Question

The method of least squares is used to find the line or curve that is closest to the data points in a mean squared error sense Suppose you have a degree four polynomial function representing an alien parasite population growth at time t. f(t)=a₁ t⁴ +a₂ t³ +a₃ t²+a₄ t+a₀

a. Data is collected at 6 different times to measure the parasite growth. Write a matrix formulation of the above equation. Your formulation should be in the form y=Ax where y and x are vectors and A is a matrix.

Answer 1

To formulate the degree four polynomial function in matrix form, use y = Ax. The vector y contains the observed populations, A is a matrix with time-powered terms, and x is the coefficients vector. The least squares solution minimizes the SSE, finding the best-fit curve to the data.

Step-by-step explanation:

To express the degree four polynomial function f(t) = a₁ t⁴ + a₂ t³ + a₃ t² + a₄ t + a₀ that represents an alien parasite population growth at different times t, in a matrix formulation for 6 different times, you can use the following form:

y = Ax, where:

• y is the vector of observed parasite populations at the 6 different times.
• A is the matrix with each row representing a time t, raised to the power of 4, 3, 2, 1, and including a column of 1s for the constant term.
• x is the vector of coefficients [a₁, a₂, a₃, a₄, a₀].

To find the least squares solution, where A is known and y is observed, you calculate the vector x that minimizes the sum of squared errors (SSE) between the observed populations and those predicted by the polynomial.