Calculator
Paste CSV, or upload a file. Configure options, then submit to project data into principal components.
Example data table
This example uses columns A–D as numeric features.
| ID | A | B | C | D |
|---|---|---|---|---|
| 1 | 2.5 | 3.1 | 0.9 | 4.2 |
| 2 | 2.7 | 3.0 | 1.1 | 4.0 |
| 3 | 2.9 | 3.5 | 1.4 | 4.4 |
| 4 | 3.2 | 3.7 | 1.3 | 4.8 |
| 5 | 3.0 | 3.2 | 1.0 | 4.1 |
Formula used
Let X be the data matrix with n rows and p variables.
After centering (and optionally scaling), PCA forms a covariance matrix:
S = (1/(n-1)) · Xᵀ X
PCA finds eigenpairs (λ, v) of S:
S v = λ v
Scores (projected coordinates) for the top k components are:
T = X · Vₖ
Explained variance for PCj is λj / trace(S), with cumulative sums across components.
How to use this calculator
- Paste a CSV dataset or upload a CSV file.
- Set delimiter and header options to match your data.
- Select the feature columns you want to include.
- Choose missing handling, transform, and scaling preferences.
- Pick the number of components
kand run Submit. - Review variance, loadings, scores, then export CSV or PDF.
Data quality rules for reliable projection
PCA depends on numeric consistency, because the covariance structure reflects scale and noise. This calculator accepts up to 5,000 rows and 20 selected variables, which helps keep results responsive. Outliers can dominate covariance, so review extremes and cap or winsorize when appropriate. When using power iteration, a tighter tolerance such as 1e-9 and 1,500 iterations generally stabilizes the top components, but very flat spectra may need more iterations to converge fully. For stable components, remove identifiers, keep comparable units, and consider a transform when values are heavily skewed. If missing values occur, mean imputation preserves row count, while row dropping reduces bias from imputed noise but can shrink sample size quickly.
Centering, scaling, and matrix choice
Centering subtracts each variable mean so the first component describes variation, not level. Z-score scaling divides by standard deviation, making variables contribute comparably when units differ. The covariance option keeps natural units and is best when scales are meaningful. The correlation option forces z-scoring and is preferred for mixed units or when variance magnitude is not inherently important.
Eigenvalues, variance, and component selection
Each eigenvalue estimates variance captured by a component. The calculator reports variance percentage as λj divided by the trace of the covariance matrix, plus cumulative totals to guide selection. Practical workflows often target 80–95% cumulative variance, then validate interpretability using loadings. Increasing k improves reconstruction but may reduce clarity when components start modeling noise.
Loadings interpretation and directional meaning
Loadings are the component directions, showing which variables push scores positive or negative. Large absolute loadings indicate dominant influence, while near-zero loadings suggest weak contribution. Signs can flip without changing meaning, so compare patterns rather than raw sign. Use domain knowledge to label components, and check for correlated groups that move together.
Projection, reconstruction, and export practice
Scores are computed by multiplying the centered or scaled matrix by the top-k loading vectors. The optional reconstruction RMSE summarizes how well k components approximate the standardized data; lower values indicate tighter compression. For deployment, project new points using the same learned means and standard deviations, then export scores as CSV and share the PDF report for audit trails.
FAQs
1) What dataset structure works best for this projector?
Use a rectangular table where each row is an observation and each selected column is numeric. Remove IDs from selected columns, keep consistent units, and include at least two complete rows after missing-value handling.
2) When should I pick covariance instead of correlation?
Choose covariance when variable scales are meaningful and comparable. Choose correlation when units differ or variances vary widely. Correlation forces z-score scaling so each variable contributes based on standardized variation.
3) How do I choose the number of components k?
Start with k that reaches 80–95% cumulative variance, then verify interpretability using loadings. If your goal is compression, prefer smaller k; if reconstruction accuracy matters, increase k and compare RMSE changes.
4) Why do loading signs sometimes flip between runs?
Eigenvectors are defined up to a sign, so multiplying a loading vector by −1 produces the same component subspace. Interpret relative magnitudes and variable groupings, not the sign itself. This tool aligns signs for readability.
5) How does projecting new points work?
New rows are centered and scaled using the dataset’s learned means and standard deviations, then multiplied by the loading vectors. Keep the same variable order and delimiter, and avoid missing values in new rows.
6) What does the reconstruction RMSE tell me?
RMSE is computed in the scaled space between the standardized data and its k-component reconstruction. Lower RMSE indicates that the chosen components preserve more structure. Use it to compare k values consistently.