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diff --git a/posts/2019-01-03-discriminant-analysis.md b/posts/2019-01-03-discriminant-analysis.md deleted file mode 100644 index 47e2cc8..0000000 --- a/posts/2019-01-03-discriminant-analysis.md +++ /dev/null @@ -1,278 +0,0 @@ ---- -title: Discriminant analysis -date: 2019-01-03 -template: post -comments: true ---- - -In this post I talk about the theory and implementation of linear and -quadratic discriminant analysis, classical methods in statistical -learning. - -**Acknowledgement**. Various sources were of great -help to my understanding of the subject, including Chapter 4 of [The -Elements of Statistical -Learning](https://web.stanford.edu/~hastie/ElemStatLearn/), [Stanford -CS229 Lecture notes](http://cs229.stanford.edu/notes/cs229-notes2.pdf), -and [the scikit-learn -code](https://github.com/scikit-learn/scikit-learn/blob/7389dba/sklearn/discriminant_analysis.py). -Research was done while working at KTH mathematics department. - -_If you are reading on a mobile device, you may need to "request desktop site" -for the equations to be properly displayed. This post is licensed under CC BY-SA -and GNU FDL._ - -Theory ------- - -Quadratic discriminant analysis (QDA) is a classical classification -algorithm. It assumes that the data is generated by Gaussian -distributions, where each class has its own mean and covariance. - -$$(x | y = i) \sim N(\mu_i, \Sigma_i).$$ - -It also assumes a categorical class prior: - -$$\mathbb P(y = i) = \pi_i$$ - -The log-likelihood is thus - -$$\begin{aligned} -\log \mathbb P(y = i | x) &= \log \mathbb P(x | y = i) \log \mathbb P(y = i) + C\\ -&= - {1 \over 2} \log \det \Sigma_i - {1 \over 2} (x - \mu_i)^T \Sigma_i^{-1} (x - \mu_i) + \log \pi_i + C', \qquad (0) -\end{aligned}$$ - -where $C$ and $C'$ are constants. - -Thus the prediction is done by taking argmax of the above formula. - -In training, let $X$, $y$ be the input data, where $X$ is of shape -$m \times n$, and $y$ of shape $m$. We adopt the convention that each -row of $X$ is a sample $x^{(i)T}$. So there are $m$ samples and $n$ -features. Denote by $m_i = \#\{j: y_j = i\}$ be the number of samples in -class $i$. Let $n_c$ be the number of classes. - -We estimate $\mu_i$ by the sample means, and $\pi_i$ by the frequencies: - -$$\begin{aligned} -\mu_i &:= {1 \over m_i} \sum_{j: y_j = i} x^{(j)}, \\ -\pi_i &:= \mathbb P(y = i) = {m_i \over m}. -\end{aligned}$$ - -Linear discriminant analysis (LDA) is a specialisation of QDA: it -assumes all classes share the same covariance, i.e. $\Sigma_i = \Sigma$ -for all $i$. - -Guassian Naive Bayes is a different specialisation of QDA: it assumes -that all $\Sigma_i$ are diagonal, since all the features are assumed to -be independent. - -### QDA - -We look at QDA. - -We estimate $\Sigma_i$ by the variance mean: - -$$\begin{aligned} -\Sigma_i &= {1 \over m_i - 1} \sum_{j: y_j = i} \hat x^{(j)} \hat x^{(j)T}. -\end{aligned}$$ - -where $\hat x^{(j)} = x^{(j)} - \mu_{y_j}$ are the centred $x^{(j)}$. -Plugging this into (0) we are done. - -There are two problems that can break the algorithm. First, if one of -the $m_i$ is $1$, then $\Sigma_i$ is ill-defined. Second, one of -$\Sigma_i$\'s might be singular. - -In either case, there is no way around it, and the implementation should -throw an exception. - -This won\'t be a problem of the LDA, though, unless there is only one -sample for each class. - -### Vanilla LDA - -Now let us look at LDA. - -Since all classes share the same covariance, we estimate $\Sigma$ using -sample variance - -$$\begin{aligned} -\Sigma &= {1 \over m - n_c} \sum_j \hat x^{(j)} \hat x^{(j)T}, -\end{aligned}$$ - -where $\hat x^{(j)} = x^{(j)} - \mu_{y_j}$ and ${1 \over m - n_c}$ comes -from [Bessel\'s -Correction](https://en.wikipedia.org/wiki/Bessel%27s_correction). - -Let us write down the decision function (0). We can remove the first -term on the right hand side, since all $\Sigma_i$ are the same, and we -only care about argmax of that equation. Thus it becomes - -$$- {1 \over 2} (x - \mu_i)^T \Sigma^{-1} (x - \mu_i) + \log\pi_i. \qquad (1)$$ - -Notice that we just walked around the problem of figuring out -$\log \det \Sigma$ when $\Sigma$ is singular. - -But how about $\Sigma^{-1}$? - -We sidestep this problem by using the pseudoinverse of $\Sigma$ instead. -This can be seen as applying a linear transformation to $X$ to turn its -covariance matrix to identity. And thus the model becomes a sort of a -nearest neighbour classifier. - -### Nearest neighbour classifier - -More specifically, we want to transform the first term of (0) to a norm -to get a classifier based on nearest neighbour modulo $\log \pi_i$: - -$$- {1 \over 2} \|A(x - \mu_i)\|^2 + \log\pi_i$$ - -To compute $A$, we denote - -$$X_c = X - M,$$ - -where the $i$th row of $M$ is $\mu_{y_i}^T$, the mean of the class $x_i$ -belongs to, so that $\Sigma = {1 \over m - n_c} X_c^T X_c$. - -Let - -$${1 \over \sqrt{m - n_c}} X_c = U_x \Sigma_x V_x^T$$ - -be the SVD of ${1 \over \sqrt{m - n_c}}X_c$. Let -$D_x = \text{diag} (s_1, ..., s_r)$ be the diagonal matrix with all the -nonzero singular values, and rewrite $V_x$ as an $n \times r$ matrix -consisting of the first $r$ columns of $V_x$. - -Then with an abuse of notation, the pseudoinverse of $\Sigma$ is - -$$\Sigma^{-1} = V_x D_x^{-2} V_x^T.$$ - -So we just need to make $A = D_x^{-1} V_x^T$. When it comes to -prediction, just transform $x$ with $A$, and find the nearest centroid -$A \mu_i$ (again, modulo $\log \pi_i$) and label the input with $i$. - -### Dimensionality reduction - -We can further simplify the prediction by dimensionality reduction. -Assume $n_c \le n$. Then the centroid spans an affine space of dimension -$p$ which is at most $n_c - 1$. So what we can do is to project both the -transformed sample $Ax$ and centroids $A\mu_i$ to the linear subspace -parallel to the affine space, and do the nearest neighbour -classification there. - -So we can perform SVD on the matrix $(M - \bar x) V_x D_x^{-1}$ where -$\bar x$, a row vector, is the sample mean of all data i.e. average of -rows of $X$: - -$$(M - \bar x) V_x D_x^{-1} = U_m \Sigma_m V_m^T.$$ - -Again, we let $V_m$ be the $r \times p$ matrix by keeping the first $p$ -columns of $V_m$. - -The projection operator is thus $V_m$. And so the final transformation -is $V_m^T D_x^{-1} V_x^T$. - -There is no reason to stop here, and we can set $p$ even smaller, which -will result in a lossy compression / regularisation equivalent to doing -[principle component -analysis](https://en.wikipedia.org/wiki/Principal_component_analysis) on -$(M - \bar x) V_x D_x^{-1}$. - -Note that as of 2019-01-04, in the [scikit-learn implementation of LDA](https://github.com/scikit-learn/scikit-learn/blob/master/sklearn/discriminant_analysis.py), the prediction is done without -any lossy compression, even if the parameter `n_components` is set to be smaller -than dimension of the affine space spanned by the centroids. -In other words, the prediction does not change regardless of `n_components`. - -### Fisher discriminant analysis - -The Fisher discriminant analysis involves finding an $n$-dimensional -vector $a$ that maximises between-class covariance with respect to -within-class covariance: - -$${a^T M_c^T M_c a \over a^T X_c^T X_c a},$$ - -where $M_c = M - \bar x$ is the centred sample mean matrix. - -As it turns out, this is (almost) equivalent to the derivation above, -modulo a constant. In particular, $a = c V_m^T D_x^{-1} V_x^T$ where -$p = 1$ for arbitrary constant $c$. - -To see this, we can first multiply the denominator with a constant -${1 \over m - n_c}$ so that the matrix in the denominator becomes the -covariance estimate $\Sigma$. - -We decompose $a$: $a = V_x D_x^{-1} b + \tilde V_x \tilde b$, where -$\tilde V_x$ consists of column vectors orthogonal to the column space -of $V_x$. - -We ignore the second term in the decomposition. In other words, we only -consider $a$ in the column space of $V_x$. - -Then the problem is to find an $r$-dimensional vector $b$ to maximise - -$${b^T (M_c V_x D_x^{-1})^T (M_c V_x D_x^{-1}) b \over b^T b}.$$ - -This is the problem of principle component analysis, and so $b$ is first -column of $V_m$. - -Therefore, the solution to Fisher discriminant analysis is -$a = c V_x D_x^{-1} V_m$ with $p = 1$. - -### Linear model - -The model is called linear discriminant analysis because it is a linear -model. To see this, let $B = V_m^T D_x^{-1} V_x^T$ be the matrix of -transformation. Now we are comparing - -$$- {1 \over 2} \| B x - B \mu_k\|^2 + \log \pi_k$$ - -across all $k$s. Expanding the norm and removing the common term -$\|B x\|^2$, we see a linear form: - -$$\mu_k^T B^T B x - {1 \over 2} \|B \mu_k\|^2 + \log\pi_k$$ - -So the prediction for $X_{\text{new}}$ is - -$$\text{argmax}_{\text{axis}=0} \left(K B^T B X_{\text{new}}^T - {1 \over 2} \|K B^T\|_{\text{axis}=1}^2 + \log \pi\right)$$ - -thus the decision boundaries are linear. - -This is how scikit-learn implements LDA, by inheriting from -`LinearClassifierMixin` and redirecting the classification there. - -Implementation --------------- - -This is where things get interesting. How do I validate my understanding -of the theory? By implementing and testing the algorithm. - -I try to implement it as close as possible to the natural language / -mathematical descriptions of the model, which means clarity over -performance. - -How about testing? Numerical experiments are harder to test than -combinatorial / discrete algorithms in general because the output is -less verifiable by hand. My shortcut solution to this problem is to test -against output from the scikit-learn package. - -It turned out to be harder than expected, as I had to dig into the code -of scikit-learn when the outputs mismatch. Their code is quite -well-written though. - -The result is -[here](https://github.com/ycpei/machine-learning/tree/master/discriminant-analysis). - -### Fun facts about LDA - -One property that can be used to test the LDA implementation is the fact -that the scatter matrix $B(X - \bar x)^T (X - \bar X) B^T$ of the -transformed centred sample is diagonal. - -This can be derived by using another fun fact that the sum of the -in-class scatter matrix and the between-class scatter matrix is the -sample scatter matrix: - -$$X_c^T X_c + M_c^T M_c = (X - \bar x)^T (X - \bar x) = (X_c + M_c)^T (X_c + M_c).$$ - -The verification is not very hard and left as an exercise. |