#top 6 machine learning libraries in python

To tackle data analysis effectively in Python, it's crucial to become familiar with several libraries that streamline the process of data manipulation, exploration, and visualization. Here's a breakdown of the essential libraries:

 1. NumPy

   - Purpose: Numerical computing.

   - Why Learn It: NumPy provides support for large multi-dimensional arrays and matrices, along with a collection of mathematical functions to operate on these arrays efficiently.

   - Key Features:

     - Fast array processing.

     - Mathematical operations on arrays (e.g., sum, mean, standard deviation).

     - Linear algebra operations.

 2. Pandas

   - Purpose: Data manipulation and analysis.

   - Why Learn It: Pandas offers data structures like DataFrames, making it easier to handle and analyze structured data.

   - Key Features:

     - Reading/writing data from CSV, Excel, SQL databases, and more.

     - Handling missing data.

     - Powerful group-by operations.

     - Data filtering and transformation.

 3. Matplotlib

   - Purpose: Data visualization.

   - Why Learn It: Matplotlib is one of the most widely used plotting libraries in Python, allowing for a wide range of static, animated, and interactive plots.

   - Key Features:

     - Line plots, bar charts, histograms, scatter plots.

     - Customizable charts (labels, colors, legends).

     - Integration with Pandas for quick plotting.

 4. Seaborn

   - Purpose: Statistical data visualization.

   - Why Learn It: Built on top of Matplotlib, Seaborn simplifies the creation of attractive and informative statistical graphics.

   - Key Features:

     - High-level interface for drawing attractive statistical graphics.

     - Easier to use for complex visualizations like heatmaps, pair plots, etc.

     - Visualizations based on categorical data.

 5. SciPy

   - Purpose: Scientific and technical computing.

   - Why Learn It: SciPy builds on NumPy and provides additional functionality for complex mathematical operations and scientific computing.

   - Key Features:

     - Optimized algorithms for numerical integration, optimization, and more.

     - Statistics, signal processing, and linear algebra modules.

 6. Scikit-learn

   - Purpose: Machine learning and statistical modeling.

   - Why Learn It: Scikit-learn provides simple and efficient tools for data mining, analysis, and machine learning.

   - Key Features:

     - Classification, regression, and clustering algorithms.

     - Dimensionality reduction, model selection, and preprocessing utilities.

 7. Statsmodels

   - Purpose: Statistical analysis.

   - Why Learn It: Statsmodels allows users to explore data, estimate statistical models, and perform tests.

   - Key Features:

     - Linear regression, logistic regression, time series analysis.

     - Statistical tests and models for descriptive statistics.

 8. Plotly

   - Purpose: Interactive data visualization.

   - Why Learn It: Plotly allows for the creation of interactive and web-based visualizations, making it ideal for dashboards and presentations.

   - Key Features:

     - Interactive plots like scatter, line, bar, and 3D plots.

     - Easy integration with web frameworks.

     - Dashboards and web applications with Dash.

 9. TensorFlow/PyTorch (Optional)

   - Purpose: Machine learning and deep learning.

   - Why Learn It: If your data analysis involves machine learning, these libraries will help in building, training, and deploying deep learning models.

   - Key Features:

     - Tensor processing and automatic differentiation.

     - Building neural networks.

 10. Dask (Optional)

   - Purpose: Parallel computing for data analysis.

   - Why Learn It: Dask enables scalable data manipulation by parallelizing Pandas operations, making it ideal for big datasets.

   - Key Features:

     - Works with NumPy, Pandas, and Scikit-learn.

     - Handles large data and parallel computations easily.

Python has already made a significant impact in the tech world, and its trajectory for the future is even more promising. From its simplicity and versatility to its widespread use in cutting-edge technologies, Python is expected to continue thriving in the coming years. Considering the kind support of Python Course in Chennai Whatever your level of experience or reason for switching from another programming language, learning Python gets much more fun.

Let's explore why Python will remain at the forefront of software development and what trends and applications will contribute to its ongoing dominance.

1. Artificial Intelligence and Machine Learning

Python is already the go-to language for AI and machine learning, and its role in these fields is set to expand further. With powerful libraries such as TensorFlow, PyTorch, and Scikit-learn, Python simplifies the development of machine learning models and artificial intelligence applications. As more industries integrate AI for automation, personalization, and predictive analytics, Python will remain a core language for developing intelligent systems.

2. Data Science and Big Data

Data science is one of the most significant areas where Python has excelled. Libraries like Pandas, NumPy, and Matplotlib make data manipulation and visualization simple and efficient. As companies and organizations continue to generate and analyze vast amounts of data, Python’s ability to process, clean, and visualize big data will only become more critical. Additionally, Python’s compatibility with big data platforms like Hadoop and Apache Spark ensures that it will remain a major player in data-driven decision-making.

3. Web Development

Python’s role in web development is growing thanks to frameworks like Django and Flask, which provide robust, scalable, and secure solutions for building web applications. With the increasing demand for interactive websites and APIs, Python is well-positioned to continue serving as a top language for backend development. Its integration with cloud computing platforms will also fuel its growth in building modern web applications that scale efficiently.

4. Automation and Scripting

Automation is another area where Python excels. Developers use Python to automate tasks ranging from system administration to testing and deployment. With the rise of DevOps practices and the growing demand for workflow automation, Python’s role in streamlining repetitive processes will continue to grow. Businesses across industries will rely on Python to boost productivity, reduce errors, and optimize performance. With the aid of Best Online Training & Placement Programs, which offer comprehensive training and job placement support to anyone looking to develop their talents, it’s easier to learn this tool and advance your career.

5. Cybersecurity and Ethical Hacking

With cyber threats becoming increasingly sophisticated, cybersecurity is a critical concern for businesses worldwide. Python is widely used for penetration testing, vulnerability scanning, and threat detection due to its simplicity and effectiveness. Libraries like Scapy and PyCrypto make Python an excellent choice for ethical hacking and security professionals. As the need for robust cybersecurity measures increases, Python’s role in safeguarding digital assets will continue to thrive.

6. Internet of Things (IoT)

Python’s compatibility with microcontrollers and embedded systems makes it a strong contender in the growing field of IoT. Frameworks like MicroPython and CircuitPython enable developers to build IoT applications efficiently, whether for home automation, smart cities, or industrial systems. As the number of connected devices continues to rise, Python will remain a dominant language for creating scalable and reliable IoT solutions.

7. Cloud Computing and Serverless Architectures

The rise of cloud computing and serverless architectures has created new opportunities for Python. Cloud platforms like AWS, Google Cloud, and Microsoft Azure all support Python, allowing developers to build scalable and cost-efficient applications. With its flexibility and integration capabilities, Python is perfectly suited for developing cloud-based applications, serverless functions, and microservices.

8. Gaming and Virtual Reality

Python has long been used in game development, with libraries such as Pygame offering simple tools to create 2D games. However, as gaming and virtual reality (VR) technologies evolve, Python’s role in developing immersive experiences will grow. The language’s ease of use and integration with game engines will make it a popular choice for building gaming platforms, VR applications, and simulations.

9. Expanding Job Market

As Python’s applications continue to grow, so does the demand for Python developers. From startups to tech giants like Google, Facebook, and Amazon, companies across industries are seeking professionals who are proficient in Python. The increasing adoption of Python in various fields, including data science, AI, cybersecurity, and cloud computing, ensures a thriving job market for Python developers in the future.

10. Constant Evolution and Community Support

Python’s open-source nature means that it’s constantly evolving with new libraries, frameworks, and features. Its vibrant community of developers contributes to its growth and ensures that Python stays relevant to emerging trends and technologies. Whether it’s a new tool for AI or a breakthrough in web development, Python’s community is always working to improve the language and make it more efficient for developers.

Conclusion

Machine learning technology has truly transformed multiple industries and continues to hold enormous potential for future development. If you're considering incorporating machine learning into your business or are simply eager to learn more about this transformative field, seeking advice from experts or enrolling in specialized courses is a wise step. For instance, the ACTE Institute offers comprehensive machine learning training programs that equip you with the knowledge and skills necessary for success in this rapidly evolving industry. Recognizing the potential of machine learning can unlock numerous avenues for data analysis, automation, and informed decision-making.

Now, let me share my successful journey in machine learning, which I believe can benefit everyone. These 10 steps have proven to be incredibly effective in helping me become a proficient machine learning practitioner:

Step 1: Understand the Basics

Develop a strong grasp of fundamental mathematics, particularly linear algebra, calculus, and statistics.

Learn a programming language like Python, which is widely used in machine learning and provides a variety of useful libraries.

Step 2: Learn Machine Learning Concepts

Enroll in online courses from reputable platforms like Coursera, edX, and Udemy. Notably, the ACTE machine learning course is a stellar choice, offering comprehensive education, job placement, and certification.

Supplement your learning with authoritative books such as "Hands-On Machine Learning with Scikit-Learn, Keras, and TensorFlow" by Aurélien Géron and "Pattern Recognition and Machine Learning" by Christopher Bishop.

Step 3: Hands-On Practice:

Dive into real-world projects using both simple and complex datasets. Practical experience is invaluable for gaining proficiency.

Participate in machine learning competitions on platforms like Kaggle to challenge yourself and learn from peers.

Step 4: Explore Advanced Topics

Delve into deep learning, a critical subset of machine learning that focuses on neural networks. Online resources like the Deep Learning Specialisation on Coursera are incredibly informative.

For those intrigued by language-related applications, explore Natural Language Processing (NLP) using resources like the "Natural Language Processing with Python" book by Steven Bird and Ewan Klein.

Step 5: Learn from the Community

Engage with online communities such as Reddit's r/Machine Learning and Stack Overflow. Participate in discussions, seek answers to queries, and absorb insights from others' experiences.

Follow machine learning blogs and podcasts to stay updated on the latest advancements, case studies, and best practices.

Step 6: Implement Advanced Projects

Challenge yourself with intricate projects that stretch your skills. This might involve tasks like image recognition, building recommendation systems, or even crafting your own AI-powered application.

Step 7: Stay updated

Stay current by reading research papers from renowned conferences like NeurIPS, ICML, and CVPR to stay on top of cutting-edge techniques.

Consider advanced online courses that delve into specialized topics such as reinforcement learning and generative adversarial networks (GANs).

Step 8: Build a Portfolio

Showcase your completed projects on GitHub to demonstrate your expertise to potential employers or collaborators.

Step 9: Network and Explore Career Opportunities

Attend conferences, workshops, and meetups to network with industry professionals and stay connected with the latest trends.

Explore job opportunities in data science and machine learning, leveraging your portfolio and projects to stand out during interviews.

Our world is gradually evolving to become more technology-reliant. And one such technology expected to revolutionize the future is Machine learning.

With its advent; engineers program the computer to learn and provide data on its own with limited or no manual instructions.

Skill set for Machine Learning Engineer

Following skill set would  be required for the best machine learning course

1.Programming language- Proficiency in many languages like Java programming for Map Reduce coding, R for visualization purpose, and Python for Machine learning.

2. Calculus and statistics- Most of the Machine learning algorithms are mathematical. Hence you should know your high school mathematics efficiently.

3. Signal Processing- You can expect exponential growth in your career; when you have the knowledge of signal processing.

4. Neural Networks- Due to their efficiency they have gained huge popularity in recent years.

5. Language processing- Machine learning deals with text data, voice data, and visual data. Hence the ability to program such data is mandatory.

Depending on their level of expertise ML Engineers may:

1. Study and transform the Data Sciences prototype.

2. Design machine learning system

3. Research and implement appropriate ML Algorithm and tools.

4. Develop Machine Learning Applications according to the rule of the industry

5. Look for Appropriate data sets and Data Representation techniques

6. Run machine learning Tests and experiments.

7. Execute Statistical Analysis and fine-tuning using test results

8. Train and Re-train systems when necessary.

9. Extend the existing ML libraries and framework.

Machine Learning Career Paths

MACHINE LEARNING ENGINEERS- Most coveted and Promising field of ML. They perform sophisticated programming to develop machines and systems. They learn and apply knowledge without specifications.

They; shape and develop efficient self-learning ML applications.   And perform statistical analysis and fine-tuning using test results.

They also conduct experiments using programming languages like Java, R, Scala, and C++.

SKILLS REQUIRED:   Must have a strong foundational knowledge of Mathematics, Statistics, and programming. Accomplished in software architecture, system design, data structures, data modeling, and ML algorithm.

DATA SCIENTIST-

Hailed as the “Sexiest Job of the 21st Century” by a Harvard Business review article. These are high-profile experts who leverage advanced technologies (like Big Data, AI, ML, Deep Learning, etc.) and analytical tools on a daily basis to collect, store, process, analyze, and interpret massive amounts of data. Their primary goal is to extract valuable insights from large datasets and convert them into business value.  

SKILLS REQUIRED: like the MLE; they should acclaim Mathematics, Statistics, and programming skills.

And also have thorough experience in data mining and how to apply various statistical research techniques and use Big Data platforms (Hadoop, Pig, Hive, Spark, Flume, etc.).  

NLP SCIENTIST

Natural Learning Process functions to coach machines the ability to understand natural human languages. NLP Scientists hold the core responsibility of designing and developing machines and applications. These machines can learn the patterns of speech of a human language and also translate spoken words into other languages.

SKILLS REQUIRED- They teach machines how to understand the nuances of human languages. Hence, they must be fluent in the syntax, spelling, and grammar of at least one language (the more, the better).

HUMAN-CENTRED MACHINE LEARNING DESIGNERS

This is an exclusive branch dedicated to designing ML algorithms centered around humans. These designers are responsible for creating intelligent systems that can “learn” the preferences and behavior patterns of individual humans through information processing and pattern recognition.

SKILLS REQUIRED- they must have an in-depth understanding of various ML concepts, algorithms, and how they function. And also have a good base in Mathematics and Statistics along with coding skills.  

An excellent example is Amazon and Netflix's Recommendation Engine.

 INDUSTRIES LEVERAGING MACHINE LEARNING ARE:

The Health care Industry

The Retail Industry

The Financial Service Industry

The Automotive Industry

Government Agencies

Transport Industries

Oil and Gas Industries.

AN AVERAGE INCOME- A machine learning engineer receives a clean slip of a roughly 7lkh per annum income in India.

CONCLUSION

While these are few careers in Machine Learning. Other profiles like Data Analyst, Data Architect, Cloud Architect, and Business Intelligence Developer also exist in this domain.

Every organization wants to capitalize on its data to gain insights, improve customer relations, increase the sale, or be competitive. ML has enough capability to help them achieve their goal.

Yet, Machine Learning's ability to automate, expect, and evolve is powerful, but that doesn't mean computers will take over the world. They still need human operators to provide context, to set parameters of operations, and to continue improving the algorithms.

Why You Should Learn Python

Python is one of the most cherished programming dialects by designers, information researchers, programming engineers, and even programmers on account of its adaptability, adaptability, and item situated highlights. 

Huge numbers of the web and portable applications we appreciate today is a direct result of Python's plentiful libraries, different systems, immense assortments of modules, and record expansions. Not just that, Python is extraordinary for building miniature task to large scale venture web benefits just as for supporting different sorts of programming dialects.

Even though it's an elevated level of language and can-do complex undertakings, Python is anything but difficult to learn and has a spotless linguistic structure. Consequently, it's recommendable for the two tenderfoots and experienced developers. Likewise, there are a lot of ways on the best way to learn Python; you can do self-study, enlist yourself in coding Bootcamp or school or watch online instructional exercises or take Python course London. The decision is yours to make! 

Top 7 Uses of Python

With its close flawlessly excellence, Python is named as one of the top-picked programming dialects of most colleges and businesses. What makes it overwhelming is its wide applications to different fields and boundless conceivable outcomes to make astounding things out of scratch.

As per TechRepublic, Python engineer is one of the "10 Most In-Demand Tech Jobs of 2019". This is because of the ascent of AI and Machine Learning innovations in the worldwide market. As of February 23, 2019, the normal compensation for a Python designer is $123,201 every year in the United States, making it one of the most famous and worthwhile professions today.

In reality, there are numerous reasons why you ought to learn Python.

Furthermore, to comprehend further its degree and advantages, here are the Top 7 employments of Python.

Python can be utilized on the accompanying:

Table of Contents

1. Why PyGame?

2. Why Not?

3. Explicit Is Better Than Implicit

4. Three Pain Points

5. Other Cool Libraries

6. More To Come

1 Why PyGame?

PyGame is a nice library for getting started with game development. It is a good way to learn programming in a "real programming language" while making games1. There is very "magic" going on in PyGame, and you can go from print("Hello World") to drawing a triangle on the screen and then to Pong or Tetris with a smooth learning curve.

PyGame is also an easy way to get your feet wet in game development if you know and use Python already, for example in machine learning or web development. That's how I came to learn PyGame.

Exploring Phantasms of the Living (1886) through Machine Learning: Presentiment, Crisis Apparitions and Thought Transference

NOTE: Click to open graphics for an expanded and clearer view of the findings they contain  

Phantasms of the Living, published in 1886 by the Society for Psychical Research (SPR), was a landmark ESP study that presented the case for “telepathy" or thought transference from mind to mind.  The study consisted of 702 cases spanning over 1400 pages that considered several varieties of telepathic experiences collectively referred to as “phantasms of the living”  

The case collection examined non-sensory and internalized impressions, many of which were presentiment experiences involving dreams, clairvoyance, visions, feelings or an awareness in connection with the deaths of family members or friends.  These experiences often coincided with the approximate time of death

Cases also considered sensory and externalized impressions, in particular apparitional representations of living persons, who were perceived to be in moments of crisis or danger.  These situations appeared evidential of shock-induced forms of thought transference from a distressed agent to a percipient in the form of telepathic hallucinations

As a follow-on to the earlier wordcloud project, we wondered whether unsupervised machine learning could discover main topics within Phantasms of the Living.  For the project, two varieties of generative topic models were used: Latent Dirichlet Allocation (LDA) and probabilistic Latent Semantic Analysis (pLSA)

Both models view documents as having a latent semantic structure of topics that can be inferred from co-occurrences of words in documents. The mathematics underlying both models are beyond the scope of this post, but on an intuitive level there are key differences between the two methods

pLSA views topics as probability distributions over words.  Topics are seen as conditionally independent across the documents that produced them.  Non-Negative Matrix Factorization (NMF) is a method for finding topic clusters that equates to pLSA

LDA by contrast views documents as probability distributions over topics and topics as probability distributions over words.  All documents share the same collection of topics, but each document contains those topics in different proportions. The LDA algorithm samples words across topics until it arrives at topics and word selections that most likely generated the documents

The project used various packages and libraries for natural language processing within the Python programming platform to include: the Natural Language ToolKit (NLTK) for processing the data set; scikit-learn to prepare and fit the LDA and NMF models; pyLDAvis was used to display the results and t-Distributed Stochastic Neighbor Embedding (t-SNE) to map topic distances

The end-to-end project pipeline involved: data set processing; conversion of words and documents into matrix and vector space; fitting the LDA and NMF models; and then displaying the results

Processing. The book was decomposed into several documents from its constituent sections, chapters and volumes for the data set. Stopwords were removed such as common prepositions and conjunctions using the wordcloud application   

Since telepathic experiences are spontaneous and can occur at any time or place, words conveying times and locations were removed as well as ordinal and cardinal types of numeric rankings

Nouns or titles representing persons were removed (e.g. man, woman, Mr., Mrs., etc.); however, interpersonal relationships were preserved (i.e. family, friends, acquaintances or strangers)  

Conversion. Vector transformations converted the data set into a document-term matrix for mathematical processing  

The rows of the matrix correspond to documents with columns corresponding to the frequency of a term.  Count vectorizers count word frequencies.  Term Frequency-Inverse Document Frequency (TF-IDF) vectorizers normalize (divide) word counts by their frequency in the documents 

Both vectorizers converted words to lower case and removed non-word expressions. The vectorizers were also instructed to look for bigrams (or words that were often used together) such as "thought-transference" and "telepathic hallucination"

Model Fit/Display. The LDA and NMF models were fitted using ten topics.  Words within topics were sorted and ranked with respect to their frequency in and relevance within a topic

The LDA model was fitted with using Count and TF-IDF vectorization and ran with a maximum of 10 iterations.  LDA model results were displayed using pyLDAvis and t-SNE to map topic distances

The NMF model was fitted with TF-IDF vectorization only and ran with a maximum of 200 iterations. NMF model results were displayed via spreadsheet

Results. The topics produced from the models are unlabeled.  However words within topics often can be woven into a coherent theme

The first two pyLDAvis graphs provide the top 30 words and bigrams in Topics 1 and 2 using Count vectorization  

Words in Topic 1 include: “dreams”; “visions”; “impressions”; and “experiences” in connection with the “death”(s) of family members and friends.  This can be considered a presentiment topic and it generated 67% of the content.  This mirrors results from the prior wordcloud project  

Words and bigrams in Topic 2 include: “thought-transference”, “hallucination(s)”, “phantasms”, “mind(s)”, “percipients”, “agent” and “telepathy.”  This can be considered a telepathic hallucinations topic and it produced 27% of the content 

The third pyLDAvis graph provides the top 30 words in Topic 1 using TF-IDF vectorization

Topic 1 combines all the aforementioned words into one topic.  This can be considered a “presentiment and telepathic hallucinations ” topic and it accounts for 95% of the content, rendering all other topics practically insignificant in influence.  The reason for this consolidation is that TF-IDF vectorization lowers the contribution weight of commonly used words

The spreadsheets compare LDA and NMF model runs using TF-IDF vectorizations with results limited to the top 10 words.  Although topic weights and distances are not available, some topics appear more meaningful and cohesive, and are likely more impactful than others

There is overlap between topics 5 and 6 in the LDA model and together they form the presentiment and telepathic hallucinations topic. Topics 0 and 1 in the NMF model respectively appear to correspond to presentiment and crisis apparitions topics

The bigram “thought-transference” arises in both the LDA and NMF results and appears associated with the “Society” for “Psychical” Research and the late F.W.H. “Myers” who invented the term “telepathy”

This project had an extended preparation and production pipeline.  The results indicate that unsupervised machine learning using LDA and NMF effectively and comprehensively summarized topical content in Phantasms of the Living.  Moreover, key topics approximately corresponded to the types of internalized and externalized telepathic experiences described in the book  

This project demonstrates the usefulness of topic generation models for finding meaningful patterns in masses of unlabeled or unstructured data.  Moreover, visualization and graphing tools are essential for fully comprehending these patterns. Elsewhere in parapsychology LDA or NMF could also be applied to survey data, case collections, web or social media content of interest.  

REFERENCES

Anaya, L. A. Comparing Latent Dirichlet Allocation and Latent Semantic Analysis as Classifiers. University of North Texas, 2011.

Blei, D. M., Ng, A. Y., & Jordan, M. I. (2003). Latent dirichlet allocation. Journal of machine Learning research, 3(Jan), 993-1022.

Christou, D. (2016). Feature extraction using Latent Dirichlet Allocation and Neural Networks: A case study on movie synopses. arXiv preprint arXiv:1604.01272.

Deerwester, S. (1988). Improving information retrieval with latent semantic indexing.

Gurney, E., Myers, F. W., & Podmore, F. (1886). Phantasms of the Living (2 vols.). London: Trübner.  Reprinted at the Esalen Center.

Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., ... & Vanderplas, J. (2011). Scikit-learn: Machine learning in Python. Journal of machine learning research, 12(Oct), 2825-2830.

Sievert, C., & Shirley, K. (2014). LDAvis: A method for visualizing and interpreting topics. In Proceedings of the workshop on interactive language learning, visualization, and interfaces (pp. 63-70).

IMAGES

pyLDAvis Graph of Topic 1 (Count Vectorization) from Phantasms of the Living Corpus. (2018, Mar 24). © Maryland Paranormal Research ®.  All rights reserved.

pyLDAvis Graph of Topic 2 (Count Vectorization) from Phantasms of the Living Corpus. (2018, Mar 24). © Maryland Paranormal Research ®.  All rights reserved.

pyLDAvis Graph of Topic 1 (TF-IDF Vectorization) from Phantasms of the Living Corpus. (2018, Mar 24). © Maryland Paranormal Research ®.  All rights reserved.

Top Words List: Latent Dirichlet Allocation (TF-IDF Vectorization) from Phantasms of the Living Corpus. (2018, Mar 24). © Maryland Paranormal Research ®.  All rights reserved.

Implementing ML and AI algorithms necessitates a well-structured and well-tested framework to enable developers to provide high-quality code solutions. A Python library or framework is a ready-to-use software for typical coding tasks.

Let's get to know the Best Python Libraries For Machine Learning: 

TensorFlow is a complete Python machine learning toolkit for high-end numerical computations.

Keras Python is a popular open-source Python framework for building neural networks and machine learning projects.

Theano Python is a well-known scientific computing toolkit that allows you to design, optimize, and evaluate mathematical equations that deal with multidimensional arrays..