Quadratic Discriminant Analysis Calculator

Calculator Input

Example Data Table

Formula Used

Quadratic discriminant score:

g_k(x) = -0.5 ln|Σ_k| - 0.5 (x - μ_k)^T Σ_k^-1 (x - μ_k) + ln(π_k)

• x is the sample feature vector.
• μ_k is the class mean vector.
• Σ_k is the class covariance matrix.
• |Σ_k| is the determinant of that covariance matrix.
• π_k is the prior probability for the class.

The calculator evaluates both classes, converts the scores into posterior probabilities, and returns the class with the larger posterior value.

How to Use This Calculator

1. Enter two chemistry class labels. These may represent two sample families.
2. Set three feature labels. Use lab variables such as absorbance, conductivity, or chromatographic response.
3. Enter the mean vector for each class.
4. Enter a symmetric 3 × 3 covariance matrix for each class.
5. Provide prior probabilities. They can be raw values. The calculator normalizes them.
6. Enter the sample vector for the unknown chemistry observation.
7. Run the calculator to view posterior probabilities, score differences, and the predicted class.
8. Export the output as CSV or use the PDF button for print to PDF saving.

Quadratic Discriminant Analysis in Chemistry

Why QDA matters in analytical chemistry

Quadratic discriminant analysis helps chemists separate classes when variance patterns differ. Many laboratory datasets show unequal spread. Spectroscopy, chromatography, and sensor arrays often behave this way. A linear boundary can miss that structure. QDA handles class specific covariance matrices. That makes the decision surface curved instead of fixed. In chemistry, this is useful when one material family has tight measurements and another has broad variation. The method supports smarter classification of unknown samples. It also helps chemometric workflows stay grounded in measurable statistics.

How the calculator supports lab decisions

This calculator converts three chemical features into discriminant scores and posterior probabilities. It accepts two class means, two covariance matrices, and prior probabilities. That lets a chemist reflect real process knowledge. Priors matter when one sample class appears more often. Covariance matters when variables move together. For example, absorbance and peak area may rise together for one formulation but not for another. QDA captures that relationship. The result is a stronger classification view for quality control, method development, and routine sample screening.

Where QDA fits in chemometric modeling

Chemometric models often compare known reference groups. A lab may separate compliant batches from suspect batches. Another team may distinguish mineral rich extracts from synthetic blends. QDA is valuable when the data clouds are not equally shaped. That is common in chemical production and environmental testing. The calculator also reports Mahalanobis style distance terms, determinants, and score differences. These values help users inspect model behavior. They show whether the final class came from strong separation or from a narrow margin.

Practical value for reproducible classification

Reproducibility matters in chemistry. Teams need clear formulas, visible inputs, and portable outputs. This page supports that process with example data, a formula section, export tools, and a simple layout. A chemist can review the matrix values, adjust priors, and rerun the sample in seconds. The CSV export supports documentation. The PDF option supports reports and audits. Together, these features make the quadratic discriminant analysis calculator useful for teaching, bench work, and controlled laboratory decision support.

FAQs

1. What does QDA do in a chemistry workflow?

QDA classifies an unknown chemistry sample into one of two groups. It uses class means, covariance matrices, and prior probabilities to estimate the most likely class.

2. When is QDA better than a linear classifier?

QDA is often better when the two classes have different covariance structures. That creates curved boundaries and can improve classification for uneven laboratory data.

3. Why must the covariance matrix be symmetric?

A covariance matrix represents shared variance between variables. By definition, covariance from variable A to B matches covariance from B to A. Symmetry preserves valid matrix behavior.

4. Why does the determinant need to be positive?

A positive determinant helps confirm the covariance matrix is usable and invertible for classification. If the determinant is zero or negative, the score formula breaks down.

5. What do prior probabilities change?

Priors shift the model toward classes that are more expected before measuring the sample. They help when one chemistry class appears more often in real production.

6. Can I use spectroscopy or chromatography variables here?

Yes. The three features can represent absorbance values, peak areas, retention metrics, conductivity ratios, or similar numerical chemistry measurements.

7. What does the posterior probability mean?

The posterior probability is the model based probability that the sample belongs to a specific class after considering the input values and class statistics.

8. How do the CSV and PDF options help?

The CSV file supports data logging and spreadsheet review. The PDF option supports printing, reporting, and record keeping for method checks or audits.