LinearFit¶

linear fit must pass (x0, y0)¶

create a linear fit that must pass a specified point (x0, y0).

• shift the data points so that the desired point (x0, y0) becomes the new origin (0, 0)

• we only need to calculate the slope (linear regression), as the intercept of the line passing through the new origin is zero

• get the intercept using (x0, y0): y - y0 = k (x - x0) => y = kx + (y0 - kx0)

  import numpy as np
  import matplotlib.pyplot as plt
  
  def constrained_linear_fit(x, y, x0, y0):
      """
      Performs a linear fit (y = kx + b) that must pass through the point (x0, y0).
  
      Args:
          x: Array of x-values.
          y: Array of y-values.
          x0: x-coordinate of the point the line must pass through.
          y0: y-coordinate of the point the line must pass through.
  
      Returns:
          tuple: (slope, intercept) of the fitted line.
      """
  
      if len(x) != len(y):
          raise ValueError('x and y arrays must have the same length.')
  
      # Modified x and y values for fitting
      x_ = x - x0
      y_ = y - y0
  
      # Calculate the slope (k) using a simplified linear regression
      k = np.sum(x_ * y_) / np.sum(x_**2)
  
      # Calculate the intercept (b) using the constraint point
      b = y0 - k * x0
  
      return k, b
  
  # Example Usage
  x = np.array([-300, -150, 0, 150, 300])
  y = np.array([450, 290, 250, 180, 120])
  x0 = 0    # Desired x-coordinate
  y0 = 250  # Desired y-coordinate
  
  slope, intercept = constrained_linear_fit(x, y, x0, y0)
  print(f'Slope (k): {slope}')
  print(f'Intercept (b): {intercept}')
  
  # Generate points for plotting the fitted line
  x_fit = np.linspace(min(x) - 1, max(x) + 1, 100)
  y_fit = slope * x_fit + intercept
  
  # Plotting
  plt.scatter(x, y, label='Data Points')
  plt.plot(x_fit, y_fit, color='red', label=f'Fit: y = {slope:.2f}x + {intercept:.2f}')
  plt.scatter(x0, y0, color='green', label='Constraint Point')
  plt.xlabel('x')
  plt.ylabel('y')
  plt.title('Constrained Linear Fit')
  plt.legend()
  plt.grid(True)
  plt.show()