Yi Shuo School District (45) SPSS Statistical Analysis (55) Seasonal Decomposition of Time Series

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【思维导图】

【基础知识】

在实际工作中，人们经常按月（或年、季度、小时等）记录资料，如每个月的出生人口数、死亡率、某种疾病的发病率、某产品的销售额等，这些资料可能符合某种季节性分布，但这些数字的大小往往受多种因素的影响，从原始数据中很难看出季节趋势。

In practical work, people often record data on a monthly basis (or annual, quarterly, hourly, etc.), such as the number of births, mortality, the incidence of a certain disease, the sales of a certain product, etc. These data may conform to a certain seasonal distribution, but the size of these figures is often affected by many factors, and it is difficult to see the seasonal trend from the original data.

季节分解法将时间序列分解成三个组成部分，或称三个变量，即“趋势分量”，“季节分量”和“随机波动”，趋势分量采用多项式拟合，季节分量用傅里叶变换估计，其数学表达式为：Yt=f(Tt,St,It)。式中，Tt代表长期趋势（可以是线性趋势，也可以是周期性波动或长周波动），St是季节因子（幅度和周期固定的波动，日历效应为常见的季节因子），It为随机波动（可视为误差）。

Seasonal decomposition method decomposes time series into three components, or three variables, namely "trend component", "seasonal component" and "random fluctuation". The trend component is fitted by polynomial, and the seasonal component is estimated by Fourier transform. Its mathematical expression is Yt=f (Tt, St, It). In the formula, Tt represents long-term trend (which can be linear trend, periodic fluctuation or long-term fluctuation), St is seasonal factor (fluctuation with fixed amplitude and period, calendar effect is common seasonal factor), and It is random fluctuation (which can be regarded as error).

常见的时间序列分解模型有加法和乘法两种，加法模型为：Yt=Tt+St+It，乘法模型为Yt=Tt*St*It。相对而言，乘法模型比加法模型用的更多，在乘法模型中，时间序列值和长期趋势用绝对值表示，季节变动和不规则变动用相对值（百分数）表示。季节性分解要求无缺失数据，在处理前数据已经定义好日期变量并指定周期。

There are two common time series decomposition models: addition and multiplication. The addition model is Yt=Tt+St + It, and the multiplication model is Yt=Tt*St * It. Multiplicative models, in which time series values and long-term trends are expressed in absolute terms, and seasonal and irregular variations are expressed in relative terms (percentages), are used more often than additive models. Seasonal decomposition requires no missing data, and the data has defined date variables and specified periods before processing.

【实例分析】

下面我们来看一个季节性分解的实例分析：

Let's look at an example analysis of seasonal decomposition:

如下图所示，对某企业的销售数据进行季节性分解。

As shown in the following figure, the sales data of an enterprise are decomposed seasonally.

第一步，分析并组织数据。定义“年份，月份”格式的日期变量。

The first step is to analyze and organize the data. Define date variables in the format of Year and Month.

第二步，观察数据序列的性质。

The second step is to observe the properties of the data sequence.

如下图所示，对销售额做时序图，从该时序图可以看出销售额总的趋势是增长的，但增长并不是单调上升的，而是有涨有落。这种升降不是杂乱无章的，和季节或月份的季节因素有关，当然，除了增长的趋势和季节影响之外，还有些无规律的随机因素的作用。

As shown in the following figure, make a timing chart for sales. From this timing chart, we can see that the general trend of sales is increasing, but the growth is not monotonous, but ups and downs. This rise and fall is not disorderly, but related to the seasonal factors of seasons or months. Of course, besides the growth trend and seasonal influence, there are also some irregular random factors.

第三步，季节性分解设置。按下图所示进行设置。

The third step is seasonal decomposition setting. Set as shown in the following figure.

第四步，主要结果及分析。

The fourth step, the main results and analysis.

下期预告：本期，我们学习了季节性分解的理论基础和实例分析。下一期，我们将会学习信度分析的基础知识。

Forecast for the next issue: In this issue, we learned the theoretical basis and case analysis of seasonal decomposition. In the next issue, we will learn the basics of reliability analysis.

如果您对今天的文章有独特的想法，欢迎给我们留言，让我们相约明天，祝您今天过得开心快乐！

If you have a unique idea of today's article, welcome to leave us a message, let us meet tomorrow, I wish you a happy today!

参考资料：《SPSS23（中文版）统计分析实用教程》、百度百科

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