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International trade

 International trade is the exchange of capital, goods, and services across international borders or territories because there is a need or want of goods or services. (see: World economy)

In most countries, such trade represents a significant share of gross domestic product (GDP). While international trade has existed throughout history (for example Uttarapatha, Silk Road, Amber Road, salt roads), its economic, social, and political importance has been on the rise in recent centuries.

Carrying out trade at an international level is a complex process when compared to domestic trade. When trade takes place between two or more states, factors like currency, government policies, economy, judicial system, laws, and markets influence trade.

To ease and justify the process of trade between countries of different economic standing in the modern era, some international economic organizations were formed, such as the World Trade Organization. These organizations work towards the facilitation and growth of international trade. Statistical services of intergovernmental and supranational organizations and governmental statistical agencies publish official statistics on international trade.

Characteristics of global trade
A product that is transferred or sold from a party in one country to a party in another country is an export from the originating country, and an import to the country receiving that product. Imports and exports are accounted for in a country's current account in the balance of payments.

Trading globally may give consumers and countries the opportunity to be exposed to new markets and products. Almost every kind of product can be found in the international market, for example: food, clothes, spare parts, oil, jewellery, wine, stocks, currencies, and water. Services are also traded, such as in tourism, banking, consulting, and transportation.

Advanced technology (including transportation), globalization, industrialization, outsourcing and multinational corporations have major impacts on the international trade systems

Differences from domestic trade
International trade is, in principle, not different from domestic trade as the motivation and the behavior of parties involved in a trade do not change fundamentally regardless of whether trade is across a border or not.

However, in practical terms, carrying out trade at an international level is typically a more complex process than domestic trade. The main difference is that international trade is typically more costly than domestic trade. This is due to the fact that cross-border trade typically incurs additional costs such as explicit tariffs as well as explicit or implicit non-tariff barriers such as time costs (due to border delays), language and cultural differences, product safety, the legal system, and so on.

Another difference between domestic and international trade is that factors of production such as capital and labor are often more mobile within a country than across countries. Thus, international trade is mostly restricted to trade in goods and services, and only to a lesser extent to trade in capital, labour, or other factors of production. Trade in goods and services can serve as a substitute for trade in factors of production. Instead of importing a factor of production, a country can import goods that make intensive use of that factor of production and thus embody it. An example of this is the import of labor-intensive goods by the United States from China. Instead of importing Chinese labor, the United States imports goods that were produced with Chinese labor. One report in 2010, suggested that international trade was increased when a country hosted a network of immigrants, but the trade effect was weakened when the immigrants became assimilated into their new country.

History
The history of international trade chronicles notable events that have affected trading among various economies.

Theories and models
There are several models that seek to explain the factors behind international trade, the welfare consequences of trade and the pattern of trade.

Largest countries or regions by total international trade

The following table is a list of the 25 largest trading states according to the World Trade Organization in 2021 and 2022.

Top traded commodities by value (exports)
Source: International Trade Centre

Observances
In the US, the various U.S. Presidents have held observances to promote big and small companies to be more involved with the export and import of goods and services. President George W. Bush observed World Trade Week on May 18, 2001, and May 17, 2002. On May 13, 2016, President Barack Obama proclaimed May 15 through May 21, 2016, World Trade Week, 2016. On May 19, 2017, President Donald Trump proclaimed May 21 through May 27, 2017, World Trade Week, 2017. World Trade Week is the third week of May. Every year the President declares that week to be World Trade Week.

Food security
The trade-offs between local food production and distant food production are controversial with limited studies comparing environmental impact and scientists cautioning that regionally specific environmental impacts should be considered. A 2020 study indicated that local food crop production alone cannot meet the demand for most food crops with "current production and consumption patterns" and the locations of food production at the time of the study for 72–89% of the global population and 100–km radiuses as of early 2020. Studies found that food miles are a relatively minor factor of carbon emissions, albeit increased food localization may also enable additional, more significant, environmental benefits such as recycling of energy, water, and nutrients. For specific foods regional differences in harvest seasons may make it more environmentally friendly to import from distant regions than more local production and storage or local production in greenhouses.

Qualitative differences and economic aspects
Qualitative differences between substitutive products of different production regions may exist due to different legal requirements and quality standards or different levels of controllability by local production- and governance-systems which may have aspects of security beyond resource security, environmental protection, product quality and product design and health. The process of transforming supply as well as labor rights may differ as well.

Local production has been reported to increase local employment in many cases. A 2018 study claimed that international trade can increase local employment. A 2016 study found that local employment and total labor income in both manufacturing and nonmanufacturing were negatively affected by rising exposure to imports.

Local production in high-income countries, rather than distant regions may require higher wages for workers. Higher wages incentivize automation which could allow for automated workers' time to be reallocated by society and its economic mechanisms or be converted into leisure-like time.

Specialization, production efficiency and regional differences
Local production may require knowledge transfer, technology transfer and may not be able to compete in efficiency initially with specialized, established industries and businesses, or in consumer demand without policy measures such as eco-tariffs. Regional differences may cause specific regions to be more suitable for a specific production, thereby increasing the advantages of specific trade over specific local production. Forms of local products that are highly localized may not be able to meet the efficiency of more large-scale, highly consolidated production in terms of efficiency, including environmental impact.

Resource security
A systematic, and possibly first large-scale, cross-sectoral analysis of water, energy and land in security in 189 countries that links total and sectorial consumption to sources showed that countries and sectors are highly exposed to over-exploited, insecure, and degraded such resources with economic globalization having decreased security of global supply chains. The 2020 study finds that most countries exhibit greater exposure to resource risks via international trade – mainly from remote production sources – and that diversifying trading partners is unlikely to help countries and sectors to reduce these or to improve their resource self-sufficiency.

Illegal gold trade
A number of people in Africa, including children, were using informal or "artisanal" methods to produce gold. While millions were making a livelihood through the small-scale mining, governments of Ghana, Tanzania and Zambia complaint about the increase in illegal production and gold smuggling. Sometimes the procedure involved criminal operations and even human and environmental cost. Investigative reports based on Africa's export data revealed that gold in large quantities is smuggled out of the country through the United Arab Emirates, without any taxes being paid to the producing states. Analysis also reflected discrepancies in the amount exported from Africa and the total gold imported into the UAE.

In July 2020, a report by Swissaid highlighted that the Dubai-based precious metal refining firms, including Kaloti Jewellery International Group and Trust One Financial Services (T1FS), received most of their gold from poor African states like Sudan. The gold mines in Sudan were seldom under the militias involved in war crimes and human rights abuses. The Swissaid report also highlighted that the illicit gold coming into Dubai from Africa is imported in large quantities by the world's largest refinery in Switzerland, Valcambi.

Another report in March 2022 revealed the contradiction between the lucrative gold trade of West African countries and the illicit dealings. Like Sudan, Democratic Republic of Congo (DRC), Ghana and other states, differences were recorded in the gold production in Mali and its trade with Dubai, UAE. The third largest gold exporter in Africa, Mali imposed taxes only on first 50kg gold exports per month, which allowed several small-scale miners to enjoy tax exemptions and smuggle gold worth millions. In 2014, Mali's gold production was of 45.8 tonnes, while the UAE's gold import were at 59.9 tonnes.

In May 2024, a report released by Swissaid revealed that the prime recipient of tens of billions of dollars of smuggled African gold each year had been the UAE. Between the 2012 and 2022, 2596 tonnes of gold undeclared was exported out of Africa to the UAE. In 2022 alone, 435 tonnes (worth about $31 billion) of gold not declared was exported from Africa, and the UAE accounted for 93% of the undeclared exports. Following the Emirates, next two prime importers were Switzerland and India. Majority of the artisanal gold extracted in the 18 African countries ends up in the Emirates. As per Swiss regulations, gold’s place of origin is usually where it is last refined, because of which smuggled gold linked to conflicts and human rights violations legally enters Switzerland.

Statistics from intergovernmental sources
Data on the value of exports and imports and their quantities often broken down by detailed lists of products are available in statistical collections on international trade published by the statistical services of intergovernmental and supranational organisations and national statistical institutes. The definitions and methodological concepts applied for the various statistical collections on international trade often differ in terms of definition (e.g. special trade vs. general trade) and coverage (reporting thresholds, inclusion of trade in services, estimates for smuggled goods and cross-border provision of illegal services). Metadata providing information on definitions and methods are often published along with the data.
 * United Nations Commodity Trade Database
 * Trade Map, trade statistics for international business development
 * WTO Statistics Portal
 * Statistical Portal: OECD
 * European Union International Trade in Goods Data
 * Food and Agricultural Trade Data by FAO

Other external links

 * The MIT Observatory of Economic Complexity
 * The McGill Faculty of Law runs a Regional Trade Agreements Database that contains the text of almost all preferential and regional trade agreements in the world. ptas.mcgill.ca
 * Historical documents on international trade available on FRASER (St Louis Fed)

title: International trade


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Interpolation

In the mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing (finding) new data points based on the range of a discrete set of known data points.

In engineering and science, one often has a number of data points, obtained by sampling or experimentation, which represent the values of a function for a limited number of values of the independent variable. It is often required to interpolate; that is, estimate the value of that function for an intermediate value of the independent variable.

A closely related problem is the approximation of a complicated function by a simple function. Suppose the formula for some given function is known, but too complicated to evaluate efficiently. A few data points from the original function can be interpolated to produce a simpler function which is still fairly close to the original. The resulting gain in simplicity may outweigh the loss from interpolation error and give better performance in calculation process.

Example
This table gives some values of an unknown function $$f(x)$$. Interpolation provides a means of estimating the function at intermediate points, such as $$x=2.5.$$

We describe some methods of interpolation, differing in such properties as: accuracy, cost, number of data points needed, and smoothness of the resulting interpolant function.

Piecewise constant interpolation

The simplest interpolation method is to locate the nearest data value, and assign the same value. In simple problems, this method is unlikely to be used, as linear interpolation (see below) is almost as easy, but in higher-dimensional multivariate interpolation, this could be a favourable choice for its speed and simplicity.

Linear interpolation

One of the simplest methods is linear interpolation (sometimes known as lerp). Consider the above example of estimating f(2.5). Since 2.5 is midway between 2 and 3, it is reasonable to take f(2.5) midway between f(2) = 0.9093 and f(3) = 0.1411, which yields 0.5252.

Generally, linear interpolation takes two data points, say (xa,ya) and (xb,yb), and the interpolant is given by:
 * $$ y = y_a + \left( y_b-y_a \right) \frac{x-x_a}{x_b-x_a} \text{ at the point } \left( x,y \right) $$

 * $$ \frac{y-y_a}{y_b-y_a} = \frac{x-x_a}{x_b-x_a} $$

 * $$ \frac{y-y_a}{x-x_a} = \frac{y_b-y_a}{x_b-x_a} $$

This previous equation states that the slope of the new line between $$ (x_a,y_a) $$ and $$ (x,y) $$ is the same as the slope of the line between $$ (x_a,y_a) $$ and $$ (x_b,y_b) $$

Linear interpolation is quick and easy, but it is not very precise. Another disadvantage is that the interpolant is not differentiable at the point xk.

The following error estimate shows that linear interpolation is not very precise. Denote the function which we want to interpolate by g, and suppose that x lies between xa and xb and that g is twice continuously differentiable. Then the linear interpolation error is

 * $$ |f(x)-g(x)| \le C(x_b-x_a)^2 \quad\text{where}\quad C = \frac18 \max_{r\in[x_a,x_b]} |g''(r)|. $$

In words, the error is proportional to the square of the distance between the data points. The error in some other methods, including polynomial interpolation and spline interpolation (described below), is proportional to higher powers of the distance between the data points. These methods also produce smoother interpolants.

Polynomial interpolation

Polynomial interpolation is a generalization of linear interpolation. Note that the linear interpolant is a linear function. We now replace this interpolant with a polynomial of higher degree.

Consider again the problem given above. The following sixth degree polynomial goes through all the seven points:
 * $$ f(x) = -0.0001521 x^6 - 0.003130 x^5 + 0.07321 x^4 - 0.3577 x^3 + 0.2255 x^2 + 0.9038 x. $$

Substituting x = 2.5, we find that f(2.5) = ~0.59678.

Generally, if we have n data points, there is exactly one polynomial of degree at most n−1 going through all the data points. The interpolation error is proportional to the distance between the data points to the power n. Furthermore, the interpolant is a polynomial and thus infinitely differentiable. So, we see that polynomial interpolation overcomes most of the problems of linear interpolation.

However, polynomial interpolation also has some disadvantages. Calculating the interpolating polynomial is computationally expensive (see computational complexity) compared to linear interpolation. Furthermore, polynomial interpolation may exhibit oscillatory artifacts, especially at the end points (see Runge's phenomenon).

Polynomial interpolation can estimate local maxima and minima that are outside the range of the samples, unlike linear interpolation. For example, the interpolant above has a local maximum at x ≈ 1.566, f(x) ≈ 1.003 and a local minimum at x ≈ 4.708, f(x) ≈ −1.003. However, these maxima and minima may exceed the theoretical range of the function; for example, a function that is always positive may have an interpolant with negative values, and whose inverse therefore contains false vertical asymptotes.

More generally, the shape of the resulting curve, especially for very high or low values of the independent variable, may be contrary to commonsense; that is, to what is known about the experimental system which has generated the data points. These disadvantages can be reduced by using spline interpolation or restricting attention to Chebyshev polynomials.

Spline interpolation

Linear interpolation uses a linear function for each of intervals [xk,xk+1]. Spline interpolation uses low-degree polynomials in each of the intervals, and chooses the polynomial pieces such that they fit smoothly together. The resulting function is called a spline.

For instance, the natural cubic spline is piecewise cubic and twice continuously differentiable. Furthermore, its second derivative is zero at the end points. The natural cubic spline interpolating the points in the table above is given by

 * $$ f(x) = \begin{cases}

-0.1522 x^3 + 0.9937 x, & \text{if } x \in [0,1], \\ -0.01258 x^3 - 0.4189 x^2 + 1.4126 x - 0.1396, & \text{if } x \in [1,2], \\ 0.1403 x^3 - 1.3359 x^2 + 3.2467 x - 1.3623, & \text{if } x \in [2,3], \\ 0.1579 x^3 - 1.4945 x^2 + 3.7225 x - 1.8381, & \text{if } x \in [3,4], \\ 0.05375 x^3 -0.2450 x^2 - 1.2756 x + 4.8259, & \text{if } x \in [4,5], \\ -0.1871 x^3 + 3.3673 x^2 - 19.3370 x + 34.9282, & \text{if } x \in [5,6]. \end{cases} $$

In this case we get f(2.5) = 0.5972.

Like polynomial interpolation, spline interpolation incurs a smaller error than linear interpolation, while the interpolant is smoother and easier to evaluate than the high-degree polynomials used in polynomial interpolation. However, the global nature of the basis functions leads to ill-conditioning. This is completely mitigated by using splines of compact support, such as are implemented in Boost.Math and discussed in Kress.

Mimetic interpolation
Depending on the underlying discretisation of fields, different interpolants may be required. In contrast to other interpolation methods, which estimate functions on target points, mimetic interpolation evaluates the integral of fields on target lines, areas or volumes, depending on the type of field (scalar, vector, pseudo-vector or pseudo-scalar).

A key feature of mimetic interpolation is that vector calculus identities are satisfied, including Stokes' theorem and the divergence theorem. As a result, mimetic interpolation conserves line, area and volume integrals. Conservation of line integrals might be desirable when interpolating the electric field, for instance, since the line integral gives the electric potential difference at the endpoints of the integration path. Mimetic interpolation ensures that the error of estimating the line integral of an electric field is the same as the error obtained by interpolating the potential at the end points of the integration path, regardless of the length of the integration path.

Linear, bilinear and trilinear interpolation are also considered mimetic, even if it is the field values that are conserved (not the integral of the field). Apart from linear interpolation, area weighted interpolation can be considered one of the first mimetic interpolation methods to have been developed.

Function approximation
Interpolation is a common way to approximate functions. Given a function $$f:[a,b] \to \mathbb{R}$$ with a set of points $$x_1, x_2, \dots, x_n \in [a, b]$$ one can form a function $$s: [a,b] \to \mathbb{R}$$ such that $$f(x_i)=s(x_i)$$ for $$i=1, 2, \dots, n$$ (that is, that $$s$$ interpolates $$f$$ at these points). In general, an interpolant need not be a good approximation, but there are well known and often reasonable conditions where it will. For example, if $$f\in C^4([a,b])$$ (four times continuously differentiable) then cubic spline interpolation has an error bound given by $$\|f-s\|_\infty \leq C \|f^{(4)}\|_\infty h^4$$ where $$h \max_{i=1,2, \dots, n-1} |x_{i+1}-x_i|$$ and $$C$$ is a constant.

Via Gaussian processes
Gaussian process is a powerful non-linear interpolation tool. Many popular interpolation tools are actually equivalent to particular Gaussian processes. Gaussian processes can be used not only for fitting an interpolant that passes exactly through the given data points but also for regression; that is, for fitting a curve through noisy data. In the geostatistics community Gaussian process regression is also known as Kriging.

Other forms
Other forms of interpolation can be constructed by picking a different class of interpolants. For instance, rational interpolation is interpolation by rational functions using Padé approximant, and trigonometric interpolation is interpolation by trigonometric polynomials using Fourier series. Another possibility is to use wavelets.

The Whittaker–Shannon interpolation formula can be used if the number of data points is infinite or if the function to be interpolated has compact support.

Sometimes, we know not only the value of the function that we want to interpolate, at some points, but also its derivative. This leads to Hermite interpolation problems.

When each data point is itself a function, it can be useful to see the interpolation problem as a partial advection problem between each data point. This idea leads to the displacement interpolation problem used in transportation theory.

In higher dimensions
Multivariate interpolation is the interpolation of functions of more than one variable. Methods include bilinear interpolation and bicubic interpolation in two dimensions, and trilinear interpolation in three dimensions. They can be applied to gridded or scattered data. Mimetic interpolation generalizes to $$n$$ dimensional spaces where $$n > 3$$.

In digital signal processing
In the domain of digital signal processing, the term interpolation refers to the process of converting a sampled digital signal (such as a sampled audio signal) to that of a higher sampling rate (Upsampling) using various digital filtering techniques (for example, convolution with a frequency-limited impulse signal). In this application there is a specific requirement that the harmonic content of the original signal be preserved without creating aliased harmonic content of the original signal above the original Nyquist limit of the signal (that is, above fs/2 of the original signal sample rate). An early and fairly elementary discussion on this subject can be found in Rabiner and Crochiere's book Multirate Digital Signal Processing.

Related concepts
The term extrapolation is used to find data points outside the range of known data points.

In curve fitting problems, the constraint that the interpolant has to go exactly through the data points is relaxed. It is only required to approach the data points as closely as possible (within some other constraints). This requires parameterizing the potential interpolants and having some way of measuring the error. In the simplest case this leads to least squares approximation.

Approximation theory studies how to find the best approximation to a given function by another function from some predetermined class, and how good this approximation is. This clearly yields a bound on how well the interpolant can approximate the unknown function.

Generalization
If we consider $$x$$ as a variable in a topological space, and the function $$f(x)$$ mapping to a Banach space, then the problem is treated as "interpolation of operators". The classical results about interpolation of operators are the Riesz–Thorin theorem and the Marcinkiewicz theorem. There are also many other subsequent results.

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

title: Interpolation


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Intension

In any of several fields of study that treat the use of signs—for example, in linguistics, logic, mathematics, semantics, semiotics, and philosophy of language—an intension is any property or quality connoted by a word, phrase, or another symbol. In the case of a word, the word's definition often implies an intension. For instance, the intensions of the word plant include properties such as "being composed of cellulose (not always true)", "alive", and "organism", among others. A comprehension is the collection of all such intensions.

Overview
The meaning of a word can be thought of as the bond between the idea the word means and the physical form of the word. Swiss linguist Ferdinand de Saussure (1857–1913) contrasts three concepts:

 * 1) the signifier – the "sound image" or the string of letters on a page that one recognizes as the form of a sign
 * 2) the signified – the meaning, the concept or idea that a sign expresses or evokes
 * 3) the referent – the actual thing or set of things a sign refers to. See Dyadic signs and Reference (semantics).

Without intension of some sort, a word has no meaning. For instance, the terms rantans or brillig have no intension and hence no meaning. Such terms may be suggestive, but a term can be suggestive without being meaningful. For instance, ran tan is an archaic onomatopoeia for chaotic noise or din and may suggest to English speakers a din or meaningless noise, and brillig though made up by Lewis Carroll may be suggestive of 'brilliant' or 'frigid'. Such terms, it may be argued, are always intensional since they connote the property 'meaningless term', but this is only an apparent paradox and does not constitute a counterexample to the claim that without intension a word has no meaning. Part of its intension is that it has no extension. Intension is analogous to the signified in the Saussurean system, extension to the referent.

In philosophical arguments about dualism versus monism, it is noted that thoughts have intensionality and physical objects do not (S. E. Palmer, 1999), but rather have extension in space and time.

Statement forms
A statement-form is simply a form obtained by putting blanks into a sentence where one or more expressions with extensions occur—for instance, "The quick brown ___ jumped over the lazy ___'s back." An instance of the form is a statement obtained by filling the blanks in.

Intensional statement form
An intensional statement-form is a statement-form with at least one instance such that substituting co-extensive expressions into it does not always preserve logical value. An intensional statement is a statement that is an instance of an intensional statement-form. Here co-extensive expressions are expressions with the same extension.

That is, a statement-form is intensional if it has, as one of its instances, a statement for which there are two co-extensive expressions (in the relevant language) such that one of them occurs in the statement, and if the other one is put in its place (uniformly, so that it replaces the former expression wherever it occurs in the statement), the result is a (different) statement with a different logical value. An intensional statement, then, is an instance of such a form; it has the same form as a statement in which substitution of co-extensive terms fails to preserve logical value.

Examples

 * 1) Everyone who has read Huckleberry Finn knows that Mark Twain wrote it.
 * 2) It is possible that Aristotle did not tutor Alexander the Great.
 * 3) Aristotle was pleased that he had a sister.

To see that these are intensional, make the following substitutions: (1) "Mark Twain" → "The author of 'Corn-pone Opinions'"; (2) "Aristotle" → "the tutor of Alexander the Great"; (3) can be seen to be intensional given "had a sister" → "had a female sibling."

The intensional statements above feature expressions like "knows", "possible", and "pleased". Such expressions always, or nearly always, produce intensional statements when added (in some intelligible manner) to an extensional statement, and thus they (or more complex expressions like "It is possible that") are sometimes called intensional operators. A large class of intensional statements, but by no means all, can be spotted from the fact that they contain intensional operators.

Extensional statement form
An extensional statement is a non-intensional statement. Substitution of co-extensive expressions into it always preserves logical value. A language is intensional if it contains intensional statements, and extensional otherwise. All natural languages are intensional. The only extensional languages are artificially constructed languages used in mathematical logic or for other special purposes and small fragments of natural languages.

Examples

 * 1) Mark Twain wrote Huckleberry Finn.
 * 2) Aristotle had a sister.

Note that if "Samuel Clemens" is put into (1) in place of "Mark Twain", the result is as true as the original statement. It should be clear that no matter what is put for "Mark Twain", so long as it is a singular term picking out the same man, the statement remains true. Likewise, we can put in place of the predicate any other predicate belonging to Mark Twain and only to Mark Twain, without changing the logical value. For (2), likewise, consider the following substitutions: "Aristotle" → "The tutor of Alexander the Great"; "Aristotle" → "The author of the 'Prior Analytics'"; "had a sister" → "had a sibling whose body was capable of producing egg cells"; "had a sister" → "had a parent who had a female child".

title: Intension