Cumulative variance

Definition

Cumulative variance refers to the total variance captured by a set of principal components in Principal Component Analysis (PCA). It helps in understanding how many principal components are needed to explain the variability in the data and assists in determining the optimal number of components for data reduction while retaining significant information.

5 Must Know Facts For Your Next Test

1. Cumulative variance is calculated by summing the explained variances of all selected principal components up to a certain point.
2. A common approach is to plot cumulative variance against the number of components to visualize how many are needed to reach a desired threshold, such as 95% of total variance.
3. Cumulative variance helps in making decisions about dimensionality reduction by indicating how much information can be retained with fewer dimensions.
4. In PCA, retaining components that collectively explain at least 70-90% of the cumulative variance is often considered sufficient for analysis.
5. The concept of cumulative variance is crucial when assessing trade-offs between reducing dimensions and maintaining data integrity.

Review Questions

• How does cumulative variance assist in determining the number of principal components to retain in PCA?
  • Cumulative variance provides insight into how much variability in the data is explained by a given number of principal components. By plotting cumulative variance against the number of components, one can visually assess where the curve levels off, indicating diminishing returns in explained variance. This allows for an informed decision on how many components to keep for effective data representation without excessive dimensionality.
• Discuss the relationship between cumulative variance and explained variance ratio in the context of PCA.
  • Cumulative variance and explained variance ratio are closely related concepts in PCA. The explained variance ratio shows the proportion of total variance attributed to each individual principal component. When these ratios are summed over multiple components, they yield the cumulative variance, which reveals how much overall variability is captured as more components are included. This relationship is essential for understanding data reduction and ensuring that significant information is preserved.
• Evaluate how cumulative variance impacts the interpretation and usability of reduced datasets derived from PCA.
  • Cumulative variance plays a crucial role in evaluating the effectiveness of dimensionality reduction through PCA. By analyzing cumulative variance, researchers can ensure that their reduced datasets retain a meaningful amount of original variability, which is vital for accurate interpretation. If too few components are retained, important patterns may be lost; if too many are kept, it may introduce noise. Thus, cumulative variance guides practitioners in striking a balance between simplicity and preserving essential information in their analyses.

Related terms

Principal Component: A principal component is a linear combination of original variables that captures the maximum variance in the data, serving as a new axis in reduced-dimensional space.

Eigenvalue: An eigenvalue indicates the amount of variance explained by its corresponding eigenvector, which is associated with a principal component in PCA.

Explained Variance Ratio: The explained variance ratio is the proportion of variance explained by each principal component relative to the total variance in the dataset.