### Assessment 1

1. Reducing observation noise has a simlar effect to which of the following?

Need to clear the relationships between reducing noise and sampling density.

As noise goes down the constraint formed by the targets is stronger (we don’t have to worry about noise) or equivalently the targets become more informative so we don’t require such closely spaced samples (of course it is best if we do have them).

2. The accuracy of approximation by a polynomial of high enough degree is guaranteed by the Stone-Weierstrass theorem for [x1] functions on a [x2] interval.

Feedback: [x1]: the key concept is continuity (no jumps) of the function.

[x2]: the key concept is finiteness of the interval (any interval bounded at each end by a finite real number i.e. excludes +/- infinity).

You need to undertand this basic idea about the nature of function approximation to know what is reasonable. Since polynomials and the other basis functions we consider are continuous functions themselves, it is reasonable the expect that they can be combined to fit continuous functions well, however they always go to plus or minus infinity for large absolute input values so can only do so over a finite range. This can all be proved mathematically but it is only the idea that is important here.

1. The least-squares solution is unique if which one of the following is true.

$\Psi_{T}\Psi$ has to be invertible otherwise there are multiple solutions and the pseudo inverse picks the shortest one

2. You have been given a set of data that your company’s IT department has collected from customer hits on your website. It is your job to try to relate it to the volume of sales of your main product using a data model. Which of the following best describes the task?

this is a classic black box datamodelling problem - unknown model, observational data

1. Compute the number of unique terms of a degree 5 polynomial of 9 variables.

The answer is 2002. Here $d=9,\delta=5$, then $$m=\frac{(9+5)!}{9!5!}=2002$$

For example: $16\times16$ bitmap rearranged into vector, $d=256$, assume it is a cubic polynomial, $$m=\frac{259!}{256!3!}=\frac{259\times258\times257\times256!}{256!\times3\times2\times1}=2862209$$

So, $m\timesm$ matrix inversion for least-squares.

2. You are given the task of modelling the cause of acute pain in a patient’s chest. You will use “blood-pressure”, “ECG results” and “past-history of heart problems” as inputs and “acute coronary syndrome”, “stable angina”, “musculo-skeletal” and “other” as targets. Which of the following best describes the task?

using the same inputs to model multiple outputs is known as multi-response regression.

--- The following is a wasteland without any knowledge.---