# Simple smoothers

A set of smoothers for a single column of evenly spaced data. See also the spline and LOESS smoothers in the Model menu. Missing data are supported.

#### Moving average

Simple n-point, centered moving average (n must be odd). Commonly used, but has unfortunate properties such as a non-monotonic frequency response.

#### Gaussian

Weighted moving average using a Gaussian kernel with standard deviation set to 1/5 of the window size (of n points). This is an overall good method.

#### Moving median

Similar to moving average but takes the median instead of the mean. This method is more robust to outliers but produces a “blocky” appearance.

#### AR 1 (exponential)

Recursive (autoregressive) filter, *y _{i}* =

*y*-1 + (1-α)

_{i}*x*with α a smoothing coefficient from 0 to 1. This corresponds to weighted averaging with exponentially decaying weights. Gives a phase delay and also a transient in the beginning of the series. Included for completeness.

_{i}#### Savitzky-Golay

The Savitzky-Golay method implements least-squares fit to a polynomial inside a moving window of size n points. Second-order (m=2) and fourth-order (m=4) polynomials are included. These are “optimal” smoothers in the sense that they preserve all moments up to m. This tends to give better preservation of peak values and peak widths than other smoothers.

#### Non-local means

Non-local means is a relatively new, powerful smoothing method, mostly used for image denoising but also effective for time series (Tracey & Miller 2012). It is an averaging method like the moving average and Gaussian methods, but the average is taken not over neighboring points but over points in similar regions, which can be far away. This tends to preserve peaks and transitions better than local averaging. In Past, the size of the local regions (patch size) can be selected; it could be set to e.g. N=7 or N=13. The search radius is fixed at half the length of the time series. The value of lambda controls the degree of smoothing. Tracey & Miller (2012) suggest a value of about 0.6 times the standard deviation of the noise (which is usually unknown, but can be estimated by eye).

Tracey, B. & Miller, E. 2012. Nonlocal means denoising of ECG signals. *IEEE Transactions on Biomedical Engineering* 59: 2383-2386.