Explained variance

Definition

Explained variance is a statistical measure that indicates how much of the total variability in a dataset is accounted for by a specific model or factor. It is crucial in understanding how well a model, such as those derived from Principal Component Analysis (PCA), captures the underlying structure of the data. By quantifying the proportion of variance explained by different components, it helps to identify the most significant dimensions of variability in a dataset.

5 Must Know Facts For Your Next Test

1. In PCA, explained variance helps to determine how many principal components are needed to effectively summarize the dataset.
2. The total explained variance can be expressed as a percentage, showing how much of the total variance is attributed to each principal component.
3. Higher explained variance indicates that a model captures more information from the data, making it more effective for analysis.
4. Explained variance is calculated using eigenvalues obtained from the covariance matrix of the dataset during PCA.
5. Interpreting explained variance helps in dimensionality reduction by identifying which components contribute most to the overall structure of the data.

Review Questions

• How does explained variance contribute to selecting the number of principal components in PCA?
  • Explained variance is essential in deciding how many principal components to retain in PCA because it quantifies how much information each component captures from the dataset. By examining the cumulative explained variance associated with each principal component, one can determine a threshold for retaining components that significantly contribute to understanding the data's variability. This helps prevent overfitting and allows for an efficient representation of complex datasets.
• Discuss the relationship between eigenvalues and explained variance in the context of PCA.
  • In PCA, each principal component corresponds to an eigenvalue that reflects its contribution to explained variance. The larger the eigenvalue, the more variance that component accounts for. By ranking principal components according to their eigenvalues, analysts can assess which components explain most of the variability in the data. This relationship highlights why interpreting eigenvalues is crucial for understanding and applying PCA effectively.
• Evaluate the implications of low explained variance on data analysis outcomes when using PCA.
  • Low explained variance in PCA suggests that the chosen principal components do not adequately capture the underlying structure or variability within the dataset. This can lead to misleading conclusions, as significant patterns may be overlooked or misrepresented. Consequently, relying on models with low explained variance may result in ineffective predictions and decisions based on incomplete information. Hence, ensuring sufficient explained variance is critical for robust data analysis and reliable interpretations.

Related terms

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

Eigenvalue: An eigenvalue represents the amount of variance captured by its corresponding eigenvector in PCA, indicating the importance of that principal component.

Variance:

Variance measures the spread of a set of data points around their mean, reflecting how much the values differ from one another.