1.6 Correlation maps CHAPTER 1. DATA ANALYSIS 47

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1 CHAPTER 1. DATA ANALYSIS Correlation maps Correlation analysis can be a very powerful tool to establish a statistical relationship between the two variables. Section 1.4 showed that a correlation coe cient between apairoftwovariablesx and y is defined as the covariance divided by the standard deviation of x and y (Eq. 1.24). Then section 1.5 further developed the method to evaluate the statistical significance of the correlation coe cient. Consider the example of surface air temperature and the index for El-Nino. The surface air temperature has the spatial and temporal variation, T = T (t, x, y), where x is longitude, y is latitude and t is time. Then it is possible to construct a x y map of correlation coe cient by performing the calculation of the correlation coe cient many times, for each position in (x, y) space. Repetitive calculations can be done by the loop statement in MATLAB. For example, you can define a new function correlate that calculates the correlation coe cient and its statistical significance, where m(t) istheindexofel- Nino and T (t, x, y) isthesurfaceairtemperature. >> for i=1:nx >> for j=1:ny >> [r,s] = correlate(m,t(:,i,j)); >> rxy(i, j)=r ; >> sxy(i, j)=s ;

2 CHAPTER 1. DATA ANALYSIS 48 >> end >> end Here, we have to be careful that both the indices of El-Nino and temperature must be anomalies from the climatology. It usually means that the mean seasonal cycle is subtracted from the original data. El-Nino events have strong interannual variability (2 7 year timescale) and this step allows us to focus on the year-to-year variability. The raw data often includes strong signature of seasonality, and it should be removed before calculating the correlation. Fig 1.11 shows the resulting pattern of rxy. Exercises We will generate correlation maps between surface air temperature and the index of North Pacific Gyre Oscillation 1. Develop a MATLAB function that calculates the correlation coe cient between two variables. 2. Download the NPGO index from Prof. Di Lorenzo s website ( and load it on MATLAB. 3. Remove the mean seasonal cycle from the data and calculate the e ective sample size of the air temperature data for each grid cell for the 66 year period ( ).

3 CHAPTER 1. DATA ANALYSIS o N o N o 60 o E 120 o E 180 o W 120 o W 60o W 0 o o S o S Figure 1.11: Correlation map. Surface air temperature anomalies are correlated with the Nino 3.4 index. The data period is from 1950 to Calculate the e ective sample size of the NPGO index for the 66 year period ( ). 5. Calculate correlation coe cient for each grid cell. 6. Generate a map similar to Fig 1.11 for the NPGO. 7. Publish the MATLAB script that performs all of the activity above, and submit it as a report in the PDF format. Reference

4 CHAPTER 1. DATA ANALYSIS 50 Di Lorenzo et al., 2008, North Pacific Gyre Oscillation links ocean climate and ecosystem change, Geophysical Research Letters.

5 CHAPTER 1. DATA ANALYSIS Multiple Linear Regression Figure 1.12 shows the recent history of atmospheric CO 2 observed at the Mauna Loa observatory in Hawaii. This observatory is the world s oldest continuous CO 2 monitoring station, providing a benchmark for measurement of the greenhouse gases. The concentration of CO 2 in the atmosphere has been measured here since 1950s, showing prominent increase due to the anthropogenic emission of fossil fuel CO 2,knownasthe KeelingCurve. In addition to the long-term increase, there is a stron annual cycle associated with the seasonal photosynthesis and respiration on land. Atmospheric CO 2 tends to increase during cool seasons and decrease during warm seasons, resulting in the maximum in the spring and minimum in the fall. Let s consider a simple statistical model for the evolution of atmospheric CO 2 including a linear trend and a sinusoidal seasonal cycle. y = a 0 + a 1 t + a 2 sin(2 t)+a 3 cos(2 t) (1.36) y consists of atmospheric CO 2, as a function of time (t). The periodicity is set to 1 year as it represents qthe seasonal cycle, and its amplitude is determined by a a 2 3. Section 1.4 showed that the linear regression can be expressed in the vector-matrix problem, y = Ex, where y vector contains the observable and E is a N 2matrix

6 CHAPTER 1. DATA ANALYSIS Mauna Loa CO 2, ppmv time Figure 1.12: Atmospheric CO 2 level observed at the Mauna Loa observatory in Hawaii from 1991 to The data is provided by the Global Monitoring Division of NOAA and the Scripps CO 2 program ( ucsd.edu/data/atmospheric_co2/mlo). The data in MATLAB format is downloadable from courses/eas2655/mlo_co2.mat.

7 CHAPTER 1. DATA ANALYSIS 53 consisting of two column vectors with ones and another variable (e.g. time) used to explain the variations in y. Then we calculated a particular choice of x that gives us the best linear fit to the observation, containing the slope and intercept of the regression line. In this framework, it is possible to have more than two columns in E. 0 y 1 y 2. y N 1 C = A y = 0 0 a 0 + a 1 t 1 + a 2 sin(2 t 1 )+a 3 cos(2 t 1 ) a 0 + a 1 t 2 + a 2 sin(2 t 2 )+a 3 cos(2 t 2 ). a 0 + a 1 t N + a 2 sin(2 t N )+a 3 cos(2 t N ) 1 t 1 sin(2 t 1 ) cos(2 t 1 ) 1 t 2 sin(2 t 2 ) cos(2 t 2 ). 1 t N sin(2 t N ) cos(2 t N ) {z } E 1 0 C B a 0 a 1 a 2 a 3 1 C A {z } x y = Ex (1.37) Here the objective is to find x that provides the best fit to the observation for y. This is exactly the same mathematical problem as Eq So, the pseudo inverse of E can be used to find the regression coe cient, x. x = E T E 1 E T y obs (1.38) This is called the multiple linear regression (MLR). When this calculation is applied to the Mauna Loa CO 2 data, we find the regression coe cients of x(1) = 1 C A

8 CHAPTER 1. DATA ANALYSIS [ppm], x(2) = 1.97[ppm/yr], x(3) = 3.02[ppm] and x(4) = 0.80[ppm]. The resulting multiple regression line is shown in Fig It also compares the raw data, simple linear regression and the multiple linear regression. r 2 value is for simple linear regression, and it increases to for the multiple linear regression. In this particular example, MLR can explain more than 99% of the variance. Exercises We will perform MLR fit to observed greenhouse gas concentrations. 1. Download the monthly CO 2 timeseries from 1958 to present from the Scripps CO 2 program website ( co2/primary_mlo_co2_record). 2. Plot the raw data of pco 2 as a function of time (t). 3. Perform simple linear regression including t only. Examine the regression coe cients and r 2 value. 4. Perform multiple linear regression including t and t 2. Note that t 2 is required for fitting the accelerating rate of CO 2 increase. Examine the regression coe cients and r 2 value. 5. Perform multiple linear regression including t, t 2, sin and cosine of t. Examine the regression coe - cients and r 2 value.

9 CHAPTER 1. DATA ANALYSIS Observation MLR fit Simple Regr atmospheric pco2, ppm time Figure 1.13: Atmospheric CO 2 level observed at the Mauna Loa observatory in Hawaii from 1991 to 2016 (black), simple linear regression (blue) and multiple linear regression (red). 6. Plot everything together similar to Fig Publish your MATLAB script that performs above tasks.