Add Mean Shift clustering algorithm in machine_learning/

machine_learning/mean_shift.py

@@ -0,0 +1,260 @@
+"""
+Mean Shift Clustering
+
+A non-parametric, centroid-based clustering algorithm that does not require
+specifying the number of clusters in advance. It works by iteratively shifting
+each data point toward the mean of points within a given bandwidth (radius),
+until convergence.
+
+How it works:
+    1. Each point starts as its own candidate centroid.
+    2. For each candidate, compute the mean of all points within `bandwidth`
+       distance (the "window").
+    3. Shift the candidate to that mean.
+    4. Repeat until candidates stop moving (convergence).
+    5. Merge candidates that are closer than `bandwidth` to each other.
+    6. Assign each original point to its nearest final centroid.
+
+Key Properties:
+    - No need to specify number of clusters (unlike K-Means)
+    - Can find arbitrarily shaped clusters (like DBSCAN)
+    - Sensitive to the `bandwidth` parameter
+    - Deterministic (no random initialization)
+
+Time Complexity:  O(n² * iterations) with brute-force window search
+Space Complexity: O(n)
+
+References:
+    - https://en.wikipedia.org/wiki/Mean_shift
+    - Comaniciu, D. & Meer, P. "Mean Shift: A Robust Approach Toward
+      Feature Space Analysis." IEEE TPAMI, 2002.
+      https://doi.org/10.1109/34.1000236
+"""
+
+
+def euclidean_distance(point_a: list[float], point_b: list[float]) -> float:
+    """
+    Compute the Euclidean distance between two n-dimensional points.
+
+    >>> euclidean_distance([0.0, 0.0], [3.0, 4.0])
+    5.0
+    >>> euclidean_distance([1.0, 1.0], [1.0, 1.0])
+    0.0
+    >>> euclidean_distance([0.0], [5.0])
+    5.0
+    >>> euclidean_distance([0.0, 0.0], [1.0])
+    Traceback (most recent call last):
+        ...
+    ValueError: Both points must have the same number of dimensions.
+    """
+    if len(point_a) != len(point_b):
+        raise ValueError("Both points must have the same number of dimensions.")
+    return sum((a - b) ** 2 for a, b in zip(point_a, point_b)) ** 0.5
+
+
+def get_points_within_bandwidth(
+    data: list[list[float]], center: list[float], bandwidth: float
+) -> list[list[float]]:
+    """
+    Return all points in data that lie within `bandwidth` distance of `center`.
+
+    >>> data = [[0.0, 0.0], [0.5, 0.5], [5.0, 5.0]]
+    >>> get_points_within_bandwidth(data, [0.0, 0.0], 1.0)
+    [[0.0, 0.0], [0.5, 0.5]]
+    >>> get_points_within_bandwidth(data, [5.0, 5.0], 1.0)
+    [[5.0, 5.0]]
+    >>> get_points_within_bandwidth(data, [0.0, 0.0], 10.0)
+    [[0.0, 0.0], [0.5, 0.5], [5.0, 5.0]]
+    """
+    return [point for point in data if euclidean_distance(point, center) <= bandwidth]
+
+
+def compute_mean(points: list[list[float]]) -> list[float]:
+    """
+    Compute the element-wise mean of a list of points.
+
+    >>> compute_mean([[1.0, 2.0], [3.0, 4.0]])
+    [2.0, 3.0]
+    >>> compute_mean([[0.0, 0.0, 0.0]])
+    [0.0, 0.0, 0.0]
+    >>> compute_mean([])
+    Traceback (most recent call last):
+        ...
+    ValueError: Cannot compute mean of empty list.
+    """
+    if not points:
+        raise ValueError("Cannot compute mean of empty list.")
+    n_dims = len(points[0])
+    return [sum(point[dim] for point in points) / len(points) for dim in range(n_dims)]
+
+
+def shift_point(
+    point: list[float], data: list[list[float]], bandwidth: float
+) -> list[float]:
+    """
+    Shift a single point to the mean of all data points within `bandwidth`.
+
+    If no points fall within the bandwidth, the point remains unchanged.
+
+    >>> data = [[1.0, 1.0], [1.5, 1.5], [10.0, 10.0]]
+    >>> shift_point([1.0, 1.0], data, 2.0)
+    [1.25, 1.25]
+    >>> shift_point([10.0, 10.0], data, 1.0)
+    [10.0, 10.0]
+    """
+    neighbors = get_points_within_bandwidth(data, point, bandwidth)
+    if not neighbors:
+        return point
+    return compute_mean(neighbors)
+
+
+def has_converged(
+    old_point: list[float], new_point: list[float], tolerance: float
+) -> bool:
+    """
+    Check whether a point has converged (moved less than `tolerance`).
+
+    >>> has_converged([1.0, 1.0], [1.0000001, 1.0000001], 1e-4)
+    True
+    >>> has_converged([1.0, 1.0], [1.5, 1.5], 1e-4)
+    False
+    """
+    return euclidean_distance(old_point, new_point) < tolerance
+
+
+def merge_centroids(
+    centroids: list[list[float]], bandwidth: float
+) -> list[list[float]]:
+    """
+    Merge centroids that are within `bandwidth` distance of each other.
+
+    Iterates through centroids and greedily merges any that are close enough,
+    keeping the first encountered as the representative.
+
+    >>> centroids = [[1.0, 1.0], [1.1, 1.1], [10.0, 10.0]]
+    >>> merged = merge_centroids(centroids, 1.0)
+    >>> len(merged)
+    2
+    >>> centroids = [[0.0, 0.0], [5.0, 5.0], [10.0, 10.0]]
+    >>> len(merge_centroids(centroids, 1.0))
+    3
+    """
+    merged: list[list[float]] = []
+    for centroid in centroids:
+        if all(
+            euclidean_distance(centroid, existing) >= bandwidth for existing in merged
+        ):
+            merged.append(centroid)
+    return merged
+
+
+def mean_shift(
+    data: list[list[float]],
+    bandwidth: float,
+    max_iterations: int = 300,
+    tolerance: float = 1e-4,
+) -> list[int]:
+    """
+    Perform Mean Shift clustering on a dataset.
+
+    Args:
+        data:           List of n-dimensional data points.
+        bandwidth:      Radius of the window used to compute the mean.
+                        Must be greater than 0.
+        max_iterations: Maximum number of shift iterations per point.
+                        Must be at least 1.
+        tolerance:      Convergence threshold — stop shifting when movement
+                        is smaller than this value. Must be greater than 0.
+
+    Returns:
+        A list of integer cluster labels, one per input point.
+        Cluster IDs start from 0.
+
+    Raises:
+        ValueError: If data is empty.
+        ValueError: If bandwidth is not positive.
+        ValueError: If max_iterations is less than 1.
+        ValueError: If tolerance is not positive.
+
+    Example — two well-separated clusters:
+    >>> data = [
+    ...     [1.0, 1.0], [1.2, 1.0], [1.0, 1.2],
+    ...     [9.0, 9.0], [9.2, 9.0], [9.0, 9.2],
+    ... ]
+    >>> labels = mean_shift(data, bandwidth=2.0)
+    >>> len(set(labels))  # two clusters
+    2
+    >>> labels[0] == labels[1] == labels[2]  # first group same cluster
+    True
+    >>> labels[3] == labels[4] == labels[5]  # second group same cluster
+    True
+    >>> labels[0] != labels[3]               # different clusters
+    True
+
+    Example — single cluster (all points close together):
+    >>> data = [[0.0, 0.0], [0.1, 0.0], [0.0, 0.1], [0.1, 0.1]]
+    >>> labels = mean_shift(data, bandwidth=2.0)
+    >>> len(set(labels))
+    1
+
+    Example — invalid inputs:
+    >>> mean_shift([], bandwidth=1.0)
+    Traceback (most recent call last):
+        ...
+    ValueError: Data must not be empty.
+    >>> mean_shift([[1.0, 2.0]], bandwidth=0.0)
+    Traceback (most recent call last):
+        ...
+    ValueError: Bandwidth must be greater than 0.
+    >>> mean_shift([[1.0, 2.0]], bandwidth=1.0, max_iterations=0)
+    Traceback (most recent call last):
+        ...
+    ValueError: max_iterations must be at least 1.
+    >>> mean_shift([[1.0, 2.0]], bandwidth=1.0, tolerance=0.0)
+    Traceback (most recent call last):
+        ...
+    ValueError: Tolerance must be greater than 0.
+    """
+    if not data:
+        raise ValueError("Data must not be empty.")
+    if bandwidth <= 0:
+        raise ValueError("Bandwidth must be greater than 0.")
+    if max_iterations < 1:
+        raise ValueError("max_iterations must be at least 1.")
+    if tolerance <= 0:
+        raise ValueError("Tolerance must be greater than 0.")
+
+    # each point starts as its own candidate centroid
+    candidates = [point[:] for point in data]
+
+    for _ in range(max_iterations):
+        new_candidates = [
+            shift_point(candidate, data, bandwidth) for candidate in candidates
+        ]
+        if all(
+            has_converged(old, new, tolerance)
+            for old, new in zip(candidates, new_candidates)
+        ):
+            break
+        candidates = new_candidates
+
+    centroids = merge_centroids(candidates, bandwidth)
+
+    # assign each original point to its nearest centroid
+    labels = [
+        min(
+            range(len(centroids)),
+            key=lambda centroid_index: euclidean_distance(
+                point, centroids[centroid_index]
+            ),
+        )
+        for point in data
+    ]
+
+    return labels
+
+
+if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod(verbose=True)