How to find curve equation from data?

Solution 1:

From plotting you data (Wolfram|Alpha link), it does not look linear. So it better be fit by a polynomial. I assume you want to fit the data:

X   Y
1   4
2   8
3   13
4   18
5   24
..

using a quadratic polynomial $y = ax^2 + bx + c.$ If so, then put your data in a matrix form (note that $x^0, x^1, x^2, y$ below are not actually in the matrix. They're just comments for your understanding): $$ \begin{pmatrix} \color{red}{x^0} & {\color{red} x} & \color{red}{x^2} \\ 1 & 1 & 1 \\ 1 & 2 & 4 \\ 1 & 3 & 9 \\ & \ldots & \end{pmatrix} \begin{pmatrix} c \\ b \\ a \end{pmatrix} = \begin{pmatrix} {\color{red} y} \\ 4 \\ 8 \\ 13 \\ \ldots \end{pmatrix} \tag{1} $$

Now, equation $(1)$ is really of the following form: $$ Xv = y \tag{2}$$ where each row encodes $cx^0 + bx+ax^2 = y$ for a particular pair of $(x,y)$ values. And we're looking for a solution vector $v^{T} = \begin{pmatrix} c & b & a \end{pmatrix}$ that gives the coefficients of that best fitting polynomial $ax^2 + bx+c$.

To solve for $v$, multiply $(2)$ both sides by $X^{T},$ we have $X^{T}Xv= X^{T}y,$ or $$ v = (X^{T}X)^{-1} X^{T}y.$$

This is a called linear least squares method because best means minimize the squared error. Several software packages can handle that for you. Luckily, Wolfram|Alpha can do.

To repeat for a polynomial of degree, say 4, construct $$ \begin{pmatrix} \color{red}{x^0} & {\color{red} x} & \color{red}{x^2} & \color{red}{x^3} & \color{red}{x^4} \\ 1 & 1 & 1 & 1 & 1\\ 1 & 2 & 4 & 8 & 16 \\ 1 & 3 & 9 & 27 & 81 \\ & \ldots & \end{pmatrix} \begin{pmatrix} e \\ d\\ c \\ b \\ a \end{pmatrix} = \begin{pmatrix} {\color{red} y} \\ 4 \\ 8 \\ 13 \\ \ldots \end{pmatrix} \tag{1} $$ and solve for $v$, this should give you the parameters $a,b,c,d,e$ s.t. $$ax^4+bx^3+cx^2+dx+e$$ best fits your data.