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Update Improvedgekpls.md
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Spinachboul authored Jan 4, 2024
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# GEKPLS Function

Gradient Enhanced Kriging with Partial Least Squares Method (GEKPLS) is a surrogate modelling technique that brings down computation time and returns improved accuracy for high-dimensional problems. The Julia implementation of GEKPLS is adapted from the Python version by [SMT](https://github.com/SMTorg) which is based on this [paper](https://arxiv.org/pdf/1708.02663.pdf).

# Modifications for Improved GEKPLS Function:

To enhance the GEKPLS function, sampling method was changed from ```SobolSample()``` to ```HaltonSample()```.


```@example gekpls_water_flow
using Surrogates
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n = 1000
lb = [0.05,100,63070,990,63.1,700,1120,9855]
ub = [0.15,50000,115600,1110,116,820,1680,12045]
x = sample(n,lb,ub,HaltonSample())
x = sample(n,lb,ub,SobolSample())
grads = gradient.(water_flow, x)
y = water_flow.(x)
n_test = 100
Expand All @@ -43,13 +36,3 @@ y_pred = g.(x_test)
rmse = sqrt(sum(((y_pred - y_true).^2)/n_test)) #root mean squared error
println(rmse) #0.0347
```

<br>
<br>



| **Sampling Method** | **RMSE** | **Differences** |
|----------------------|--------------------------|---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|
| **Sobol Sampling** | 0.021472963465423097 | Utilizes digital nets to generate quasi-random numbers, offering low discrepancy points for improved coverage. - Requires careful handling, especially in higher dimensions. |
| **Halton Sampling** | 0.02144270998045834 | Uses a deterministic sequence based on prime numbers to generate points, allowing for quasi-random, low-discrepancy sampling. - Simpler to implement but may exhibit correlations in some dimensions affecting coverage.

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