Table 1.
Nonlinear autoregressive structure of time series on snowshoe hare and lynx in the Canadian boreal forest: Dimension and complexity
| No. | Time series | Years | General model: Xt = F(Xt−1, Xt−2, ⋯ , Xt−d) + ɛt
|
Additivity (P value) | The General Additive Model: Xt = f1(Xt−1) + f2(Xt−2) + ⋯ + fd(Xt−d) + ∂t
|
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|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Dimension
|
Linearity (P value)
|
Dimension
|
Slope at equilibrium of the GAM (linear coefficients ± 2 SEM)
|
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| N-W* | LL reg* | H-T | Tsay | Linear* | GAM* | lag1 (Xt−1) | lag2 (Xt−2) | lag3 (Xt−1) | ||||
| H1 | Snowshoe hare | 1844–1904 | 1 (0.67;0.74) | 1 (0.70;0.79) | √ (0.18) | √ (0.07) | √ (0.11) | 4 (0.69;0.76) | 4 (0.59;0.66) | 0.82 (0.62 ± 0.26) | 0.13 (0.1 ± 0.30) | −0.42 (−0.23 ± 0.26) |
| H2 | Snowshoe hare | 1905–1935 | 3 (0.39) | 3 (0.33) | √ (0.39) | √ (0.52) | √ (0.13) | 3 (0.30) | 3 (0.22) | 1.02 (0.95 ± 0.33) | 0.14 (0.01 ± 0.49) | −0.56 (−0.49 ± 0.33) |
| L1 | West | 1825–1856 | 2 (0.38) | 3 (0.36;0.38) | √ (0.09) | √ (0.31) | √ (0.77) | 4 (0.35;0.39) | 4 (0.34;0.44) | 1.00 (1.12 ± 0.33) | −0.93 (−0.55 ± 0.33) | |
| L2† | West | 1897–1934 | 2 (0.28) | 2 (0.25) | √ (0.87) | √ (0.16) | √ (0.08) | 2 (0.24) | 4 (0.23;0.25) | 1.38 (1.34 ± 0.26) | −073 (−0.67 ± 0.26) | |
| L3‡ | MacKenzie River | 1821–1934 | 2 (0.19) | 4 (0.15;0.16) | (0.01) | (0.00) | √ (0.40) | 4 (0.17;0.18) | 4 (0.14;0.15) | 1.38 (1.38 ± 0.13) | −0.95 (−0.75 ± 0.13) | |
| L4 | Athabasca Basin | 1821–1891 | 2 (0.31) | 2 (0.24) | (0.05) | (0.00) | √ (0.88) | 3 (0.35;0.41) | 3 (0.26;0.26) | 1.16 (1.03 ± 0.22) | −0.72 (−0.41 ± 0.22) | |
| L5† | Athabasca Basin | 1897–1934 | 4 (0.47;0.50) | 2 (0.20) | √ (0.44) | (0.00) | √ (0.20) | 2 (0.22) | 2 (0.18) | 1.23 (1.33 ± 0.26) | −0.67 (−0.74 ± 0.26) | |
| L6 | West Central | 1821–1891 | 2 (0.15) | 2 (0.09) | √ (0.76) | √ (0.08) | √ (0.92) | 2 (0.10) | 2 (0.09) | 1.43 (1.46 ± 0.14) | −1.03 (−0.82 ± 0.14) | |
| L7† | West Central | 1897–1934 | 4 (0.38;0.48) | 4 (0.23;0.30) | √ (0.32) | (0.01) | √ (0.11) | 2 (0.30) | 2 (0.32) | 0.92 (1.09 ± 0.34) | −0.45 (−0.45 ± 0.34) | |
| L8 | Upper Saskatch. | 1821–1891 | 2 (0.21) | 3 (0.17;0.20) | √ (0.94) | √ (0.27) | √ (0.37) | 3 (0.19;0.21) | 3 (0.20;0.21) | 1.30 (1.32 ± 0.18) | −0.83 (−0.68 ± 0.18) | |
| L9 | Winnipeg Basin | 1821–1891 | 3 (0.19;0/21) | 4 (0.14;0.16) | √ (0.86) | √ (0.98) | √ (0.31) | 4 (0.14;0.15) | 4 (0.15;0.16) | 1.42 (1.40 ± 0.15) | −0.91 (−0.78 ± 0.15) | |
| L10 | North Central | 1821–1891 | 2 (0.35) | 2 (0.35) | (0.01) | (0.00) | √ (0.85) | 4 (0.39;0.44) | 3 (0.28;0.29) | 1.15 (1.02 ± 0.21) | −0.71 (−0.52 ± 0.21) | |
| L11† | James Bay | 1895–1939 | 2 (0.20) | 2 (0.11) | √ (0.74) | √ (0.86) | √ (0.11) | 2 (0.12) | 3 (0.11;0.12) | 1.37 (1.47 ± 0.20) | −0.82 (−0.84 ± 0.20) | |
| L12† | Lakes | 1897–1939 | 2 (0.26) | 2 (0.20) | √ (0.41) | √ (0.21) | (0.03) | 2 (0.18) | 2 (0.20) | 1.26 (1.34 ± 0.26) | −0.79 (−0.56 ± 0.26) | |
| L13† | James B.+Lakes | 1897–1939 | 2 (0.20) | 2 (0.13) | √ (0.76) | √ (0.49) | √ (0.65) | 2 (0.13) | 2 (0.14) | 1.50 (1.46 ± 0.20) | −0.85 (−0.76 ± 0.20) | |
| L14† | Gulf | 1897–1939 | 2 (0.55) | 2 (0.39) | √ (0.80) | √ (0.60) | √ (0.20) | 2 (0.37) | 2 (0.40) | 1.02 (1.05 ± 0.29) | −0.73 (−0.47 ± 0.29) | |
Dimension refers to the optimal order using CV, adopting the Nadaraya–Watson kernel (N-W), the locally linear model (LL-reg), the additive model (GAM), or the linear autoregression model (Linear). The corresponding CV value are given in parentheses. The CV value for dimension two (and three) for all lynx (and hare) series are given as the second number in the parentheses whenever not optimal (not boldface). Two methods were used when testing for nonlinearity [H-T (21); Tsay (22)]. For both methods, P values are given in parentheses. Additivity gives P values from a Lagrange multiplier test for additivity (23). The estimated partial derivatives of the various fi functions (see text) at equilibrium of the skeleton model is given under lag1, lag2, and lag3. The corresponding linear autoregressive parameters (±2 SEM) are given in parentheses. No detrending has been carried out in spite of possible trends in some of the time series. Check marks indicate tests for which the null hypothesis was not rejected. Boldface indicates order estimates corresponding to the structure and complexity hypothesized in the main text of the paper.
CV value in parenthesis. First number gives the CV for the optimal dimension; second number (if present) gives the CV for dimension three for hare, and dimension two for lynx. The degrees of freedom for each lag is restricted to be 0 (the lag is not present), 1 (linear), 2 (nonlinear), or 4 (strongly nonlinear).
Series has been interpolated for the missing observation in year 1914.
The L3 series in the linear case is selected to have dimension 11 based on the AIC (24) criterion when there is no upper limit for the possible dimensions to be selected. This is exceptional compared with the other lynx series.