two examples. canadian lynx data 1821-1934 annual trappings of canadian lynx
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Two examples
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Canadian Lynx data1821-1934
Annual trappings of Canadian Lynx
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The Data (an example of a prey-predator relationship). Note sharp peaks and wide minima
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Try Log Scale
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Its ACF. MA models not likely. Looks like AR model with complex roots
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PACF is confusing though. Should we try AR(2) or AR(4) or AR(7) or even AR(11)?
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AR(2)?
• m = 6.686 p-value 0.0000• a1 = 1.39 p-value 0.0000
• a2 = -0.7528 p-value 0.0000• Portmanteau test: p-value 0.0999
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ACF for residuals in AR(2) model. Looks good? But-
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AR(4)?
• m = 6.684 p-value 0.0000• a1 = 1.272 p-value 0.0000
• a2 = -0.7005 p-value 0.0000
• a3 = 0.1413 p-value 0.3604
• a4 = -0.2061 p-value 0.0318
• Portmanteau test: p-value 0.0350
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ACF for residuals in AR(4) model, Portmanteau test is no good any longer.
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AR(7)?
• m = 6.699 p-value 0.0000• a1 = 1.269 p-value 0.0000
• a2 = -0.6901 p-value 0.0000
• a3 = 0.2776 p-value 0.1018
• a4 = -0.3588 p-value 0.0335
• a5 = 0.1836 p-value 0.2778
• a6 = -0.213 p-value 0.1728
• a7 = 0.2316 p-value 0.0177• Portmanteau test: p-value 0.0184
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For AR(7) model, Portmanteau test is even worse: p-value 0.018
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AR(10)-first model with good Portmanteau test
• m = 6.697 p-value 0.0000• a1 = 1.243 p-value 0.0000
• a2 = -0.6605 p-value 0.0000
• a3 = 0.2863 p-value 0.0881
• a4 = -0.3504 p-value 0.0398
• a5 = 0.2112 p-value 0.2192
• a6 = -0.2084 p-value 0.2269
• a7 = 0.174 p-value 0.3072
• a8 = -0.1253 p-value 0.4589
• a9 = 0.3683 p-value 0.0194
• a10 = -0.2184 p-value 0.0335• Portmanteau test: p-value 0.2812
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ACF for residuals in AR(10) model
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Since φ(11) looked significant, we try AR(11) as well
• m = 6.697 p-value 0.0000• a1 = 1.168 p-value 0.0000
• a2 = -0.5346 p-value 0.0005
• a3 = 0.2515 p-value 0.1121
• a4 = -0.2963 p-value 0.0661
• a5 = 0.1409 p-value 0.3881
• a6 = -0.1397 p-value 0.3938
• a7 = 0.05 p-value 0.7618
• a8 = -0.0288 p-value 0.8595
• a9 = 0.1458 p-value 0.3633
• a10 = 0.2216 p-value 0.1503
• a11 = -0.3758 p-value 0.0003• Portmanteau test: p-value 0.8189
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ACF for residuals in AR(11) model
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AR(11) has the best AIC value, followed by AR(12)-AR(19) (??)Remember, the data set is 114
points long.
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AIC values
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Sunspots data
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As of Yesterday …
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We have seen this series before.We clearly see 11 years cycle. The data is too short to
see longer (100+ years) cycles
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Here is another view:
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and another …
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Yet another view (reversed in time)
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Let’s consider this, once again
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ACF. Typical behavior for AR(2) model with complex roots.
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Its PACF. AR(2)?
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Now, let us look at monthly data,778 points, since December 1944. Note steep accents and not as
steep drops. This is a clear evidence of non-linearity in the model.
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Its ACF (200 points, about 19 years)
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Its PACF. First model that has reasonable Portmanteau test, is AR(13)
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Since the data is only 60+ years long, we can see only the 11 years cycle here though longer
cycles definitely exist. The model below predicts next grand minimum around 2050
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Have a Nice Break!