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mathematics

Heat map

Heat map is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Heat map rather than just read about it. In short: A heat map (or heatmap) is a two-dimensional data visualization technique that represents the magnitude of individual values within a dataset as a color. The variation in color may be by hue or intensity.

Heat map — main illustration
Heat map — illustration

Key takeaways

  • Heat map belongs to mathematics; place it in that map before memorising details.
  • Learn the definition first, then one example that makes the definition concrete.
  • Connect Heat map to a quantity you can measure, compute or draw — that is where exam questions come from.
  • Reproduce the core statement of Heat map from memory before moving on to harder problems.

Reference excerpt

A heat map (or heatmap) is a two-dimensional data visualization technique that represents the magnitude of individual values within a dataset as a color. The variation in color may be by hue or intensity. In some applications such as crime analytics or website click-tracking, color is used to represent the density of data points rather than a value associated with each point. "Heat map" is a relatively new term, but the practice of shading matrices has existed for over a century.

History Heat maps originated in 2D displays of the values in a data matrix. Larger values were represented by small dark gray or black squares (pixels) and smaller values by lighter squares. The earliest known example dates to 1873, when Toussaint Loua used a hand-drawn and colored shaded matrix to visualize social statistics across the districts of Paris. The idea of reordering rows and columns to reveal structure in a data matrix, known as seriation, was introduced by Flinders Petrie in 1899. In 1950, Louis Guttman developed the Scalogram, a method for ordering binary matrices to expose a one-dimensional scale structure. In 1957, Peter Sneath displayed the results of a cluster analysis by permuting the rows and the columns of a matrix to place similar values near each other according to the clustering. This idea was implemented by Robert Ling in 1973 with a computer program called SHADE. Ling used overstruck printer characters to represent different shades of gray, one character-width per pixel. Leland Wilkinson developed the first computer program in 1994 (SYSTAT) to produce cluster heat maps with high-resolution color graphics. The Eisen et al. display shown in the figure is a replication of the earlier SYSTAT design. Software designer Cormac Kinney trademarked the term 'heat map' in 1991 to describe computer software used to display real-time financial market information. In 1998 the trademark was acquired by SS&C Technologies, Inc., but the company did not extend the license, so it was annulled in 2006.

Types There are two primary categories of heat maps: spatial and grid. Additionally, there are over ten various types of heat maps. A spatial heat map displays the magnitude of a spatial phenomenon as color, usually cast over a map. In the image labeled "Spatial Heat Map Example," temperature is displayed by color range across a map of the world. Color ranges from blue (cold) to red (hot). A grid heat map displays magnitude as color in a two-dimensional matrix, with each dimension representing a category of trait and the color representing the magnitude of some measurement on the combined traits from each of the two categories. For example, one dimension might represent year, and the other dimension might represent month, and the value measured might be temperature. This heat map would show how temperature changed over the years in each month. Grid heat maps are further categorized into two different types of matrices: clustered, and correlogram.

Clustered heat map: The example of the monthly temperature by year is a clustered heat map. Correlogram: A correlogram is a clustered heat map that has the same trait for each axis in order to display how the traits in the set of traits interact with each other. The correlogram is a triangle instead of a square because the combination of A-B is the same as the combination of B-A and so does not need to be expressed twice. In a grid heat map, colors are presented in a grid of a fixed size, with every cell in the grid also being an equal size and shape. The goal is to detect clustering, or suggest the presence of clusters. A spatial heat map is often used on maps or satellite imagery (see GIS), where there is no concept of cells, and instead the colors vary continuously.

Uses Heat maps have a wide range of possibilities amongst applications due to their ability to simplify data and make for visually appealing to read data analysis. Many applications using different types of heat maps are listed below.

Business Analysis: Heat maps are used in business analytics to give a visual representation about a company's current functioning, performance, and the need for improvements. Heat maps are a way to analyze a company's existing data and update it to reflect growth and other specific efforts. Heat maps visually appeal to team members and clients of the business or company. Websites: There are many different ways heat maps are used within websites to determine a visiting users actions. Typically, there are multiple heat maps used together to determine insight to a website on what are the best and worst performing elements on the page. Some specific heat maps used for website analysis are listed below.

Mouse Tracking: Mouse tracking heat maps or hover maps, are used to visualize where the user of the site hovers their cursor. Eye tracking: Eye tracking heat maps measure the eye position of the website's users and gathers measurements such as eye fixation volume, eye fixation duration, and areas of interest. Click Tracking: Click tracking heat maps or touch maps, are similar to mouse tracking heat maps, but instead of hover actions, these types of heat maps help visualize the users click actions. Click tracking heat maps not only allow for visual cues on clickable components on a webpage, such as buttons or dropdown menus, but these heat maps also allow for tracking on non-clickable objects anywhere on the page. AI-Generation Attention: AI-generated attention heat maps help visualize where the visiting user's attention will go on a certain section of a webpage. These types of heat maps are implemented using a created software algorithm to determine and predict the attention actions of the user. Scroll Tracking: Scroll tracking heat maps are used to represent the scrolling behavior of the website's users. This helps produce visual cues to what section on the website the user spends the most time at.

… excerpt ends here. Continue reading the full article.

Illustrations

Heat map: Heat map generated from DNA microarray data reflecting gene expression values in several conditions (Eisen et al.)
Heat map generated from DNA microarray data reflecting gene expression values in several conditions (Eisen et al.)
Heat map: Shaded matrix display from Toussaint Loua (1873)
Shaded matrix display from Toussaint Loua (1873)
Heat map: A heat map showing the RF coverage of a drone detection system
A heat map showing the RF coverage of a drone detection system
Heat map: Spatial Heat Map Example: Displays atmospheric carbon across a world image with red being the highest and blue being the lowest concentration of CO2 (5 April 2019).
Spatial Heat Map Example: Displays atmospheric carbon across a world image with red being the highest and blue being the lowest concentration of CO2 (5 April 2019).
Heat map: Warming stripes graphics are one-dimensional heat maps that show global warming's yearly progress—in this example, since 1850. To represent cooler and warmer temperatures, the design's originator, climatologist Ed Hawkins, chose the 8 most saturated blues and reds of the ColorBrewer 9-class single hue palette.
Warming stripes graphics are one-dimensional heat maps that show global warming's yearly progress—in this example, since 1850. To represent cooler and warmer temperatures, the design's originator, climatologist Ed Hawkins, chose the 8 most saturated blues and reds of the ColorBrewer 9-class single hue palette.

Worked examples

Example 1 — a first encounter with Heat map

Start with the simplest possible case. Write down what Heat map claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Heat map before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Heat map ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Heat map

In research
Heat map appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry
Engineering practice reuses Heat map in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom
Heat map is common in secondary-school and first-year university syllabi. It links to neighbouring topics Bioinformatics, Color scales, Data and information visualization, so understanding it makes those chapters shorter.
In everyday life
Look for Heat map outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Heat map in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Heat map means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Heat map out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Heat map in simple terms?

A heat map (or heatmap) is a two-dimensional data visualization technique that represents the magnitude of individual values within a dataset as a color. The variation in color may be by hue or intensity.

Why does Heat map matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Heat map?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Heat map.

Tags

  • Bioinformatics
  • Color scales
  • Data and information visualization
  • Numerical function drawing
  • Thematic maps
  • Usability

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