[jarenmf]

I'm struggling to see the advantage over the arithmetic mean if I don't have a model of the true value. How is the confidence estimated for each value? given the example from the article where the variance of the temperature sensor is considered to be constant.

[michaelt]

Imagine you want to know the depth of liquid in a tank you're filling up - and you've got a noisy depth gauge and a noisy flow-rate meter.

If you merely take the mean of the last 10 depth gauge readings, your smoothed depth reading will always lag behind the true depth, being about 5 readings out-of-date.

By fusing together the noisy depth gauge, and the noisy flow-rate meter, and a model saying how fast depth rises with flow, you can average out the noise without creating the same level of lag.

This is useful in applications like GPS receivers - there's noise so you do need filtering, but for driving through complex junctions, the last thing you want is a 5 second delay!

[em500]

If you have separate estimates / noisy measurements of the same quantity, and also have some estimate of their variance (e.g. pollsters or weather forecasters with different accuracy track records), you'd probably want a weighted mean (inversely by their variance) rather than a simple mean. If you have a system that runs for some time, you'd usually (but not always) want an average with higher weights placed on the recent past than on the distant past. (The arithmetic mean is the optimal estimate in a model where the true value is unknown but time-constant.)

The Kalman filter is "just" a recipe to calculate the weights for optimal estimation, given different model assumptions. The difficult part is not applying the formulas, but mostly in coming up with a good model of the movements of the things you want to measure/estimate.

[quietbritishjim]

At a high level, the Kalman filter offers two advantages over a simple arithmetic mean (which correspond to its two steps, update and observe):

* The Kalman filter can account for changes in state. This is useful the value you're measuring is changing over time (e.g. the position of a moving vehicle, or water level in a bucket being filled mentioned in another comment). The type of variation that can be handled by the Kalman filter will depend on what assumptions you make - how you configure it, if you like. Often you would account for velocity, but you can go a step further and account for acceleration too. A moving average will always lag behind a bit, even when movement is totally linear and there is no error in the measurement at all.

* A Kalman filter effectively gives more weight to recent measurements and less weight to older ones. e.g. a moving average with period 4 will have weights of (..., 0, 0, 0.25, 0.25, 0.25, 0.25), whereas a Kalman filter might effectively have weights of (..., 0.125, 0.25, 0.5). You can even have a continuously-varying time gap between measurements, so if a new measurement comes in then it will update the estimate a lot if the previous measurement was a long time ago, whereas if the previous measurement was extremely recent then it will effectively be averaged with the new one. One downside of this is that if the state changes a lot very suddenly then the Kalman filter will remember the old state, to some extent, whereas a moving average will forget it entirely once it drops out of the window; but this is offset to some extent by the previous point.

If you don't have a model of the various confidences then you'll need to guess them - if you do a really bad job then it might work out worse than a moving average. But if the Kalman filter's working better then you can always make use of the state estimate while ignoring the computed uncertainty.