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| '''Fourier amplitude sensitivity testing (FAST)''' is a variance-based global [[sensitivity analysis]] method. The sensitivity value is defined based on [[conditional variance]]s which indicate the individual or joint effects of the uncertain inputs on the output.
| | Weapons and Knives are a must, the extra the better. The weapons really don’t even have to shoot in any respect. Perhaps a few of them, but for essentially the most part they’re just for intimidation. Similar idea for the knives. It is best to have knives strapped to your self all over the place. Huge knives. None of these tine ten greenback lock blade pocket knives from the boy scouts. They must be massive and conspicuous. It's crucial that they are visible so that everybody can see them. Their also handy for grooming your hair or scratching stick figures onto the table on the local bar.<br><br>The enterprise mannequin for wholesale dropshipping for wholesale knives is sort of simple. The web knife store or the specialty knife retail outlet markets the wholesale knives as their stock, then takes orders for wholesale knives from clients, selling them at retail costs. The pocket knife ecommerce store or retail store will accept fee, document and promise delivery. The proprietor of the knife retailer then contacts the dropshipping wholesale knife distributor and locations and pays for the orders simply obtained from the purchasers with the money already received by the purchasers.<br><br>Whether you might have a helper or not, having a second ladder is an asset when hanging ceiling drywall. By positioning each ladder so that each finish of the drywall sheet might be accessed quickly, the job will go a lot faster. In the event you can master them, drywall stilts are even a better plan. These adjustable stilts will make you the proper top to work on the drywall with out the ladder. Another various to the ladder would be a superb bench made sturdy enough to work from, lengthy enough to reach the whole length of a bit of drywall and tall sufficient to entry the ceiling.<br><br>The folding knife is very fashionable because it may be transformed into a hard and fast bladed knife as and if you want and could be transformed again into a folding knife once the job is done. That is achieved by inserting a lock on the pivot. There are various kinds [http://www.thebestpocketknifereviews.com/spyderco-tenacious-review/ Spyderco Tenacious Maintenance] of locks placed in the folding knife There are numerous types and kinds of Swiss Army knives each with its personal tool mixture for a selected tasks. The Swiss Military's knife has a corrugated metallic floor with a purple emblem and adorns a blade, a reamer, a bottle-opener/screwdriver/wire stripper, and a can-opener/screwdriver.<br><br>Wholesale knives and pocket knives is usually a very worthwhile area of interest for the small business proprietor, so long as they're purchasing their wholesale knives from a wholesale knife distributor that understands the small enterprise proprietor's distinctive needs. Do not let high portions of wholesale knives and retail costs cut into your income. Do enterprise with a wholesale knife [http://ner.vse.cz80/wiki/index.php/Pimped_Spyderco_Tenacious distributor] that dropships and understands your enterprise wants, then you'll put the profits again into your pockets. Primary Whittling Techniques. It goes without saying but you've got to be certain. Keep fingers and all other physique parts away from knife edges. Take no possibilities.<br><br>What are the various kinds of blades for pocket knives and fixed blade knives ? There are lots of of them, for example there is the clip level, drop point or the trailing point. This article will describe a few of the types of knives you will encounter when buying pocket knives or fastened blade knives Some fastened knives are made for a selected goal akin to fishing, looking and skinning. Others are designed to do a number of tasks. Evaluation the descriptions of the knife blades beneath, so you perceive to types and purposes of each one. |
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| FAST first represents conditional variances via coefficients from the multiple [[Fourier series]] expansion of the output function. Then the [[ergodic theorem]] is applied to transform the multi-dimensional integral to a one-dimensional integral in evaluation of the Fourier coefficients. A set of incommensurate frequencies is required to perform the transform and most frequencies are irrational. To facilitate computation a set of integer frequencies is selected instead of the irrational frequencies. The integer frequencies are not strictly incommensurate, resulting in an error between the multi-dimensional integral and the transformed one-dimensional integral. However, the integer frequencies can be selected to be incommensurate to any order so that the error can be controlled meeting any precision requirement in theory. Using integer frequencies in the integral transform, the resulted function in the one-dimensional integral is periodic and the integral only needs to evaluate in a single period. Next, since the continuous integral function can be recovered from a set of finite sampling points if the [[Nyquist–Shannon sampling theorem]] is satisfied, the one-dimensional integral is evaluated from the summation of function values at the generated sampling points.
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| FAST is more efficient to calculate sensitivities than other variance-based global sensitivity analysis methods via [[Monte Carlo integration]]. However the calculation by FAST is usually limited to sensitivities referring to “main effect” or “total effect”.
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| == History ==
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| The FAST method originated in study of coupled chemical reaction systems in 1973<ref>Cukier, R.I., C.M. Fortuin, K.E. Shuler, A.G. Petschek and J.H. Schaibly (1973). Study of the sensitivity of coupled reaction systems to uncertainties in rate coefficients. I Theory. ''Journal of Chemical Physics'', '''59''', 3873–3878.</ref><ref>Schaibly, J.H. and K.E. Shuler (1973). Study of the sensitivity of coupled reaction systems to uncertainties in rate coefficients. II Applications. ''Journal of Chemical Physics'', '''59''', 3879–3888.</ref> and the detailed analysis of the computational error was presented latter in 1975.<ref>Cukier, R.I., J.H. Schaibly, and K.E. Shuler (1975). Study of the sensitivity of coupled reaction systems to uncertainties in rate coefficients. III. Analysis of the approximations. ''Journal of Chemical Physics'', '''63''', 1140–1149.</ref> Only the first order sensitivity indices referring to “main effect” were calculated in the original method. A [[FORTRAN]] computer program capable of analyzing either algebraic or differential equation systems was published in 1982.<ref>McRae, G.J., J.W. Tilden and J.H. Seinfeld (1982). Global sensitivity analysis—a computational implementation of the Fourier Amplitude Sensitivity Test (FAST). ''Computers & Chemical Engineering'', '''6''', 15–25.</ref> In 1990s, the relationship between FAST sensitivity indices and Sobol’s ones calculated from [[Monte-Carlo simulation]] was revealed in the general framework of [[ANOVA]]-like decomposition <ref>Archer G.E.B., A. Saltelli and I.M. Sobol (1997). Sensitivity measures, ANOVA-like techniques and the use of bootstrap. ''Journal of Statistical Computation and Simulation'', '''58''', 99–120.</ref> and an extended FAST method able to calculate sensitivity indices referring to “total effect” was developed.<ref>Saltelli A., S. Tarantola and K.P.S. Chan (1999). A quantitative model-independent method for global sensitivity analysis of model output. ''Technometrics'', '''41''', 39–56.</ref>
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| == Foundation ==
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| === Variance-based sensitivity ===
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| {{Main|Variance-based sensitivity analysis}}
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| Sensitivity indices of a variance-based method are calculated via ANOVA-like decomposition of the function for analysis. Suppose the function is <math> Y = f\left(\mathbf{X}\right)=f\left(X_1,X_2,\dots,X_n\right) </math> where <math> 0 \leq X_j \leq 1, j=1, \dots, n</math>. The ANOVA-like decomposition is
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| :<math>f\left(X_1,X_2,\ldots,X_n\right)=f_0+\sum_{j=1}^nf_j\left(X_j\right)+\sum_{j=1}^{n-1}\sum_{k=j+1}^n f_{jk}\left(X_j,X_k\right)+ \cdots +f_{12 \dots n}</math>
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| provided that <math> f_0 </math> is a constant and the integral of each term in the sums is zero, i.e.
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| :<math> \int_0^1 f_{j_1 j_2 \dots j_r}\left(X_{j_1},X_{j_2},\dots,X_{j_r}\right)dX_{j_k}=0, \text{ } 1 \leq k \leq r.</math>
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| The conditional variance which characterizes the contribution of each term to the total variance of <math> f\left(\mathbf{X}\right) </math> is
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| :<math> V_{j_1 j_2 \dots j_r}=\int_0^1 \cdots \int_0^1 f_{j_1 j_2 \dots j_r}^2\left(X_{j_1},X_{j_2},\dots,X_{j_r}\right)dX_{j_1}dX_{j_2}\dots dX_{j_r}.</math>
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| The total variance is the sum of all conditional variances
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| :<math> V = \sum_{j=1}^n V_j + \sum_{j=1}^{n-1} \sum_{k=j+1}^n V_{jk} + \cdots + V_{12\dots n}.</math>
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| The sensitivity index is defined as the normalized conditional variance as
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| :<math> S_{j_1 j_2 \dots j_r} = \frac{V_{j_1 j_2 \dots j_r}}{V} </math>
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| especially the first order sensitivity
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| :<math> S_j=\frac{V_j}{V} </math>
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| which indicates the main effect of the input <math> X_j </math>.
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| === Multiple Fourier series ===
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| One way to calculate the ANOVA-like decomposition is based on multiple Fourier series. The function <math> f\left(\mathbf{X}\right) </math> in the unit hyper-cube can be extended to a multiply periodic function and the multiple Fourier series expansion is
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| :<math> f\left(X_1,X_2,\dots,X_n\right) = \sum_{m_1=-\infty}^{\infty} \sum_{m_2=-\infty}^{\infty} \cdots \sum_{m_n=-\infty}^{\infty} C_{m_1m_2...m_n}\exp\bigl[2\pi i\left( m_1X_1 + m_2X_2 + \cdots + m_nX_n \right) \bigr], \text{ for integers }m_1, m_2, \dots, m_n</math>
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| where the Fourier coefficient is
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| :<math> C_{m_1m_2...m_n} = \int_0^1 \cdots \int_0^1 f\left(X_1,X_2,\dots,X_n\right) \exp\bigl[-2\pi i \left( m_1X_1+m_2X_2+\dots+m_nX_n \right) \bigr].</math>
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| The ANOVA-like decomposition is | |
| :<math>
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| \begin{align}
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| f_0 &= C_{00 \dots 0} \\
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| f_j &= \sum_{m_j \neq 0} C_{0 \dots m_j \dots 0} \exp\bigl[2\pi i m_jX_j \bigr] \\
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| f_{jk} &= \sum_{m_j \neq 0} \sum_{m_k \neq 0} C_{0 \dots m_j \dots m_k \dots 0} \exp\bigl[2\pi i \left( m_jX_j + m_kX_k \right) \bigr] \\
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| f_{12 \dots n} &= \sum_{m_1 \neq 0} \sum_{m_2 \neq 0} \cdots \sum_{m_n \neq 0} C_{m_1 m_2 \dots m_n} \exp\bigl[ 2\pi i \left( m_1X_1+m_2X_2+\cdots+m_nX_n \right) \bigr].
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| \end{align}
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| </math>
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| The first order conditional variance is
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| :<math>
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| \begin{align}
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| V_j &= \int_0^1 f_j^2\left(X_j\right)dX_j\\
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| &= \sum_{ m_j \neq 0 } \left| C_{0 \dots m_j \dots 0} \right|^2\\
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| &= 2\sum_{m_j=1}^{\infty} \left( A_{m_j}^2+B_{m_j}^2 \right)
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| \end{align}</math>
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| where <math> A_{m_j} </math> and <math> B_{m_j} </math> are the real and imaginary part of <math> C_{0 \dots m_j \dots 0} </math> respectively
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| :<math>
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| \begin{align}
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| A_{m_j} &= \int_0^1 \cdots \int_0^1 f \left(X_1, X_2, \dots, X_n\right) \cos\left(2\pi m_jX_j\right)dX_1dX_2 \dots dX_n \\
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| B_{m_j} &= \int_0^1 \cdots \int_0^1 f \left(X_1, X_2, \dots, X_n\right) \sin\left(2\pi m_jX_j\right)dX_1dX_2 \dots dX_n
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| \end{align}
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| </math>
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| === Ergodic theorem ===
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| A multi-dimensional integral is required to evaluate for calculating the Fourier coefficients. One way is to transform the multi-dimensional integral into a one-dimensional integral by expressing every input as a function of a new independent variable <math> s </math> as
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| :<math> X_j \left( s \right) = \frac{1}{2\pi}\left(\omega_j s \text{ mod } 2\pi \right), j = 1,2,\dots,n </math>
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| where <math> \left\{\omega_j\right\} </math> is a set of incommensurate frequencies, i.e.
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| :<math> \sum_{j=1}^n \gamma_j\omega_j = 0 </math> | |
| for an integer set of <math> \left\{\gamma_j\right\} </math> if and only if <math> \gamma_j = 0 </math> for every <math> j </math>.
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| Then the Fourier coefficients can be calculated by a one-dimensional integral according to the ergodic theorem <ref>Weyl, H. (1938). Mean motion. ''American Journal of Mathematics'', '''60''', 889–896.</ref>
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| :<math>
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| \begin{align}
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| A_{m_j} &= \lim_{T \to \infty} \frac{1}{2T} \int_{-T}^T f\bigl(X_1\left(s\right),X_2\left(s\right),\dots,X_n\left(s\right)\bigr)\cos\bigl(2\pi m_jX_j\left(s\right)\bigr)ds\\
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| B_{m_j} &= \lim_{T \to \infty} \frac{1}{2T} \int_{-T}^T f\bigl(X_1\left(s\right),X_2\left(s\right),\dots,X_n\left(s\right)\bigr)\sin\bigl(2\pi m_jX_j\left(s\right)\bigr)ds
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| \end{align}
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| </math>
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| == Implementation ==
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| === Integer frequencies ===
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| At most one of the incommensurate frequencies <math> \left\{\omega_j\right\} </math> can be rational with all others being irrational. Since the numerical value of an irrational number cannot be stored exactly in a computer, an approximation of the incommensurate frequencies by all rational numbers is required in implementation. Without loss of any generality the frequencies can be set as integers instead of any rational numbers. A set of integers <math> \left\{\omega_j\right\} </math> is approximately incommensurate to the order of <math> M </math> if
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| :<math> \sum_{j=1}^n \gamma_j\omega_j \neq 0 </math>
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| for | |
| :<math> \sum_{j=1}^n \left| \gamma_j \right| \leq M + 1 </math>
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| where <math> M </math> is an integer. The exact incommensurate condition is a extreme case when <math> M \to \infty </math>.
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| Using the integer frequencies the function in the transformed one-dimensional integral is periodic so only the integration over a period of <math> 2\pi </math> is required. The Fourier coefficients can be approximately calculated as
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| :<math>
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| \begin{align}
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| A_{m_j} &\approx \frac{1}{2\pi} \int_{-\pi}^{\pi} f\bigl(X_1\left(s\right),X_2\left(s\right),\dots,X_n\left(s\right)\bigr)\cos\left(m_j\omega_j s\right)ds := \hat{A}_{m_j}\\
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| B_{m_j} &\approx \frac{1}{2\pi} \int_{-\pi}^{\pi} f\bigl(X_1\left(s\right),X_2\left(s\right),\dots,X_n\left(s\right)\bigr)\sin\left(m_j\omega_j s\right)ds := \hat{B}_{m_j}
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| \end{align}
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| </math>
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| The approximation of the incommensurate frequencies for a finite <math> M </math> results in a discrepancy error between the true Fourier coefficients <math> A_{m_j} </math>, <math> B_{m_j} </math> and their estimates <math> \hat{A}_{m_j} </math>, <math> \hat{B}_{m_j} </math>. The larger the order <math> M </math> is the smaller the error is but the more computational efforts are required to calculate the estimates in the following procedure. In practice <math> M </math> is frequently set to 4 and a table of resulted frequency sets which have up to 50 frequencies is available. (McRae et al., 1982)
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| === Search curve ===
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| The transform, <math> X_j \left( s \right) = \frac{1}{2\pi}\left(\omega_j s \text{ mod } 2\pi \right)</math>, defines a search curve in the input space. If the frequencies, <math> \omega_j, j = 1,\dots,n </math>, are incommensurate, the search curve can pass through every point in the input space as <math> s </math> varies from 0 to <math>\infty</math> so the multi-dimensional integral over the input space can be accurately transformed to a one-dimensional integral along the search curve. However, if the frequencies are approximately incommensurate integers, the search curve cannot pass through every point in the input space. If fact the search is repeated since the transform function is periodic, with a period of <math>2\pi</math>. The one-dimensional integral can be evaluated over a single period instead of the infinite interval for incommensurate frequencies; However, a computational error arises due to the approximation of the incommensuracy.
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| <gallery caption="Search curve" widths="320px" heights="320px" perrow="3">
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| File:Search_curve_1.gif | The search curve in the case of ω<sub>1</sub>=π and ω<sub>2</sub>=7. Since the frequencies are incommensurate, the search curve is not repeated and can pass through every point on the square
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| File:Search_curve_2.gif | The search curve in the case of ω<sub>1</sub>=3 and ω<sub>2</sub>=7. Since the frequencies are integers, which are approximately incommensurate, the search curve is repeated and cannot pass through every point on the square
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| File:Search_curve_3.gif | The search curve in the case of ω<sub>1</sub>=11 and ω<sub>2</sub>=7. Since the frequencies are integers, which are approximately incommensurate, the search curve is repeated and cannot pass through every point on the square
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| </gallery>
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| === Sampling ===
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| The approximated Fourier can be further expressed as
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| :<math>
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| \hat{A}_{m_j}=
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| \begin{cases}
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| 0 & m_j \text{ odd} \\
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| \frac{1}{\pi}\int_{-\pi/2}^{\pi/2}f\bigl(\mathbf X(s)\bigr)\cos\left(m_j\omega_js\right)ds & m_j \text{ even}
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| \end{cases}
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| </math>
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| and | |
| :<math>
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| \hat{B}_{m_j}=
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| \begin{cases}
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| \frac{1}{\pi}\int_{-\pi/2}^{\pi/2}f\bigl(\mathbf X(s)\bigr)\sin\left(m_j\omega_js\right)ds & m_j \text{ odd} \\
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| 0 & m_j \text{ even}
| |
| \end{cases}
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| </math>
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| The non-zero integrals can be calculated from sampling points
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| :<math>
| |
| \begin{align}
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| \hat{A}_{m_j} &= \frac{1}{2q+1}\sum_{k=-q}^q f\bigl(\mathbf X(s_k)\bigr)\cos\left( m_j \omega_j s_k\right), m_j \text{ even}\\
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| \hat{B}_{m_j} &= \frac{1}{2q+1}\sum_{k=-q}^q f\bigl(\mathbf X(s_k)\bigr)\sin\left( m_j \omega_j s_k\right), m_j \text{ odd }
| |
| \end{align}
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| </math>
| |
| where the uniform sampling point in <math> \left[-\pi/2, \pi/2\right] </math> is
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| :<math> s_k = \frac{\pi k}{2q+1}, k=-q,\dots,-1,0,1,\dots,q. </math>
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| The total number of sampling points is <math> 2q+1 </math> which should satisfy the Nyquist sampling criterion, i.e.
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| :<math> 2q+1 \geq N\omega_{max}+1 </math>
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| where <math> \omega_{max} </math> is the largest frequency in <math> \left\{\omega_k\right\} </math> and <math> N </math> is the maximum order of the calculated Fourier coefficients.
| |
| | |
| === Partial sum ===
| |
| After calculating the estimated Fourier coefficients, the first order conditional variance can be approximated by
| |
| :<math>
| |
| \begin{align}
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| V_j &= 2\sum_{m_j=1}^{\infty} \left( A_{m_j}^2+B_{m_j}^2 \right) \\
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| &\approx 2\sum_{m_j=1}^{\infty} \left( \hat{A}_{m_j}^2+\hat{B}_{m_j}^2 \right) \\
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| &\approx 2\sum_{m_j=1}^{2} \left( \hat{A}_{m_j}^2+\hat{B}_{m_j}^2 \right) \\
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| &= 2\left( \hat{A}_{m_j=2}^2 + \hat{B}_{m_j=1}^2 \right)
| |
| \end{align}</math>
| |
| where only the partial sum of the first two terms is calculated and <math> N=2 </math> for determining the number of sampling points. Using the partial sum can usually return an adequately good approximation of the total sum since the terms corresponding to the fundamental frequency and low order frequencies usually contribute most to the total sum. Additionally, the Fourier coefficient in the summation are just an estimate of the true value and adding more higher order terms will not help improve the computational accuracy significantly. Since the integer frequencies are not exactly incommensurate there are two integers <math> m_j </math> and <math> m_k </math> such that <math> m_j\omega_j = m_k\omega_k. </math> Interference between the two frequencies can occur if higher order terms are included in the summation.
| |
| | |
| Similarly the total variance of <math> f\left( \mathbf X \right) </math> can be calculated as
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| :<math> V \approx \hat{A}_0\left[ f^2 \right] - \hat{A}_0\left[ f \right]^2 </math>
| |
| where <math> \hat{A}_0\left[ f^2 \right] </math> denotes the estimated Fourier coefficient of the function of <math> f^2 </math> inside the bracket and <math> \hat{A}_0\left[ f \right]^2 </math> is the squared Fourier coefficient of the function <math> f </math>. Finally the sensitivity referring to the main effect of an input can be calculated by dividing the conditional variance by the total variance.
| |
| | |
| == References ==
| |
| <references/>
| |
| | |
| [[Category:Sensitivity analysis]]
| |
| [[Category:Fourier series]]
| |
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