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One of the big differences between working on purely numerical solutions and working with data from experiments is that the experimental signals are often more complex than expected. This is not improved by the additional noise usually found in measured signals. Analyzing this data often requires that trends in the data be extracted and ideally expressed as a simple mathematical function. Determining an appropriate mathematical function is the domain of the many curve-fitting techniques that are commonly used.
Complex curves can also be created from many simpler mathematical functions by adding them together. The individual components of these synthesized curves can be explored separately. We have already shown that there are ways to introduce noise into numerical functions and that the noise in data can be filtered to reduce its impact. Adding noise to a synthesized curve can make the curve more like an experimental data set.
The process of determining whether a curve fits a data set requires the development of metrics to use for comparison. These metrics provide a measure of the quality of the fit between the curve and the data. One simple metric that we will develop will provide a “goodness of the fit” test. This simple test relies on the curve through the data being, on average, equidistant from the curve so that the sum is zero.
In this chapter we will introduce some simple tools for working with complex signals contaminated with unwanted information. The appropriate statistics such as the mean and the standard deviation will be used, and we will go farther by showing that we can find the trend in the data and remove this bias so that we can look in more detail at the noise signal.
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