IAML – INFR10069
计算机科学课业代写 1.1 (9 POINTS)We want to see how each feature in Xtrn is distributed for each class. Since there are nine attributes, we plot a total of nine
Question 1 : (70 total points) 计算机科学课业代写
Experiments ona binary-classification data set
1.1 (9 points)
We want to see how each feature in Xtrn is distributed for each class. Since there are nine attributes, we plot a total of nine figures in a 3-by-3 grid, where the top-left figure shows the histograms for attribute ’A0’ and the bottom-right ’A8’. In each figure, you show histograms of instances of class 0 and those of class 1 using pyplot.hist([Xa, Xb], bins=15), where Xa corresponds to instances of class 0 and Xb to those of class 1, and you set the number of bins to 15. Use grid lines. Based on the results you obtain, discuss and explain your findings.
1.2 (9 points) 计算机科学课业代写
Calculate the correlation coefficient between each attribute of Xtrn and the label Ytrn, so that you calculate nine correlation coefficients. Answer the following questions.
(a) Report the correlation coefficients in a table.
(b) Discuss if it is a good idea to use the attributes that have large correlations with the label for classification tasks.
(c) Discuss if it is a good idea to ignore the attributes that have small correlations with the label for classification tasks.
1.4 (5 points)
Calculate the unbiased sample variance of each attribute of Xtrn, and sort the variances in decreasing order. Answer the following questions.
(a) Report the sum of all the variances.
(b) Plot the following two graphs side-by-side. Use grid lines in each plot.
- A graph of the amount of variance explained by each of the (sorted) attributes, where you indicate attribute numbers on the x-axis.
- A graph of the cumulative variance ratio against the number of attributes, where the range of y-axis should be [0, 1].
1.5 (8 points) 计算机科学课业代写
Apply Principal Component Analysis (PCA) to Xtrn, where you should not rescale Xtrn. Use Sklearn’s PCA with default parameters, i.e. specifying no parameters.
(a) Report the total amount of unbiased sample variance explained by the whole set of principal components.
(b) Plot the following two graphs side-by-side. Use grid lines in each plot.
- A graph of the amount of variance explained by each of the principal components.
- A graph of the cumulative variance ratio, where the range of y-axis should be [0, 1].
(c) Mapping all the instances in Xtrn on to the 2D space spanned with the first two principal components, and plot a scatter graph of the instances on the space, where instances of class 0 are displayed in blue and those of class 1 in red. Use grid lines. Note that the mapping should be done directly using the eigen vectors obtained in PCA – you should not use Sklearn’s functions, e.g. transform().
(d) Calculate the correlation coefficient between each attribute and each of the first and second principal components, report the result in a table.
1.6 (4 points) 计算机科学课业代写
We now standardise the data by mean and standard deviation using the method described below, and look into how the standardisation has impacts on PCA.
Create the standardised training data Xtrn_s and test data Xtst_s in your code in the following manner.
from sklearn . preprocessing import Standard Scaler
scaler = StandardScaler( ). fit( Xtrn )
Xtrn_s = scaler . transform ( Xtrn ) # standardised training data
Xtst_s = scaler . transform ( Xtst ) # standardised test data
Using the standardised data Xtrn_s instead of Xtrn, answer the questions (a), (b), (c), and (d) in 1.5.
1.7 (7 points)
Based on the results you obtained in 1.4, 1.5, and 1.6, answer the following questions.
(a) Comparing the results of 1.4 and 1.5, discuss and explain your findings.
(b) Comparing the results of 1.5 and 1.6, discuss and explain your findings and discuss (using your own words) whether you are strongly advised to standardise this particular data set before PCA.
1.8 (12 points) 计算机科学课业代写
We now want to run experiments on Support Vector Machines (SVMs) with a RBF kernel, where we try to optimise the penalty parameter C. By using 5-fold CV on the standardised training data Xtrn_s described above, estimate the classification accuracy, while you vary the penalty parameter C in the range 0.01 to 100 – use 13 values spaced equally in log space, where the logarithm base is 10. Use Sklearn’s SVC and StratifiedKFold with default parameters unless specified. Do not shuffle the data.
Answer the following questions.
(a) Calculate the mean and standard deviation of cross-validation classification accuracy for each C, and plot them against C by using a log-scale for the x-axis, where standard deviations are shown with error bars. On the same figure, plot the same information (i.e. the mean and standard deviation of classification accuracy) for the training set in the cross validation.
(b) Comment (in brief) on any observations.
(c) Report the highest mean cross-validation accuracy and the value of C which yielded it.
(d) Using the best parameter value you found, evaluate the corresponding best classifier on the test set { Xtst_s, Ytst }. Report the number of instances correctly classified and classification accuracy.
1.9 (5 points) 计算机科学课业代写
We here consider a two-dimensional (2D) Gaussian distribution for a set of two-dimensional vectors, which we form by selecting a pair of attributes, A4 and A7, in Xtrn (NB: not
Xtrn_s) whose label is 0. To make the distribution of data simpler, we ignore the instances whose A4 value is less than 1. Save the resultant set of 2D vectors to a Numpy array, Ztrn, where the first dimension corresponds to A4 and the second to A7. You will find 318 instances in Ztrn.
Using Numpy’s libraries, estimate the sample mean vector and unbiased sample covariance matrix of a 2D Gaussian distribution for Ztrn. Answer the following questions.
(a) Report the mean vector and covariance matrix of the Gaussian distribution.
(b) Make a scatter plot of the instances and display the contours of the estimated distribution on it using Matplotlib’s contour. Note that the first dimension of Ztrn should correspnd to the x-axis and the second to y-axis. Use the same scaling (i.e. equal aspect) for the x-axis and y-axis, and show grid lines.
1.10 (7 points) 计算机科学课业代写
Assuming naive-Bayes, estimate the model parameters of a 2D Gaussian distribution for the data Ztrn you created in 1.9, and answer the following questions.
(a) Report the sample mean vector and unbiased sample covariance matrix of the Gaussian distribution.
(b) Make a new scatter plot of the instances in Ztrn and display the contours of the estimated distribution on it. Note that you should always correspond the first dimension of Ztrn to x-axis and the second dimension to y-axis. Use the same scaling (i.e. equal aspect) for x-axis and y-axis, and show grid lines.
(c) Comparing the result with the one you obtained in 1.9, discuss and explain your findings, and discuss if it is a good idea to employ the naive Bayes assumption for this data Ztrn.
Question 2 : (75 total points) 计算机科学课业代写
Experiments on an image data set of handwritten letters
2.1 (5 points)
(a) Report (using a table) the minimum, maximum, mean, and standard deviation of pixel values for each Xtrn and Xtst. (Note that we mean a single value of each of min, max, etc. for each Xtrn and Xtst.)
(b) Display the gray-scale images of the first two instances in Xtrn properly, clarifying the class number for each image. The background colour should be white and the foreground colour black.
2.2 (4 points)
(a) Xtrn_m is a mean-vector subtracted version of Xtrn. Discuss if the Euclidean distance between a pair of instances in Xtrn_m is the same as that in Xtrn.
(b) Xtst_m is a mean-vector subtracted version of Xtst, where the mean vector of Xtrn was employed in the subtraction instead of the one of Xtst. Discuss whether we should instead use the mean vector of Xtst in the subtraction.
2.3 (7 points) 计算机科学课业代写
Apply k-means clustering to the instances of each of class 0, 5, 8 (i.e. ’A’, ’F’, ’I’) in Xtrn with k = 3, 5, for which use Sklearn’s KMeans with n_clusters=k and random_state=0 while using default values for the other parameters. Note that you should apply the clustering to each class separately. Make sure you use Xtrn rather than Xtrn_m. Answer the following questions.
(a) Display the images of cluster centres for each k, so that you show two plots, one for k = 3 and the other for k = 5. Each plot displays the grayscale images of cluster centres in a 3-by-k grid, where each row corresponds to a class and each column to cluster number, so that the top-left grid item corresponds to class 0 and the first cluster, and the bottom-right one to class 8 and the last cluster.
(b) Discuss and explain your findings, including discussions if there are any concerns of using this data set for classification tasks.
2.4 (5 points) 计算机科学课业代写
Explain (using your own words) why the sum of square error (SSE) in k-means clustering does not increase for each of the following cases.
(a) Clustering with k + 1 clusters compared with clustering with k clusters.
(b) The update step at time t + 1 compared with the update step at time t when clustering with k clusters.
2.5 (11 points)
Here we apply multi-class logistic regression classification to the data. You should use Sklearn’s LogisticRegression with parameters ’max_iter=1000’ and ’random_state=0’ while use default values for the other parameters. Use Xtrn_m for training and Xtst_m for testing. We do not employ cross validation here. Carry out a classification experiment.
(a) Report the classification accuracy for each of the training set and test set.
(b) Find the top five classes that were misclassified most in the test set. You should provide the class numbers, corresponding alphabet letters (e.g. A,B,. . .), and the numbers of misclassifications.
(c) For each class that you identified in the above, make a quick investigation and explain possible reasons for the misclassifications.
2.6 (20 points) 计算机科学课业代写
Without changing the learning algorithm (i.e. use logistic regression), your task here is to improve the classification performance of the model in 2.5. Any training and optimisation (e.g. hyper parameter tuning) should be done within the training set only. Answer the following questions.
(a) Discuss (using your own wards) three possible approaches to improve classification accuracy, decide which one(s) to implement, and report your choice.
(b) Briefly describe your implemented approach/algorithm so that other people can understand it without seeing your code. If any optimisation (e.g. parameter searching) is involved, clarify and describe how it was done.
(c) Carry out experiments using the new classification system, and report the results, including results of parameter optimisation (if any) and classification accuracy for the test set. Comments on the results.
2.7 (9 points) 计算机科学课业代写
Using the training data of class 0 (’A’) from the training set Xtrn_m, calculate the sample mean vector, and unbiased sample covariance matrix using Numpy’s functions, and answer the following.
(a) Report the minimum, maximum, and mean values of the elements of the covariance matrix.
(b) Report the minimum, maximum, and mean values of the diagonal elements of the covariance matrix.
(c) Show the histogram of the diagonal values of the covariance matrix. Set the number of bins to 15, and use grid lines in your plot.
(d) Using Scipy’s multivariate_normal with the mean vector and covariance matrix you obtained, try calculating the likelihood of the first element of class 0 in the test set (Xtst_m). You will receive an error message. Report the main part of error message, i.e. the last line of the message, and explain why you received the error, clarifying the problem with the data you used.
(e) Discuss (using your own words) three possible options you would employ to avoid the error.
Note that your answer should not include using a different data set.
2.8 (8 marks) 计算机科学课业代写
Instead of Scipy’s multivariate_normal we used in 2.7, we now use Sklearn’s GaussianMixture with parameters, n_components=1, covariance_type=’full’, so that there is a single Gaussian distribution fitted to the data. Use { Xtrn_m, Ytrn } as the training set and { Xtst_m, Ytst } as the test set.
(a) Train the model using the data of class 0 (’A’) in the training set, and report the log-likelihood of the first instance in the test set with the model. Explain why you could calculate the value this time.
(b) We now carry out a classification experiment considering all the 26 classes, for which we assign a separate Gaussian distribution to each class. Train the model for each class on the training set, run a classification experiment using a multivariate Gaussian classifier, and report the number of correctly classified instances and classification accuracy for each training set and test set.
(c) Briefly comment on the result you obtained.
2.9 (6 points) 计算机科学课业代写
Answer the following question on Gaussian Mixture Models (GMMs).
(a) Explain (using your own words) why Maximum Likelihood Estimation (MLE) cannot be applied to the training of GMMs directly.
(b) The Expectation Maximisation (EM) algorithm is normally used for the training of GMMs, but another training algorithm is possible, in which you employ k-means clustering to split the training data into clusters and apply MLE to estimate model parameters of a Gaussian distribution for each cluster. Explain the difference between the two algorithms in terms of parameter estimation of GMMs.
