Analysis of vector-valued curves
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Analysis ( : analyses) is the process of breaking a complex topic or
substance Substance may refer to: * Matter, anything that has mass and takes up space Chemistry * Chemical substance, a material with a definite chemical composition * Drug substance ** Substance abuse, drug-related healthcare and social policy diagnosis ...
into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
and logic since before Aristotle (384–322 B.C.), though ''analysis'' as a formal concept is a relatively recent development. The word comes from the Ancient Greek ἀνάλυσις (''analysis'', "a breaking-up" or "an untying;" from ''ana-'' "up, throughout" and ''lysis'' "a loosening"). From it also comes the word's plural, ''analyses''. As a formal concept, the method has variously been ascribed to
Alhazen Ḥasan Ibn al-Haytham, Latinized as Alhazen (; full name ; ), was a medieval mathematician, astronomer, and physicist of the Islamic Golden Age from present-day Iraq.For the description of his main fields, see e.g. ("He is one of the prin ...
, René Descartes ('' Discourse on the Method''), and Galileo Galilei. It has also been ascribed to Isaac Newton, in the form of a practical method of physical discovery (which he did not name). The converse of analysis is synthesis: putting the pieces back together again in new or different whole.


Applications


Science

The field of
chemistry Chemistry is the science, scientific study of the properties and behavior of matter. It is a natural science that covers the Chemical element, elements that make up matter to the chemical compound, compounds made of atoms, molecules and ions ...
uses analysis in three ways: to identify the components of a particular chemical compound (qualitative analysis), to identify the proportions of components in a
mixture In chemistry, a mixture is a material made up of two or more different chemical substances which are not chemically bonded. A mixture is the physical combination of two or more substances in which the identities are retained and are mixed in the ...
(quantitative analysis), and to break down chemical processes and examine chemical reactions between
elements Element or elements may refer to: Science * Chemical element, a pure substance of one type of atom * Heating element, a device that generates heat by electrical resistance * Orbital elements, parameters required to identify a specific orbit of ...
of matter. For an example of its use, analysis of the concentration of elements is important in managing a nuclear reactor, so nuclear scientists will analyze neutron activation to develop discrete measurements within vast samples. A
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
can have a considerable effect on the way a chemical analysis is conducted and the quality of its results. Analysis can be done manually or with a
device A device is usually a constructed tool. Device may also refer to: Technology Computing * Device, a colloquial term encompassing desktops, laptops, tablets, smartphones, etc. * Device file, an interface of a device driver * Peripheral, any devi ...
. Types of Analysis: A) Qualitative Analysis: It is concerned with which components are in a given sample or compound. Example: Precipitation reaction B) Quantitative Analysis: It is to determine the quantity of individual component present in a given sample or compound. Example: To find concentration by uv-spectrophotometer.


Isotopes

Chemists can use
isotope analysis Isotope analysis is the identification of isotopic signature, abundance of certain stable isotopes of chemical elements within organic and inorganic compounds. Isotopic analysis can be used to understand the flow of energy through a food web ...
to assist analysts with issues in anthropology,
archeology Archaeology or archeology is the scientific study of human activity through the recovery and analysis of material culture. The archaeological record consists of artifacts, architecture, biofacts or ecofacts, sites, and cultural landscap ...
, food chemistry,
forensics Forensic science, also known as criminalistics, is the application of science to criminal and civil laws, mainly—on the criminal side—during criminal investigation, as governed by the legal standards of admissible evidence and crimina ...
, geology, and a host of other questions of physical science. Analysts can discern the origins of natural and man-made isotopes in the study of
environmental radioactivity Environmental radioactivity is produced by radioactive materials in the human environment. While some radioisotopes, such as strontium-90 (90Sr) and technetium-99 (99Tc), are only found on Earth as a result of human activity, and some, like po ...
.


Business

* Financial statement analysis – the analysis of the accounts and the economic prospects of a firm *
Financial analysis Financial analysis (also known as financial statement analysis, accounting analysis, or analysis of finance) refers to an assessment of the viability, stability, and profitability of a business, sub-business or project. It is performed by profes ...
 – refers to an assessment of the viability, stability, and profitability of a
business Business is the practice of making one's living or making money by producing or Trade, buying and selling Product (business), products (such as goods and Service (economics), services). It is also "any activity or enterprise entered into for pr ...
, sub-business or
project A project is any undertaking, carried out individually or collaboratively and possibly involving research or design, that is carefully planned to achieve a particular goal. An alternative view sees a project managerially as a sequence of even ...
*
Gap analysis In management literature, gap analysis involves the comparison of actual performance with potential or desired performance. If an organization does not make the best use of current resources, or forgoes investment in capital or technology, it m ...
– involves the comparison of actual performance with potential or desired performance of an organization * Business analysis – involves identifying the needs and determining the solutions to business problems *
Price analysis Price analysis is the study of how a price relates to other things such as product demand. Its specfici meaning varies in contexts such as marketing and general business. Marketing In marketing, price analysis refers to the analysis of consum ...
– involves the breakdown of a price to a unit figure * Market analysis – consists of suppliers and customers, and price is determined by the interaction of
supply and demand In microeconomics, supply and demand is an economic model of price determination in a Market (economics), market. It postulates that, Ceteris paribus, holding all else equal, in a perfect competition, competitive market, the unit price for a ...
* Sum-of-the-parts analysis – method of valuation of a multi-divisional company * Opportunity analysis – consists of customers trends within the industry, customer demand and experience determine purchasing behavior


Computer science

* Requirements analysis – encompasses those tasks that go into determining the needs or conditions to meet for a new or altered product, taking account of the possibly conflicting requirements of the various stakeholders, such as beneficiaries or users. *
Competitive analysis (online algorithm) Competitive analysis is a method invented for analyzing online algorithms, in which the performance of an online algorithm (which must satisfy an unpredictable sequence of requests, completing each request without being able to see the future) is co ...
– shows how online algorithms perform and demonstrates the power of randomization in algorithms * Lexical analysis – the process of processing an input sequence of characters and producing as output a sequence of symbols * Object-oriented analysis and design – à la Booch *
Program analysis (computer science) In computer science, program analysis is the process of automatically analyzing the behavior of computer programs regarding a property such as correctness, robustness, safety and liveness. Program analysis focuses on two major areas: program op ...
 – the process of automatically analysing the behavior of computer programs *
Semantic analysis (computer science) In computing, a compiler is a computer program that Translator (computing), translates computer code written in one programming language (the ''source'' language) into another language (the ''target'' language). The name "compiler" is primarily ...
 – a pass by a compiler that adds semantical information to the parse tree and performs certain checks * Static code analysis – the analysis of computer software that is performed without actually executing programs built from that * Structured systems analysis and design methodology – à la
Yourdon Edward Nash Yourdon (April 30, 1944 – January 20, 2016) was an American software engineer, computer consultant, author and lecturer, and software engineering methodology pioneer. He was one of the lead developers of the structured analysis tech ...
*
Syntax analysis In linguistics, syntax () is the study of how words and morphemes combine to form larger units such as phrases and sentences. Central concerns of syntax include word order, grammatical relations, hierarchical sentence structure (constituency), ...
 – a process in compilers that recognizes the structure of programming languages, also known as parsing * Worst-case execution time – determines the longest time that a piece of software can take to run


Economics

*
Agroecosystem analysis Agroecosystem analysis is a thorough analysis of an agricultural environment which considers aspects from ecology, sociology, economics, and politics with equal weight. There are many aspects to consider; however, it is literally impossible to acco ...
*
Input–output model In economics, an input–output model is a quantitative economic model that represents the interdependencies between different sectors of a national economy or different regional economies.Thijs Ten Raa, Input–Output Economics: Theory and App ...
if applied to a region, is called Regional Impact Multiplier System


Engineering

Analysts in the field of engineering look at
requirements In product development and process optimization, a requirement is a singular documented physical or functional need that a particular design, product or process aims to satisfy. It is commonly used in a formal sense in engineering design, includi ...
, structures, mechanisms,
systems A system is a group of interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenced by its environment, is described by its boundaries, structure and purpose and express ...
and dimensions. Electrical engineers analyse
systems A system is a group of interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenced by its environment, is described by its boundaries, structure and purpose and express ...
in electronics. Life cycles and system failures are broken down and studied by engineers. It is also looking at different factors incorporated within the design.


Intelligence

The field of intelligence employs analysts to break down and understand a wide array of questions. Intelligence agencies may use heuristics, inductive and
deductive reasoning Deductive reasoning is the mental process of drawing deductive inferences. An inference is deductively valid if its conclusion follows logically from its premises, i.e. if it is impossible for the premises to be true and the conclusion to be fals ...
, social network analysis,
dynamic network analysis Dynamic network analysis (DNA) is an emergent scientific field that brings together traditional social network analysis (SNA), link analysis (LA), social simulation and multi-agent systems (MAS) within network science and network theory. Dynamic ne ...
, link analysis, and
brainstorming Brainstorming is a group creativity technique by which efforts are made to find a conclusion for a specific problem by gathering a list of ideas spontaneously contributed by its members. In other words, brainstorming is a situation where a grou ...
to sort through problems they face. Military intelligence may explore issues through the use of
game theory Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has appli ...
,
Red Team A red team or team red are a group that plays the role of an enemy or competitor to provide security feedback from that perspective. Red teams are used in many fields, especially in cybersecurity, airport security, law enforcement, the military a ...
ing, and wargaming.
Signals intelligence Signals intelligence (SIGINT) is intelligence-gathering by interception of ''signals'', whether communications between people (communications intelligence—abbreviated to COMINT) or from electronic signals not directly used in communication ( ...
applies
cryptanalysis Cryptanalysis (from the Greek ''kryptós'', "hidden", and ''analýein'', "to analyze") refers to the process of analyzing information systems in order to understand hidden aspects of the systems. Cryptanalysis is used to breach cryptographic sec ...
and
frequency analysis In cryptanalysis, frequency analysis (also known as counting letters) is the study of the frequency of letters or groups of letters in a ciphertext. The method is used as an aid to breaking classical ciphers. Frequency analysis is based on t ...
to break codes and
cipher In cryptography, a cipher (or cypher) is an algorithm for performing encryption or decryption—a series of well-defined steps that can be followed as a procedure. An alternative, less common term is ''encipherment''. To encipher or encode i ...
s. Business intelligence applies theories of competitive intelligence analysis and competitor analysis to resolve questions in the marketplace. Law enforcement intelligence applies a number of theories in crime analysis.


Linguistics

Linguistics explores individual languages and language in general. It breaks language down and analyses its component parts: theory, sounds and their meaning, utterance usage, word origins, the history of words, the meaning of words and word combinations, sentence construction, basic construction beyond the sentence level,
stylistics Stylistics, a branch of applied linguistics, is the study and interpretation of texts of all types and/or spoken language in regard to their linguistic and tonal style, where style is the particular variety of language used by different individu ...
, and
conversation Conversation is interactive communication between two or more people. The development of conversational skills and etiquette is an important part of socialization. The development of conversational skills in a new language is a frequent focus ...
. It examines the above using statistics and modeling, and semantics. It analyses language in context of anthropology, biology, evolution, geography, history, neurology, psychology, and sociology. It also takes the applied approach, looking at individual language development and clinical issues.


Literature

Literary criticism Literary criticism (or literary studies) is the study, evaluation, and interpretation of literature. Modern literary criticism is often influenced by literary theory, which is the philosophical discussion of literature's goals and methods. Th ...
is the analysis of literature. The focus can be as diverse as the analysis of Homer or
Freud Sigmund Freud ( , ; born Sigismund Schlomo Freud; 6 May 1856 – 23 September 1939) was an Austrian neurologist and the founder of psychoanalysis, a clinical method for evaluating and treating pathologies explained as originating in conflicts in ...
. While not all literary-critical methods are primarily analytical in nature, the main approach to the teaching of literature in the west since the mid-twentieth century, literary formal analysis or close reading, is. This method, rooted in the academic movement labelled
The New Criticism New Criticism was a formalist movement in literary theory that dominated American literary criticism in the middle decades of the 20th century. It emphasized close reading, particularly of poetry, to discover how a work of literature functioned as ...
, approaches texts – chiefly short poems such as sonnets, which by virtue of their small size and significant complexity lend themselves well to this type of analysis – as units of discourse that can be understood in themselves, without reference to biographical or historical frameworks. This method of analysis breaks up the text linguistically in a study of prosody (the formal analysis of meter) and phonic effects such as alliteration and
rhyme A rhyme is a repetition of similar sounds (usually, the exact same phonemes) in the final stressed syllables and any following syllables of two or more words. Most often, this kind of perfect rhyming is consciously used for a musical or aesthetic ...
, and cognitively in examination of the interplay of syntactic structures, figurative language, and other elements of the poem that work to produce its larger effects.


Mathematics

Modern mathematical analysis is the study of infinite processes. It is the branch of mathematics that includes calculus. It can be applied in the study of classical concepts of mathematics, such as
real numbers In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
,
complex variables Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebra ...
, trigonometric functions, and algorithms, or of non-classical concepts like constructivism, harmonics,
infinity Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol . Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions amo ...
, and vectors. Florian Cajori explains in ''A History of Mathematics'' (1893) the difference between modern and ancient mathematical analysis, as distinct from logical analysis, as follows:
The terms ''synthesis'' and ''analysis'' are used in mathematics in a more special sense than in logic. In ancient mathematics they had a different meaning from what they now have. The oldest definition of mathematical analysis as opposed to synthesis is that given in ppended to Euclid, XIII. 5, which in all probability was framed by Eudoxus: "Analysis is the obtaining of the thing sought by assuming it and so reasoning up to an admitted truth; synthesis is the obtaining of the thing sought by reasoning up to the inference and proof of it."
The analytic method is not conclusive, unless all operations involved in it are known to be reversible. To remove all doubt, the Greeks, as a rule, added to the analytic process a synthetic one, consisting of a reversion of all operations occurring in the analysis. Thus the aim of analysis was to aid in the discovery of synthetic proofs or solutions.
James Gow uses a similar argument as Cajori, with the following clarification, in hi
''A Short History of Greek Mathematics''
(1884):
The synthetic proof proceeds by shewing that the proposed new truth involves certain admitted truths. An analytic proof begins by an assumption, upon which a synthetic reasoning is founded. The Greeks distinguished ''theoretic'' from ''problematic'' analysis. A theoretic analysis is of the following kind. To ''prove'' that A is B, ''assume'' first that A is B. If so, then, since B is C and C is D and D is E, therefore A is E. If this be known a falsity, A is not B. But if this be a known truth and all the intermediate propositions be convertible, then the reverse process, A is E, E is D, D is C, C is B, therefore A is B, constitutes a synthetic proof of the original theorem. Problematic analysis is applied in all cases where it is proposed to construct a figure which is assumed to satisfy a given condition. The problem is then converted into some theorem which is involved in the condition and which is proved synthetically, and the steps of this synthetic proof taken backwards are a synthetic solution of the problem.


Music

*
Musical analysis Musical analysis is the study of musical structure in either compositions or performances. According to music theorist Ian Bent, music analysis "is the means of answering directly the question 'How does it work?'". The method employed to answer ...
 – a process attempting to answer the question "How does this music work?" **Musical Analysis is a study of how the composers use the notes together to compose music. Those studying music will find differences with each composer's musical analysis, which differs depending on the culture and history of music studied. An analysis of music is meant to simplify the music for you. *
Schenkerian analysis Schenkerian analysis is a method of analyzing tonal music based on the theories of Heinrich Schenker (1868–1935). The goal is to demonstrate the organic coherence of the work by showing how it relates to an abstracted deep structure, the ''Ursatz' ...
**Schenkerian analysis is a collection of music analysis that focuses on the production of the graphic representation. This includes both analytical procedure as well as the notational style. Simply put, it analyzes tonal music which includes all chords and tones within a composition.


Philosophy

* Philosophical analysis – a general term for the techniques used by philosophers **Philosophical analysis refers to the clarification and composition of words put together and the entailed meaning behind them. Philosophical analysis dives deeper into the meaning of words and seeks to clarify that meaning by contrasting the various definitions. It is the study of reality, justification of claims, and the analysis of various concepts. Branches of philosophy include logic, justification, metaphysics, values and ethics. If questions can be answered empirically, meaning it can be answered by using the senses, then it is not considered philosophical. Non-philosophical questions also include events that happened in the past, or questions science or mathematics can answer. * '' Analysis'' is the name of a prominent journal in philosophy.


Psychotherapy

* Psychoanalysis – seeks to elucidate connections among unconscious components of patients' mental processes * Transactional analysis ** Transactional analysis is used by therapists to try to gain a better understanding of the unconscious. It focuses on understanding and intervening human behavior.


Policy

*
Policy analysis Policy analysis is a technique used in the public administration sub-field of political science to enable civil servants, nonprofit organizations, and others to examine and evaluate the available options to implement the goals of laws and elected ...
 – The use of statistical data to predict the effects of policy decisions made by governments and agencies ** Policy analysis includes a systematic process to find the most efficient and effective option to address the current situation. * Qualitative analysis – The use of anecdotal evidence to predict the effects of policy decisions or, more generally, influence policy decisions


Signal processing

*
Finite element analysis The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat ...
 – a computer simulation technique used in engineering analysis * Independent component analysis *
Link quality analysis In adaptive high-frequency (HF) radio, link quality analysis (LQA) is the overall process by which measurements of signal quality are made, assessed, and analyzed. In LQA, signal quality is determined by measuring, assessing, and analyzing link p ...
 – the analysis of signal quality *
Path quality analysis Path quality analysis: In a telecommunication, communications path, an analysis that (a) includes the overall evaluation of the electronic component, component quality measures, the individual data link, link quality measures, and the aggregate pat ...
*
Fourier analysis In mathematics, Fourier analysis () is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Josep ...


Statistics

In
statistics Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
, the term ''analysis'' may refer to any method used for
data analysis Data analysis is a process of inspecting, cleansing, transforming, and modeling data with the goal of discovering useful information, informing conclusions, and supporting decision-making. Data analysis has multiple facets and approaches, enco ...
. Among the many such methods, some are: * Analysis of variance (ANOVA) – a collection of statistical models and their associated procedures which compare means by splitting the overall observed variance into different parts *
Boolean analysis Boolean analysis was introduced by Flament (1976).Flament, C. (1976). "L'analyse booleenne de questionnaire", Paris: Mouton. The goal of a Boolean analysis is to detect deterministic dependencies between the items of a questionnaire or similar data ...
 – a method to find deterministic dependencies between variables in a sample, mostly used in exploratory data analysis *
Cluster analysis Cluster analysis or clustering is the task of grouping a set of objects in such a way that objects in the same group (called a cluster) are more similar (in some sense) to each other than to those in other groups (clusters). It is a main task of ...
 – techniques for finding groups (called clusters), based on some measure of proximity or similarity * Factor analysis – a method to construct models describing a data set of observed variables in terms of a smaller set of unobserved variables (called factors) * Meta-analysis – combines the results of several studies that address a set of related research hypotheses *
Multivariate analysis Multivariate statistics is a subdivision of statistics encompassing the simultaneous observation and analysis of more than one outcome variable. Multivariate statistics concerns understanding the different aims and background of each of the dif ...
 – analysis of data involving several variables, such as by factor analysis, regression analysis, or principal component analysis *
Principal component analysis Principal component analysis (PCA) is a popular technique for analyzing large datasets containing a high number of dimensions/features per observation, increasing the interpretability of data while preserving the maximum amount of information, and ...
 – transformation of a sample of correlated variables into uncorrelated variables (called principal components), mostly used in exploratory data analysis * Regression analysis – techniques for analysing the relationships between several predictive variables and one or more outcomes in the data * Scale analysis (statistics) – methods to analyse survey data by scoring responses on a numeric scale * Sensitivity analysis – the study of how the variation in the output of a model depends on variations in the inputs * Sequential analysis – evaluation of sampled data as it is collected, until the criterion of a stopping rule is met * Spatial analysis – the study of entities using geometric or geographic properties *
Time-series analysis In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Exa ...
 – methods that attempt to understand a sequence of data points spaced apart at uniform time intervals


Other

*
Aura analysis According to spiritual beliefs, an aura or energy field is a colored emanation said to enclose a human body or any animal or object. In some esoteric positions, the aura is described as a subtle body. Psychics and holistic medicine practitioner ...
 – a technique in which supporters of the method claim that the body's aura, or energy field is analysed * Bowling analysis – Analysis of the performance of
cricket Cricket is a bat-and-ball game played between two teams of eleven players on a field at the centre of which is a pitch with a wicket at each end, each comprising two bails balanced on three stumps. The batting side scores runs by striki ...
players * Lithic analysis – the analysis of stone tools using basic scientific techniques **Lithic analysis is most often used by archeologists in determining which types of tools were used at a given time period pertaining to current artifacts discovered. *
Protocol analysis Protocol analysis is a psychological research method that elicits verbal reports from research participants. Protocol analysis is used to study thinking in cognitive psychology (Crutcher, 1994), cognitive science (Simon & Kaplan, 1989), and beha ...
– a means for extracting persons' thoughts while they are performing a task


See also

* Formal analysis * Metabolism in biology * Methodology * Scientific method


References


External links

* * * {{Authority control Abstraction Critical thinking skills Emergence Empiricism Epistemological theories Intelligence Mathematical modeling Metaphysics of mind Methodology Ontology Philosophy of logic Rationalism Reasoning Research methods Scientific method Theory of mind