edX assignment link: http://bit.ly/2KE2g00
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.
In this problem, 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, 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 (hint: use subset). 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, 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). Use the training set to build the model.
Enter the model R2 (the “Multiple R-squared” value):
climate = read.csv("climate_change.csv")
trainingSet = subset(climate,Year <= 2006)
testingSet = subset(climate, Year > "2006")
model1 = lm(Temp ~ MEI + CO2 + CH4 + N2O + CFC.11 + CFC.12 + TSI + Aerosols, data = trainingSet )
summary(model1)
Call:
lm(formula = Temp ~ MEI + CO2 + CH4 + N2O + CFC.11 + CFC.12 +
TSI + Aerosols, data = trainingSet)
Residuals:
Min 1Q Median 3Q Max
-0.25888 -0.05913 -0.00082 0.05649 0.32433
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -1.246e+02 1.989e+01 -6.265 1.43e-09 ***
MEI 6.421e-02 6.470e-03 9.923 < 2e-16 ***
CO2 6.457e-03 2.285e-03 2.826 0.00505 **
CH4 1.240e-04 5.158e-04 0.240 0.81015
N2O -1.653e-02 8.565e-03 -1.930 0.05467 .
CFC.11 -6.631e-03 1.626e-03 -4.078 5.96e-05 ***
CFC.12 3.808e-03 1.014e-03 3.757 0.00021 ***
TSI 9.314e-02 1.475e-02 6.313 1.10e-09 ***
Aerosols -1.538e+00 2.133e-01 -7.210 5.41e-12 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.09171 on 275 degrees of freedom
Multiple R-squared: 0.7509, Adjusted R-squared: 0.7436
F-statistic: 103.6 on 8 and 275 DF, p-value: < 2.2e-16# Multiple R-squared : 0.7509Which variables are significant in the model? We will consider a variable signficant only if the p-value is below 0.05. (Select all that apply.)
# MEI, CO2, CFC.11, CFC.12, TSI, AerosolsCurrent 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.
Which of the following is the simplest correct explanation for this contradiction?
Climate scientists are wrong that N2O and CFC-11 are greenhouse gases - this regression analysis constitutes part of a disproof.
There is not enough data, so the regression coefficients being estimated are not accurate.
All of the gas concentration variables reflect human development - N2O and CFC.11 are correlated with other variables in the data set.
# All of the gas concentration variables reflect human development - N2O and CFC.11 are correlated with other variables in the data set.Compute the correlations between all the variables in the training set. Which of the following independent variables is N2O highly correlated with (absolute correlation greater than 0.7)? Select all that apply.
cor(climate) Year Month MEI CO2 CH4 N2O CFC.11 CFC.12 TSI Aerosols
Year 1.00000000 -0.025789103 -0.14534485 0.98537870 0.91056328 0.99484971 0.46096457 0.870067456 0.02235316 -0.36188438
Month -0.02578910 1.000000000 -0.01634543 -0.09628668 0.01755804 0.01239521 -0.01491372 -0.001084139 -0.03275430 0.01484519
MEI -0.14534485 -0.016345434 1.00000000 -0.15291104 -0.10555472 -0.16237531 0.08817074 -0.039835666 -0.07682560 0.35235073
CO2 0.98537870 -0.096286676 -0.15291104 1.00000000 0.87225311 0.98113544 0.40128447 0.823210310 0.01786672 -0.36926514
CH4 0.91056328 0.017558035 -0.10555472 0.87225311 1.00000000 0.89440921 0.71350408 0.958237181 0.14633495 -0.29038142
N2O 0.99484971 0.012395210 -0.16237531 0.98113544 0.89440921 1.00000000 0.41215475 0.839295454 0.03989183 -0.35349882
CFC.11 0.46096457 -0.014913724 0.08817074 0.40128447 0.71350408 0.41215475 1.00000000 0.831381310 0.28462884 -0.03230227
CFC.12 0.87006746 -0.001084139 -0.03983567 0.82321031 0.95823718 0.83929545 0.83138131 1.000000000 0.18927009 -0.24378508
TSI 0.02235316 -0.032754296 -0.07682560 0.01786672 0.14633495 0.03989183 0.28462884 0.189270090 1.00000000 0.08323812
Aerosols -0.36188438 0.014845187 0.35235073 -0.36926514 -0.29038142 -0.35349882 -0.03230227 -0.243785082 0.08323812 1.00000000
Temp 0.75573115 -0.098015821 0.13529168 0.74850465 0.69969658 0.74324183 0.38011134 0.688944109 0.18218561 -0.39206945
Temp
Year 0.75573115
Month -0.09801582
MEI 0.13529168
CO2 0.74850465
CH4 0.69969658
N2O 0.74324183
CFC.11 0.38011134
CFC.12 0.68894411
TSI 0.18218561
Aerosols -0.39206945
Temp 1.00000000# CO2, CH4, CFC.12Which of the following independent variables is CFC.11 highly correlated with? Select all that apply.
# CH4, CFC.12Given 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. Remember to use the training set to build the model.
Enter the coefficient of N2O in this reduced model:
model2 = lm( Year ~ MEI+TSI+Aerosols+N2O , data=trainingSet )
summary(model2)
Call:
lm(formula = Year ~ MEI + TSI + Aerosols + N2O, data = trainingSet)
Residuals:
Min 1Q Median 3Q Max
-2.12248 -0.43341 -0.02631 0.48569 1.67431
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 2145.49515 154.71103 13.868 < 2e-16 ***
MEI 0.10422 0.05089 2.048 0.041497 *
TSI -0.43704 0.11380 -3.840 0.000152 ***
Aerosols -3.03116 1.66772 -1.818 0.070206 .
N2O 1.43178 0.01003 142.710 < 2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.7304 on 279 degrees of freedom
Multiple R-squared: 0.9888, Adjusted R-squared: 0.9886
F-statistic: 6146 on 4 and 279 DF, p-value: < 2.2e-16# the coefficient of N2O = 1.43178(How does this compare to the coefficient in the previous model with all of the variables?)
Enter the model R2:
# 0.9888We 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 (HINT: If your initial full model was called “climateLM”, you could create a new model with the step function by typing step(climateLM). Be sure to save your new model to a variable name so that you can look at the summary. For more information about the step function, type ?step in your R console.)
Enter the R2 value of the model produced by the step function:
simplifiedModel = step(model1)Start: AIC=-1348.16
Temp ~ MEI + CO2 + CH4 + N2O + CFC.11 + CFC.12 + TSI + Aerosols
Df Sum of Sq RSS AIC
- CH4 1 0.00049 2.3135 -1350.1
<none> 2.3130 -1348.2
- N2O 1 0.03132 2.3443 -1346.3
- CO2 1 0.06719 2.3802 -1342.0
- CFC.12 1 0.11874 2.4318 -1335.9
- CFC.11 1 0.13986 2.4529 -1333.5
- TSI 1 0.33516 2.6482 -1311.7
- Aerosols 1 0.43727 2.7503 -1301.0
- MEI 1 0.82823 3.1412 -1263.2
Step: AIC=-1350.1
Temp ~ MEI + CO2 + N2O + CFC.11 + CFC.12 + TSI + Aerosols
Df Sum of Sq RSS AIC
<none> 2.3135 -1350.1
- N2O 1 0.03133 2.3448 -1348.3
- CO2 1 0.06672 2.3802 -1344.0
- CFC.12 1 0.13023 2.4437 -1336.5
- CFC.11 1 0.13938 2.4529 -1335.5
- TSI 1 0.33500 2.6485 -1313.7
- Aerosols 1 0.43987 2.7534 -1302.7
- MEI 1 0.83118 3.1447 -1264.9summary(model1)
Call:
lm(formula = Temp ~ MEI + CO2 + CH4 + N2O + CFC.11 + CFC.12 +
TSI + Aerosols, data = trainingSet)
Residuals:
Min 1Q Median 3Q Max
-0.25888 -0.05913 -0.00082 0.05649 0.32433
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -1.246e+02 1.989e+01 -6.265 1.43e-09 ***
MEI 6.421e-02 6.470e-03 9.923 < 2e-16 ***
CO2 6.457e-03 2.285e-03 2.826 0.00505 **
CH4 1.240e-04 5.158e-04 0.240 0.81015
N2O -1.653e-02 8.565e-03 -1.930 0.05467 .
CFC.11 -6.631e-03 1.626e-03 -4.078 5.96e-05 ***
CFC.12 3.808e-03 1.014e-03 3.757 0.00021 ***
TSI 9.314e-02 1.475e-02 6.313 1.10e-09 ***
Aerosols -1.538e+00 2.133e-01 -7.210 5.41e-12 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.09171 on 275 degrees of freedom
Multiple R-squared: 0.7509, Adjusted R-squared: 0.7436
F-statistic: 103.6 on 8 and 275 DF, p-value: < 2.2e-16summary(simplifiedModel)
Call:
lm(formula = Temp ~ MEI + CO2 + N2O + CFC.11 + CFC.12 + TSI +
Aerosols, data = trainingSet)
Residuals:
Min 1Q Median 3Q Max
-0.25770 -0.05994 -0.00104 0.05588 0.32203
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -1.245e+02 1.985e+01 -6.273 1.37e-09 ***
MEI 6.407e-02 6.434e-03 9.958 < 2e-16 ***
CO2 6.402e-03 2.269e-03 2.821 0.005129 **
N2O -1.602e-02 8.287e-03 -1.933 0.054234 .
CFC.11 -6.609e-03 1.621e-03 -4.078 5.95e-05 ***
CFC.12 3.868e-03 9.812e-04 3.942 0.000103 ***
TSI 9.312e-02 1.473e-02 6.322 1.04e-09 ***
Aerosols -1.540e+00 2.126e-01 -7.244 4.36e-12 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.09155 on 276 degrees of freedom
Multiple R-squared: 0.7508, Adjusted R-squared: 0.7445
F-statistic: 118.8 on 7 and 276 DF, p-value: < 2.2e-16# R2 = 0.7508Which of the following variable(s) were eliminated from the full model by the step function? Select all that apply.
# CH4It 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, calculate temperature predictions for the testing data set, using the predict function.
Enter the testing set R2:
predictTest1 = predict(simplifiedModel, newdata = testingSet)
SSE = sum((testingSet$Temp - predictTest1)^2)
SST = sum((testingSet$Temp - mean(trainingSet$Temp))^2)
R2 = 1 - SSE/SST
R2[1] 0.6286051