To emphasize that the nature of the functions g i really is irrelevant, consider the following example. The resulting best-fit function minimizes the sum of the squares of the vertical distances from the graph of y = f ( x ) to our original data points. , B m-once we evaluate the g i, they just become numbers, so it does not matter what they are-and we find the least-squares solution. We evaluate the above equation on the given data points to obtain a system of linear equations in the unknowns B 1, B 2. Indeed, in the best-fit line example we had g 1 ( x )= x and g 2 ( x )= 1 in the best-fit parabola example we had g 1 ( x )= x 2, g 2 ( x )= x, and g 3 ( x )= 1 and in the best-fit linear function example we had g 1 ( x 1, x 2 )= x 1, g 2 ( x 1, x 2 )= x 2, and g 3 ( x 1, x 2 )= 1 (in this example we take x to be a vector with two entries). ![]() That best approximates these points, where g 1, g 2. This is denoted b Col ( A ), following this notation in Section 6.3. Hence, the closest vector of the form Ax to b is the orthogonal projection of b onto Col ( A ). This is used to predict the unknown value of variable Y when value of variable X is known. The line of regression of Y on X is given by Y a + bX where a and b are unknown constants known as intercept and slope of the equation. In other words, Col ( A ) is the set of all vectors of the form Ax. There are two lines of regression- that of Y on X and X on Y. Recall from this note in Section 2.3 that the column space of A is the set of all other vectors c such that Ax = c is consistent. Suppose that the equation Ax = b is inconsistent. ![]() In other words, a least-squares solution solves the equation Ax = b as closely as possible, in the sense that the sum of the squares of the difference b − Ax is minimized. So a least-squares solution minimizes the sum of the squares of the differences between the entries of A K x and b. The term “least squares” comes from the fact that dist ( b, Ax )= A b − A K x A is the square root of the sum of the squares of the entries of the vector b − A K x. Recall that dist ( v, w )= A v − w A is the distance between the vectors v and w. Hints and Solutions to Selected Exercises. ![]() 3 Linear Transformations and Matrix Algebra
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |