## 2.6 Scatterplots

The graphs discussed so far are useful for visualising individual time series. It is also useful to explore relationships between time series.

Figure 2.7 shows two time series: half-hourly electricity demand (in Gigawatts) and temperature (in degrees Celsius), for 2014 in Victoria, Australia. The temperatures are for Melbourne, the largest city in Victoria, while the demand values are for the entire state.

autoplot(elecdemand[,c("Demand","Temperature")], facets=TRUE) +
xlab("Year: 2014") + ylab("") +
ggtitle("Half-hourly electricity demand: Victoria, Australia")

(The actual code for this plot is a little more complicated than what is shown in order to include the months on the x-axis.)

We can study the relationship between demand and temperature by plotting one series against the other.

qplot(Temperature, Demand, data=as.data.frame(elecdemand)) +
ylab("Demand (GW)") + xlab("Temperature (Celsius)")
#> Warning: qplot() was deprecated in ggplot2 3.4.0.
#> This warning is displayed once every 8 hours.
#> Call lifecycle::last_lifecycle_warnings() to see where this warning was generated.

This scatterplot helps us to visualise the relationship between the variables. It is clear that high demand occurs when temperatures are high due to the effect of air-conditioning. But there is also a heating effect, where demand increases for very low temperatures.

### Correlation

It is common to compute correlation coefficients to measure the strength of the relationship between two variables. The correlation between variables $$x$$ and $$y$$ is given by $r = \frac{\sum (x_{t} - \bar{x})(y_{t}-\bar{y})}{\sqrt{\sum(x_{t}-\bar{x})^2}\sqrt{\sum(y_{t}-\bar{y})^2}}.$ The value of $$r$$ always lies between $$-1$$ and 1 with negative values indicating a negative relationship and positive values indicating a positive relationship. The graphs in Figure 2.9 show examples of data sets with varying levels of correlation.

The correlation coefficient only measures the strength of the linear relationship, and can sometimes be misleading. For example, the correlation for the electricity demand and temperature data shown in Figure 2.8 is 0.28, but the non-linear relationship is stronger than that.

The plots in Figure 2.10 all have correlation coefficients of 0.82, but they have very different relationships. This shows how important it is to look at the plots of the data and not simply rely on correlation values.

### Scatterplot matrices

When there are several potential predictor variables, it is useful to plot each variable against each other variable. Consider the five time series shown in Figure 2.11, showing quarterly visitor numbers for five regions of New South Wales, Australia.

autoplot(visnights[,1:5], facets=TRUE) +
ylab("Number of visitor nights each quarter (millions)")

To see the relationships between these five time series, we can plot each time series against the others. These plots can be arranged in a scatterplot matrix, as shown in Figure 2.12. (This plot requires the GGally package to be installed.)

GGally::ggpairs(as.data.frame(visnights[,1:5]))

For each panel, the variable on the vertical axis is given by the variable name in that row, and the variable on the horizontal axis is given by the variable name in that column. There are many options available to produce different plots within each panel. In the default version, the correlations are shown in the upper right half of the plot, while the scatterplots are shown in the lower half. On the diagonal are shown density plots.

The value of the scatterplot matrix is that it enables a quick view of the relationships between all pairs of variables. In this example, the second column of plots shows there is a strong positive relationship between visitors to the NSW north coast and visitors to the NSW south coast, but no detectable relationship between visitors to the NSW north coast and visitors to the NSW south inland. Outliers can also be seen. There is one unusually high quarter for the NSW Metropolitan region, corresponding to the 2000 Sydney Olympics. This is most easily seen in the first two plots in the left column of Figure 2.12, where the largest value for NSW Metro is separate from the main cloud of observations.