*Dedicated to Roxanne Halitsky, Wife and Mother.*

**Theta Criteria** are based on operator's theory and matrices decompositions. The main idea is to evaluate differences between sets of operators' eigenvalues & eigenvectors or eigenfunctions.

**Theta Criteria** will lead to more efficient analysis, optimization and prognosis of multivariate systems and applications. **Theta Criteria** can be used for data set mining and image processing. The book is designed as a complementary tool for applied mathematics methods.

We assume the Reader has standard undergraduate knowledge of advanced calculus, matrix theory, operator theory, approximation methods and measure theory.

The book consists of Introduction and four Chapters.

In **Chapter I: Existing Statistical Methods for Multivariate Data Processing** we briefly review existing methods and software packages for multivariate systems analysis. These methods are based upon Multivariate General Linear Hypothesis (MGLH).

In line with MGLH, all dataset variables are linear, additive and relationships models are linear series of weighted terms.

We also mention the following methods: Multiple Regression, Discriminant Function Analysis, Canonical Analysis, Principle Components Analysis and formal linear algebra methods. Lastly, we discuss our methods, **Theta Criteria**, which are constructed on norms of weighted differences of matrices ordered eigenvectors.

In **Chapter II: Theta Criteria Formal Study**, we placed our main formal results. At this moment we considered only positively-defined matrices.

In **Chapter III: Theta Criteria Numerical Study** we have numerically studied **Theta Criteria** on sets of varied matrices. We compare **Theta Criteria** accuracy with existing matrices' norms and invariants.

In **Chapter IV: Theta Criteria Applications**, we briefly discuss **Theta Criteria** application areas.

At this moment we are only able to "scratch the surface" of **Theta Criteria** universe. We are convinced that our methods will find their places in various complex applications.

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Finally, you may contact us anytime.

- Multivariate General Linear Hypothesis
- Multiple Regression Equation
- Discriminant Function Analysis
- Canonical Analysis
- Factor Structure Method
- Principle Components Analysis (PCA)
- Formal Linear Algebra Methods
- Theta Criteria
- Theta Criteria Properties
- Summary
- References

- 2.1. Formal Definitions
- 2.2. Known Matrices Closeness Criteria
- 2.3. The Hilbert Theorem
- 2.4. Theta Criteria or Θ Criteria Construction
- 2.5. Theta Criteria Properties
- References

Mathematics Subject Classification:

46G12 Measures and integration on abstract linear spaces

58C40 Spectral theory; eigenvalue problems

58D30 Applications

BISAC:

Mathematics / Probability & Statistics / Multivariate Analysis

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