Finally, the K-OPLS, NPLS-DA, and unfolded PLS-DA models were developed based on the synchronous 2D correlation spectra of adulterated milk and pure milk.
About JMPJMP produces interactive statistical discovery software. We want \(m\) (the number of factors) to be much smaller than \(p\) (the number of measured attributes)Restrictions on the model\(E(\epsilon) = \mathbf{0}\)\(var(\epsilon) = \Psi_{p \times p} = diag( \psi_1, \dots, \psi_p)\)\(\mathbf{\epsilon}, \mathbf{f}\) are independentAdditional assumption could be \(E(\mathbf{f}) = \mathbf{0}, var(\mathbf{f}) = \mathbf{I}_{m \times m}\) (known as the orthogonal factor model) , which imposes the following covariance structure on \(\mathbf{y}\)\[
\begin{aligned}
var(\mathbf{y}) = \mathbf{\Sigma} = var(\mathbf{Lf} + \mathbf{\epsilon}) \\
= var(\mathbf{Lf}) + var(\epsilon) \\
= \mathbf{L} var(\mathbf{f}) \mathbf{L}’ + \mathbf{\Psi} \\
= \mathbf{LIL}’ + \mathbf{\Psi} \\
= \mathbf{LL}’ + \mathbf{\Psi}
\end{aligned}
\]Since \(\mathbf{\Psi}\) is diagonal, the off-diagonal elements of \(\mathbf{LL}’\) are \(\sigma_{ij}\), the co variances in \(\mathbf{\Sigma}\), which means \(cov(y_i, y_j) = \sum_{k=1}^m l_{ik}l_{jk}\) and the covariance of \(\mathbf{y}\) is completely determined by the m factors ( \(m p\))\(var(y_i) = \sum_{k=1}^m l_{ik}^2 + \psi_i\) where \(\psi_i\) is the specific variance and the summation term is the i-th communality (i. 83934119 -0. It covers a wide range of topics in classical multivariate analysis and presents some deep theoretical results. e.

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106 (496), December, 2011)“This book provides an overview of the theory of multivariate nonparametric methods based on spatial signs and ranks. 04526811 -0. , n_i\) and \(\epsilon_{ij} \sim N_p(\mathbf{0, \Sigma})\)However, the above model is over-parameterized (i. Let \(\mathbf{y} = \mathbf{Cx}\) , where \(\mathbf{C}\) is any diagonal matrix, then \(\mathbf{L}_y = \mathbf{CL}_x\) and \(\mathbf{\Psi}_y = \mathbf{C\Psi}_x\mathbf{C}\)Hence, check these guys out can see that factor analysis is also invariant to changes in scaleTo estimate \(\mathbf{L}\)Spectral decomposition\[
\begin{aligned}
\mathbf{\Sigma} = \lambda_1 \mathbf{a}_1 \mathbf{a}_1′ + \dots + \lambda_p \mathbf{a}_p \mathbf{a}_p’ \\
= \mathbf{A\Lambda A}’ \\
= \sum_{k=1}^m \lambda+k \mathbf{a}_k \mathbf{a}_k’ + \sum_{k= m+1}^p \lambda_k \mathbf{a}_k \mathbf{a}_k’ \\
= \sum_{k=1}^m l_k l_k’ + \sum_{k=m+1}^p \lambda_k \mathbf{a}_k \mathbf{a}_k’
\end{aligned}
\]where \(l_k = \mathbf{a}_k \sqrt{\lambda_k}\) and the second term is not diagonal in general.

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If the data are not normal, we canignore ituse nonparametric methodsuse models based upon an approximate distribution (e. e. . These are the orthogonal projections of the data onto the “new principal component axesVariances of \(y_1\) are greater than those for any other possible projectionCovariance matrix decomposition and projection onto orthogonal axes = PCA\(p \times 1\) vectors \(\mathbf{x}_1, \dots , \mathbf{x}_n\) which are iid with \(var(\mathbf{x}_i) = \mathbf{\Sigma}\)The first PC is the linear combination \(y_1 = \mathbf{a}_1′ \mathbf{x} = a_{11}x_1 + \dots + a_{1p}x_p\) with \(\mathbf{a}_1′ \mathbf{a}_1 = 1\) such that \(var(y_1)\) is the maximum of all linear combinations of \(\mathbf{x}\) which have unit lengthThe second PC is the linear combination \(y_1 = \mathbf{a}_2′ \mathbf{x} = a_{21}x_1 + \dots + a_{2p}x_p\) with \(\mathbf{a}_2′ \mathbf{a}_2 = 1\) such that \(var(y_1)\) is the maximum of all linear combinations of \(\mathbf{x}\) which have unit length and uncorrelated with \(y_1\) (i.

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And reject the null if \(|T|\) is large relative to \(t_{(1-\alpha/2,n-1)}\) because it means that seeing a value as large as what we observed is rare if the null is trueEquivalently,\[
T^2 = \frac{(\bar{y}- \mu_0)^2}{s^2/n} = n(\bar{y}- \mu_0)(s^2)^{-1}(\bar{y}- \mu_0) \sim f_{(1,n-1)}
\]\[
H_0: \mathbf{\mu} = \mathbf{\mu}_0 \\
H_a: \mathbf{\mu} \neq \mathbf{\mu}_0
\]Define Hotelling’s \(T^2\) by\[
T^2 = n(\bar{\mathbf{y}} – \mathbf{\mu}_0)’\mathbf{S}^{-1}(\bar{\mathbf{y}} – \mathbf{\mu}_0)
\]which can be viewed as a generalized distance between \(\bar{\mathbf{y}}\) and \(\mathbf{\mu}_0\)Under the assumption of normality,\[
F = \frac{n-p}{(n-1)p} T^2 \sim f_{(p,n-p)}
\]and reject the null hypothesis when \(F f_{(1-\alpha, p, n-p)}\)The \(T^2\) test is invariant to changes in measurement units. A weaker hypothesis is that the profiles for the two groups are parallel. discover this \mathbf{\Sigma}_1 = \mathbf{\Sigma}_2 = \ldots = \mathbf{\Sigma}_k = \mathbf{\Sigma} \\
H_a: \text{at least 2 are different}
\]Assume \(H_0\) is true, we would use a pooled estimate of the common covariance matrix, \(\mathbf{\Sigma}\)\[
\mathbf{S} = \frac{\sum_{i=1}^k (n_i -1)\mathbf{S}_i}{\sum_{i=1}^k (n_i – 1)}
\]with \(\sum_{i=1}^k (n_i -1)\)(a modification of the likelihood ratio test). , \mu_6\)\[
H_0: \mu_1 = \mu_2 = . Then, the maximum likelihood (linear) discriminant rule allocates \(\mathbf{y}\) to population j where j minimizes the squared Mahalanobis distance\[
(\mathbf{y} – \mathbf{\mu}_j)’ \mathbf{\Sigma}^{-1} (\mathbf{y} – \mathbf{\mu}_j)
\]If we know that population j has prior probabilities \(\pi_j\) (assume \(\pi_j 0\)) we can form the Bayes discriminant rule. .

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