回归与内插

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Simple Linear Regression

  • A bunch of data points (xi, yi) are collected
  • Assume x and y are linearly correlated

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Linear Regression Formulation

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想要求解得β0和β1使SSE最小，就要做微分

Solving Least-squares Problem

  • SSE is minimized when its gradient with respect to each parameter is equal to zero:

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Least-squares Solution

  • Suppose there exists N data points:

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Polynomial Curve Fitting: polyfit()

  • Curve fitting for polynomials of different orders
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x = [-1.2 -0.5 0.3 0.9 1.8 2.6 3.0 3.5];
y = [-15.6 -8.5 2.2 4.5 6.6 8.2 8.9 10.0];
fit = polyfit(x,y,1);

1 代表一次方得polynomial， 即f(x) = ax + b
上例中 fit(1) -> a, fit(2) -> b 

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xfit = [x(1):0.1:x(end)];
yfit = fit(1)*xfit + fit(2);
plot(x, y, 'ro', xfit, yfit);
set(gca, 'FontSize', 14);
legend('data points', 'best-fit');

Are x and y Linearly Correlated?

  • if not, the line may not well describe their relationship
  • Check the linearity by using:
  • scatter(): scatterplot 画散点图
  • corrcoef(): correlaition coefficient, -1<= r <= 1 相关系数 很强正相关 趋近1 ，很强负相关 趋近-1
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x = [-1.2 -0.5 0.3 0.9 1.8 2.6 3.0 3.5];
y = [-15.6 -8.5 2.2 4.5 6.6 8.2 8.9 10.0];
scatter(x, y); box on; axis square;
corrcoef(x, y)

corrcoef(x, y) 得到一个矩阵，对应x跟x、x跟y、y跟x、y跟y的相关系数

Higher Order Polynomials

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x = [-1.2 -0.5 0.3 0.9 1.8 2.6 3.0 3.5];
y = [-15.6 -8.5 2.2 4.5 6.6 8.2 8.9 10.0];
figure('Position', [50 50 1500 400]);
for i=1:3
    subplot(1,3,i); p = polyfit(x, y, i);
    xfit = x(1):0.1:x(end); yfit = polyval(p, xfit);
    poly(x, y, 'ro', xfit, yfit); set(gca, 'FontSize', 14);
    ylim([-17, 11]); legend('Location', 'SouthEast','Data points', 'Fitted curve');
end

is it better to use higher order polynomials?

no ,过拟合

What if There Exists More Variables?

  • Equations associated with more than one explanatory variables:

y = β0 + β1x1 + β2x2

  • Multiple linear regression: regress()
  • Note: the function gives you more statistics of the regression model

Multiple Linear Regression: regress()

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What if the Equations Are Not Linear?

  • How do we do curve fitting using nonlinear equations?

使用cftool工具

Interpolation内插 vs Regression回归

  • Interpolation:
    • The process of finding an approximation of a function
    • The fit does traverse all known points

  • Regression:

    • The process of finding a curve of best fit
    • The fit generally does not pass through the data points

Linear Interpolation: interpl()

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x = linspace(0, 2*pi, 40); x_m = x;
x_m([11:13, 28:30]) = NaN; y_m = sin(x_m);
plot(x_m, y_m, 'ro', 'MarkerFaceColor', 'r');
xlim([0, 2*pi]); ylim([-1.2, 1.2]); box on;
set(gca, 'FontName', 'symbol', 'FontSize', 16);
set(gca, 'XTick', 0:pi/2:2*pi);
set(gca, 'XTickLabel', {'0','p/2','p','3p/2','2p'});

m_i = ~isnan(x_m);
y_i = interpl(x_m(m_i), y_m(m_i), x);
hold on;
plot(x, y_i, '-b', 'LineWidth', 2);
hold off;

Spline Interpolation: spline()

更圆滑的内插

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m_i = ~isnan(x_m);
y_i = spline(x_m(m_i), y_m(m_i), x);
hold on; plot(x, y_i, '-g', 'LineWidth', 2); hold off;
h = legend('Original', 'Linear', 'Spline');
set(h, 'FontName', 'Times New Roman');

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