How to perform seasonal adjustment to a time series?

Question

Assume following data set representing each month of the year 2013 with the corresponding consumption of natrual gas to heat my flat and the respective mean temperature.

How can I seasonal adjust/normalize this time series?

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1   156 m³  1,4 °C
2   199 m³  0,3 °C
3   173 m³  2,3 °C
4   69 m³   9,6 °C
5   63 m³   12,2 °C
6   9 m³    16,9 °C
7   9 m³    20,8 °C
8   9 m³    18,5 °C
9   15 m³   14,3 °C
10  19 m³   11,4 °C
11  83 m³   5,2 °C
12  62 m³   4,2 °C

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I would like to take into account that it is colder in winter and warmer in summer and know if I needed more or less energy in comparison. (especially in terms of comparing with the year 2014 for example)

I know how to use R, if that makes it easier for you to explain.

Answer 1

Two observations per month is not much information to carry out seasonal adjustment.

While you gather more data, you may be interested in measuring how consumption changes with temperature. For that, you can fit the following model (a quadratic relationship between consumption and temperature seems appropriate at first glance but see comment by whuber below):

cons <- structure(c(156, 199, 173, 69, 63, 9, 9, 9, 15, 19, 83, 62), .Tsp = c(1, 
  1.91666666666667, 12), class = "ts")
temp <- structure(c(1.4, 0.3, 2.3, 9.6, 12.2, 16.9, 20.8, 18.5, 14.3, 
  11.4, 5.2, 4.2), .Tsp = c(1, 1.91666666666667, 12), class = "ts")

fit <- lm(c(cons) ~ poly(c(temp), 2))
summary(fit)
#Residuals:
#    Min      1Q  Median      3Q     Max 
#-52.340  -7.920  -2.102  17.963  34.661 
#Coefficients:
#                  Estimate Std. Error t value Pr(>|t|)    
#(Intercept)         72.167      7.451   9.686 4.66e-06 ***
#poly(c(temp), 2)1 -201.911     25.810  -7.823 2.65e-05 ***
#poly(c(temp), 2)2   69.987     25.810   2.712   0.0239 *  
#---
#Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#Residual standard error: 25.81 on 9 degrees of freedom
#Multiple R-squared:  0.8839,   Adjusted R-squared:  0.8582 
#F-statistic: 34.28 on 2 and 9 DF,  p-value: 6.18e-05

Display observed data (black points) and fitted values (in blue):

plot(c(temp), c(cons), pch = 16)
lines(c(sort(temp)), predict(fit)[order(temp)], type = "b", col = "blue")

Upon this regression you can obtain the predicted value for consumption given for example temperatures of 5 and 15 Celsius degrees (along with 95% confidence intervals).

p <- predict(fit, newdata = data.frame(temp = c(5, 15)), se.fit = TRUE)
res <- cbind(p$fit - 1.96 * p$se, p$fit, p$fit + 1.96 * p$se)
colnames(res) <- c("lower limit", "pred", "upper limit")
res
#  lower limit     pred upper limit
#1   83.072304 102.5629   122.05356
#2   -5.487951  14.9201    35.32816

As you obtain more data you can think of further methods. See, for instance, the documentation and examples for function decompose. You could also include seasonal dummies in the above regression. These dummies are indicator variables pointing each one to observations recorded at a given month, e.g. [1 0 0 0 0 0 0 0 0 0 0 0], [0 1 0 0 0 0 0 0 0 0 0 0] and so on. (Note: one of the the 12 seasonal dummies should be omitted in the regression if an intercept is included.)

Answer 2

In your example, you have less than three observations per month, i.e. you only have one value for January, one for February, etc. In a case with less than three per month, you cannot perform established seasonal adjustment methods such as X-13ARIMA-SEATS by the US Census Bureau. In order to apply X-13ARIMA-SEATS you need at least three observations per month, i.e. you need a time series that is at least three years long. In case you do not have three years of data, you could do year over year comparison.

In your case, you should have more than three years of data by now. Hence, you could perform X-13ARIMA-SEATS. The following site introduces seasonal adjustment:

https://economictheoryblog.com/2017/05/02/seasonal-adjustment

Furthermore, and relevant for your case, I guess, the site provides a hands-on example of how to perform seasonal adjustment in R:

https://economictheoryblog.com/2017/05/16/seasonal-adjustment-in-r