Temperature monitoring and time series analysis (part 1)

Exploratory analysis

Abstract

Time series analysis applies whenever you have available the evolution with time of some physical quantity. The aim of such analysis is to understand the structure of the time series, in order to describe it, to simulate or even make predictions.

In the present and next post I am going to run some basic exploratory analysis, showing plots and summarisations on a given time series.

Analysis

I am going to analyze the dataset reporting the monthly temperature of the Nottingham castle from January 1920 up to December 1939. Such time series can be found at ref. [1]. as part of the Time Series Data Library (ref. [2]).

The data has been downloaded in the local file-system by selecting the Export option on the left pane and then CSV file (comma separated) format. The dataset file is named as mean-monthly-air-temperature-deg.csv.

After having load it, I cut out its last record as it reports just a description string without any temperature data associated.

suppressPackageStartupMessages(library(xts))
suppressPackageStartupMessages(library(TSA))
suppressPackageStartupMessages(library(knitr))
suppressPackageStartupMessages(library(ggplot2))
suppressPackageStartupMessages(library(tidyr))
suppressPackageStartupMessages(library(dplyr))
suppressPackageStartupMessages(library(Rmisc))
suppressPackageStartupMessages(library(lubridate))

Sys.setlocale("LC_TIME", "english")

## [1] "English_United States.1252"

mean.monthly.temp <- read.csv("mean-monthly-air-temperature-deg.csv", header=TRUE, stringsAsFactors = FALSE)
colnames(mean.monthly.temp) <- c("Date", "Temperature")
kable(head(mean.monthly.temp))

Date	Temperature
1920-01	40.6
1920-02	40.8
1920-03	44.4
1920-04	46.7
1920-05	54.1
1920-06	58.5

kable(tail(mean.monthly.temp))

	Date	Temperature
236	1939-08	61.8
237	1939-09	58.2
238	1939-10	46.7
239	1939-11	46.6
240	1939-12	37.8
241	Mean monthly air temperature (Deg. F) Nottingham Castle 1920-1939	NA

mmt <- mean.monthly.temp[complete.cases(mean.monthly.temp),]

Our dataset reports monthly data, so the frequency to be used in building the timeseries should be 12 (see ref. [3]). Anyway I want to verify that by means of the periodogram function made available by the TSA package. That could be useful in case you are uncertain about the frequency value to be specified.

temp.pdgram <- periodogram(mmt$Temperature, plot=TRUE)

(max.freq <- which.max(temp.pdgram$spec))

## [1] 20

(temp.frequency = 1/temp.pdgram$freq[max.freq])

## [1] 12

As we can see the frequency is confirmed to be equal to 12. I then create an xts time series object. The xts package made available some useful functions to summarise per specific time periods, such as week, month and year.

mmt$Date <- as.Date(paste(mmt$Date,1,sep="-"),"%Y-%m-%d")
mmt <- xts(mmt$Temperature, order.by=mmt$Date, frequency=temp.frequency)
plot(mmt, ylab="Temperature", main="Average Monthly Temperature")
points(mmt, pch=20)

Since the data we have available can be thought as indexed by a compound key as identified by the {year, month} pair, it is interesting to compare:

• same month data related to different years

• same year data related to all months

At the purpose I arrange the original time series in a data frame where columns store observations associated to each specific month of the year while rows are monthly observations associated to each specific year. All that for ease of consultation and to facilitate the computation of summary statistics.

months_idx <- sort(unique(month(mmt))) - 1
years <- sort(unique(year(mmt)))

mmt.by_month <- lapply(months_idx, function(x) { mmt[.indexmon(mmt) == x]})
names(mmt.by_month) <- format(ISOdate(2016, 1:12, 1),"%B")

mmt_df <- data.frame()
mmt_df <- sapply(mmt.by_month, function(x) {coredata(x)})
mmt_df <- data.frame(Year = years, mmt_df)
mmt_df <- as.data.frame(mmt_df)

kable(mmt_df)

Year	January	February	March	April	May	June	July	August	September	October	November	December
1920	40.6	40.8	44.4	46.7	54.1	58.5	57.7	56.4	54.3	50.5	42.9	39.8
1921	44.2	39.8	45.1	47.0	54.1	58.7	66.3	59.9	57.0	54.2	39.7	42.8
1922	37.5	38.7	39.5	42.1	55.7	57.8	56.8	54.3	54.3	47.1	41.8	41.7
1923	41.8	40.1	42.9	45.8	49.2	52.7	64.2	59.6	54.4	49.2	36.6	37.6
1924	39.3	37.5	38.3	45.5	53.2	57.7	60.8	58.2	56.4	49.8	44.4	43.6
1925	40.0	40.5	40.8	45.1	53.8	59.4	63.5	61.0	53.0	50.0	38.1	36.3
1926	39.2	43.4	43.4	48.9	50.6	56.8	62.5	62.0	57.5	46.7	41.6	39.8
1927	39.4	38.5	45.3	47.1	51.7	55.0	60.4	60.5	54.7	50.3	42.3	35.2
1928	40.8	41.1	42.8	47.3	50.9	56.4	62.2	60.5	55.4	50.2	43.0	37.3
1929	34.8	31.3	41.0	43.9	53.1	56.9	62.5	60.3	59.8	49.2	42.9	41.9
1930	41.6	37.1	41.2	46.9	51.2	60.4	60.1	61.6	57.0	50.9	43.0	38.8
1931	37.1	38.4	38.4	46.5	53.5	58.4	60.6	58.2	53.8	46.6	45.5	40.6
1932	42.4	38.4	40.3	44.6	50.9	57.0	62.1	63.5	56.2	47.3	43.6	41.8
1933	36.2	39.3	44.5	48.7	54.2	60.8	65.5	64.9	60.1	50.2	42.1	35.6
1934	39.4	38.2	40.4	46.9	53.4	59.6	66.5	60.4	59.2	51.2	42.8	45.8
1935	40.4	42.6	43.5	47.1	50.0	60.5	64.6	64.0	56.8	48.6	44.2	36.4
1936	37.3	35.0	44.0	43.9	52.7	58.6	60.0	61.1	58.1	49.6	41.6	41.3
1937	40.8	41.0	38.4	47.4	54.1	58.6	61.4	61.8	56.3	50.9	41.4	37.1
1938	42.1	41.2	47.3	46.6	52.4	59.0	59.6	60.4	57.0	50.7	47.8	39.2
1939	39.4	40.9	42.4	47.8	52.4	58.0	60.7	61.8	58.2	46.7	46.6	37.8

I then turn out the data frame above in a long format to ease the plots based on two types of abovementioned data slicing.

mmt_df_long <- gather(data=mmt_df, key=Month, value=Temperature, January:December)
mmt_df_long$Temperature <- unlist(mmt_df_long$Temperature)
month_idx <- as.vector(sapply(1:12, function(x) { rep(x,nrow(mmt_df)) }))
mmt_df_long <- data.frame(month_idx = month_idx, mmt_df_long)
kable(head(mmt_df_long))

month_idx	Year	Month	Temperature
1	1920	January	40.6
1	1921	January	44.2
1	1922	January	37.5
1	1923	January	41.8
1	1924	January	39.3
1	1925	January	40.0

Finally the plots.

ggplot(data=mmt_df_long, aes(x = Year, y=Temperature, colour = Month)) + 
  geom_line() + 
  geom_point() + 
  ggtitle("Temperature along years for each month")

mmt_df_long_2 <- mmt_df_long %>% arrange(Year) %>% mutate(Year = factor(Year)) %>% mutate(Month = factor(Month))

ggplot(data=mmt_df_long_2, aes(x = month_idx, y=Temperature, colour = Year)) +
  geom_line() + 
  geom_point() + 
  scale_x_discrete(limits=mmt_df_long_2$Month[1:12]) +
  theme(axis.text.x = element_text(face="bold", angle=90)) +
  ggtitle("Temperature along months for each year")

I save some of the variables in order to be able to go on next post with the exploratory analysis.

save(months_idx, years, mmt, mmt_df, mmt_df_long, mmt_df_long_2, file="nottingham.RData")

References

1. Mean monthly air temperature (Deg. F) Nottingham Castle 1920-1939

2. Time Series Data Library

3. Seasonal Periods

Note

Original post published on January 1st 2015 has been updated by substantial reworking and enhancing the analysis. Due to blog limits in post size, I had also to split it in two separate posts.