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Regression in the 21st Century

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Presentation on theme: "Regression in the 21st Century"β€” Presentation transcript:

1 Regression in the 21st Century
Modern Statistical Methods

2 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.

3 Constant Nature of A Variable
𝛼π‘₯=1 π‘₯=πœ‡ 𝑅 2 = 𝑛 π‘₯ 2 π‘₯ 2 π‘₯ 2 = π‘₯βˆ’ π‘₯ 2 +𝑛 π‘₯ 2

4 Constant nature of π‘₯~𝑁(πœ‡,𝜎)

5 Constant nature of π‘₯~π‘ˆ(πœ‡βˆ’3𝜎,πœ‡+3𝜎)

6 Non-response Analysis
Given data related by a coβˆ’dependent relationship, equation 1; balanced by an unknown measure, 𝑧 that is assumed to be relatively constant in nature with a mean πœ‡ and a deviation 𝜎, 𝑧~𝑁(πœ‡,𝜎). 𝑧=𝑔 π‘₯,𝑦

7 Unity To fit the data to the non-response model, consider the scaled model where 𝑒= 𝑧 πœ‡ 𝑧 and therefore, the unitized variable, 𝑒 is normally distribution with a mean of one, πœ‡ 𝑒 =1; and standard error equal to the coefficient of variation, 𝜎 𝑒 = 𝜎 𝑧 πœ‡ 𝑧 , Equation 2. 𝑒=β„Ž(π‘₯,𝑦)

8 Fitted model with parameters 𝑧~𝑁(500,15)
π‘₯𝑦=𝑧,𝑧~𝑁 πœ‡,𝜎 𝑦= 𝛽 0 + 𝛽 1 π‘₯ 𝑦= 𝛽 0 + 𝛽 1 1 π‘₯ 𝑒= 𝛼 0 π‘₯𝑦

9 Law of cosine Fit the outlined relationship estimating unity as one; that is, let 𝑒 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 βˆ’ 𝑦 2 ,𝑆𝑆𝑅= 𝑦 𝑖 βˆ’ 𝑦 𝑖 2 ,and 𝑆𝑆𝐸= ( 𝑦 𝑖 βˆ’ 𝑦 𝑖 ), equation 3 and the measured degree of separation, equation 4..

10 Degree of separation 𝑆𝑆𝑇=𝑆𝑆𝑅+π‘†π‘†πΈβˆ’2 𝑆𝑆𝑅×𝑆𝑆𝐸 π‘π‘œπ‘ πœƒ πœƒ=π‘Žπ‘π‘œπ‘  π‘†π‘†π‘‡βˆ’π‘†π‘†π‘…βˆ’π‘†π‘†πΈ βˆ’2 𝑆𝑆𝑅×𝑆𝑆𝐸

11 Detecting Conic Sections
𝑑~π‘ˆ π‘Ž,𝑏 π‘₯= 𝛾 0 + 𝛾 1 cos⁑(2πœ‹π‘‘) 𝑦= 𝛽 0 + 𝛽 1 sin 2πœ‹π‘‘ 𝑧= π‘₯βˆ’ 𝛾 0 𝛾 π‘¦βˆ’ 𝛽 0 𝛽 1 2

12 Detecting Circles 𝛼 1 π‘₯ 2 + 𝛼 2 π‘₯+ 𝛼 3 π‘₯𝑦 + 𝛼 4 𝑦+ 𝛼 5 𝑦 2 =1

13 Hurricane data 𝑒= 𝛼 1 𝑀+ 𝛼 2 𝑝+ 𝛼 3 𝑀𝑝 𝑒~𝑁 1, 𝜎 πœ‡

14 Bivariate Probability distribution
𝑓 𝑀,𝑝 = 1 𝜎 𝑒 2πœ‹ 𝑒 βˆ’ 𝛼 1 𝑀+ 𝛼 2 𝑝+ 𝛼 3 π‘€π‘βˆ’ πœ‡ 𝑒 𝜎 𝑒 2

15 Conditional Marginal Probabilities

16 Conditional bivariate probability density function

17 The End of Presentation
Thank you


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