HyperKnee Finder

This tool will found the optimal values for two inter-dependant parameters using the well known knee/elbow method.

The method prescribe to search for the point where the curvature is at the maximum (Here an example).

A more formal definition (from Satopää, Albrecht, Irwin, and Raghavan, 2011, p.1) states that the knee/elbow point is the point after which:

relative costs to increase [or decrease, NdC] some tunable parameter is no longer worth the corresponding performance benefit

While usually this method is used for tuning one single parameter, nothing impeach to the same for multiple, inter-dependant parameters. This tool is anyway able to find the optimal value only for two parameters.

In facts, HyperKnee Finder is a 2-d generalisation of the KneeFinder tool.

Install and usage

to install ust use PIP:

pip install HyperkneeFinder

The data has to be shaped in a proper way, i,e, X and Y should be 1-dimensional arrays, while Z should be a 2-dimensional array. Each Z[i,j] should contain the Z value for X[i] and Y[j].

To discover the HyperKnee point just call

hkf= HyperKneeFinder(X, Y, Z)
hyperknee_coordinates = hfk.get_hyperknee_point()

You can also plot the data for a crosscheck:

hkf.visualise_hyperknee()

Please check the examples in this documentation.

Motivations for HyperKnee Finder

In many situations the parameters of an algorithm depends on each other. What you usually do is to ignore this dependency, and so you optimise the first parameter, then you use the found value to optimise the second parameter.

Acting like this, in general, does not guarantee you to land on the optimal combination for the two parameters.

Indeed, you have to evaluate all the possible combination of the two parameters, and at that point you can make your choice.

What's behind the scenes

Similarly to KneeFinder, this tool fits a 2 dimensional linear function (i.e., a plane) to the (cleaned) data, then search for the point with maximum distance to the plane. Note that only the points internal to the pseudo-convexity of the curve will be accounted in this last step.

Examples

The following example show how to find the HyperKnee point in a double exponential decay.

from hyperkneefinder import HyperKneeFinder
import numpy as np
%matplotlib ipympl

#double exponential decay plus noise, clipped
X = np.arange(1, 8, 0.1)

Y = np.arange(6, 10, 0.1)
Z = np.zeros((len(X), len(Y)))
for i in range(len(X)):
    for j in range(len(Y)):
        Z[i, j] = np.exp(-X[i]/3) + np.exp(-(Y[j]-5)) + np.random.rand()/45

Z = np.clip(Z, a_min= 0.5, a_max=2)
hkf= HyperKneeFinder(X, Y, Z, name_x='parameter_1', name_y='parameter_2', clean_data=True, clean_threshold=0.8)
hkf.visualise_hyperknee()

Also with different pseudo-convexity:

X = np.arange(1, 8, 0.1)

Y = np.arange(6, 10, 0.1)
Z = np.zeros((len(X), len(Y)))
for i in range(len(X)):
    for j in range(len(Y)):
        Z[i, j] = np.exp(-2/X[i]) + np.exp(-3/(Y[j]-5)) # + np.random.rand()/15
        
Z = np.clip(Z, a_min= 0, a_max=1)
hkf= HyperKneeFinder(X, Y, Z, name_x='parameter_1', name_y='parameter_2', clean_data=True, clean_threshold=0.8)
hkf.visualise_hyperknee()