Optimum Fitting of Richards Growth Model in Random Environment

Abstract

Richards four-parameter nonlinear growth model is a very versatile model for describing many growth processes. Two limitations of the corresponding Richards nonlinear statistical model are discussed. Accordingly, in this article, the general approach of ‘Stochastic differential equations’ is considered. Ghosh and Prajneshu (J Indian Soc Agric Stat 71:127–138, 2017) developed the methodology for fitting Richards growth model in random environment, when one of the parameters, viz. m takes a particular value. Purpose of the present article is to extend this type of work by proposing the methodology, which is valid for all values of m. Relevant computer programs for its application are written and the same are included as an “Appendix”. Finally, as an illustration, pig growth data are considered and superiority of our proposed model is shown over the Richards nonlinear statistical model for given data.

Appendix

• (1) SAS Code for Estimation of Parameters

• %macropigdata;

• proc optmodel;

• ods output PrintTable=parms_&kk.;

• set l={1..n1}; /*n1= total number of ages in data*/

• set j=2..n2; /*value for last age */

• number y{l};

• number yt{l};

• read data abc_&kk. into [_n_] y;

• read data abcd_&kk. into [_n_] yt;

• number n init n1; /* In the following, z1 = r, z2 = α = K^−m, z3 = m and z4 = σ */

• var z1>=0<=1;

• var z2>=0<=MAXIMUM_CARRYING_CAPACITY;

• var z3>=0<=1;

• var z4>=0<=1;

• max f=sum{i in j}((abs(z3)*(y[i])**((z3+1)/z3))*(1+((y[i]-(z2+(y[i-1]-z2)*exp(-z1)))/(exp(-z1*i)*(1+((y[i-1]-(z2+(y[1]-z2)*exp(-z1*(i-1))))/exp(-z1*(i-1)))))))**-1*((exp(z1*i)*( (1+((y[i-1]-(z2+(y[1]-z2)*exp(-z1*(i-1))))/exp(-z1*(i-1))))))**-1)*(1/sqrt(6.28*z4*z4)*exp(-0.5*z4*z4*(log(1+((y[i]-(z2+(y[i-1]-z2)*exp(-z1)))/(1+((y[i-1]-(z2+(y[1]-z2)*exp(-z1*(i-1))))/exp(-z1*(i-1)))))))**2)));

• solve;

• print f z1 z2 z3 z4;run;quit;

• %mend;

• %macro iml;

• %let count=1;

• %let pp=1;

• %let qq=SIZE_OF_DATA; /* number of columns in data */

• %loop:

• proc iml;

• x={DATA_FILE}; /*a matrix where each column corresponds to data of a particular pig*/

• %do j=-10 %to 10 %by 1;

• yy=x[,1:SIZE_OF_DATA];

• vv=&j/10;

• ty=yy##-vv;

• kt=kt||ty;%end;

• aa=kt[,&pp:&qq];

• %let pp=%eval(&pp+SIZE_OF_DATA);

• %let qq=%eval(&qq+SIZE_OF_DATA);

• %do i=1 %to SIZE_OF_DATA;

• y1=aa[,&i];

• y_&i.=(y1);

• varnames={y};

• create abc_&i. from y_&i.[colname=varnames];

• append from y_&i.;

• close abc_&i.;%end;

• %do i=1 %to SIZE_OF_DATA;

• ym=yy[,&i];

• yt_&i.=(ym);

• varnames={yt};

• create abcd_&i. from yt_&i.[colname=varnames];

• append from yt_&i.;

• close abcd_&i.;%end;

• create x1 from x;

• append from x;

• close x1;

• %let count=%eval(&count+1);

• %do kk=1 %to SIZE_OF_DATA;

• %pigdata;%end;

• proc iml;

• %do kk=1 %to SIZE_OF_DATA;

• use abc_&kk.;

• read all into y_&kk.;

• use parms_&kk.;

• read all into z_&kk.;

• use x1;

• read all into x;

• zz=z_&kk.;

• zz1=zz1//zz;

• %end;print zz1;

• %if &count<=NUMBER_OF_ITERATIONS; %then %do;

• %goto loop;

• %end;quit;

• %mend;%iml;

• (2) Program for Fitting and Forecasting Purposes

• proc iml; /*a matrix where each column corresponds to estimates of parameter of a particular pig */

• z={ESTIMATES_OF_PARAMETERS};

• y={DATA_FILE}; /*a matrix where each column corresponds to data of a particular pig*/

• do j=1 to SIZE_OF_DATA; /* number of columns in data */

• yyy=(y[,j])##(-1*z[3,j]);

• sum1=0;fit=0;

• do i=2 to LAST_AGE; /* last age to perform fitting */

• fittedw=(z[2,j]+(yyy[i-1]-z[2,j])*exp(-z[1,j])+exp(-z[1,j]*i)*(1+((yyy[i-1]+(z[2,j]+(yyy[1]-z[2,j])*exp(-z[1,j]*(i-1))))/exp(-z[1,j]*(i-1))))*(exp(0.5*z[4,j]*z[4,j])-1))**(-1/z[3,j]);

• /* fitting performed with time span of 1 and forecasting with time span of 4*/

• fit=fit//fittedw;mm=y[i,j];

• diff=(mm-fittedw)##2;

• sum1=sum1+diff;end;

• av=sum1/(NUMBER_OF_AGES);

• av1=av1||av;

• fitting=fitting||fit;

• end;print fitting;

• av2=av1##0.5;

• print av2;quit;