Explained variance

Definition

Explained variance refers to the portion of the total variance in a dataset that is accounted for by a statistical model, such as a regression model or Principal Component Analysis (PCA). It indicates how well the model captures the information present in the data and helps determine the effectiveness of dimensionality reduction techniques like PCA.

5 Must Know Facts For Your Next Test

1. Explained variance is typically expressed as a percentage, representing how much of the total variance is captured by each principal component in PCA.
2. In PCA, higher explained variance for the first few principal components suggests that these components contain the most significant information about the data structure.
3. When choosing the number of principal components to retain, explained variance helps decide how many components are needed to represent the data adequately.
4. The sum of the explained variances of all retained components should ideally be close to 100%, indicating that most of the information has been preserved.
5. A common threshold for explained variance is to retain enough components that cumulatively explain at least 70-90% of the total variance in the dataset.

Review Questions

• How does explained variance help in evaluating the effectiveness of a PCA model?
  • Explained variance is crucial for evaluating PCA because it quantifies how much of the original data's variability is captured by each principal component. By examining the explained variances, one can determine which components are most informative and decide how many should be retained for effective data representation. This evaluation ensures that the model simplifies the data while retaining essential features, helping avoid overfitting.
• Discuss how you would use explained variance to decide on the number of principal components to keep in a PCA analysis.
  • To decide on the number of principal components to retain based on explained variance, one would create a scree plot showing the explained variance against each component. By identifying an 'elbow' point where additional components contribute less significantly to total explained variance, a practical cutoff can be established. Typically, one might aim to retain enough components that account for at least 70-90% of total explained variance, balancing between simplicity and sufficient data representation.
• Evaluate how changes in sample size or feature selection might impact the explained variance in a PCA model and its implications.
  • Changes in sample size can significantly impact explained variance in PCA; larger samples often lead to more stable estimates of variance and more reliable principal components. Conversely, if irrelevant features are included or important features omitted, it may distort the true structure of the data, affecting how much variance is explained. This can result in misleading interpretations and choices regarding dimensionality reduction. Careful feature selection ensures that PCA reflects meaningful data patterns and retains relevant information while minimizing noise.

Related terms

Variance: Variance measures how much the data points differ from the mean value, providing insights into the distribution and spread of the data.

Principal Component: Principal components are the directions in which the data varies the most, derived from PCA, and are used to reduce dimensionality while preserving as much information as possible.

Singular Value Decomposition (SVD): SVD is a mathematical method used in PCA to decompose a dataset into its constituent components, allowing for efficient computation of explained variance.