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Published byDevon McKenna Modified over 2 years ago

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Two examples

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Canadian Lynx data 1821-1934 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 a 1 = 1.39 p-value 0.0000 a 2 = -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 a 1 = 1.272 p-value 0.0000 a 2 = -0.7005 p-value 0.0000 a 3 = 0.1413 p-value 0.3604 a 4 = -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 a 1 = 1.269p-value 0.0000 a 2 = -0.6901p-value 0.0000 a 3 = 0.2776 p-value 0.1018 a 4 = -0.3588 p-value 0.0335 a 5 = 0.1836p-value 0.2778 a 6 = -0.213 p-value 0.1728 a 7 = 0.2316p-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 a 1 = 1.243p-value 0.0000 a 2 = -0.6605p-value 0.0000 a 3 = 0.2863p-value 0.0881 a 4 = -0.3504p-value 0.0398 a 5 = 0.2112p-value 0.2192 a 6 = -0.2084 p-value 0.2269 a 7 = 0.174p-value 0.3072 a 8 = -0.1253p-value 0.4589 a 9 = 0.3683p-value 0.0194 a 10 = -0.2184p-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 a 1 = 1.168p-value 0.0000 a 2 = -0.5346p-value 0.0005 a 3 = 0.2515p-value 0.1121 a 4 = -0.2963p-value 0.0661 a 5 = 0.1409p-value 0.3881 a 6 = -0.1397 p-value 0.3938 a 7 = 0.05p-value 0.7618 a 8 = -0.0288p-value 0.8595 a 9 = 0.1458p-value 0.3633 a 10 = 0.2216p-value 0.1503 a 11 = -0.3758p-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!

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ARIMA-models for non-stationary time series

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