There have been many studies documenting that the average global temperature has been increasing over the last century. The consequences of a continued rise in global temperature will be dire. Rising sea levels and an increased frequency of extreme weather events will affect billions of people.

Today, we will attempt to study the relationship between average global temperature and several other factors.

The file climate_change.csv contains climate data from May 1983 to December 2008. The available variables include:

• Year: the observation year.
• Month: the observation month.
• Temp: the difference in degrees Celsius between the average global temperature in that period and a reference value. This data comes from the Climatic Research Unit at the University of East Anglia.
• CO2, N2O, CH4, CFC.11, CFC.12: atmospheric concentrations of carbon dioxide (CO2), nitrous oxide (N2O), methane (CH4), trichlorofluoromethane (CCl3F; commonly referred to as CFC-11) and dichlorodifluoromethane (CCl2F2; commonly referred to as CFC-12), respectively. This data comes from the ESRL/NOAA Global Monitoring Division. CO2, N2O and CH4 are expressed in ppmv (parts per million by volume -- i.e., 397 ppmv of CO2 means that CO2 constitutes 397 millionths of the total volume of the atmosphere) CFC.11 and CFC.12 are expressed in ppbv (parts per billion by volume).
• Aerosols: the mean stratospheric aerosol optical depth at 550 nm. This variable is linked to volcanoes, as volcanic eruptions result in new particles being added to the atmosphere, which affect how much of the sun's energy is reflected back into space. This data is from the Godard Institute for Space Studies at NASA.
• TSI: the total solar irradiance (TSI) in W/m2 (the rate at which the sun's energy is deposited per unit area). Due to sunspots and other solar phenomena, the amount of energy that is given off by the sun varies substantially with time. This data is from the SOLARIS-HEPPA project website.
• MEI: multivariate El Nino Southern Oscillation index (MEI), a measure of the strength of the El Nino/La Nina-Southern Oscillation (a weather effect in the Pacific Ocean that affects global temperatures). This data comes from the ESRL/NOAA Physical Sciences Division.

We are interested in how changes in these variables affect future temperatures, as well as how well these variables explain temperature changes so far. To do this, first read the dataset climate_change.csv into R.

Then, we split the data into a training set, consisting of all the observations up to and including 2006, and a testing set consisting of the remaining years. A training set refers to the data that will be used to build the model (this is the data we give to the lm() function), and a testing set refers to the data we will use to test our predictive ability.

Next, we build a linear regression model to predict the dependent variable Temp, using MEI, CO2, CH4, N2O, CFC.11, CFC.12, TSI, and Aerosols as independent variables (Year and Month should NOT be used in the model).

climate = read.csv("climate_change.csv")
train = subset(climate, Year <= 2006)
test = subset(climate, Year > 2006)

Then, we can create the model using the command:

climatelm = lm(Temp ~ MEI + CO2 + CH4 + N2O + CFC.11 + CFC.12 + TSI + Aerosols, data=train)

Lastly, look at the model using summary(climatelm). The Multiple R-squared value is 0.7509.

In order to know, which variables are significant in the model, We will consider a variable signficant only if the p-value is below 0.05. If we look at the model we created in the previous problem using summary(climatelm), all of the variables have at least one star except for CH4 and N2O. So MEI, CO2, CFC.11, CFC.12, TSI, and Aerosols are all significant.

Current scientific opinion is that nitrous oxide and CFC-11 are greenhouse gases: gases that are able to trap heat from the sun and contribute to the heating of the Earth. However, the regression coefficients of both the N2O and CFC-11 variables are negative, indicating that increasing atmospheric concentrations of either of these two compounds is associated with lower global temperatures. The linear correlation of N2O and CFC.11 with other variables in the data set is quite large. All of the gas concentration variables reflect human development - N2O and CFC.11 are correlated with other variables in the data set.

We can calculate all correlations at once using cor(train) where train is the name of the training data set. CO2, CH4, CFC-12 are the variables, N2O is highly correlated with (absolute correlation greater than 0.7). CH4, CFC-12 are the variables, CFC-11 is highly correlated with.

Given that the correlations are so high, let us focus on the N2O variable and build a model with only MEI, TSI, Aerosols and N2O as independent variables. We can create this simplified model with the command:

LinReg = lm(Temp ~ MEI + N2O + TSI + Aerosols, data=train)

We can get the coefficient for N2O and the model R-squared by typing summary(LinReg).

We have observed that, for this problem, when we remove many variables the sign of N2O flips. The model has not lost a lot of explanatory power (the model R2 is 0.7261 compared to 0.7509 previously) despite removing many variables. This type of behavior is typical when building a model where many of the independent variables are highly correlated with each other. Here many of the variables (CO2, CH4, N2O, CFC.11 and CFC.12) are highly correlated, since they are all driven by human industrial development.

We have many variables in this problem, and as we have seen above, dropping some from the model does not decrease model quality. R provides a function, step, that will automate the procedure of trying different combinations of variables to find a good compromise of model simplicity and R2. This trade-off is formalized by the Akaike information criterion (AIC) - it can be informally thought of as the quality of the model with a penalty for the number of variables in the model.

The step function has one argument - the name of the initial model. It returns a simplified model. Use the step function in R to derive a new model, with the full model as the initial model Our initial full model was called "climateLM", we can create a new model with the step function by typing step(climateLM). We have to save your new model to a variable name so that we can look at the summary.

StepModel = step(climateLM)

where "climateLM" is the name of the full model. If we look at the summary of the model with summary(StepModel), we can see that the R-squared value is 0.75, and only CH4 was removed.

It is interesting to note that the step function does not address the collinearity of the variables, except that adding highly correlated variables will not improve the R2 significantly. The consequence of this is that the step function will not necessarily produce a very interpretable model - just a model that has balanced quality and simplicity for a particular weighting of quality and simplicity (AIC).

We have developed an understanding of how well we can fit a linear regression to the training data, but does the model quality hold when applied to unseen data? Using the model produced from the step function, we can calculate temperature predictions for the testing data set, using the predict function.

tempPredict = predict(climateStep, newdata = test)

SSE = sum((tempPredict - test$Temp)^2)
SST = sum( (mean(train$Temp) - test$Temp)^2)
R2 = 1 - SSE/SST

Testing set R2 is 0.6286051.