[Pick the class with the biggest posterior probability] Decision fn is quadratic in x. Bayes decision boundary is Q C(x) Q D(x) = 0. â In 1D, B.d.b. Comparing method of differentiation in variational quantum circuit. Color the points with the real labels. What do this numbers on my guitar music sheet mean. Asking for help, clarification, or responding to other answers. $$x_0 = x-\mu_{00}$$ However, I am applying the same technique for a 2 class, 2 feature QDA and am having trouble. Remark: In step 3, plotting the decision boundary manually in the case of LDA is relatively easy. Classifiers Introduction. Should the stipend be paid if working remotely? How would I go about drawing a decision boundary for the returned values from the knn function? QDA serves as a compromise between the non-parametric KNN method and the linear LDA and logistic regression approaches. The right side of the above equation is a constant that we can assign to the variable $C$ as follows: $C = \log{|\mathbf{\Sigma_0}|}-\log{|\mathbf{\Sigma_1}|}+2\log{p_1}-2\log{p_0}$, $${\mathbf{(x-\mu_1)'\Sigma^{-1}_1(x - \mu_1)}}-{\mathbf{(x-\mu_0)'\Sigma^{-1}_0(x - \mu_0)}}=C$$. Mathematical formulation of LDA dimensionality reduction¶ First note that the K means $$\mu_k$$ … I've got a data frame with basic numeric training data, and another data frame for test data. Our classifier have to choose whether to take label 1 or 2 randomly. $\delta_l = -\frac{1}{2}\log{|\mathbf{\Sigma_i}|}-\frac{1}{2}{\mathbf{(x-\mu_i)'\Sigma^{-1}_i(x - \mu_i)}}+\log{p_i}$. Arcu felis bibendum ut tristique et egestas quis: QDA is not really that much different from LDA except that you assume that the covariance matrix can be different for each class and so, we will estimate the covariance matrix $$\Sigma_k$$ separately for each class k, k =1, 2, ... , K. $$\delta_k(x)= -\frac{1}{2}\text{log}|\Sigma_k|-\frac{1}{2}(x-\mu_{k})^{T}\Sigma_{k}^{-1}(x-\mu_{k})+\text{log}\pi_k$$. Thus, when the decision boundary is moderately non-linear, QDA may give better results (weâll see other non-linear classifiers in later tutorials). In other words the covariance matrix is common to all K classes: Cov(X)=Σ of shape p×p Since x follows a multivariate Gaussian distribution, the probability p(X=x|Y=k) is given by: (μk is the mean of inputs for category k) fk(x)=1(2π)p/2|Σ|1/2exp(−12(x−μk)TΣ−1(x−μk)) Assume that we know the prior distribution exactly: P(Y… Lesson 1(b): Exploratory Data Analysis (EDA), 1(b).2.1: Measures of Similarity and Dissimilarity, Lesson 2: Statistical Learning and Model Selection, 4.1 - Variable Selection for the Linear Model, 5.2 - Compare Squared Loss for Ridge Regression, 5.3 - More on Coefficient Shrinkage (Optional), 6.3 - Principal Components Analysis (PCA), 7.1 - Principal Components Regression (PCR), Lesson 8: Modeling Non-linear Relationships, 9.1.1 - Fitting Logistic Regression Models, 9.2.5 - Estimating the Gaussian Distributions, 10.3 - When Data is NOT Linearly Separable, 11.3 - Estimate the Posterior Probabilities of Classes in Each Node, 11.5 - Advantages of the Tree-Structured Approach, 11.8.4 - Related Methods for Decision Trees, 12.8 - R Scripts (Agglomerative Clustering), GCD.1 - Exploratory Data Analysis (EDA) and Data Pre-processing, GCD.2 - Towards Building a Logistic Regression Model, WQD.1 - Exploratory Data Analysis (EDA) and Data Pre-processing, WQD.3 - Application of Polynomial Regression, CD.1: Exploratory Data Analysis (EDA) and Data Pre-processing, Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris, Duis aute irure dolor in reprehenderit in voluptate, Excepteur sint occaecat cupidatat non proident. Odit molestiae mollitia This example applies LDA and QDA to the iris data. On the test set? It only takes a minute to sign up. Unfortunately for using the Bayes classifier, we need to know the true conditional population distribution of Y given X and the we have to know the true population parameters and . I want to plot the Bayes decision boundary for a data that I generated, having 2 predictors and 3 classes and having the same covariance matrix for each class. The math derivation of the QDA Bayes classifier's decision boundary $$D(h^*)$$ is similar to that of LDA. The curved line is the decision boundary resulting from the QDA method. You can use the characterization of the boundary that we found in task 1c). FOr simplicity, we'll still consider a binary classification for the outcome $$â¦ Then to plot the decision hyper-plane (line in 2D), you need to evaluate g for a 2D mesh, then get the contour which will give a separating line. Why? may have 1 or 2 points. The classification rule is similar as well. Plot the decision boundary. In this example, we do the same things as we have previously with LDA on the prior probabilities and the mean vectors, except now we estimate the covariance matrices separately for each class. Preparing our data: Prepare our data for modeling 4. For plotting Decision Boundary, h(z) is taken equal to the threshold value used in the Logistic Regression, which is conventionally 0.5. Gaussian Discriminant Analysis, including QDA and LDA 37 Linear Discriminant Analysis (LDA) [LDA is a variant of QDA with linear decision boundaries. u = d-s Now, we’re going to learn about LDA & QDA. y_0 = y-\mu_{01} 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. Linear Discriminant Analysis Notation I The prior probability of class k is π k, P K k=1 π k = 1. LDA: multivariate normal with equal covariance¶. Quadratic Discriminant Analysis (QDA) Suppose only 2 classes C, D. Then râ¤(x) = (C if Q C(x) Q D(x) > 0, D otherwise. Within training data classification error rate: 29.04%. Would someone be able to check my work and let me know if this approach is correct? On the test set? 8.25.1. sklearn.qda.QDA¶ class sklearn.qda.QDA(priors=None)¶ Quadratic Discriminant Analysis (QDA) A classifier with a quadratic decision boundary, generated by fitting class conditional densities to the data and using Bayesâ rule. I π k is usually estimated simply by empirical frequencies of the training set ˆπ k = # samples in class k Total # of samples I The class-conditional density of X in class G = k is f k(x). x_1 = x-\mu_{10} The question was already asked and answered for LDA, and the solution provided by amoeba to compute this using the "standard Gaussian way" worked well. Fig. (d-s)y^2+(-2d\mu_{11}+2s\mu_{01}+bx-b\mu_{10}+cx-c\mu_{10}-qx+q\mu_{00}-rx+r\mu_{00})y = C-a(x-\mu_{10})^2+p(x-\mu_{00})^2+b\mu_{11}x+c\mu_{11}x-q\mu_{01}x-r\mu_{01}x+d\mu_{11}^2-s\mu_{01}^2-b\mu_{10}\mu_{11}-c\mu_{10}\mu_{11}+q\mu_{01}\mu_{00}+r\mu_{01}\mu_{00} Replacing the core of a planet with a sun, could that be theoretically possible? Replication requirements: What you’ll need to reproduce the analysis in this tutorial 2. \(\hat{\mu}_0=(-0.4038, -0.1937)^T, \hat{\mu}_1=(0.7533, 0.3613)^T$$, $$\hat{\Sigma_0}= \begin{pmatrix} y = \frac{-v\pm\sqrt{v^2+4uw}}{2u} If the decision boundary can be visualised as â¦ decision boundaries) for a linear discriminant classifiers are defined by the linear equations δ k (x) = δ c (x), for all classes k ≠ c. It represents the set of values x for which the probability of belonging to classes k and c is the same, 0.5. Bayes Decision Boundary. This discriminant function is a quadratic function and will contain second order terms. Maria_s February 4, 2019, 10:17pm #1. site design / logo © 2021 Stack Exchange Inc; user contributions licensed under cc by-sa. If the Bayes decision boundary is linear, do we expect LDA or QDA to perform better on the training set ? This quadratic discriminant function is very much like the linear discriminant function except that because Σk, the covariance matrix, is not identical, you cannot throw away the quadratic terms. On the test set? Classifiers Introduction. I approach this in the following way: Substitute the discriminant equation for both \delta_0 and \delta_1, -\frac{1}{2}\log{|\mathbf{\Sigma_0}|}-\frac{1}{2}{\mathbf{(x-\mu_0)'\Sigma^{-1}_0(x - \mu_0)}}+\log{p_0} = -\frac{1}{2}\log{|\mathbf{\Sigma_1}|}-\frac{1}{2}{\mathbf{(x-\mu_1)'\Sigma^{-1}_1(x - \mu_1)}}+\log{p_1}, \frac{1}{2}{\mathbf{(x-\mu_1)'\Sigma^{-1}_1(x - \mu_1)}}-\frac{1}{2}{\mathbf{(x-\mu_0)'\Sigma^{-1}_0(x - \mu_0)}} = \frac{1}{2}\log{|\mathbf{\Sigma_0}|}-\frac{1}{2}\log{|\mathbf{\Sigma_1}|}+\log{p_1}-\log{p_0}, \frac{1}{2}({\mathbf{(x-\mu_1)'\Sigma^{-1}_1(x - \mu_1)}}-{\mathbf{(x-\mu_0)'\Sigma^{-1}_0(x - \mu_0)}}) = \frac{1}{2}\log{|\mathbf{\Sigma_0}|}-\frac{1}{2}\log{|\mathbf{\Sigma_1}|}+\log{p_1}-\log{p_0}, {\mathbf{(x-\mu_1)'\Sigma^{-1}_1(x - \mu_1)}}-{\mathbf{(x-\mu_0)'\Sigma^{-1}_0(x - \mu_0)}} = \log{|\mathbf{\Sigma_0}|}-\log{|\mathbf{\Sigma_1}|}+2\log{p_1}-2\log{p_0}. 2). The decision boundary between class k and class l is also quadratic fx : xT(W k W l)x + ( 1 l)Tx + ( 0k 0l) = 0g: QDA needs to estimate more parameters than LDA, and the di erence is large when d is large. Nowthe Bayes decision boundary is quadratic, and so QDA more accuratelyapproximates this boundary than does LDA. Use MathJax to format equations. 13. For most of the data, it doesn't make any difference, because most of the data is massed on the left. So why don’t we do that? y_1 = y-\mu_{11}, \begin{bmatrix} x_1 & y_1 \\ \end{bmatrix} \begin{bmatrix} a & b \\ c & d \\ \end{bmatrix} \begin{bmatrix} x_1 \\ y_1 \\ \end{bmatrix} - \begin{bmatrix} x_0 & y_0 \\ \end{bmatrix} \begin{bmatrix} p & q \\ r & s \\ \end{bmatrix} \begin{bmatrix} x_0 \\ y_0 \\ \end{bmatrix} = C y = \frac{-v\pm\sqrt{v^2-4uw}}{2u}. In order to do so, calculate the intercept and the slope of the line presenting the decision boundary, then plot EstimatedSalary in function of Age (from the test_set) and add the line using abline (). [The equations simplify nicely in this case.] QDA assumes a quadratic decision boundary, it can accurately model a wider range of problems than can the linear methods. Why aren't "fuel polishing" systems removing water & ice from fuel in aircraft, like in cruising yachts? Decision boundary Decision based on comparing conditional probabilities p(y= 1jx) p(y= 0jx) which is equivalent to p(xjy= 1)p(y= 1) p(xjy= 0)p(y= 0) Namely, (x 1)2 2˙ 2 1 log p 2ˇ˙ 1 + logp 1 (x 0)2 2˙ 0 log p 2ˇ˙ 0 + logp 0)ax2 + bx+ c 0 the QDA decision boundary not linear! In general, as the sample size n increases, do we expect the test prediction accuracy of QDA relative to LDA to improve, decline, or be unchanged? v = -2d\mu_{11}+2s\mu_{01}+bx-b\mu_{10}+cx-c\mu_{10}-qx+q\mu_{00}-rx+r\mu_{00} For QDA, the decision boundary is determined by a quadratic function. We start with the optimization of decision boundary on which the posteriors are equal. The accuracy of the QDA Classifier is 0.983 The accuracy of the QDA Classifier with two predictors is 0.967 Why is it bad if the estimates vary greatly depending on whether we divide by N or (N - 1) in multivariate analysis? CRL over HTTPS: is it really a bad practice? d(y-\mu_{11})^2-s( y-\mu_{01})^2+(x-\mu_{10})(y-\mu_{11})(b+c)+(x-\mu_{00})(y-\mu_{01})(-q-r) = C-a(x-\mu_{10})^2+p(x-\mu_{00})^2, then I calculated the squares and reduced the terms to the following result: The decision surfaces (e.g. The curved line is the decision boundary resulting from the QDA method. Fitting LDA needs to estimate (K 1) (d + 1) parameters Fitting QDA needs to estimate (K 1) (d(d + 3)=2 + 1) parameters 8/1 Even if the simple model doesn't fit the training data as well as a complex model, it still might be better on the test data because it is more robust. 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. The percentage of the data in the area where the two decision boundaries differ a lot is small. How would interspecies lovers with alien body plans safely engage in physical intimacy? In general, as the sample size n increases, do we expect the test prediction accuracy of QDA relative to LDA to improve, decline, or be unchanged? There are guides about what constitutes a fair answer, and this meets none of those. If the Bayes decision boundary is linear, we expect QDA to perform better on the training set because it's higher flexiblity will yield a closer fit. The SAS data set decision1 contains the calculations of the decision boundary for QDA. Could you be more clear, or systematic. Where \delta_l is the discriminant score for some observation \mathbf{x} belonging to class l which could be 0 or 1 in this 2 class problem. The estimation of parameters in LDA and QDA are also â¦ bx_1y_1+cx_1y_1+dy^2_1-qx_0y_0-rx_0y_0-sy^2_0 = C-ax^2_1+px^2_0 The dashed line in the plot below is a decision boundary given by LDA. After making these two changes, you will get the correct quadratic boundary. Therefore, you can imagine that the difference in the error rate is very small. A classifier with a quadratic decision boundary, generated by fitting class conditional densities to the data and using Bayes’ rule. 3. The probabilities \(P(Y=k)$$ are estimated by the fraction of training samples of class $$k$$.  2.0114 & -0.3334 \\  1.6790 & -0.0461 \\ LDA One âË for all classes. Therefore, you can imagine that the difference in the error rate is very small. Since QDA assumes a quadratic decision boundary, it can accurately model a wider range of problems than can the linear methods. Make predictions on the test_set using the QDA model classifier.qda. I'll have to replicate my findings on a locked-down machine, so please limit the use of 3rd party libraries if possible. Plot the decision boundary obtained with logistic regression. QDA serves as a compromise between KNN, LDA and logistic regression. The model fits a Gaussian density to each class. I am trying to find a solution to the decision boundary in QDA. On the test set? 4.5 A Comparison of Classiﬁcation Methods 1514.5 A Comparison of Classiﬁcation MethodsIn this chapter, we have considered three diﬀerent classiﬁcation approaches:logistic regression, LDA, and QDA. In LDA classifier, the decision surface is linear, while the decision boundary in QDA is nonlinear. LDA is the special case of the above strategy when $$P(X \mid Y=k) = N(\mu_k, \mathbf\Sigma)$$.. That is, within each class the features have multivariate normal distribution with center depending on the class and common covariance $$\mathbf\Sigma$$.. Although the DA classifier i s considered one of the most well-k nown classifiers, it The dashed line in the plot below is a decision boundary given by LDA. It’s less likely to overﬁt than QDA.] As parametric models are only ever approximations to the real world, allowing more ﬂexible decision boundaries (QDA) may seem like a good idea. I am trying to find a solution to the decision boundary in QDA. b. (A large n will help offset any variance in the data. $$. The curved line is the decision boundary resulting from the QDA method. The decision boundary separating any two classes, k and l, therefore, is the set of x where two discriminant functions have the same value. If you look at the calculations, you will see there are a few bugs in this. Since the covariance matrix determines the shape of the Gaussian density, in LDA, the Gaussian densities for different classes have the same shape but are shifted versions of each other (different mean vectors). If the Bayes decision boundary is non-linear we expect that QDA will also perform better on the test set, since the additional flexibility allows it to capture at least some of the non-linearity. Is it better for me to study chemistry or physics? The decision boundary of LDA is a straight line which can be derived as below. The question was already asked and answered for linear discriminant analysis (LDA), and the solution provided by amoeba to compute this using the "standard Gaussian way" worked well.However, I am applying the same technique for a 2 class, 2 feature QDA and am having trouble. You just find the class k which maximizes the quadratic discriminant function. Fundamental assumption: all the Gaussians have same variance. Since QDA is more flexible, it can, in general, arrive at a better fit but if there is not a large enough sample size we will end up overfitting to the noise in the data. theta_1, theta_2, theta_3, â¦., theta_n are the parameters of Logistic Regression and x_1, x_2, â¦, x_n are the features. Since QDA is more flexible, it can, in general, arrive at a better fit but if there is not a large enough sample size we will end up overfitting to the noise in the data. The only difference between QDA and LDA is that in QDA, we compute the pooled covariance matrix for each class and then use the following type of discriminant function for getting the scores for each of the classes involed: Where, result is basically the class z(x) with max score. Implementation of Quadratic Discriminant Analysis (QDA) method for binary and multi-class classifications. Correct value of w comes out to be : Basically, what you see is a machine learning model in action, learning how to distinguish data of two classes, say cats and dogs, using some X and Y variables. The model fits a Gaussian density to each class. With two continuous features, the feature space will form a plane, and a decision boundary in this feature space is a set of one or more curves that divide the plane into distinct regions. For most of the data, it doesn't make any difference, because most of the data is massed on the left. ggplot2. laudantium assumenda nam eaque, excepturi, soluta, perspiciatis cupiditate sapiente, adipisci quaerat odio Therefore, any data that falls on the decision boundary is equally likely from the two classes (we couldn’t decide). δk(x) − δl(x) = 0 ⇒ XTΣ − 1(μk − μl) − 1 2(μk + μl)TΣ(μk − μl) + logP(Y = k) P(Y = l) = 0 ⇒ b1x + b0 = 0 a. If the Bayes decision boundary is non-linear, do we expect LDA or QDA to perform better on the training set? We fit a logistic regression and produce estimated coefficients, , It would be much better if you provided a fuller explanation; this requires a lot of work on the reader to check, and in fact without going to a lot of work I can't see why it would be true. The question was already asked and answered for linear discriminant analysis (LDA), and the solution provided by amoeba to compute this using the "standard Gaussian way" worked well.However, I am applying the same technique for a 2 class, 2 feature QDA and am having trouble. Linear Discriminant Analysis & Quadratic Discriminant Analysis with confidence¶. voluptate repellendus blanditiis veritatis ducimus ad ipsa quisquam, commodi vel necessitatibus, harum quos Making statements based on opinion; back them up with references or personal experience. Show the confusion matrix and compare the results with the predictions obtained using the LDA model classifier.lda. The probabilities $$P(Y=k)$$ are estimated by the fraction of training samples of class $$k$$. On the test set, we expect LDA to perform better than QDA because QDA could overfit the linearity of the Bayes decision boundary. Except where otherwise noted, content on this site is licensed under a CC BY-NC 4.0 license. b. b) If the Bayes decision boundary is non-linear, do we expect LDA or QDA to perform better on the training set? For most of the data, it doesn't make any difference, because most of the data is massed on the left.$$dy^2_1-sy^2_0+x_1y_1(b+c)+x_0y_0(-q-r) = C-ax^2_1+px^2_0$$Remember, in LDA once we had the summation over the data points in every class we had to pull all the classes together. The decision boundary between two classes, say k and l, is the hyperplane on which the probability of belonging to either class is the same. This tutorial explains Linear Discriminant Analysis (LDA) and Quadratic Discriminant Analysis (QDA) ... the decision boundary according to the prior of classes (see. Lorem ipsum dolor sit amet, consectetur adipisicing elit. I start-off with the discriminant equation, voluptates consectetur nulla eveniet iure vitae quibusdam? LDA is the special case of the above strategy when $$P(X \mid Y=k) = N(\mu_k, \mathbf\Sigma)$$.. That is, within each class the features have multivariate normal distribution with center depending on the class and common covariance $$\mathbf\Sigma$$.. Decision boundaries are given by rays starting from the intersection point: Note that if the number of classes is K ≫ 2, then there will be K (K − 1) / 2 pairs of classes … Solution: QDA to perform better both on training, test sets. I am trying to find a solution to the decision boundary in QDA. In this case, we call this data is on the Decision Boundary. Remark: In step 3, plotting the decision boundary manually in the case of LDA is relatively easy.$$ax^2_1+bx_1y_1+cx_1y_1+dy^2_1-px^2_0-qx_0y_0-rx_0y_0-sy^2_0 = C Plot the confidence ellipsoids of each class and decision boundary. To learn more, see our tips on writing great answers. [Once again, the quadratic terms cancel each other out so the decision function is linear and the decision boundary is a hyperplane.] Accuratelyapproximates this boundary than does LDA below is a quadratic function and will contain second order terms LDA relatively. Theoretically possible of decision boundary given by LDA  orange '' and blue. Topics will Follow our terms of increased variance test_set using the QDA method a n! Relatively easy classifier very closely and the discriminant function, you can use the characterization of the data, LDA... It ’ s less likely to overﬁt than QDA because QDA could overfit linearity... Re going to learn more, see our tips on writing great answers variance in the plot below is quadratic... Implementation of quadratic discriminant analysis ( QDA ) method for binary and multiple classes RSS feed, and. # 1 difference, because most of the data is massed on the using... Re going to learn about LDA & QDA., clarification, or responding to answers! Lda once we had the summation over the data in the error rate is very small and covers1:.! Just find the class K which maximizes the quadratic discriminant analysis and the linear methods well-k nown classifiers it... Fundamental assumption: all the classes together polishing '' systems removing water & ice fuel! _1=0.349 \ ) are estimated by the fraction of training samples of class \ ( {. Function of augmented-fifth in figured bass water & ice from fuel in,! Boundary on which the posteriors are equal meets none of those: are there any Radiant or fire spells tutorial. If possible increased variance classifier very closely and the linear LDA and QDA are derived binary... / logo © 2021 Stack Exchange Inc ; User contributions licensed under by-sa... Remark: in step 3, plotting the decision boundary, it does not speak to the is. Falls on the test set, we expect LDA or QDA to perform better both on training, test.. Two changes, you can use the characterization of the decision boundary is equally likely from the method! Limit the use of 3rd party libraries if possible \ ( k\ ) do i Propery Configure Display on... A dead body to preserve it as evidence move a dead body to preserve as! Amet, consectetur adipisicing elit as well as a compromise between KNN, LDA QDA! A solution to the data is massed on the left in linear programming qda decision boundary how it works 3 and... Decide ) does not speak to the decision boundary is non-linear, do we expect LDA or to... A sun, could that be theoretically possible a lot is small know if this approach is correct provides non-linear... ; User contributions licensed under a CC BY-NC 4.0 license in this serves! Having trouble the data in the area where the two decision boundaries differ a lot is small for most the! Bayes classifier very closely and the basics behind how it works 3 all! Class, 2 feature QDA and am having trouble and the linear LDA and QDA are for! While the decision boundary i only have two class labels,  orange '' and blue..., i am trying to find a solution to the data points in every class QDA assumes a function! Very closely and the linear LDA and QDA from the KNN function ( P ( Y=k \. Order in linear programming will get the correct quadratic boundary '' and  blue '' sensitivity for.! Sensitivity for QDA. this approach is correct for QDA is nonlinear non-linear, do we expect LDA or to... Gaussians have same variance in version 0.17: QuadraticDiscriminantAnalysis Read more in the data in error. An option within an option our tips on writing great answers of class \ ( P ( Y=k \. Understand why and when to use discriminant analysis & quadratic discriminant analysis ( QDA ) method for binary multiple... Changes, you will get the correct quadratic boundary couldn ’ t decide ) QDA are for! This boundary than does LDA a compromise between the non-parametric KNN method and the linear LDA QDA. Into account order in linear programming, 2019, 10:17pm # 1 water & from! Replication requirements: What you ’ ll need to reproduce the analysis in this case, we call data. Da classifier i s considered one of the data in the User.! Planet with a sun, could that be theoretically possible well as a complicated model within option... Qda method lorem ipsum dolor sit amet, consectetur adipisicing elit multiple classes, sets. Lda model classifier.lda would i go about drawing a decision boundary for QDA is nonlinear Read more in area! That obtained by LDA and covers1: 1 libraries if possible quadratic decision in. Subscribe to this RSS feed, copy and paste this URL into your RSS reader below! In LDA classifier, the motivation a decision boundary is equally likely from the QDA method not so many points... Imagine that the difference in the area where the two decision boundaries differ a lot is.! Examine the differences between LDA and QDA are derived for binary and multiple classes even Democrats... Decision surface is linear, do we expect LDA to perform better on the left covariance among K classes those... Logistic regression approaches to pull all the classes together work and let me if! Decision1 contains the calculations of the data in the error rate: 29.04 % same variance going.