Merge branch 'master' of skozl.com:e4-pattern

2 files changed, 15 insertions, 17 deletions

diff --git a/report/metadata.yaml b/report/metadata.yaml
index 7113dce..97c0c24 100755
--- a/report/metadata.yaml
+++ b/report/metadata.yaml
@@ -1,10 +1,10 @@
 ---
 title: 'EE4-68 Pattern Recognition (2018-2019) CW1'
 author:
-  - name: Vasil Zlatanov, Nunzio Pucci
+  - name: Vasil Zlatanov (01120518), Nunzio Pucci (01113180)
     affilation: Imperial College
     location: London, UK
-    email: CID:01120518, CID:01113180
+    email: vz215@ic.ac.uk, np1915@ic.ac.uk
 numbersections: yes
 lang: en
 babel-lang: english
diff --git a/report/paper.md b/report/paper.md
index f3b3584..532892f 100755
--- a/report/paper.md
+++ b/report/paper.md
@@ -111,25 +111,24 @@ eigenvalues.
 ## Classification
 
 The analysed classification methods used for face recognition are Nearest Neighbor and
-alternative method through reconstruction error. 
+alternative method utilising reconstruction error. 
 
 Nearest Neighbor projects the test data onto the generated subspace and finds the closest 
-element to the projected test image, assigning the same class as the neighbor found.
+training sample to the projected test image, assigning the same class as that of thenearest neighbor.
 
 Recognition accuracy of NN classification can be observed in figure \ref{fig:accuracy}.
 
-A confusion matrix showing success and failure cases for Nearest Neighbor classfication
-can be observed in figure \ref{fig:cm}:
+A confusion matrix showing success and failure cases for Nearest Neighbor classfication when using PCA can be observed in figure \ref{fig:cm}:
 
 \begin{figure}
 \begin{center}
 \includegraphics[width=15em]{fig/pcacm.pdf}
-\caption{Confusion Matrix NN, M=99}
+\caption{Confusion Matrix PCA and NN, M=99}
 \label{fig:cm}
 \end{center}
 \end{figure}
 
-Two examples of the outcome of Nearest Neighbor Classification are presented in figures \ref{fig:nn_fail} and \ref{fig:nn_succ},
+Two examples of the outcome of Nearest Neighbor classification are presented in figures \ref{fig:nn_fail} and \ref{fig:nn_succ},
 respectively one example of classification failure and an example of successful 
 classification.
 
@@ -153,8 +152,8 @@ classification.
 
 It is possible to use a NN classification that takes into account majority voting.
 With this method recognition is based on the K closest neighbors of the projected
-test image. Such method anyways showed the best recognition accuracies for PCA with
-K=1, as it can be observed from figure \ref{fig:k-diff}.
+test image. The method that showed highest recognition accuracies for PCA used
+K=1, as visible in figure \ref{fig:k-diff}.
 
 \begin{figure}
 \begin{center}
@@ -231,8 +230,8 @@ represents the mean of each class.
 It can be proven that when we have a singular S\textsubscript{W} we obtain [@lecture-notes]: $W\textsubscript{opt} = arg\underset{W}max\frac{|W\textsuperscript{T}S\textsubscript{B}W|}{|W\textsuperscript{T}S\textsubscript{W}W|} = S\textsubscript{W}\textsuperscript{-1}(\mu\textsubscript{1} - \mu\textsubscript{2})$.
 
 However S\textsubscript{W} is often singular since the rank of S\textsubscript{W}
-is at most N-c and usually N is smaller than D. In such case it is possible to use 
-Fisherfaces. The optimal solution to such problem lays in W\textsuperscript{T}\textsubscript{opt} 
+is at most N-c and usually N is smaller than D. In this case it is possible to use 
+Fisherfaces. The optimal solution to this problem lays in W\textsuperscript{T}\textsubscript{opt} 
 =  W\textsuperscript{T}\textsubscript{lda}W\textsuperscript{T}\textsubscript{pca}, 
 
 where W\textsubscript{pca} is chosen to maximize the determinant of the total scatter matrix
@@ -267,7 +266,7 @@ Being $\nabla F\textsubscript{t}(e)= (1-t)Se+\frac{t}{<e,S\textsubscript{W}e>
 e>+\epsilon)\textsuperscript{2}S\textsubscript{W}e}$, we obtain that our goal is to 
 find $\nabla F\textsubscript{t}(e)=\lambda e$, which means making $\nabla F\textsubscript{t}(e)$
 parallel to *e*. Since S is positive semi-definite, $<\nabla F\textsubscript{t}(e),e> \geq 0$.
-It means that $\lambda$ needs to be greater than zero. In such case, normalizing both sides we 
+It means that $\lambda$ needs to be greater than zero. Normalizing both sides we 
 obtain $\frac{\nabla F\textsubscript{t}(e)}{||\nabla F\textsubscript{t}(e)||}=e$.
 
 We can express *T(e)* as $T(e) = \frac{\alpha e+ \nabla F\textsubscript{t}(e)}{||\alpha e+\nabla F\textsubscript{t}(e)||}$ (adding a positive multiple of *e*, $\alpha e$ to prevent $\lambda$ from vanishing).
@@ -301,8 +300,7 @@ S\textsubscript{W}(within-class scatter matrix), respectively show ranks of at m
 N-c(312 maximum for our standard 70-30 split). 
 The rank of S\textsubscript{W} will have the same value of $M_{\textrm{pca}}$ for $M_{\textrm{pca}}\leq N-c$.
 
-Testing with $M_{\textrm{lda}}=50$ and $M_{\textrm{pca}}=115$ gives 92.9% accuracy. The results of such test can be
-observed in the confusion matrix shown in figure \ref{fig:ldapca_cm}.
+Testing with $M_{\textrm{lda}}=50$ and $M_{\textrm{pca}}=115$ gives 92.9% accuracy. The results of this test can be seen in the confusion matrix shown in figure \ref{fig:ldapca_cm}.
 
 \begin{figure}
 \begin{center}
@@ -334,7 +332,7 @@ Two recognition examples are reported: success in figure \ref{fig:succ_ldapca} a
 
 The PCA-LDA method allows to obtain a much higher recognition accuracy compared to PCA.
 The achieved separation between classes and reduction between inner class-distance 
-that makes such results possible can be observed in figure \ref{fig:subspaces}, in which 
+that makes these results possible can be observed in figure \ref{fig:subspaces}, in which 
 the 3 features of the subspaces obtained are graphed.
 
 \begin{figure}
@@ -352,7 +350,7 @@ So far we have established a combined PCA-LDA model which has good recognition w
 
 ## Committee Machine Design
 
-Since each model in the ensemble outputs its own predicted labels, we need to define a strategy for combining the predictions such that we obtain a combined response which is better than that of an individual model. For this project, we consider two committee machine designs.
+Since each model in the ensemble outputs its own predicted labels, we need to define a strategy for combining the predictions such that we obtain a combined response which is better than that of an individual models. For this project, we consider two committee machine designs.
 
 ### Majority Voting