How does PCA and eigenvectors help in the actual analysis of data? PCA can be used to reduce the dimensions of a data set. There are 3 variables so it is a 3D data set. Now imagine that the data forms into an oval like the ones above, but that this oval is on a plane. Like this:. The first two eigenvectors will show the width and depth of the data, but because there is no height on the data it is on a piece of paper the third eigenvalue will be zero.
On the picture below ev1 is the first eignevector the one with the biggest eigenvalue, the principal component , ev2 is the second eigenvector which has a non-zero eigenvalue and ev3 is the third eigenvector, which has an eigenvalue of zero. We can now rearrange our axes to be along the eigenvectors, rather than age, hours on internet and hours on mobile. However we know that the ev3, the third eigenvector, is pretty useless.
Therefore instead of representing the data in 3 dimensions, we can get rid of the useless direction and only represent it in 2 dimensions, like before:. This is dimension reduction. We have reduced the problem from a 3D to a 2D problem, getting rid of a dimension. Reducing dimensions helps to simplify the data and makes it easier to visualise. Imagine we did the example again, except instead of the oval being on a 2D plane, it had a tiny amount of height to it.
There would still be 3 eigenvectors, however this time all the eigenvalues would not be zero. The values would be something like 10, 8 and 0.
The eigenvectors corresponding to 10 and 8 are the dimensions where there is alot of information, the eigenvector corresponding to 0. The OxIS report asked around people a set of questions about their internet use.
It then identified 4 principal components in the data. This is an example of dimension reduction. There are therefore 50 variables, making it a dimension data set. There are lots of eigenvalues, but there are only 4 which have big values — indicating along those four directions there is alot of information.
These are then identified as the four principal components of the data set which in the report were labelled as enjoyable escape, instrumental efficiency, social facilitator and problem generator , the data set can then be reduced from 50 dimensions to only 4 by ignoring all the eigenvectors that have insignificant eigenvalues. So dimension reduction using PCA helped simplify this data set by finding the dominant dimensions within it.
Thank you so much for this simplified explanation of PCA, I am now no longer in despair of trying to understand eigenvalues and eigenvectors. However, just to make sure, I am currently working with a two dimensional data where the dimension is 58 x 54, If I were to reduce its dimensionality, which should I pick as the x-axis and y-axis or is it really up to me whether to reduce the columns or the row for my data?
Thank you. Thanks so much for this article. This is a clear and useful explanation that has made my life as an already-stressed grad student much easier! Einstein Thanks for the simple explanation. You are commenting using your WordPress. You are commenting using your Google account. You are commenting using your Twitter account. You are commenting using your Facebook account.
Notify me of new comments via email. Notify me of new posts via email. In simpler words, eigenvalue can be seen as the scaling factor for eigenvectors. Here is the formula for what is called eigenequation. Note that the new vector Ax has different direction than vector x. Many disciplines traditionally represent vectors as matrices with a single column rather than as matrices with a single row. Whenever there is a complex system having large number of dimensions with a large number of data, eigenvectors and eigenvalues concepts help in transforming the data in a set of most important dimensions principal components.
This will result in processing the data in a faster manner. Your email address will not be published. Time limit is exhausted. Thank you for visiting our site today. We welcome all your suggestions in order to make our website better. Please feel free to share your thoughts. Data Analytics. Table of Contents. Author Recent Posts. Follow me. Ajitesh Kumar. I would love to connect with you on Linkedin and Twitter.
In the following sections we will determine the eigenvectors and eigenvalues of a matrix , by solving equation 3. Matrix in this example, is defined by: 4. To determine the eigenvalues for this example, we substitute in equation 3 by equation 4 and obtain:.
To solve this quadratic equation in , we find the discriminant:. Since the discriminant is strictly positive, this means that two different values for exist: 7. We have now determined the two eigenvalues and. Note that a square matrix of size always has exactly eigenvalues, each with a corresponding eigenvector. The eigenvalue specifies the size of the eigenvector. We can now determine the eigenvectors by plugging the eigenvalues from equation 7 into equation 1 that originally defined the problem.
The eigenvectors are then found by solving this system of equations. We first do this for eigenvalue , in order to find the corresponding first eigenvector:. Since this is simply the matrix notation for a system of equations, we can write it in its equivalent form:.
Since an eigenvector simply represents an orientation the corresponding eigenvalue represents the magnitude , all scalar multiples of the eigenvector are vectors that are parallel to this eigenvector, and are therefore equivalent If we would normalize the vectors, they would all be equal. Thus, instead of further solving the above system of equations, we can freely chose a real value for either or , and determine the other one by using equation 9. For this example, we arbitrarily choose , such that.
Therefore, the eigenvector that corresponds to eigenvalue is. Calculations for the second eigenvector are similar to those needed for the first eigenvector; We now substitute eigenvalue into equation 1 , yielding:. Solving the first equation as a function of resuls in:. We then arbitrarily choose , and find. In this article we reviewed the theoretical concepts of eigenvectors and eigenvalues. These concepts are of great importance in many techniques used in computer vision and machine learning, such as dimensionality reduction by means of PCA, or face recognition by means of EigenFaces.
We all hate spam. Your email address will not be sold or shared with anyone else. Great post! Great writing it is such a cool and nice idea thanks for sharing your post. I like your post very much. Thanks for your post. Very helpful piece though. Very nice article! I have a question. I was also confused about this. It would also have an area of zero if one of the vectors is a null-vector. This is just my understanding after googling some stuff. It helped me a lot to get things clear in my mind.
Please go on! BTW typo: Eq. Thanks, Sebastian. I have a question for you. But this statement was confusing for me. Can you please explain what do you mean by this statement? Hi Nrupatunga, Usually, we normalize the eigenvector such that its magnitude is one.
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