Mathematical Tools for Real-World Applications by Alexandr Draganov

Author:Alexandr Draganov
Language: eng
Format: epub, pdf
Tags: applied mathematics; problem solving; mathematical intuition; dimensionality; limiting cases; symmetry; scaling; order of magnitude estimates; successive approximations
Publisher: MIT Press

This scaling is illustrated in figure 4.5. As we stretch the plot horizontally, the slope of the straight line fit decreases, but the intersection of that line with the vertical axis stays intact.

3. Finally, we consider a very common scenario when a researcher processes some data and then gets additional data and repeats the processing. If our model is sound, we do not expect the parameters to vary significantly when the new data become available. This should show as an invariance with respect to scaling the number of data points N. In practice, this invariance is not perfect because the data contain random measurement errors, which are different from one batch of data to another. To prove this invariance mathematically, we consider a hypothetical case when the measurements are fully repeatable. We compare two data sets: one with N points xi, yi and another with 2N points , such that in the second data set all points can be grouped in pairs i and i + 1, where the variables are identical in each pair: . A plot for the second data set with 2N points would look exactly as a plot for the first data set with N points, but with plot markers pairwise overlaid on top of each other. For this idealized 2N data set we expect to get the same values of a and b. Indeed, let us consider the scaling of various sums in equations (4.33). In each sum, we group the terms by the pairs and scale the sum accordingly. All such sums will scale linearly with N. For example, we see that

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