#### Local Weighted Learning

p
```"""
Locally weighted linear regression, also called local regression, is a type of
non-parametric linear regression that prioritizes data closest to a given
prediction point. The algorithm estimates the vector of model coefficients β
using weighted least squares regression:

β = (XᵀWX)⁻¹(XᵀWy),

where X is the design matrix, y is the response vector, and W is the diagonal
weight matrix.

This implementation calculates wᵢ, the weight of the ith training sample, using
the Gaussian weight:

wᵢ = exp(-‖xᵢ - x‖²/(2τ²)),

where xᵢ is the ith training sample, x is the prediction point, τ is the
"bandwidth", and ‖x‖ is the Euclidean norm (also called the 2-norm or the L²
norm). The bandwidth τ controls how quickly the weight of a training sample
decreases as its distance from the prediction point increases. One can think of
the Gaussian weight as a bell curve centered around the prediction point: a
training sample is weighted lower if it's farther from the center, and τ
controls the spread of the bell curve.

Other types of locally weighted regression such as locally estimated scatterplot
smoothing (LOESS) typically use different weight functions.

References:
- https://en.wikipedia.org/wiki/Local_regression
- https://en.wikipedia.org/wiki/Weighted_least_squares
- https://cs229.stanford.edu/notes2022fall/main_notes.pdf
"""

import matplotlib.pyplot as plt
import numpy as np

def weight_matrix(point: np.ndarray, x_train: np.ndarray, tau: float) -> np.ndarray:
"""
Calculate the weight of every point in the training data around a given
prediction point

Args:
point: x-value at which the prediction is being made
x_train: ndarray of x-values for training
tau: bandwidth value, controls how quickly the weight of training values
decreases as the distance from the prediction point increases

Returns:
m x m weight matrix around the prediction point, where m is the size of
the training set
>>> weight_matrix(
...     np.array([1., 1.]),
...     np.array([[16.99, 10.34], [21.01,23.68], [24.59,25.69]]),
...     0.6
... )
array([[1.43807972e-207, 0.00000000e+000, 0.00000000e+000],
[0.00000000e+000, 0.00000000e+000, 0.00000000e+000],
[0.00000000e+000, 0.00000000e+000, 0.00000000e+000]])
"""
m = len(x_train)  # Number of training samples
weights = np.eye(m)  # Initialize weights as identity matrix
for j in range(m):
diff = point - x_train[j]
weights[j, j] = np.exp(diff @ diff.T / (-2.0 * tau**2))

return weights

def local_weight(
point: np.ndarray, x_train: np.ndarray, y_train: np.ndarray, tau: float
) -> np.ndarray:
"""
Calculate the local weights at a given prediction point using the weight
matrix for that point

Args:
point: x-value at which the prediction is being made
x_train: ndarray of x-values for training
y_train: ndarray of y-values for training
tau: bandwidth value, controls how quickly the weight of training values
decreases as the distance from the prediction point increases
Returns:
ndarray of local weights
>>> local_weight(
...     np.array([1., 1.]),
...     np.array([[16.99, 10.34], [21.01,23.68], [24.59,25.69]]),
...     np.array([[1.01, 1.66, 3.5]]),
...     0.6
... )
array([[0.00873174],
[0.08272556]])
"""
weight_mat = weight_matrix(point, x_train, tau)
weight = np.linalg.inv(x_train.T @ weight_mat @ x_train) @ (
x_train.T @ weight_mat @ y_train.T
)

return weight

def local_weight_regression(
x_train: np.ndarray, y_train: np.ndarray, tau: float
) -> np.ndarray:
"""
Calculate predictions for each point in the training data

Args:
x_train: ndarray of x-values for training
y_train: ndarray of y-values for training
tau: bandwidth value, controls how quickly the weight of training values
decreases as the distance from the prediction point increases

Returns:
ndarray of predictions
>>> local_weight_regression(
...     np.array([[16.99, 10.34], [21.01, 23.68], [24.59, 25.69]]),
...     np.array([[1.01, 1.66, 3.5]]),
...     0.6
... )
array([1.07173261, 1.65970737, 3.50160179])
"""
y_pred = np.zeros(len(x_train))  # Initialize array of predictions
for i, item in enumerate(x_train):
y_pred[i] = np.dot(item, local_weight(item, x_train, y_train, tau)).item()

return y_pred

dataset_name: str, x_name: str, y_name: str
) -> tuple[np.ndarray, np.ndarray, np.ndarray]:
"""
Load data from seaborn and split it into x and y points
>>> pass    # No doctests, function is for demo purposes only
"""
import seaborn as sns

x_data = np.array(data[x_name])
y_data = np.array(data[y_name])

one = np.ones(len(y_data))

# pairing elements of one and x_data
x_train = np.column_stack((one, x_data))

return x_train, x_data, y_data

def plot_preds(
x_train: np.ndarray,
preds: np.ndarray,
x_data: np.ndarray,
y_data: np.ndarray,
x_name: str,
y_name: str,
) -> None:
"""
Plot predictions and display the graph
>>> pass    # No doctests, function is for demo purposes only
"""
x_train_sorted = np.sort(x_train, axis=0)
plt.scatter(x_data, y_data, color="blue")
plt.plot(
x_train_sorted[:, 1],
preds[x_train[:, 1].argsort(0)],
color="yellow",
linewidth=5,
)
plt.title("Local Weighted Regression")
plt.xlabel(x_name)
plt.ylabel(y_name)
plt.show()

if __name__ == "__main__":
import doctest

doctest.testmod()

# Demo with a dataset from the seaborn module
training_data_x, total_bill, tip = load_data("tips", "total_bill", "tip")
predictions = local_weight_regression(training_data_x, tip, 5)
plot_preds(training_data_x, predictions, total_bill, tip, "total_bill", "tip")
```  