(PCA): - Dimension reduction analysis
-> It is a technique for feature extraction from a given data set.
-> There are N-number of Principal Component corresponding to the N-number of data.
-> 95% of the features of the extracted data belong to the first principal component.
-> Therefore, we have to select the first n-number of the principal component corresponding to the N-number of data; the choice of the n-number principal component is determined by the precision we are aiming for.
-> So, PCA reduces the N-number of the principal components corresponding to the N-number of data into the n-number of the features; N >> n.
-> Consequently, another name for this method is a dimension reduction analysis.
-> Example: - Considering the situation for the data sets X and Y.
X = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.
Y = 1, 4, 9, 16, 25, 36, 49, 64, 81, 100.
Here, the number of features = 2 and the number of samples = 10.
The steps for computing PCA are given as follows:
Step 1: Generate the covariance matrix for datasets X and Y.
| Cov (X, X) Cov (X, Y) |
$A _{(2 X 2)}$ =
| Cov (Y, X) Cov (Y, Y) |
$\begin{align}Cov{(X, Y)}=\sum_{i=1}^N\frac{(x_i-\mu_X)(y_i-\mu_Y)}{N}\end{align}$
$Where, \; \mu_X$ and $\mu_Y$ are the mean of the given data sets $X$ and $Y$ respectively.
| 8.25 90.75 |
$A _{(2 X 2)}$ =
| 90.75 1051.05 |
Step 2: Generate the characteristics equation by using covariance matrix $A_{(2 X 2)}$.
Note:- det ($A_{(2 X 2)}$ - $ \lambda $ I ) = 0; represents the characteristics equation and I = unit matrix.
| 8.25 - $\lambda$ 90.75 |
$\det$ = 0
| 90.75 1051.05 - $\lambda$ |
$\implies (8.25 - \lambda) (1051.05 - \lambda) - 90.75 * 90.75 = 0 $
$\implies \lambda^{2} - 1059.3 \lambda + 435.6 = 0 $ . . . (1)
$ \implies \lambda_{1}=1058.89, \lambda_{2}=0.411375$ .
The $ \lambda_{1}, \lambda_{2}$ represents the Eigen Values of the matrix $A_{(2 X 2)}$.
The first principal component is defined by the largest eigenvalue, the second principal component by the second-largest eigenvalue, and so on.
Step 3: The computation of the Eigen Vectors corresponding to the Eigen Values.
$(A_{(2 X 2)} - \lambda_{i} I) U_{i} = 0$ . . . (2)
When $\lambda_{1}$ = 1058.89, then the $(A_{(2 X 2)} - \lambda_{i} I) U_{i}$ =
| -1050.64 90.75 | | $u_{1}$ | | 0 |
=
| 90.75 -7.84 | | $u_{2}$ | | 0 |
Now equating the matrix on both sides, we get.
$-1050.64 * u_{1} + 90.75 * u_{2} = 0 $ . . . (3)
and $ 90.75 * u_{1} - 7.84 * u_{2} = 0 $ ... (4)
The Eigen Vectors corresponding to equations (3) and (4) are as follows:
|$u_{1}$| | $90.75 * k$| |$7.84*k$|
= OR
|$u_{2}$| |$1050.64*k$| |$90.75*k$|
Where 'k=1' is a constant.
When $\lambda_{2}$ = 0.411375, then the $(A_{(2 X 2)} - \lambda_{i} I) U_{i}$ =
| 7.838625 90.75 | | $u_{1}$ | | 0 |
=
| 90.75 1050.638625 | | $u_{2}$ | | 0 |
Now equating the matrix on both sides, we get.
$7.838625 * u_{1} + 90.75 * u_{2} = 0 $ . . . (5)
and
$ 90.75 * u_{1} + 1050.638625 * u_{2} = 0 $ . . . (6)
The Eigen Vectors corresponding to equations (5) and (6) are as follows:
|$u_{1}$| |$90.75*k$| |$1050.64*k$|
= OR
|$u_{2}$| |$-7.84*k$| |$-90.75*k$|
Where 'k=1' is a constant.
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