MATLAB Curve Segment Fitting: Handling Complex Curves to Improve Accuracy

# Introduction to MATLAB Curve Fitting

MATLAB curve fitting is a powerful technique used to fit complex curves to a set of data points. It approximates data by finding one or more functions, revealing underlying patterns and trends. Curve fitting is widely applied in various scientific and engineering fields, such as data analysis, modeling, and forecasting.

MATLAB provides various curve fitting tools, including piecewise fitting. Piecewise fitting is an advanced technique that divides a curve into several segments and applies different functions to each segment. This approach can significantly improve the accuracy of fitting for complex curves, especially where nonlinearities or discontinuities are present.

# Theoretical Basis of Piecewise Fitting

### Advantages and Principles of Piecewise Fitting

Piecewise fitt***pared to traditional global fitting, piecewise fitting has the following advantages:

- **Increased Precision:** By dividing the curve into smaller segments, piecewise fitting can more accurately capture the local characteristics of the curve, thus improving fitting accuracy.
- **Enhanced Robustness:** Piecewise fitting can reduce the impact of noise and outliers, enhancing the robustness of the fitting results.
- **Improved Efficiency:** After dividing the curve into smaller segments, each segment's fitting can be performed in parallel, thus improving the efficiency of fitting.
- **Enhanced Interpretability:** Piecewise fitting can break down the curve into several smaller segments, making it easier to analyze and understand the curve's changing patterns.

### Piecewise Fitting Algorithms

Piecewise fitting algorithms are primarily divided into two categories:

#### Least Squares Method

The least squares method is a commonly used piecewise fitting algorithm. The basic idea is: for each segment, find a curve that minimizes the sum of squared distances from all data points on the segment to the curve.

**Algorithm Steps:**

1. Divide the curve into multiple smaller segments.
2. Fit a curve to each segment using the least squares method.
3. Piece together all segment's fitted curves to obtain the piecewise fitted curve.

**Parameter Description:**

- `x`: The x-coordinates of data points
- `y`: The y-coordinates of data points
- `f(x)`: The fitted curve
- `n`: The number of data points
- `w`: Weight factor

**Code Block:**

% Data points
x = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10];
y = [1, 2, 4, 8, 16, 32, 64, 128, 256, 512];

% Piecewise fitting
segments = 3;
[p, S] = polyfit(x, y, segments);
f = polyval(p, x);

% Plot original data and fitted curve
plot(x, y, 'o');
hold on;
plot(x, f, 'r-');
hold off;

**Logical Analysis:**

The code first defines the data points `x` and `y`. Then, it uses the `polyfit` function for piecewise fitting, where the `segments` parameter specifies the number of segments. The `polyfit` function returns the fitting coefficients `p` and the covariance matrix `S`. Finally, the `polyval` function is used to calculate the fitted curve `f` and plot it.

#### Weighted Least Squares Method

The weighted least squares method is an improved version of the least squares method that can assign different weights to different data points. The weight factor can be determined based on the reliability or importance of the data points.

**Algorithm Steps:**

1. Divide the curve into multiple smaller segments.
2. Use the weighted least squares method to fit a curve to each segment.
3. Piece together all segment's fitted curves to obtain the piecewise fitted curve.

**Parameter Description:**

- `x`: The x-coordinates of data points
- `y`: The y-coordinates of data points
- `f(x)`: The fitted curve
- `n`: The number of data points
- `w`