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125-416: In model-based FDI techniques some model of the system is used to decide about the occurrence of fault. The system model may be mathematical or knowledge based. Some of the model-based FDI techniques include observer-based approach, parity-space approach, and parameter identification based methods. There is another trend of model-based FDI schemes, which is called set-membership methods. These methods guarantee

250-503: A paradigm shift offers radical simplification. For example, when modeling the flight of an aircraft, we could embed each mechanical part of the aircraft into our model and would thus acquire an almost white-box model of the system. However, the computational cost of adding such a huge amount of detail would effectively inhibit the usage of such a model. Additionally, the uncertainty would increase due to an overly complex system, because each separate part induces some amount of variance into

375-400: A prior probability distribution (which can be subjective), and then update this distribution based on empirical data. An example of when such approach would be necessary is a situation in which an experimenter bends a coin slightly and tosses it once, recording whether it comes up heads, and is then given the task of predicting the probability that the next flip comes up heads. After bending

500-508: A satisfiability modulo theories problem solvable by brute force (Haynal & Haynal, 2011). Most of the attempts to lower or prove the complexity of FFT algorithms have focused on the ordinary complex-data case, because it is the simplest. However, complex-data FFTs are so closely related to algorithms for related problems such as real-data FFTs, discrete cosine transforms , discrete Hartley transforms , and so on, that any improvement in one of these would immediately lead to improvements in

625-402: A DFT as a convolution, but this time of the same size (which can be zero-padded to a power of two and evaluated by radix-2 Cooley–Tukey FFTs, for example), via the identity Hexagonal fast Fourier transform (HFFT) aims at computing an efficient FFT for the hexagonally-sampled data by using a new addressing scheme for hexagonal grids, called Array Set Addressing (ASA). In many applications,

750-413: A DFT of power-of-two length n = 2 m {\displaystyle n=2^{m}} . Moreover, explicit algorithms that achieve this count are known (Heideman & Burrus , 1986; Duhamel, 1990 ). However, these algorithms require too many additions to be practical, at least on modern computers with hardware multipliers (Duhamel, 1990; Frigo & Johnson , 2005). A tight lower bound

875-453: A bound on a measure of the FFT algorithm's "asynchronicity", but the generality of this assumption is unclear. For the case of power-of-two n , Papadimitriou (1979) argued that the number n log 2 ⁡ n {\textstyle n\log _{2}n} of complex-number additions achieved by Cooley–Tukey algorithms is optimal under certain assumptions on the graph of

1000-436: A common approach is to split the data into two disjoint subsets: training data and verification data. The training data are used to estimate the model parameters. An accurate model will closely match the verification data even though these data were not used to set the model's parameters. This practice is referred to as cross-validation in statistics. Defining a metric to measure distances between observed and predicted data

1125-493: A complex DFT of half the length (whose real and imaginary parts are the even/odd elements of the original real data), followed by O ( n ) {\displaystyle O(n)} post-processing operations. It was once believed that real-input DFTs could be more efficiently computed by means of the discrete Hartley transform (DHT), but it was subsequently argued that a specialized real-input DFT algorithm (FFT) can typically be found that requires fewer operations than

1250-546: A computer, a model that is computationally feasible to compute is made from the basic laws or from approximate models made from the basic laws. For example, molecules can be modeled by molecular orbital models that are approximate solutions to the Schrödinger equation. In engineering , physics models are often made by mathematical methods such as finite element analysis . Different mathematical models use different geometries that are not necessarily accurate descriptions of

1375-1062: A few feature values from signals, causing a dimensionality reduction from the original signal . By using Convolutional neural networks , the continuous wavelet transform scalogram can be directly classified to normal and faulty classes. Such a technique avoids omitting any important fault message and results in a better performance of fault detection and diagnosis. In addition, by transforming signals to image constructions, 2D Convolutional neural networks can be implemented to identify faulty signals from vibration image features. Deep belief networks , Restricted Boltzmann machines and Autoencoders are other deep neural networks architectures which have been successfully used in this field of research. In comparison to traditional machine learning , due to their deep architecture, deep learning models are able to learn more complex structures from datasets , however, they need larger samples and longer processing time to achieve higher accuracy. Fault Recovery in FDIR

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1500-440: A field where calculation of Fourier transforms presented a formidable bottleneck. While many methods in the past had focused on reducing the constant factor for O ( n 2 ) {\textstyle O(n^{2})} computation by taking advantage of "symmetries", Danielson and Lanczos realized that one could use the "periodicity" and apply a "doubling trick" to "double [ n ] with only slightly more than double

1625-451: A given model involving a variety of abstract structures. In general, mathematical models may include logical models . In many cases, the quality of a scientific field depends on how well the mathematical models developed on the theoretical side agree with results of repeatable experiments. Lack of agreement between theoretical mathematical models and experimental measurements often leads to important advances as better theories are developed. In

1750-763: A given signal into normal and faulty segments. Machine fault diagnosis is a field of mechanical engineering concerned with finding faults arising in machines. A particularly well developed part of it applies specifically to rotating machinery, one of the most common types encountered. To identify the most probable faults leading to failure, many methods are used for data collection, including vibration monitoring, thermal imaging , oil particle analysis, etc. Then these data are processed utilizing methods like spectral analysis , wavelet analysis , wavelet transform, short term Fourier transform, Gabor Expansion, Wigner-Ville distribution (WVD), cepstrum, bispectrum, correlation method, high resolution spectral analysis, waveform analysis (in

1875-401: A human system, we know that usually the amount of medicine in the blood is an exponentially decaying function, but we are still left with several unknown parameters; how rapidly does the medicine amount decay, and what is the initial amount of medicine in blood? This example is therefore not a completely white-box model. These parameters have to be estimated through some means before one can use

2000-415: A large- n example ( n = 2 ) using a probabilistic approximate algorithm (which estimates the largest k coefficients to several decimal places). FFT algorithms have errors when finite-precision floating-point arithmetic is used, but these errors are typically quite small; most FFT algorithms, e.g. Cooley–Tukey, have excellent numerical properties as a consequence of the pairwise summation structure of

2125-431: A priori information on the system is available. A black-box model is a system of which there is no a priori information available. A white-box model (also called glass box or clear box) is a system where all necessary information is available. Practically all systems are somewhere between the black-box and white-box models, so this concept is useful only as an intuitive guide for deciding which approach to take. Usually, it

2250-521: A rotating machine are strongly related to its rotational speed, it can be said that they are time-variant signals in nature. These time-variant features carry the machine fault signatures. Consequently, how these features are extracted and interpreted is important to research and industrial applications. The most common method used in signal analysis is the FFT , or Fourier transform. The Fourier transform and its inverse counterpart offer two perspectives to study

2375-441: A row-column algorithm. Other, more complicated, methods include polynomial transform algorithms due to Nussbaumer (1977), which view the transform in terms of convolutions and polynomial products. See Duhamel and Vetterli (1990) for more information and references. An O ( n 5 / 2 log ⁡ n ) {\textstyle O(n^{5/2}\log n)} generalization to spherical harmonics on

2500-464: A sequence, or its inverse (IDFT). Fourier analysis converts a signal from its original domain (often time or space) to a representation in the frequency domain and vice versa. The DFT is obtained by decomposing a sequence of values into components of different frequencies. This operation is useful in many fields, but computing it directly from the definition is often too slow to be practical. An FFT rapidly computes such transformations by factorizing

2625-467: A set of d nested summations (over n j = 0 … N j − 1 {\textstyle n_{j}=0\ldots N_{j}-1} for each j ), where the division n / N = ( n 1 / N 1 , … , n d / N d ) {\textstyle \mathbf {n} /\mathbf {N} =\left(n_{1}/N_{1},\ldots ,n_{d}/N_{d}\right)}

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2750-466: A signal: via the time domain or via the frequency domain. The FFT -based spectrum of a time signal shows us the existence of its frequency contents. By studying these and their magnitude or phase relations, we can obtain various types of information, such as harmonics , sidebands , beat frequency , bearing fault frequency and so on. However, the FFT is only suitable for signals whose frequency contents do not change over time; however, as mentioned above,

2875-555: A simple procedure checking the linearity, impulse-response, and time-shift properties of the transform on random inputs (Ergün, 1995). The values for intermediate frequencies may be obtained by various averaging methods. As defined in the multidimensional DFT article, the multidimensional DFT transforms an array x n with a d -dimensional vector of indices n = ( n 1 , … , n d ) {\textstyle \mathbf {n} =\left(n_{1},\ldots ,n_{d}\right)} by

3000-495: A thousand times less than with direct evaluation. In practice, actual performance on modern computers is usually dominated by factors other than the speed of arithmetic operations and the analysis is a complicated subject (for example, see Frigo & Johnson , 2005), but the overall improvement from O ( n 2 ) {\textstyle O(n^{2})} to O ( n log ⁡ n ) {\textstyle O(n\log n)} remains. By far

3125-475: Is time domain reflectometry where a signal is sent down a cable or electrical line and the reflected signal is compared mathematically to original signal to identify faults. Spread Spectrum Time Domain Reflectometry, for instance, involves sending down a spread spectrum signal down a wire line to detect wire faults. Several clustering methods have also been proposed to identify the novel fault and segment

3250-514: Is a large part of the field of operations research . Mathematical models are also used in music , linguistics , and philosophy (for example, intensively in analytic philosophy ). A model may help to explain a system and to study the effects of different components, and to make predictions about behavior. Mathematical models can take many forms, including dynamical systems , statistical models , differential equations , or game theoretic models . These and other types of models can overlap, with

3375-538: Is a useful tool for assessing model fit. In statistics, decision theory, and some economic models , a loss function plays a similar role. While it is rather straightforward to test the appropriateness of parameters, it can be more difficult to test the validity of the general mathematical form of a model. In general, more mathematical tools have been developed to test the fit of statistical models than models involving differential equations . Tools from nonparametric statistics can sometimes be used to evaluate how well

3500-403: Is already known from direct investigation of the phenomenon being studied. An example of such criticism is the argument that the mathematical models of optimal foraging theory do not offer insight that goes beyond the common-sense conclusions of evolution and other basic principles of ecology. It should also be noted that while mathematical modeling uses mathematical concepts and language, it

3625-451: Is based on interpreting the FFT as a recursive factorization of the polynomial z n − 1 {\displaystyle z^{n}-1} , here into real-coefficient polynomials of the form z m − 1 {\displaystyle z^{m}-1} and z 2 m + a z m + 1 {\displaystyle z^{2m}+az^{m}+1} . Another polynomial viewpoint

3750-584: Is based on the compressibility (rank deficiency) of the Fourier matrix itself rather than the compressibility (sparsity) of the data. Conversely, if the data are sparse—that is, if only k out of n Fourier coefficients are nonzero—then the complexity can be reduced to O ( k log ⁡ n log ⁡ n / k ) {\displaystyle O(k\log n\log n/k)} , and this has been demonstrated to lead to practical speedups compared to an ordinary FFT for n / k > 32 in

3875-533: Is common to use idealized models in physics to simplify things. Massless ropes, point particles, ideal gases and the particle in a box are among the many simplified models used in physics. The laws of physics are represented with simple equations such as Newton's laws, Maxwell's equations and the Schrödinger equation . These laws are a basis for making mathematical models of real situations. Many real situations are very complex and thus modeled approximately on

Fault detection and isolation - Misplaced Pages Continue

4000-423: Is defined by the formula where e i 2 π / n {\displaystyle e^{i2\pi /n}} is a primitive n 'th root of 1. Evaluating this definition directly requires O ( n 2 ) {\textstyle O(n^{2})} operations: there are n outputs X k  , and each output requires a sum of n terms. An FFT is any method to compute

4125-526: Is described by Rokhlin and Tygert. The fast folding algorithm is analogous to the FFT, except that it operates on a series of binned waveforms rather than a series of real or complex scalar values. Rotation (which in the FFT is multiplication by a complex phasor) is a circular shift of the component waveform. Various groups have also published "FFT" algorithms for non-equispaced data, as reviewed in Potts et al. (2001). Such algorithms do not strictly compute

4250-506: Is exploited by the Winograd FFT algorithm, which factorizes z n − 1 {\displaystyle z^{n}-1} into cyclotomic polynomials —these often have coefficients of 1, 0, or −1, and therefore require few (if any) multiplications, so Winograd can be used to obtain minimal-multiplication FFTs and is often used to find efficient algorithms for small factors. Indeed, Winograd showed that

4375-437: Is not even) (see Frigo and Johnson, 2005). Still, this remains a straightforward variation of the row-column algorithm that ultimately requires only a one-dimensional FFT algorithm as the base case, and still has O ( n log ⁡ n ) {\displaystyle O(n\log n)} complexity. Yet another variation is to perform matrix transpositions in between transforming subsequent dimensions, so that

4500-640: Is not itself a branch of mathematics and does not necessarily conform to any mathematical logic , but is typically a branch of some science or other technical subject, with corresponding concepts and standards of argumentation. Mathematical models are of great importance in the natural sciences, particularly in physics . Physical theories are almost invariably expressed using mathematical models. Throughout history, more and more accurate mathematical models have been developed. Newton's laws accurately describe many everyday phenomena, but at certain limits theory of relativity and quantum mechanics must be used. It

4625-599: Is not known on the number of required additions, although lower bounds have been proved under some restrictive assumptions on the algorithms. In 1973, Morgenstern proved an Ω ( n log ⁡ n ) {\displaystyle \Omega (n\log n)} lower bound on the addition count for algorithms where the multiplicative constants have bounded magnitudes (which is true for most but not all FFT algorithms). Pan (1986) proved an Ω ( n log ⁡ n ) {\displaystyle \Omega (n\log n)} lower bound assuming

4750-429: Is not rigorously proved whether DFTs truly require Ω ( n log ⁡ n ) {\textstyle \Omega (n\log n)} (i.e., order n log ⁡ n {\displaystyle n\log n} or greater) operations, even for the simple case of power of two sizes, although no algorithms with lower complexity are known. In particular, the count of arithmetic operations

4875-765: Is often advantageous for cache locality to group the dimensions recursively. For example, a three-dimensional FFT might first perform two-dimensional FFTs of each planar "slice" for each fixed n 1 , and then perform the one-dimensional FFTs along the n 1 direction. More generally, an asymptotically optimal cache-oblivious algorithm consists of recursively dividing the dimensions into two groups ( n 1 , … , n d / 2 ) {\textstyle (n_{1},\ldots ,n_{d/2})} and ( n d / 2 + 1 , … , n d ) {\textstyle (n_{d/2+1},\ldots ,n_{d})} that are transformed recursively (rounding if d

5000-439: Is performed element-wise. Equivalently, it is the composition of a sequence of d sets of one-dimensional DFTs, performed along one dimension at a time (in any order). This compositional viewpoint immediately provides the simplest and most common multidimensional DFT algorithm, known as the row-column algorithm (after the two-dimensional case, below). That is, one simply performs a sequence of d one-dimensional FFTs (by any of

5125-415: Is preferable to use as much a priori information as possible to make the model more accurate. Therefore, the white-box models are usually considered easier, because if you have used the information correctly, then the model will behave correctly. Often the a priori information comes in forms of knowing the type of functions relating different variables. For example, if we make a model of how a medicine works in

Fault detection and isolation - Misplaced Pages Continue

5250-411: Is that this reactive controller can also be connected to a continuous-time model of the actuator hydraulics, allowing the study of switching transients. In signal processing based FDI, some mathematical or statistical operations are performed on the measurements, or some neural network is trained using measurements to extract the information about the fault. A good example of signal processing based FDI

5375-446: Is the action taken after a failure has been detected and isolated to return the system to a stable state. Some examples of fault recoveries are: Mathematical model A mathematical model is an abstract description of a concrete system using mathematical concepts and language . The process of developing a mathematical model is termed mathematical modeling . Mathematical models are used in applied mathematics and in

5500-408: Is the total number of data points transformed. In particular, there are n / n 1 transforms of size n 1 , etc., so the complexity of the sequence of FFTs is: In two dimensions, the x k can be viewed as an n 1 × n 2 {\displaystyle n_{1}\times n_{2}} matrix , and this algorithm corresponds to first performing the FFT of all

5625-425: Is therefore limited to power-of-two sizes, but any factorization can be used in general (as was known to both Gauss and Cooley/Tukey ). These are called the radix-2 and mixed-radix cases, respectively (and other variants such as the split-radix FFT have their own names as well). Although the basic idea is recursive, most traditional implementations rearrange the algorithm to avoid explicit recursion. Also, because

5750-615: Is to minimize the total number of real multiplications and additions, sometimes called the "arithmetic complexity" (although in this context it is the exact count and not the asymptotic complexity that is being considered). Again, no tight lower bound has been proven. Since 1968, however, the lowest published count for power-of-two n was long achieved by the split-radix FFT algorithm , which requires 4 n log 2 ⁡ ( n ) − 6 n + 8 {\textstyle 4n\log _{2}(n)-6n+8} real multiplications and additions for n > 1 . This

5875-535: Is used on large datasets . Since k NN is not able to automatically extract the features to overcome the curse of dimensionality , so often some data preprocessing techniques like Principal component analysis (PCA), Linear discriminant analysis (LDA) or Canonical correlation analysis (CCA) accompany it to reach a better performance. In many industrial cases, the effectiveness of k NN has been compared with other methods, specially with more complex classification models such as Support Vector Machines (SVMs), which

6000-712: Is usually the focus of such questions, although actual performance on modern-day computers is determined by many other factors such as cache or CPU pipeline optimization. Following work by Shmuel Winograd (1978), a tight Θ ( n ) {\displaystyle \Theta (n)} lower bound is known for the number of real multiplications required by an FFT. It can be shown that only 4 n − 2 log 2 2 ⁡ ( n ) − 2 log 2 ⁡ ( n ) − 4 {\textstyle 4n-2\log _{2}^{2}(n)-2\log _{2}(n)-4} irrational real multiplications are required to compute

6125-407: Is where all of the radices are equal (e.g. vector-radix-2 divides all of the dimensions by two), but this is not necessary. Vector radix with only a single non-unit radix at a time, i.e. r = ( 1 , … , 1 , r , 1 , … , 1 ) {\textstyle \mathbf {r} =\left(1,\ldots ,1,r,1,\ldots ,1\right)} , is essentially

6250-414: Is widely used in this field. Thanks to their appropriate nonlinear mapping using kernel methods , SVMs have an impressive performance in generalization, even with small training data. However, general SVMs do not have automatic feature extraction themselves and just like k NN , are often coupled with a data pre-processing technique. Another drawback of SVMs is that their performance is highly sensitive to

6375-529: The O ( n log ⁡ n ) {\textstyle O(n\log n)} scaling. Tukey came up with the idea during a meeting of President Kennedy 's Science Advisory Committee where a discussion topic involved detecting nuclear tests by the Soviet Union by setting up sensors to surround the country from outside. To analyze the output of these sensors, an FFT algorithm would be needed. In discussion with Tukey, Richard Garwin recognized

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6500-593: The DFT matrix into a product of sparse (mostly zero) factors. As a result, it manages to reduce the complexity of computing the DFT from O ( n 2 ) {\textstyle O(n^{2})} , which arises if one simply applies the definition of DFT, to O ( n log ⁡ n ) {\textstyle O(n\log n)} , where n is the data size. The difference in speed can be enormous, especially for long data sets where n may be in

6625-474: The discrete cosine / sine transform(s) ( DCT / DST ). Instead of directly modifying an FFT algorithm for these cases, DCTs/DSTs can also be computed via FFTs of real data combined with O ( n ) {\displaystyle O(n)} pre- and post-processing. A fundamental question of longstanding theoretical interest is to prove lower bounds on the complexity and exact operation counts of fast Fourier transforms, and many open problems remain. It

6750-446: The natural sciences (such as physics , biology , earth science , chemistry ) and engineering disciplines (such as computer science , electrical engineering ), as well as in non-physical systems such as the social sciences (such as economics , psychology , sociology , political science ). It can also be taught as a subject in its own right. The use of mathematical models to solve problems in business or military operations

6875-649: The physical sciences , a traditional mathematical model contains most of the following elements: Mathematical models are of different types: In business and engineering , mathematical models may be used to maximize a certain output. The system under consideration will require certain inputs. The system relating inputs to outputs depends on other variables too: decision variables , state variables , exogenous variables, and random variables . Decision variables are sometimes known as independent variables. Exogenous variables are sometimes known as parameters or constants . The variables are not independent of each other as

7000-463: The prime-factor (Good–Thomas) algorithm (PFA), based on the Chinese remainder theorem , to factorize the DFT similarly to Cooley–Tukey but without the twiddle factors. The Rader–Brenner algorithm (1976) is a Cooley–Tukey-like factorization but with purely imaginary twiddle factors, reducing multiplications at the cost of increased additions and reduced numerical stability ; it was later superseded by

7125-425: The root mean square (rms) errors are much better than these upper bounds, being only O ( ε log ⁡ n ) {\textstyle O(\varepsilon {\sqrt {\log n}})} for Cooley–Tukey and O ( ε n ) {\textstyle O(\varepsilon {\sqrt {n}})} for the naïve DFT (Schatzman, 1996). These results, however, are very sensitive to

7250-403: The speed of light , and we study macro-particles only. Note that better accuracy does not necessarily mean a better model. Statistical models are prone to overfitting which means that a model is fitted to data too much and it has lost its ability to generalize to new events that were not observed before. Any model which is not pure white-box contains some parameters that can be used to fit

7375-572: The split-radix variant of Cooley–Tukey (which achieves the same multiplication count but with fewer additions and without sacrificing accuracy). Algorithms that recursively factorize the DFT into smaller operations other than DFTs include the Bruun and QFT algorithms. (The Rader–Brenner and QFT algorithms were proposed for power-of-two sizes, but it is possible that they could be adapted to general composite n . Bruun's algorithm applies to arbitrary even composite sizes.) Bruun's algorithm , in particular,

7500-496: The Cooley–Tukey algorithm (Welch, 1969). Achieving this accuracy requires careful attention to scaling to minimize loss of precision, and fixed-point FFT algorithms involve rescaling at each intermediate stage of decompositions like Cooley–Tukey. To verify the correctness of an FFT implementation, rigorous guarantees can be obtained in O ( n log ⁡ n ) {\textstyle O(n\log n)} time by

7625-465: The Cooley–Tukey algorithm breaks the DFT into smaller DFTs, it can be combined arbitrarily with any other algorithm for the DFT, such as those described below. There are FFT algorithms other than Cooley–Tukey. For n = n 1 n 2 {\textstyle n=n_{1}n_{2}} with coprime n 1 {\textstyle n_{1}} and n 2 {\textstyle n_{2}} , one can use

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7750-410: The DFT (which is only defined for equispaced data), but rather some approximation thereof (a non-uniform discrete Fourier transform , or NDFT, which itself is often computed only approximately). More generally there are various other methods of spectral estimation . The FFT is used in digital recording, sampling, additive synthesis and pitch correction software. The FFT's importance derives from

7875-548: The DFT can be computed with only O ( n ) {\displaystyle O(n)} irrational multiplications, leading to a proven achievable lower bound on the number of multiplications for power-of-two sizes; this comes at the cost of many more additions, a tradeoff no longer favorable on modern processors with hardware multipliers . In particular, Winograd also makes use of the PFA as well as an algorithm by Rader for FFTs of prime sizes. Rader's algorithm , exploiting

8000-428: The DFT's sums directly involves n 2 {\textstyle n^{2}} complex multiplications and n ( n − 1 ) {\textstyle n(n-1)} complex additions, of which O ( n ) {\textstyle O(n)} operations can be saved by eliminating trivial operations such as multiplications by 1, leaving about 30 million operations. In contrast,

8125-614: The FFT as "the most important numerical algorithm of our lifetime", and it was included in Top 10 Algorithms of 20th Century by the IEEE magazine Computing in Science & Engineering . The best-known FFT algorithms depend upon the factorization of n , but there are FFTs with O ( n log ⁡ n ) {\displaystyle O(n\log n)} complexity for all, even prime , n . Many FFT algorithms depend only on

8250-498: The NARMAX (Nonlinear AutoRegressive Moving Average model with eXogenous inputs) algorithms which were developed as part of nonlinear system identification can be used to select the model terms, determine the model structure, and estimate the unknown parameters in the presence of correlated and nonlinear noise. The advantage of NARMAX models compared to neural networks is that NARMAX produces models that can be written down and related to

8375-458: The above algorithms): first you transform along the n 1 dimension, then along the n 2 dimension, and so on (actually, any ordering works). This method is easily shown to have the usual O ( n log ⁡ n ) {\textstyle O(n\log n)} complexity, where n = n 1 ⋅ n 2 ⋯ n d {\textstyle n=n_{1}\cdot n_{2}\cdots n_{d}}

8500-615: The accuracy of the twiddle factors used in the FFT (i.e. the trigonometric function values), and it is not unusual for incautious FFT implementations to have much worse accuracy, e.g. if they use inaccurate trigonometric recurrence formulas. Some FFTs other than Cooley–Tukey, such as the Rader–Brenner algorithm, are intrinsically less stable. In fixed-point arithmetic , the finite-precision errors accumulated by FFT algorithms are worse, with rms errors growing as O ( n ) {\textstyle O({\sqrt {n}})} for

8625-645: The algorithm (his assumptions imply, among other things, that no additive identities in the roots of unity are exploited). (This argument would imply that at least 2 N log 2 ⁡ N {\textstyle 2N\log _{2}N} real additions are required, although this is not a tight bound because extra additions are required as part of complex-number multiplications.) Thus far, no published FFT algorithm has achieved fewer than n log 2 ⁡ n {\textstyle n\log _{2}n} complex-number additions (or their equivalent) for power-of-two n . A third problem

8750-431: The algorithms. The upper bound on the relative error for the Cooley–Tukey algorithm is O ( ε log ⁡ n ) {\textstyle O(\varepsilon \log n)} , compared to O ( ε n 3 / 2 ) {\textstyle O(\varepsilon n^{3/2})} for the naïve DFT formula, where 𝜀 is the machine floating-point relative precision. In fact,

8875-421: The bearing's damaged state is not enough for precision maintenance purposes. The root cause needs to be identified and remedied. If this is not done, the replacement bearing will soon wear out for the same reason and the machine will suffer more damage, remaining dangerous. Of course, the cause may also be visible as a result of the spectral analysis undertaken at the data-collection stage, but this may not always be

9000-545: The case. The most common technique for detecting faults is the time-frequency analysis technique. For a rotating machine, the rotational speed of the machine (often known as the RPM ), is not a constant, especially not during the start-up and shutdown stages of the machine. Even if the machine is running in the steady state, the rotational speed will vary around a steady-state mean value, and this variation depends on load and other factors. Since sound and vibration signals obtained from

9125-408: The coin, the true probability that the coin will come up heads is unknown; so the experimenter would need to make a decision (perhaps by looking at the shape of the coin) about what prior distribution to use. Incorporation of such subjective information might be important to get an accurate estimate of the probability. In general, model complexity involves a trade-off between simplicity and accuracy of

9250-446: The complex relations (which generally exist inherently in fault detection and diagnosis problems) and are easy to operate. Another advantage of ANNs is that they perform automatic feature extraction by allocating negligible weights to the irrelevant features, helping the system to avoid dealing with another feature extractor. However, ANNs tend to over-fit the training set, which will have consequences of having poor validation accuracy on

9375-423: The controller reacts to detected faults, and the state chart defines how the controller switches between the different modes of operation (passive, active, standby, off, and isolated) of each actuator. For example, if a fault is detected in hydraulic system 1, then the truth table sends an event to the state chart that the left inner actuator should be turned off. One of the benefits of this model-based FDI technique

9500-400: The corresponding DHT algorithm (FHT) for the same number of inputs. Bruun's algorithm (above) is another method that was initially proposed to take advantage of real inputs, but it has not proved popular. There are further FFT specializations for the cases of real data that have even/odd symmetry, in which case one can gain another factor of roughly two in time and memory and the DFT becomes

9625-403: The cost of what is to be avoided in the first place, requires an effective scheme of applying them. This is the subject of maintenance, repair and operations ; the different strategies include: In fault detection and diagnosis, mathematical classification models which in fact belong to supervised learning methods, are trained on the training set of a labeled dataset to accurately identify

9750-441: The data fit a known distribution or to come up with a general model that makes only minimal assumptions about the model's mathematical form. Assessing the scope of a model, that is, determining what situations the model is applicable to, can be less straightforward. If the model was constructed based on a set of data, one must determine for which systems or situations the known data is a "typical" set of data. The question of whether

9875-401: The detection of fault under certain conditions. The main difference is that instead of finding the most likely model, these techniques omit the models, which are not compatible with data. The example shown in the figure on the right illustrates a model-based FDI technique for an aircraft elevator reactive controller through the use of a truth table and a state chart. The truth table defines how

10000-452: The evolution of the conventional FFT , then quadratic time frequency analysis would be the power spectrum counterpart. Quadratic algorithms include the Gabor spectrogram, Cohen's class and the adaptive spectrogram. The main advantage of time frequency analysis is discovering the patterns of frequency changes, which usually represent the nature of the signal. As long as this pattern is identified

10125-426: The existence of a generator for the multiplicative group modulo prime n , expresses a DFT of prime size n as a cyclic convolution of (composite) size n – 1 , which can then be computed by a pair of ordinary FFTs via the convolution theorem (although Winograd uses other convolution methods). Another prime-size FFT is due to L. I. Bluestein, and is sometimes called the chirp-z algorithm ; it also re-expresses

10250-549: The fact that e − 2 π i / n {\textstyle e^{-2\pi i/n}} is an n 'th primitive root of unity , and thus can be applied to analogous transforms over any finite field , such as number-theoretic transforms . Since the inverse DFT is the same as the DFT, but with the opposite sign in the exponent and a 1/ n factor, any FFT algorithm can easily be adapted for it. The development of fast algorithms for DFT can be traced to Carl Friedrich Gauss 's unpublished 1805 work on

10375-610: The fact that it has made working in the frequency domain equally computationally feasible as working in the temporal or spatial domain. Some of the important applications of the FFT include: An original application of the FFT in finance particularly in the Valuation of options was developed by Marcello Minenna. Despite its strengths, the Fast Fourier Transform (FFT) has limitations, particularly when analyzing signals with non-stationary frequency content—where

10500-479: The fault detection and diagnosis in industries such as gearbox , machinery parts (i.e. mechanical bearings ), compressors , wind and gas turbines and steel plates . With the research advances in ANNs and the advent of deep learning algorithms using deep and complex layers, novel classification models have been developed to cope with fault detection and diagnosis. Most of the shallow learning models extract

10625-458: The frequency characteristics change over time. The FFT provides a global frequency representation, meaning it analyzes frequency information across the entire signal duration. This global perspective makes it challenging to detect short-lived or transient features within signals, as the FFT assumes that all frequency components are present throughout the entire signal. For cases where frequency information varies over time, alternative transforms like

10750-423: The frequency contents of the sound and vibration signals obtained from a rotating machine are very much time-dependent. For this reason, FFT -based spectra are unable to detect how the frequency contents develop over time. To be more specific, if the RPM of a machine is increasing or decreasing during its startup or shutdown period, its bandwidth in the FFT spectrum will become much wider than it would be simply for

10875-452: The general applicability of the algorithm not just to national security problems, but also to a wide range of problems including one of immediate interest to him, determining the periodicities of the spin orientations in a 3-D crystal of Helium-3. Garwin gave Tukey's idea to Cooley (both worked at IBM's Watson labs ) for implementation. Cooley and Tukey published the paper in a relatively short time of six months. As Tukey did not work at IBM,

11000-429: The general idea of an FFT) was popularized by a publication of Cooley and Tukey in 1965, but it was later discovered that those two authors had together independently re-invented an algorithm known to Carl Friedrich Gauss around 1805 (and subsequently rediscovered several times in limited forms). The best known use of the Cooley–Tukey algorithm is to divide the transform into two pieces of size n/2 at each step, and

11125-401: The geometry of the universe. Euclidean geometry is much used in classical physics, while special relativity and general relativity are examples of theories that use geometries which are not Euclidean. Often when engineers analyze a system to be controlled or optimized, they use a mathematical model. In analysis, engineers can build a descriptive model of the system as a hypothesis of how

11250-415: The help of a fast multipole method . A wavelet -based approximate FFT by Guo and Burrus (1996) takes sparse inputs/outputs (time/frequency localization) into account more efficiently than is possible with an exact FFT. Another algorithm for approximate computation of a subset of the DFT outputs is due to Shentov et al. (1995). The Edelman algorithm works equally well for sparse and non-sparse data, since it

11375-515: The initial parameters, particularly to the kernel methods , so in each signal dataset , a parameter tuning process is required to be conducted first. Therefore, the low speed of the training phase is a limitation of SVMs when it comes to its usage in fault detection and diagnosis cases. Artificial Neural Networks (ANNs) are among the most mature and widely used mathematical classification algorithms in fault detection and diagnosis. ANNs are well-known for their efficient self-learning capabilities of

11500-442: The input data for the DFT are purely real, in which case the outputs satisfy the symmetry and efficient FFT algorithms have been designed for this situation (see e.g. Sorensen, 1987). One approach consists of taking an ordinary algorithm (e.g. Cooley–Tukey) and removing the redundant parts of the computation, saving roughly a factor of two in time and memory. Alternatively, it is possible to express an even -length real-input DFT as

11625-411: The labor", though like Gauss they did not do the analysis to discover that this led to O ( n log ⁡ n ) {\textstyle O(n\log n)} scaling. James Cooley and John Tukey independently rediscovered these earlier algorithms and published a more general FFT in 1965 that is applicable when n is composite and not necessarily a power of 2, as well as analyzing

11750-469: The machine fault associated with this pattern can be identified. Another important use of time frequency analysis is the ability to filter out a particular frequency component using a time-varying filter. In practice, model uncertainties and measurement noise can complicate fault detection and isolation. As a result, using fault diagnostics to meet industrial needs in a cost-effective way, and to reduce maintenance costs without requiring more investments than

11875-449: The model to the system it is intended to describe. If the modeling is done by an artificial neural network or other machine learning , the optimization of parameters is called training , while the optimization of model hyperparameters is called tuning and often uses cross-validation . In more conventional modeling through explicitly given mathematical functions, parameters are often determined by curve fitting . A crucial part of

12000-467: The model describes well the properties of the system between data points is called interpolation , and the same question for events or data points outside the observed data is called extrapolation . As an example of the typical limitations of the scope of a model, in evaluating Newtonian classical mechanics , we can note that Newton made his measurements without advanced equipment, so he could not measure properties of particles traveling at speeds close to

12125-700: The model's user. Depending on the context, an objective function is also known as an index of performance , as it is some measure of interest to the user. Although there is no limit to the number of objective functions and constraints a model can have, using or optimizing the model becomes more involved (computationally) as the number increases. For example, economists often apply linear algebra when using input–output models . Complicated mathematical models that have many variables may be consolidated by use of vectors where one symbol represents several variables. Mathematical modeling problems are often classified into black box or white box models, according to how much

12250-553: The model. In black-box models, one tries to estimate both the functional form of relations between variables and the numerical parameters in those functions. Using a priori information we could end up, for example, with a set of functions that probably could describe the system adequately. If there is no a priori information we would try to use functions as general as possible to cover all different models. An often used approach for black-box models are neural networks which usually do not make assumptions about incoming data. Alternatively,

12375-493: The model. Occam's razor is a principle particularly relevant to modeling, its essential idea being that among models with roughly equal predictive power, the simplest one is the most desirable. While added complexity usually improves the realism of a model, it can make the model difficult to understand and analyze, and can also pose computational problems, including numerical instability . Thomas Kuhn argues that as science progresses, explanations tend to become more complex before

12500-427: The model. It is therefore usually appropriate to make some approximations to reduce the model to a sensible size. Engineers often can accept some approximations in order to get a more robust and simple model. For example, Newton's classical mechanics is an approximated model of the real world. Still, Newton's model is quite sufficient for most ordinary-life situations, that is, as long as particle speeds are well below

12625-408: The modeling process is the evaluation of whether or not a given mathematical model describes a system accurately. This question can be difficult to answer as it involves several different types of evaluation. Usually, the easiest part of model evaluation is checking whether a model predicts experimental measurements or other empirical data not used in the model development. In models with parameters,

12750-614: The most commonly used FFT is the Cooley–Tukey algorithm. This is a divide-and-conquer algorithm that recursively breaks down a DFT of any composite size n = n 1 n 2 {\textstyle n=n_{1}n_{2}} into n 1 {\textstyle n_{1}} smaller DFTs of size n 2 {\textstyle n_{2}} , along with O ( n ) {\displaystyle O(n)} multiplications by complex roots of unity traditionally called twiddle factors (after Gentleman and Sande, 1966). This method (and

12875-457: The orbits of asteroids Pallas and Juno . Gauss wanted to interpolate the orbits from sample observations; his method was very similar to the one that would be published in 1965 by James Cooley and John Tukey , who are generally credited for the invention of the modern generic FFT algorithm. While Gauss's work predated even Joseph Fourier 's 1822 results, he did not analyze the method's complexity , and eventually used other methods to achieve

13000-562: The others (Duhamel & Vetterli, 1990). All of the FFT algorithms discussed above compute the DFT exactly (i.e. neglecting floating-point errors). A few "FFT" algorithms have been proposed, however, that compute the DFT approximately , with an error that can be made arbitrarily small at the expense of increased computations. Such algorithms trade the approximation error for increased speed or other properties. For example, an approximate FFT algorithm by Edelman et al. (1999) achieves lower communication requirements for parallel computing with

13125-400: The patentability of the idea was doubted and the algorithm went into the public domain, which, through the computing revolution of the next decade, made FFT one of the indispensable algorithms in digital signal processing . Let x 0 , … , x n − 1 {\displaystyle x_{0},\ldots ,x_{n-1}} be complex numbers . The DFT

13250-451: The purpose of modeling is to increase our understanding of the world, the validity of a model rests not only on its fit to empirical observations, but also on its ability to extrapolate to situations or data beyond those originally described in the model. One can think of this as the differentiation between qualitative and quantitative predictions. One can also argue that a model is worthless unless it provides some insight which goes beyond what

13375-563: The quadratic method describes the energy distribution of a signal in the joint time frequency domain, which is useful for analysis, classification, and detection of signal features, phase information is lost in the quadratic time-frequency representation; also, the time histories cannot be reconstructed with this method. The short-term Fourier transform ( STFT ) and the Gabor transform are two algorithms commonly used as linear time-frequency methods. If we consider linear time-frequency analysis to be

13500-483: The radix-2 Cooley–Tukey algorithm , for n a power of 2, can compute the same result with only ( n / 2 ) log 2 ⁡ ( n ) {\textstyle (n/2)\log _{2}(n)} complex multiplications (again, ignoring simplifications of multiplications by 1 and similar) and n log 2 ⁡ ( n ) {\textstyle n\log _{2}(n)} complex additions, in total about 30,000 operations —

13625-485: The redundancies, faults and anomalous samples. During the past decades, there are different classification and preprocessing models that have been developed and proposed in this research area. K -nearest-neighbors algorithm ( k NN) is one of the oldest techniques which has been used to solve fault detection and diagnosis problems. Despite the simple logic that this instance-based algorithm has, there are some problems with large dimensionality and processing time when it

13750-401: The rows (resp. columns), grouping the resulting transformed rows (resp. columns) together as another n 1 × n 2 {\displaystyle n_{1}\times n_{2}} matrix, and then performing the FFT on each of the columns (resp. rows) of this second matrix, and similarly grouping the results into the final result matrix. In more than two dimensions, it

13875-456: The same end. Between 1805 and 1965, some versions of FFT were published by other authors. Frank Yates in 1932 published his version called interaction algorithm , which provided efficient computation of Hadamard and Walsh transforms . Yates' algorithm is still used in the field of statistical design and analysis of experiments. In 1942, G. C. Danielson and Cornelius Lanczos published their version to compute DFT for x-ray crystallography ,

14000-519: The same results in O ( n log ⁡ n ) {\textstyle O(n\log n)} operations. All known FFT algorithms require O ( n log ⁡ n ) {\textstyle O(n\log n)} operations, although there is no known proof that lower complexity is impossible. To illustrate the savings of an FFT, consider the count of complex multiplications and additions for n = 4096 {\textstyle n=4096} data points. Evaluating

14125-486: The simplest non-row-column FFT is the vector-radix FFT algorithm , which is a generalization of the ordinary Cooley–Tukey algorithm where one divides the transform dimensions by a vector r = ( r 1 , r 2 , … , r d ) {\textstyle \mathbf {r} =\left(r_{1},r_{2},\ldots ,r_{d}\right)} of radices at each step. (This may also have cache benefits.) The simplest case of vector-radix

14250-442: The speed of light. Likewise, he did not measure the movements of molecules and other small particles, but macro particles only. It is then not surprising that his model does not extrapolate well into these domains, even though his model is quite sufficient for ordinary life physics. Many types of modeling implicitly involve claims about causality . This is usually (but not always) true of models involving differential equations. As

14375-500: The sphere S with n nodes was described by Mohlenkamp, along with an algorithm conjectured (but not proven) to have O ( n 2 log 2 ⁡ ( n ) ) {\textstyle O(n^{2}\log ^{2}(n))} complexity; Mohlenkamp also provides an implementation in the libftsh library. A spherical-harmonic algorithm with O ( n 2 log ⁡ n ) {\textstyle O(n^{2}\log n)} complexity

14500-404: The state variables are dependent on the decision, input, random, and exogenous variables. Furthermore, the output variables are dependent on the state of the system (represented by the state variables). Objectives and constraints of the system and its users can be represented as functions of the output variables or state variables. The objective functions will depend on the perspective of

14625-440: The steady state. Hence, in such a case, the harmonics are not so distinguishable in the spectrum. The time frequency approach for machine fault diagnosis can be divided into two broad categories: linear methods and the quadratic methods. The difference is that linear transforms can be inverted to construct the time signal, thus, they are more suitable for signal processing, such as noise reduction and time-varying filtering. Although

14750-494: The system could work, or try to estimate how an unforeseeable event could affect the system. Similarly, in control of a system, engineers can try out different control approaches in simulations . A mathematical model usually describes a system by a set of variables and a set of equations that establish relationships between the variables. Variables may be of many types; real or integer numbers, Boolean values or strings , for example. The variables represent some properties of

14875-498: The system, for example, the measured system outputs often in the form of signals , timing data , counters, and event occurrence. The actual model is the set of functions that describe the relations between the different variables. General reference Philosophical Fast Fourier transform A fast Fourier transform ( FFT ) is an algorithm that computes the Discrete Fourier Transform (DFT) of

15000-574: The thousands or millions. In the presence of round-off error , many FFT algorithms are much more accurate than evaluating the DFT definition directly or indirectly. There are many different FFT algorithms based on a wide range of published theories, from simple complex-number arithmetic to group theory and number theory . Fast Fourier transforms are widely used for applications in engineering, music, science, and mathematics. The basic ideas were popularized in 1965, but some algorithms had been derived as early as 1805. In 1994, Gilbert Strang described

15125-499: The time domain, because spectral analysis usually concerns only frequency distribution and not phase information) and others. The results of this analysis are used in a root cause failure analysis in order to determine the original cause of the fault. For example, if a bearing fault is diagnosed, then it is likely that the bearing was not itself damaged at installation, but rather as the consequence of another installation error (e.g., misalignment) which then led to bearing damage. Diagnosing

15250-427: The transforms operate on contiguous data; this is especially important for out-of-core and distributed memory situations where accessing non-contiguous data is extremely time-consuming. There are other multidimensional FFT algorithms that are distinct from the row-column algorithm, although all of them have O ( n log ⁡ n ) {\textstyle O(n\log n)} complexity. Perhaps

15375-427: The underlying process, whereas neural networks produce an approximation that is opaque. Sometimes it is useful to incorporate subjective information into a mathematical model. This can be done based on intuition , experience , or expert opinion , or based on convenience of mathematical form. Bayesian statistics provides a theoretical framework for incorporating such subjectivity into a rigorous analysis: we specify

15500-522: The validation set. Hence, often, some regularization terms and prior knowledge are added to the ANN model to avoid over-fitting and achieve higher performance. Moreover, properly determining the size of the hidden layer needs an exhaustive parameter tuning, to avoid poor approximation and generalization capabilities. In general, different SVMs and ANNs models (i.e. Back-Propagation Neural Networks and Multi-Layer Perceptron ) have shown successful performances in

15625-494: Was recently reduced to ∼ 34 9 n log 2 ⁡ n {\textstyle \sim {\frac {34}{9}}n\log _{2}n} (Johnson and Frigo, 2007; Lundy and Van Buskirk, 2007 ). A slightly larger count (but still better than split radix for n ≥ 256 ) was shown to be provably optimal for n ≤ 512 under additional restrictions on the possible algorithms (split-radix-like flowgraphs with unit-modulus multiplicative factors), by reduction to

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