Variance explained

Definition

Variance explained refers to the proportion of the total variability in a dataset that can be accounted for by a statistical model, such as regression or principal component analysis. It provides insight into how well the model captures the underlying patterns in the data, indicating its effectiveness in summarizing and interpreting the information. This concept is particularly crucial in determining the usefulness of models in reducing dimensionality while retaining essential characteristics of the original dataset.

5 Must Know Facts For Your Next Test

1. Variance explained is often expressed as a percentage, showing how much of the total variance in data is captured by principal components.
2. In principal component analysis, the first few components usually explain most of the variance, allowing for effective dimensionality reduction.
3. The cumulative variance explained by the components can be visualized using a scree plot, which helps determine the optimal number of components to retain.
4. A higher percentage of variance explained indicates a better fit of the model, suggesting it captures more essential patterns from the data.
5. Variance explained is critical in model evaluation; it helps researchers understand if their statistical approach is adequately representing the complexity of the data.

Review Questions

• How does variance explained help assess the effectiveness of principal component analysis?
  • Variance explained provides a quantitative measure to evaluate how well principal component analysis captures the underlying structure of the data. By looking at how much variance is accounted for by each principal component, researchers can determine whether a smaller number of components can summarize the data effectively. This helps in deciding which components to retain for further analysis while ensuring significant information is not lost.
• Discuss the significance of cumulative variance explained and how it influences decisions in dimensionality reduction.
  • Cumulative variance explained shows the total proportion of variance captured by a set of principal components combined. By analyzing this cumulative percentage, researchers can make informed decisions about how many components to keep for further study. A common practice is to retain components until they explain a threshold percentage of total variance, such as 80-90%, ensuring that the retained components sufficiently summarize the original data without excessive loss of information.
• Evaluate the implications of having low variance explained in a principal component analysis context and propose potential solutions.
  • Low variance explained suggests that the selected principal components are not capturing significant patterns within the dataset, indicating potential issues with model fit. This could arise from inadequate preprocessing, inappropriate selection of components, or inherent limitations within the data itself. To address this, researchers might consider including additional relevant variables, exploring different transformation techniques, or utilizing alternative modeling methods to better capture variability and enhance interpretability.

Related terms

Eigenvalues: Eigenvalues are scalars that provide a measure of the variance captured by each principal component in principal component analysis.

Principal Components: Principal components are new variables created through linear combinations of the original variables, aiming to maximize variance explained and minimize dimensionality.

Total Variance: Total variance is the sum of variances across all observed variables in a dataset, serving as a baseline to evaluate how much variance is explained by a model.