A team of mathematical statisticians at Bristol University has developed a method of wavelet analysis which they have used to successfully de-noise and analyse time series data for applications as diverse as monitoring solar flares and infant sleep patterns.

Wavelets are relatively recent methods for representing functions (objects). They are a transform method, the most famous of which is probably the Fourier transform, discovered by Fourier in the 1800s.

In the 1950s, Fourier's transform methods were developed to perform the transform much more quickly – the Fast Fourier Transform (FFT). An FT helps answer the question, what is the frequency content of, for example, an image? If you resolved it to a greyscale image, how many changes would there be over the space of the image, ie what is the frequency of the changes?

In the 1980s, a new type of transform was discovered, wavelets, and developed by researchers Meyer, Daubechies, Mallat and others.

Ts gives you information about the frequency of changes within the function, but averaged out over the whole function. Wavelets give you information about a function at different scales and different locations simultaneously.

Wavelets also turn out to provide very sparse representations of many kinds of function, including those that contain step changes. For example, an FT might need very many coefficients to represent a 256 x 256 pixel image but, typically, a wavelet will require far fewer. Hence wavelets are very useful for applications like video compression where you need it to work in real time and without consuming vast amounts of memory and bandwidth.

Wavelets are also very good at picking out signal from noise, because the signal get well compressed and the noise does not. So non-zero signal coefficients can often be better picked out from the noise in the wavelet domain.

Piotr Fryzlewicz and Guy Nason at Bristol have recently introduced a new class of techniques called Haar-Fisz transforms (HFTs). These operate on the wide range of problems where the local signal variance is related to the local signal level. Such data occurs naturally in 'sensed objects', such as images in charged coupled devices (CCDs) or in microarrays in genetic determinations. Essentially, Haar-Fisz transforms stabilise variance in signals so that it is nearly constant at all scales and all locations.

Fryzlewicz and Nason, in conjunction with Veronique Delouille of the Royal Observatory of Belgium, applied adaptive data-driven HFT (DDHFT) methods to solar flare data collected by the solar X-ray sensor (XRS) on board the GOES-8 satellite. GOES - Geostationary Operational Environmental Satellites - are a series of geostationary satellites that form an integral part of US weather monitoring and forecasting. GOES-8 was launched on 13th April 1994 and carried a wide range of instrumentation.

As well as background X-ray information the XRS provides warning that a solar flare has occurred. Such a prediction is essential as the effects of large flares can disrupt earth-bound communications, navigation systems and even knock out power grids.

The underlying signal from the XRS appears "spiky", so it is natural to attempt to de-noise it by means of a wavelet-based method. But conventional wavelet methods yield extremely noisy estimates because they assume that the variance of the noise is constant over the support of the signal, which is not the case for extraterrestrial X-ray signals.

Fryzlewicz, Delouille and Nason used a DDHFT to first estimate the mean-variance relationship and then to variance-stabilise the XRS data in a multiscale way. The transformed XRS data could then be de-noised and the inverse DDHFT applied to obtain a successfully de-noised estimate.

A different multiscale approach was used to establish the relationship between infant sleep patterns and a time series of their heart rate measured by ECG (electrocardiogram).

Conventionally, paediatricians use video, EEG (electroencephalogram – measuring 'brain-waves') and EOG (electro-oculogram – measuring eye movements) to indicate whether the infant is in quiet sleep, active sleep or awake.

Sleep state is a useful indicator of potential problems such as apnea (sudden heart and respiratory failure) but measuring it can be expensive, potentially distressing to the infant because of the number of sensors that need to be attached, and it requires a hospital or laboratory environment. ECG, however, is relatively inexpensive and unobtrusive, since the leads are attached to the infant's chest which can be done readily at home.

So establishing the relationship between the time series of ECG and sleep state was vital to the paediatricians work in predicting infants' sleep-state-related health problems.