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Published byDevon McKenna Modified about 1 year ago

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Two examples

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Canadian Lynx data 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 = p-value a 1 = 1.39 p-value a 2 = p-value Portmanteau test: p-value

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ACF for residuals in AR(2) model. Looks good? But-

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AR(4)? m = p-value a 1 = p-value a 2 = p-value a 3 = p-value a 4 = p-value Portmanteau test: p-value

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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 = p-value a 1 = 1.269p-value a 2 = p-value a 3 = p-value a 4 = p-value a 5 = p-value a 6 = p-value a 7 = p-value Portmanteau test: p-value

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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 = p-value a 1 = 1.243p-value a 2 = p-value a 3 = p-value a 4 = p-value a 5 = p-value a 6 = p-value a 7 = 0.174p-value a 8 = p-value a 9 = p-value a 10 = p-value Portmanteau test: p-value

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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 = p-value a 1 = 1.168p-value a 2 = p-value a 3 = p-value a 4 = p-value a 5 = p-value a 6 = p-value a 7 = 0.05p-value a 8 = p-value a 9 = p-value a 10 = p-value a 11 = p-value Portmanteau test: p-value

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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 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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