A review of smoothing techniques

Recently I need to smooth the Kinect-based finger tracking result. Without smoothing, there are a lot of jitters as shown in this demo video:

In searching for the best solution, I think it's good to have a review of different smoothing techniques. 

Simple moving average

$$s_t = \frac{1}{k}\sum_{n=0}^{k-1}x_{t-n}=\frac{x_t + x_{t-1} + ...+x_{t-k+1}}{k}=s_{t-1}+\frac{x_t-x_{t-k}}{k}$$

- Disadvantage: cannot be used on the first k-1 terms.

Weighted moving average

Give more weight to the recent terms in the time series. 

- Disadvantage: same as the simple moving average method, it cannot be used on the first k-1 terms. It also requires more complicated calculation at each step as it cannot be written as a inductive formula.

Exponential moving average

$$s_1 = x_0$$

$$s_t = \alpha x_{t-1} + (1 - \alpha)s_{t-1}$$

Exponential smoothing and moving average smoothing are similar in that they both assume a stationary, not trending, time series, therefore lagging behind the trend if one exists. Exponential smoothing also take into account all past data, whereas moving average only takes into account k past data points.

Double exponential smoothing

Double exponential smoothing can be used when there is a trend in the data.

Let {\(x_t\)} be the raw data sequence, {\(s_t\)} be the smoothed value for time t, and {\(b_t\)} be the best estimate of the trend at time t. The output of the algorithm is now written as \(F_{t+m}\). The formulae are:

$$s_1 = x_0$$

$$b_1 = x_1 - x_0$$

And for t>1

$$s_t = \alpha x_t + (1 - \alpha)(s_{t-1} + b_{t-1})$$

$$b_t = \beta(s_t - s_{t-1}) + (1 - \beta)b_{t-1}$$

where \(\alpha\) is the smoothing factor, \(0 < \alpha < 1\), and \(\beta\) is the trend smoothing factor, \(0 < \beta < 1\).