MTH6139 Time Series

UK Gas consumption Analysis and Forecasting

Section 1: Introduction

1.1 What is the ‘UK gas’ dataset?

The UK gas dataset contains quarterly UK gas consumption data from 1960 (Q1) to 1986 (Q4), measured in millions of therms.

1.2 Dataset Summary (UK gas Documentation)

• Time Period: 1960 Q1 – 1986 Q4
• Source: Durbin, J. and Koopman, S. J. (2001). Time Series Analysis by State Space Methods. Oxford University Press.
• Frequency: Quarterly (4 observations per year) Total Observations: 108

This dataset is useful for analysing trends in gas consumption, identifying seasonal variations, and forecasting future energy demand.

1.3 The purpose of this project

The purpose of this project is to analyse UK quarterly gas consumption from 1960 to 1986 using the UK gas dataset. The main objectives are to:

• Examine long-term trends in gas consumption to understand whether usage is increasing or decreasing over time.
• Identify seasonal patterns to determine if gas consumption fluctuates predictably across different quarters.
• Use linear regression to quantify the rate of change in gas consumption over the years.
• Apply the Prophet forecasting model to predict future gas consumption trends.
• Provide insights for policymakers and energy analysts to help manage energy resources and plan for future demand. By analysing historical gas consumption, we can gain valuable insights into energy usage patterns, the impact of seasonality, and how future consumption may evolve.

Section 2: Exploring UK gas Trends and Patterns

2.1 Visualising the UK gas Time Series

To understand the historical trends in UK quarterly gas consumption, we plot the UKgas dataset over time. This helps identify long-term trends, seasonal variations, and potential anomalies.

library(prophet)

## Loading required package: Rcpp

## Warning: package 'Rcpp' was built under R version 4.4.3

## Loading required package: rlang

## Warning: package 'rlang' was built under R version 4.4.3

Loads the Prophet package, which is used for time series forecasting.

library(zoo)  # For handling time series dates

## Warning: package 'zoo' was built under R version 4.4.3

## 
## Attaching package: 'zoo'

## The following objects are masked from 'package:base':
## 
##     as.Date, as.Date.numeric

Loads the zoo package, which helps in handling time series data and converting dates into proper formats.

ukgas_df <- data.frame(
    ds = as.Date(as.yearmon(time(UKgas))),  # Extracts time in yearmon format
    y = as.numeric(UKgas)  # Converts CO2 values into numeric
)

This function converts the UKgas dataset into a format compatible with Prophet, and extracts the time indices, converts them to a year-month format, and ensures they are in the required “YYYY-MM-DD” date format, it also converts gas consumption values into numeric form for modeling.

head(ukgas_df)

##           ds     y
## 1 1960-01-01 160.1
## 2 1960-04-01 129.7
## 3 1960-07-01  84.8
## 4 1960-10-01 120.1
## 5 1961-01-01 160.1
## 6 1961-04-01 124.9

ukgas_model <- prophet::prophet(ukgas_df)

## Disabling weekly seasonality. Run prophet with weekly.seasonality=TRUE to override this.

## Disabling daily seasonality. Run prophet with daily.seasonality=TRUE to override this.

Stores the trained model, which learns patterns from past data (trends & seasonality).

future_dates<- prophet::make_future_dataframe(ukgas_model, periods = 8, freq = "quarter")
tail(future_dates)

##             ds
## 111 1987-07-01
## 112 1987-10-01
## 113 1988-01-01
## 114 1988-04-01
## 115 1988-07-01
## 116 1988-10-01

ukgas_forecast <- predict(ukgas_model, future_dates)

This function uses the trained Prophet model to forecast gas consumption for the next 2 years (8 quarters), storing predicted values along with uncertainty intervals.

plot(ukgas_model, ukgas_forecast,type ="1",col = "blue", xlab= "Year", ylab ="Gas Consumption (Millions of Therms)", main = expression("UK Quarterly Gas Consumption (1960-1986)"))

The figure above shows the quarterly UK gas consumption from 1960 to 1986, along with a forecasted trend using the Prophet model.

• Trend: There is a clear upward trend, meaning gas consumption has steadily increased over time.
• Seasonality: The regular cycles indicate seasonal variations in gas usage, likely higher in winter and lower in summer.
• Uncertainty: The forecasted values have a wider confidence interval as time progresses, indicating increased uncertainty in long-term predictions.
• Key Insights: Around 1980–1986, gas consumption rises sharply, possibly due to economic growth, energy demand changes, or policy shifts.

This visualization helps in understanding historical gas usage trends and predicting future consumption, which is valuable for energy planning and policy decisions.

prophet::prophet_plot_components(ukgas_model, ukgas_forecast)

• Plots trend and seasonal patterns in gas consumption based on the Prophet forecast.

• Shows long-term trends, quarterly seasonality, and other components of the time series.

ukgas_df$year <- as.numeric(format(ukgas_df$ds, "%Y"))

lm_model <- lm(y ~ year, data = ukgas_df)

• Fits a linear regression model to analyze the relationship between gas consumption and time .
• Helps measure whether gas consumption is increasing or decreasing over time.

summary(lm_model)

## 
## Call:
## lm(formula = y ~ year, data = ukgas_df)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -368.74  -84.20    2.89   74.27  513.32 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -47159.213   4074.600  -11.57   <2e-16 ***
## year            24.073      2.065   11.66   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 167.2 on 106 degrees of freedom
## Multiple R-squared:  0.5618, Adjusted R-squared:  0.5576 
## F-statistic: 135.9 on 1 and 106 DF,  p-value: < 2.2e-16

2.2 Analysis of UK Gas Consumption 📈

Trend analysis

1. Steady increase in Gas consumption:

• The overall trend shows that gas consumption has increased over time.
• The growth appears to accelerate in the later years (1980s).

2. Linear Regression for growth rate The linear regression shows us that:

• It has a positive slope which implies that the gas consumption is increasing each year.
• The R-squared value is 0.5618 which means that 56.18% of the variation in gas consumption is explained by the year variable, which suggests a moderate correlation. -Th P-value (<2e-16) is very small indicating that the relationship between year and gas consumption is statistically significant.

Seasonality

1. Prophet’s seasonality Breakdown (the Prophet model helps to visualise seasonal effects):

• It shows us how gas usage fluctuates within each year.
• It also portrays the long-term gas consumption over the years.

2. Quarterly Fluctuations:

• Viewing the prophets plot we can see that the data rises and falls within each year, implying that there are seasonal variations.
• The gas consumption is typically higher in (Q1 and Q4) and low in the summer (Q2 and Q3), which could be due to the demand of heating.

2.3 More visualisations

Comparing Actual vs Predicted Values

# Predict gas consumption using the linear model
ukgas_df$predicted <- predict(lm_model)

# Plot actual vs. predicted values
plot(ukgas_df$ds, ukgas_df$y, type = "l", col = "blue",
     xlab = "Year", ylab = "Gas Consumption (Millions of Therms)",
     main = "Actual vs. Predicted Gas Consumption")
lines(ukgas_df$ds, ukgas_df$predicted, col = "red", lwd = 2)

legend("topleft", legend = c("Actual", "Predicted"),
       col = c("blue", "red"), lty = 1, lwd = 2)

The graph above compares the actual UK gas consumption (blue lines) with the predicted values from the linear regression model (red line) over the period 1960 to o1986.

Interpretation

• The blue line represents the actual gas consumption quarterly. The trend of it going up and down indicates the strong seasonality usage. Overall the blue line shows an increase in the consumption of gas over the years.

• The red line represents the predicted values from the linear regression model.The upward slope captures the overall increasing of the gas usage, however, the predicted values do not capture the seasonal fluctuations. It captures the general increase of gas usage in the long term but not the seasonal variations.

This comparison evaluates the accuracy of the regression model, showcasing its strengths and limitations in predicting UK gas consumption trends.

Checking Residuals

plot(ukgas_df$ds, residuals(lm_model), type = "p", col = "red",
     xlab = "Year", ylab = "Residuals",
     main = "Residuals of Linear Regression Model")
abline(h = 0, col = "black", lwd = 2)

The residual plot above displays the difference between the actual and predicted gas consumption values over time. Residuals represent the errors in the model—how far off the predictions are from the actual values.

Interpretations

• The plot shows a clear trend in the residuals, which suggests that the linear model does not fully capture the pattern in the data.
• In the graph you can see that after 1970 the residuals become more spread out with larger errors appearing in the 1980’s, suggesting that the model performs worse later in the years. Implying that gas consumption may not be increasing at a constant rate.
• The U-shape pattern that is forming in the residuals indicates that the simple linear regression model is not sufficient.

2.4 Key Numbers Inside R Objects

head(ukgas_df)

##           ds     y year predicted
## 1 1960-01-01 160.1 1960  24.67619
## 2 1960-04-01 129.7 1960  24.67619
## 3 1960-07-01  84.8 1960  24.67619
## 4 1960-10-01 120.1 1960  24.67619
## 5 1961-01-01 160.1 1961  48.74960
## 6 1961-04-01 124.9 1961  48.74960

• This confirms that ds contains dates and y represents gas consumption (millions of therms).

str(ukgas_df)

## 'data.frame':    108 obs. of  4 variables:
##  $ ds       : Date, format: "1960-01-01" "1960-04-01" ...
##  $ y        : num  160.1 129.7 84.8 120.1 160.1 ...
##  $ year     : num  1960 1960 1960 1960 1961 ...
##  $ predicted: num  24.7 24.7 24.7 24.7 48.7 ...

• This lets us know about the data type of each column.

summary(ukgas_df$y)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##    84.8   153.3   220.9   337.6   469.9  1163.9

• This gives a useful summary statistics about gas consumption.

head(ukgas_forecast[, c("ds", "yhat", "yhat_lower", "yhat_upper")])

##           ds        yhat yhat_lower yhat_upper
## 1 1960-01-01  197.081175   56.77199 344.376825
## 2 1960-04-01   -4.817374 -151.80553 137.093884
## 3 1960-07-01 -138.025678 -269.92465   3.290908
## 4 1960-10-01   77.187060  -72.36734 214.878425
## 5 1961-01-01  219.609022   88.28171 370.301920
## 6 1961-04-01   18.177628 -122.06914 163.698510

• The model predicts steady growth in gas consumption. The confidence interval widens over time, indicating increasing uncertainty.

coef(lm_model)  # Extracts slope & intercept of the regression model

##  (Intercept)         year 
## -47159.21270     24.07341

• Confirms gas consumption grows by ~24 million therms per year.

2.5 Comments on Results

• Gas consumption has consistently increased over time, at a rate of ~24 million therms per year.
• Linear regression captures the overall trend but fails to account for seasonality.
• Prophet forecasting suggests continued growth, but the uncertainty increases in long-term predictions.
• The residual plot shows a pattern, suggesting the model does not fully capture changes in trend growth.
• Gas consumption growth was slower in the 1960s but accelerated in the 1980s, requiring more advanced modeling.

While the model captures the general increase in gas consumption, a more advanced time series approach may improve predictions.

References

• Datacamp course: https://www.datacamp.com/courses/reporting-with-rmarkdown
• RStudio reference: https://rmarkdown.rstudio.com/lesson-1.html
• More sophisticated reference: https://bookdown.org/yihui/rmarkdown/