Case Studies in Time Series II Case Studies in Time Series II Periodic Behavior and Related issues Periodic Behavior and Related issues David A. Dickey David A. Dickey Professor of Statistics Professor of Statistics N. C. State University N. C. State University
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Case Studies in Time Series II Periodic Behavior and Related issues David A. Dickey Professor of Statistics N. C. State University.
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Case Studies in Time Series IICase Studies in Time Series IIPeriodic Behavior and Related issuesPeriodic Behavior and Related issues
Rollers are to even out fabric coating Rollers are to even out fabric coating Offset roller => Periodicity in thicknessOffset roller => Periodicity in thickness Period = Roller circumferencePeriod = Roller circumference YYtt = thickness at time = thickness at time ARIMA model does not help!ARIMA model does not help!
““Frequency Domain”Frequency Domain”
Coating Thickness every ¼ inchCoating Thickness every ¼ inch
PeriodicityPeriodicity ??????
One “radian”One “radian”
Fun with radians Fun with radians Half circle is Half circle is radians radians Assume all angles in [ - Assume all angles in [ - Rotation slides along sine waveRotation slides along sine wave
-1 x
Fun with trigonometry Fun with trigonometry
A Sin(A Sin(t+t+) = A Sin() = A Sin(t)Cos(t)Cos( A Cos( A Cos(t)Sin(t)Sin())
Two of our rollers are likely out of line. Two of our rollers are likely out of line. circumferences 5, 6, 8, 11, 16 and 20 inches circumferences 5, 6, 8, 11, 16 and 20 inches
One roller that is out of line would rotate once in 32 One roller that is out of line would rotate once in 32 observations. With 4 observations per inch, that roller observations. With 4 observations per inch, that roller would have circumference 8 inches. would have circumference 8 inches.
A roller of circumference 19.69 quarter inches (4.9 A roller of circumference 19.69 quarter inches (4.9 inches) is not one of our known circumferences. Perhaps inches) is not one of our known circumferences. Perhaps it is really the 5 inch one.it is really the 5 inch one.
Important Frequencies ??
data waves; pi=4*atan(1); data waves; pi=4*atan(1); set fabric; set fabric; s5 = sin(2*pi*location/5); c5 = cos(2*pi*location/5);s5 = sin(2*pi*location/5); c5 = cos(2*pi*location/5); s6 = sin(2*pi*location/6); c6 = cos(2*pi*location/6);s6 = sin(2*pi*location/6); c6 = cos(2*pi*location/6); s8 = sin(2*pi*location/8); c8 = cos(2*pi*location/8);s8 = sin(2*pi*location/8); c8 = cos(2*pi*location/8); s11 = sin(2*pi*location/11); c11 = cos(2*pi*location/11);s11 = sin(2*pi*location/11); c11 = cos(2*pi*location/11); s16 = sin(2*pi*location/16); c16 = cos(2*pi*location/16);s16 = sin(2*pi*location/16); c16 = cos(2*pi*location/16); s20 = sin(2*pi*location/20); c20 = cos(2*pi*location/20);s20 = sin(2*pi*location/20); c20 = cos(2*pi*location/20); proc reg; model Y = s5--c20/ss1 ss2; proc reg; model Y = s5--c20/ss1 ss2; T_5: test s5=0, c5=0; T_6: test s6=0, c6=0; T_5: test s5=0, c5=0; T_6: test s6=0, c6=0; T_8: test s8=0, c8=0; T_11: test s11=0, c11=0; T_8: test s8=0, c8=0; T_11: test s11=0, c11=0; T_16: test s16=0, c16=0; T_20: test s20=0, c20=0; T_16: test s16=0, c16=0; T_20: test s20=0, c20=0;
The TEST statement results areThe TEST statement results are
5 inch5 inch 6 inch 6 inch 8 inch8 inch 37.0 37.0 0.22 0.22 20.0820.08 Pr > F = Pr > F = <.0001<.0001 .8018.8018 <.0001<.0001
11 inch 16 inch 20 inch11 inch 16 inch 20 inch 3.27 2.72 1.363.27 2.72 1.36 Pr > F = .0387 .0670 .2586Pr > F = .0387 .0670 .2586
proc spectra whitetest; proc spectra whitetest; The SPECTRA ProcedureThe SPECTRA Procedure Test for White Noise for Variable YTest for White Noise for Variable Y M-1 14M-1 14 Max(P(*)) 11932.89Max(P(*)) 11932.89 Sum(P(*)) 34590.36Sum(P(*)) 34590.36 Fisher's Kappa: (M-1)*Max(P(*))/Sum(P(*))Fisher's Kappa: (M-1)*Max(P(*))/Sum(P(*)) Kappa 4.829682Kappa 4.829682 Bartlett's Kolmogorov-Smirnov Statistic:Bartlett's Kolmogorov-Smirnov Statistic: Maximum absolute difference of the standardizedMaximum absolute difference of the standardized partial sums of the periodogram and the CDF of apartial sums of the periodogram and the CDF of a uniform(0,1) random variable.uniform(0,1) random variable. Test Statistic 0.255984Test Statistic 0.255984 Approximate P-Value 0.3180Approximate P-Value 0.3180
4.877 at the 5% level4.877 at the 5% level (e.g. Fuller 1996)(e.g. Fuller 1996)
Lesson learned Lesson learned
F test F test waswas significant (regression) significant (regression) Spectral tests not significant.Spectral tests not significant.
Price paid (power) for not knowing Price paid (power) for not knowing
frequency !!frequency !!
Researcher Researcher expectedexpected this frequency in this frequency in