Combat readiness mathematical modeling 31 data interpolation and curve fitting 3

Catalog

One 、 Classic cases of data interpolation

Two 、 Curve fitting classic case

1- Polynomial fitting

2- Nonlinear fitting

---

One 、 Classic cases of data interpolation

The following is pool 1 1,3,5...15 Various indicators of the week , Interpolate the indicators in the pool , And all the images drawn after interpolation are displayed on one graph .

More data , It is troublesome to select each row for interpolation , Circular interpolation possible , then subplot Function to draw multiple graphs , Of course , For one-dimensional interpolation , We prefer cubic spline interpolation .

Concrete MATLAB The code is as follows ：

clear; clc
load('Z.mat')
x = Z(1,:) ; % Get a row vector , Weeks 
[n,m] = size(Z) ; % Get the number of rows and columns of the original data 
% Each picture y The label of the shaft 
ylab={' Weeks ',' rotifer ',' Dissolved oxygen ','COD',' The water temperature ','PH value ',' Salinity ',' transparency ',' Total alkalinity ',' Chloride ion ',' transparency ',' Biomass '};
P = zeros(11,15) ;

for i = 2 : n % From 2 Row start interpolation 
    y = Z(i,:) ;
    new_x = 1:15 ;
    p1 = interp1(x,y,new_x,'spline') ; % Cubic spline interpolation 
    subplot(4,3,i-1) ;
    plot(x,y,'ro',new_x,p1,'-');
    axis([0 15,-inf,inf]) ;
    xlabel(' Weeks ') ;
    ylabel(ylab(i)) ;
    P(i-1,:) = p1 ;
end
legend(' Raw data ',' Cubic spline interpolation data ','Location','SouthEast'); % legend 
disp(P) ;

    

The drawing is as follows ：

Two 、 Curve fitting classic case

1- Polynomial fitting

Let's look at a simple example , The following data are fitted by polynomial , Of course, it can be directly fitted and solved , That is to solve the linear regression equation of one variable , It can also be solved by the least square method , In addition to drawing the fitting curve , The goodness of fit is also calculated .

MATLAB The code is as follows ：
Method 1, Direct polynomial fitting

clear; clc
load('nihe.mat');
%plot(x,y,'o') ;
xlabel('x Value ') ;
ylabel('y Value ') ;
n = size(x,1) ; % Number of sample points 
x = x' ;
y = y' ;
xp = 2.7 : 0.1 : 6.9 ;
p = polyfit(x,y,1) ;
yp = polyval(p,xp);
plot(x,y,'o',xp,yp) ;
legend(' Sample data ',' Fit function ','location','SouthEast') ;

Method 2, Least squares parameter estimation , Then draw according to the expression , The goodness of fit is calculated here .

clear; clc
load('nihe.mat');
plot(x,y,'o') ;
xlabel('x Value ') ;
ylabel('y Value ') ;
n = size(x,1) ; % Number of sample points 
% Least square method for parameter estimation 
k = (n*sum(x.*y)-sum(x)*sum(y))/(n*sum(x.*x)-sum(x)*sum(x)) ;
b = (sum(x.*x)*sum(y)-sum(x)*sum(x.*y))/(n*sum(x.*x)-sum(x)*sum(x)) ;
hold on ;
grid on ;
f = @(x) k*x + b ; % Anonymous functions 
fplot(f,[1,7.5]) ;
legend(' Sample data ',' Fit function ','location','SouthEast') ;
y_hat = k * x + b ;
SSR = sum((y_hat-mean(y)).^2) ; % Sum of regression squares 
SSE = sum((y_hat-y).^2) ; % The sum of the squares of the errors 
SST = sum((y-mean(y)).^2) ; % Total sum of squares 
disp(' Goodness of fit values are as follows ：') ;
R_2= SSR / SST ;
disp(R_2) ;

The explanation of goodness of fit is as follows ： The goodness of fit is closer to 1, The smaller the sum of squared errors , The better the fitting effect is . 

The drawing is as follows ：

2- Nonlinear fitting

Let's take a look at the envious population forecast , The nonlinear fitting function can be used directly , You can also use the fitting toolbox cftool

MATLAB The code is as follows , Here, nonlinear fitting is performed directly , Anonymous functions are required .

clear; clc
x = 1790:10:2000;
y = [3.9,5.3,7.2,9.6,12.9,17.1,23.2,31.4,38.6,50.2,62.9,76.0,92.0,106.5,123.2,131.7,150.7,179.3,204.0,226.5,251.4,281.4];
plot(x,y,'o')
y1 = @(b,t) b(1) ./ (1 + (b(1) / 3.9 - 1)*exp(-b(2)*(t-1790))); % Anonymous functions 
b0 = [100 0.1] ; % The selection of initial value is very important 
a = nlinfit(x,y,y1,b0)  ; % Coefficient after nonlinear fitting 
xp = 1790 : 10 : 2030 ;
yp = y1(a,xp) ; % Substitute coefficients and values into the expression for evaluation 
hold on ;
plot(xp, yp, '-') ;
legend(' Original scatter ',' Fit the curve ') ;

The drawing is as follows ：