Linear Regression — Linearisable Models

Fit your (x, y) dataset by least squares: linear, exponential, logarithmic, or power — with R², RMSE, and a LaTeX-rendered equation.

6 / 7

Tip: paste from Excel or drop a .csv file

Fitted equationn = 6

y = 1.986\,x + 0.1

Slope (a)

1.986

Intercept (b)

0.1

R² Coefficient

0.9993

fit

RMSE

0.09258

y units

Fundamentals & Explanation

Ordinary least squares

Linear regression is the workhorse method for describing the relationship between two numerical variables. It assumes y depends on x proportionally. We pick the line that best represents the trend by reducing the accumulated squared error of all points to its minimum.

General Model

\hat{y}_i = a\,x_i + b

Residual Minimization

\min_{a,b}\;\sum_{i=1}^{n}\bigl(y_i - a\,x_i - b\bigr)^2

We seek the estimators a and b that make the vertical distances (residuals) between each observed datum and the curve as small as possible.

Closed-Form Coefficients

Since regression (or Ordinary Least Squares) is a strictly quadratic problem, it has a closed-form analytical solution; meaning, it does not require an iterative approximation to deduce the coefficients:

Slope (a)

a = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}

Intercept (b)

b = \bar{y} - a\,\bar{x}

Linearising non-linear models

Although born as a straight-line method, it is possible to adapt families of curves (such as exponential growth, decay, or logarithmic saturation) so that they are solved as a classical regression plane. This is achieved by introducing logarithms in the independent axis, the dependent one, or both.

\text{Exponential: } y = a\,e^{b x} \;\Rightarrow\; \ln y = \ln a + b\,x

\text{Logarithmic: } y = a + b\,\ln x

\text{Power: } y = a\,x^{b} \;\Rightarrow\; \ln y = \ln a + b\,\ln x

Distorted Error Consideration: By clearing using logarithms, we convert a curve fit into traditional least squares on transformed variables. The constraint here is that the organic error structure is deformed; statistically giving a different weight to outliers than they correspond to on the natural linear scale.

Goodness of Fit

Every model needs tools to validate how closely the synthetic numbers describe the natural case: R-Squared ( $R^2$ ) and the Root Mean Square Error ( $\text{RMSE}$ ).

R^2 = 1 - \dfrac{\sum (y_i - \hat{y}_i)^2}{\sum (y_i - \bar{y})^2}

\text{RMSE} = \sqrt{\dfrac{1}{n}\sum (y_i - \hat{y}_i)^2}

$R^2 \approx 1$ indicates that the model explains almost all the variability of your data; instead, an R² close to 0 will mathematically warn you about the inability of the algorithm to predict on that point cloud.

On the other side, $\text{RMSE}$ rescues the idea measuring in "the same units as y". So the evaluator has a natural physical human understanding of the average error.

Linear Regression — Linearisable Models

Fundamentals & Explanation

Ordinary least squares

Closed-Form Coefficients

Linearising non-linear models

Goodness of Fit

References & Further Reading

You may also like:

Ordinary least squares

Closed-Form Coefficients

Linearising non-linear models

Goodness of Fit

References & Further Reading