Mathematical Data Science — DU B.A. (Prog) Mathematics as Major Previous Year Question Papers

: Mathematical Data science : B.A/B.Sc. (Programme) Mathematics 1. (a) Differentiate between structured and unstructured data. Give examples of each type. (b) What are the common problems faced with data? Explain the issues of dirty and missing data in tabular for mats. (c) Consider the following dataset collected from a fitness study : (i) Type of Exercise (Yoga, Running, Swimming) (ii) Gender (Male/Female) (iv) Satisfaction Level (Low, Medium, High) Classify each variable according to its data type and explain why. 2. (a) Discuss anomaly detection techniques and their applications in real-world data science problems. (b) Define Data quality and Data hygiene. Describe methods used for assessing and improving data quality. (c) Explain meta-data and its role in data description. Differentiate between XML and JSON for mats with examples. 3. (a) Define the Probably Approximately Correct (PAC) and explain its key parameters. Compare how Chemoff-Hoeffding inequality provide bounds on the deviation between empirical and true error, and relate these bounds to error tolerance and probability failure. (b) Consider Jri,X2 ,..., X2o iid random variables with expected value E[Xi] = 18 and variance Var(Xi) = 4. Assume we also know that each Xi must satisfy 12 S Xi S 22. i. Use the chebyshev inequality to find the upper-bound p[ff > 19]. ii. Use the Chemoff-Hoeffding inequality to find the upper-bound (c) Show how Mahalanobis distance generalizes the Euclidean distance. Under what condition do the two coincide? Given two vectors cl = b- •, cmd M - . Compute the Mahalanobis distance CZM(a, b). (i) On the above data, run polynomial regression forp = 3. Report polynomial models for each. With each of these models, predict the value of y for a new x values of 0.4 and 0.6. (fi) Use cross-validation to predict an error, with error of a single data point (j;, y) from a model A4 as (j`41/x/ -y)2. Describe how you did this, and which data was used for what. Solve for the coefficient Ridge Regression crs = (XTX + s2/)-1xTy with s = 2; that is generate the ridge regression solution. For this, report the error in the estimate 9 of y as ||y -Xcrsll2 = lly -Sllz. (c) Consider the function Starting with (x, y) = (0,1) run the gradient descent algorithm for each function. Run for two iterations, and report the function value at the end of each step. (i) First, run with a fixed learning rate of y = 0.1 for f. (ii) Second, run with any variant of gradient descent you want. Try to get the smallest function value after two steps. (b) Find linear Discriminant projection vector for 2-dimensional data of two classes as follows : (c) Compute the flrst Principal Component for the following 2-dimensional data : 6. (a) What are differences between principal component Analysis (PCA) and Linear discriminant analysis. (b) Draw Voronoi diagram with four points A (0,1), 8 (0,4), C (2,0) and D