时间序列里边的一个小问题

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用时间序列分解法预测复合型序列的第一步要分离季节成分，算季节指数。用“移动平均趋势剔出法”算季节指数要算“中心化移动平均值”（CMA），没看懂这个CMA怎么算的……怎么算的啊？

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关键词：时间序列 小问题 中心化移动平均 移动平均值 移动平均 时间 序列

For monthly data, it is 13 points moving average: in Additive model X(t) = Trend+Seasonal+ Irregular

Trend = T(t') = (1/24)*X(t-6) + (1/12)*X(t-5) + .....+ (1/12)*X(t) + ................(1/12)*X(t+5) + (1/24)*X(t+6)

Detrended = SI(t') = X(t) - T(t')

The preliminary estimate of the seasonal component is then obtained by averaging each
monthly sub-series of SI(t') using a 5 point moving average:

Shat(t') = (1/9)*SI(t'-24) + (2/9)*SI(t'-12) + (3/9)*SI(t') + (2/9)*SI(t'+12) + (1/9)*SI(t'+24)

This estimate needs to be centred around the trend to get the initial estimate of the
seasonal component:

Seasonal = S(t') = Shat(t') - [(1/24)*Shat(t'-6) + (1/12) *Shat(t'-5) + .............+(1/12)*Shat(t'+5) + (1/24)*Shat(t'+6)]

Then Seasonally adjusted series Ad(t') = X(t) - S(t')

However with this approach, there are 30 months lost at each ends. Therefore it is only considered as a preliminary estimate of decomposition.

This is the additive form of X-12, and can be achieved by using X-12 decomposition, which will not lost information at both ends