Variance Explained

Definition

Variance explained refers to the proportion of the total variance in a dataset that is accounted for by a statistical model, specifically in the context of dimensionality reduction techniques like PCA. Understanding variance explained helps in determining how much information is retained when reducing the dimensionality of the data, guiding decisions on how many components to keep for analysis without losing significant information.

5 Must Know Facts For Your Next Test

1. Variance explained is typically expressed as a percentage, indicating how much of the total variability in the data is represented by each principal component.
2. In PCA, components are ordered such that the first few components account for a large portion of the variance, allowing for effective dimensionality reduction.
3. Cumulative variance explained is calculated by summing the variance explained by each principal component, helping to determine how many components to retain.
4. A common threshold for deciding how many components to keep is around 70-90% of total variance explained, balancing simplicity and information retention.
5. Variance explained can also help in assessing model performance and ensuring that models built on reduced datasets still capture significant patterns from the original data.

Review Questions

• How does variance explained relate to the decision-making process when determining the number of principal components to retain?
  • Variance explained is crucial in deciding how many principal components to keep because it quantifies how much of the total data variation is captured by each component. By analyzing the percentage of variance each principal component explains, one can identify a cutoff point where adding more components provides diminishing returns on explanatory power. Typically, retaining enough components to explain 70-90% of total variance ensures that most of the original information is preserved while reducing complexity.
• Discuss how eigenvalues contribute to understanding variance explained in PCA and their role in selecting principal components.
  • Eigenvalues directly indicate how much variance each principal component captures in PCA. A higher eigenvalue means that its corresponding component accounts for more variability within the dataset. By examining the eigenvalues associated with each principal component, one can prioritize which components are most significant based on their ability to explain variance. This process informs decisions about retaining essential components while discarding those that contribute little to understanding the overall structure of the data.
• Evaluate how using a scree plot can enhance one's understanding of variance explained and its implications for data analysis.
  • A scree plot visually represents eigenvalues against principal components, making it easier to assess variance explained. By observing where the curve flattens, analysts can identify an optimal number of components to retain while ensuring sufficient explanation of variance. This evaluation aids in striking a balance between maintaining data integrity and simplifying analysis. As a result, using scree plots effectively informs better choices regarding dimensionality reduction strategies and subsequent modeling efforts.

Related terms

Eigenvalues: Eigenvalues are scalars associated with a linear transformation that provide information about the amount of variance captured by each principal component in PCA.

Principal Components: Principal components are the new dimensions created from the original variables in PCA, which capture the most variance in the data.

Scree Plot: A scree plot is a graphical representation that shows the eigenvalues on the y-axis against the number of components on the x-axis, helping to visualize variance explained and identify optimal component numbers.