Regression in the 21st Century Modern Statistical Methods

Abstract This presentation introduces modern techniques of regression used to fit co-dependent measures: conic sections, indirect relationships and implicit equations; and how to fit bivariate probability distributions to co- dependent variables using these methods.

Constant Nature of A Variable 𝛼𝑥=1 𝑥=𝜇 𝑅 2 = 𝑛 𝑥 2 𝑥 2 𝑥 2 = 𝑥− 𝑥 2 +𝑛 𝑥 2

Constant nature of 𝑥~𝑁(𝜇,𝜎)

Constant nature of 𝑥~𝑈(𝜇−3𝜎,𝜇+3𝜎)

Non-response Analysis Given data related by a co−dependent relationship, equation 1; balanced by an unknown measure, 𝑧 that is assumed to be relatively constant in nature with a mean 𝜇 and a deviation 𝜎, 𝑧~𝑁(𝜇,𝜎). 𝑧=𝑔 𝑥,𝑦

Unity To fit the data to the non-response model, consider the scaled model where 𝑢= 𝑧 𝜇 𝑧 and therefore, the unitized variable, 𝑢 is normally distribution with a mean of one, 𝜇 𝑢 =1; and standard error equal to the coefficient of variation, 𝜎 𝑢 = 𝜎 𝑧 𝜇 𝑧 , Equation 2. 𝑢=ℎ(𝑥,𝑦)

Fitted model with parameters 𝑧~𝑁(500,15) 𝑥𝑦=𝑧,𝑧~𝑁 𝜇,𝜎 𝑦= 𝛽 0 + 𝛽 1 𝑥 𝑦= 𝛽 0 + 𝛽 1 1 𝑥 𝑢= 𝛼 0 𝑥𝑦

Law of cosine Fit the outlined relationship estimating unity as one; that is, let 𝑢 be represented by a column of ones. The degree of separation between the measures in the developed model can be measured using the law of cosines, where 𝑆𝑆𝑇= 𝑦 𝑖 − 𝑦 2 ,𝑆𝑆𝑅= 𝑦 𝑖 − 𝑦 𝑖 2 ,and 𝑆𝑆𝐸= ( 𝑦 𝑖 − 𝑦 𝑖 ), equation 3 and the measured degree of separation, equation 4..

Degree of separation 𝑆𝑆𝑇=𝑆𝑆𝑅+𝑆𝑆𝐸−2 𝑆𝑆𝑅×𝑆𝑆𝐸 𝑐𝑜𝑠𝜃 𝜃=𝑎𝑐𝑜𝑠 𝑆𝑆𝑇−𝑆𝑆𝑅−𝑆𝑆𝐸 −2 𝑆𝑆𝑅×𝑆𝑆𝐸

Detecting Conic Sections 𝑡~𝑈 𝑎,𝑏 𝑥= 𝛾 0 + 𝛾 1 cos⁡(2𝜋𝑡) 𝑦= 𝛽 0 + 𝛽 1 sin 2𝜋𝑡 𝑧= 𝑥− 𝛾 0 𝛾 1 2 + 𝑦− 𝛽 0 𝛽 1 2

Detecting Circles 𝛼 1 𝑥 2 + 𝛼 2 𝑥+ 𝛼 3 𝑥𝑦 + 𝛼 4 𝑦+ 𝛼 5 𝑦 2 =1

Hurricane data 𝑢= 𝛼 1 𝑤+ 𝛼 2 𝑝+ 𝛼 3 𝑤𝑝 𝑢~𝑁 1, 𝜎 𝜇

Bivariate Probability distribution 𝑓 𝑤,𝑝 = 1 𝜎 𝑢 2𝜋 𝑒 − 1 2 𝛼 1 𝑤+ 𝛼 2 𝑝+ 𝛼 3 𝑤𝑝− 𝜇 𝑢 𝜎 𝑢 2

Conditional Marginal Probabilities

Conditional bivariate probability density function

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