Linearity is privileged mostly because it is the simplest type of relationship. This may seem arbitrary, but:
It generalizes well. For example, if y is a polynomial function of x then y can be viewed as a linear function of the powers of x;
Simple relationships have fewer free parameters and so should be favoured when selecting between models of comparable explanatory power;
Simple relationships and linearity in particular have very nice mathematical properties that allow much deeper analysis than one might expect from their simplicity;
A huge range of relationships are locally linear, in the sense of differentiability (which is a linear concept).
The first and last points in particular are used very widely in practice to great effect.
A researcher seeing the top-right chart is absolutely going to look for a suitable family of functions (such as polynomials where the relationship is a linear function of coefficients) and then use something like a least-squares method (which is based on linear error models) to find the best parameters and check how much variance (also based on linear foundations) remains unexplained by the proposed relationship.
Linearity is privileged mostly because it is the simplest type of relationship. This may seem arbitrary, but:
It generalizes well. For example, if y is a polynomial function of x then y can be viewed as a linear function of the powers of x;
Simple relationships have fewer free parameters and so should be favoured when selecting between models of comparable explanatory power;
Simple relationships and linearity in particular have very nice mathematical properties that allow much deeper analysis than one might expect from their simplicity;
A huge range of relationships are locally linear, in the sense of differentiability (which is a linear concept).
The first and last points in particular are used very widely in practice to great effect.
A researcher seeing the top-right chart is absolutely going to look for a suitable family of functions (such as polynomials where the relationship is a linear function of coefficients) and then use something like a least-squares method (which is based on linear error models) to find the best parameters and check how much variance (also based on linear foundations) remains unexplained by the proposed relationship.
This hit the spot for me, thanks.