Data Science For Dummies. Lillian Pierson

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      FIGURE 4-2: An example of a non-linear relationship between watch time and % viewership.

Reducing Data Dimensionality with Linear Algebra

      Any intermediate-level data scientist should have a good understanding of linear algebra and how to do math using matrices. Array and matrix objects are the primary data structure in analytical computing. You need them in order to perform mathematical and statistical operations on large and multidimensional datasets — datasets with many different features to be tracked simultaneously. In this section, you see exactly what is involved in using linear algebra and machine learning methods to reduce a dataset’s dimensionality — in other words, to reduce a dataset’s feature count, without losing the important information the dataset contains, by compressing its features’ information into synthetic variables that you can subsequently utilize to make predictions or as input into another machine learning model.

      Decomposing data to reduce dimensionality

Okay, what can you do with all this theory? Well, for starters, using a linear algebra method called singular value decomposition (SVD), you can reduce the dimensionality of your dataset — reduce the number of features that you track when carrying out an analysis, in other words. Dimension reduction algorithms are ideal options if you need to compress your dataset while also removing redundant information and noise. In data science, SVD is applied to analyze principal components from with large, noisy, sparse data sets — an approach machine learning folks call Principal Component Analysis (PCA). Since the linear algebra involved in PCA is rooted in SVD, let’s look at how SVD works.

The SVD linear algebra method decomposes the data matrix into the three resultant matrices shown in Figure 4-4. The product of these matrices, when multiplied together, gives you back your original matrix. SVD is handy when you want to compress or clean your dataset. Using SVD enables you to uncover latent variables — inferred variables hidden within your dataset that affect how that dataset behaves. Two main ways to use the SVD algorithm include

      FIGURE 4-3: Applying SVD to compress a sparse, clean dataset.

      FIGURE 4-4: Applying SVD to clean and compress a sparse, dirty dataset.

The lower the CVE, the more you should take your model’s results with a grain of salt.

      

If you remove some components, then when you go to reconstruct your matrix, you'll probably notice that the resulting matrix isn’t an exact match to your original dataset. Worry not! That is the data that remains after much of the information redundancy and noise was filtered out by SVD and removed by you.

Getting a little nitty-gritty about SVD, let’s look at the formula for SVD, but keep in mind — this is linear algebra not regular algebra, so we are looking at matrix math not regular math. To take it from the beginning, you need to understand the concept of eigenvector. To do that, think of a matrix called A. Now consider a nonzero vector called x and that Ax = λx for a scalar λ. In this scenario, scalar λ is what’s called an eigenvalue of matrix A. It’s permitted to take on a value of 0. Furthermore, x is the eigenvector that corresponds to λ, and again, it’s not permitted to be a zero value. λ is simply the scale factor of the eigenvector. SVD decomposes the matrix down into three resultant matrices shown in Figure 4-5. The product of these matrices, when multiplied together, gives you back your original matrix.

      Take a closer look at Figure 4-5:

      A = u * S * v

       A: This is the matrix that holds all your original data.

       u: This is a left-singular vector (an eigenvector) of A, and it holds all the important, nonredundant information about your data’s observations.

       v: СКАЧАТЬ