text:
Urban design

Urban design is an approach to the design of buildings and the spaces between them that focuses on specific design processes and outcomes. In addition to designing and shaping the physical features of towns, cities, and regional spaces, urban design considers 'bigger picture' issues of economic, social and environmental value and social design. The scope of a project can range from a local street or public space to an entire city and surrounding areas. Urban designers connect the fields of architecture, landscape architecture and urban planning to better organize physical space and community environments.

Some important focuses of urban design on this page include its historical impact, paradigm shifts, its interdisciplinary nature, and issues related to urban design.

Theory
Urban design deals with the larger scale of groups of buildings, infrastructure, streets, and public spaces, entire neighbourhoods and districts, and entire cities, with the goal of making urban environments that are equitable, beautiful, performative, and sustainable.

Urban design is an interdisciplinary field that utilizes the procedures and the elements of architecture and other related professions, including landscape design, urban planning, civil engineering, and municipal engineering. It borrows substantive and procedural knowledge from public administration, sociology, law, urban geography, urban economics and other related disciplines from the social and behavioral sciences, as well as from the natural sciences. In more recent times different sub-subfields of urban design have emerged such as strategic urban design, landscape urbanism, water-sensitive urban design, and sustainable urbanism. Urban design demands an understanding of a wide range of subjects from physical geography to social science, and an appreciation for disciplines, such as real estate development, urban economics, political economy, and social theory.

Urban design theory deals primarily with the design and management of public space (i.e. the 'public environment', 'public realm' or 'public domain'), and the way public places are used and experienced. Public space includes the totality of spaces used freely on a day-to-day basis by the general public, such as streets, plazas, parks, and public infrastructure. Some aspects of privately owned spaces, such as building facades or domestic gardens, also contribute to public space and are therefore also considered by urban design theory. Important writers on urban design theory include Christopher Alexander, Peter Calthorpe, Gordon Cullen, Andrés Duany, Jane Jacobs, Jan Gehl, Allan B. Jacobs, Kevin Lynch, Aldo Rossi, Colin Rowe, Robert Venturi, William H. Whyte, Camillo Sitte, Bill Hillier (space syntax), and Elizabeth Plater-Zyberk.

History
Although contemporary professional use of the term 'urban design' dates from the mid-20th century, urban design as such has been practiced throughout history. Ancient examples of carefully planned and designed cities exist in Asia, Africa, Europe, and the Americas, and are particularly well known within Classical Chinese, Roman, and Greek cultures. Specifically, Hippodamus of Miletus was a famous ancient Greek architect and urban planner, and all around academic that is often considered to be a "father of European urban planning", and the namesake of the "Hippodamian plan", also known as the grid plan of a city layout.

European Medieval cities are often, and often erroneously, regarded as exemplars of undesigned or 'organic' city development. There are many examples of considered urban design in the Middle Ages. In England, many of the towns listed in the 9th-century Burghal Hidage were designed on a grid, examples including Southampton, Wareham, Dorset and Wallingford, Oxfordshire, having been rapidly created to provide a defensive network against Danish invaders. 12th century western Europe brought renewed focus on urbanisation as a means of stimulating economic growth and generating revenue. The burgage system dating from that time and its associated burgage plots brought a form of self-organising design to medieval towns.

Throughout history, the design of streets and deliberate configuration of public spaces with buildings have reflected contemporaneous social norms or philosophical and religious beliefs. Yet the link between designed urban space and the human mind appears to be bidirectional. Indeed, the reverse impact of urban structure upon human behaviour and upon thought is evidenced by both observational study and historical records. There are clear indications of impact through Renaissance urban design on the thought of Johannes Kepler and Galileo Galilei. Already René Descartes in his Discourse on the Method had attested to the impact Renaissance planned new towns had upon his own thought, and much evidence exists that the Renaissance streetscape was also the perceptual stimulus that had led to the development of coordinate geometry.

Early modern era
The beginnings of modern urban design in Europe are associated with the Renaissance but, especially, with the Age of Enlightenment. Spanish colonial cities were often planned, as were some towns settled by other imperial cultures. These sometimes embodied utopian ambitions as well as aims for functionality and good governance, as with James Oglethorpe's plan for Savannah, Georgia. In the Baroque period the design approaches developed in French formal gardens such as Versailles were extended into urban development and redevelopment. In this period, when modern professional specializations did not exist, urban design was undertaken by people with skills in areas as diverse as sculpture, architecture, garden design, surveying, astronomy, and military engineering. In the 18th and 19th centuries, urban design was perhaps most closely linked with surveyors engineers and architects. The increase in urban populations brought with it problems of epidemic disease, the response to which was a focus on public health, the rise in the UK of municipal engineering and the inclusion in British legislation of provisions such as minimum widths of street in relation to heights of buildings in order to ensure adequate light and ventilation.

Much of Frederick Law Olmsted's work was concerned with urban design, and the newly formed profession of landscape architecture also began to play a significant role in the late 19th century.

Modern urban design

In the 19th century, cities were industrializing and expanding at a tremendous rate. Private businesses largely dictated the pace and style of this development. The expansion created many hardships for the working poor and concern for public health increased. However, the laissez-faire style of government, in fashion for most of the Victorian era, was starting to give way to a New Liberalism. This gave more power to the public. The public wanted the government to provide citizens, especially factory workers, with healthier environments. Around 1900, modern urban design emerged from developing theories on how to mitigate the consequences of the industrial age.

The first modern urban planning theorist was Sir Ebenezer Howard. His ideas, although utopian, were adopted around the world because they were highly practical. He initiated the garden city movement. in 1898. His garden cities were intended to be planned, self-contained communities surrounded by parks. Howard wanted the cities to be proportional with separate areas of residences, industry, and agriculture. Inspired by the Utopian novel Looking Backward and Henry George's work Progress and Poverty, Howard published his book Garden Cities of To-morrow in 1898. His work is an important reference in the history of urban planning. He envisioned the self-sufficient garden city to house 32,000 people on a site of 6000 acres. He planned on a concentric pattern with open spaces, public parks, and six radial boulevards, 120 ft wide, extending from the center. When it reached full population, Howard wanted another garden city to be developed nearby. He envisaged a cluster of several garden cities as satellites of a central city of 50,000 people, linked by road and rail. His model for a garden city was first created at Letchworth and Welwyn Garden City in Hertfordshire. Howard's movement was extended by Sir Frederic Osborn to regional planning.

20th century
In the early 1900s, urban planning became professionalized. With input from utopian visionaries, civil engineers, and local councilors, new approaches to city design were developed for consideration by decision-makers such as elected officials. In 1899, the Town and Country Planning Association was founded. In 1909, the first academic course on urban planning was offered by the University of Liverpool. Urban planning was first officially embodied in the Housing and Town Planning Act of 1909 Howard's 'garden city' compelled local authorities to introduce a system where all housing construction conformed to specific building standards. In the United Kingdom following this Act, surveyor, civil engineers, architects, and lawyers began working together within local authorities. In 1910, Thomas Adams became the first Town Planning Inspector at the Local Government Board and began meeting with practitioners. In 1914, The Town Planning Institute was established. The first urban planning course in America was not established until 1924 at Harvard University. Professionals developed schemes for the development of land, transforming town planning into a new area of expertise.

In the 20th century, urban planning was changed by the automobile industry. Car-oriented design impacted the rise of 'urban design'. City layouts now revolved around roadways and traffic patterns.

In June 1928, the International Congresses of Modern Architecture (CIAM) was founded at the Chateau de la Sarraz in Switzerland, by a group of 28 European architects organized by Le Corbusier, Hélène de Mandrot, and Sigfried Giedion. The CIAM was one of many 20th century manifestos meant to advance the cause of "architecture as a social art".

Postwar
Team X was a group of architects and other invited participants who assembled starting in July 1953 at the 9th Congress of the International Congresses of Modern Architecture (CIAM) and created a schism within CIAM by challenging its doctrinaire approach to urbanism.

In 1956, the term "Urban design" was first used at a series of conferences hosted by Harvard University. The event provided a platform for Harvard's Urban Design program. The program also utilized the writings of famous urban planning thinkers: Gordon Cullen, Jane Jacobs, Kevin Lynch, and Christopher Alexander.

In 1961, Gordon Cullen published The Concise Townscape. He examined the traditional artistic approach to city design of theorists including Camillo Sitte, Barry Parker, and Raymond Unwin. Cullen also created the concept of 'serial vision'. It defined the urban landscape as a series of related spaces. Also in 1961, Jane Jacobs published The Death and Life of Great American Cities. She critiqued the modernism of CIAM (International Congresses of Modern Architecture). Jacobs also claimed crime rates in publicly owned spaces were rising because of the Modernist approach of 'city in the park'. She argued instead for an 'eyes on the street' approach to town planning through the resurrection of main public space precedents (e.g. streets, squares).

In the same year, Kevin Lynch published The Image of the City. He was seminal to urban design, particularly with regards to the concept of legibility. He reduced urban design theory to five basic elements: paths, districts, edges, nodes, landmarks. He also made the use of mental maps to understand the city popular, rather than the two-dimensional physical master plans of the previous 50 years.

Other notable works:

 * Architecture of the City by Aldo Rossi (1966)
 * Learning from Las Vegas by Robert Venturi and Denise Scott Brown (1972)
 * Collage City by Colin Rowe (1978)
 * The Next American Metropolis by Peter Calthorpe (1993)
 * The Social Logic of Space by Bill Hillier and Julienne Hanson (1984)

The popularity of these works resulted in terms that become everyday language in the field of urban planning. Aldo Rossi introduced 'historicism' and 'collective memory' to urban design. Rossi also proposed a 'collage metaphor' to understand the collection of new and old forms within the same urban space. Peter Calthorpe developed a manifesto for sustainable urban living via medium-density living. He also designed a manual for building new settlements in his concept of Transit Oriented Development (TOD). Bill Hillier and Julienne Hanson introduced Space Syntax to predict how movement patterns in cities would contribute to urban vitality, anti-social behaviour, and economic success. 'Sustainability', 'livability', and 'high quality of urban components' also became commonplace in the field.

Current trends
Today, urban design seeks to create sustainable urban environments with long-lasting structures, buildings, and overall livability. Walkable urbanism is another approach to practice that is defined within the Charter of New Urbanism. It aims to reduce environmental impacts by altering the built environment to create smart cities that support sustainable transport. Compact urban neighborhoods encourage residents to drive less. These neighborhoods have significantly lower environmental impacts when compared to sprawling suburbs. To prevent urban sprawl, Circular flow land use management was introduced in Europe to promote sustainable land use patterns.

As a result of the recent New Classical Architecture movement, sustainable construction aims to develop smart growth, walkability, architectural tradition, and classical design. It contrasts with modernist and globally uniform architecture. In the 1980s, urban design began to oppose the increasing solitary housing estates and suburban sprawl. Managed Urbanisation with the view to making the urbanising process completely culturally and economically, and environmentally sustainable, and as a possible solution to the urban sprawl, Frank Reale has submitted an interesting concept of Expanding Nodular Development (E.N.D.) that integrates many urban designs and ecological principles, to design and build smaller rural hubs with high-grade connecting freeways, rather than adding more expensive infrastructure to existing big cities and the resulting congestion.

Paradigm shifts
Throughout the young existence of the Urban Design discipline, many paradigm shifts have occurred that have affected the trajectory of the field regarding theory and practice. These paradigm shifts cover multiple subject areas outside of the traditional design disciplines.

 * Team 10 - The first major paradigm shift was the formation of Team 10 out of CIAM, or the Congres Internationaux d'Architecture Moderne. They believed that Urban Design should introduce ideas of 'Human Association', which pivots the design focus from the individual patron to concentrating on the collective urban population.
 * The Brundtland Report and Silent Spring - Another paradigm shift was the publication of the Brundtland Report and the book Silent Spring by Rachel Carson. These writings introduced the idea that human settlements could have detrimental impacts on ecological processes, as well as human health, which spurred a new era of environmental awareness in the field.
 * The Planner's Triangle - The Planner's Triangle, created by Scott Cambell, emphasized three main conflicts in the planning process. This diagram exposed the complex relationships between Economic Development, Environmental Protection, and Equity and Social Justice. For the first time, the concept of Equity and Social Justice was considered as equally important as Economic Development and Environmental Protection within the design process.
 * Death of Modernism (Demolition of Pruitt Igoe) - Pruitt Igoe was a spatial symbol and representation of Modernist theory regarding social housing. In its failure and demolition, these theories were put into question and many within the design field considered the era of Modernism to be dead.
 * Neoliberalism & the election of Reagan - The election of President Reagan and the rise of Neoliberalism affected the Urban Design discipline because it shifted the planning process to emphasize capitalistic gains and spatial privatization. Inspired by the trickle-down approach of Reaganomics, it was believed that the benefits of a capitalist emphasis within design would positively impact everyone. Conversely, this led to exclusionary design practices and to what many consider as "the death of public space".
 * Right to the City - The spatial and political battle over our citizens' rights to the city has been an ongoing one. David Harvey, along with Dan Mitchell and Edward Soja, discussed rights to the city as a matter of shifting the historical thinking of how spatial matter was determined in a critical form. This change of thinking occurred in three forms: ontologically, sociologically, and the combination of this socio-spatial dialect. Together the aim shifted to be able to measure what matters in a socio-spatial context.
 * Black Lives Matter (Ferguson) - The Black Lives Matter movement challenged design thinking because it emphasized the injustices and inequities suffered by people of color in urban space, as well as emphasized their right to public space without discrimination and brutality. It claims that minority groups lack certain spatial privileges and that this deficiency can result in matters of life and death. In order to reach an equitable state of urbanism, there needs to be equal identification of socio-economic lives within our urbanscapes.

New approaches
There have been many different theories and approaches applied to the practice of urban design.

New Urbanism is an approach that began in the 1980s as a place-making initiative to combat suburban sprawl. Its goal is to increase density by creating compact and complete towns and neighborhoods. The 10 principles of new urbanism are walkability, connectivity, mixed-use and diversity, mixed housing, quality architecture and urban design, traditional neighborhood structure, increased density, smart transportation, sustainability, and quality of life. New urbanism and the developments that it has created are sources of debates within the discipline, primarily with the landscape urbanist approach but also due to its reproduction of idyllic architectural tropes that do not respond to the context. Andres Duany, Elizabeth Plater-Zyberk, Peter Calthorpe, and Jeff Speck are all strongly associated with New Urbanism and its evolution over the years.

Landscape Urbanism is a theory that first surfaced in the 1990s, arguing that the city is constructed of interconnected and ecologically rich horizontal field conditions, rather than the arrangement of objects and buildings. Charles Waldheim, Mohsen Mostafavi, James Corner, and Richard Weller are closely associated with this theory. Landscape urbanism theorises sites, territories, ecosystems, networks, and infrastructures through landscape practice according to Corner, while applying a dynamic concept to cities as ecosystems that grow, shrink or change phases of development according to Waldheim.

Everyday Urbanism is a concept introduced by Margaret Crawford and influenced by Henry Lefebvre that describes the everyday lived experience shared by urban residents including commuting, working, relaxing, moving through city streets and sidewalks, shopping, buying, eating food, and running errands. Everyday urbanism is not concerned with aesthetic value. Instead, it introduces the idea of eliminating the distance between experts and ordinary users and forces designers and planners to contemplate a 'shift of power' and address social life from a direct and ordinary perspective.

Tactical Urbanism (also known as DIY Urbanism, Planning-by-Doing, Urban Acupuncture, or Urban Prototyping) is a city, organizational, or citizen-led approach to neighborhood-building that uses short-term, low-cost, and scalable interventions and policies to catalyze long term change.

Top-up Urbanism is the theory and implementation of two techniques in urban design: top-down and bottom-up. Top-down urbanism is when the design is implemented from the top of the hierarchy - normally the government or planning department. Bottom-up or grassroots urbanism begins with the people or the bottom of the hierarchy. Top-up means that both methods are used together to make a more participatory design, so it is sure to be comprehensive and well regarded in order to be as successful as possible.

Infrastructural Urbanism is the study of how the major investments that go into making infrastructural systems can be leveraged to be more sustainable for communities. Instead of the systems being solely about efficiency in both cost and production, infrastructural urbanism strives to utilize these investments to be more equitable for social and environmental issues as well. Linda Samuels is a designer investigating how to accomplish this change in infrastructure in what she calls "next-generation infrastructure" which is "multifunctional; public; visible; socially productive; locally specific, flexible, and adaptable; sensitive to the eco-economy; composed of design prototypes or demonstration projects; symbiotic; technologically smart; and developed collaboratively across disciplines and agencies".

Sustainable Urbanism is the study from the 1990s of how a community can be beneficial for the ecosystem, the people, and the economy for which it is associated. It is based on Scott Campbell's planner's triangle which tries to find the balance between economy, equity, and the environment. Its main concept is to try and make cities as self-sufficient as possible while not damaging the ecosystem around them, today with an increased focus on climate stability. A key designer working with sustainable urbanism is Douglas Farr.

Feminist Urbanism is the study and critique of how the built environment affects genders differently because of patriarchal social and political structures in society. Typically, the people at the table making design decisions are men, so their conception about public space and the built environment relates to their life perspectives and experiences, which do not reflect the same experiences of women or children. Dolores Hayden is a scholar who has researched this topic from 1980 to the present day. Hayden's writing says, “when women, men, and children of all classes and races can identify the public domain as the place where they feel most comfortable as citizens, Americans will finally have homelike urban space.”

Educational Urbanism is an emerging discipline, at the crossroads of urban planning, educational planning, and pedagogy. An approach that tackles the notion that economic activities, the need for new skills at the workplace, and the spatial configuration of the workplace rely on the spatial reorientation in the design of educational spaces and the urban dimension of educational planning.

Black Urbanism is an approach in which black communities are active creators, innovators, and authors of the process of designing and creating the neighborhoods and spaces of the metropolitan areas they have done so much to help revive over the past half-century. The goal is not to build black cities for black people but to explore and develop the creative energy that exists in so-called black areas: that has the potential to contribute to the sustainable development of the whole city.

Debates in urbanism
Underlying the practice of urban design are the many theories about how to best design the city. Each theory makes a unique claim about how to effectively design thriving, sustainable urban environments. Debates over the efficacy of these approaches fill the urban design discourse. Landscape Urbanism and New Urbanism are commonly debated as distinct approaches to addressing suburban sprawl. While Landscape Urbanism proposes landscape as the basic building block of the city and embraces horizontality, flexibility, and adaptability, New Urbanism offers the neighborhood as the basic building block of the city and argues for increased density, mixed uses, and walkability. Opponents of Landscape Urbanism point out that most of its projects are urban parks, and as such, its application is limited. Opponents of New Urbanism claim that its preoccupation with traditional neighborhood structures is nostalgic, unimaginative, and culturally problematic. Everyday Urbanism argues for grassroots neighborhood improvements rather than master-planned, top-down interventions. Each theory elevates the roles of certain professions in the urban design process, further fueling the debate. In practice, urban designers often apply principles from many urban design theories. Emerging from the conversation is a universal acknowledgement of the importance of increased interdisciplinary collaboration in designing the modern city.

Urban design as an integrative profession

Urban designers work with architects, landscape architects, transportation engineers, urban planners, and industrial designers to reshape the city. Cooperation with public agencies, authorities and the interests of nearby property owners is necessary to manage public spaces. Users often compete over the spaces and negotiate across a variety of spheres. Input is frequently needed from a wide range of stakeholders. This can lead to different levels of participation as defined in Arnstein's Ladder of Citizen Participation.

While there are some professionals who identify themselves specifically as urban designers, a majority have backgrounds in urban planning, architecture, or landscape architecture. Many collegiate programs incorporate urban design theory and design subjects into their curricula. There is an increasing number of university programs offering degrees in urban design at the post-graduate level.

Urban design considers:

 * Pedestrian zones
 * Incorporation of nature within a city
 * Aesthetics
 * Urban structure – arrangement and relation of business and people
 * Urban typology, density, and sustainability - spatial types and morphologies related to the intensity of use, consumption of resources, production, and maintenance of viable communities
 * Accessibility – safe and easy transportation
 * Legibility and wayfinding – accessible information about travel and destinations
 * Animation – Designing places to stimulate public activity
 * Function and fit – places support their varied intended uses
 * Complimentary mixed uses – Locating activities to allow constructive interaction between them
 * Character and meaning – Recognizing differences between places
 * Order and incident – Balancing consistency and variety in the urban environment
 * Continuity and change – Locating people in time and place, respecting heritage and contemporary culture
 * Civil society – people are free to interact as civic equals, important for building social capital
 * Participation/engagement – including people in the decision-making process can be done at many different scales.

Relationships with other related disciplines
The original urban design was thought to be separated from architecture and urban planning. Urban Design has developed to a certain extent, and comes from the foundation of engineering. In Anglo-Saxon countries, it is often considered as a branch under the architecture, urban planning, and landscape architecture and limited as the construction of the urban physical environment. However Urban Design is more integrated into the social science-based, cultural, economic, political, and other aspects. Not only focus on space and architectural group, but also look at the whole city from a broader and more holistic perspective to shape a better living environment. Compared to architecture, the spatial and temporal scale of Urban Design processing is much larger. It deals with neighborhoods, communities, and even the entire city.

The urban design education
The University of Liverpool's Department of Civic Design is the first urban design school in the world founded in 1909. Following the 1956 Urban Design conference, Harvard University established the first graduate program with urban design in its title, The Master of Architecture in Urban Design, although as a subject taught in universities its history in Europe is far older. Urban design programs explore the built environment from diverse disciplinary backgrounds and points of view. The pedagogically innovative combination of interdisciplinary studios, lecture courses, seminars, and independent study creates an intimate and engaging educational atmosphere in which students thrive and learn. Soon after in 1961, Washington University in St. Louis founded their Master of Urban Design program. Today, twenty urban design programs exist in the United States:

 * Andrews University, Berrien Springs, MI
 * Clemson University - Charleston, SC
 * Columbia Graduate School of Architecture, Planning and Preservation - New York, NY
 * City College of New York - New York, NY
 * Estopinal College of Architecture and Planning at Ball State University - Muncie, IN
 * Georgia Institute of Technology College of Design - Atlanta, GA
 * Harvard Graduate School of Design - Cambridge, MA
 * Iowa State University - Ames, IA
 * New York Institute of Technology - New York, NY
 * Notre Dame School of Architecture - Notre Dame, IN
 * Pratt Institute - Brooklyn, NY
 * Sam Fox School of Design & Visual Arts at Washington University in St. Louis - St. Louis, MO
 * Savannah College of Art and Design - Savannah, GA
 * Stuart Weitzman School of Design at University of Pennsylvania - Philadelphia, PA
 * Taubman College of Architecture and Urban Planning at University of Michigan - Ann Arbor, MI
 * University of California, Berkeley - Berkeley, CA
 * University of Colorado Denver - Denver, CO
 * University of Maryland - College Park, MD
 * University of Miami - Miami, FL
 * Stuart Weitzman School of Design at University of Pennsylvania - Philadelphia, PA
 * University of Texas at Austin School of Architecture - Austin, TX
 * University of North Carolina at Charlotte - Charlotte, NC

In the United Kingdom, Master's programmes in Urban Design at University of Manchester or University of Sheffield and Cardiff University or London South Bank University and City Design at the Royal College of Art or Queen's University Belfast are offered.

Issues
The field of urban design holds enormous potential for helping us address today's biggest challenges: an expanding population, mass urbanization, rising inequality, and climate change. In its practice as well as its theories, urban design attempts to tackle these pressing issues. As climate change progresses, urban design can mitigate the results of flooding, temperature changes, and increasingly detrimental storm impacts through a mindset of sustainability and resilience. In doing so, the urban design discipline attempts to create environments that are constructed with longevity in mind, such as zero-carbon cities. Cities today must be designed to minimize resource consumption, waste generation, and pollution while also withstanding the unprecedented impacts of climate change. To be truly resilient, our cities need to be able to not just bounce back from a catastrophic climate event but to bounce forward to an improved state.

Another issue in this field is that it is often assumed that there are no mothers of planning and urban design. However, this is not the case, many women have made proactive contributions to the field, including the work of Mary Kingsbury Simkhovitch, Florence Kelley, and Lillian Wald, to name a few of whom were prominent leaders in the City Social movement. The City Social was a movement that steamed between the commonly known City Practical and City Beautiful movements. It was a movement mainly concerning lay with the economic and social equalities regarding urban issues.

Justice is and will always be a key issue in urban design. As previously mentioned, past urban strategies have caused injustices within communities incapable of being remedied via simple means. As urban designers tackle the issue of justice, they often are required to look at the injustices of the past and must be careful not to overlook the nuances of race, place, and socioeconomic status in their design efforts. This includes ensuring reasonable access to basic services, transportation, and fighting against gentrification and the commodification of space for economic gain. Organizations such as the Divided Cities Initiatives at Washington University in St. Louis and the Just City Lab at Harvard work on promoting justice in urban design.

Until the 1970s, the design of towns and cities took little account of the needs of people with disabilities. At that time, disabled people began to form movements demanding recognition of their potential contribution if social obstacles were removed. Disabled people challenged the 'medical model' of disability which saw physical and mental problems as an individual 'tragedy' and people with disabilities as 'brave' for enduring them. They proposed instead a 'social model' which said that barriers to disabled people result from the design of the built environment and attitudes of able-bodied people. 'Access Groups' were established composed of people with disabilities who audited their local areas, checked planning applications, and made representations for improvements. The new profession of 'access officer' was established around that time to produce guidelines based on the recommendations of access groups and to oversee adaptations to existing buildings as well as to check on the accessibility of new proposals. Many local authorities now employ access officers who are regulated by the Access Association. A new chapter of the Building Regulations (Part M) was introduced in 1992. Although it was beneficial to have legislation on this issue the requirements were fairly minimal but continue to be improved with ongoing amendments. The Disability Discrimination Act 1995 continues to raise awareness and enforce action on disability issues in the urban environment.

The issue of walkability has gained prominence in recent years, not only with the concerns of the aforementioned climate change, but also the health outcomes of residents. Car-centric urban design has an invariably negative effect on such outcomes. With proximity to internal combustion engines, residents tend to suffer from dangerous levels of air pollution which lead to cardiovascular complications ranging from the acute, in hypertension and alterations in heart rate, and the chronic, the outright development of atherosclerosis. More people die from air pollution each year than from car accidents. This issue has been used to fuel movements for alternative forms of long to mid range transportation such as trains and bicycles, with walking as the primary means of short-range travel. This would bring benefits from two simultaneous avenues. The physical activity from walking, and the lack of particulate matter (carbon dioxide, sulfur dioxide, nitrogen dioxide, etc.) has shown to alleviate and lower the risk of many maladies such as diabetes, hypertension and cardiovascular disease. Physical activity levels from walking are closely related to the abundance of open public spaces, commercial shops, greenery, among others. These attributes also have been stated to contribute to stronger social and emotional health as the open public spaces facilitate more social interaction within communities. This issue is most prevalent in the United States, where the rise of neoliberalism directly and intentionally caused the car-centric infrastructure.

title: Urban design


text:
Museum of Science and Industry (Chicago)

The Griffin Museum of Science and Industry (MSI), formerly known as the Museum of Science and Industry, is a science museum located in Chicago, Illinois, in Jackson Park, in the Hyde Park neighborhood between Lake Michigan and The University of Chicago. It is housed in the Palace of Fine Arts from the 1893 World's Columbian Exposition. Initially endowed by Julius Rosenwald, the Sears, Roebuck and Company president and philanthropist, it was supported by the Commercial Club of Chicago and opened in 1933 during the Century of Progress Exposition. It was renamed for benefactor and financier Kenneth C. Griffin on May 19, 2024.

Among the museum's exhibits are the command module of Apollo 8, a full-size replica coal mine, GS U-505 captured during World War II, the first diesel-powered streamlined stainless-steel passenger train (Pioneer Zephyr), and a 3500 sqft model railroad, as well as, exhibits on flight, computers, physics, the brain and chemistry and mechanics of the human body.

History
The Palace of Fine Arts was built for the 1893 World's Columbian Exposition and was designed by Charles B. Atwood for D. H. Burnham & Co. During the fair, the palace displayed paintings, prints, drawing, sculpture, and metalwork from around the world. Unlike the other "White City" buildings, it was constructed with a brick substructure under its plaster facade.

After the World's Fair, the palace initially housed the Columbian Museum, largely displaying collections left from the fair, which evolved into the Field Museum of Natural History. When the Field Museum moved to a new building five miles north in the Near South Side in 1920, the palace was left vacant.

School of the Art Institute of Chicago professor Lorado Taft led a public campaign to restore the building and turn it into another art museum, one devoted to sculpture. The South Park Commissioners (now part of the Chicago Park District) won approval in a referendum to sell $5 million in bonds to pay for restoration costs, hoping to turn the building into a sculpture museum, a technical trade school, and other things. However, after a few years, the building was selected as the site for a new science museum.

At this time, the Commercial Club of Chicago was interested in establishing a science museum in Chicago. Julius Rosenwald, the Sears, Roebuck and Company president and philanthropist, energized his fellow club members by pledging to pay $3 million towards the cost of converting the Palace of Fine Arts (Rosenwald eventually contributed more than $5 million to the project). During its conversion into the MSI, the building's exterior was re-cast in limestone to retain its 1893 Beaux Arts look. The interior was replaced with a new one in Art Moderne style designed by Alfred P. Shaw.

Rosenwald established the museum organization in 1926 but declined to have his name on the building. For the first few years, the museum was often called the Rosenwald Industrial Museum. In 1928, the name of the museum was officially changed to the Museum of Science and Industry. Rosenwald's vision was to create a museum in the style of the Deutsches Museum in Munich, which he had visited in 1911 while in Germany with his family.

Sewell Avery, another businessman, had supported the museum within the Commercial Club and was selected as its first president of the board of directors. The museum conducted a nationwide search for the first director. MSI's Board of Directors selected Waldemar Kaempffert, then the science editor of The New York Times, because he shared Rosenwald's vision.

He assembled the museum's curatorial staff and directed the organization and construction of the exhibits. In order to prepare the museum, Kaempffert and his staff visited the Deutsches Museum in Munich, the Science Museum in Kensington, and the Technical Museum in Vienna, all of which served as models. Kaempffert was instrumental in developing close ties with the science departments of the University of Chicago, which supplied much of the scholarship for the exhibits. Kaempffert resigned in early 1931 amid growing disputes with the second president of the board of directors; they disagreed over the objectivity and neutrality of the exhibits and Kaempffert's management of the staff.

The new Museum of Science and Industry opened to the public in three stages between 1933 and 1940. The first opening ceremony took place during the Century of Progress Exposition. Two of the museum's presidents, a number of curators and other staff members, and exhibits came to MSI from the Century of Progress event.

For years, visitors entered the museum through its original main entrance, but an increasing volume of visitors required larger facilities. The newer main entrance is a structure detached from the main museum building, through which visitors descend into an underground area and then re-ascend into the main building, similar to the Louvre Pyramid in Paris.

In 1992, due to increased attendance, the museum started planning its underground parking lot, located in three levels below the front lawn. Construction of the underground parking lot was finished in July 1998.

For over 55 years, admission to the MSI was free, although some exhibits such as the Coal Mine and U-505 required small fees. General entrance fees were first charged in the early 1990s, with general admission rates increasing from $13 in 2008 to $18 in 2015 and $25.95 in 2024. Many "free days"—for Illinois residents only—are offered throughout the year.

On October 3, 2019, the museum announced that it intended to change its name to the Kenneth C. Griffin Museum of Science and Industry, after a donation of $125 million from billionaire Kenneth C. Griffin. It is the largest single gift in the museum's history, effectively doubling its endowment. However, president and chief executive officer David Mosena said the formal name change could take some time, due to the complexity of the process. He also said part of the gift will go into funding "a state-of-the-art digital gallery and performance space that will be the only experience of its kind in North America." Chevy Humphrey became president and CEO of the private, non-profit museum in January 2021. The new name was officially unveiled on May 19, 2024.

Exhibits
The museum has over 2,000 exhibits, displayed in 75 major halls. The museum has several major permanent exhibits. Access to several of the exhibits (including the Coal Mine and U-505) requires the payment of an additional fee.

Entry Hall
The first diesel-powered, streamlined stainless-steel train, the Pioneer Zephyr, is on permanent display in the Great Hall, renamed the Entry Hall in 2008. The train was once displayed outdoors, but it was restored and placed in the former Great Hall during the construction of the museum's underground parking lot.

U-505
GS U-505 is one of just six German submarines captured by the Allies during World War II, and, since its arrival in 1954, the only one on display in the Western Hemisphere. The U-boat was newly restored beginning in 2004 after 50 years of being displayed outdoors, and was then moved indoors as The New U-505 Experience on June 5, 2005. Displayed in an underground shed, it remains as a popular exhibit for visitors, as well as a memorial to all the casualties of the Battle of the Atlantic during World War II. Guided tours of the submarine are offered for an additional fee. Near the U-505 there is both a Mold-A-Rama machine and a penny flattening device. Both have U-505 designs.

Henry Crown Space Center
MSI's Henry Crown Space Center includes the Apollo 8 spacecraft, which flew the first mission beyond low earth orbit to the Moon, enabling its crew, Frank Borman, James Lovell and William Anders, to become the first human beings to see the Earth as a whole, as well as becoming the first to view the Moon up close (as well as the first to view its far side). Other exhibits include Scott Carpenter's Mercury-Atlas 7 spacecraft, a lunar module trainer, and a SpaceX Dragon 1 cargo spacecraft.

Located in the Henry Crown Space Center is a domed theater, considered to be the only domed theater in Chicago. The screen of the theater is made of perforated aluminum, allowing the speakers mounted behind the screen to be heard throughout the theater.

FarmTech
The FarmTech exhibit showcases modern agricultural techniques and how farmers use modern technology like GPS systems to improve work on the farm, and includes a tractor and a combine harvester from John Deere. The exhibit also showcases a greenhouse, a mock up of a kitchen showcasing how much of the food we eat comes from soybeans, and how we use cows, from energy to what we drink.

Other
A transportation gallery is located on the museum's west wing, containing models of "Ships Through the Ages" and several historic racing cars.

Future Energy Chicago shows alternative resources, housing developments, and the future of Chicago. The exhibit requires an additional fee.

Some areas in the museum aim for younger children, including the Swiss Jollyball, the world's largest pinball machine built by a British man from Switzerland using nothing but salvaged junk; the Idea Factory, a toddler water table play area; the Eye Spy gallery; and the former Circus exhibit, featuring animated dioramas of a miniature circus, as well as a shadow garden and several funhouse mirrors. The circus exhibit was closed in September 2022 and is now replaced with a Mold-a-Rama exhibit.

Silent-film star and stock-market investor Colleen Moore's Fairy Castle "doll's house" is on display.

Numbers in Nature: A Mirror Maze
Numbers in Nature: A Mirror Maze contains an interactive theater and stations to learn about patterns in nature, including the Golden Ratio, spirals, fractal branching, and Voronoi patterns.

It also contains a mirror maze to help emphasize the geometric patterns that can be utilized.

Transportation Gallery
The Transportation Gallery contains several permanent exhibits.

The Great Train Story is a 3500 sqft HO-scale model railroad and recounts the story of transportation from Chicago to Seattle.

The museum includes a replica of Stephenson's Rocket, which was the first steam locomotive to exceed 25 miles per hour.

The 999 Empire State Express steam locomotive was alleged to be the first vehicle to exceed 100 mph in 1893, although no reliable measurement ever took place (and such a speed was likely impossible). Designed to win the battle of express trains to the World's Columbian Exhibition, it was donated to the museum by the New York Central in 1962. The locomotive was located outside the museum until 1993, when extensive restoration took place and it was moved indoors as an exhibit in the Transportation Zone.

A replica of the Wright Brothers first airplane, the Wright Flyer, is on display.

Two World War II warplanes are also exhibited. Both were donated by the British government: a German Ju 87 R-2/Trop. Stuka divebomber—one of only two intact Stukas left in the world—and a British Supermarine Spitfire. Also on display is the museum's Travel Air Type R Mystery Ship, nicknamed "Texaco 13", which set many world records in flying.

Take Flight features the first Boeing 727 jet plane in commercial service, donated by United Airlines, with one wing removed and holes cut on the fuselage to facilitate visitor access.

Science Storms
In March 2010, the museum opened Science Storms in the Allstate Court, as a permanent exhibit. This multilevel exhibit features a 40 ft water vapor tornado, tsunami tank, Tesla coil, heliostat system, and a Wimshurst machine built by James Wimshurst in the late 19th century. Also housed are Newton's Cradle, the color spectrum, and Foucault pendulum. All artifacts allow guests to explore the physics and chemistry of the natural world.

Genetics: Decoding Life
In keeping with Rosenwald's vision, many of the exhibits are interactive. Genetics: Decoding Life, looks at how genetics affect human and animal development as well as containing a chick hatchery composed of an incubator where baby chickens hatch from their eggs and a chick pen for those that have already hatched, as well as housing genetically modified frogs, mice, and drought resistant plants.

The chick hatchery has been part of the museum since 1956. About 20 chicks are hatched a day, around 140 hatch in a week, and up to 8000 hatch in a year. A week after emerging from their shells, the chicks are sent to the Lincoln Park Zoo to be fed to various animals, including lions, crocodiles, snakes, vultures, owls and tigers. This partnership between the museum and the zoo has been operating for decades, with about 7000 chicks being sent to the zoo each year. Some of the chicks hatched are of the Java species of chicken, and these chicks are sent to a farm in La Fox, Illinois that works to preserve the rare breed. There have been numerous efforts to shut down the exhibit, as early as 1998 and as recent as 2017.

ToyMaker 3000
ToyMaker 3000, is a working assembly line which lets visitors order a toy top and watch as it is made. The interactive Fab Lab MSI is intended as an interactive lab where members can "build anything".

Coal Mine
The Coal Mine re-creates a working deep-shaft, bituminous coal mine inside the museum's Central Pavilion, using original equipment from Old Ben #17, circa 1933. It is one of the oldest exhibits at the museum. In this unique exhibit, visitors go underground and ride a mine train to different parts of the mine and learn the basics of its operation. The experience takes around 30 minutes and requires an additional fee.

Yesterday's Main Street
Yesterday's Main Street is a mock-up of a Chicago street from the early 20th century, complete with a cobblestone roadway, old-fashioned light fixtures, fire hydrants, and several shops, including the precursors to several Chicago-based businesses. Included are: • The Berghoff restaurant

• Chicago Post Office

• Commonwealth Edison

• Gossard Corset Shop

• Jewel Tea Company grocery

• Jenner and Block Law office

• Lytton's Clothing Store

• Dr. John B. Murphy's office

• The Nickelodeon Cinema

• Chas. A. Stevens & Co.

• Walgreens Drug Company

Unlike the other shops, The Nickelodeon Cinema can be entered and is functional, and plays silent films throughout the day

Other
In spring 2013, the Art of the Bicycle exhibit opened, showcasing the history of the bicycle and how modern bikes continue to evolve.

Reusable City focuses on recycling and other methods that could cut down harmful pollution and especially climate change and the Regenstein Hall of Science, containing a giant periodic table of the elements. Other main level exhibits include: Fast Forward, which features some aspects of how technology will change in the future; Earth Revealed, featuring a "Science on a Sphere" holographic globe; and a "Whispering Gallery".

YOU! The Experience
The museum is also known for unique and quirky permanent exhibits, such as a walk-through model of the human heart, which was removed in 2009 for the construction of ''YOU! the Experience'', which replaced it with a 13 ft, interactive, 3D heart. Also well known are the "Body Slices" (two cadavers exhibited in 1/2 in slices) in the exhibit.

Other
Several US Navy warship models are also on display in the museum, and flight simulators including of the new F-35 Lightning II are featured.

Former exhibits
An F-104 Starfighter on loan to MSI from the US Air Force since 1978 was sent to the Mid-America Air Museum in Liberal, Kansas, in 1993.

In March 1995, Santa Fe Steam Locomotive 2903 was moved from outside the museum to the Illinois Railway Museum.

Telefun Town, a hall dedicated to the wonders of telephone communication, sponsored by the company then known as the Bell Telephone Company, no longer exists.

Exhibitions
In addition to its three floors of standing exhibits, the museum hosts temporary and traveling exhibitions. Exhibitions last for five months or less and usually require a separate paid admission fee. Exhibitions at MSI have included Titanic: The Exhibition, which was the largest display of relics from the wreck of RMS Titanic; Gunther von Hagens' Body Worlds, a view into the human body through use of plastinated human specimens; Game On, which featured the history and culture of video games; Leonardo da Vinci: Man, Inventor, Genius; CSI: The Experience; Robots Like Us; City of the Future; Star Wars: Where Science Meets Imagination; The Glass Experience; Harry Potter: The Exhibition; Robot Revolution, which was sponsored by Google and featured numerous hands-on demonstrations and advice from experts for prospective future robot scientists and engineers; and four installments of Smart Home: Green + Wired, featuring the work of green architect Michelle Kaufmann. The Science Behind Pixar exhibit opened May 24, 2018. The Wired to Wear exhibit opened on March 21, 2019. The touring exhibit Marvel: Universe of Super Heroes officially opened to the public on March 7, 2021.

Yearly, from late November to early January, the museum hosts its Christmas Around the World and Holidays of Light exhibit, featuring Christmas trees from different cultures from around the world. Started in 1942 with just one tree to honor soldiers fighting in World War Two, the tradition spawned into more than 50 trees.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/


text:
Multivariate normal distribution

In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions. One definition is that a random vector is said to be k-variate normally distributed if every linear combination of its k components has a univariate normal distribution. Its importance derives mainly from the multivariate central limit theorem. The multivariate normal distribution is often used to describe, at least approximately, any set of (possibly) correlated real-valued random variables, each of which clusters around a mean value.

Notation and parametrization
The multivariate normal distribution of a k-dimensional random vector $$\mathbf{X} = (X_1,\ldots,X_k)^{\mathrm T}$$ can be written in the following notation:

\mathbf{X}\ \sim\ \mathcal{N}(\boldsymbol\mu,\, \boldsymbol\Sigma), $$ or to make it explicitly known that X is k-dimensional,

\mathbf{X}\ \sim\ \mathcal{N}_k(\boldsymbol\mu,\, \boldsymbol\Sigma), $$ with k-dimensional mean vector
 * $$ \boldsymbol\mu = \operatorname{E}[\mathbf{X}] = ( \operatorname{E}[X_1], \operatorname{E}[X_2], \ldots, \operatorname{E}[X_k] ) ^ \mathrm{T}, $$

and $$k \times k$$ covariance matrix
 * $$ \Sigma_{i,j} = \operatorname{E} [(X_i - \mu_i)( X_j - \mu_j)] = \operatorname{Cov}[X_i, X_j] $$

such that $$1 \le i \le k$$ and $$1 \le j \le k$$. The inverse of the covariance matrix is called the precision matrix, denoted by $$\boldsymbol{Q}=\boldsymbol\Sigma^{-1}$$.

Standard normal random vector
A real random vector $$\mathbf{X} = (X_1,\ldots,X_k)^{\mathrm T}$$ is called a standard normal random vector if all of its components $$X_i$$ are independent and each is a zero-mean unit-variance normally distributed random variable, i.e. if $$X_i \sim\ \mathcal{N}(0,1)$$ for all $$i=1\ldots k$$.

Centered normal random vector
A real random vector $$\mathbf{X} = (X_1,\ldots,X_k)^{\mathrm T}$$ is called a centered normal random vector if there exists a deterministic $$k \times \ell$$ matrix $$\boldsymbol{A}$$ such that $$\boldsymbol{A} \mathbf{Z}$$ has the same distribution as $$\mathbf{X}$$ where $$\mathbf{Z}$$ is a standard normal random vector with $$\ell$$ components.

Normal random vector
A real random vector $$\mathbf{X} = (X_1,\ldots,X_k)^{\mathrm T}$$ is called a normal random vector if there exists a random $$\ell$$-vector $$\mathbf{Z}$$, which is a standard normal random vector, a $$k$$-vector $$\mathbf{\mu}$$, and a $$k \times \ell$$ matrix $$\boldsymbol{A}$$, such that $$\mathbf{X}=\boldsymbol{A} \mathbf{Z} + \mathbf{\mu}$$.

Formally:

Here the covariance matrix is $$\boldsymbol\Sigma = \boldsymbol{A} \boldsymbol{A}^{\mathrm T}$$.

In the degenerate case where the covariance matrix is singular, the corresponding distribution has no density; see the section below for details. This case arises frequently in statistics; for example, in the distribution of the vector of residuals in the ordinary least squares regression. The $$X_i$$ are in general not independent; they can be seen as the result of applying the matrix $$\boldsymbol{A}$$ to a collection of independent Gaussian variables $$\mathbf{Z}$$.

Equivalent definitions
The following definitions are equivalent to the definition given above. A random vector $$\mathbf{X} = (X_1, \ldots, X_k)^\mathrm{T}$$ has a multivariate normal distribution if it satisfies one of the following equivalent conditions. \varphi_\mathbf{X}(\mathbf{u}) = \exp\Big( i\mathbf{u}^\mathrm{T}\boldsymbol\mu - \tfrac{1}{2} \mathbf{u}^\mathrm{T}\boldsymbol\Sigma \mathbf{u} \Big). $$
 * Every linear combination $$Y=a_1 X_1 + \cdots + a_k X_k$$ of its components is normally distributed. That is, for any constant vector $$\mathbf{a} \in \mathbb{R}^k$$, the random variable $$Y=\mathbf{a}^{\mathrm T}\mathbf{X}$$ has a univariate normal distribution, where a univariate normal distribution with zero variance is a point mass on its mean.
 * There is a k-vector $$\mathbf{\mu}$$ and a symmetric, positive semidefinite $$k \times k$$ matrix $$\boldsymbol\Sigma$$, such that the characteristic function of $$\mathbf{X}$$ is $$

The spherical normal distribution can be characterised as the unique distribution where components are independent in any orthogonal coordinate system.

Non-degenerate case
The multivariate normal distribution is said to be "non-degenerate" when the symmetric covariance matrix $$\boldsymbol\Sigma$$ is positive definite. In this case the distribution has density

where $${\mathbf x}$$ is a real k-dimensional column vector and $$|\boldsymbol\Sigma|\equiv \det\boldsymbol\Sigma$$ is the determinant of $$\boldsymbol\Sigma$$, also known as the generalized variance. The equation above reduces to that of the univariate normal distribution if $$\boldsymbol\Sigma$$ is a $$1 \times 1$$ matrix (i.e. a single real number).

The circularly symmetric version of the complex normal distribution has a slightly different form.

Each iso-density locus — the locus of points in k-dimensional space each of which gives the same particular value of the density — is an ellipse or its higher-dimensional generalization; hence the multivariate normal is a special case of the elliptical distributions.

The quantity $$\sqrt{({\mathbf x}-{\boldsymbol\mu})^\mathrm{T}{\boldsymbol\Sigma}^{-1}({\mathbf x}-{\boldsymbol\mu})}$$ is known as the Mahalanobis distance, which represents the distance of the test point $${\mathbf x}$$ from the mean $${\boldsymbol\mu}$$. The squared Mahalanobis distance $$({\mathbf x}-{\boldsymbol\mu})^\mathrm{T}{\boldsymbol\Sigma}^{-1}({\mathbf x}-{\boldsymbol\mu})$$ is decomposed into a sum of k terms, each term being a product of three meaningful components. Note that in the case when $$k = 1$$, the distribution reduces to a univariate normal distribution and the Mahalanobis distance reduces to the absolute value of the standard score. See also Interval below.

Bivariate case
In the 2-dimensional nonsingular case ($$k = \operatorname{rank}\left(\Sigma\right) = 2$$), the probability density function of a vector $$\text{[XY]}\prime$$ is: $$ f(x,y) = \frac{1}{2 \pi \sigma_X \sigma_Y \sqrt{1-\rho^2}} \exp \left( -\frac{1}{2\left[1 - \rho^2\right]}\left[         \left(\frac{x-\mu_X}{\sigma_X}\right)^2 -          2\rho\left(\frac{x - \mu_X}{\sigma_X}\right)\left(\frac{y - \mu_Y}{\sigma_Y}\right) +          \left(\frac{y - \mu_Y}{\sigma_Y}\right)^2         \right]       \right) $$ where $$\rho$$ is the correlation between $$X$$ and $$Y$$ and where $$ \sigma_X>0 $$ and $$ \sigma_Y>0 $$. In this case,

\boldsymbol\mu = \begin{pmatrix} \mu_X \\ \mu_Y \end{pmatrix}, \quad \boldsymbol\Sigma = \begin{pmatrix} \sigma_X^2 & \rho \sigma_X \sigma_Y \\ \rho \sigma_X \sigma_Y & \sigma_Y^2 \end{pmatrix}. $$ In the bivariate case, the first equivalent condition for multivariate reconstruction of normality can be made less restrictive as it is sufficient to verify that a countably infinite set of distinct linear combinations of $$X$$ and $$Y$$ are normal in order to conclude that the vector of $$\text{[XY]}\prime$$ is bivariate normal.

The bivariate iso-density loci plotted in the $$x,y$$-plane are ellipses, whose principal axes are defined by the eigenvectors of the covariance matrix $$\boldsymbol\Sigma$$ (the major and minor semidiameters of the ellipse equal the square-root of the ordered eigenvalues).

As the absolute value of the correlation parameter $$\rho$$ increases, these loci are squeezed toward the following line :

y(x) = \sgn (\rho)\frac{\sigma_Y}{\sigma_X} (x - \mu _X) + \mu_Y. $$

This is because this expression, with $$\sgn(\rho)$$ (where sgn is the Sign function) replaced by $$\rho$$, is the best linear unbiased prediction of $$Y$$ given a value of $$X$$.

Degenerate case
If the covariance matrix $$\boldsymbol\Sigma$$ is not full rank, then the multivariate normal distribution is degenerate and does not have a density. More precisely, it does not have a density with respect to k-dimensional Lebesgue measure (which is the usual measure assumed in calculus-level probability courses). Only random vectors whose distributions are absolutely continuous with respect to a measure are said to have densities (with respect to that measure). To talk about densities but avoid dealing with measure-theoretic complications it can be simpler to restrict attention to a subset of $$\operatorname{rank}(\boldsymbol\Sigma)$$ of the coordinates of $$\mathbf{x}$$ such that the covariance matrix for this subset is positive definite; then the other coordinates may be thought of as an affine function of these selected coordinates.

To talk about densities meaningfully in singular cases, then, we must select a different base measure. Using the disintegration theorem we can define a restriction of Lebesgue measure to the $$\operatorname{rank}(\boldsymbol\Sigma)$$-dimensional affine subspace of $$\mathbb{R}^k$$ where the Gaussian distribution is supported, i.e. $$\left\{\boldsymbol\mu + \boldsymbol{\Sigma^{1/2}}\mathbf{v} : \mathbf{v} \in \mathbb{R}^k\right\}$$. With respect to this measure the distribution has the density of the following motif:

 * $$f(\mathbf{x}) = \frac{\exp\left( -\frac{1}{2}\left(\mathbf{x} - \boldsymbol\mu\right)^\mathrm{T} \boldsymbol\Sigma^+\left(\mathbf{x} - \boldsymbol\mu\right)\right) }\sqrt{\det\nolimits^*(2\pi\boldsymbol\Sigma)}$$

where $$\boldsymbol\Sigma^+$$ is the generalized inverse and $$\det\nolimits^*$$ is the pseudo-determinant.

Cumulative distribution function
The notion of cumulative distribution function (cdf) in dimension 1 can be extended in two ways to the multidimensional case, based on rectangular and ellipsoidal regions.

The first way is to define the cdf $$F(\mathbf{x})$$ of a random vector $$\mathbf{X}$$ as the probability that all components of $$\mathbf{X}$$ are less than or equal to the corresponding values in the vector $$\mathbf{x}$$:

 * $$ F(\mathbf{x}) = \mathbb{P}(\mathbf{X}\leq \mathbf{x}), \quad \text{where } \mathbf{X} \sim \mathcal{N}(\boldsymbol\mu,\, \boldsymbol\Sigma).$$

Though there is no closed form for $$F(\mathbf{x})$$, there are a number of algorithms that estimate it numerically.

Another way is to define the cdf $$F(r)$$ as the probability that a sample lies inside the ellipsoid determined by its Mahalanobis distance $$r$$ from the Gaussian, a direct generalization of the standard deviation. In order to compute the values of this function, closed analytic formulae exist, as follows.

Interval
The interval for the multivariate normal distribution yields a region consisting of those vectors x satisfying

 * $$({\mathbf x}-{\boldsymbol\mu})^\mathrm{T}{\boldsymbol\Sigma}^{-1}({\mathbf x}-{\boldsymbol\mu}) \leq \chi^2_k(p).$$

Here $${\mathbf x}$$ is a $$k$$-dimensional vector, $${\boldsymbol\mu}$$ is the known $$k$$-dimensional mean vector, $$\boldsymbol\Sigma$$ is the known covariance matrix and $$\chi^2_k(p)$$ is the quantile function for probability $$p$$ of the chi-squared distribution with $$k$$ degrees of freedom. When $$k = 2,$$ the expression defines the interior of an ellipse and the chi-squared distribution simplifies to an exponential distribution with mean equal to two (rate equal to half).

Complementary cumulative distribution function (tail distribution)
The complementary cumulative distribution function (ccdf) or the tail distribution is defined as $$\overline F(\mathbf{x}) = 1 - \mathbb P\left(\mathbf X\leq \mathbf x\right)$$. When $$ \mathbf{X} \sim \mathcal{N}(\boldsymbol\mu,\, \boldsymbol\Sigma)$$, then the ccdf can be written as a probability the maximum of dependent Gaussian variables:

 * $$ \overline F(\mathbf{x}) = \mathbb P\left(\bigcup_i\{X_i\geq x_i\}\right) = \mathbb P\left(\max_i Y_i\geq 0\right),

\quad \text{where } \mathbf{Y} \sim \mathcal{N}\left(\boldsymbol\mu - \mathbf{x},\, \boldsymbol\Sigma\right).$$ While no simple closed formula exists for computing the ccdf, the maximum of dependent Gaussian variables can be estimated accurately via the Monte Carlo method.

Probability in different domains
The probability content of the multivariate normal in a quadratic domain defined by $$q(\boldsymbol{x}) = \boldsymbol{x}' \mathbf{Q_2} \boldsymbol{x} + \boldsymbol{q_1}' \boldsymbol{x} + q_0>0$$ (where $$\mathbf{Q_2}$$ is a matrix, $$\boldsymbol{q_1}$$ is a vector, and $$q_0$$ is a scalar), which is relevant for Bayesian classification/decision theory using Gaussian discriminant analysis, is given by the generalized chi-squared distribution. The probability content within any general domain defined by $$f(\boldsymbol{x})>0$$ (where $$f(\boldsymbol{x})$$ is a general function) can be computed using the numerical method of ray-tracing (Matlab code).

Higher moments
The kth-order moments of x are given by

 * $$\mu_{1,\ldots,N}(\mathbf{x})

\mathrel\stackrel{\mathrm{def}}{=} \mu _{r_1,\ldots,r_N}(\mathbf{x}) \mathrel\stackrel{\mathrm{def}}{=} \operatorname E\left[ \prod_{j=1}^N X_j^{r_j} \right] $$

where $r_{1} + r_{2} + ⋯ + r_{N} = k.$

The kth-order central moments are as follows

1. If k is odd, $μ_{1, ..., N}(x − μ) = 0$.

2. If k is even with $k = 2λ$, then $ \mu_{1,\dots,2\lambda}(\mathbf{x}-\boldsymbol\mu ) = \sum \left( \sigma_{ij} \sigma_{k\ell} \cdots \sigma_{XZ}\right) $ where the sum is taken over all allocations of the set $$\left\{ 1,\ldots,2\lambda \right\}$$ into λ (unordered) pairs. That is, for a kth $(= 2λ = 6)$ central moment, one sums the products of covariances (the expected value μ is taken to be 0 in the interests of parsimony):

 * $$\begin{align}

& \operatorname E[X_1 X_2 X_3 X_4 X_5 X_6] \\[8pt] = {} & \operatorname E[X_1 X_2]\operatorname E[X_3 X_4]\operatorname E[X_5 X_6] + \operatorname E[X_1 X_2]\operatorname E[X_3 X_5]\operatorname E[X_4 X_6] + \operatorname E[X_1 X_2]\operatorname E[X_3 X_6] \operatorname E[X_4 X_5] \\[4pt] & {} + \operatorname E[X_1 X_3]\operatorname E[X_2 X_4]\operatorname E[X_5 X_6] + \operatorname E[X_1 X_3]\operatorname E[X_2 X_5]\operatorname E[X_4 X_6] + \operatorname E[X_1 X_3]\operatorname E[X_2 X_6] \operatorname E[X_4 X_5] \\[4pt] & {} + \operatorname E[X_1 X_4]\operatorname E[X_2 X_3]\operatorname E[X_5 X_6] + \operatorname E[X_1 X_4]\operatorname E[X_2 X_5]\operatorname E[X_3 X_6]+ \operatorname E[X_1 X_4]\operatorname E[X_2 X_6] \operatorname E[X_3 X_5] \\[4pt] & {} + \operatorname E[X_1 X_5]\operatorname E[X_2 X_3]\operatorname E[X_4 X_6] + \operatorname E[X_1 X_5]\operatorname E[X_2 X_4]\operatorname E[X_3 X_6] + \operatorname E[X_1 X_5]\operatorname E[X_2 X_6] \operatorname E[X_3 X_4] \\[4pt] & {} + \operatorname E[X_1 X_6]\operatorname E[X_2 X_3]\operatorname E[X_4 X_5] + \operatorname E[X_1 X_6]\operatorname E[X_2 X_4]\operatorname E[X_3 X_5] + \operatorname E[X_1 X_6] \operatorname E[X_2 X_5]\operatorname E[X_3 X_4]. \end{align} $$

This yields $$\tfrac{(2\lambda -1)!}{2^{\lambda -1}(\lambda -1)!}$$ terms in the sum (15 in the above case), each being the product of λ (in this case 3) covariances. For fourth order moments (four variables) there are three terms. For sixth-order moments there are $3 × 5 = 15$ terms, and for eighth-order moments there are $3 × 5 × 7 = 105$ terms.

The covariances are then determined by replacing the terms of the list $$[ 1, \ldots, 2\lambda]$$ by the corresponding terms of the list consisting of r1 ones, then r2 twos, etc.. To illustrate this, examine the following 4th-order central moment case:

\begin{align} \operatorname E \left [ X_i^4 \right ] & = 3\sigma_{ii}^2 \\[4pt] \operatorname E \left[ X_i^3 X_j \right ] & = 3\sigma_{ii} \sigma_{ij} \\[4pt] \operatorname E \left[ X_i^2 X_j^2 \right ] & = \sigma_{ii}\sigma_{jj}+2 \sigma _{ij}^2 \\[4pt] \operatorname E \left[ X_i^2 X_j X_k \right ] & = \sigma_{ii}\sigma _{jk}+2\sigma _{ij}\sigma_{ik} \\[4pt] \operatorname E \left [ X_i X_j X_k X_n \right ] & = \sigma_{ij}\sigma _{kn} + \sigma _{ik} \sigma_{jn} + \sigma_{in} \sigma _{jk}. \end{align} $$

where $$\sigma_{ij}$$ is the covariance of Xi and Xj. With the above method one first finds the general case for a kth moment with k different X variables, $$E\left[ X_i X_j X_k X_n\right]$$, and then one simplifies this accordingly. For example, for $$\operatorname E[ X_i^2 X_k X_n ]$$, one lets $X_{i} = X_{j}$ and one uses the fact that $$\sigma_{ii} = \sigma_i^2$$.

Functions of a normal vector

A quadratic form of a normal vector $$\boldsymbol{x}$$, $$q(\boldsymbol{x}) = \boldsymbol{x}' \mathbf{Q_2} \boldsymbol{x} + \boldsymbol{q_1}' \boldsymbol{x} + q_0$$ (where $$\mathbf{Q_2}$$ is a matrix, $$\boldsymbol{q_1}$$ is a vector, and $$q_0$$ is a scalar), is a generalized chi-squared variable. The direction of a normal vector follows a projected normal distribution.

If $$f(\boldsymbol{x})$$ is a general scalar-valued function of a normal vector, its probability density function, cumulative distribution function, and inverse cumulative distribution function can be computed with the numerical method of ray-tracing (Matlab code).

Likelihood function
If the mean and covariance matrix are known, the log likelihood of an observed vector $$\boldsymbol{x}$$ is simply the log of the probability density function:

 * $$\ln L (\boldsymbol{x})= -\frac{1}{2} \left[ \ln (|\boldsymbol\Sigma|\,) + (\boldsymbol{x}-\boldsymbol\mu)'\boldsymbol\Sigma^{-1}(\boldsymbol{x}-\boldsymbol\mu) + k\ln(2\pi) \right]$$,

The circularly symmetric version of the noncentral complex case, where $$\boldsymbol{z}$$ is a vector of complex numbers, would be

 * $$\ln L (\boldsymbol{z}) = -\ln (|\boldsymbol\Sigma|\,) -(\boldsymbol{z}-\boldsymbol\mu)^\dagger\boldsymbol\Sigma^{-1}(\boldsymbol{z}-\boldsymbol\mu) -k\ln(\pi)$$

i.e. with the conjugate transpose (indicated by $$\dagger$$) replacing the normal transpose (indicated by $$'$$). This is slightly different than in the real case, because the circularly symmetric version of the complex normal distribution has a slightly different form for the normalization constant.

A similar notation is used for multiple linear regression.

Since the log likelihood of a normal vector is a quadratic form of the normal vector, it is distributed as a generalized chi-squared variable.

Differential entropy
The differential entropy of the multivariate normal distribution is

$$ \begin{align} h\left(f\right) & = -\int_{-\infty}^\infty \int_{-\infty}^\infty \cdots\int_{-\infty}^\infty f(\mathbf{x}) \ln f(\mathbf{x})\,d\mathbf{x}\\ & = \frac12 \ln{} \left|2\pi e\boldsymbol\Sigma \right| = \frac{k}{2} + \frac{k}{2} \ln{} 2\pi + \frac{1}{2} \ln{} \left|\boldsymbol\Sigma \right|\\ \end{align} $$,

where the bars denote the matrix determinant, $k$ is the dimensionality of the vector space, and the result has units of nats.

Kullback–Leibler divergence
The Kullback–Leibler divergence from $$\mathcal{N}_1(\boldsymbol\mu_1, \boldsymbol\Sigma_1)$$ to $$\mathcal{N}_0(\boldsymbol\mu_0, \boldsymbol\Sigma_0)$$, for non-singular matrices Σ1 and Σ0, is:

D_\text{KL}(\mathcal{N}_0 \parallel \mathcal{N}_1) = { 1 \over 2 } \left\{ \operatorname{tr} \left( \boldsymbol\Sigma_1^{-1} \boldsymbol\Sigma_0 \right) + \left( \boldsymbol\mu_1 - \boldsymbol\mu_0\right)^{\rm T} \boldsymbol\Sigma_1^{-1} ( \boldsymbol\mu_1 - \boldsymbol\mu_0 ) - k +\ln { | \boldsymbol \Sigma_1 | \over | \boldsymbol\Sigma_0 | } \right\}, $$ where $$|\cdot|$$ denotes the matrix determinant, $$tr(\cdot)$$ is the trace, $$ln(\cdot)$$ is the natural logarithm and $$k$$ is the dimension of the vector space.

The logarithm must be taken to base e since the two terms following the logarithm are themselves base-e logarithms of expressions that are either factors of the density function or otherwise arise naturally. The equation therefore gives a result measured in nats. Dividing the entire expression above by loge 2 yields the divergence in bits.

When $$\boldsymbol\mu_1 = \boldsymbol\mu_0$$,

D_\text{KL}(\mathcal{N}_0 \parallel \mathcal{N}_1) = { 1 \over 2 } \left\{ \operatorname{tr} \left( \boldsymbol\Sigma_1^{-1} \boldsymbol\Sigma_0 \right) - k +\ln { |  \boldsymbol \Sigma_1 | \over | \boldsymbol\Sigma_0 | }\right\}. $$

Mutual information
The mutual information of a distribution is a special case of the Kullback–Leibler divergence in which $$P$$ is the full multivariate distribution and $$Q$$ is the product of the 1-dimensional marginal distributions. In the notation of the Kullback–Leibler divergence section of this article, $$\boldsymbol\Sigma_1$$ is a diagonal matrix with the diagonal entries of $$\boldsymbol\Sigma_0$$, and $$\boldsymbol\mu_1 = \boldsymbol\mu_0$$. The resulting formula for mutual information is:

I(\boldsymbol{X}) = - { 1 \over 2 } \ln |  \boldsymbol \rho_0 |, $$

where $$\boldsymbol \rho_0$$ is the correlation matrix constructed from $$\boldsymbol \Sigma_0$$.

In the bivariate case the expression for the mutual information is:

I(x;y) = - { 1 \over 2 } \ln (1 - \rho^2). $$

Normally distributed and independent
If $$X$$ and $$Y$$ are normally distributed and independent, this implies they are "jointly normally distributed", i.e., the pair $$(X,Y)$$ must have multivariate normal distribution. However, a pair of jointly normally distributed variables need not be independent (would only be so if uncorrelated, $$ \rho = 0$$ ).

Two normally distributed random variables need not be jointly bivariate normal
The fact that two random variables $$X$$ and $$Y$$ both have a normal distribution does not imply that the pair $$(X,Y)$$ has a joint normal distribution. A simple example is one in which X has a normal distribution with expected value 0 and variance 1, and $$Y=X$$ if $$|X| > c$$ and $$Y=-X$$ if $$|X| < c$$, where $$c > 0$$. There are similar counterexamples for more than two random variables. In general, they sum to a mixture model.

Correlations and independence
In general, random variables may be uncorrelated but statistically dependent. But if a random vector has a multivariate normal distribution then any two or more of its components that are uncorrelated are independent. This implies that any two or more of its components that are pairwise independent are independent. But, as pointed out just above, it is not true that two random variables that are (separately, marginally) normally distributed and uncorrelated are independent.

Conditional distributions
If N-dimensional x is partitioned as follows

\mathbf{x} = \begin{bmatrix} \mathbf{x}_1 \\ \mathbf{x}_2 \end{bmatrix} \text{ with sizes }\begin{bmatrix} q \times 1 \\ (N-q) \times 1 \end{bmatrix}$$

and accordingly μ and Σ are partitioned as follows

\boldsymbol\mu = \begin{bmatrix} \boldsymbol\mu_1 \\ \boldsymbol\mu_2 \end{bmatrix} \text{ with sizes }\begin{bmatrix} q \times 1 \\ (N-q) \times 1 \end{bmatrix}$$

\boldsymbol\Sigma = \begin{bmatrix} \boldsymbol\Sigma_{11} & \boldsymbol\Sigma_{12} \\ \boldsymbol\Sigma_{21} & \boldsymbol\Sigma_{22} \end{bmatrix} \text{ with sizes }\begin{bmatrix} q \times q & q \times (N-q) \\ (N-q) \times q & (N-q) \times (N-q) \end{bmatrix}$$

then the distribution of x1 conditional on x2 = a is multivariate normal (x1 where

\bar{\boldsymbol\mu} = \boldsymbol\mu_1 + \boldsymbol\Sigma_{12} \boldsymbol\Sigma_{22}^{-1} \left( \mathbf{a} - \boldsymbol\mu_2 \right) $$

and covariance matrix

\overline{\boldsymbol\Sigma} = \boldsymbol\Sigma_{11} - \boldsymbol\Sigma_{12} \boldsymbol\Sigma_{22}^{-1} \boldsymbol\Sigma_{21}. $$

Here $$\boldsymbol\Sigma_{22}^{-1}$$ is the generalized inverse of $$\boldsymbol\Sigma_{22}$$. The matrix $$\overline{\boldsymbol\Sigma}$$ is the Schur complement of Σ22 in Σ. That is, the equation above is equivalent to inverting the overall covariance matrix, dropping the rows and columns corresponding to the variables being conditioned upon, and inverting back to get the conditional covariance matrix.

Note that knowing that alters the variance, though the new variance does not depend on the specific value of a; perhaps more surprisingly, the mean is shifted by $$\boldsymbol\Sigma_{12} \boldsymbol\Sigma_{22}^{-1} \left(\mathbf{a} - \boldsymbol\mu_2 \right)$$; compare this with the situation of not knowing the value of a, in which case x1 would have distribution $$\mathcal{N}_q \left(\boldsymbol\mu_1, \boldsymbol\Sigma_{11} \right)$$.

An interesting fact derived in order to prove this result, is that the random vectors $$\mathbf{x}_2$$ and $$\mathbf{y}_1=\mathbf{x}_1-\boldsymbol\Sigma_{12}\boldsymbol\Sigma_{22}^{-1}\mathbf{x}_2$$ are independent.

The matrix Σ12Σ22−1 is known as the matrix of regression coefficients.

Bivariate case
In the bivariate case where x is partitioned into $$X_1$$ and $$X_2$$, the conditional distribution of $$X_1$$ given $$X_2$$ is

 * $$X_1\mid X_2=a \ \sim\ \mathcal{N}\left(\mu_1+\frac{\sigma_1}{\sigma_2}\rho( a - \mu_2),\, (1-\rho^2)\sigma_1^2\right) $$

where $$\rho$$ is the correlation coefficient between $$X_1$$ and $$X_2$$.

In the general case

\begin{pmatrix} X_1 \\ X_2 \end{pmatrix} \sim \mathcal{N} \left( \begin{pmatrix} \mu_1 \\ \mu_2 \end{pmatrix}, \begin{pmatrix} \sigma^2_1 &  \rho \sigma_1 \sigma_2 \\ \rho \sigma_1 \sigma_2 &  \sigma^2_2 \end{pmatrix} \right) $$

The conditional expectation of X1 given X2 is:

 * $$\operatorname{E}(X_1 \mid X_2=x_2) = \mu_1 + \rho \frac{\sigma_1}{\sigma_2}(x_2 - \mu_2)$$

Proof: the result is obtained by taking the expectation of the conditional distribution $$X_1\mid X_2$$ above.

In the centered case with unit variances

\begin{pmatrix} X_1 \\ X_2 \end{pmatrix} \sim \mathcal{N} \left( \begin{pmatrix} 0 \\ 0 \end{pmatrix}, \begin{pmatrix} 1 & \rho \\ \rho & 1 \end{pmatrix} \right) $$

The conditional expectation of X1 given X2 is

 * $$\operatorname{E}(X_1 \mid X_2=x_2)= \rho x_2 $$

and the conditional variance is

 * $$ \operatorname{var}(X_1 \mid X_2 = x_2) = 1-\rho^2; $$

thus the conditional variance does not depend on x2.

The conditional expectation of X1 given that X2 is smaller/bigger than z is:

\operatorname{E}(X_1 \mid X_2 < z) = -\rho { \varphi(z) \over \Phi(z) } , $$

\operatorname{E}(X_1 \mid X_2 > z) = \rho { \varphi(z) \over (1- \Phi(z)) } , $$

where the final ratio here is called the inverse Mills ratio.

Proof: the last two results are obtained using the result $$\operatorname{E}(X_1 \mid X_2=x_2)= \rho x_2 $$, so that

\operatorname{E}(X_1 \mid X_2 < z) = \rho E(X_2 \mid X_2 < z)$$ and then using the properties of the expectation of a truncated normal distribution.

Marginal distributions
To obtain the marginal distribution over a subset of multivariate normal random variables, one only needs to drop the irrelevant variables (the variables that one wants to marginalize out) from the mean vector and the covariance matrix. The proof for this follows from the definitions of multivariate normal distributions and linear algebra.

Example

Let be multivariate normal random variables with mean vector  and covariance matrix Σ (standard parametrization for multivariate normal distributions). Then the joint distribution of is multivariate normal with mean vector  and covariance matrix $$ \boldsymbol\Sigma' = \begin{bmatrix} \boldsymbol\Sigma_{11} & \boldsymbol\Sigma_{13} \\ \boldsymbol\Sigma_{31} & \boldsymbol\Sigma_{33} \end{bmatrix} $$.

Affine transformation
If is an affine transformation of $$\mathbf{X}\ \sim \mathcal{N}(\boldsymbol\mu, \boldsymbol\Sigma),$$ where c is an $$M \times 1$$ vector of constants and B is a constant $$M \times N$$ matrix, then Y has a multivariate normal distribution with expected value c + Bμ and variance BΣBT i.e., $$\mathbf{Y} \sim \mathcal{N} \left(\mathbf{c} + \mathbf{B} \boldsymbol\mu, \mathbf{B} \boldsymbol\Sigma \mathbf{B}^{\rm T}\right)$$. In particular, any subset of the Xi has a marginal distribution that is also multivariate normal. To see this, consider the following example: to extract the subset (X1, X2, X4)T, use

\mathbf{B} = \begin{bmatrix} 1 & 0 & 0 & 0 & 0 & \ldots & 0 \\ 0 & 1 & 0 & 0 & 0 & \ldots & 0 \\ 0 & 0 & 0 & 1 & 0 & \ldots & 0 \end{bmatrix} $$

which extracts the desired elements directly.

Another corollary is that the distribution of, where b is a constant vector with the same number of elements as X and the dot indicates the dot product, is univariate Gaussian with $$Z\sim\mathcal{N}\left(\mathbf{b}\cdot\boldsymbol\mu, \mathbf{b}^{\rm T}\boldsymbol\Sigma \mathbf{b}\right)$$. This result follows by using

\mathbf{B}=\begin{bmatrix} b_1   & b_2    & \ldots & b_n \end{bmatrix} = \mathbf{b}^{\rm T}. $$ Observe how the positive-definiteness of Σ implies that the variance of the dot product must be positive.

An affine transformation of X such as 2X is not the same as the sum of two independent realisations of X.

Geometric interpretation
The equidensity contours of a non-singular multivariate normal distribution are ellipsoids (i.e. affine transformations of hyperspheres) centered at the mean. Hence the multivariate normal distribution is an example of the class of elliptical distributions. The directions of the principal axes of the ellipsoids are given by the eigenvectors of the covariance matrix $$\boldsymbol\Sigma$$. The squared relative lengths of the principal axes are given by the corresponding eigenvalues.

If is an eigendecomposition where the columns of U are unit eigenvectors and Λ is a diagonal matrix of the eigenvalues, then we have

 * $$\mathbf{X}\ \sim \mathcal{N}(\boldsymbol\mu, \boldsymbol\Sigma) \iff \mathbf{X}\ \sim \boldsymbol\mu+\mathbf{U}\boldsymbol\Lambda^{1/2}\mathcal{N}(0, \mathbf{I}) \iff \mathbf{X}\ \sim \boldsymbol\mu+\mathbf{U}\mathcal{N}(0, \boldsymbol\Lambda).$$

Moreover, U can be chosen to be a rotation matrix, as inverting an axis does not have any effect on N(0, Λ), but inverting a column changes the sign of U's determinant. The distribution N(μ, Σ) is in effect N(0, I) scaled by Λ1/2, rotated by U and translated by μ.

Conversely, any choice of μ, full rank matrix U, and positive diagonal entries Λi yields a non-singular multivariate normal distribution. If any Λi is zero and U is square, the resulting covariance matrix UΛUT is singular. Geometrically this means that every contour ellipsoid is infinitely thin and has zero volume in n-dimensional space, as at least one of the principal axes has length of zero; this is the degenerate case.

"The radius around the true mean in a bivariate normal random variable, re-written in polar coordinates (radius and angle), follows a Hoyt distribution."

In one dimension the probability of finding a sample of the normal distribution in the interval $$\mu\pm \sigma$$ is approximately 68.27%, but in higher dimensions the probability of finding a sample in the region of the standard deviation ellipse is lower.

Parameter estimation
The derivation of the maximum-likelihood estimator of the covariance matrix of a multivariate normal distribution is straightforward.

In short, the probability density function (pdf) of a multivariate normal is

 * $$f(\mathbf{x})= \frac{1}{\sqrt { (2\pi)^k|\boldsymbol \Sigma| } } \exp\left(-{1 \over 2} (\mathbf{x}-\boldsymbol\mu)^{\rm T} \boldsymbol\Sigma^{-1} ({\mathbf x}-\boldsymbol\mu)\right)$$

and the ML estimator of the covariance matrix from a sample of n observations is

 * $$\widehat{\boldsymbol\Sigma} = {1 \over n}\sum_{i=1}^n ({\mathbf x}_i-\overline{\mathbf x})({\mathbf x}_i-\overline{\mathbf x})^\mathrm{T} $$

which is simply the sample covariance matrix. This is a biased estimator whose expectation is

 * $$E\left[\widehat{\boldsymbol\Sigma}\right] = \frac{n-1}{n} \boldsymbol\Sigma.$$

An unbiased sample covariance is

 * $$\widehat{\boldsymbol\Sigma} = \frac1{n-1}\sum_{i=1}^n (\mathbf{x}_i-\overline{\mathbf{x}})(\mathbf{x}_i-\overline{\mathbf{x}})^{\rm T}

= \frac1{n-1} \left[X'\left(I - \frac{1}{n} \cdot J\right) X\right] $$ (matrix form; $$I$$ is the $$K\times K$$ identity matrix, J is a $$K \times K$$ matrix of ones; the term in parentheses is thus the $$K \times K$$ centering matrix)

The Fisher information matrix for estimating the parameters of a multivariate normal distribution has a closed form expression. This can be used, for example, to compute the Cramér–Rao bound for parameter estimation in this setting. See Fisher information for more details.

Bayesian inference
In Bayesian statistics, the conjugate prior of the mean vector is another multivariate normal distribution, and the conjugate prior of the covariance matrix is an inverse-Wishart distribution $$\mathcal{W}^{-1}$$. Suppose then that n observations have been made
 * $$\mathbf{X} = \{\mathbf{x}_1,\dots,\mathbf{x}_n\} \sim \mathcal{N}(\boldsymbol\mu,\boldsymbol\Sigma)$$

and that a conjugate prior has been assigned, where
 * $$p(\boldsymbol\mu,\boldsymbol\Sigma)=p(\boldsymbol\mu\mid\boldsymbol\Sigma)\ p(\boldsymbol\Sigma),$$

where
 * $$p(\boldsymbol\mu\mid\boldsymbol\Sigma) \sim\mathcal{N}(\boldsymbol\mu_0,m^{-1}\boldsymbol\Sigma) ,$$

and
 * $$p(\boldsymbol\Sigma) \sim \mathcal{W}^{-1}(\boldsymbol\Psi,n_0).$$

Then

\begin{array}{rcl} p(\boldsymbol\mu\mid\boldsymbol\Sigma,\mathbf{X}) & \sim & \mathcal{N}\left(\frac{n\bar{\mathbf{x}} + m\boldsymbol\mu_0}{n+m},\frac{1}{n+m}\boldsymbol\Sigma\right),\\ p(\boldsymbol\Sigma\mid\mathbf{X}) & \sim & \mathcal{W}^{-1}\left(\boldsymbol\Psi+n\mathbf{S}+\frac{nm}{n+m}(\bar{\mathbf{x}}-\boldsymbol\mu_0)(\bar{\mathbf{x}}-\boldsymbol\mu_0)', n+n_0\right), \end{array} $$ where

\begin{align} \bar{\mathbf{x}} & = \frac{1}{n}\sum_{i=1}^{n} \mathbf{x}_i ,\\ \mathbf{S} & = \frac{1}{n}\sum_{i=1}^{n} (\mathbf{x}_i - \bar{\mathbf{x}})(\mathbf{x}_i - \bar{\mathbf{x}})'. \end{align} $$

Multivariate normality tests
Multivariate normality tests check a given set of data for similarity to the multivariate normal distribution. The null hypothesis is that the data set is similar to the normal distribution, therefore a sufficiently small p-value indicates non-normal data. Multivariate normality tests include the Cox–Small test and Smith and Jain's adaptation of the Friedman–Rafsky test created by Larry Rafsky and Jerome Friedman.

Mardia's test is based on multivariate extensions of skewness and kurtosis measures. For a sample {x1, ..., xn} of k-dimensional vectors we compute
 * $$\begin{align}

& \widehat{\boldsymbol\Sigma} = {1 \over n} \sum_{j=1}^n \left(\mathbf{x}_j - \bar{\mathbf{x}}\right)\left(\mathbf{x}_j - \bar{\mathbf{x}}\right)^\mathrm{T} \\

& A = {1 \over 6n} \sum_{i=1}^n \sum_{j=1}^n \left[ (\mathbf{x}_i - \bar{\mathbf{x}})^\mathrm{T}\;\widehat{\boldsymbol\Sigma}^{-1} (\mathbf{x}_j - \bar{\mathbf{x}}) \right]^3 \\

& B = \sqrt{\frac{n}{8k(k+2)}}\left\{{1 \over n} \sum_{i=1}^n \left[ (\mathbf{x}_i - \bar{\mathbf{x}})^\mathrm{T}\;\widehat{\boldsymbol\Sigma}^{-1} (\mathbf{x}_i - \bar{\mathbf{x}}) \right]^2 - k(k+2) \right\} \end{align}$$ Under the null hypothesis of multivariate normality, the statistic A will have approximately a chi-squared distribution with ⋅k(k + 1)(k + 2) degrees of freedom, and B will be approximately standard normal N(0,1).

Mardia's kurtosis statistic is skewed and converges very slowly to the limiting normal distribution. For medium size samples $$(50 \le n < 400)$$, the parameters of the asymptotic distribution of the kurtosis statistic are modified For small sample tests ($$n<50$$) empirical critical values are used. Tables of critical values for both statistics are given by Rencher for k = 2, 3, 4.

Mardia's tests are affine invariant but not consistent. For example, the multivariate skewness test is not consistent against symmetric non-normal alternatives.

The BHEP test computes the norm of the difference between the empirical characteristic function and the theoretical characteristic function of the normal distribution. Calculation of the norm is performed in the L2(μ) space of square-integrable functions with respect to the Gaussian weighting function $$ \mu_\beta(\mathbf{t}) = (2\pi\beta^2)^{-k/2} e^{-|\mathbf{t}|^2/(2\beta^2)}$$. The test statistic is
 * $$\begin{align}

T_\beta &= \int_{\mathbb{R}^k} \left| {1 \over n} \sum_{j=1}^n e^{i\mathbf{t}^\mathrm{T}\widehat{\boldsymbol\Sigma}^{-1/2}(\mathbf{x}_j - \bar{\mathbf{x})}} - e^{-|\mathbf{t}|^2/2} \right|^2 \; \boldsymbol\mu_\beta(\mathbf{t}) \, d\mathbf{t} \\ &= {1 \over n^2} \sum_{i,j=1}^n e^{-{\beta^2 \over 2}(\mathbf{x}_i-\mathbf{x}_j)^\mathrm{T}\widehat{\boldsymbol\Sigma}^{-1}(\mathbf{x}_i-\mathbf{x}_j)} - \frac{2}{n(1 + \beta^2)^{k/2}}\sum_{i=1}^n e^{ -\frac{\beta^2}{2(1+\beta^2)} (\mathbf{x}_i-\bar{\mathbf{x}})^\mathrm{T}\widehat{\boldsymbol\Sigma}^{-1}(\mathbf{x}_i-\bar{\mathbf{x}})} + \frac{1}{(1 + 2\beta^2)^{k/2}} \end{align}$$ The limiting distribution of this test statistic is a weighted sum of chi-squared random variables.

A detailed survey of these and other test procedures is available.

Gaussian Discriminant Analysis
Suppose that observations (which are vectors) are presumed to come from one of several multivariate normal distributions, with known means and covariances. Then any given observation can be assigned to the distribution from which it has the highest probability of arising. This classification procedure is called Gaussian discriminant analysis. The classification performance, i.e. probabilities of the different classification outcomes, and the overall classification error, can be computed by the numerical method of ray-tracing (Matlab code).

Drawing values from the distribution
A widely used method for drawing (sampling) a random vector x from the N-dimensional multivariate normal distribution with mean vector μ and covariance matrix Σ works as follows:

 * 1) Find any real matrix A such that . When Σ is positive-definite, the Cholesky decomposition is typically used, and the extended form of this decomposition can always be used  (as the covariance matrix may be only positive semi-definite) in both cases a suitable matrix A is obtained. An alternative is to use the matrix A = UΛ1/2 obtained from a spectral decomposition Σ = UΛU−1 of Σ. The former approach is more computationally straightforward but the matrices A change for different orderings of the elements of the random vector, while the latter approach gives matrices that are related by simple re-orderings. In theory both approaches give equally good ways of determining a suitable matrix A, but there are differences in computation time.
 * 2) Let  be a vector whose components are N independent standard normal variates (which can be generated, for example, by using the Box–Muller transform).
 * 3) Let x be μ + Az. This has the desired distribution due to the affine transformation property.

title: Multivariate normal distribution


text:
Wikipedia:Internationalization bug reports

'''This is an archive only of bug reports from Phase II of the Wikipedia software (used before June 20, 2002). Please see Bug reports for instructions on adding bug reports for the current system.'''

return to Bug reports/Phase II

Slovak(Slovakia) language is actually Slovenian(Slovenia)

Chinese (unicode?) article title created, breaks interface The Anome, Thursday, April 4, 2002

Some one has managed to create f (link removed)

with all sorts of nasty consequences (try to edit it, for example).

 * Removed it manually. This (the link problems) should already be fixed in CVS. Brion VIBBER, Thursday, April 4, 2002

This may be the same issue as above. Go to Poincaré conjecture, then click on "edit this link". On IE 6, I get a page "PoincarÃ© conjecture". In order to actually edit the article, or access its history, you'll have to manually enter something like
 * http://www.wikipedia.com/wiki/Poincar%E9_conjecture&action=edit

I suppose the links produced by the script should always use these % escapes for non-ascii characters. AxelBoldt, Tuesday, April 9, 2002
 * They have done for several days in CVS. Jimbo, can we see an upgrade soon, please? Brion VIBBER, Tuesday, April 9, 2002
 * The problem has largely been fixed in the latest update but still persists in "Redirected from" contexts. Try going to Poincaré conjecture (which currently redirects to Poincare conjecture) and trying to see where you were redirected from. -- Toby Bartels, Sunday, May 19, 2002

Non-ascii in titles

 * (2002/1/21) -- ISO-8859-2 characters cannot be in title of article - number of international wikipedias need this ...
 * I'll wait for the English script and the bomis CVS to go online, there are some people who know the character encoding things better than me... --Magnus Manske

Japanese Wikipedia marked as ISO 8859-1 2001-12-31

http://ja.wikipedia.com is illegible with IE 5.0 (Mac) because its HTTP (MIME) header includes the line

Content-Type: text/html; charset=ISO-8859-1

The charset should be changed to something appropriate (Shift_JIS or UTF8), or removed and replaced by the equivalent META tag.

 * The Japanese wikipedia now puts out "x-sjis" in its headers, Shift-JIS is allowed in page titles, and it is more or less usable. However, some characters are being mysteriously screwed up (katakana RU for instance); please test and report if you can figure out exactly what's going on. 2002-03-01 Brion VIBBER

 * Found it, the darn Perl CGI module was at fault with bad defaults for escaping characters. Sent fix to Jimbo. 2002-03-01 Brion VIBBER

On a similar note, visitors to http://www.wikipedia.com should be automatically redirected to the Wikipedia written in the language of their choice, as expressed in their browser language preferences. -- poslfit
 * Then how would Dutch/English bilinguals switch to the English version? en.wikipedia.com doesn't seem to have any content. --User:Damian Yerrick

The same problem occurs in Netscape 4.77. However, IE5 works fine. This is very similar and may be related to the problem reported at the bottom of this page. See talk:Ranma 1 for details.

wrong Slovak webside I'm sorry but your hyperlink has send me to slovenia not slovakia!

Technical: IPA
Moved from Village Pump

Why is it that on IE 5, while this web site can display every single IPA letters and diacritics correctly, none of the Wikipedias can display any, except the standard 26 English letters? Something looks wrong. We should better WP. --Menchi 07:13 May 7, 2003 (UTC)

 * It's the difference between <meta http-equiv="Content-Type" content="text/html; charset=utf-8"> and <meta http-equiv="Content-type" content="text/html; charset=iso-8859-1">, I suspect.
 * --Paul A 08:10 May 7, 2003 (UTC)

 * That reminds me, are there plans afoot for following LiveJournal and using UTF-8? (LJ did so, and arranged for server-side on-the-fly translation of "legacy" entries to UTF-8 - and LJ is a good example because that's mostly user-contributed content too.  Most users were unware of the change to UTF-8, but those that maintained journals in non-english charsets were overjoyed.)  Tenbaset 23:21 May 7, 2003 (UTC)

 * We already run in UTF-8 on all phase 3 wikis except for English, Danish, German, French, Dutch, Spanish, Swedish. The reason the western European languages haven't been moved over yet is that when you edit a page on a wiki, your browser has a chance to screw up not just your contributions but the entire contents of the page. It happens fairly regularly on meta and wiktionary that someone (usually our dear Anthere :) destroys every non-ASCII character in a page by making an edit with a browser that has broken UTF-8 support, and someone else has to go in and fix them. Until we either banish all such browsers ;) or make tweaks to make safe round-trip edits in those browsers without inconveniencing the rest of us with too much armor, the western european languages will stay in Latin-1.

 * Note that you can use all the Unicode characters you want, including IPA, with &#12345; etc. See Special characters. --Brion 00:15 May 8, 2003 (UTC)

Suggestion: keep using SAMPA in markup, but provide an option to have this "translated" into IPA characters if the user chooses. -- Tarquin 08:23 May 7, 2003 (UTC)


text: Talk:Frontier

title: Talk:Frontier