i got a missing information issue but this is all!!

11.8 Atmospheric 002 Nonlinear Data Fitting It may come as a surprise that we can use linear programming to solve many nonlinear data tting problems. The key aspect is that the model of the data is expressible as a linear combination of nonparametrized but otherwise arbitrary functions. Con sider the monthly mean atmospheric carbon dioxide (002) measurement data taken at -na Loa in Hawaii Data from the last several years is shown in Figure m 415 4':- h o _a 01 O Atmospheric 002 Concentration (ppm) 4:. O O 2010 2012 2014 2016 2018 Calendar Year Figure 11.2: Monthly average 002 concentration as measured in the atmosphere on Mauna Loa (red) and a veparameter least deviation t to the data (blue). The data suggests two major timedependent features. First, there is a general upward trend that appears roughly linear. Second, there is a cyclic annual variation. A possible model that captures these features is y = a0 + (1133 + (12332 + a3 sin (27m) + (.14 cos (27m) . Here, m = T 2010 is the number of years since the beginning of 2010, where T is the calendar year. 3; is the 002 concentration in parts per million. The coeicients a0, a1, a2, a3, G4 are the undetermined coeicients of the model our decision variables. This model allows for a quadratic trend and annual cyclic variation of variable phase shift. We want to choose coefcients a. so that yin = a0 + 0.13:1\" + (12$: + a3 sin (27min) + (.14 cos (271335;) for all data points (Xk, yk). Using the method of least deviation, we have the opti- mization problem N min z = > do + allck + a2xx + a3 sin (27xk) + a4 cos (2Tack) - yk K = 1 and the equivalent linear formulation N min Z = a,d C 8k K = 1 s.t. ok > tao + allk + a2xx + a3 sin (27Xk) + 04 cos (27Xk) - yk k = 1, 2, ..., N ok 2 -do - allk - a2Xk - a3 sin (27Xk) - a4 Cos (2Tack) + yk k = 1,2, ..., N. Upon some rearrangement of terms, we have N min 2 = a,d K = 1 s.t. do - allk - a2xx - a3 sin (27XK) - a4 COS (27XK) + 6k 2 -yk K = 1, 2, ..., N tao + allk + a2xx + a3 sin (27XK) + a4 cos (2TXCK) + k 2 tyk k = 1,2, ..., N. which we can write in matrix form min z = c'w w s.t. Aw > B w 2 0 W E RN+5 w. = ao al a2 a3 a4 61 62 ... ON = 0 0 0 0 0 1 1 ... A = -IN -x -X -S -C IN +IN +x +X +S +C IN B = -y + 2where IN is the column vector of NV ones, IN is the N X N identity matrix, X is the column vector of squared x values, S and C are the column vectors of values sin (27x) and cos(27x), respectively. The following Octave code solves the CO2 data fitting problem (assuming vectors x and y are defined. N=length (x) ; c= [zeros (5, 1) ; ones (N, 1) ] ; w=ones (N, 1) ; X=x. 2; S=sin (2*pi*x) ; C=cos (2*pi*x) ; A=[-w -x -X -S -C eye(N) ; w x X S C eye(N) ]; B= [-y ; y] ; 1b=-inf (N+5, 1) ; ub= ; ct=repmat ( "L", 1, 2*N) ; vt=repmat ("C" , 1, N+5) ; sense=1 ; [xstar, zstar]=glpk(c, A, B, 1b, ub, ct , vt, sense) ; The optimal model parameters are a* = [388.6 2.114 0.0314 2.976 - 0.840 ] . We can interpret these values as follows. . The seasonally adjusted CO2 level at the beginning of year 2010 was 389 ppm. . The annual increase in CO2 level is 2.11 ppm at the beginning of year 2010. . The annual increase in CO2 level is 2.36 ppm at year 2018 (a1 + 2a2(8) = 2.36). . The phase of annual variation is close to zero, most of the amplitude being in the sine term. . The seasonal variation in CO2 concentration has amplitude 3.09 ppm (3.09 = V (2.976)2 + (0.840)2).8. Based on the veparameter model t to the Mauna Loa (302 data, suggest a different model with no more than seven parameters that may more accurately represent the data set. Carefully justify your model. You do not need to solve this new