Sum Of The Squared Deviation

Stochastic Analysis

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

8.2.1.two Sample Variance

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

$s_j^{\mathrm{two}}=\frac{1}{\mathrm{North}}{\displaystyle \sum _{t=\mathrm{ane}}^N{\left({\mathrm{Ten}}_j^t-{\overline{\mathrm{Ten}}}_j\right)}^{\mathrm{ii}}}.$

I of the North degrees of liberty is "used up" in that all Northward observations are required to summate ${\overline{X}}_j.$ If N is replaced by N − 1, then $s_j^2$ becomes an unbiased reckoner for ${\sigma }_j^2:E\left(s_j^{\mathrm{two}}\right)={\sigma }_j^{\mathrm{ii}}.$

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Measures of key tendency

J. Hayavadana , in Statistics for Cloth and Clothes Management, 2012

3 The principle of least squares

The sum of the squared deviations of all the scores almost the mean is less than the sum of the squared deviations about whatever other value. This is called the principle of least squares. For case, referring to the to a higher place tabular array the sum of the squared deviations most the mean is 10. If however, 3 and half-dozen are taken every bit arbitrary values of the mean, the ∑  x² becomes 15 and 30, respectively. Thus we can say that when hateful is iv the value ∑   x² is 10 which is less than 15 and 30. From this we can say that, the essential belongings of mean is that information technology is closer to the private scores over the entire group than any other unmarried 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 7: f_i (x_i − $\overline{x}$ )², the variance and the standard departure

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

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Analysis of variance

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

One-way classification

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

(a)

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

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

(i)

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

${\displaystyle \sum \left[\frac{{\mathit{X}}^{\mathrm{two}}}{\mathit{Five}}\right]-\frac{{\left({\displaystyle \sum \mathit{ten}}\right)}^{\mathrm{two}}}{\mathit{n}}}$

where X is the sum of observations in each sample (i.east. 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

$\begin{array}{l}=\mathrm{Total}\mathrm{sum}\mathrm{of}\mathrm{squares}–\mathrm{Sum}\mathrm{of}\mathrm{squares}\mathrm{between}\mathrm{samples}.\\ =\left[{\displaystyle \sum {\mathit{x}}^2-\frac{{\left({\displaystyle \sum \mathit{x}}\right)}^2}{\mathit{n}}}\right]-{\displaystyle \sum \left[\frac{{\mathit{X}}^2}{\mathit{V}}\right]-\frac{{\left({\displaystyle \sum \mathit{x}}\right)}^{\mathrm{ii}}}{\mathit{due\; north}}}\\ ={\displaystyle \sum {\mathit{10}}^2-}{\displaystyle \sum \left[\frac{{\mathit{X}}^2}{\mathit{V}}\right]}\end{array}$

(b)

Variance for each role is obtained by dividing the sum of the squares by the respective degrees of freedom.

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

(i)

df   for   between   samples   =   number   of   samples   ‐   one   =   N   ‐   i

(ii)

$\begin{array}{ll}\mathrm{df}\mathrm{for}\mathrm{inside}\mathrm{samples}\hfill & =\mathrm{Total}\mathrm{df}-\mathrm{df}\mathrm{for}\mathrm{betwixt}\mathrm{samples}=\hfill \\ \hfill & =\left(\mathrm{n}-\mathrm{i}\right)-(\mathrm{N}-1)=\mathrm{n}-\mathrm{N}.\hfill \end{array}$

An analysis of variance tabular array is ready as follows:

Source of variation	Sum of squares	df	Variance
(a) Between samples	${\displaystyle \sum \left[\frac{{\mathit{X}}^2}{\mathit{5}}\right]-\frac{{\left({\displaystyle \sum \mathit{Ten}}\right)}^2}{\mathit{due\; north}}}$	N−1	$\frac{{\displaystyle \sum \left[\frac{{\mathit{X}}^2}{\mathit{V}}\right]-\frac{{\left({\displaystyle \sum \mathit{X}}\right)}^{\mathrm{ii}}}{\mathit{n}}}}{\mathit{N}-1}={\mathit{v}}_{\mathrm{ii}}$
(b) Within sample	${\displaystyle \sum {\mathit{X}}^2-{\displaystyle \sum \left[\frac{{\mathit{Ten}}^{\mathrm{two}}}{\mathit{5}}\right]}}$	north−North	$\frac{{\displaystyle \sum {\mathit{X}}^2-{\displaystyle \sum \left[\frac{{\mathit{X}}^{\mathrm{two}}}{\mathit{V}}\right]}}}{\mathit{n}-\mathit{N}}={\mathit{v}}_1$

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

The variance ratio or F   = v _ii/v _i

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

Joachim Pum , in Advances in Clinical Chemistry, 2019

3.1.1.two.2 Deming regression

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

A major advantage of Deming regression is that it makes provision for measurement errors in both methods. In society for these to exist taken into business relationship, λ must be known beforehand. This is either estimated from duplicate measurements or taken from quality control data. While the unweighted Deming method presumes constant belittling standard deviations for both methods, the weighted modification allows for non-constant measurement errors. The ratio between the analytical standard deviations is assumed to remain abiding, however. The reward 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 probable notwithstanding perform ameliorate than OLR [35].

The start step in computing Deming regression parameters is to calculate the ordinary least squares slope with y every bit the dependent and x as the independent variables (b _yx ). Adjacent, the ordinary to the lowest degree squares slope is calculated with x as the dependent and y every bit the independent variable (b _xy ).

The mistake ratio λ is calculated as:

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

Next, f is defined every bit:

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

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

(26) $b_d=0.5\times \left(f+{\left(f^{\mathrm{ii}}+\mathrm{four}\lambda \right)}^{0.5}\right)$

(27) $a_d=\overline{y}-\left(b_d\times \mathrm{10}\right)$

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Fuel prison cell parameters estimation using optimization techniques

Ahmed Southward. Menesy , ... Francisco Jurado , in Renewable Energy Systems, 2021

22.2.two Formulation of the objective function

The PEMFC mainly depends on 7 variable parameters during its operation. The estimation of these parameters can be represented as an optimization trouble. In this trouble, the full squared deviations (TSD) between the measured concluding voltages and the estimated ones are considered every bit the main objective part (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). Still, this OF is represented as follows:

(22.xvi) $\mathrm{OF}=\min \mathrm{TSD}(\mathrm{Ten})=\sum _{i=\mathrm{one}}^N{\left({\mathrm{Five}}_{\mathrm{one\; thousand}eas}(i)-V_{cal}(i)\right)}^2$

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

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Statistics, Probability and Dissonance

Steven W. Smith , in Digital Bespeak Processing: A Applied Guide for Engineers and Scientists, 2003

Signal vs. Underlying Process

Statistics is the science of interpreting numerical data, such as acquired signals. In comparison, probability is used in DSP to sympathize the processes that generate signals. Although they are closely related, the distinction between the acquired signal and the underlying process is cardinal to many DSP techniques.

For example, imagine creating a 1000-betoken signal by flipping a coin 1000 times. If the money flip is heads, the respective sample is fabricated a value of one. On tails, the sample is set to zero. The procedure that created this signal has a mean of exactly 0.5, determined by the relative probability of each possible outcome: 50% heads, 50% tails. All the same, it is unlikely that the actual thousand-point signal will have a mean of exactly 0.v. Random chance will make the number of ones and zeros slightly unlike each time the signal is generated. The probabilities of the underlying process are constant, but the statistics of the acquired betoken change each time the experiment is repeated. This random irregularity establish in actual data is chosen by such names as: statistical variation, statistical fluctuation, and statistical noise.

This presents a bit of a dilemma. When you meet the terms: mean and standard deviation, how exercise yous know if the writer is referring to the statistics of an bodily signal, or the probabilities of the underlying process that created the signal? Unfortunately, the only way you can tell is by the context. This is not so for all terms used in statistics and probability. For example, the histogram and probability mass part (discussed in the side by side section) are matching concepts that are given separate names.

Now, dorsum to Eq. 2-2, adding of the standard deviation. Every bit previously mentioned, this equation divides by N −1 in calculating the average of the squared deviations, rather than simply past N. To empathize why this is and then, imagine that yous want to find the mean and standard deviation of some process that generates signals. Toward this end, y'all acquire a signal of N samples from the procedure, and calculate the mean of the signal via Eq. 2.one. You can and so use this as an judge of the mean of the underlying process; withal, y'all know there will exist an mistake due to statistical noise. In particular, for random signals, the typical error between the mean of the Northward points, and the mean of the underlying process, is given by:

EQUATION two-4

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

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

If N is small-scale, the statistical noise in the calculated mean will be very large. In other words, you exercise 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 Law of Big Numbers, guarantees that the error becomes cipher as N approaches infinity.

In the next step, we would like to summate the standard deviation of the acquired indicate, and apply it as an estimate of the standard divergence of the underlying process. Herein lies the problem. Earlier yous can calculate the standard deviation using Eq. 2-2, you need to already know the mean, μ. However, you don't know the hateful of the underlying process, only the mean of the Northward point signal, which contains an error due to statistical noise. This error tends to reduce the calculated value of the standard deviation. To recoup for this, N is replaced by N−1. If N is large, the deviation doesn't thing. If Northward is pocket-size, this replacement provides a more accurate estimate of the standard divergence of the underlying procedure. In other words, Eq. two-ii is an judge of the standard departure of the underlying process. If we divided by North in the equation, it would provide the standard difference of the acquired signal.

As an illustration of these ideas, look at the signals in Fig. 2-iii, and inquire: are the variations in these signals a result of statistical noise, or is the underlying process irresolute? It probably isn't hard to convince yourself that these changes are too big for random run a risk, and must exist related to the underlying procedure. Processes that change their characteristics in this fashion are chosen nonstationary. In comparison, the signals previously presented in Fig. 2-1 were generated from a stationary process, and the variations event completely from statistical noise. Figure 2-3b illustrates a common problem with nonstationary signals: the slowly changing hateful interferes with the adding of the standard deviation. In this instance, the standard departure of the signal, over a short interval, is i. However, the standard deviation of the entire signal is 1.sixteen. This error can exist nearly eliminated by breaking the point into brusque sections, and computing the statistics for each section individually. If needed, the standard deviations for each of the sections can be averaged to produce a single value.

FIGURE 2-3. Examples of signals generated from nonstationary processes. In (a), both the mean and standard deviation alter. In (b), the standard deviation remains a abiding value of ane, while the hateful changes from a value of zero to two. It is a common analysis technique to pause these signals into brusk segments, and calculate the statistics of each segment individually.

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Cost Models

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

Bated: Nonlinear To the lowest degree-Squares Estimation

For reasons stated earlier, 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 be maximum likelihood (as well as nonlinear least-squares estimators). Considering the first-order conditions will yield a set of nonlinear equations to which there will not be explicit solutions, an iterative procedure 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 most efficient (except in the case of ordinarily distributed errors). An excerpt from Greene (1993) illustrates this point nicely:

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

(five.threescore) $plik1/\mathrm{northward}\mathit{X}\prime \mathit{X}=\mathit{Q},$

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

(5.61) $(1/\sqrt{n})\mathit{Ten}\prime \epsilon \to \mathrm{North}[0,{\sigma }^2\mathit{Q}],$

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

(v.62) $pli\mathrm{thou}(\mathrm{one}\mathrm{/}\mathrm{north})\underset{¯}{\mathit{Ten}}\prime \underset{¯}{\mathit{X}}=pli\mathrm{yard}(1/\mathrm{north}){\sum }_i[\partial h(x_i,{\mathit{\beta }}^0)/\partial {\mathit{\beta }}^0][\partial h(x_i,{\mathit{\beta }}^0)/\partial {\mathit{\beta }}^0{}^{\prime }]=\underset{¯}{\mathit{Q}}$

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

The nonlinear least-squares benchmark part is given by

(5.63) $S(\mathit{b})={\displaystyle {\sum }_i{[y_i-h({\mathit{10}}_i,b)]}^2}={\displaystyle {\sum }_i{e}_i{}^2},$

where b , which will exist the solution value, has been inserted. First-guild conditions for a minimum are

(5.64) $\mathrm{thou}(b)=-\mathrm{ii}{\displaystyle {\sum }_i[y_i-h({\mathit{x}}_i,\mathit{b})][\partial h({\mathit{x}}_i,\mathit{b})/\partial \mathit{b}]}=0$

or

(5.65) $\mathrm{thou}(b)=-\mathrm{ii}\underset{¯}{\mathit{Ten}}\prime \mathit{e}.$

This is a standard problem in nonlinear estimation, which can be solved by a number of methods. One of the most oft used is that of Gauss–Newton, which, at its last iteration, the estimate of Q ⁻ⁱ will provide the correct approximate of the asymptotic covariance matrix for the parameter estimates. A consistent calculator of σ^two tin be computed using the residuals

(five.66) ${\sigma }^{\mathrm{ii}}=(1/\mathrm{north}){\displaystyle {\sum }_i{[y_i-h({\mathit{x}}_i,\mathit{b})]}^{\mathrm{ii}}}.$

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

(5.67) $\mathit{b}\to N[\mathit{\beta },{\sigma }^{\mathrm{ii}}/n{\mathit{Q}}^{-\mathrm{i}}],$

where

(v.68) $\mathit{Q}=pli\mathrm{thousand}{(\underset{¯}{\mathit{X}}\prime \underset{¯}{\mathit{X}})}^{-\mathrm{i}}.$

The sample estimate of the asymptotic covariance matrix is

(v.69) $\mathrm{Eastward}st.Asy.Var[b]={\underset{¯}{\sigma }}^{\mathrm{ii}}{(\underset{¯}{\mathit{X}}\prime \underset{¯}{\mathit{X}})}^{-1}.$

From these, inference and hypothesis tests can go on accordingly.

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Time-Frequency Methods in Radar, Sonar & Acoustics

In Fourth dimension Frequency Analysis, 2003

xiv.3.iv.i Narrowband Source in Level Flight with Constant Velocity: Microphone in Air

The source parameters {f ₀, 5, 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 [1]. Specifically, the NLS estimates of {α, β, τ _c , s} are given past

(14.3.9) $\left\{\widehat{\alpha },\widehat{\beta },{\widehat{\tau }}_c,\widehat{s}\right\}=\arg ╡\left\{\underset{\left\{\alpha \text{'},\beta \text{'},{\tau }_c\text{'},\mathrm{southward}\text{'}\right\}}{\min ╡}{\displaystyle \sum _{k=\mathrm{i}}^{\mathrm{Thou}}{\left[{\alpha }^{\prime }+{\beta }^{\prime }p\left(t_m;{{\tau }^{\prime }}_c,s^{\prime }\right)-g\left(t_k\right)\right]}^2}\right\}$

where g(tk) is the IF gauge at sensor time t = t_k and 1000 is the number of IF estimates. The four-dimensional minimization in (14.3.9) tin exist reduced to a two-dimensional maximization [1]:

(14.3.10) $\left\{{\widehat{\tau }}_c,\widehat{\mathrm{southward}}\right\}=\arg \left\{\underset{\left\{{\tau }_c\text{'},s\text{'}\right\}}{\max }\frac{{\left|{\Sigma }_{k=1}^K\left[g\left(t_{\mathrm{1000}}\right)-\overline{g}\right]p\left(t_{\mathrm{yard}}\right)\right|}^2}{{\Sigma }_{\mathrm{one\; thousand}=1}^{\mathrm{Thousand}}{\left[p\left(t_{\mathrm{thousand}}\right)-\overline{p}\right]}^2}\right\}$

(14.3.11) $\widehat{\beta }=\frac{{\Sigma }_{k=1}^{\mathrm{One\; thousand}}\left[g\left(t_k\right)-\overline{g}\right]\widehat{p}\left(t_k\right)}{{\Sigma }_{k=\mathrm{ane}}^G{\left[\widehat{p}\left(t_{\mathrm{yard}}\right)-\overline{\widehat{p}}\right]}^2}$

(14.3.12) $\widehat{\alpha }=\overline{g}-\widehat{\beta }\overline{\widehat{p}}$

where $\overline{g}=\frac{\mathrm{one}}{\mathrm{Grand}}{\Sigma }_m\mathrm{chiliad}\left(t_k\right),p\left(t_k\right)=p\left(t_k;{{\tau }^{\prime }}_c,s^{\prime }\right),\overline{p}=\frac{1}{\mathrm{Yard}}{\Sigma }_{k}p\left(t_k\right),\widehat{p}\left(t_k\right)=p\left(t_k;{\widehat{\tau }}_c,\widehat{s}\right)$ , and $\overline{\widehat{p}}=\frac{1}{\mathrm{Chiliad}}{\Sigma }_k\widehat{p}\left(t_k\right)$ . Solving (14.three.2) and (14.iii.iii) using the estimated values for α and β gives the estimates of the source speed v and source frequency f ₀ every bit

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

(14.iii.14) ${\widehat{f}}_0=\widehat{\alpha }\left(\mathrm{i}-{\widehat{\upsilon }}^2/c_a^{\mathrm{ii}}\right).$

From (fourteen.three.4), the estimate of the CPA slant range R_c is given by

(14.3.xv) ${\widehat{R}}_c=\widehat{s}\widehat{\upsilon }{c}_a/\sqrt{c_a^{\mathrm{ii}}-{\widehat{\upsilon }}^2.}$

The maximization in (14.3.10) is performed using the quasi-Newton method where the initial estimates of τ _c and s are given by the method described in [1]. The results of applying the source parameter estimation method to experimental data (represented by the circles) are shown at the top of Figs. 14.three.i(a) and 14.3.ane(b). The estimates closely match the actual values of the aircraft'southward speed, distance, and propeller or master rotor blade rate.

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Time-Frequency Methods in Radar, Sonar, and Acoustics

In Time-Frequency Signal Analysis and Processing (2nd 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 past minimizing the sum of the squared deviations of the noisy IF estimates from their predicted values [ 16]. Specifically, the NLS estimates of {α,β,τ _c ,s} are given past

(xiv.3.9) $\begin{array}{l}\hfill \{\widehat{\alpha },\widehat{\beta },\widehat{{\tau }_c},ŝ\}=arg\left\{\underset{\{{\alpha }^{\prime }{\beta }^{\prime },{\tau }_c^{\prime },{\mathrm{south}}^{\prime }\}}{min}\sum _{\mathrm{one\; thousand}=\mathrm{ane}}^K{\left[{\alpha }^{\prime }+{\beta }^{\prime }p(t_{\mathrm{grand}};{\tau }_c^{\prime },{\mathrm{southward}}^{\prime })-\mathrm{grand}(t_{\mathrm{thousand}})\right]}^{\mathrm{ii}}\right\},\end{array}$

where chiliad(t _k ) is the IF estimate at sensor time t = t _k and G is the number of IF estimates. The four-dimensional minimization in Eq. (fourteen.3.9) tin be reduced to a 2-dimensional maximization [16]:

(14.3.x) $\begin{array}{ll}\hfill \{{\widehat{\tau }}_c,ŝ\}=& arg\left\{\underset{\{{\tau }_c^{\prime },s^{\prime }\}}{max}\frac{{\left|\sum _{k=\mathrm{i}}^K[g(t_{\mathrm{grand}})-\stackrel{-}{g}]p(t_{\mathrm{one\; thousand}})\right|}^2}{\sum _{k=1}^{\mathrm{Thou}}{[p(t_{\mathrm{chiliad}})-\stackrel{-}{p}]}^2}\right\},\hfill \end{array}$

(14.three.11) $\begin{array}{ll}\hfill \widehat{\beta }=& \frac{\sum _{\mathrm{thousand}=1}^M[\mathrm{chiliad}(t_{\mathrm{grand}})-\stackrel{-}{g}]\widehat{p}(t_{\mathrm{thousand}})}{\sum _{k=1}^K{[\widehat{p}(t_k)-\stackrel{-}{\widehat{p}}]}^2},\hfill \end{array}$

(14.three.12) $\begin{array}{ll}\hfill \widehat{\alpha }=& \stackrel{-}{g}-\widehat{\beta }\stackrel{-}{\widehat{p}},\hfill \end{array}$

where $\stackrel{-}{g}=\frac{1}{K}{\sum }_{k}g(t_{\mathrm{grand}})$ , $p(t_k)=p(t_{\mathrm{yard}};{\tau }_c^{\prime },{\mathrm{southward}}^{\prime })$ , $\stackrel{-}{p}=\frac{1}{\mathrm{Chiliad}}{\sum }_{g}p(t_m)$ , $\widehat{p}(t_k)=p(t_{\mathrm{yard}};{\widehat{\tau }}_c,ŝ)$ , and $\stackrel{-}{\widehat{p}}=\frac{\mathrm{i}}{K}{\sum }_{\mathrm{thou}}\widehat{p}(t_k)$ . Solving Eqs. (14.3.2) and (fourteen.iii.3) using the estimated values for α and β gives the estimates of the source speed v and source frequency f ₀ equally

(14.3.13) $\begin{array}{ll}\hfill \widehat{\mathrm{five}}=& -(\widehat{\beta }/\widehat{\alpha })c_{\mathrm{a}},\hfill \end{array}$

(14.3.14) $\begin{array}{ll}\hfill {\widehat{f}}_0=& \widehat{\alpha }(\mathrm{i}-{\widehat{v}}^{\mathrm{two}}/c_{\mathrm{a}}^{\mathrm{two}}).\hfill \end{array}$

From Eq. (xiv.3.4), the approximate of the CPA camber range R _c is given by

(xiv.3.fifteen) $\begin{array}{l}\hfill {\widehat{R}}_c=ŝ\widehat{v}{c}_{\mathrm{a}}/\sqrt{c_{\mathrm{a}}^{\mathrm{ii}}-{\widehat{\mathrm{five}}}^{\mathrm{ii}}}.\end{array}$

The maximization in Eq. (fourteen.three.10) is performed using the quasi-Newton method where the initial estimates of τ _c and s are given past the method described in Ref. [sixteen]. The results of applying the source parameter estimation method to experimental data (represented past the circles) are shown at the meridian of Fig. 14.3.1 (a) and (b). The estimates closely match the actual values of the aircraft's speed, altitude, and propeller or master rotor blade charge per unit.

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Sum Of The Squared Deviation,

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