The method of least squares can be extended to fit various types of curves, including polynomials, exponential curves, logarithmic curves, and other nonlinear functions. The general approach remains the same: minimize the sum of squared differences between observed and predicted values. Here’s how it can be applied to different types of curves:

1. Polynomial Curve Fitting:
   • To fit a polynomial curve to a set of data points, the model becomes:

     $$

     y=a0​+a1​x+a2​x2+…+an​xn

   • Here,

     $$

     n is the degree of the polynomial, and
     ${}_{}$

     a0​,a1​,…,an​ are the coefficients to be estimated.

   • The method of least squares is applied similarly: calculate residuals, square them, and minimize the sum of squared residuals by adjusting the coefficients.
   • Once the coefficients are estimated, the polynomial curve of degree $n$ is determined.
2. Exponential Curve Fitting:
   • Exponential curves follow the general form: y=aebx

   • $b$ is the growth rate.

   • To fit an exponential curve, take the natural logarithm of both sides to linearize the equation:

     $\ln$

     ln(y)=ln(a)+bx

   • Now, the model is linear in terms of the parameters

     $\ln$

     ln(a) and
     $$

     b, and the method of least squares can be applied as before.

3. Logarithmic Curve Fitting:
   • Logarithmic curves follow the general form:

     $$

     y=aln(x)+b

   • Similar to exponential curves, logarithmic curves can be linearized by transforming the equation:

     $$

     y−b=aln(x)

   • Now, the model is linear in terms of the parameters

     $$

     a and
     $$

     b, and the method of least squares can be applied to estimate them.

4. Other Nonlinear Curve Fitting:
   • For other types of curves, such as power functions, sigmoid curves, or trigonometric functions, similar techniques can be applied.
   • Linearize the equation if possible by transforming it to a linear form.
   • Apply the method of least squares to estimate the parameters of the linearized model.
   • Use the estimated parameters to reconstruct the original curve.

the method of least squares is a versatile tool for curve fitting, allowing for the estimation of parameters in various types of functions. By minimizing the sum of squared residuals, it provides a robust way to find the best-fitting curve to a given set of data points, regardless of the functional form of the curve.