Python Lab 7: Projection and least squares

Open 07_Least_Squares.ipynb and enter a=$a and k=$k. Fit a line y\approx c_0+c_1t to the data (0,$a),\qquad(1,$a+$k),\qquad(2,$a+2*$k+1).

1. Enter the design matrix for this model by choosing the correct NumPy construction.

   NumPy construction of the line fitting design matrix np.column_stack((np.ones_like(t), t)) np.array([1, t]) np.diag(t) np.outer(t, t) $designChoice.selectedIndex=1

2. Use np.linalg.lstsq to compute \widehat c=\begin{bmatrix}c_0\\c_1\end{bmatrix}: c_0= least squares intercept , c_1= least squares slope .

   $c0Input=$cZero and $c1Input=$cOne

3. Enter the residual r=y-X\widehat c. First component: first residual component ; second component: second residual component ; third component: third residual component .

   $res1=1/6 and $res2=-1/3 and $res3=1/6

4. Compute the squared error \lVert r\rVert^2= squared norm of the least squares residual .

   $squaredError=1/6

5. What does the notebook's calculation X^Tr\approx0 mean geometrically?

   geometric meaning of the least squares residual The residual lies in the column space of X. The residual is orthogonal to the column space of X. The data vector is orthogonal to every fitted vector. The design matrix is singular. $geometryChoice.selectedIndex=2