The random walk was mentioned
in the section on signals and noise as a type of
low-frequency ("pink") noise. Wikipedia says:
random walk is a mathematical formalization of a path that
consists of a succession of random steps. For example, the path
traced by a molecule as it travels in a liquid or a gas, the
search path of a foraging animal, "superstring" behavior, the
price of a fluctuating stock and the financial status of a
gambler can all be modeled as random walks, although they may
not be truly random in reality."
Random walks describe and serve as a model for many kinds of unstable behavior. Whereas white, 1/f, and blue noises are anchored to a mean value to which they tend to return, random walks tend to be more aimless and often drift off on one or another direction, possibly never to return. Mathematically, a random walk can be modeled as the cumulative sum of some random process, for example the 'randn' function. The graph on the right compares a 200-point sample of white noise (computed as 'randn' and shown in blue) to a random walk (computed as a cumulative sum, 'cumsum', and shown in red). Both samples are scaled to have exactly the same standard deviation, but even so their behavior is vastly different. The random walk has much more low frequency behavior, in this case wandering off beyond the amplitude range of the white noise. This type of random behavior is very disruptive to the measurement process, distorting the shapes of peaks and causing baselines to shift and making them hard to define, and it can not be reduced significantly by smoothing (See NoiseColorTest.m). In this particular example, the random walk has an overall positive slope and a "bump" near the middle that could be confused for a real signal peak (it's not; it's just noise). But another sample might have very different behavior. Unfortunately, it is not uncommon to observe this behavior in experimental signals.
To demonstrate the measurement difficulties, the script RandomWalkBaseline.m simulates a Gaussian peak with randomly variable position and width, on a random walk baseline, with a S/N ratio is 15. The peak is measured by least-squares curve fitting methods using peakfit.m with two different methods of baseline correction in an attempt to handle the random walk:
(a) a single-component Gaussian model (shape 1) with autozero set to 1 (meaning a linear baseline is first interpolated from the edges of the data segment and subtracted from the signal): peakfit([x;y],0,0,1,1,0,10,1);
(b) a 2-component model, the first being a Gaussian (shape 1) and the second a linear slope (shape 26), with autozero set to 1: peakfit([x;y],0,0,2,[1 26],[0 0],10,0).
In this particular case the fitting error is lower for the second method, especially if the peak falls near the edges of the data range.
relative percent errors of the peak parameters show that the first method gives a
lower error for position and width, at least in this case. On
average, the peak parameters are about the same.
Height Error Width Error
Method a: 0.2772 3.0306 0.0125
Method b: 0.4938 2.3085 1.5418
You can compare this to WhiteNoiseBaseline.m which has a similar signal and S/N ratio, except that the noise is white. Interestingly, the fitting error with white noise is greater, but the parameter errors (peak position, height, width, and area) are lower, and the residuals are more random and less likely to produce false noise peaks. This is because the random walk noise is very highly concentrated at low frequencies where the signal frequencies usually lie, whereas white noise also has considerable power at higher frequencies, which increases the fitting error but does comparatively little damage to signal measurement accuracy. This may be slightly counter-intuitive, but it's important to realize that fitting error does not always correlate with peak parameter error. Bottom line: random walk is troublesome.
Depending on the type of experiment, an instrumental design based on modulation techniques may help, and ensemble averaging multiple measurements can help with any type of unpredictable random noise, which is discussed in the very next section.
This page is part of "A Pragmatic Introduction to Signal
Processing", created and maintained by Prof. Tom O'Haver ,
Department of Chemistry and Biochemistry, The University of
Maryland at College Park. Comments, suggestions and questions
should be directed to Prof. O'Haver at email@example.com. Updated July, 2022.