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墨西哥帽小波变换的影响域和计算方案新探讨
New Research on Cone of Influence and Computing Scheme of Mexican Hat Wavelet Transform
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摘要: 该文分析了Torrence等人研究的墨西哥帽 (Mexican Hat) 小波变换的影响域和高频失真问题并探讨了解决方案。平移点b处、小波尺度为a的墨西哥帽小波函数的有效定义域为[ b-2.12a, b+2.12a], 以墨西哥帽为母函数的小波变换的真正影响域是2.12a, 小波尺度a的最大值应为N/4.24(N为时间序列长度)。提出了充分利用小波函数速降性质和立方样条插值小波系数的新计算方案, 消除了该计算方案产生虚假的显著高频周期振荡问题。利用正弦函数型时间序列小波变换的解析式检验了文中提出的新计算方案的合理性。利用新计算方案对冬季Niño3.4指数进行分析, 冬季Niño3.4指数的平均整体小波功率谱表明, 存在约12年的年代际变化和准4年的年际变化, 不存在显著的准两年周期振荡。Abstract: Using the effective domain of wavelet function, the problems of cone of influence (COI) and high-frequency distortion of Mexican Hat wavelet transform in the paper of Torrence and Compo (1998) are analyzed and the solution are explored. The analytic expressions of wavelet coefficient and global wavelet power spectrum for time series of sinusoid and the relationship between period and wavelet scale are deduced. The effective domain of Mexican Hat wavelet transform is [ b-2.12a, b+2.12a] for localized time b and wavelet scale a. The authentic cone of influence for Mexican Hat wavelet transform is 2.12a. The maximum value of wavelet scale is N/4.24 (N is the length of time series). When wavelet scale approaches zero, wavelet function is singular. So the simple sum scheme replacing the integration for wavelet coefficient does not include the singularity of wavelet function, therefore the pseudo high-frequency oscillation is deduced. A new computing scheme for wavelet coefficient is designed. The major elements include improved-resolution and located time b in the new grids. The difference of effective domain of wavelet function for small different wavelet scales can be distinguished by the improved-resolution. Located time b has great contribution to its wavelet coefficient because the wavelet function is the biggest at the located point b. The pseudo significant high-frequency oscillation that is produced by the computing scheme proposed by Torrence and Compo (1998) can be eliminated by the new computing scheme of wavelet coefficient, well using sharply descending characteristics of wavelet function and cubic spline interpolation. The rationality of new computing scheme is tested by the analytic solution of wavelet transform of sinusoid time sequence. The global wavelet power spectrum from the new computing scheme indicates that the variability of winter Niño3.4 index exhibits interdecadal oscillation of about 12 years and interannual oscillation of quasi-four years, and no significant quasi-biennial oscillation. The questions of cone of influence and pseudo high-frequency oscillation have two adverse aftermaths. First is that the correct judgment of interdecadal variation can be affected and second is that pseudo quasi-biennial variation can be deduced. Therefore, the results about COI and the new computing scheme for wavelet coefficient make the conclusions induced from wavelet analysis more credible.
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Key words:
- wavelet transform;
- cone of influence;
- computing scheme;
- Niño3.4 index
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图 1 函数g(x)=(1-x2)· e-x2/2在x＞0时的分布 (实、虚直线分别表示1/e, -1/e)
Fig. 1 The distribution of function g(x)=(1-x2)·e-x2/2 for x > 0(the solid and dashed lines are for the value of 1/e and-1/e, respectively)
图 2 墨西哥帽小波函数有效定义域及新计算方案网格示意图
(a) 小波尺度a=0.47(实线) 和a=0.16(虚线) 所对应的小波函数 (浅色水平线对应到这两种小波尺度上的有效定义域) 及3个相邻时间的距平序列 (柱状), (b) 小波尺度a=0.47所对应的小波函数 (实线) 及新的高分辨率网格 (竖虚线)
Fig. 2 Schematic diagram of effective domain of Mexican wavelet function and new computing scheme
(a) wavelet functions for wavelet scale a=0.47(solid line) and a=0.16(dashed line) (light horizontal lines stand for the effective domains for these wavelet scales) and anomalous series of 3 sequential time (bar), (b) wavelet function for wavelet scale a=0.47(solid line) and new high reso lution grid (vertical dashed line)
图 3 时间序列f(t)=sin (2πt/4) (t=0, …, 40) 的平均整体小波功率谱
(点线、断线和实线分别为式 (9)、文献[7]计算方案和本文新计算方案的结果)
Fig. 3 Mean global wavelet power spectrum of time series f(t)=sin (2πt/4)(t=0, …, 40)
(dotted line, dashed line and solid line stand for the results from analytic formula (9), computing scheme in reference [7] and new computing scheme in this paper)
图 4 1951—2007年冬季Niño3.4指数距平序列及其小波变换
(a)1951— 2007年冬季Niño3.4指数距平序列, (b) 本文新计算方案计算的小波系数和影响域, (c) 文献[7]方案计算的小波系数和影响域 (图中两边的斜交叉线区域为影响域, 阴影分别表示通过α=0.1的显著性水平检验 (Monte-Carlo方法))
Fig. 4 Time evolution of winter Niño3.4 index from 1951 to 2007 and wavelet transform
(a) anomalous series of winter Niño3.4 index from 1951 to 2007, (b) wavelet coefficient computed by new computing scheme and COI in this paper, (c) wavelet coefficient computed by computing scheme and COI in reference [7] (cross-hatched regions on either end indicate the COI; shaded areas represent regions pass the test of α=0.1 level (Monte-Carlo method))
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