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Regression in the 21st Century
Modern Statistical Methods
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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.
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Constant Nature of A Variable
πΌπ₯=1 π₯=π π
2 = π π₯ 2 π₯ 2 π₯ 2 = π₯β π₯ 2 +π π₯ 2
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Constant nature of π₯~π(π,π)
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Constant nature of π₯~π(πβ3π,π+3π)
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Non-response Analysis
Given data related by a coβdependent relationship, equation 1; balanced by an unknown measure, π§ that is assumed to be relatively constant in nature with a mean π and a deviation π, π§~π(π,π). π§=π π₯,π¦
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Unity To fit the data to the non-response model, consider the scaled model where π’= π§ π π§ and therefore, the unitized variable, π’ is normally distribution with a mean of one, π π’ =1; and standard error equal to the coefficient of variation, π π’ = π π§ π π§ , Equation 2. π’=β(π₯,π¦)
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Fitted model with parameters π§~π(500,15)
π₯π¦=π§,π§~π π,π π¦= π½ 0 + π½ 1 π₯ π¦= π½ 0 + π½ 1 1 π₯ π’= πΌ 0 π₯π¦
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Law of cosine Fit the outlined relationship estimating unity as one; that is, let π’ 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 πππ= π¦ π β π¦ 2 ,πππ
= π¦ π β π¦ π 2 ,and πππΈ= ( π¦ π β π¦ π ), equation 3 and the measured degree of separation, equation 4..
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Degree of separation πππ=πππ
+πππΈβ2 πππ
ΓπππΈ πππ π π=ππππ πππβπππ
βπππΈ β2 πππ
ΓπππΈ
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Detecting Conic Sections
π‘~π π,π π₯= πΎ 0 + πΎ 1 cosβ‘(2ππ‘) π¦= π½ 0 + π½ 1 sin 2ππ‘ π§= π₯β πΎ 0 πΎ π¦β π½ 0 π½ 1 2
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Detecting Circles πΌ 1 π₯ 2 + πΌ 2 π₯+ πΌ 3 π₯π¦ + πΌ 4 π¦+ πΌ 5 π¦ 2 =1
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Hurricane data π’= πΌ 1 π€+ πΌ 2 π+ πΌ 3 π€π π’~π 1, π π
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Bivariate Probability distribution
π π€,π = 1 π π’ 2π π β πΌ 1 π€+ πΌ 2 π+ πΌ 3 π€πβ π π’ π π’ 2
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Conditional Marginal Probabilities
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Conditional bivariate probability density function
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The End of Presentation
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