Constructing a regression model

A multiple linear regression model is of the form:

(1)
\begin{align} Y = A_0 + A_1\times X_1 + A_2\times X_1 + ... + A_n\times X_n \end{align}

where:
$Y$ is the data vector to be fitted. Most often Y is a time series of measurements or output from a model.
$A_m$ are the regression model fit coefficients.
$X_m$ are the basis functions on which the data vector is thought to depend.

This form of regression model is often used when we want to quantify relationships that we believe exist, or in some cases to look for relationships that may exist. Examples of linear regression include: deriving correlation coefficients between two variables, identifying trends or quantifying relationships between two or more measured variables such as the response of ozone to halogen concentrations. For the case when the quantity of interest depends on only one other variable e.g. the dependence of temperature on time, or the dependence of methane on humidity, this is often referred to as simple linear regression, or SLR for short. When the quantity of interest depends on two or more parameters, it falls under the more general category of multiple linear regression or MLR for short.

Constructing your regression model essentially comes down to selecting your basis functions. When you select the basis functions, you are defining a physical model which you believe represents the key real-world process that link changes in your basis functions to changes in the time series being fitted (the data vector). Things to consider when choosing your basis functions:

  1. Does $Y$ depend linearly on the basis function? If not, you may want to include higher order terms e.g. an additional basis function where the basis function values are squared (see here).
  2. Does the dependence of $Y$ on a specific $X_n$ depend on anything else? For example, does it depend on season? If yes, see here. Does it depend on latitude and/or longitude? If yes, see here. Does it depend on the value of some other basis function? If yes, see here)

You will also need to give some consideration as to how to 'pre-treat' your basis functions.

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