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ML4ENG Coursework
创建日期：2024-05-16 16:14:01
所属分类：数学mathematics
标签： ML4ENG
functions

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ML4ENG Coursework – Part 1

1 GENERAL GUIDELINES
This coursework concerns the problem of binary prediction for a heart attack using the data
set [1]. To start, download the data set dataset_heart_attack.mat and the template
file template_cw1.m from the module’s Keats website. Once this is done:
1. Change the template_cw1.m to your k number. In the following, we will refer to
this file as k12345678.m.
2. Open the k12345678.m file with your MATLAB editor. Note that the file contains a
preamble, referred to as main body, which you should not modify, and the definition
of several functions.
3. Follow the Instructions (Section 3 of this document) to fill in the details of the
functions in the template file. The main body of k12345678.m has been divided into
sections, with each section containing one or more functions to be completed. The
functions in the k12345678.m file have been numbered according to the numbered
list below in Section 3 (Instructions).
4. Once you have written the functions, verify k12345678.m runs without errors when
the file is included in a folder containing only the file itself and the data set
dataset_heart_attack.mat.
5. Check that no MATLAB toolbox was used: the output of the last main body line should
be matlab with no further toolboxes.
6. Submit only the k12345678.m file on Keats. No other files are allowed.

IMPORTANT: Excessive printouts (caused by omitting ‘;’) will incur a mark loss. The use of
MATLAB toolboxes will also cause the subtraction of mark points. Please carefully follow
matrix sizes and vector dimension (row or column).
2

2 DATA SET
The file dataset_heart_attack.mat contains a data set = {, }=1
, which consists
of = 303 examples. Each example consists of:
1. Input vector in ℝ
13, encompassing = 13 medical features.
2. Its corresponding binary label ∈ {0,1}, where 1 stands for high chance of heart
attack and 0 for low chance as diagnosed by a medical expert.
The data is loaded into the workspace to have
Name Size Type Description
t × 1 Logical Diagnosis (binary label): 1 = high chance of heart
attack and 0 = low chance.
X × Double Data matrix (samples vectors as rows)
x_titles 1 × String Description for the d features in

The input sample vector is denoted as
= [(1) … ()]
,
and the inputs of the data sets are given by stacking up samples
= [
1

⋮


] = [
1
(1)
⋯ 1
()
⋮ ⋱ ⋮

(1)
⋯
()
].
The labels are also stacked up, forming the vector
= [
1
⋮

].
The entries of the vector x_titles annotate the features.
3

3 INSTRUCTIONS FOR COMPLETING THE COURSEWORK
Section 1
Assume you are given the sensitivity and specificity values of a heart attack hard predictor
̂(⋅). Furthermore, the prior of having a heart attack ( = 1) is also known.
1. [10 points] Design the function
function tn= true_negative(sens, spec, prior)
that calculates the probability of a negative test to be correct, meaning
( = 0|̂ = 0), by using Bayes’ rule. All three arguments are scalars representing
ratios in the interval [0,1]: sens is the sensitivity, spec is the specificity and prior
is the prior.
Section 2
In this section, we split the full data set into training tr and test set te using splitting ratio
∈ (0,1).
2. [10 points] Design the function
function [X_tr, t_tr, X_te, t_te]= split_tr_te(X, t, eta)
that splits the input data {X,t} set into two disjoint data set. The training set
{X_tr,t_tr} should have the last tr = round() samples and labels, and the
test set {X_te,t_te} the first te = − tr. Here stands for the ratio of the
training data set size from the entire data set. Note that this partition involves no
randomness.
The main body of code splits the data set using = 0.7.
Section 3
3. [10 points] Design the function
function loss = detection_error_loss(t_hat, t)
that computes the empirical detection-error loss = (x,t)~(,)[1(t≠̂(x))]
of binary predictions t_hat (as a vector ̂ which operated over some input ̂(x)
which is not given here) with respect to the true targets t, both vectors of the
same length.
4

In the main code, this function runs over two suggested hard predictors: the one following the
sex feature and the other following fbs feature, which is the binary variable 1(fasting blood
sugar > 120 mg/dl), with 1() being the indicator function.
Section 4
We wish to operate over the next loss function ℓ(, ̂)
\ ̂ 0 1
0 0 10
1 3 0
4. [10 points] Design the function
function loss = loss_func(t_hat, t)
that computes the empirical loss = (x,t)~(,)[ℓ(t, ̂(x))] of binary
predictions t_hat (as a vector ̂ which operated over some input ̂(x) which is
not given here) with respect to the true targets t, both vectors of the same
length.
Section 5
In this section, we train hard predictors based on the available training data. To this end, we
consider linear predictors using a different number of features ∈ {0,1, … ,13}. A predictor
using M features selects the first features of the inputs
() = [1,
(1), (2), … , ()]

∈ ℝ+1
as feature vector. Recall that is the = 13-dimensional input feature vector. The model
class is accordingly defined as
ℋ = {̂(⋅ |) =
⋅ ()| ∈ ℝ
+1}.
To train the predictors for a given order , we optimize the model parameter vectors
using the quadratic loss by solving a standard least squares problem over training data matrix.
The LS function is provided.
5. [10 points] Design the function
function out = X_M(X,M)
with input data matrix X of size × ( is input depended, can be extracted by the
dimensionality of X) and order M ∈ ℝ that produces the data matrix of size
× ( + 1) using the feature mapping (⋅)
5

= [
((1))

⋮
(())

].
Section 6
This section visualises the predictor ̂2 on the two-dimensional space of input variables
(1)
and (2). To this end, it spans the space using a grid and it predicts for each sample in that
grid X_gr the outcome of the two predictors. Since the LS prediction 2
⋅ 2() is
continuous and not binary, clipping to the interval [0,1] is done, and hard thresholding as
̂hard(|2) = {
1, 2
⋅ 2() > 0.5
0, otherwise

is applied to determine the decision region. The labelled test set is illustrated on top of the
predictors’ outcomes.
6. [10 points] Design the function
function out = linear_combiner(X, theta)
that applies the predictor ⋅ () (with theta for theta of arbitrary length +
1 and the data matrix X of size × ( + 1)) to each input features sample in data
matrix X (i.e., to each row () of the matrix).
Section 7
We further evaluate the mean square error (MSE) loss on its binary targets of a predictor. We
use the test set for this purpose.
7. [10 points] Design the function
function out = mse_loss(t_hat, t)
that computes the empirical MSE loss of prediction t_hat using the true labels t,
both vectors of the same length.
Section 8
We now wish to see the dependency of the MSE loss over the order .
8. [20 points] Design the function
function out = mse_vs_M(X_tr, t_tr, X_te, t_te)
that uses all given samples in the split data sets (see section 2 for arguments details.
For each = 0,1, … ,13, it trains using the entire training data (solving an LS
problem) a model set using model class of order M, and then computes the empirical
6

MSE test loss of the predictor using the true test labels t_te. The output is a
column vector out∈ ℝ14, with the test losses of [te(0), te(1), … , te(13)]
.
Once coded, a graph will be shown.

Section 9
In this section, the input matrix features are reversed, and will follow the same steps.
9. [10 points] Add a two-line comments in function discussion() why the MSE
test loss of the reversed feature is not identical to the original ordering. Infer which
feature group is more useful for heart attack prediction - is it the lower indexed
features or the higher ones?
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