Application of Linear Algebra in Data Science
“Astronomy, that micrography of heaven, is the most magnificent of the sciences… Astronomy has its clear side and its luminous side; on its clear side it is tinctured with algebra, on its luminous side with poetry” – Victor Hugo
Data is the life-giving aspect of every business. Data mining is a process of collecting huge chunks of data, often not with the intention of the use it will be put into. Hence such unstructured data will have to undergo a refining process to be ready for use. Linear algebra is the core of many data science algorithms such as Machine learning, Natural Language processing and Computer Vision etc and it powers many data science algorithms and applications.
The digital universe is doubling and trebling at an exponential rate and data is growing at the rate of the pandemic!
Applications of Linear Algebra
Linear algebra is the mathematics of data and hence used in many domains. Here we see below how linear algebra can be infused into different areas
1. Cryptography – It is the study of encoding and decoding messages.
2. Game Theory – A two-player model where decisions may result in payoffs or penalties
3. Markov Analysis – Linear algebra is used to predict future states using preceding observations
3. Computer Graphics – Linear transformations and coordinate matrices are used to draw objects from different views
4. Economic Models – Matrix theory can be utilised in describing interrelations between price, output and demand
5. Genetics – It is used in the study of the inheritance of dominant and recessive traits in individuals and also to predict the future distribution of traits
6. Fractals – It is used in computer graphics to replicate ferns, trees or coastal lines
7. Computed tomography – It helps to create and analyse cross-sectional view of x rays and scans
8. Electrical Networks – Helps to create models of electrical circuits
9. Equilibrium temperature distribution – Helps to plot heat distribution on a plate given the temperature at the end of a plate.
Linear Algebra and Data Analytics
In big data analysis, linear algebra relies heavily on linear equations, eigen value problems, principal component analysis, singular value decomposition, quadratic forms, linear inequalities, modelling and prediction. Matrix algorithms are the core of big data analytical application as they provide a convenient mathematical structure for modelling a wide range of applications. Some of the practical big data applications that use linear algebra are
1. Page Rank Algorithm
2. Recommender Systems
3. Topic Modeling
4. Social Network Analysis
5. Internet Search
6. Complex System Analysis
7. Image Segmentation
8. Graph Clustering
9. Link Prediction
10. Cellular Networks
Eigenvalues and Eigen vectors are used in page rank algorithm, network clustering and weather system modelling. Spectral decomposition is used in spectral clustering and link prediction in social networks. It can be utilised in production planning, portfolio selection, transportation problem, minimisation of production cost and environmental damage and maximisation of profits. Linear algebra can also be utilised to more practical applications like face recognition, fingerprint recognition, plagiarism detection.
“When you have eliminated all which is the impossible, then whatever remains, however improbable, must be the truth” – Sherlock Holmes
Authors – Benila Jacob