Description
1. Using the χ
2 goodness-of-fit test determine whether the data set contained in the files Data1 and Data2 can be reasonably modeled by a
Gaussian distribution at an α = 0.05 confidence level.
(a) Plot the histogram for each data set.
(b) Overlay the best fit Gaussian on this histogram plot.
(c) Provide the results of the χ
2 goodness-of-fit test for each data sets,
inclusive of its computed p-value.
(d) For the data set(s) that fail the χ
2 goodness-of-fit test for Gaussian
p(x) determine which p(x) distribution does provide a reasonable
model for the data.
• i.e., use the χ
2 goodness-of-fit test to determine statistically
what other distribution could be used to model the data.
• Hint: the shape of the data’s histogram can provide a a good
indication of which analytical p(x)’s you should test.
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2. Write a function to generate N random data samples with mean µ =
[µx1 µx2
]
T and Σ =
σ
2
x1
σx1 σx2
σx2 σx1 σ
2
x2
(a) Use this function to generate three sets of data all with N = 1000,
µ = [5, 5]T
, with ρ = −0.8, ρ = 0.2 and ρ = 0.9, σx1 = 2 and
σx2 = 1.
(b) Produce a 2-D scatter plot for each generated data set.
(c) On these scatter plots overlay the eigenvectors of Σ and draw the
1-, 2-, and 3-σ ellipses that are associated with the generated data.
3. The data in the file Data3 belongs to 3 classes, with the third column
of the file denoting which class each data item belongs to.
(a) For each class estimate its mean and covariance, assuming that
the data within each class follows p(x) ∼ N(µ, Σ).
(b) For each of the following unclassified data point use the Mahalanobis distance to assigned each of the unclassified data points
x1, x2, x3 and x4 to one of the three classes.
x1 = [10, 2], x2 = [−3, 4], x3 = [2, 2], and x4 = [5 − 7]
(c) Provide scatter plots of each of the three clusters (all on the same
plot but in different colours), place the x1 through x4 data points
on this plot, and use your eigenvector and ellipse drawing routing
from Question 2 to draw the principal axes and 1-, 2-, and 3-σ
ellipses associated with each pattern class.
(d) How would your estimations of the three classes’ statistics change
if you did not have a priori knowledge of the class labels?
• i.e., If you were given the file Data3 with just its first 2
columns and not its third column.
• This denotes the distinction between supervised learning (with
column 3) and unsupervised learning (without column 3).
(e) For each pair of classes plot (on the same plot as generated from
3(c) above) the 2-Class decision boundaries defined by where pxi
(x) =
pxj
(x), for i 6= j and i, j ∈ {1, 2, 3} and provide the formulas for
these decisions boundaries.
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