了解傅立叶的季节性 [英] Understanding Fourier for Seasonality

问题描述

我正在使用R中预测包中的auto.arima来确定傅立叶级数的最佳K项.

I am using the auto.arima from the forecast package in R to determine the optimal K-terms for fourier series.

执行完此操作后，我想计算季节性并将该一个季节性变量插入多元回归模型中.

After I do that, I want to then calculate the seasonality and plug that one seasonality variable into a multiple regression model.

使用预测包中的数据集，我能够提取最佳数量的傅立叶项:

Using the dataset from the forecast package, I was able to extract the optimal amount of fourier terms:

library(forecast)

##Public dataset from the forecast package
head(gas)

##Choose Optimal Amount of K-Terms
bestfit <- list(aicc=Inf)
for(i in 1:6)
{
  fit <- auto.arima(gas, xreg=fourier(gas, K=i), seasonal=FALSE)
  if(fit$aicc < bestfit$aicc)
    bestfit <- fit
  else break;
  optimal_k_value<-max(i)
  print(i)
}

##Extract Fourier Terms 
seasonality<-data.frame(fourier(gas, K=optimal_k_value))

##Convert Gas TS Data to Dataframe
gas_df <- data.frame(gas, year = trunc(time(gas)), 
                 month = month.abb[cycle(gas)])

##Extract True Seasonality by Taking Sum of Rows
seasonality$total<- rowSums(seasonality)

##Combine Seasonality to Month and Year
final_df<-cbind(gas_df, seasonality$total)

seasonality$total列是否应被季节变量"考虑以用于以后的建模，还是需要在其中添加系数?

Would the seasonality$total column be considered by "seasonality variable" for later modelling or do I need to add coefficients to it?

推荐答案

否，seasonality$total不是季节性变量.要看到这一点，请注意fourier(gas, K = optimal_k_value)的每一列都是从-1到1的季节性分量，因此它们只是sin(...)和cos(...)，没有任何系数.显然，不同的季节成分必须具有不同的系数，所以您不应该将它们相加.

No, seasonality$total is not the seasonality variable. To see that, note that each column of fourier(gas, K = optimal_k_value) is just a seasonal component going from -1 to 1 so that they are just sin(...) and cos(...) without any coefficients. Clearly, different seasonal components must have different coefficients, so you shouldn't just sum them up.

侧面注释1 :由于i始终只是一个数字，因此使用max(i)毫无意义，只需optimal_k_value <- i就足够了.

Side comment 1: since i is always just a single number, there is no point in using max(i), just optimal_k_value <- i is enough.

旁注2 :我建议检查

plot(resid(auto.arima(gas, xreg = fourier(gas, K = optimal_k_value), seasonal = FALSE)))

其中一个可能是季节性低于年度频率(似乎fourier不允许考虑)，尽管您可能会将其单独建模为趋势.另外，将数据拆分为1970年之前和之后的数据可能是个好主意.

For one, there may be seasonality of lower than yearly frequency (it seems like fourier doesn't allow to consider that), although perhaps you are going to model it separately as a trend. Also, it may be a good idea to split the data to something like before and after 1970.

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