Curve Fitting and Regression

Points: $(1,15), (2,22), (3,33), (4,48), (5,70)$, with $n = 5$. Taking natural log: $$\ln y = \ln a + bx$$ Let $Y = \ln y$, $A = \ln a$. Then $Y = A + bx$ (linear). Normal equations: $$\sum Y = nA + b\sum x$$ $$\sum xY = A\sum x + b\sum x^2$$ $x$ $y$ $Y = \...

Relation: $T = be^{t/4} + a$ t (min) 1 2 3 4 --------------- T (°C) 70 83 100 124 Estimate: T at t = 6. Substitute $u = e^{t/4}$, giving the linear form $T = a + bu$. Compute u values: t T $u = e^{t/4}$ --------- 1 70 $e^{0.25} = 1.28403$ 2 83 $e^{0.50} = 1...

X 1 2 3 4 5 ------------------ f(X) 2 6 12 20 30 Model: $f(x) = a0 + a1 x + a2 x^2$, with $n = 5$. x f x² x³ x⁴ xf x²f --------------------------- 1 2 1 1 1 2 2 2 6 4 8 16 12 24 3 12 9 27 81 36 108 4 20 16 64 256 80 320 5 30 25 125 625 150 750 Σ 70 55 225 9...

$x$ 1 3 4 5 6 -------------------- $y$ 2 7 8 7 5 $n = 5$. Fit $y = a0 + a1 x + a2 x^2$, estimate $f(2)$. $x$ $y$ $x^2$ $x^3$ $x^4$ $xy$ $x^2y$ --------------------------------------------- 1 2 1 1 1 2 2 3 7 9 27 81 21 63 4 8 16 64 256 32 128 5 7 25 125 625 ...

The least squares method finds the best-fit curve for a set of data points by minimizing the sum of squares of residuals (differences between observed values $yi$ and fitted values $f(xi)$): $$S = \sum{i=1}^{n} [yi - f(xi)]^2 \to \text{minimum}$$ $$x: 1, 2,...