Given two data points in 2 dimensions:
\displaystyle \displaystyle \mathbf{x}^{(1)} \displaystyle = \displaystyle (x^{(1)}, y^{(1)})= (0,1)
\displaystyle \mathbf{x}^{(2)} \displaystyle = \displaystyle (x^{(2)}, y^{(2)})=(0,-1)
Notice that this sample is already centered, with sample mean 0 in both x- and y- coordinates.
Without computation, find the direction of largest variance. (This is the first principal component \text {PC1}.
(1,0)
(0,1)
(1,1)
We can determine the direction of largest variance without computation by visually inspecting the data points.
In this case, the two data points are (0,1) and (0,-1), which lie on a vertical line. Since the points are symmetric about the x-axis, the direction of largest variance is also vertical, along the y-axis.
Therefore, the direction of largest variance (PC1) is (0,1), which is the second option.
To determine the direction of the largest variance without computation, we can visually analyze the given data points.
From the provided data points, we observe that both points lie on the x-axis. This implies that the variance in the y-coordinate is zero.
Therefore, the direction of largest variance is along the x-axis, which corresponds to the vector (1,0).
To find the direction of the largest variance without computation, we can visually analyze the given data points in the 2-dimensional space.
The direction of largest variance corresponds to the direction along which the data points are spread out the most. In other words, it represents the direction in which the data points exhibit the greatest variability.
Looking at the given data points: (0, 1) and (0, -1), we can see that these points lie on a vertical line along the y-axis.
Since the points are perfectly aligned vertically, it means that the data points have no variance in the horizontal direction (x-axis). Therefore, the direction of the largest variance is along the vertical axis (y-axis).
So, the correct answer is (0, 1), which represents the direction of the largest variance (PC1).