Homework1

# Homework1 - MAE118A Introduction to Energy Systems Homework...

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Unformatted text preview: MAE118A: Introduction to Energy Systems Homework 1 Kowsik Bodi October 7, 2009 Problem 1 The yearly data can be obtained by interpolation from the plot, or from the online databases at 1. EIA website : http://tonto.eia.doe.gov/cfapps/ipdbproject/IEDIndex3. cfm?tid=44&pid=44&aid=2 2. BP website : http://www.bp.com/liveassets/bp_internet/globalbp/ globalbp_uk_english/reports_and_publications/statistical_energy_ review_2008/STAGING/local_assets/2009_downloads/statistical_ review_of_world_energy_full_report_2009.xls 0.1 Linear Least–squares Fit We want to fit a straight line of the form f ( x ) = a + bx (1) to the N data points ( x i , y i ), i = 1 , 2 , . . . N . In order to quantify the idea of “best–fit”, we define an error function that needs to be minimized. We define the error as the sum of squares of the deviation of data points from the line–fit S ( a, b ) = N X i =1 [ y i- f ( x i )] 2 (2) with respect to the coefficients a, b . This leads to the constraints ∂S ∂a = 0 , ∂S ∂b = 0 (3) 1 If we define P = N X i =1 x i , R = N X i x 2 i ζ = N X i =1 y i , η = N X i x i y i These constraints lead to the equations that help us determine a, b Na + Pb = ζ Pa + Rb = η Thus, our coefficients are a = Rζ- Pη RN- P 2 , b = ηN- ζP RN- P 2 (4) 0.2 Industrialized Nations years: [1970 1971 1972 1973 1974 1975 1976 1977 1978 1979...
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Homework1 - MAE118A Introduction to Energy Systems Homework...

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