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A group of climate scientists are running a climate model that outputs the temperature at every location on Earth for every 6-hour period in the years 2006 and 21001. The climate model is deterministic, and given the athmospheric starting conditions and model parameters, you will always get the same result. The challenge is that the parameters of the climate model must be selected so that the output provides as realistic evolution in time as possible. This is immensely difficult because running the model only once may require one month of computation time. For the sake of this project, assume that the only way to choose these parameters are to run the climate model for different parameter values and compare to observed temperatures.

We limit the focus to one parameter, “the albedo of sea ice”, which is a measure how much sun light is reflected by sea ice. We call this parameter θ, and we decide to choose this parameter so that the temperature observed on October 18, 2019, at 12:00–18:00 best matches the output of the climate model. The fit is measured through a score y(θ) calculated based on the model output generated with parameter value θ.

The group of climate scientists have spent the last month running the model in five computing centres and provides you with five evaluation points of (θ, y(θ)): (0.30, 0.5), (0.35, 0.32), (0.39, 0.40), (0.41, 0.35), and (0.45, 0.60).

Use a Gaussian process model {Y(θ) : θ ∈ [0,1]} to model the unknown relation- ship between the parameter value and the score. Use E[Y(θ)] ≡ 0.5, Var[Y(θ)] ≡ 0.52, and Corr[Y (θ1), Y (θ2)] = (1 + 15|θ1 − θ2|) exp(−15|θ1 − θ2|) for θ1, θ2 ∈ [0, 1].

a) Define a regular grid of parameter values from θ = 0.25 to θ = 0.50 with spacing 0.005 (n = 51 points). Construct the mean vector and the covariance matrix required to compute the condi- tional means and covariances of the process at the 51 points conditional on the five evaluation points. Display the prediction as a function of θ, along with 90% prediction intervals.

b) The scientists’ goal is to achieve y(θ) < 0.30. Use the predictions from a) to compute the conditional probability that y(θ) < 0.30 given the 5 evalution points. Plot the probability as a function of θ.

I'll admit I am very new to Gaussian processes, but from what I know a Gaussian process is completely determined by a mean vector E(Y(θ)) and a covariance function Cov[Y(θ1), Y(θ2)]. E(Y(θ)) is given, and we have the correlation, which is just the covariance divided by Var(θ1)*Var(θ2).The dimension of the mean vector and covariance matrix I believe is 51 (equal to the number of unknown parameter values).

So it seems I have everything I need, but I don't know how to go about computing these things in Matlab.

We were given the following algorithm by our lecturer:

Code:

```
1. calculate Cholesky decomposition Σ = LLT
2. for i = 1...n
3. draw zi ∼ N(0,1)
4. end
5. set ⃗x = μ⃗ + L⃗z
```

Here ⃗x is a draw from Nn(μ⃗, Σ). However I am not sure how to use this to calculate the information requested in a) and b).