| Introduction of Time Series

library(ggplot2)
library(readxl)

# Set directory for data file
# load data Table B.4 for Exercise 2.2
AppendixB_4 <- read_excel("B4_Data.xlsx")
# change data as data frame
my.data <- as.data.frame(AppendixB_4)
# Check the column name then replaced with suitable one
colnames(my.data)

## [1] "Year"                     "Production, thousand lbs"

# column name changed into a small and suitable one, which is easy to type in coding
colnames(my.data) = c("Year", "Production")

# load data Table B.6 for Exercise 2.2
AppendixB_6 <- read_excel("B6_Data.xlsx")
# change data as data frame
my.data2 <- as.data.frame(AppendixB_6)
colnames(my.data2) # Check the column name then replaced with suitable one

## [1] "Year"       "Anomaly, C" "CO2, ppmv"

colnames(my.data2) = c("Year", "Anomaly","CO2" ) # column name changed into a small and suitable one, which is easy to type in coding
colnames(my.data2)

## [1] "Year"    "Anomaly" "CO2"

# Define a function variogram
# Variogram estimation by using define a function variogram
variogram <- function (x, lag ){
    Lag <- NULL
    vark <- NULL
    vario <- NULL
  for (k in 1: lag ){
    Lag[k] <- k
    vark [k] = sd( diff (x,k ))^2
  vario [k] = vark [k]/ vark [1]
  }
return (as.data.frame( cbind (Lag , vario )))
}

#*********************************************************************************************

Example related to Autocorrelation(ACF) and Variogram.

Consider the data on US production of blue and gorgonzola cheeses in Table B.4. a. Find the sample autocorrelation function and the variogram for these data. Is the time series stationary or nonstationary?

Answer: A time series is a series of data points indexed in time. The fact that time series data is ordered makes it unique in the data space because it often displays serial dependence. Serial dependence occurs when the value of a datapoint at one time is statistically dependent on another datapoint in another time. Autocorrelation is a type of serial dependence. Specifically, autocorrelation is when a time series is linearly related to a lagged version of itself. By contrast, correlation is simply when two independent variables are linearly related.

At first, we visualize the data using ggplot tool to have a time series plot along with the histogram.From the figure, we can clearly say that the trend is stationary for the US production of blue and gorgonzola cheeses.

#plotting time series data using ggplot
p <- ggplot (my.data, aes(x= Year, y=Production))
p + geom_point()+
  geom_line()+
  geom_smooth(method = "lm", se= FALSE, col="red", linetype = 2)+
  labs(x = "Year", y = "Production, 10000lb", title = "Variation of US cheeses production from 1950 -1997")+
  theme_classic()+
  theme(plot.title = element_text(size = rel(1.5), face = "bold", hjust = 0.5),
        axis.title = element_text(size = rel(1.5)),
        legend.position = "bottom")

## `geom_smooth()` using formula 'y ~ x'

Figure 1: Variation of US blue and gorgonzola cheeses production from 1950 -1997. The red dot line shows the fitted regression line.

ggplot(my.data, aes(x = Production))+
  geom_histogram(bins = 10, color="black", fill="white")+
  labs(x = "Year", y = "Frequency", title = "Histogram of US cheeses production")+
  theme_classic()+
  theme(plot.title = element_text(size = rel(1.5), face = "bold", hjust = 0.5),
        axis.title = element_text(size = rel(1.5)),
        legend.position = "bottom")

Figure 2: Histogram of US blue and gorgonzola cheeses production for 1950 -1997

Figure 3: Sample Autocorrelation funciton (ACF) for the US production of blue and gorgonzola cheeses for 1950 -1997

Conclusion: As we know that a time series can be called stationary if the mean is fixed and the variance is a constant number. From the ACF plot depicted above, we can clearly say that, this time series is not stationary as the value of ACF were not decreased to zero over time. In addition, there is no presence of mild sinusoidal pattern.

# call variogram funciton for my.data [,2]; for column two

variogram_production <- as.data.frame(variogram(my.data[,2], length (my.data[ ,2])/4))

ggplot(variogram_production, aes(x=Lag, y=vario))+
  geom_col()+
  labs(x = "Lag", y = "Variogram", title = "Variogram of US cheeses production")+
  theme_classic()+
  theme(plot.title = element_text(size = rel(1.5), face = "bold", hjust = 0.5),
        axis.title = element_text(size = rel(1.5)),
        legend.position = "bottom")

Figure 4: variogram of US blue and gorgonzola cheeses production for 1950 -1997

Conclusion: If we are unable to predict the data to be either stationary or not by means of ACF, then we can analyze further with the help of variogram analysis. If the variogram varies around a constant number then we can say that, time series is stationary. From the obtained value of variogram, we can clearly say that the values of variogram were increased over time. So, the time series is not stationary.

We have got similar results using JUMP software. From ACF value, we assumed that time series was weakly stationary but with further analysis with Variogram, we came to a clear conclusion that, this is a non-stationary time series.

Example 2

Table B.6 contains two time series: the global mean surface air temperature anomaly and the global CO2 concentration. Find the sample autocorrelation function and the variogram for both of these time series. Is either one of the time series stationary?

p <- ggplot (my.data2, aes(x= Year, y=Anomaly))
p + geom_point()+
  geom_line()+
  geom_smooth(method = "lm", se= FALSE, col="red", linetype = 2)+
  labs(x = "Year", y = "Production, 10000lb", title = "Variation of Global air temperature anomaly")+
  theme_classic()+
  theme(plot.title = element_text(size = rel(1.5), face = "bold", hjust = 0.5),
        axis.title = element_text(size = rel(1.5)),
        legend.position = "bottom")

## `geom_smooth()` using formula 'y ~ x'

Figure 5: Variation of the global mean surface air temperature anomaly in oC from 1880 -2004. The red dot line shows the fitted regression line.

ggplot(my.data2, aes(x = Anomaly))+
  geom_histogram(bins = 10, color="black", fill="white")+
  labs(x = "Year", y = "Frequency", title = "Histogram of air temperature anomaly")+
  theme_classic()+
  theme(plot.title = element_text(size = rel(1.5), face = "bold", hjust = 0.5),
        axis.title = element_text(size = rel(1.5)),
        legend.position = "bottom")

Figure 6: Histogram of Global air temperature anomaly

# Autocorrelation function code
acf(my.data2[,2], lag.max=length (my.data2[ ,2])/4,type="correlation", main="ACF of global temp anomaly")

Figure 7: Sample Autocorrelation funciton (ACF) for the global surface temperature anomaly for 1880 -2004

Conclusion: As we know that a time series can be called stationary if the mean is fixed and the variance is a constant number. From the ACF plot depicted above, we can clearly say that this time series is stationary as the value of ACF were decreased to zero over time. In addition, there is a presence of mild sinusoidal pattern which is one of the criteria of a typical time series plot.

# call variogram funciton for my.data [,2]; for column two
variogram_Anomaly <- as.data.frame(variogram(my.data2[,2], length (my.data2[ ,2])/4))

ggplot(variogram_Anomaly, aes(x=Lag, y=vario))+
  geom_col()+
  labs(x = "Lag", y = "Variogram", title = "Variogram of US cheeses production")+
  theme_classic()+
  theme(plot.title = element_text(size = rel(1.5), face = "bold", hjust = 0.5),
        axis.title = element_text(size = rel(1.5)),
        legend.position = "bottom")

Figure 8: variogram of the global surface temperature anomaly

Conclusion: If we are unable to predict the data to be either stationary or not by means of ACF, then we can analyze further with the help of variogram analysis. If the variogram varies around a constant number then we can say that, time series is stationary. From the obtained value of variogram, we can clearly say that the values of variogram were constant over time. So, the time series is stationary.

We have got similar results using JUMP software. From ACF value, we assumed that time series was stationary and with further analysis with Variogram, we came to a clear conclusion that, this is a stationary time series.

p <- ggplot (my.data2, aes(x= Year, y=CO2))
p + geom_point()+
  geom_line()+
  geom_smooth(method = "lm", se= FALSE, col="red", linetype = 2)+
  labs(x = "Year", y = "Production, 10000lb", title = "Variation of the global CO2 concentration")+
  theme_classic()+
  theme(plot.title = element_text(size = rel(1.5), face = "bold", hjust = 0.5),
        axis.title = element_text(size = rel(1.5)),
        legend.position = "bottom")

## `geom_smooth()` using formula 'y ~ x'

Figure 9: Variation of the global CO2 concentration from 1880 -2004. The red dot line shows the fitted regression line.

ggplot(my.data2, aes(x = CO2))+
  geom_histogram(bins = 10, color="black", fill="white")+
  labs(x = "Year", y = "Frequency", title = "Histogram of the global CO2 concentration")+
  theme_classic()+
  theme(plot.title = element_text(size = rel(1.5), face = "bold", hjust = 0.5),
        axis.title = element_text(size = rel(1.5)),
        legend.position = "bottom")

Figure 10: Histogram of the global CO2 concentration

# Autocorrelation function code
acf(my.data2[,3], lag.max=length (my.data2[ ,3])/4,type="correlation", main="ACF of global CO2 concentration")

Figure 11: Sample Autocorrelation funciton (ACF) for the the global CO2 concentration

Conclusion: As we know that a time series can be called stationary if the mean is fixed and variance is a constant number. From the ACF plot depicted above, we can clearly say that this time series is not stationary as the value of ACF were not decreased to zero over time. In addition, there is no presence of mild sinusoidal pattern.

# call variogram funciton for my.data [,2]; for column two
variogram_CO2 <- as.data.frame(variogram(my.data2[,3], length (my.data2[ ,3])/4))

ggplot(variogram_CO2, aes(x=Lag, y=vario))+
  geom_col()+
  labs(x = "Lag", y = "Variogram", title = "Variogram of US cheeses production")+
  theme_classic()+
  theme(plot.title = element_text(size = rel(1.5), face = "bold", hjust = 0.5),
        axis.title = element_text(size = rel(1.5)),
        legend.position = "bottom")

Figure 12: variogram of US blue and gorgonzola cheeses production for 1950 -1997

We have got similar results using JUMP software. From ACF value, we assumed that time series was not stationary and with further analysis with Variogram, we came to a clear conclusion that, this is a non-stationary time series.

| Introduction of Time Series

Md Mominul Islam

Date: 08/24/2020

Example 2