(noun) A mathematical function used to reduce noise, eliminate sharp transitions, or approximate data with a continuous curve that follows the general trend while reducing fluctuations.
The machine learning model used smoothing functions to eliminate outliers from the training data.
Smoothing functions are fundamental in signal processing, data analysis, and machine learning. They help create more stable models by reducing the impact of noise and temporary variations in data.
A simple moving average smoothing function:
$$ S_t = \frac{1}{n} \sum_{i=0}^{n-1} x_{t-i} $$
where $S_t$ is the smoothed value at time $t$, $n$ is the window size, and $x_{t-i}$ are the original data points.
Example with 3-point moving average:
| $t$ | $x_t$ | $S_t$ |
|---|---|---|
| 1 | 2 | - |
| 2 | 8 | 3.7 |
| 3 | 1 | 6.0 |
| 4 | 9 | 4.3 |
| 5 | 3 | 6.3 |
| 6 | 7 | 4.7 |
| 7 | 4 | 5.7 |
| 8 | 6 | - |
where $S_2 = \frac{2+8+1}{3} = 3.7$, $S_3 = \frac{8+1+9}{3} = 6.0$, etc.
Examples:
- Gaussian smoothing for image processing
- Spline interpolation for curve fitting
- Kernel smoothing in statistical estimation
- Exponential smoothing for forecasting