> ## Documentation Index
> Fetch the complete documentation index at: https://aegean.ai/llms.txt
> Use this file to discover all available pages before exploring further.

# One-Dimensional Kalman Filters

> Introduction to Kalman filtering through one-dimensional examples, building intuition for state estimation under Gaussian assumptions.

This section introduces the Kalman filter through one-dimensional examples. Starting from simple cases, it builds intuition about how the filter combines predictions with noisy measurements to estimate hidden state.

## Key Concepts

* **Gaussian Distributions**: The Kalman filter's representation of uncertainty
* **Prediction and Update**: The two-step recursive estimation process
* **Kalman Gain**: Optimal weighting between prediction and measurement
* **Tracking Examples**: Position estimation from noisy sensor data

<Note>
  This section is adapted from the excellent book [Kalman and Bayesian Filters in Python](https://github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python) by Roger Labbe.
</Note>

<Card title="Run the Tutorial" icon="play" href="https://colab.research.google.com/github/rlabbe/Kalman-and-Bayesian-Filters-in-Python/blob/master/04-One-Dimensional-Kalman-Filters.ipynb">
  Open the original interactive notebook in Google Colab to execute the code and explore the concepts.
</Card>

## Notebook Preview

<Frame caption="One-Dimensional Kalman Filters Tutorial (read-only preview via nbviewer)">
  <iframe src="https://nbviewer.jupyter.org/github/rlabbe/Kalman-and-Bayesian-Filters-in-Python/blob/master/04-One-Dimensional-Kalman-Filters.ipynb" title="One-Dimensional Kalman Filters Tutorial" className="w-full rounded-lg border border-gray-200 dark:border-gray-700" style={{height: "800px"}} />
</Frame>

**Key references**: (Wilson et al., 2011; Raissi et al., 2017; Frazier, 2018; Lukežič et al., 2016)

## References

* Frazier, P. (2018). *A Tutorial on Bayesian Optimization*.
* Lukežič, A., Vojíř, T., Čehovin, L., Matas, J., Kristan, M. (2016). *Discriminative correlation filter with channel and Spatial Reliability*.
* Raissi, M., Perdikaris, P., Karniadakis, G. (2017). *Numerical Gaussian Processes for Time-dependent and Non-linear Partial Differential Equations*.
* Wilson, A., Knowles, D., Ghahramani, Z. (2011). *Gaussian Process Regression Networks*.

***

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