How To Find Sum Of Squared Deviations

Stochastic Analysis

Donald W. Boyd , in Systems Analysis and Modeling, 2001

8.2.1.ii Sample Variance

The mean squared deviation of observations from the sample mean, ${\mathrm{southward}}_j^2={\widehat{\sigma }}_j^2,$ is called the sample variance and provides an estimate of the 2d moment of inertia

$s_j^{\mathrm{ii}}=\frac{\mathrm{one}}{N}{\displaystyle \sum _{t=\mathrm{ane}}^{\mathrm{Northward}}{\left(X_j^t-{\overline{\mathrm{10}}}_j\right)}^2}.$

One of the Due north degrees of freedom is "used up" in that all Northward observations are required to calculate ${\overline{X}}_j.$ If N is replaced by N − 1, and then $s_j^2$ becomes an unbiased estimator for ${\sigma }_j^2:E\left({\mathrm{south}}_j^2\right)={\sigma }_j^2.$

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Measures of key trend

J. Hayavadana , in Statistics for Textile and Apparel Direction, 2012

3 The principle of to the lowest degree squares

The sum of the squared deviations of all the scores about the mean is less than the sum of the squared deviations nearly whatever other value. This is chosen the principle of least squares. For example, referring to the above tabular array the sum of the squared deviations about the mean is 10. If however, iii and 6 are taken equally arbitrary values of the hateful, the ∑  x² becomes 15 and 30, respectively. Thus we tin can say that when hateful is four the value ∑   x² is 10 which is less than xv and 30. From this we can say that, the essential holding of hateful is that information technology is closer to the individual scores over the entire grouping than any other single value. This concept is used in regression and prediction.

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Probability and statistics

Mary Attenborough , in Mathematics for Electrical Engineering and Computing, 2003

Cavalcade vii: f_i (x_i − $\overline{x}$ )^two, the variance and the standard deviation

For each grade we multiply the frequency by the squared deviation, calculated in column half-dozen. This gives an approximation to the total squared deviation for that form. For the 6th class we multiply the squared difference of 54 149 past the frequency 103 to get 5 577 347. The sum of this column gives the total squared divergence from the mean for the whole sample. Dividing by the number of sample points gives an idea of the average squared divergence. This is called the variance. It is found by summing column 7 and dividing by thousand, the number in the sample, giving a variance of 39 120. A amend measure of the spread of the data is given by the square root of this number, called the standard deviation and usually represented by σ. Here $\sigma =\sqrt{39120}\approx 198$ .

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Analysis of variance

J. Hayavadana , in Statistics for Textile and Wearing apparel Management, 2012

One-fashion classification

The value of variance ratio or F tin can be computed as follows:

(a)

Find the sum of squared deviations (total sum of squares) using the formula ${\displaystyle \sum {\mathit{x}}^2-\frac{{\left({\displaystyle \sum \mathit{ten}}\right)}^{\mathrm{two}}}{\mathit{n}}.}$

We are required to dissever the sum of squares into two parts.

(i)

Sum of squares due to 'variation between samples'. This sum can be obtained past using the formula

${\displaystyle \sum \left[\frac{{\mathit{X}}^2}{\mathit{Five}}\right]-\frac{{\left({\displaystyle \sum \mathit{x}}\right)}^2}{\mathit{n}}}$

where X is the sum of observations in each sample (i.e. sum total) and Five is the number of observations (values) in each sample.

(ii)

Sum of squares due to variation within samples, i.e., sum of squares inside samples

$\begin{array}{l}=\mathrm{Full}\mathrm{sum}\mathrm{of}\mathrm{squares}–\mathrm{Sum}\mathrm{of}\mathrm{squares}\mathrm{betwixt}\mathrm{samples}.\\ =\left[{\displaystyle \sum {\mathit{x}}^2-\frac{{\left({\displaystyle \sum \mathit{x}}\right)}^2}{\mathit{n}}}\right]-{\displaystyle \sum \left[\frac{{\mathit{X}}^{\mathrm{two}}}{\mathit{V}}\right]-\frac{{\left({\displaystyle \sum \mathit{x}}\right)}^{\mathrm{two}}}{\mathit{northward}}}\\ ={\displaystyle \sum {\mathit{x}}^2-}{\displaystyle \sum \left[\frac{{\mathit{X}}^2}{\mathit{V}}\right]}\end{array}$

(b)

Variance for each part is obtained by dividing the sum of the squares past the corresponding degrees of freedom.

$\text{Total number of df}=\text{Total number of items}-1=-1.$

(i)

df   for   between   samples   =   number   of   samples   ‐   1   =   North   ‐   ane

(ii)

$\begin{array}{ll}\mathrm{df}\mathrm{for}\mathrm{within}\mathrm{samples}\hfill & =\mathrm{Total}\mathrm{df}-\mathrm{df}\mathrm{for}\mathrm{between}\mathrm{samples}=\hfill \\ \hfill & =\left(\mathrm{due\; north}-\mathrm{one}\right)-(\mathrm{N}-\mathrm{i})=\mathrm{north}-\mathrm{North}.\hfill \end{array}$

An analysis of variance table is prepare as follows:

Source of variation	Sum of squares	df	Variance
(a) Between samples	${\displaystyle \sum \left[\frac{{\mathit{10}}^2}{\mathit{V}}\right]-\frac{{\left({\displaystyle \sum \mathit{Ten}}\right)}^{\mathrm{two}}}{\mathit{north}}}$	N−1	$\frac{{\displaystyle \sum \left[\frac{{\mathit{X}}^2}{\mathit{Five}}\right]-\frac{{\left({\displaystyle \sum \mathit{X}}\right)}^2}{\mathit{n}}}}{\mathit{Due\; north}-1}={\mathit{v}}_2$
(b) Within sample	${\displaystyle \sum {\mathit{X}}^2-{\displaystyle \sum \left[\frac{{\mathit{X}}^{\mathrm{two}}}{\mathit{V}}\right]}}$	n−N	$\frac{{\displaystyle \sum {\mathit{X}}^2-{\displaystyle \sum \left[\frac{{\mathit{X}}^2}{\mathit{V}}\right]}}}{\mathit{due\; north}-\mathit{N}}={\mathit{v}}_1$

The variance for inside samples is known as error variance besides.

The variance ratio or F   = v ₂/v _i

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A applied guide to validation and verification of analytical methods in the clinical laboratory

Joachim Pum , in Advances in Clinical Chemistry, 2019

3.i.1.2.2 Deming regression

With Deming regression (Fig. 2 C), the line-of-best-fit is estimated by minimizing the sum of the squared deviations between actual and observed values at an angle determined by the ratio between the analytical standard deviations of both measurement methods ( λ). If λ  =   1, the angle is 90°. Although diverse procedures are bachelor for calculating standard errors of slope and intercept [31,32], the jackknife method is preferred, as hypothesis testing becomes more than accurate, when standard errors are estimated with this technique [31]. The jackknife procedure consists of creating a number of subsets with n  −   ane samples, by iteratively removing 1 sample from the consummate gear up, calculating slope and intercept for each subset and then determining the hateful differences for slope and intercept between the total and censored subsets. The standard errors are then calculated, using these mean differences, rather than the calculated gradient and intercept [33,34].

A major advantage of Deming regression is that it makes provision for measurement errors in both methods. In club for these to exist taken into business relationship, λ must be known beforehand. This is either estimated from indistinguishable measurements or taken from quality command information. While the unweighted Deming method presumes constant analytical standard deviations for both methods, the weighted modification allows for non-constant measurement errors. The ratio betwixt the analytical standard deviations is assumed to remain constant, however. The advantage of applying weighted Deming regression, if both methods display a proportional fault, increases with an increasing range ratio [32]. In spite of the importance of correctly estimating λ, the model is surprisingly robust and, even with an incorrectly specified error ratio, will likely all the same perform better than OLR [35].

The first pace in calculating Deming regression parameters is to calculate the ordinary to the lowest degree squares slope with y as the dependent and ten every bit the independent variables (b _yx ). Next, the ordinary least squares slope is calculated with 10 as the dependent and y as the independent variable (b _xy ).

The mistake ratio λ is calculated every bit:

(24) $\lambda =\frac{{\mathit{SD}}_{\mathrm{10}}^2}{{\mathit{SD}}_y^2}$

Next, f is divers equally:

(25) $f=\frac{1}{b_{\mathit{xy}}}-\frac{\lambda }{b_{\mathit{yx}}}$

The Deming slope (b _d ) and intercept (a _d ) are and so calculated as:

(26) $b_d=0.5\times \left(f+{\left(f^2+4\lambda \right)}^{\mathrm{0.v}}\right)$

(27) $a_d=\overline{y}-\left(b_d\times x\right)$

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Fuel prison cell parameters estimation using optimization techniques

Ahmed Due south. Menesy , ... Francisco Jurado , in Renewable Energy Systems, 2021

22.2.2 Formulation of the objective part

The PEMFC mainly depends on seven variable parameters during its performance. The interpretation of these parameters can be represented every bit an optimization problem. In this trouble, the total squared deviations (TSD) between the measured final voltages and the estimated ones are considered as the main objective function (OF) (Ali, El-Hameed, & Farahat, 2017; Chen & Wang, 2019; El-Fergany, 2017; Menesy, Sultan, et al., 2020; Menesy, Sultan, Korashy, et al. 2020; Rao et al., 2019; Sultan et al., 2020; Turgut and Coban, 2016). Yet, this OF is represented equally follows:

(22.16) $\mathrm{OF}=\min \mathrm{TSD}(\mathrm{10})=\sum _{i=1}^{\mathrm{Due\; north}}{\left({\mathrm{Five}}_{meas}(i)-V_{cal}(i)\right)}^2$

where X is a vector of the seven parameters, N denotes the measured points number, i is an iteration counter, V _meas represents the measured voltage, and Five _cal denotes the calculated PEMFC voltage.

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Statistics, Probability and Racket

Steven Westward. Smith , in Digital Signal Processing: A Practical Guide for Engineers and Scientists, 2003

Signal vs. Underlying Procedure

Statistics is the science of interpreting numerical data, such equally caused signals. In comparing, probability is used in DSP to empathize the processes that generate signals. Although they are closely related, the distinction betwixt the caused indicate and the underlying process is key to many DSP techniques.

For example, imagine creating a chiliad-bespeak signal by flipping a coin 1000 times. If the coin flip is heads, the respective sample is made a value of ane. On tails, the sample is set up to nix. The process that created this signal has a hateful of exactly 0.v, adamant by the relative probability of each possible outcome: 50% heads, 50% tails. Still, information technology is unlikely that the actual 1000-indicate bespeak will have a hateful of exactly 0.5. Random chance volition make the number of ones and zeros slightly different each time the signal is generated. The probabilities of the underlying procedure are constant, simply the statistics of the caused signal change each time the experiment is repeated. This random irregularity plant in actual data is chosen past such names equally: statistical variation, statistical fluctuation, and statistical racket.

This presents a flake of a dilemma. When you see the terms: mean and standard deviation, how do you know if the writer is referring to the statistics of an actual signal, or the probabilities of the underlying process that created the signal? Unfortunately, the only way you can tell is past the context. This is not so for all terms used in statistics and probability. For example, the histogram and probability mass function (discussed in the next section) are matching concepts that are given dissever names.

Now, back to Eq. 2-2, calculation of the standard deviation. As previously mentioned, this equation divides by Northward −1 in computing the average of the squared deviations, rather than only by N. To understand why this is so, imagine that you lot desire to find the mean and standard deviation of some process that generates signals. Toward this end, you learn a point of N samples from the process, and calculate the mean of the signal via Eq. 2.ane. Yous can then use this as an estimate of the hateful of the underlying process; however, you know there will exist an error due to statistical noise. In particular, for random signals, the typical mistake betwixt the hateful of the North points, and the mean of the underlying process, is given by:

EQUATION 2-4

Typical error in calculating the hateful of an underlying process by using a finite number of samples, Northward. The parameter, σ, is the standard divergence.

$Typicalerror=\frac{\sigma }{N^{1/2}}$

If Due north is pocket-size, the statistical noise in the calculated mean will be very large. In other words, you do not have access to enough data to properly characterize the process. The larger the value of N, the smaller the expected error will become. A milestone in probability theory, the Strong Police force of Large Numbers, guarantees that the error becomes zero as Due north approaches infinity.

In the next pace, nosotros would like to calculate the standard deviation of the acquired betoken, and use it as an estimate of the standard deviation of the underlying process. Herein lies the problem. Before you tin can summate the standard deviation using Eq. 2-2, you need to already know the hateful, μ. However, y'all don't know the mean of the underlying process, only the mean of the Due north point signal, which contains an fault due to statistical racket. This error tends to reduce the calculated value of the standard difference. To compensate for this, N is replaced by N−i. If N is big, the divergence doesn't matter. If N is small, this replacement provides a more accurate estimate of the standard deviation of the underlying process. In other words, Eq. 2-2 is an estimate of the standard departure of the underlying process. If we divided past Due north in the equation, information technology would provide the standard deviation of the caused betoken.

As an illustration of these ideas, expect at the signals in Fig. 2-3, and ask: are the variations in these signals a upshot of statistical racket, or is the underlying process changing? It probably isn't hard to convince yourself that these changes are too large for random take a chance, and must exist related to the underlying process. Processes that change their characteristics in this mode are chosen nonstationary. In comparison, the signals previously presented in Fig. ii-1 were generated from a stationary process, and the variations result completely from statistical noise. Figure 2-3b illustrates a common problem with nonstationary signals: the slowly changing mean interferes with the calculation of the standard divergence. In this case, the standard deviation of the point, over a short interval, is one. However, the standard deviation of the unabridged signal is one.xvi. This error can be virtually eliminated by breaking the signal into short sections, and computing the statistics for each section individually. If needed, the standard deviations for each of the sections tin can be averaged to produce a unmarried value.

Figure ii-3. Examples of signals generated from nonstationary processes. In (a), both the hateful and standard divergence change. In (b), the standard deviation remains a constant value of one, while the mean changes from a value of nada to ii. It is a common analysis technique to break these signals into brusque segments, and calculate the statistics of each segment individually.

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Cost Models

Monica Greer Ph.D , in Electricity Marginal Toll Pricing, 2012

Aside: Nonlinear Least-Squares Interpretation

For reasons stated before, this model must be estimated using a nonlinear interpretation procedure, namely nonlinear least squares. In this case, values of the parameters that minimize the sum of squared deviations will exist maximum likelihood (as well as nonlinear least-squares estimators). Because the kickoff-order conditions volition yield a set of nonlinear equations to which there will not be explicit solutions, an iterative process is required, ⁹ such every bit the Gauss–Newton method, which is the preferred method.

Probably the greatest concern here is that the estimators produced by this nonlinear least-squares procedure are not necessarily the virtually efficient (except in the example of ordinarily distributed errors). An excerpt from Greene (1993) illustrates this indicate nicely:

In the classical regression model, in order to obtain the requisite asymptotic results, it is assumed that the sample moment matrix, (1/n) X′10, converges to a positive definite matrix, Q. By analogy, the same status is imposed on the regressors in the linearized model when they are computed at the true parameter values. That is, if:

(5.60) $pli\mathrm{one\; thousand}1/\mathrm{due\; north}\mathit{X}\prime \mathit{X}=\mathit{Q},$

a positive definite matrix, then the coefficient estimates are consistent estimators. In add-on, if

(five.61) $(\mathrm{i}/\sqrt{n})\mathit{X}\prime \epsilon \to \mathrm{Northward}[0,{\sigma }^2\mathit{Q}],$

and so the estimators are asymptotically normal as well. Under nonlinear estimation, this is analogous to:

(five.62) $plim(1\mathrm{/}n)\underset{¯}{\mathit{X}}\prime \underset{¯}{\mathit{X}}=pli\mathrm{1000}(1/n){\sum }_i[\partial h(x_i,{\mathit{\beta }}^0)/\partial {\mathit{\beta }}^0][\partial h(x_i,{\mathit{\beta }}^0)/\partial {\mathit{\beta }}^0{}^{\prime }]=\underset{¯}{\mathit{Q}}$

where Q is a positive definite matrix. In add-on, in this case the derivatives in X play the part of the regressors.

The nonlinear least-squares criterion function is given by

(5.63) $\mathrm{South}(\mathit{b})={\displaystyle {\sum }_i{[y_i-h({\mathit{x}}_i,b)]}^2}={\displaystyle {\sum }_i{\mathrm{due\; east}}_i{}^2},$

where b , which will be the solution value, has been inserted. Get-go-order conditions for a minimum are

(5.64) $g(b)=-2{\displaystyle {\sum }_i[y_i-h({\mathit{x}}_i,\mathit{b})][\partial h({\mathit{10}}_i,\mathit{b})/\partial \mathit{b}]}=0$

or

(5.65) $g(b)=-2\underset{¯}{\mathit{X}}\prime \mathit{east}.$

This is a standard trouble in nonlinear estimation, which tin can be solved past a number of methods. Ane of the most often used is that of Gauss–Newton, which, at its final iteration, the approximate of Q ⁻¹ will provide the right approximate of the asymptotic covariance matrix for the parameter estimates. A consistent estimator of σ² can be computed using the residuals

(5.66) ${\sigma }^2=(\mathrm{i}/n){\displaystyle {\sum }_i{[y_i-h({\mathit{x}}_i,\mathit{b})]}^2}.$

In addition, it has been shown that ( Amemiya, 1985 )

(5.67) $\mathit{b}\to N[\mathit{\beta },{\sigma }^2/\mathrm{northward}{\mathit{Q}}^{-\mathrm{one}}],$

where

(five.68) $\mathit{Q}=pli\mathrm{thou}{(\underset{¯}{\mathit{X}}\prime \underset{¯}{\mathit{X}})}^{-\mathrm{one}}.$

The sample guess of the asymptotic covariance matrix is

(5.69) $E\mathrm{due\; south}t.Asy.Var[b]={\underset{¯}{\sigma }}^{\mathrm{two}}{(\underset{¯}{\mathit{X}}\prime \underset{¯}{\mathit{X}})}^{-\mathrm{i}}.$

From these, inference and hypothesis tests tin proceed accordingly.

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Time-Frequency Methods in Radar, Sonar & Acoustics

In Time Frequency Analysis, 2003

14.3.four.1 Narrowband Source in Level Flight with Abiding Velocity: Microphone in Air

The source parameters {f ₀, 5, τ _c , R_c }, or equivalently {α, β, τ _c , due south }, are estimated by minimizing the sum of the squared deviations of the noisy IF estimates from their predicted values [1]. Specifically, the NLS estimates of {α, β, τ _c , s} are given by

(14.iii.9) $\left\{\widehat{\alpha },\widehat{\beta },{\widehat{\tau }}_c,\widehat{\mathrm{south}}\right\}=\arg ╡\left\{\underset{\left\{\alpha \text{'},\beta \text{'},{\tau }_c\text{'},\mathrm{due\; south}\text{'}\right\}}{\min ╡}{\displaystyle \sum _{\mathrm{one\; thousand}=1}^M{\left[{\alpha }^{\prime }+{\beta }^{\prime }p\left(t_m;{{\tau }^{\prime }}_c,s^{\prime }\right)-\mathrm{1000}\left(t_k\right)\right]}^2}\right\}$

where g(tk) is the IF estimate at sensor time t = t_k and K is the number of IF estimates. The four-dimensional minimization in (fourteen.three.9) tin exist reduced to a two-dimensional maximization [one]:

(14.three.10) $\left\{{\widehat{\tau }}_c,\widehat{s}\right\}=\arg \left\{\underset{\left\{{\tau }_c\text{'},s\text{'}\right\}}{\max }\frac{{\left|{\Sigma }_{\mathrm{chiliad}=\mathrm{ane}}^{\mathrm{Thou}}\left[\mathrm{thousand}\left(t_k\right)-\overline{g}\right]p\left(t_k\right)\right|}^{\mathrm{two}}}{{\Sigma }_{\mathrm{one\; thousand}=\mathrm{ane}}^{\mathrm{Thou}}{\left[p\left(t_{\mathrm{one\; thousand}}\right)-\overline{p}\right]}^2}\right\}$

(fourteen.3.11) $\widehat{\beta }=\frac{{\Sigma }_{\mathrm{chiliad}=\mathrm{one}}^{\mathrm{Thousand}}\left[g\left(t_k\right)-\overline{g}\right]\widehat{p}\left(t_{\mathrm{one\; thousand}}\right)}{{\Sigma }_{k=1}^{\mathrm{Thou}}{\left[\widehat{p}\left(t_{\mathrm{one\; thousand}}\right)-\overline{\widehat{p}}\right]}^2}$

(fourteen.3.12) $\widehat{\alpha }=\overline{g}-\widehat{\beta }\overline{\widehat{p}}$

where $\overline{g}=\frac{1}{K}{\Sigma }_k\mathrm{grand}\left(t_k\right),p\left(t_k\right)=p\left(t_{\mathrm{thousand}};{{\tau }^{\prime }}_c,{\mathrm{southward}}^{\prime }\right),\overline{p}=\frac{1}{\mathrm{One\; thousand}}{\Sigma }_{k}p\left(t_{\mathrm{chiliad}}\right),\widehat{p}\left(t_k\right)=p\left(t_k;{\widehat{\tau }}_c,\widehat{s}\right)$ , and $\overline{\widehat{p}}=\frac{1}{\mathrm{Thousand}}{\Sigma }_k\widehat{p}\left(t_{\mathrm{1000}}\right)$ . Solving (14.three.2) and (14.3.3) using the estimated values for α and β gives the estimates of the source speed v and source frequency f ₀ as

(14.3.13) $\widehat{\upsilon }=-\left(\widehat{\beta }/\widehat{\alpha }\right)c_a$

(14.three.xiv) ${\widehat{f}}_0=\widehat{\alpha }\left(1-{\widehat{\upsilon }}^{\mathrm{ii}}/c_a^{\mathrm{two}}\right).$

From (14.3.4), the judge of the CPA slant range R_c is given by

(xiv.iii.15) ${\widehat{R}}_c=\widehat{\mathrm{south}}\widehat{\upsilon }{c}_a/\sqrt{c_a^2-{\widehat{\upsilon }}^2.}$

The maximization in (xiv.three.ten) is performed using the quasi-Newton method where the initial estimates of τ _c and south are given by the method described in [1]. The results of applying the source parameter interpretation method to experimental data (represented by the circles) are shown at the top of Figs. 14.3.i(a) and 14.3.1(b). The estimates closely friction match the bodily values of the aircraft's speed, distance, and propeller or master rotor blade rate.

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Fourth dimension-Frequency Methods in Radar, Sonar, and Acoustics

In Time-Frequency Signal Analysis and Processing (Second Edition), 2016

fourteen.3.4.1 Narrowband source in level flight with abiding velocity: Microphone in air

The source parameters {f ₀,v,τ _c ,R _c }, or equivalently {α,β,τ _c ,s }, are estimated by minimizing the sum of the squared deviations of the noisy IF estimates from their predicted values [ sixteen]. Specifically, the NLS estimates of {α,β,τ _c ,s} are given by

(fourteen.3.9) $\begin{array}{l}\hfill \{\widehat{\alpha },\widehat{\beta },\widehat{{\tau }_c},ŝ\}=arg\left\{\underset{\{{\alpha }^{\prime }{\beta }^{\prime },{\tau }_c^{\prime },s^{\prime }\}}{min}\sum _{\mathrm{1000}=\mathrm{i}}^{\mathrm{Yard}}{\left[{\alpha }^{\prime }+{\beta }^{\prime }p(t_{\mathrm{chiliad}};{\tau }_c^{\prime },s^{\prime })-k(t_k)\right]}^2\right\},\end{array}$

where g(t _k ) is the IF estimate at sensor time t = t _grand and K is the number of IF estimates. The iv-dimensional minimization in Eq. (fourteen.3.ix) tin can be reduced to a two-dimensional maximization [16]:

(14.three.10) $\begin{array}{ll}\hfill \{{\widehat{\tau }}_c,ŝ\}=& arg\left\{\underset{\{{\tau }_c^{\prime },s^{\prime }\}}{max}\frac{{\left|\sum _{\mathrm{one\; thousand}=1}^K[g(t_k)-\stackrel{-}{g}]p(t_k)\right|}^2}{\sum _{k=1}^{\mathrm{Grand}}{[p(t_k)-\stackrel{-}{p}]}^2}\right\},\hfill \end{array}$

(14.3.eleven) $\begin{array}{ll}\hfill \widehat{\beta }=& \frac{\sum _{k=1}^K[\mathrm{chiliad}(t_k)-\stackrel{-}{\mathrm{1000}}]\widehat{p}(t_k)}{\sum _{k=\mathrm{ane}}^K{[\widehat{p}(t_{\mathrm{thousand}})-\stackrel{-}{\widehat{p}}]}^2},\hfill \end{array}$

(14.3.12) $\begin{array}{ll}\hfill \widehat{\alpha }=& \stackrel{-}{\mathrm{chiliad}}-\widehat{\beta }\stackrel{-}{\widehat{p}},\hfill \end{array}$

where $\stackrel{-}{\mathrm{1000}}=\frac{\mathrm{i}}{K}{\sum }_{\mathrm{chiliad}}m(t_k)$ , $p(t_k)=p(t_k;{\tau }_c^{\prime },{\mathrm{southward}}^{\prime })$ , $\stackrel{-}{p}=\frac{1}{M}{\sum }_{k}p(t_k)$ , $\widehat{p}(t_g)=p(t_m;{\widehat{\tau }}_c,ŝ)$ , and $\stackrel{-}{\widehat{p}}=\frac{1}{K}{\sum }_k\widehat{p}(t_{\mathrm{thousand}})$ . Solving Eqs. (14.three.ii) and (14.3.three) using the estimated values for α and β gives the estimates of the source speed 5 and source frequency f ₀ as

(xiv.3.xiii) $\begin{array}{ll}\hfill \widehat{v}=& -(\widehat{\beta }/\widehat{\alpha })c_{\mathrm{a}},\hfill \end{array}$

(14.three.14) $\begin{array}{ll}\hfill {\widehat{f}}_0=& \widehat{\alpha }(1-{\widehat{v}}^2/c_{\mathrm{a}}^2).\hfill \end{array}$

From Eq. (14.three.four), the estimate of the CPA camber range R _c is given by

(14.3.15) $\begin{array}{l}\hfill {\widehat{R}}_c=ŝ\widehat{v}{c}_{\mathrm{a}}/\sqrt{c_{\mathrm{a}}^2-{\widehat{v}}^2}.\end{array}$

The maximization in Eq. (14.3.ten) is performed using the quasi-Newton method where the initial estimates of τ _c and southward are given past the method described in Ref. [16]. The results of applying the source parameter estimation method to experimental information (represented past the circles) are shown at the tiptop of Fig. xiv.iii.one (a) and (b). The estimates closely friction match the actual values of the aircraft's speed, distance, and propeller or main rotor blade charge per unit.

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