Denoising Techniques in Time-series Forecasting

Often, before being able to use a time-series, it is necessary to make it more smooth, or remove the noise that is always present in time-series.
Classical approaches to this problem have always been to apply a moving average or an exponential moving average to the time-series to obtain the denoised or smoothed one. As you can read in this article it is however very dangerous to use moving averages on time-series because they normally change the statistical properties of the time-series.

Another approach has recently emerged that is based on Wavelet Transforms. This method has a very interesting property that is, together with denoising and smoothing the time-series it maintains the time synchronization with the original time-series.

Let's consider the following dataset:

this could be representing a classical business time-series (demand for a particular good, price of a stock).

On this time-series we will apply an adaptive response rate exponential smoothing that is classically used to smooth the time-series and a Daubechies denoising algorithm. In the following picture you can see the graphical effect of both:

As you can see from the picture the Adaptive response rate algorithm (as any moving average) is partially lagged and "late" with respect to the original time-series while the Daubechies denoising is perfectly on time.

Of course both algorithms are changing the statistical properties of the time-series as already described in another article.