Narrow Results

Various algorithms have been devised to mathematically model the dynamic mechanism of the gene expression data. Gillespie’s stochastic simulation (GSSA) has been exceptionally primal for chemical reaction synthesis with future ameliorations. Several other mathematical techniques such as differential equations, thermodynamic models and Boolean models have been implemented to optimally and effectively represent the gene functioning. We present a novel mathematical framework of gene expression, undertaking the mathematical modeling of the transcription and translation phases, which is a detour from conventional modeling approaches. These subprocesses are inherent to every gene expression, which is implicitly an experimental outcome. As we foresee, there can be modeled a generality about some basal translation or transcription values that correspond to a particular assay.

The feasibility of function of errors with fractional exponent for solving of a problem of optimization and tutoring of neural networks was theoretically explored. The analytical expressions for estimation of parameters of the models or weight factors were obtained. The algorithms were designed and the numerical experiment on actual economic datas was held, where the efficiency of an offered procedure is shown.

The following sections are included:

• INTRODUCTION

• POPULATION PARAMETERS AND THE REGRESSION MODELS
  • Data Description
  • Building the Population Regression Model
  • Sample Versus Population Regression Model

• THE LEAST-SQUARES ESTIMATION OF α AND β
  • Scatter Diagram
  • The Method of Least Squares
  • Estimation of Intercept and Slope

• STANDARD ASSUMPTIONS FOR LINEAR REGRESSION

• THE STANDARD ERROR OF ESTIMATE AND THE COEFFICIENT OF DETERMINATION
  • Variance Decomposition
  • Standard Error of Residuals (Estimate)
  • The Coefficient of Determination

• THE BIVARIATE NORMAL DISTRIBUTION AND CORRELATION ANALYSIS
  • The Sample Correlation Coefficient
  • The Relationship Between r and b
  • The Relationship Between r and R²

• Summary

• Appendix 13A Derivation of Normal Equations and Optimal Portfolio Weights

• Appendix 13B The Derivation of Equation 13.16

• Appendix 13C The Bivariate Normal Density Function
  • Using a Mathematics Aptitude Test to Predict Grade in Statistics

• Appendix 13D American Call Option and the Bivariate Normal CDF
  • Valuating American Option

• Questions and Problems