# methods of discriminant analysis

Here the basic assumption is that all the variables are independent given the class label. Basically, if you are given an x above the line, then we would classify this x into the first-class. However, both are quite different in the approaches they use to reduce… How do we estimate the covariance matrices separately? $$\ast \text{Decision boundary: } 5.56-2.00x_1+3.56x_2=0.0$$. $$\hat{\mu}_2$$ = 0.8224, You can see that in the upper right the red and blue are very well mixed, however, in the lower left the mix is not as great. From: Olives and Olive Oil in Health and Disease Prevention, 2010, A.M. Pustjens, ... S.M. Then multiply its transpose. Discriminant analysis is a valuable tool in statistics. It has a fairly steep learning curve, but is extremely powerful. One final method for cross-validation is the leave-one-out method. Alkarkhi, Wasin A.A. Alqaraghuli, in, Encyclopedia of Forensic Sciences (Second Edition), Chemometrics for Food Authenticity Applications. Therefore, LDA is well suited for nontargeted metabolic profiling data, which is usually grouped. $Pr(G=1|X=x) =\frac{e^{- 0.3288-1.3275x}}{1+e^{- 0.3288-1.3275x}}$. In practice, logistic regression and LDA often give similar results. Rayens, in Comprehensive Chemometrics, 2009. Lavine, W.S. \end {align} \). The dashed or dotted line is the boundary obtained by linear regression of an indicator matrix. Interpretation. & = \text{arg } \underset{k}{\text{max}}\left[-\text{log}((2\pi)^{p/2}|\Sigma|^{1/2})-\frac{1}{2}(x-\mu_k)^T\Sigma^{-1}(x-\mu_k)+\text{log}(\pi_k)  \right] \\ First of all the within the class of density is not a single Gaussian distribution, instead, it is a mixture of two Gaussian distributions. Let's look at what the optimal classification would be based on the Bayes rule. And we will talk about how to estimate this in a moment. Here are some examples that might illustrate this. Below, in the plots, the black line represents the decision boundary. -0.3334 & 1.7910 Discriminant analysis is a way to build classifiers: that is, the algorithm uses labelled training data to build a predictive model of group membership which can then be applied to new cases. When the classification model is applied to a new data set, the error rate would likely be much higher than predicted. For example, this method could be used to separate four types of flour prepared from green and ripe Cavendish bananas based on physicochemical properties (green peel (Gpe), ripe peel (Rpe), green pulp (Gpu), and ripe pulp (Rpu)). Remember this is the density of X conditioned on the class k, or class G = k denoted by$$f _ { k } ( x )$$. Descriptive analysis is an insight into the past. DA defines the distance of a sample from the center of a class, and creates a new set of axes to place members of the same group as close together as possible, and move the groups as far apart from one another as possible. 2.16A. 1. Then, if we apply LDA we get this decision boundary (above, black line), which is actually very close to the ideal boundary between the two classes. The vector x and the mean vector $$\mu_k$$ are both column vectors. & = \begin{cases} The Wide Linear method is an efficient way to fit a Linear model when the number of covariates is large. Discriminant analysis is a very popular tool used in statistics and helps companies improve decision making, processes, and solutions across diverse business lines. LDA is another dimensionality reduction technique. Because we have equal weights and because the covariance matrix two classes are identical, we get these symmetric lines in the contour plot. Within training data classification error rate: 29.04%. Within training data classification error rate: 28.26%. Remember, K is the number of classes. Are some groups different than the others? In this case, we are doing matrix multiplication. Next, we normalize by the scalar quantity, N - K. When we fit a maximum likelihood estimator it should be divided by N, but if it is divided by N – K, we get an unbiased estimator. Discriminant analysis (DA) is a multivariate technique used to separate two or more groups of observations (individuals) based on k variables measured on each experimental unit (sample) and find the contribution of each variable in separating the groups. According to the Bayes rule, what we need is to compute the posterior probability: $$Pr(G=k|X=x)=\frac{f_k(x)\pi_k}{\sum^{K}_{l=1}f_l(x)\pi_l}$$. \end{pmatrix}  \]. Actually, for linear discriminant analysis to be optimal, the data as a whole should not be normally distributed but within each class the data should be normally distributed. Based on the true distribution, the Bayes (optimal) boundary value between the two classes is -0.7750 and the error rate is 0.1765. The first type has a prior probability estimated at 0.651. This discriminant function is a quadratic function and will contain second order terms. Discriminant analysis is used to predict the probability of belonging to a given class (or category) based on one or multiple predictor variables. & =  \text{log }\frac{\pi_k}{\pi_K}-\frac{1}{2}(\mu_k+\mu_K)^T\Sigma^{-1}(\mu_k-\mu_K) \\ However, other classification approaches exist and are listed in the next section. In the DA, objects are separated into classes, minimizing the variance within the class and maximizing the variance between classes, and finding the linear combination of the original variables (directions). Descriptive Analysis. DA is often applied to the same sample types as is PCA, where the latter technique can be used to reduce the number of variables in the data set and the resultant PCs are then used in DA to define and predict classes. There are many different times during a particular study when the researcher comes face to face with a lot of questions which need answers at best. In the above example,  the blue class breaks into two pieces, left and right. Furthermore, this model will enable one to assess the contributions of different variables. Also, acquiring enough data to have appropriately sized training and test sets may be time-consuming or difficult due to resources. Largely you will find out that LDA is not appropriate and you want to take another approach. The boundary may be linear or nonlinear; in this example both a linear and a quadratic line are fitted. J.S. No assumption is made about $$Pr(X)$$; while the LDA model specifies the joint distribution of X and G. $$Pr(X)$$ is a mixture of Gaussians: $Pr(X)=\sum_{k=1}^{K}\pi_k \phi (X; \mu_k, \Sigma)$. Therefore, for maximization, it does not make a difference in the choice of k. The MAP rule is essentially trying to maximize $$\pi_k$$times $$f_k(x)$$. If we were looking at class k, for every point we subtract the corresponding mean which we computed earlier. If the result is greater than or equal to zero, then claim that it is in class 0, otherwise claim that it is in class 1. Remember x is a column vector, therefore if we have a column vector multiplied by a row vector, we get a square matrix, which is what we need. The null hypothesis, which is statistical lingo for what would happen if the treatment does nothing, is that there is no relationship between consumer age/income and website format preference. LDA assumes that the various classes collecting similar objects (from a given area) are described by multivariate normal distributions having the same covariance but different location of centroids within the variable domain (Leardi, 2003). As we talked about at the beginning of this course, there are trade-offs between fitting the training data well and having a simple model to work with. 1.7949 & -0.1463\\ Figure 4 shows the results of such a treatment on the same set of data shown in Figure 3. DA works by finding one or more linear combinations of the k selected variables. Within-center retrospective discriminant analysis methods to differentiate subjects with early ALS from controls have resulted in an overall classification accuracy of 90%–95% (2,4,10). The class membership of every sample is then predicted by the model, and the cross-validation determines how often the rule correctly classified the samples. The decision boundaries are quadratic equations in x. QDA, because it allows for more flexibility for the covariance matrix, tends to fit the data better than LDA, but then it has more parameters to estimate. The error rate on the test data set is 0.2205. The means and variance of the two classes estimated by LDA are: $$\hat{\mu}_1$$ = -1.1948, LDA makes some strong assumptions. If it is below the line, we would classify it into the second class. Linear Discriminant Analysis (LDA) is, like Principle Component Analysis (PCA), a method of dimensionality reduction. The Bayes rule is applied. It is a fairly small data set by today's standards. Here is the formula for estimating the $$\pi_k$$'s and the parameters in the Gaussian distributions. Discriminant analysis attempts to identify a boundary between groups in the data, which can then be used to classify new observations. The classification rule is similar as well. Consequently, the ellipses of different categories differ not only for their position in the plane but also for eccentricity and axis orientation (Geisser, 1964). If more than two or two observation groups are given having measurements on various interval variables, a linear combin… For this reason, SWLDA is widely used as classification method for P300 BCI. This makes the computation much simpler. However, discriminant analysis is surprising robust to violation of these assumptions, and is usually a good first choice for classifier development. This means that for this data set about 65% of these belong to class 0 and the other 35% belong to class 1. Krusienski et al. The contour plot for the density for class 1 would be similar except centered above and to the right. Resubstitution has a major drawback, however. It is always a good practice to plot things so that if something went terribly wrong it would show up in the plots. \[ \begin{align*}\hat{G}(x) Notice that the denominator is identical no matter what class k you are using. 1 & otherwise Each within-class density of X is a mixture of two normals: The class-conditional densities are shown below. Abbas F.M. Alkarkhi, Wasin A.A. Alqaraghuli, in Easy Statistics for Food Science with R, 2019. The loading from LDA shows the significance of metabolite in differentiating the groups. The resulting models are evaluated by their predictive ability to predict new and unknown samples (Varmuza and Filzmoser, 2009). Be bad when the classification of the determinant of this, we do have similar data sets which follow the. In Easy statistics for Food authenticity Applications we would classify this X the! In the next step ( forward SWLDA ) discriminant functions and their is... 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