Regression mathematics#

Trendline overview is the practical starting point. This page summarises the shared numerical methods behind the available models.

Least squares#

GraphX fits selected calibrated graph points by minimising residual error, where

$r_i=y_i-\hat y_i,\qquad SSE=\sum_i r_i^2.$

Linear regression#

Linear regression fits $y=ax+b$ by least squares. For Linear models, an optional fixed Y intercept constrains the fit to the supplied value rather than estimating $b$. See Linear regression.

Polynomial regression#

Polynomial models use orders 2–6. Direct powers of a large or offset X value can be poorly conditioned, so GraphX centres and scales X with $t=(x-c)/s$, then constructs a polynomial design matrix. It solves the least-squares problem with modified Gram–Schmidt QR decomposition:

$A=QR,\qquad Rc=Q^Ty.$

A fixed intercept uses a constrained basis. GraphX evaluates the stable centred/scaled basis; when expanding it into original-X coefficients would lose meaningful precision, it displays the centred/scaled representation instead. See Polynomial regression.

Transformed models#

• Logarithmic regression uses $z=\ln(x)$ and fits $y=a+bz$; $x>0$ is required.
• Exponential regression fits $y=ae^{bx}$ by fitting $\ln(y)=\ln(a)+bx$; $y>0$ is required.
• Power-law regression fits $y=ax^b$ by fitting $\ln(y)=\ln(a)+b\ln(x)$; both $x>0$ and $y>0$ are required.

R²#

GraphX reports original-Y-space $R^2$ from

$R^2=1-\frac{SSE}{SST},\qquad SST=\sum_i(y_i-\bar y)^2.$

Exponential and power models additionally report transformed-space $R^2$ for their transformed least-squares calculation. The two answer different questions; see Understanding R². Fit and display domains remain separate as described in Fit and display ranges.