# Sensitivity Analysis¶

This module could be easily taken off of this library. But one could take advantage of sensitivity analysis to reduce the complexity of expensive models. Moreover, observations in practice shows significant gains in performance by employing data preprocessing based on sensitivity analysis.

Sensitivity analysis is defined as the study of how the uncertainty in the output of a model can be apportioned to sources of uncertainty in inputs.

Given a model $$y=f(x_1,\dots,x_n)$$, the following are standard sensitivity measures quantifying sensitivity of the model with respect to $$x_1,\dots,x_n$$:

## Morris¶

The Morris method facilitates a global sensitivity analysis by making a number of local changes at different points of the possible range of input values. The following quantities are usually measured regarding Morris method:

$\begin{split}\mu_i=\int\frac{\partial f}{\partial x_i}dx_1\cdots dx_n,\\ \mu^*_i=\int|\frac{\partial f}{\partial x_i}|dx_1\cdots dx_n,\\ \sigma_i=Var(\frac{\partial f}{\partial x_i}).\end{split}$

Generally, $$\mu^*$$ is used to detect input factors with an important overall influence on the output. $$\sigma$$ is used to detect factors involved in interaction with other factors or whose effect is non-linear.

## Sobol¶

The Sobol method (aka variance-based sensitivity analysis) works by decomposing the variance of the output of the model into fractions which can be attributed to inputs or sets of inputs. The first-order indices are defined as:

$S_i=\frac{D_i(y)}{Var(y)},\quad S_{ij}=\frac{D_{ij}(y)}{Var(y)},\dots$

where

$D_i(y)=Var_{x_i}(E_{x_{-i}}(y|x_i)), ~ D_{ij}(y)=Var_{x_{ij}}(E_{x_{-ij}}(y|x_i,x_j))-(D_i(y)+D_j(y)),\dots,$

and the total-effect index:

$S_{T_i}=\frac{E_{x_{-i}}(Var_{x_i}(y|x_{-i}))}{Var(y)}= 1-\frac{Var_{x_{-i}}(E_{x_i}(y|x_{-i}))}{Var(y)}.$

## Moment-Independent $$\delta$$ Index¶

Let $$g_Y(y)$$ be the distribution of the values of $$y$$ and denote by $$g_{Y|x_i}(y)$$ the distribution of values of $$y$$ when the value of $$x_i$$ is fixed. Let $$s(x_i)=\int|g_Y(y)-g_{Y|x_i}(y)|dy$$, then the delta index of $$x_i$$ is defined as:

$\delta_i=\frac{1}{2}\int s(x_i)g_{x_i}dx_i,$

where $$g_{x_i}(x_i)$$ is the distribution of the values of $$x_i$$.

Note

The class SensAprx acts as a scikit-learn wrapper as a transformer based on the sensitivity analysis library SALib.

It accepts a scikit-learn compatible regressor at initiation, fits the regressor on the $$X, y$$ arguments of SensAprx.fit and performs sensitivity analysis on the regressor.

• The type of analysis can be determined at initiation by choosing method among [‘sobol’, ‘morris’, ‘delta-mmnt’] (default: ‘sobol’).
• After calling the fit method, coefficients are stored in SensAprx.weights_.
• After calling SensAprx.fit by calling SensAprx.transform(X) selects the top n features where n is given at initiation through n_features_to_select.
• It is easier to do sensitivity analysis on functions using SALib’s ui, but if one prefers using scikit-learn’s wrapper, then the function should be modified to resemble a scikit-learn regressor which simply ignores training data.

# Eliminate features based on Pearson correlation¶

The Pearson correlation is widely used to eliminate some of highly correlated features to reduce the number of features. Surprisingly, there is no scikit-learn compatible code implementing feature selection according to a given correlation threshold (at the time publishing this library).

Although, it sound like an easy task to do, it is not clear how to select a minimal set of features with low correlation and which ones can be safely excluded.

SKSurrogate.sensapprx.CorrelationThreshold implements the following algorithm to select a minimal set of features with correlation below a given threshold:

Note

Input: the set of all variables $$V$$, a positive threshold $$t$$

Output a subset $$W\subseteq V$$ where $$\forall a,b\in W\quad |corr(a,b)|<t$$

set $$P:=$$ the set of all pairs $$(a, b)\in V\times V$$ where $$|corr(a, b)|\ge t$$

set $$W:=\{v\in V:\forall x\in V |corr(v,x)|<t\}$$

while $$P\neq\emptyset$$ do:
form the undirected graph $$G=G(V\setminus W, P)$$ find a node $$w\in V\setminus W$$ with highest degree update $$W:=W\cup\{w\}$$ remove all pairs from $$P$$ involving $$w$$

return $$W$$

The above procedure selects those features which has high (positive or negative) correlation with higher number of other features and omits the other features. Repeats this process until no more pair with high correlation remains.