We gain knowledge of the world around us through measurements. Measurements tell us about the world, but these measurements are often noisy, which can make extracting information harder. If we have a system we can only observe through noisy measurements, we need a way in which we can gain knowledge of the system to understand the true underlying process of the data. A Kalman filter is one of the ways in which we can go about better understanding noisy data.

Here, we will consider measurements relevant for air quality. Figure 1 shows a time series of hourly concentrations of PM2.5, which is particulate matter smaller than 2.5µm in diameter (PM_2.5), recorded, over 4 years at the (UK government) Automatic Urban and Rural Network (AURN) site in Preston. From this data we may be able to get X and Y, but it’s noisy and complicated

Figure 1: Hourly PM_2.5 at Preston

Applying a Kalman filter to the data, we are able to ‘filter out’ the noise associated with the system and the measurement as well as easily capture the uncertainty in our new estimated value. For cases in which data is normally distributed or Gaussian, where data is tends to be centred around a central value with no bias left or right a Kalman filter can be used to estimate the unknown state of the system. When defining a Gaussian we can describe the data with just the mean and the variance, which tells us the spread of the data and how far it is from the mean. A Gaussian in a Kalman filter gives the estimated value along with its error/uncertainty as the variance at each time step. In Figure 2, we have applied the Kalman filter to the previous shown AURN data and the Kalman filter is quickly able to capture the trends of the data.

Figure 2: Kalman filter applied to daily PM_2.5 at Preston

The Kalman filter allows us to produce estimates for a system, and its associated uncertainty, that tends to be more accurate than the measurement alone. While the Kalman filter is a simple and powerful tool when dealing with noisy data caution should be used around assuming the distribution of the data. Further applications in Kalman filtering can be seen in forecasting using historical data and also correction of air quality forecasts reducing errors and biases in the forecast.

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