http___en.wikipedia.org_wiki_Interpolation.xml
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Интерполация Интерполация в числовия анализ е метод на конструиране на нови числови стойности в област от множество на изолирани точки от известни числови стойности. В инженерните и други науки може да са налице брой от числови стойности, които са придобити чрез проби или експеримент и да е необходимо да се създаде функция, която много близко да покрива тези стойности. Това се нарича създаване на крива или регресивен анализ. Така че интерполацията е специфичен случай на създаване на крива, при който функцията трябва да мине точно през дадените числови стойности. Друг вид задача, която се решава с интерполация е апроксимацията (приближаването) на сложна функция до проста функция. Например може да имаме фунция, която е твърде сложна, за да се оцени ефикасно. В този случай може да се изберат някои точки за създаване на таблица и след това от тях да се състави по-опростена функция. Има и друг вид интерполация в математиката - "интерполация на оператори" като класически резултат от интерполация на оператори са Теорема на Рисц-Торин and the Теорема на Маркинкиевиц, както и много други подрезултати. С други думи е метод, при който функция, зададена таблично (чрез стойностите в отделни точки), се замества с аналитична функция y=f(x), така че стойностите на функцията f(x) във възлите на интерполиране да бъдат равни на съответните таблични стойности на наблюденията. В геометричен смисъл графиката на f(x) минава през точките на интерполиране в координатната равнина. Според вида на функцията f(x) в участъците между възлите, интерполацията може да бъде линейна, параболична, билинейна, бикубична и други, в зависимост от избраната функция за интерполиране. Съдържание Линейна интерполация[редактиране | редактиране на кода] Нека функцията, която ще интерполираме, се нарича f(x). Отсечката от тази функция [X0 ; X1] се заменя с линейна функция, стойностите на която при X=X0 и X=X1 съвпадат със стойностите на дадената функция f(X0) и f(X1). Тоест в отрязъка [X0;X1] имаме два възела на интерполация X0 и X1, а интерполационният многочлен е от първа степен. Означаваме f(X0)=Y0, f(X1)=Y1, f(X)=Y. Интерполиращата линейна функция се намира, като се напише уравнението на права, минаваща през две точки (X0,Y0) и (X1,Y1): - (Лагранж) което може да се представи също Полиномна интерполация[редактиране | редактиране на кода] Полиномна интерполация e генерализация на линеарната интерполация. Всъщност линеарната интерполация е линеарна функция и тук заместваме този интерполант с полином от по-висока степен. Приложение в икономиката[редактиране | редактиране на кода] Методът се използва в икономиката при изчисляването на някои оптимизационни задачи, като например намирането на вътрешната норма на възвръщаемост на инвестиция. Пример: Фирма има възможност да избира между два варианта за инвестиции: - Първоначален инвестиционен разход – 20 млн. лева, очаквани нетни парични потоци 4 млн. лева годишно в продължение на четири години. - Начална инвестиция – 15 млн. лева, нетни парични потоци по 3,8 млн. лева годишно през следващите пет години. Кой от двата варианта е по-изгоден за фирмата? Необходими формули: – вътрешна норма на възвращаемост – нетна настояща стойност – очаквани приходи – първоначална инвестиция – търсената дискантова ставка, която привежда към нула Обработка на изображения[редактиране | редактиране на кода] Интерполацията е основен метод за увеличение при цифрови изображения. Стойностите на яркостта на всеки пиксел са дискретните стойности, които служат за интерполационни възли. Литература[редактиране | редактиране на кода] Поляк, „Численные методы анализа наблюдений“
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Interpolation In the mathematical field of numerical analysis, interpolation is a method of constructing new data points within 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 (i.e. estimate) the value of that function for an intermediate value of the independent variable. This may be achieved by curve fitting or regression analysis. A different problem which is closely related to interpolation is the approximation of a complicated function by a simple function. Suppose the formula for some given function is known, but too complex to evaluate efficiently. A few known data points from the original function can be used to create an interpolation based on a simpler function. Of course, when a simple function is used to estimate data points from the original, interpolation errors are usually present; however, depending on the problem domain and the interpolation method used, the gain in simplicity may be of greater value than the resultant loss in accuracy. There is also another very different kind of interpolation in mathematics, namely the "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. Contents - 1 Example - 2 Interpolation via Gaussian processes - 3 Other forms of interpolation - 4 Interpolation in digital signal processing - 5 Related concepts - 6 See also - 7 References - 8 External links Example[edit] For example, suppose we have a table like this, which gives some values of an unknown function f. Interpolation provides a means of estimating the function at intermediate points, such as x = 2.5. There are many different interpolation methods, some of which are described below. Some of the concerns to take into account when choosing an appropriate algorithm are: How accurate is the method? How expensive is it? How smooth is the interpolant? How many data points are needed? Piecewise constant interpolation[edit] 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[edit] 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: 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 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[edit] 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: Substituting x = 2.5, we find that f(2.5) = 0.5965. 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). More generally, the shape of the resulting curve, especially for very high or low values of the independent variable, may be contrary to commonsense, i.e. 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[edit] Remember that 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 In this case we get f(2.5) = 0.5972. Like polynomial interpolation, spline interpolation incurs a smaller error than linear interpolation and the interpolant is smoother. However, the interpolant is easier to evaluate than the high-degree polynomials used in polynomial interpolation. It also does not suffer from Runge's phenomenon. Interpolation via Gaussian processes[edit] 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, i.e., for fitting a curve through noisy data. In the geostatistics community Gaussian process regression is also known as Kriging. Other forms of interpolation[edit] Other forms of interpolation can be constructed by picking a different class of interpolants. For instance, rational interpolation is interpolation by rational functions, and trigonometric interpolation is interpolation by trigonometric polynomials. Another possibility is to use wavelets. The Whittaker–Shannon interpolation formula can be used if the number of data points is infinite. 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. 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. Interpolation in digital signal processing[edit] 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 a higher sampling rate (Upsampling) using various digital filtering techniques (e.g., 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 (i.e., 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.[1] Related concepts[edit] The term extrapolation is used if we want 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. See also[edit] References[edit] - ^ R.E. Crochiere and L.R. Rabiner. Multirate Digital Signal Processing. Englewood Cliffs, NJ: Prentice–Hall.