Exercise 16.1.3#
Question 1#
Implement the linear least squares solution for three variables to find the constants in the Cepheid variable functional relation:
\[
M = a_0 + a_1 \log P + a_2 (B - V)
\]
using the same data file as before: ./data/cepheid_data.csv
.
(Answer: a_0 = -2.15 mag, a1 = -3.12 mag, a2 = 1.49)
Question 2#
Package the multivariate least squares solution into a function, taking the file path (and possibly the delimeter used as well as number of lines to skip) and return the \(\boldsymbol{A}\) matrix. Unlike in the case with the Cepheid variable data, you can assume that the first column of the file will contain \(y\) data and the rest \(x_j\) data.
Test your function on the data sets ./exercise-data/data02.csv
and ./exercise-data/data03.csv
.
Answers:
\[\begin{eqnarray*}
\text{data02.csv} & \quad \quad \quad & \text{data03.csv}\\
\boldsymbol{A} =
\begin{pmatrix}
-0.402\\
0.201\\
1.222\\
0.341\\
0.134\\
\end{pmatrix}
& \quad \quad \quad &
\boldsymbol{A} =
\begin{pmatrix}
2.81\\
0.55\\
1.63\\
2.71\\
1.33\\
-1.16\\
2.39\\
1.12\\
0.65\\
4.05\\
-2.81\\
\end{pmatrix}\\
\end{eqnarray*}\]