PCA Input and Options
Formula Used
Centered data: Xc = X − μ
Standardized data: Zij = (Xij − μj) / σj
Covariance matrix: C = XcTXc / (n − 1)
Eigendecomposition: C vk = λkvk
Scores: T = XcV
Explained variance: 100 × λk / Σλ
Reconstruction: X̂ = TkVkT + μ
PCA rotates correlated features into orthogonal directions. Each direction maximizes remaining variance. Components are ordered from strongest to weakest.
Eigenvalues measure component variance. Eigenvectors define component directions. Loadings connect original features with components.
Scores locate observations in component space. Reconstruction estimates original values from retained components. Error falls as components increase.
How to Use
- Paste numeric rows or upload a CSV file.
- Choose row orientation, headers, labels, and missing-value handling.
- Select centering, standardization, covariance, correlation, or SVD-equivalent analysis.
- Choose manual, variance, eigenvalue, Kaiser, or complete component retention.
- Calculate and inspect variance, loadings, scores, plots, and reconstruction.
- Copy results or export scores, matrices, loadings, charts, and PDF reports.
Standardization is usually best when units differ. Mean-centering suits similarly scaled features. Correlation PCA standardizes automatically.
Use the scree plot to locate an elbow. Check cumulative variance before reducing dimensions. Confirm important variables through absolute loadings.
Use score plots to find clusters and outliers. Compare reconstruction error across component counts. Retain enough information for your task.
Example Dataset
| Observation | Sepal length | Sepal width | Petal length | Petal width |
|---|---|---|---|---|
| 1 | 5.1 | 3.5 | 1.4 | 0.2 |
| 2 | 4.9 | 3.0 | 1.4 | 0.2 |
| 3 | 6.2 | 3.4 | 5.4 | 2.3 |
| 4 | 5.9 | 3.0 | 5.1 | 1.8 |
| 5 | 6.7 | 3.1 | 4.7 | 1.5 |
This example contains four related measurements. PCA summarizes their shared variation. Petal measurements often influence the leading direction strongly.
The reduced scores can support visualization and clustering. Loadings explain which measurements create separation. Reconstruction measures information lost during reduction.
Try different thresholds to compare retained dimensions. Standardized and centered results may differ. Choose preprocessing that matches feature units.
Advantages and Limitations
PCA reduces dimensionality and multicollinearity. It supports visualization, compression, and preprocessing. Orthogonal components simplify many downstream models.
PCA is linear and scale-sensitive. Components may be difficult to name. Outliers can strongly alter directions.
Variance does not always equal predictive value. Supervised targets are ignored completely. Interpret results within the application context.
Frequently Asked Questions
What does PCA calculate?
PCA calculates orthogonal feature combinations that capture maximum variance. It returns eigenvalues, eigenvectors, loadings, scores, and explained variance.
Should data be standardized first?
Standardize when features use different units or scales. Mean-centering may be enough when measurements are already comparable.
How many components should be retained?
Use cumulative variance, a scree-plot elbow, eigenvalue rules, reconstruction error, and downstream performance. No single rule fits every dataset.
What is explained variance?
Explained variance is the percentage of total analyzed variation represented by one component. Cumulative variance combines successive components.
What do PCA loadings mean?
Loadings describe feature relationships with components. Larger absolute values indicate stronger influence, while signs indicate direction.
Why can eigenvector signs change?
Both an eigenvector and its negative describe the same axis. Scores and loadings reverse together without changing geometry.
Can PCA handle missing values?
This calculator can remove incomplete observations or impute feature means or medians. Advanced production workflows may require stronger imputation models.
What does whitening do?
Whitening rescales retained scores to approximately unit variance. It removes component scale differences but may amplify low-variance noise.
Is PCA suitable for categorical data?
Standard PCA expects numeric continuous variables. Encoded categories can distort distances, so specialized methods may be more suitable.