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# Introduction to Curve Fitting

Last modified: November 6, 2022

## 1. Overview

In this tutorial, **we’ll briefly introduce curve fitting.** First, we’ll present the basic terminology and the main categories of curve fitting, and then we’ll present the least-squares algorithm for curve fitting along with a detailed example.

## 2. Preliminaries

Let’s suppose that we are given a set of measured data points. **Curve fitting is the process of finding a mathematical function in an analytic form that best fits this set of data.** The first question that may arise is **why do we need that.** There are many cases that curve fitting can prove useful:

- quantify a general trend of the measured data
- remove noise from a function
- extract meaningful parameters from the learning curve
- summarize the relationships among two or more variables

## 3. Categories

**Generally, curve fitting falls into two main categories based on how we handle the observed data:**

### 3.1. Best Fit

**Here, we assume that the measured data points are noisy.** So, we don’t want to fit a curve that intercepts every data point. **Our goal is to learn a function that minimizes some predefined error on the given data points.**

The simplest best fit method is **linear regression, where the curve is a straight line.** More formally, we have the parametric function were is the slope and is the intercept and a set of samples . Our goal is to learn the values of and to minimize an error criterion on the given samples.

More generally, **polynomial regression refers to the case that we want to fit a polynomial of a specific order to our data:**

- linear when
- quadratic when
- cubic when

In the functions above, we can observe that each time **the parameters we want to learn are equal to .** Below, we can see some examples of curve fitting using different functions:

Also, we can fit any other function to our data like:

- trigonometric functions
- Gaussian functions
- sigmoid functions

### 3.2. Exact Fit

**In this case, we assume that the given samples are not noisy, and we want to learn a curve that passes through each point.** It is useful in cases we want to derive finite-difference approximations or to find minimums, maximums, and zero crossings of a function.

In the figure below, we can see an example of an exact fit using polynomial interpolation.

## 4. Least-Squares Algorithm

There are many proposed algorithms for curve fitting. **The most well-known method is least squares, where we search for a curve such that the sum of squares of the residuals is minimum.** By saying residual, we refer to the difference between the observed sample and the estimation from the fitted curve. It can be used both for linear and non-linear relationships.

In the linear case when , the function we want to fit is . So, we want to find the parameters so as to minimize (sum of residuals).

In order to minimize the above term there are two conditions:

**In non-linear least-squares problems, a very famous algorithm is the Gauss-Newton method.** While we don’t have theoretical proof that the algorithm converges, it finds a proper solution in practice and is easy to implement.

## 5. Example

**Here we’ll see an example of fitting a straight line in a set of samples using the least-squares method.** Let’s suppose we have the following data:

In the graph below, we can see the data in a scatter plot:

We want to fit the linear function . If we replace our data in the equations we derived in the previous section we have the following results:

- 26 = 5b + 15a
- 90 = 15b + 55a

We solve the above system of two equations and two variables, and we find that and . So, we have the function :

In the graph above, the red line corresponds to the fitted curve on the input data (blue dots). **As we expected, the fitted line is the one that has the minimum distance from the input points.**

## 6. Conclusion

In this tutorial, we talked about curve fitting. First, we made an introduction to the basic terminology and the categories of curve fitting. Then, we presented the least-squares algorithm both theoretically and through an example.