Least Squares Data Fitting with Applications

1. The Linear Data Fitting Problem

Chapter 1 The Linear Data Fitting Problem This chapter gives an introduction to the linear data fitting problem: how it is defined, its mathematical aspects and how it is analyzed. We also give important statistical background that provides insight into the data fitting problem. Anyone with more interest in the subject is encouraged to consult the pedagogical expositions by Bevington [13], Rust [213], Strutz [233] and van den Bos [242]. We start with a couple of simple examples that introduce the basic concepts of data fitting. Then we move on to a more formal definition, and we discuss some statistical aspects. Throughout the first chapters of this book we will return to these data fitting problems in order to illustrate the ensemble of numerical methods and techniques available to solve them. 1.1 Parameter estimation, data approximation Example 1. Parameter estimation. In food-quality analysis, the amount and mobility of water in meat has been shown to affect quality attributes like appearance, texture and storage stability. The water contents can be measured by means of nuclear magnetic resonance (NMR) techniques, in which the measured signal reflects the amount and properties of different types of water environments in the meat. Here we consider a simplified example involving frozen cod, where the ideal time signal φ(t) from NMR is a sum of two damped exponentials plus a constant background, φ(t) = x1 e−λ1t + x2 e−λ2t + x3, λ1, λ2 > 0. In this example we assume that we know the parameters λ1 and λ2 that control the decay of the two exponential components. In practice we do not 1 2 LEAST SQUARES DATA FITTING WITH APPLICATIONS Figure 1.1.1: Noisy measurements of the time signal φ(t) from NMR, for the example with frozen cod meat. measure this pure signal, but rather a noisy realization of it as shown in Figure 1.1.1. The parameters λ1 = 27 s−1 and λ2 = 8 s−1 characterize two different types of proton environments, responsible for two different water mobilities. The amplitudes x1 and x2 are proportional to the amount of water contained in the two kinds of proton environments. The constant x3 accounts for an undesired background (bias) in the measurements. Thus, there are three unknown parameters in this model, namely, x1, x2 and x3. The goal of data fitting in relation to this problem is to use the measured data to estimate the three unknown parameters and then compute the different kinds of water contents in the meat sample. The actual fit is presented in Figure 1.2.1. In this example we used the technique of data fitting for the purpose of estimating unknown parameters in a mathematical model from measured data. The model was dictated by the physical or other laws that describe the data. Example 2. Data approximation. We are given measurements of air pollution, in the form of the concentration of NO, over a period of 24 hours, on a busy street in a major city. Since the NO concentration is mainly due to the cars, it has maximum values in the morning and in the afternoon, when the traffic is most intense. The data is shown in Table 1.1 and the plot in Figure 1.2.2. For further analysis of the air pollution we need to fit a smooth curve to the measurements, so that we can compute the concentration at an arbitrary time between 0 and 24 hours. For example, we can use a low-degree polynomial to model the data, i.e., we assume that the NO concentration can be approximated by f(t) = x1 tp + x2 tp−1 + · · · + xp t + xp+1, THE LINEAR DATA FITTING PROBLEM 3 ti yi ti yi ti yi ti yi ti yi 0 110.49 5 29.37 10 294.75 15 245.04 20 216.73 1 73.72 6 74.74 11 253.78 16 286.74 21 185.78 2 23.39 7 117.02 12 250.48 17 304.78 22 171.19 3 17.11 8 298.04 13 239.48 18 288.76 23 171.73 4 20.31 9 348.13 14 236.52 19 247.11 24 164.05 Table 1.1: Measurements of NO concentration yi as a function of time ti. The units of yi and ti are μg/m3 and hours, respectively. where t...