### Install Dependencies with requirements.txt

Source: https://scikit-fuzzy-c-means.readthedocs.io/en/latest/installation.html

Install all project dependencies listed in the requirements.txt file using pip.

```bash
pip install -r requirements.txt
```

--------------------------------

### Clone Repository and Install Dependencies (Manual)

Source: https://scikit-fuzzy-c-means.readthedocs.io/en/latest/_sources/installation.rst.txt

Clone the FuzzyCMeans repository and install its dependencies using pip and a requirements file.

```bash
git clone https://github.com/your-repo/fuzzy-c-means
cd fuzzy-c-means
```

```bash
pip install -r requirements.txt
```

--------------------------------

### Install Dependencies with Poetry

Source: https://scikit-fuzzy-c-means.readthedocs.io/en/latest/installation.html

Use Poetry to install project dependencies and automatically create a virtual environment.

```bash
poetry install
```

--------------------------------

### Install Poetry

Source: https://scikit-fuzzy-c-means.readthedocs.io/en/latest/installation.html

Install the Poetry dependency manager using pip if it is not already installed.

```bash
pip install poetry
```

--------------------------------

### Install FuzzyCMeans with pip

Source: https://scikit-fuzzy-c-means.readthedocs.io/en/latest/installation.html

Install the FuzzyCMeans package using pip after activating your virtual environment.

```bash
pip install scikit-fuzzy-c-means
```

--------------------------------

### Install Dependencies with Poetry

Source: https://scikit-fuzzy-c-means.readthedocs.io/en/latest/_sources/installation.rst.txt

Use Poetry to install project dependencies and automatically create a virtual environment. This command should be run after cloning the repository.

```bash
git clone https://github.com/your-repo/fuzzy-c-means
cd fuzzy-c-means
```

```bash
poetry install
```

--------------------------------

### Activate Poetry Shell

Source: https://scikit-fuzzy-c-means.readthedocs.io/en/latest/installation.html

Activate the virtual environment managed by Poetry to start working on the project.

```bash
poetry shell
```

--------------------------------

### Clone Repository and Navigate

Source: https://scikit-fuzzy-c-means.readthedocs.io/en/latest/installation.html

Clone the FuzzyCMeans repository from GitHub and navigate into the project directory.

```bash
git clone https://github.com/your-repo/fuzzy-c-means
cd fuzzy-c-means
```

--------------------------------

### K-Means++ Initialization for Cluster Centers

Source: https://scikit-fuzzy-c-means.readthedocs.io/en/latest/algorithm.html

Initializes cluster centers using the K-Means++ algorithm to ensure better initial placement and faster convergence. It selects initial centers probabilistically based on squared distances to existing centers.

```python
def _kmeans_plusplus_init(self, X):
    np.random.seed(self.random_state)
    centers = []
    initial_idx = np.random.choice(n_samples)
    centers.append(X[initial_idx])

    for _ in range(1, self.n_clusters):
        distances = np.min(cdist(X, np.array(centers)), axis=1)
        probabilities = distances**2 / np.sum(distances**2)
        next_idx = np.random.choice(n_samples, p=probabilities)
        centers.append(X[next_idx])

    return np.array(centers)

```

--------------------------------

### Create Virtual Environment (venv)

Source: https://scikit-fuzzy-c-means.readthedocs.io/en/latest/installation.html

Use this command to create a virtual environment for isolating project dependencies. Activate it using the appropriate command for your operating system.

```bash
python -m venv .venv
source .venv/bin/activate  # On macOS/Linux
# OR
.venv\Scripts\activate     # On Windows
```

--------------------------------

### Fuzzy C-Means Constructor Initialization

Source: https://scikit-fuzzy-c-means.readthedocs.io/en/latest/algorithm.html

Initializes the FuzzyCMeans class with parameters for number of clusters, fuzziness coefficient, maximum iterations, convergence tolerance, and random state for reproducibility.

```python
def __init__(self, n_clusters=3, m=2, max_iter=150, tol=1e-4, random_state=None):
    self.n_clusters = n_clusters
    self.m = m
    self.max_iter = max_iter
    self.tol = tol
    self.random_state = random_state

```

--------------------------------

### Select Best Parameters and Evaluate Clustering

Source: https://scikit-fuzzy-c-means.readthedocs.io/en/latest/usage.html

This snippet demonstrates how to select the optimal parameters for Fuzzy C-Means by sorting results based on the silhouette score, fitting the model, predicting memberships, and evaluating the clustering using Silhouette, Davies-Bouldin, and Calinski-Harabasz indices.

```python
best_params = results_df.sort_values("silhouette_score", ascending=False).iloc[0]
fcm_best = FuzzyCMeans(
    n_clusters=int(best_params["n_clusters"]), m=best_params["m"], random_state=42
)
fcm_best.fit(X_train)

# Predict memberships for the validation data
U_val_best = fcm_best._compute_memberships(X_val)
y_pred_best = U_val_best.argmax(axis=1)

# Evaluate using all metrics
silhouette = silhouette_score(X_val, y_pred_best)
davies_bouldin = davies_bouldin_score(X_val, y_pred_best)
calinski_harabasz = calinski_harabasz_score(X_val, y_pred_best)

print(f"Silhouette Score: {silhouette:.3f}")
print(f"Davies-Bouldin Index: {davies_bouldin:.3f}")
print(f"Calinski-Harabasz Index: {calinski_harabasz:.3f}")

```

--------------------------------

### Fit Fuzzy C-Means Model and Visualize Centers

Source: https://scikit-fuzzy-c-means.readthedocs.io/en/latest/_sources/usage.rst.txt

Initializes and fits a Fuzzy C-Means model to the training data. Visualizes the computed cluster centers on the training data.

```python
# Initialize and fit the Fuzzy C-Means model
fcm = FuzzyCMeans(n_clusters=3, m=2, random_state=42)
fcm.fit(X_train)

# Visualize the fuzzy cluster centers
sns.scatterplot(x=X_train[:, 0], y=X_train[:, 1], color="gray", alpha=0.6)
sns.scatterplot(
    x=fcm.cluster_centers_[:, 0],
    y=fcm.cluster_centers_[:, 1],
    color="red",
    s=100,
    label="Cluster Centers",
)
plt.title("Fuzzy Cluster Centers on Training Data")
plt.xlabel("Feature 1")
plt.ylabel("Feature 2")
plt.legend()
plt.show()
```

--------------------------------

### Select Best Parameters and Evaluate Final Model

Source: https://scikit-fuzzy-c-means.readthedocs.io/en/latest/_sources/usage.rst.txt

Selects the best hyperparameters based on the silhouette score from the tuning results. Fits the Fuzzy C-Means model with the best parameters and evaluates its performance on the validation data, printing the key metrics.

```python
# Select best parameters
best_params = results_df.sort_values("silhouette_score", ascending=False).iloc[0]
fcm_best = FuzzyCMeans(
    n_clusters=int(best_params["n_clusters"]),
    m=best_params["m"],
    random_state=42,
)
fcm_best.fit(X_train)

# Predict memberships for the validation data
U_val_best = fcm_best._compute_memberships(X_val)
y_pred_best = U_val_best.argmax(axis=1)

# Evaluate using all metrics
silhouette = silhouette_score(X_val, y_pred_best)
davies_bouldin = davies_bouldin_score(X_val, y_pred_best)
calinski_harabasz = calinski_harabasz_score(X_val, y_pred_best)

print(f"Silhouette Score: {silhouette:.3f}")
print(f"Davies-Bouldin Index: {davies_bouldin:.3f}")
print(f"Calinski-Harabasz Index: {calinski_harabasz:.3f}")
```

--------------------------------

### Split Data into Training and Validation Sets

Source: https://scikit-fuzzy-c-means.readthedocs.io/en/latest/_sources/usage.rst.txt

Splits the generated dataset into training and validation sets using train_test_split. The test_size parameter is set to 0.2, meaning 20% of the data will be used for validation.

```python
from sklearn.model_selection import train_test_split

# Split data into training and validation sets
X_train, X_val = train_test_split(X, test_size=0.2, random_state=42)
```

--------------------------------

### Fuzzy C-Means Model Fitting

Source: https://scikit-fuzzy-c-means.readthedocs.io/en/latest/algorithm.html

Fits the Fuzzy C-Means model to the input data X. It initializes cluster centers using K-Means++, then iteratively updates cluster centers and the membership matrix until convergence or max_iter is reached.

```python
def fit(self, X, y=None):
    n_samples, n_features = X.shape
    self.X_ = X

    # Step 1: Initialize cluster centers
    self.cluster_centers_ = self._kmeans_plusplus_init(X)

    # Step 2: Initialize membership matrix
    self.U_ = np.random.dirichlet(np.ones(self.n_clusters), size=n_samples)

    for iteration in range(self.max_iter):
        U_prev = self.U_.copy()

        # Step 3: Update cluster centers
        self.cluster_centers_ = self._compute_centers(X)

        # Step 4: Update membership matrix
        self.U_ = self._update_memberships(X)

        # Step 5: Check for convergence
        if np.linalg.norm(self.U_ - U_prev) < self.tol:
            break

    return self

```

--------------------------------

### Generate Synthetic Dataset with Overlapping Clusters

Source: https://scikit-fuzzy-c-means.readthedocs.io/en/latest/_sources/usage.rst.txt

Generates a synthetic dataset using make_blobs with increased standard deviation to create overlapping clusters. Visualizes the generated data.

```python
from sklearn.datasets import make_blobs
import matplotlib.pyplot as plt
import seaborn as sns
import pandas as pd
import numpy as np

# Generate synthetic dataset with overlapping clusters
X, y_true = make_blobs(
    n_samples=300,
    centers=3,
    cluster_std=2.5,  # Increased std for overlap
    random_state=42,
)

# Visualize the dataset
sns.scatterplot(x=X[:, 0], y=X[:, 1], hue=y_true, palette="deep", legend=None)
plt.title("Generated Data with Overlap")
plt.xlabel("Feature 1")
plt.ylabel("Feature 2")
plt.show()
```

--------------------------------

### Perform Hyperparameter Tuning for Fuzzy C-Means

Source: https://scikit-fuzzy-c-means.readthedocs.io/en/latest/_sources/usage.rst.txt

Performs a grid search over specified hyperparameters (n_clusters, m) for the Fuzzy C-Means model. Evaluates clustering performance on the validation set using silhouette, Davies-Bouldin, and Calinski-Harabasz scores.

```python
from sklearn.model_selection import ParameterGrid
from sklearn.metrics import (
    silhouette_score,
    davies_bouldin_score,
    calinski_harabasz_score,
)
import numpy as np

# Define hyperparameter grid
param_grid = {"n_clusters": [2, 3, 4], "m": [1.5, 2, 2.5]}

results = []

# Perform grid search
for params in ParameterGrid(param_grid):
    fcm = FuzzyCMeans(n_clusters=params["n_clusters"], m=params["m"], random_state=42)
    fcm.fit(X_train)

    # Predict membership on validation data
    U_val = fcm._compute_memberships(X_val)
    y_pred = U_val.argmax(axis=1)

    # Evaluate clustering performance
    silhouette = silhouette_score(X_val, y_pred)
    davies_bouldin = davies_bouldin_score(X_val, y_pred)
    calinski_harabasz = calinski_harabasz_score(X_val, y_pred)

    results.append(
        {
            "n_clusters": params["n_clusters"],
            "m": params["m"],
            "silhouette_score": silhouette,
            "davies_bouldin_score": davies_bouldin,
            "calinski_harabasz_score": calinski_harabasz,
        }
    )
```

--------------------------------

### Predicting Cluster Labels with Fuzzy C-Means

Source: https://scikit-fuzzy-c-means.readthedocs.io/en/latest/_sources/algorithm.rst.txt

Assigns hard cluster labels to input data X by computing membership degrees and selecting the cluster with the highest degree for each data point. Requires the `_compute_memberships` method to be implemented.

```python
def predict(self, X):
    memberships = self._compute_memberships(X)
    return memberships.argmax(axis=1)
```

--------------------------------

### Update Membership Matrix Calculation

Source: https://scikit-fuzzy-c-means.readthedocs.io/en/latest/algorithm.html

Updates the membership matrix U based on the Euclidean distances between data points and cluster centers. It applies the formula uij = 1 / sum((||xi - cj|| / ||xi - ck||)^(2/(m-1))) for each data point and cluster.

```python
def _update_memberships(self, X):
    distances = cdist(X, self.cluster_centers_, metric="euclidean")
    distances = np.fmax(distances, 1e-10)
    inv_distances = 1.0 / distances
    power = 2 / (self.m - 1)
    return inv_distances**power / np.sum(inv_distances**power, axis=1, keepdims=True)

```

--------------------------------

### Update Cluster Centers Calculation

Source: https://scikit-fuzzy-c-means.readthedocs.io/en/latest/algorithm.html

Computes the updated cluster centers based on the current membership matrix U and the fuzziness coefficient m. The formula used is cj = sum(uij^m * xi) / sum(uij^m).

```python
def _compute_centers(self, X):
    um = self.U_**self.m
    return (um.T @ X) / np.sum(um.T, axis=1, keepdims=True)

```

--------------------------------

### Fuzzy C-Means Algorithm Class Definition

Source: https://scikit-fuzzy-c-means.readthedocs.io/en/latest/algorithm.html

Defines the FuzzyCMeans class, inheriting from scikit-learn's BaseEstimator and ClusterMixin. Requires numpy and scipy.spatial.distance.cdist.

```python
import numpy as np
from sklearn.base import BaseEstimator, ClusterMixin
from scipy.spatial.distance import cdist


class FuzzyCMeans(BaseEstimator, ClusterMixin):
    """
    Fuzzy C-Means clustering algorithm.
    """

```

--------------------------------

### Predict Cluster Labels

Source: https://scikit-fuzzy-c-means.readthedocs.io/en/latest/algorithm.html

Assigns hard cluster labels to new data points based on the maximum membership degree calculated by the _compute_memberships method. This provides a crisp assignment from the fuzzy memberships.

```python
def predict(self, X):
    memberships = self._compute_memberships(X)
    return memberships.argmax(axis=1)

```

--------------------------------

### Visualize Fuzzy C-Means Membership Degrees

Source: https://scikit-fuzzy-c-means.readthedocs.io/en/latest/_sources/usage.rst.txt

This code snippet visualizes the membership degrees of data points in a Fuzzy C-Means clustering using Plotly. It converts the membership matrix to a long format suitable for interactive scatter plots, where point size represents the membership degree.

```python
import plotly.express as px

# Convert membership matrix to long format
df_membership = pd.DataFrame(
    U_val_best, columns=[f"Cluster {i+1}" for i in range(fcm_best.n_clusters)]
)
df_membership["Max Cluster"] = df_membership.idxmax(axis=1)
df_membership["X"] = X_val[:, 0]
df_membership["Y"] = X_val[:, 1]

# Long format for interactive plot
df_membership_long = df_membership.melt(
    id_vars=["X", "Y", "Max Cluster"],
    value_vars=[f"Cluster {i+1}" for i in range(fcm_best.n_clusters)],
    var_name="Cluster",
    value_name="Membership Degree",
)

# Interactive plot
fig = px.scatter(
    df_membership_long,
    x="X",
    y="Y",
    color="Cluster",
    size="Membership Degree",
    title="Membership Degrees per Data Point",
    labels={"Membership Degree": "Degree"},
)
fig.show()
```

--------------------------------

### Update Membership Matrix

Source: https://scikit-fuzzy-c-means.readthedocs.io/en/latest/_sources/algorithm.rst.txt

Updates the membership matrix U based on the current cluster centers and data points. It calculates the inverse distances raised to a power to determine the degree of membership for each data point in each cluster.

```python
def _update_memberships(self, X):
    distances = cdist(X, self.cluster_centers_, metric="euclidean")
    distances = np.fmax(distances, 1e-10) # Avoid division by zero
    inv_distances = 1.0 / distances
    power = 2 / (self.m - 1)
    return inv_distances**power / np.sum(inv_distances**power, axis=1, keepdims=True)

```

--------------------------------

### Fuzzy NOT (Complement) Operation

Source: https://scikit-fuzzy-c-means.readthedocs.io/en/latest/fuzzy_logic.html

Represents the complement of a fuzzy set, calculated as one minus the membership value. Used for negation in fuzzy logic.

```mathematics
μ¬A(x)=1−μA(x)
```

--------------------------------

### Fuzzy OR (Union) Operation

Source: https://scikit-fuzzy-c-means.readthedocs.io/en/latest/fuzzy_logic.html

Represents the union of two fuzzy sets, calculated as the maximum of their membership values. Used in fuzzy rule evaluation.

```mathematics
μA∪B(x)=max(μA(x),μB(x))
```

--------------------------------

### Triangular Membership Function Formula

Source: https://scikit-fuzzy-c-means.readthedocs.io/en/latest/fuzzy_logic.html

Defines a triangular membership function using three parameters: left endpoint, peak, and right endpoint. Used to represent fuzzy sets.

```mathematics
μ(x)=max(min(x−ab−a,c−xc−b),0)
```

--------------------------------

### Trapezoidal Membership Function Formula

Source: https://scikit-fuzzy-c-means.readthedocs.io/en/latest/fuzzy_logic.html

Defines a trapezoidal membership function using four parameters: a, b, c, and d. It's a generalized form of the triangular membership function.

```mathematics
μ(x)=max(min(x−ab−a,1,d−xd−c),0)
```

--------------------------------

### Gaussian Membership Function Formula

Source: https://scikit-fuzzy-c-means.readthedocs.io/en/latest/fuzzy_logic.html

Defines a Gaussian membership function with a smooth bell shape, characterized by a mean (c) and standard deviation (sigma).

```mathematics
μ(x)=exp⁡(−(x−c)22σ2)
```

--------------------------------

### Fuzzy AND (Intersection) Operation

Source: https://scikit-fuzzy-c-means.readthedocs.io/en/latest/fuzzy_logic.html

Represents the intersection of two fuzzy sets, calculated as the minimum of their membership values. Used in fuzzy rule evaluation.

```mathematics
μA∩B(x)=min(μA(x),μB(x))
```

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