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):
0.7509
climate_change <- read.csv('C:/Users/bolin/Desktop/Rclass/Unit2/data/climate_change.csv')
climate_trainng <- subset(climate_change , Year <= 2006)
summary(climate_trainng)
Year Month MEI CO2 CH4
Min. :1983 Min. : 1.000 Min. :-1.5860 Min. :340.2 Min. :1630
1st Qu.:1989 1st Qu.: 4.000 1st Qu.:-0.3230 1st Qu.:352.3 1st Qu.:1716
Median :1995 Median : 7.000 Median : 0.3085 Median :359.9 Median :1759
Mean :1995 Mean : 6.556 Mean : 0.3419 Mean :361.4 Mean :1746
3rd Qu.:2001 3rd Qu.:10.000 3rd Qu.: 0.8980 3rd Qu.:370.6 3rd Qu.:1782
Max. :2006 Max. :12.000 Max. : 3.0010 Max. :385.0 Max. :1808
N2O CFC.11 CFC.12 TSI Aerosols
Min. :303.7 Min. :191.3 Min. :350.1 Min. :1365 Min. :0.00160
1st Qu.:307.7 1st Qu.:249.6 1st Qu.:462.5 1st Qu.:1366 1st Qu.:0.00270
Median :310.8 Median :260.4 Median :522.1 Median :1366 Median :0.00620
Mean :311.7 Mean :252.5 Mean :494.2 Mean :1366 Mean :0.01772
3rd Qu.:316.1 3rd Qu.:267.4 3rd Qu.:541.0 3rd Qu.:1366 3rd Qu.:0.01400
Max. :320.5 Max. :271.5 Max. :543.8 Max. :1367 Max. :0.14940
Temp
Min. :-0.2820
1st Qu.: 0.1180
Median : 0.2325
Mean : 0.2478
3rd Qu.: 0.4065
Max. : 0.7390 model1 <- lm(Temp ~ MEI+ CO2+ CH4+ N2O+ CFC.11+ CFC.12+ TSI + Aerosols ,data=climate_trainng )
summary(model1)
Call:
lm(formula = Temp ~ MEI + CO2 + CH4 + N2O + CFC.11 + CFC.12 +
TSI + Aerosols, data = climate_trainng)
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-16Which 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.)
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.
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.
CO2 CH4 N2O CFC.12
cor(climate_trainng)
Year Month MEI CO2 CH4 N2O
Year 1.00000000 -0.0279419602 -0.0369876842 0.98274939 0.91565945 0.99384523
Month -0.02794196 1.0000000000 0.0008846905 -0.10673246 0.01856866 0.01363153
MEI -0.03698768 0.0008846905 1.0000000000 -0.04114717 -0.03341930 -0.05081978
CO2 0.98274939 -0.1067324607 -0.0411471651 1.00000000 0.87727963 0.97671982
CH4 0.91565945 0.0185686624 -0.0334193014 0.87727963 1.00000000 0.89983864
N2O 0.99384523 0.0136315303 -0.0508197755 0.97671982 0.89983864 1.00000000
CFC.11 0.56910643 -0.0131112236 0.0690004387 0.51405975 0.77990402 0.52247732
CFC.12 0.89701166 0.0006751102 0.0082855443 0.85268963 0.96361625 0.86793078
TSI 0.17030201 -0.0346061935 -0.1544919227 0.17742893 0.24552844 0.19975668
Aerosols -0.34524670 0.0148895406 0.3402377871 -0.35615480 -0.26780919 -0.33705457
Temp 0.78679714 -0.0998567411 0.1724707512 0.78852921 0.70325502 0.77863893
CFC.11 CFC.12 TSI Aerosols Temp
Year 0.56910643 0.8970116635 0.17030201 -0.34524670 0.78679714
Month -0.01311122 0.0006751102 -0.03460619 0.01488954 -0.09985674
MEI 0.06900044 0.0082855443 -0.15449192 0.34023779 0.17247075
CO2 0.51405975 0.8526896272 0.17742893 -0.35615480 0.78852921
CH4 0.77990402 0.9636162478 0.24552844 -0.26780919 0.70325502
N2O 0.52247732 0.8679307757 0.19975668 -0.33705457 0.77863893
CFC.11 1.00000000 0.8689851828 0.27204596 -0.04392120 0.40771029
CFC.12 0.86898518 1.0000000000 0.25530281 -0.22513124 0.68755755
TSI 0.27204596 0.2553028138 1.00000000 0.05211651 0.24338269
Aerosols -0.04392120 -0.2251312440 0.05211651 1.00000000 -0.38491375
Temp 0.40771029 0.6875575483 0.24338269 -0.38491375 1.00000000Which of the following independent variables is CFC.11 highly correlated with? Select all that apply.
CH4 CFC.11 CFC.12
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. Remember to use the training set to build the model.
Enter the coefficient of N2O in this reduced model:
2.532e-02
model2 <- lm(Temp ~ MEI+ N2O+ TSI + Aerosols ,data=climate_trainng )
summary(model2)
Call:
lm(formula = Temp ~ MEI + N2O + TSI + Aerosols, data = climate_trainng)
Residuals:
Min 1Q Median 3Q Max
-0.27916 -0.05975 -0.00595 0.05672 0.34195
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -1.162e+02 2.022e+01 -5.747 2.37e-08 ***
MEI 6.419e-02 6.652e-03 9.649 < 2e-16 ***
N2O 2.532e-02 1.311e-03 19.307 < 2e-16 ***
TSI 7.949e-02 1.487e-02 5.344 1.89e-07 ***
Aerosols -1.702e+00 2.180e-01 -7.806 1.19e-13 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.09547 on 279 degrees of freedom
Multiple R-squared: 0.7261, Adjusted R-squared: 0.7222
F-statistic: 184.9 on 4 and 279 DF, p-value: < 2.2e-16(How does this compare to the coefficient in the previous model with all of the variables?)
Enter the model R2:
R2 = 0.7261
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 (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:
R2 = 0.7508
StepModel <- 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(StepModel)
Call:
lm(formula = Temp ~ MEI + CO2 + N2O + CFC.11 + CFC.12 + TSI +
Aerosols, data = climate_trainng)
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-16Which of the following variable(s) were eliminated from the full model by the step function? Select all that apply.
CH4
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, calculate temperature predictions for the testing data set, using the predict function.
Enter the testing set R2: 0.6286
climate_Test <- subset(climate_change , Year >2006)
str(climate_Test)
'data.frame': 24 obs. of 11 variables:
$ Year : int 2007 2007 2007 2007 2007 2007 2007 2007 2007 2007 ...
$ Month : int 1 2 3 4 5 6 7 8 9 10 ...
$ MEI : num 0.974 0.51 0.074 -0.049 0.183 ...
$ CO2 : num 383 384 385 386 387 ...
$ CH4 : num 1800 1803 1803 1802 1796 ...
$ N2O : num 321 321 321 321 320 ...
$ CFC.11 : num 248 248 248 248 247 ...
$ CFC.12 : num 539 539 539 539 538 ...
$ TSI : num 1366 1366 1366 1366 1366 ...
$ Aerosols: num 0.0054 0.0051 0.0045 0.0045 0.0041 0.004 0.004 0.0041 0.0042 0.0041 ...
$ Temp : num 0.601 0.498 0.435 0.466 0.372 0.382 0.394 0.358 0.402 0.362 ...PredictTest <- predict(StepModel,newdata =climate_Test)
SSE <- sum((climate_Test$Temp - PredictTest)^2)
SSE[1] 0.2176444SST <- sum((climate_Test$Temp - mean(climate_trainng$Temp))^2)
SST[1] 0.5860189R2 <- 1-SSE/SST
R2[1] 0.6286051