Chapter 6: Time Series Analysis

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ATM 552 Notes: Time Series Analysis - Section 6a Page 1126.2 Harmonic AnalysisHarmonic analysis is the interpretation of a time or space series as a summation ofcontributions from harmonic functions, each with a characteristic time or space scale.Consider that we have a set of N values of y(t i ) = y i . Then we can use a least-squaresprocedure to find the coefficients of the following expansionN2yt ( ) = A + A cos2k to k T + B sin2k t k T k =1(6.14)where T = the length of the period of record. y(t) is a continuous function of t which passesthrough all of the data points. Note that B k = 0 when k=N/2 , since you cannot determinethe phase of the wave with a wavelength of two time steps. If the data points are not evenlyspaced in time then we must be careful. The results can be very sensitive to small changesin y i . One should test for the effects of this sensitivity by imposing small variations in y iand be particularly careful where there are large gaps. Where the data are unevenly spacedit may be better to eliminate the higher harmonics. In this case one no longer achieves anexact fit, but the behavior may be much better between the data points.Useful Math Identities:It may be of use to you to have this reference list of trigonometric identities:cos( ) = cos cos + sin sin ; tan = sincos(6.15)You can use the above two relations to show that:C cos + S sin =( ) ; where A = C 2 + S 2 ; o = Arc tan S CA cos o(6.16)where you need to note the signs of S and C to get the phase in the correct quadrant.The complex forms of the trig functions also come up importantly here.Also, from this you can get;e i = cos + i sin where i = 1 (6.17)sin = ei e i2i; cos = ei + e i2(6.18)If you need more of these, check any book of standard mathematical tables.Copyright 2005 Dennis L. Hartmann 1/25/05 5:04 PM 112
ATM 552 Notes: Time Series Analysis - Section 6a Page 1136.2.1 Evenly Spaced Data Discrete Fourier TransformOn the interval 0 < t < T chosen such that t 1 =0 and t N+1 = T where N is an evennumber. The analytic functions are of the formcos( 2k it T),sin( 2k it T) (6.19)t is the (constant) spacing between the grid points. In the case of evenly spaced data wehave:a) a o = the average of y on interval 0
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Chapter 6: Time Series Analysis

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