:

const double R_POSITION_VARIANCE = 0.1
const double R_VELOCITY_VARIANCE = 0.1
const Matrix R_MEASUREMENT_NOISE = {

How do you select good Q and R values? I’m not an expert, but here’s my two tips:

  • If you’re filtering position and velocity sensors that have a normal / gaussian error, and they really are separate sensors as opposed to faking velocity by nowPosition-lastPosition, use the variance and covariance of the two sensors for R, and perhaps experiment with Q until it works the way you’d like. Variance = average[residual^2], covariance = average[positionResidual * velocityResidual], residual = measuredValue - realValue. If you don’t have real values, you could try substituting some kind of moving average.
  • If you’re filtering non-gaussian data (eg stock prices), or your velocity is faked, simply experiment with Q and R until you’re happy with the filter performance. I’d write a program to generate a million random Q+R pairs, then choose the pair that predicts ‘next’ data points best. That is a (whopping) exercise for the reader, sorry!

Next, we need our ‘state transition matrix’, aka F in Kalman articles. This is a constant matrix that, when multiplied by the current state vector (position + velocity), gives us our expected next state vector (old position + velocity, same velocity). Eg it looks like:

const Matrix F_STATE_TRANSITION = {
    tl: 1, tr: 1,
    bl: 0, br: 1,

Maybe the above needs more explaining. Say our old state is position 100m, velocity 10m/s. Multiplying this by F works out like so:

Normal matrix * vector formula:
a: * vec.a + * vec.b,
b: * vec.a + * vec.b


new position: * position + * velocity
new velocity: * position + * velocity


new position: 1 * position + 1 * velocity
new velocity: 0 * position + 1 * velocity


new position: position + velocity
new velocity: velocity


new position: 110m
new velocity: 10m/s

In other words, for a timestep of 1 second, F simply adds the velocity-per-second to the current position, and assumes no friction hence velocity stays constant.

Next, we need the ‘state error’, aka P in Kalman articles. I find it usually works fine to start filtering with this as 1. (Perhaps this should be scaled to match your typical values.)

Matrix pStateError = {
    tl: 1, tr: 1,
    bl: 1, br: 1,

Next, we need our ‘current state’, aka X in Kalman articles. If, like me, you’re faking the velocity, start the filter with this set using the first two measurements:

Vector xCurrentState = {
    a: data[1],
    b: data[1] - data[0], // Faux velocity.

Phew! Now for the interesting bit. For each new data point/measurement, we need to first predict what it’ll be, then blend that prediction with the measurement to produce an estimate. This estimate is the filter’s output. A ‘Kalman gain’ (known as K in Kalman articles) is calculated then is used to weight the prediction vs measurement when blending them to form the estimate:

// Make predictions.
Vector xPredicted = F_STATE_TRANSITION * xCurrentState

// Update it with the measurement.
Vector zMeasurement = Vector {
    a: data[current_index], // Position.
    b: data[current_index] - data[previous_index] // Faux velocity.
Matrix kKalmanGain = pPredicted * invert(pPredicted + R_MEASUREMENT_NOISE_VARIANCE)
Vector xEstimate = xPredicted + kKalmanGain * (zMeasurement - xPredicted)
Matrix pEstimate = pPredicted - kKalmanGain * pPredicted

// Store for the next iteration.
pStateError = pEstimate
xCurrentState = xEstimate // This is the output!

Thanks for reading, I hope this helped, God bless, and have a nice week :)

The Kalman Filter for Programmers

Hi all, here’s how to implement the Kalman filter from scratch, in C-ish pseudocode. The intended audience is developers, as opposed to mathematicians, so there will be no complicated formulae - just additions, subtractions, and multiplications. I won’t try to explain the theory behind the Kalman filter and its intended uses, that is explained far better elsewhere (Wikipedia is a great place to start). To oversimplify, it’s a little like a moving average.

This article explains how to implement a position-only Kalman filter, as a stepping stone to implementing a position-and-velocity filter in my next article.

Firstly, you’ll need the ‘external noise variance’. This is usually referred to as Q in Kalman literature:

const double Q_EXTERNAL_NOISE_VARIANCE = 0.1

Next, you’ll need the ‘measurement noise variance’, aka R in Kalman articles:


How do you select good Q and R values? I’m not an expert, but here’s my two tips:

  • If you’re filtering a sensor that has a normal / gaussian distribution vs the actual value, use the variance of its inaccuracy as R (variance = average[residual^2], residual = measurement - actual value), and experiment with Q until it works the way you’d like.
  • If you’re filtering non-gaussian data (eg stock prices), simply experiment with Q and R until you’re happy with it. I’d write a program to generate a million random Q+R pairs, then choose the pair that predicts the ‘next’ data points best. That is an (whopping) exercise for the reader, sorry!

Next, we need our ‘state transition matrix’, aka F in Kalman articles. Since this is the one-dimensional filter, this is a simple double that, when multiplied by a data point, gives us our expected next data point. Eg it’ll simply be 1.

const double F_STATE_TRANSITION = 1

Next, we need the ‘state error’, aka P in Kalman articles. I find it usually works fine to start filtering with this as 1:

double pStateError = 1

Next, we need our ‘current state’, aka X in Kalman articles. Start filtering with this set to the first measurement:

double xCurrentState = data[0]

Now a prerequisite: transposing a ‘matrix’. In our simple one-dimensional case, this is a no-op, but I’m leaving it here as a stepping stone to the next article:

double transpose(double a) { return a }

Another prerequisite: inverting a ‘matrix’. In our simple case, this is just 1 / a.

double invert(double a) { return 1 / a }

Now for the interesting bit. For each new data point/measurement, we need to first predict what it’ll be, then blend that prediction with the measurement to produce an estimate. This estimate is the filter’s output. A ‘Kalman gain’ (known as K in Kalman articles) is calculated that is used to weight the prediction vs measurement when blending them to form the estimate.

// Make predictions.
double xPredicted = F_STATE_TRANSITION * xCurrentState

// Update it with the measurement.
double zMeasurement = data[current_index]
double kKalmanGain = pPredicted * invert(pPredicted + R_MEASUREMENT_NOISE_VARIANCE)
double xEstimate = xPredicted + kKalmanGain * (zMeasurement - xPredicted)
double pEstimate = pPredicted - kKalmanGain * pPredicted

// Setup for the next iteration.
pStateError = pEstimate
xCurrentState = xEstimate // This is the output!

You may find my next article about position-and-velocity Kalman filters helpful!

Thanks for reading, I hope this helps, God bless, and have a nice week :)


Hi all, I’d like to share some adventures I’ve recently had with interacting with Interactive Brokers’ TWS API directly using Rust, without an SDK.

You’ll need TWS (or IB Gateway) running on your machine. Note that their API host is TWS - you don’t talk directly to their servers. You’ll also need a real account, but not a paid data subscription.

The code can be found here in completed form: gist

And without further ado…

  • Go here and download the latest TWS:
  • You must login with a real live or paper account. If you use the ‘try the demo’ mode the API does not respond.
  • You will not require a paid data subscription for the symbols we’re watching here.
  • Go File > Global Configuration > Api > Settings > check ‘Enable ActiveX and Socket Clients’
  • !!!! ENSURE “Read-Only API” IS CHECKED !!!!
  • Also take note of ‘Socket port’ on that screen (7496 is what I see).
  • Select Apply and Ok
  • Run nc -z localhost 7496 to test that TWS API is listening.

Ok lets start talking to this with Rust.

First lets make a new rust project:

  • Install rust if you need to!
  • cargo new my_tws_api_talker
  • cd my_tws_api_talker
  • cargo run

You should see a Hello message.

Lets make it connect. Open up src/

use std::net::TcpStream;

fn main() {
    let mut stream = TcpStream::connect("").expect("connect");
    stream.set_nodelay(true).expect("nodelay"); // Because we're wannabe HFT traders.
    println!("Able to connect")

Perform cargo run, it should say it was able to connect. Now lets send the greeting. Most messages to TWS need a big-endian 4-byte length prefix. So lets implement that as a prerequisite:

fn add_length_prefix(bytes: &[u8]) -> Vec<u8> {
    let len = bytes.len() as u32;
    let mut data: Vec<u8> = Vec::new();
    data.push(((len >> 24) & 0xff) as u8);
    data.push(((len >> 16) & 0xff) as u8);
    data.push(((len >> 8) & 0xff) as u8);
    data.push((len & 0xff) as u8);

And one small helper. Not sure why this isn’t part of the stdlib, unless I’m missing something:

fn concat(a: &[u8], b: &[u8]) -> Vec<u8> {
    let mut both = a.to_owned();

So now we can use these helpers to send the greeting:

use std::io::{Read, Write}; // Add this up top.

const MIN_SERVER_VER_BOND_ISSUERID: u32 = 176; // From

fn send_greeting(stream: &mut TcpStream) {
    let prefix = "API\0";
    let version = format!("v{}..{}", DESIRED_VERSION, DESIRED_VERSION);
    let version_msg = add_length_prefix(version.as_bytes());
    let both = concat(prefix.as_bytes(), &version_msg);

Now to actually call that greeting, add this to the bottom of the main function:

send_greeting(&mut stream);
println!("Greeting sent");

Then cargo run and it should say greeting sent. Great!

Next we need to read a message from TWS. These messages all look like this:

  • 4 bytes big endian message length
  • Message

A message is a list of fields. Each field is a string, with a zero after it. Numbers are strings, and bools are “0” (false) or “1” (true). Eg a simple 3-field message would be “Field A\0Field B\0Field C\0”.

First we need to read a length:

fn read_length(reader: &mut TcpStream) -> u32 {
    let mut len_buf: [u8; 4] = [0; 4];
    reader.read_exact(&mut len_buf).expect("Read length");
    let length: u32 = ((len_buf[0] as u32) << 24)
        | ((len_buf[1] as u32) << 16)
        | ((len_buf[2] as u32) << 8)
        | len_buf[3] as u32;

Next we need something to split a read message into its fields:

fn split_message(message: &[u8]) -> Vec<String> {
    // Split into an array of buffers:
    let mut components = Vec::<Vec<u8>>::new();
    let mut current_component = Vec::<u8>::new();
    for byte in message {
        if *byte == 0 {
            if !current_component.is_empty() {
        } else {
    if !current_component.is_empty() {

    // Convert the buffers into strings:
        .map(|v| String::from_utf8_lossy(&v).to_string())

Next lets bring all that together. Read the length, then read the message, then split it:

fn read_message(reader: &mut TcpStream) -> Vec<String> {
    let length = read_length(reader);
    let mut buffer: Vec<u8> = vec![0; length as usize];
    reader.read_exact(&mut buffer).expect("Read message");

To use this, add the following to the bottom of

let greeting_response = read_message(&mut stream);
println!("Greeting response: {:?}", greeting_response);

cargo run and you should see TWS respond with something like ["176", "20230826 22:07:41 AEST"] These fields are the api version then datetime. The version should match DESIRED_VERSION from our code. You should verify that it does, in production code.

Right after the greeting, we have to send the ‘Start API’ message. If you download IB’s python api and look in, you can see all the request type codes. Lets write an enum with only a few request types we’ll use:

enum OutgoingRequestType {
    ReqMktData = 1,
    StartApi   = 71,

Now we’ll need a function to compose a message:

// Join and delimit the fields, prefixing the length.
fn make_message(fields: &[String]) -> Vec<u8> {
    let mut delimited_fields = Vec::<u8>::new();
    for field in fields {
        delimited_fields.push(0); // Even goes after the last field.

Then we need our function to compose and send the ‘start api’ message:

fn send_start_api(stream: &mut TcpStream, client_id: u32) {
    let fields: Vec<String> = vec![
        (OutgoingRequestType::StartApi as u32).to_string(),
        "2".to_string(), // Version of the Start API message.
        "".to_string(), // Optional capabilities.
    let message = make_message(&fields);
    stream.write_all(&message).expect("Start api");

And let’s call it. Add this to the end of main():

send_start_api(&mut stream, 123456); // Client ID 123456.

If you cargo run now it should work.

At this stage, the API will want to send us a bunch of messages. So we’d better loop for those. Firstly lets identify some of the incoming message types:

enum IncomingRequestType {
    TickPrice = 1,
    ErrMsg    = 4,
} // From

Using the above message types, lets add a message handler:

fn message_received(fields: &[String]) {
    if fields.is_empty() { return }
    let Ok(t) = fields[0].parse::<u32>() else { return };
    if t == IncomingRequestType::ErrMsg as u32 {
        let _request_id = fields.get(2);
        let _code = fields.get(3);
        let Some(text) = fields.get(4) else { return };
        println!("Message: {}", text);
    } else if t == IncomingRequestType::TickPrice as u32 {
        let Some(tick_type) = fields.get(3) else { return };
        if tick_type != "4" { return } // We only want 4 = 'Last' from
        let Some(price) = fields.get(4) else { return };
        println!("Price: {}", price);
    } else {
        println!("Received: {:?}", fields); 

Now, lets listen for messages. Add this at the bottom of main(), but keep in mind we’ll replace it with a threaded version soon:

loop {
    let message = read_message(&mut stream);

Run it with cargo run and you should see a bunch of messages printed. You’ll need to do ctrl-c to quit.

But you won’t see any prices come in, because we haven’t requested any. This led me to some thinking around how to receive and send messages. I personally like the idea of making a reading thread, a writing thread, and a control (main) thread, to spread the load. So firstly, in main(), remove the loop above, and add this in its place, to ‘split the socket’ into a reader and writer:

// !! Ensure you remove the previous loop above !!
let mut writer = stream.try_clone().expect("Clone");
let mut reader = stream;

Next we want a mechanism for the main thread to enqueue messages for the writer thread to send:

// use std::sync::mpsc::channel;
let (writer_tx, writer_rx) = channel::<Vec<String>>();

Next lets spawn our reader thread for incoming messages:

// use std::thread;
let reader_handle = thread::Builder::new().name("Reader".into()).spawn(move || {
    loop {
        let message = read_message(&mut reader);

Next lets spawn the writer thread that listens to the channel, then writes the messages to the socket:

let writer_handle = thread::Builder::new().name("Writer".into()).spawn(move || {
    loop {
        let fields = writer_rx.recv().expect("Writer queue receive");
        let message = make_message(&fields);
        println!("Sending {:?} aka {:?} aka {:?}", 
            fields, String::from_utf8_lossy(&message), message);
        writer.write_all(&message).expect("Write all");

Now we need to get the main thread to wait for the new threads:


You can run it now, and it should receive messages and run until you perform ctrl-c.

It still isn’t showing price data however, as we need to ask TWS for that. We’ll do that now by sending a message from the main thread to the writer thread. Add the following code just above the previously-added calls to ‘join()’:

// Post-handshake delay or TWS bugs out.

let request_market_data: Vec<String> = vec![
    (OutgoingRequestType::ReqMktData as u32).to_string(),
    "11".to_string(),     // Version
    "999".to_string(),    // Request id aka Ticker id
    "".to_string(),       // Contract id
    "BTC".to_string(),    // Symbol
    "CRYPTO".to_string(), // Security type
    "".to_string(),       // LastTradeDateOrContractMonth
    "".to_string(),       // strike
    "".to_string(),       // right
    "".to_string(),       // multiplier
    "PAXOS".to_string(),  // exchange
    "".to_string(),       // primaryExchange
    "USD".to_string(),    // currency
    "".to_string(),       // localSymbol
    "".to_string(),       // tradingClass
    "0".to_string(),      // Delta neutral contract? 0=false, 1=true
    "".to_string(),       // genericTickList
    "0".to_string(),      // is a snapshot?
    "0".to_string(),      // regulatorySnapshot - costs 1c ea.
    "".to_string(),       // mktDataOptions - unsupported.
writer_tx.send(request_market_data).expect("Writer queue send");

// join calls go here...

Then run it, and you’ll get (among other messages) the real-time price of Bitcoin in USD! You can take it from here and build the trading bot of your dreams.

This code carries no warranty, use it at your own risk, and is MIT licensed!

For reference for the other messages, you’ll have to read through IB’s Python SDK. Thanks for reading, God bless, and have a nice week :)

You can see older posts in the right panel, under 'archive'.


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How to implement a position-only Kalman Filter 14 Dec 2023

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MessagePack parser for Objective-C / iPhone 30 Jun 2011

Lacquering the starboard side 25 Jun 2011

What do do with EXC_ARM_DA_ALIGN on an iPhone app 23 Jun 2011

Lacquering the hull 23 Jun 2011

Staining the boat 22 Jun 2011

NSMutableSet with weak references in objective-c 20 Jun 2011

Iphone gesture recogniser that works for baby games 20 Jun 2011

Image manipulation pixel by pixel in objective C for the iphone 19 Jun 2011

Baby Allergy Tracker 12 Jun 2011

Power sanding the deck 10 Jun 2011

Planing the edge of the deck 2 Jun 2011

Figured out the deck 2 Jun 2011

Boat bulkheads 2 Jun 2011

Simulating iOS memory warnings 31 May 2011

Putting a UIButton in a UIToolbar 29 May 2011

How to allow closing a UIActionSheet by tapping outside it 29 May 2011

Finding the currently visible view in a UITabBarController 24 May 2011

Random Chef 17 May 2011

Centered UIButton in a navigation bar on the iphone 16 May 2011

Little Orchard 13 May 2011

Boat update 13 May 2011

How to get the current time in all time zones for the iphone / obj-c 12 May 2011

Design portfolio 10 May 2011

Tricks with grand central dispatch, such as objective-c's equivalent to setTimeout 9 May 2011

How to make an iphone view controller detect left or right swipes 5 May 2011

Centered section headers on a UITableView 5 May 2011

Christmas in may 4 May 2011

Finished trimming the boat (its floatable now!) and got some parts 29 Apr 2011

How to make a multiline label with dynamic text on the iphone and get the correct height 27 Apr 2011

Forcing an image size on the image in a table view cell on an iphone 20 Apr 2011

Git on the Mac 19 Apr 2011

Build a url query string in obj-c from a dictionary of params like jquery does 12 Apr 2011

Rendering a radial gradient on the iphone / objective-c 11 Apr 2011

Skinning the port side of the boat 8 Apr 2011

Skinning the side of the boat 5 Apr 2011

Sending a UDP broadcast packet in C / Objective-C 5 Apr 2011

How to talk to a unix socket / named pipe with python 4 Apr 2011

Skinning the bottom of the boat 31 Mar 2011

Service discovery using node.js and ssdp / universal plug n play 30 Mar 2011

Extremely simple python threading 29 Mar 2011

New rescue boat 26 Mar 2011

HttpContext vs HttpContextBase vs HttpContextWrapper 5 Nov 2010

Simple C# Wiki engine 30 Sep 2010

Simple way to throttle parts of your Asp.Net web app 29 Sep 2010

How to implement DES and Triple DES from scratch 4 Aug 2010

How to use sessions with Struts 2 30 Jul 2010

How to use Cookies in Struts 2 with ServletRequest and ServletResponse 30 Jul 2010

Using Quartz Scheduler in a Java web app (servlet) 27 Jul 2010

Javascript date picker that Doesn't Suck!(tm) 27 Jul 2010

Using Oracle XE with Hibernate 20 Jul 2010

A simple implementation of AES in Ruby from scratch 29 Jun 2010

Asp.Net Forms authentication to your own database 28 May 2010

AS2805 (like ISO8583) financial message parser in C# 7 May 2010

Ruby hex dumper 4 May 2010

Using Spring to manage Hibernate sessions in Struts2 (and other web frameworks) 13 Jan 2010

Emails in C#: Delivery and Read receipts / Attachments 12 Jan 2010

Using Java libraries in a C# app with IKVM 16 Dec 2009

Learning Java tutorial 27 Nov 2009

Using generic database providers with C# 17 Nov 2009

Scheduled task executable batch babysitter 29 Oct 2009

Working with query strings in Javascript using Prototype 30 Sep 2009

Still fighting with String.Format? 9 Sep 2009

How I'd build the next Google 24 Aug 2009

Getting IIS and Tomcat to play nicely with isapi_redirect 24 Aug 2009

Using the new ODP.Net to access Oracle from C# with simple deployment 11 Aug 2009

C# Cryptography - Encrypting a bunch of bytes 14 Jul 2009

Sorting enormous files using a C# external merge sort 10 Jul 2009

Reconciling/comparing huge data sets with C# 9 Jul 2009

Some keyboard-friendly DHTML tricks 10 Jun 2009

How to figure out what/who is connected to your SQL server 18 Mar 2009

Adding a column to a massive Sql server table 16 Mar 2009

Multithreading using Delegates in C# 10 Mar 2009

Using C# locks and threads to rip through a to-do list 6 Feb 2009

Using threads and lock in C# 3 Feb 2009

Compressing using the 7Zip LZMA algorithm in C# beats GZipStream 14 Jan 2009

MS Sql Server 2005 locking 17 Dec 2008

Simple Comet demo for Ruby on Rails 19 Nov 2008

Geocoding part 2 - Plotting postcodes onto a map of Australia with C# 24 Oct 2008

Using evolutionary algorithms to make a walkthrough for the light-bot game with C# 20 Oct 2008

How to tell when memory leaks are about to kill your Asp.Net application 16 Oct 2008

C# version of isxdigit - is a character a hex digit? 15 Sep 2008

Geocoding part 1 - Getting the longitude and latitude of all australian postcodes from google maps 26 Aug 2008

Converting HSV to RGB colour using C# 14 Aug 2008

Opening a TCP connection in C# with a custom timeout 11 Aug 2008

Oracle Explorer - a very simple C# open source Toad alternative 31 Jul 2008

Linking DigitalMars' D with a C library (Mongrel's HTTP parser) 23 Jun 2008

Connecting to Oracle from C# / Winforms / without tnsnames.ora 16 Jun 2008

A simple server: DigitalMars' D + Libev 6 Jun 2008

Travelling from Rails 1 to Rails 2 9 Apr 2008

Online Rostering System 9 Apr 2008

DanceInforma 9 Apr 2008

Using RSS or Atom to keep an eye on your company's heartbeat 10 Nov 2007

Easy Integrated Active Directory Security in ASP.Net 24 Oct 2007