Linear Algebra For Data Science

What is Data Science?

Data science is the study of how to gain insightful knowledge from data for business choices, developing strategies, and other reasons utilizing state-of-the-art analytical technologies and scientific ideas. Businesses are becoming aware of its significance: among other things, data science insights assist companies in improving their marketing and sales efforts as well as operational effectiveness. They might eventually give you a competitive edge over other businesses.

Data Science combines a number of fields, including statistics, mathematics, software programming, predictive analytics, data preparation, data engineering, data mining, machine learning, and data visualization. Skilled data scientists are generally responsible for it, however, entry-level data analysts may also be engaged. Additionally, a growing number of firms now depend in part on citizen data scientists, a category that can encompass data engineers, business intelligence (BI) specialists, data-savvy business users, business analysts, and other employees without a formal experience in Data Science.

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What is Linear Algebra

Within Data Science and ML, linear algebra is a field of mathematics that is very helpful. In machine learning, linear algebra is perhaps the most crucial math concept. The vast majority of machine learning models may be written as matrices. A matrix is a common way to represent a dataset. The preprocessing, transformation, and assessment of data and models require linear algebra.

A study of linear algebra may involve the following:

• Vectors
• Matrices
• Transpose of a matrix
• The inverse of a matrix
• Determinant of a matrix
• Trace of a matrix
• Dot product
• Eigenvalues
• Eigenvectors

Why learn Linear Algebra in Data Science?

One of the fundamental building elements of Data Science is linear algebra. Without a solid foundation, you cannot erect a skyscraper, can you? Try to picture this example:

You wish to use Principal Component Analysis to minimize the dimensionality of your data (PCA). If you were unsure of how it would impact your data, how would you choose how many Principal Components to keep? Obviously, in order to make this choice, you must be familiar with the workings of the algorithm.

You will be able to gain a better sense for ML and deep learning algorithms and stop treating them as mysterious black boxes if you have a working knowledge of linear algebra. This would enable you to select suitable hyperparameters and create a more accurate model. Additionally, you would be able to develop original algorithms and algorithmic modifications.

Linear Algebra Applications for Data Scientists

We will now learn more about the most common application of linear algebra for data scientists:

Machine learning: loss functions and recommender systems

Without a question, the most well-known use of artificial intelligence is machine learning (AI). Systems automatically learn and get better with experience employing machine learning algorithms, free from human intervention. In order to detect trends and learn from them, machine learning works by creating programs that access and analyze data (whether static or dynamic). The algorithm can use this expertise to analyze fresh data sets once it has identified relationships in the data. (See this page for more information on how algorithms learn.)

Machine learning uses linear algebra in many different ways, including loss functions, regularization, support vector classification, and plenty more.

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Loss Function

Machine learning algorithms function by gathering data, interpreting it, and then creating a model via various techniques. They can then forecast upcoming data queries depending on the outcomes.

Now, we may assess the model's correctness by utilizing linear algebra, specifically loss functions. In a nutshell, loss functions provide a way to assess the precision of the prediction models. The output of the loss function will be greater if the model is completely incorrect. In contrast, a good model will cause the function to return a lower value.

Modeling a link involving a dependent variable, Y, and numerous independent variables, Xi's, is known as regression. We attempt to build a line in place on these variables upon plotting these points, and we utilize this line to forecast future values of Xi's.

The two most often used loss functions are mean squared error and mean absolute error. There are many different forms of loss functions, many of which are more complex than others.

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Recommender System

A subset of machine learning known as recommender systems provides consumers with pertinent suggestions based on previously gathered data. In order to forecast what the present user (or a new user) might like, recommender systems employ data from the user's prior interactions with the algorithm focused on their interests, demographics, and other available data. By tailoring material to each user's tastes, businesses can attract and keep customers.

The performance of recommender systems depends on two types of data being gathered: 

Characteristic data: Knowledge of things, including location, user preferences, and details like their category or price.

User-item interactions: Ratings and the volume of transactions (or purchases of related items).

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Natural language processing: word embedding

Artificial intelligence's Natural Language Processing (NLP) field focuses on how to connect with people through natural language, most frequently English. Applications for NLP encompass textual analysis, speech recognition, and chatbot.

Applications such as Grammarly, Siri, and Alexa are all based on the concept of NLP.

Word embedding

Text data cannot be understood by computers, not by its own. We use NLP algorithms on text since we need to mathematically express the test data. The use of algebra is now necessary. A sort of word representation known as word embedding enables ML algorithms to comprehend terms with comparable meanings.

With the backdrop of the words still intact, word embeddings portray words as vectors of numbers. These representations are created using the language modeling learning technique of training various neural networks on a huge corpus of text. Word2vec is among the more widely used word embedding methods.

Computer vision: image convolution

Using photos, videos, and deep learning models, the artificial intelligence discipline of computer vision teaches computers to comprehend and interpret the visual environment. This enables algorithms to correctly recognize and categorize items. 

In applications like image recognition as well as certain image processing methods like image convolution and image representation like tensors, we utilize linear algebra in computer vision.

Image Convolution

Convolution results from element-wise multiplying two matrices and then adding them together. Consider the image as a large matrix and the kernel (i.e., convolutional matrix) as just a tiny matrix used for edge recognition, blurring, as well as related image processing tasks. This is one approach to conceiving image convolution. As a result, this kernel slides over the image from top to bottom and from left to right. While doing so, it performs arithmetic operations at every image's (x, y) location to create a distorted image.

Different forms of image convolutions are performed by various kernels. Square matrices are always used as kernels. They are frequently 3x3, however, you can change the form depending on the size of the image.

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Where do we use linear algebra in Data Science?

Data Scientists often make use of Linear Algebra for various applications including:

• Vectorized Code: To create vectorized codes that are relatively more effective than their non-vectorized counterparts, linear algebra is helpful. This is so that results from vectorized codes can be produced in a single step instead of results from non-vectorized codes, which frequently involve numerous steps and loops.
• Dimensionality Reduction: In the preparation of data sets required for machine learning, dimensionality reduction is a crucial step. This is particularly true for big data sets or those with many attributes or dimensions. Many of these characteristics may occasionally have a strong correlation with one another.

The speed and effectiveness of the ML algorithm are improved by doing dimensionality reduction on a big data set. This is due to the fact that the algorithm only needs to consider a small number of features before producing a forecast.

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Concepts of linear algebra for Data Science

Linear Algebra for Data Preprocessing - Linear algebra is used for data preprocessing in the following way:

• Import the required libraries for linear algebra such as NumPy, pandas, pylab, seaborn, etc.
• Read datasets and display features
• Define column matrices to perform data visualization

Covariance Matrix- One of the most crucial matrices in Data Science and ML is the covariance matrix. It offers details on the co-movement (correlation) of characteristics. We can create a scatter pair plot to see how the features are correlated. One could construct the covariance matrix to determine the level of multicollinearity or correlation between characteristics. The covariance matrix could be written as a symmetric and real 4 x 4 matrix.
A unitary transformation, commonly known as a Principal Component Analysis (PCA) transformation, can be used to diagonalize this matrix. We note that the sum of the diagonal matrix's eigenvalues equals the total variance stored in features because the trace of a matrix stays constant during a unitary transformation.

Linear Discriminant Analysis Matrix - The Linear Discriminant Analysis (LDA) matrix is another illustration of a realistic and symmetrical matrix in Data Science. This matrix could be written as follows

where SW stands for the scatter matrix within the feature and SB for the scatter matrix between the feature. It implies that L is real and symmetric because the matrices SW & SB are also realistic and symmetrical. A feature subspace with improved class separability and decreased dimensionality is created by diagonalizing L. So, whereas PCA is not a supervised method, LDA is.

Conclusion

Often a skipped-over concept due to premeditated assumptions of difficulty, a good hold over linear algebra could help build a crucial foundation for those aspiring to have flourishing careers in Data Science.

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