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inertia_

inertia_ gives the total within-cluster sum of squares.

inertia_

inertia_ is the sum of squared distances of examples to their assigned centre. A small value shows tight clusters but doesn't measure quality alone: it always falls as k grows and depends on the data's scale.

km = KMeans(n_clusters=5, n_init=10, random_state=0).fit(X)
d = km.transform(X).min(axis=1)
print(np.isclose((d ** 2).sum(), km.inertia_))   # True

score(X) returns the negative of the same value; it is for consistency with scikit-learn's "bigger is better" rule.

Other internal criteria

There are also internal criteria that reduce scale dependence:

from sklearn.metrics import calinski_harabasz_score, davies_bouldin_score
print(calinski_harabasz_score(X, km.labels_))   # bigger is better
print(davies_bouldin_score(X, km.labels_))      # smaller is better

Calinski-Harabasz is the ratio of between-cluster to within-cluster spread; Davies-Bouldin is each cluster's average similarity to its most similar cluster. They are used together with the silhouette when choosing k.

Practice Verifying inertia

Train KMeans (5 clusters). Print with np.isclose that the sum of squares of the km.transform(X).min(axis=1) distances equals inertia_, then with np.isclose that km.score(X) is -inertia_, on separate lines.

Starter code

model.py
import numpy as np
from sklearn.datasets import make_blobs
from sklearn.cluster import KMeans

X, _ = make_blobs(n_samples=500, centers=5, cluster_std=1.0, random_state=3)

# buraya yaz

Sample solution

model.py
import numpy as np
from sklearn.datasets import make_blobs
from sklearn.cluster import KMeans

X, _ = make_blobs(n_samples=500, centers=5, cluster_std=1.0, random_state=3)

km = KMeans(n_clusters=5, n_init=10, random_state=0).fit(X)
d = km.transform(X).min(axis=1)
print(np.isclose((d ** 2).sum(), km.inertia_))
print(np.isclose(km.score(X), -km.inertia_))

Expected output

True
True

Questions and answers

Review the lesson questions together with their answers. To test yourself, solve the lesson interactively.

1 If the data is multiplied by 10 (every feature), how does inertia change?

  • It becomes 10 times larger
  • It doesn't change
  • It decreases
  • It becomes 100 times larger Correct answer

Explanation: Distances grow 10 times, squared distances 100 times.

2 Compute the internal criterion where smaller is better.

Question 
db = ____(X, km.labels_)
Answer 
db = davies_bouldin_score(X, km.labels_)

Explanation: The closer Davies-Bouldin is to 0, the better separated the clusters.

3 Match the criteria with their directions.

Term Meaning
inertia_ Smaller is tighter
silhouette_score Bigger is better (≤1)
calinski_harabasz_score Bigger is better
davies_bouldin_score Smaller is better

All lessons in Intermediate

1 scikit-learn basics

  1. What is scikit-learn?
  2. Estimator API
  3. The fit method
  4. The predict method
  5. The score method
  6. Summary - scikit-learn

2 Pipeline

  1. What is a Pipeline?
  2. Steps
  3. fit and predict
  4. Advantages
  5. ColumnTransformer
  6. Summary - Pipeline

3 Cross-validation

  1. What is cross-validation?
  2. K-fold
  3. cross_val_score
  4. Stratified
  5. Advantages
  6. Summary - Cross-validation

4 Grid search

  1. What is grid search?
  2. GridSearchCV
  3. param_grid
  4. best_params_
  5. best_score_
  6. Summary - Grid search

5 SVM

  1. What is SVM?
  2. Margin
  3. Kernel
  4. C parameter
  5. Classification
  6. Summary - SVM

6 Project: SVM

  1. Preparing data
  2. Training model
  3. Prediction
  4. Evaluation

7 Random forest

  1. What is random forest?
  2. Ensemble
  3. n_estimators
  4. Bagging
  5. Feature importance
  6. Summary - Random forest

8 Boosting

  1. What is boosting?
  2. AdaBoost
  3. Gradient boosting
  4. Learning rate
  5. Advantages
  6. Summary - Boosting

9 Clustering

  1. What is clustering?
  2. KMeans
  3. Centroid
  4. Elbow method
  5. Labels
  6. Summary - Clustering

10 KMeans in depth

  1. KMeans algorithm
  2. Initialization
  3. n_init
  4. inertia_
  5. random_state
  6. Summary - KMeans

11 Hierarchical clustering

  1. What is hierarchical clustering?
  2. Dendrogram
  3. Agglomerative
  4. Linkage
  5. Advantages
  6. Summary - Hierarchical clustering

12 Project: Clustering

  1. Preparing data
  2. Training KMeans
  3. Label analysis
  4. Printing results

13 PCA

  1. What is PCA?
  2. Variance
  3. n_components
  4. fit_transform
  5. explained_variance_
  6. Summary - PCA

14 Dimensionality reduction

  1. What is dimensionality reduction?
  2. Why reduce?
  3. PCA vs t-SNE
  4. Visualization
  5. Advantages
  6. Summary - Dimensionality reduction

15 Feature selection

  1. What is feature selection?
  2. Variance threshold
  3. SelectKBest
  4. Recursive elimination
  5. Feature importance
  6. Summary - Feature selection

16 Imbalanced data

  1. What is imbalanced data?
  2. SMOTE
  3. Class weight
  4. Oversampling
  5. Metric selection
  6. Summary - Imbalanced data

17 Project: Features

  1. Feature extraction
  2. Feature selection
  3. Training model
  4. Evaluation

18 Ensemble methods

  1. What is ensemble?
  2. Voting
  3. Bagging
  4. Boosting
  5. Stacking
  6. Summary - Ensemble

19 Regularization

  1. What is regularization?
  2. L1 (Lasso)
  3. L2 (Ridge)
  4. alpha
  5. ElasticNet
  6. Summary - Regularization

20 Intermediate final

  1. Challenge 1
  2. Challenge 2
  3. Challenge 3
  4. Challenge 4