Variance explained

Definition

Variance explained refers to the proportion of the total variability in a dataset that can be attributed to a specific model or set of predictors. In the context of dimensionality reduction techniques, it helps to understand how much of the information present in the original data is retained after transformation, allowing for insights into the importance of each principal component in summarizing the data's structure.

5 Must Know Facts For Your Next Test

1. The variance explained by a principal component is calculated as the ratio of its eigenvalue to the total sum of eigenvalues, providing a measure of its contribution to the overall variance.
2. A higher proportion of variance explained by a few principal components indicates that the data can be effectively summarized with fewer dimensions.
3. In PCA, cumulative variance explained is often used to determine how many components are necessary to represent a certain percentage of the total variability, like 80% or 90%.
4. When selecting principal components, a threshold for variance explained is typically set to retain those that contribute significantly to understanding the data structure.
5. Variance explained helps assess the quality and effectiveness of PCA by revealing how much information is preserved versus discarded during the transformation process.

Review Questions

• How does variance explained help in determining the effectiveness of principal components in summarizing a dataset?
  • Variance explained quantifies how much of the total variability in a dataset is accounted for by each principal component. By evaluating this proportion, we can identify which components capture significant information and which are less informative. This assessment allows us to choose an optimal number of components that balance simplicity and data representation without losing critical information.
• Discuss how cumulative variance explained can guide decisions on the number of principal components to retain in analysis.
  • Cumulative variance explained accumulates the variance percentages from each retained principal component, illustrating how much total variability they collectively account for. By setting a target percentage, such as 90%, analysts can decide how many components are necessary to achieve that level of data representation. This approach aids in reducing dimensionality while ensuring key patterns and trends are maintained in the analysis.
• Evaluate the implications of high versus low variance explained in PCA results and their impact on data interpretation.
  • High variance explained by a few principal components indicates that these components effectively summarize most of the information within the dataset, making it easier to interpret underlying patterns. In contrast, low variance explained suggests that many dimensions contribute equally, complicating interpretation and indicating potential redundancy in data features. Understanding these implications helps analysts make informed decisions about model complexity and clarity in presenting results.

Related terms

Principal Component: A principal component is a linear combination of the original variables in a dataset that captures the maximum variance and helps reduce dimensionality.

Eigenvalue: An eigenvalue indicates the amount of variance captured by its corresponding eigenvector, which is crucial in determining the significance of each principal component.

Scree Plot: A scree plot visually represents the eigenvalues associated with each principal component, helping to determine how many components should be retained based on the variance they explain.