% аппроксимируем зависимости
% полиномиальная регрессия дает лучший результат
y =  polynomial_reg(tt(2:length(tt))', xx(2:length(tt))', 35);
k = build_plot_disp_loss(tt(2:end)', xx(2:end)', y);
% находим коэффициенты в заданных функциях
x = tt(2:4);
y = xx(2:4);
A = [(exp(-5*x).*cos(5*x)) (exp(-5*x).*sin(5*x)) ones(3, 1)];
disp(A);
res = A \ y;
disp(res);
c1 = res(1)
c2 = res(2)
c3 = res(3)
c4 = yy(3) - c2.*exp(-5*tt(3)).*cos(5*tt(3)) + exp(-5*tt(3)).*c1.*sin(5*tt(3));
values1 = exp(-5*tt').*cos(5*tt').*c1 + exp(-5*tt').*sin(5*tt').*c2 + c3;
% вычисляем значение функции х2 в заданных точках, и строим по ним график
values2 = -exp(-5*tt').*sin(5*tt').*c1 + exp(-5*tt').*cos(5*tt').*c2 + c4;
figure(2)
plot(tt', values2, 'r');
hold on;
% отмечаем на графике значения, данные заранее
plot(tt', yy', 'bo');
% зависимость х2(х1)
val = -exp(-5*values1').*cos(5*values1').*res(1) + exp(-5*values1').*sin(5*values1').*res(2) + res(3);
figure(3)
plot(values1, val, 'r');

% линейная регрессия
function y=lin_reg(V, bf)
    a = (sum(V) * sum(bf) - length(V) * sum(V .* bf))/(sum(V)^2 - length(V)*sum(V.^2));
    b = (sum(V .* sum(bf.*V)) - sum(V.^2 .* sum(bf)))/(sum(V)^2 - length(V)*sum(V.^2));
    y = a*V + b;
end
% квадратичная регрессия
function y = square_reg(V, bf)
    A = [sum(V.^2) sum(V) length(V);
         sum(V.^3) sum(V.^2) sum(V);
         sum(V.^4) sum(V.^3) sum(V.^2)];
    B = [sum(bf) sum(V .* bf) sum(V.^2 .* bf)]';
    x = A\B;
    a = x(1);
    b = x(2);
    c = x(3);
    y = a * V.^2 + b * V + c;
end
% степенная регрессия
function y = degree_reg(V, bf)
    a = [length(V) sum(log(V));
         sum(log(V)) sum(log(V).^2)];
    b = [sum(log(bf)) sum(log(bf).*log(V))]';
    x = a\b;
    a = exp(x(1));
    b = x(2);
    y = a * V.^b;
end

function y = loss_fn(bf, res)
    y = 1/length(bf) * sum(abs((bf - res)./(bf)));
end

function loss = build_plot_disp_loss(V, bf, res)
    loss = loss_fn(bf, res);
    disp("loss is: " + loss * 100 + "%");
    figure(1);
    plot(V, bf, 'r');
    hold on;
    plot(V, res, 'b');
end
% полиномиальная регрессия
function y = polynomial_reg(V, bf, degree)
    A = [];
    list_of_degrees = [0:degree];
    for k=0:degree
        new_column = V' .^ list_of_degrees;
        A = [sum(new_column, 1)' A];
        list_of_degrees = list_of_degrees + 1;
    end
    A(1, end) = length(V);
    list_of_degrees = [0:degree];
    B = sum((V' .^ list_of_degrees).* bf', 1)';
    list_of_degrees = [degree:-1:0];
    coeffs = A\B;
    new_column = V' .^ list_of_degrees;
    y = sum(new_column .* coeffs', 2)';
end