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The Role and Contribution of Imam Khomeini In Iran's Islamic Revolution
Muhammad Arif Mustunsir
The Revolution in Iran is termed by the historians as one of the biggest historical events in the past 100 years. A revolution is a mass movement that aims to establish a new political regime by violently transforming the existing government. The Iranian Revolution of 1978–1979 violently ended the monarchy of Shah Mohammed Reza Pahlavi (1919–1980) and replaced it with an Islamic republic, the theocracy of Ayatollah Ruhollah Khomeini (1901–1989).
The shah's reign had been briefly interrupted between 1951 and 1953 with the interlude of Prime Minister Mohammed Mossedeq (1881–1967). This revolution is regarded as the main turning point of the Islamic World as it opened the eyes of those Islamic states although being powerful with the riches of oil are still being played under western powers.
We have to take a look at the historical elements that lead up to the Iranian Revolution of 1979. Post World War II created a very different world. Colonial (English, French, Dutch, and other) control of much of the Mid-East, Asia, Africa, South American (the Third World) was on the wane. Two superpowers emerged, the United States and the USSR (Russia).
The USSR leaving regimes were based on a military government with socialist tendencies and a strong tribal, regional or ethnic minority base; this was the case in Algeria, Syria, Iraq, South Yemen and Egypt.
Different to popular western misconceptions, the Iranian Revolution of the late 1970s was remarkable in that it was motivated by a large alliance of classes and religious groups within Iran; each with similar social, economic, political, and religious agendas. These groups were united in majority in their initiative to topple the Shah Kingship due to his heavy handed, and dictatorial style of rule (Friedman, 2009).
Additionally, these various groups and organizations were deeply dissatisfied with restrictions on access to the political process, the tight concentration of Iran's huge oil revenues into the hands of a very small, elite group orbiting around the Shah, as well as a growing perception that Iran was becoming too westernized and had lost her grounding in Iranian traditional values and cultures.
It explains that, in 1979, millions of Iranians succeeded in removing Muhammad Reza Shah Pahlavi (1941-1979), the Shah of Iran and forced him to leave the country. While the Shah was known as a 'darling to the West', he tried to authoritarian modernize too rapidly and did not adapt his political institutions sufficiently to the economic and social changes he had brought about.
Although inspired by hopes for democracy, economic prosperity for all classes, gender equality and a leadership that would not allow Iranian culture to be destroyed by the Western values; many Iranian women joined the rebellion.
The world was surprised and astonished by the Iranian Revolution, which was led by Ayatollah Ruhollah Khomeini in exile in Paris. The proposal relates that the Muslims believed that the modern age was 20th century remake of the Jahiliyya of ancient times that had to be destroyed as Mohammed the prophet had destroyed its predecessor. That is why Ruhollah Khomeini and his team bore the seeds of this revolution to overthrow the non-Islamic and tyrant rule of the Shah (Seyed, 2002).
The Iranian Revolution of 1979 has been described as one of the epochal events of the twentieth century, inaugurating a period of Islamic revivalism and struggles against "modernization" in many nations where Islam is the predominant religion. In discussions of the 1979 Revolution, the significance of Islamic fundamentalism, the use of political repression by the regime of Mohammed Reza Shah (heretofore referred to as the monarchist regime), particularly the violence perpetrated by SAVAK (the secret police), widespread corruption by individual public officials and members of the oligarchic economic elite, as well as official favoritism, rising income inequality, and the impact of "Western" imperialism have all played causal roles.
However, there has been a marked paucity of discussions of the way the 1979 Revolution was shaped by struggles over class processes, defined as the particular forms in which surplus labor was produced and distributed. In particular, discussions of the 1979 Revolution have failed to recognize;
i) The role of internecine conflict within the ranks of capitalist appropriators, and
ii) The importance of ancient (or self-exploiting) direct producers and their allied agents in the collapse of the monarchist regime. It is the argument of this chapter that struggles over class processes were a significant factor in shaping the crises that culminated in the 1979 Revolution and of subsequent struggles over the specific form of economic and non- economic relationships in the post-revolutionary Islamic republic. The primary thesis of this chapter is that the efforts of the monarchist regime to create a particular form of capitalism, herein described as oligarchic Capitalism that would serve as catalyst for Iran's rapid economic growth created a range of social crises that threatened the survival of ancients (or self-exploitation) and non-oligarchic capitalism.
The policies of the monarchist regime sometimes referred to as the modernization programme had a definite impact upon class processes in Iran, created and then deepened the social crises that threatened pre-existing configurations of surplus appropriation, particularly the prevalence of self- exploitation in the rural villages and urban bazaars, resulting in complex forms of resistance (Gabriel, 2001) .
Among those with self-interest in opposing the modernization program were a wide range of social agents who desired an end to these crises, including non-oligarchic capitalist appropriators, ancient direct producers, and social agents allied to one or the other or both of these groups of appropriators, including the Shi'a Islamic clergy.
In so far as the monarchist regime's modernization program was designed to displace self- exploitation, which was arguably the most widespread form of surplus appropriation in terms of numbers of direct producers involved, with oligarchic capitalist exploitation, the 1979 Revolution might better be described as a counter revolution, i.e. a change in the political relationships constituting the state designed to avert a gradually progressing economic revolution. On the other hand, capitalist exploitation was clearly dominant in terms of the total value of produced commodities in Iran (Ibid).
THE UNDISPUTED PERSONALITY OF AYATOLLAH KHOMEINI
The personality of Iran's Ayatollah Khomeini was a critical element in defining Iran's war role. Khomeini's background, charisma, and thirst for revenge on those who inflicted personal injury upon him influenced the atmosphere leading to the war. One can understand Khomeini's personality and beliefs by examining his background. Khomeini's paternal origins were not Iranian, but Indian, and he spent a substantial part of his life outside Iran.
His devotion to Shia Islam, reflected by his simplistic ascetic lifestyle focused on spiritual matters, attracted the majority of Iranian peasant-oriented society. Many of these traditional citizens flocked to the modern cities seeking opportunity, only to wind up crowded into slums.
Khomeini's Islamic ideology countered the former Shah Mohammad Reza Pahlavi's Iranian nationalism associated with the glory and affluence of the pre-Islamic Persian Empire of Cyrus the Great. Khomeini's connections outside Iran inhibited his adoption of Persian nationalist ideology in the war. But for many Iranians the Iraqi invasion reignited anti-Arab historic feelings dating from the medieval Islamic conquest. For Khomeini, his religious faith enabled him to confront stronger opponents in hostile environments (Gabriel, 2001, Shahi, 2009).
Khomeini's charisma drew followers of Islam, especially in Iran, Iraq and Lebanon, where Shia'ism dominates. The Shiites for centuries served as a low caste of underprivileged heretics throughout the Sunni dominated Moslem world. Shia'ism emphasizes martyrdom and suffering along with the belief in a messianic era entering upon the return of the Twelfth Imam, a descendant of Mohammad. Khomeini's life of suffering and final return to Iran replicated a legacy of the Shiite experience.
Khomeini's life paralleled in many ways the sect's medieval founder, Imam Hussein, whose army of followers was defeated by a mightier Sunni-led Umayyad Army. In the aftermath of his defeat, Hussein, the model of martyrdom, was beheaded and his followers were left persecuted and oppressed. Although Khomeini denied assertions of being the messianic 12th Imam returning to bring world redemption, Shiites called him "Imam," alluding to his religious stature of messiah among the illiterate poor.
Khomeini's record of overpowering enemies such as the Shah, President Jimmy Carter, and Saddam Hussein through the Iranian Revolution, the American Hostage Crisis, and the Iran-Iraq war was viewed by his followers as miraculous and a reward for his piety (Society, 2011).
There was a revengeful side to Khomeini's personality which was cast in religious terms of good overcoming evil among his faithful. Khomeini sought personal revenge on those who wronged him by equating personal enemies with enemies of the Iranian state. Khomeini's targets for revenge were the Shah, Jimmy Carter, and Saddam Hussein. Khomeini suffered harassment and expulsion both under the Shah and Saddam Hussein. He finally died in 1989.
IMAM KHOMEINI'S POLITICAL THOUGHT AND LEGACY
Imam Khomeini adamantly opposed monarchy, arguing that only rule by a leading Islamic jurist would insure Shariat-e-Muhammadi was properly followed (velayat-e faqih).
Imam Khomeini believed that Iran should strive towards self-reliance. He viewed certain elements of Western culture as being inherently decadent and a corrupting influence upon the youth. As such, he often advocated the banning of popular Western fashions, music, cinema, and literature. His ultimate vision was for Islamic nations to converge together into a single unified power, in order to avoid alignment with either side (the West or the East), and he believed that this would happen at some point in the near future.
Imam Khomeini expressed support for the Universal Declaration of Human Rights; in Sahifah-e-Noor (Vol. 2, page 242), he states: "We would like to act according to the Universal Declaration of Human Rights. We would like to be free. We would like independence."
Imam Khomeini led an ascetic lifestyle, being deeply interested in mysticism, and was against the accumulation of land and wealth by the clergy.
Many of Imam Khomeini's political and religious ideas were considered to be progressive and reformist by leftist intellectuals and activists prior to the Revolution.
Imam Khomeini's definition of democracy existed within an Islamic framework. His last will and testament largely focuses on this line of thought, encouraging both the general Iranian populace, the lower economic classes in particular, and the clergy to maintain their commitment to fulfilling Islamic revolutionary ideals (Shraghi, 2009; Wikipedia, 2012).
THE SIX MAJOR POINTS IN THIS REVOLUTION
i) First, a broad coalition of opposition forces came together to overthrow a dictatorial regime, building on longstanding social grievances but also energizing nationalist sentiment against a state and ruler seen as too compliant to foreign interests. The coalition mobilized under Ayatollah Khomeini's leadership ranged from liberal and Marxist to conservative and religious forces: in effect a classic populist alliance.
ii) Second, the victory of the revolution both required and were facilitated by the state's weakness of leadership and internal divisions. The Shah was ill, his advisers and generals were uncertain. The resemblance to other figures and regimes in a time of crisis - Louis XVI and Czar Nicholas II, as well as Charles I of England - is evident.
iii) Third, the revolution possessed the quality that distinguishes mere coups d'état or rebellions from major revolutions: namely, it was not just political (in the sense of changing the political elite and the constitution or legitimating system of the country but had profound and ongoing social and economic consequences. Because of it, Iran today has a new social order and a new set of social values - even as a new revolutionary elite, an Islamic nomenclature, united by ties of power, business and marriage, controls state revenues.
iv) Fourth, the revolution's core ideology may have propounded the need for a new, radical and egalitarian order; but it was supplemented by pre-existing ideas that were crucial to sustaining domestic support (above all nationalism and a sense of the country's historic standing and mission). Ayatollah Khomeini at first refused to use the word fatherland, and denounced secular nationalism as an insult to Islam. But with the invasion by Saddam Hussein's Iraq in 1980 all this changed, and he and other leaders adopted the Iranian version of the term used by French revolutionaries in the 1790s.
v) Fifth, the explosion of revolution at the center of a multi-ethnic country - and driven especially from within its dominant ethnic component - had profound reverberations on the relations between the Iran's different national components. In particular, it led not to the era of fraternal cooperation and solidarity anticipated in much of the political rhetoric of the time, but to conflict and war.
vi) Here again, the pattern - a revolt at the heart of a plural country and the consolidation of a new authoritarian regime provoking contrary forces in the periphery - has rich historical precedents. The Young Turk revolution of 1908, the Bolshevik revolution of 1917, and the Ethiopian revolution of 1974 are prime examples; their echo in Iran concerned, above all, the Kurds. The hopes of this significant part of the population, of an autonomous Kurdistan within a democratic Iran (and they knew the first was impossible without the second) were to be dashed.
vii) Sixth, the revolution in Iran had explosive international consequences. There were persistent attempts to export the revolution to neighboring countries, which intensified regional rivalries and fostered conditions that led to inter-state war. The Iranian revolution's efforts to promote its state interests and extend itself soon acquired resemblances to a reviving empire - with traces of France and Russia in particular, not least the contradictory trends whereby some forces in the region were inspired by the revolution while others drew on older antagonisms (such as Saddam Hussein's excoriation of Khomeini as a magus [Zoroastrian priest] and more recent concerns about a powerful new Shi'a "crescent") (Halliday, 2009).
This revolution (more than any other in history) did not rely on force, military rebellion or guerrilla war but on politics. This is true in particular with regard to the two instruments that European revolutionaries had themselves long dreamed of using The mass mobilization of people in the streets (in the Iranian case, the largest such opposition demonstrations ever recorded anywhere) and the political general strike to identify.
The non-acceptance of the Shah's decision (which, from October 1978, paralyzed the economy and foreign trade with foreign national particular American). This was perhaps the most consistent and original aspect of the Iranian revolution: in its political form and process, and despite its religious and traditional appearance, it was said to be the first modern revolution.
Iran's experience departed from the standard prescribed by both historical example and textbooks of historical sociology: namely, that a revolution's crucial requirement was the weakening of the state, usually as a result of foreign pressure - either defeat in war or by invasion, or via the withdrawal of support from an external supporter (in the case of China and Cuba, this was the United States). Some people (majorly from the West and Middle East) argue that this revolution was initially a Worker's Revolution which was later on hijacked by the extremist Islamic Clerics of the country.
This revolution was termed as the Islamic Revolution as the country had lost its Islamic Ideology throughout the Shah's Rule and struggle was to re-instate the Islamic Values and the Sharia't Law in the Country. So the accusation stands false on its ground rejected by ever concerning source.
It was the Ayatollah's effort that the country (the people and all Clerics belonging to different sects and ethics) collectively made him the country's Supreme Leader of the country (a person who is
responsible for the election of the government and the Islamic Constitution) where he remained till his death in 1989. It was said that his funeral was attended by 6 Million people approximately which is the biggest in history until now.
* ANSARI, A. M. 2001. The Myth of the White Revolution: Mohammad Reza Shah, 'Modernization' and the Consolidation of Power. Middle Eastern Studies, 37, 1-24.
* AXELROD, A., AND PHILLIPS CHARLES 1995. "Mohammad Reza Pahlavi." Dictators & Tyrants: Absolute Rulers and Would-Be Rulers in World History. Modern World History Online. 1995 Ed. New York: Fact on File Inc.
* BRITANNICA, E. 2012. Mohammad Reza Shah Pahlavi. In:MANSUR G. ABDULLAH, A. M. C. (ed.) Mohammad Reza Shah Pahlavi: Article. Chicago, USA: Encyclopedia Britannica.
* BRITANNICA, E. B 'A'. 2012. White Revolution. In: MANSUR G. ABDULLAH, A. M.
* ESHRAGHI, A. R. 2009. Khamenei vs. Khomeini [Online]. Washington DC: The New Republic. Available: http://www.newrepublic.com/article/politics/khamenei-vs-khomeini [Accessed 15-01-012 2012].
* FRIEDMAN, G. 2009. Western Misconceptions Meet Iranian Reality. Geopolitical Weekly. Austin, USA: Stratfor International.
* GABRIEL, S. J. 2001. Class Analysis of the Iranian Revolution of 1979. In: J.K. GIBSON- GRAHAM, R. S. A. W. R. (ed.) Re/Preseting Class essay in postmodern Marxism. Durham, NC: Duke University press.
* HALLIDAY, F. 2009. Iran's revolution in global history. Available: http://www.opendemocracy.net/article/iran-s-revolution-in-global-history [Accessed 23- 01-2012].
* SEYED, F. S. 2002. The Literacy Corps in Pahlavi Iran (1963-1979): Political, Social and Literary Implications, Switzerland, Lugano: Sapiens.
* SHAHI, A. 2009. Thirty years on: The Iranian revolution and its impact on the region
* SOCIETY I. C. "B" 2012. White Revolution the post-Mosaddeq era and the Shah's White Revolution In: SOCIETY, I. C. (ed.) History of Iran. Tehran: Iran Chamber Society.
* SOCIETY, I. C. "A" 2012. History of Iran:Mohammad Reza Shah Pahlavi Arya Mehr and Shahanshah (King of the Kings) In: SOCIETY, I. C. (ed.) History of Iran. Tehran, Iran: Iran Chamber Society.
* SOCIETY, I. C. 2011. History of Iran Ayatollah Khomeini Ayatollah Rouhollah Mousavi Khomeini (Imam Khomeini) In: SOCIETY, I. C. (ed.) History of Iran. Tehran, Iran: Iran Chamber Society.
* VAKILI-ZAD, C. 1990. Organization, Leadership and Revolution: Religiously-Oriented Opposition in the Iranian Revolution of 1978-1979. Journal of Conflict Studies, 10, 21.
* WATSON, K. 2006. The Shah's White Revolution—Education and Reform in Iran.
* WIKIPEDIA 2012. Political thought and legacy. In: WIKIPEDIA (ed.) Ruhollah Khomeini.Wikipedia.
* WOLFE, L. R. 2012. THE WHITE REVOLUTION IN IRAN [Online]. Washington DC: coldwarstudies.com. Available: http://www.coldwarstudies.com/2012/05/30/the-white- revolution-in-iran/ [Accessed 07-03- 2012].
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\section{Introduction}
In many data mining and machine learning problems, the goal is to extract and discover knowledge from data. One of the challenges frequently faced in these problems is the high dimensionality and the unknown relevance of features~\cite{li_feature_2017}. Ignoring these challenges will more than often result in modeling obstacles, such as sparse data, overfitting, and the curse of dimensionality. Therefore, feature selection is frequently applied, among other techniques, to effectively reduce the feature dimensionality. The smaller subset of features has the potential to explain the problem better, reduce overfitting, alleviate the curse of dimensionality, and even facilitate interpretation. Furthermore, feature selection is known to increase model performance, increase computational efficiency, and increase the robustness of many models due to the dimensionality reduction~\cite{li_feature_2017}.

In this work, we present a novel feature selection method, called \textit{powershap}, that provides an efficient and reliable feature selection approach using Shapley values, statistical tests, and power calculations.

First, in Section \ref{sec:related}, a short overview of the related work is given to show how \textit{powershap} improves upon all these methods. Subsequently, in Section \ref{sec:powershap}, the method and the design choices are explained as well as the resulting algorithm. Finally, the performance of \textit{powershap} is compared to other state-of-the-art methods in Section \ref{sec:experiments} and \ref{sec:experiments_results} using both simulation and open-source benchmark datasets and the results are discussed in Section \ref{sec:discussion}. Finally, the conclusions are summarized in Section \ref{sec:conclusion}.

\section{Related Work}\label{sec:related}
Feature selection approaches can be categorized into filter and wrapper methods.

Filter methods select features by measuring the relevance of the feature using model-agnostic measures, such as statistical tests, information gain, distance, similarity, and consistency to the dependent variable (if available).
These methods are model-independent as this category of feature selection does not rely on training machine learning models~\cite{jovic_review_2015}, resulting in a fast evaluation.
However, the disadvantages of filter methods are that they frequently impose assumptions on the data, are limited to a single type of prediction, such as classification or regression, often do not take inter-correlation between features into account, and frequently require a cut-off value or hyperparameter tuning~\cite{kumari_filter_2011}.
Examples of filter methods are rank, chi² test, f-test, and Pearson correlation~\cite{jovic_review_2015}.

Wrapper methods measure the relevance of features through training models. Depending on the wrapper technique, models are trained on either subsets of the features or on the complete feature set. The trained models are then utilized to select the resulting feature subset, using the aforementioned performance metrics, or by ranking the inferred feature importances. In general, wrapper methods tend to provide smaller and more qualitative feature subsets than filter methods, as they take the interaction between the features, and between the model and the features, into account~\cite{colaco_review_2019}. A major drawback of wrapper methods is the considerable time complexity associated with the underlying search algorithm, or in the case of feature importance ranking the hyperparameter tuning. Examples of wrapper methods are forward, backward, genetic, or rank-based feature importance feature selection.

In the interpretable machine learning field, one of the emerging and proven techniques to explain model predictions is SHAP~\cite{NIPS2017_7062}. This technique aims at quantifying the impact of features on the output. To do so, SHAP uses a game-theory inspired additive feature-attribution method based on Shapley Regression Values~\cite{NIPS2017_7062}. This method is model-agnostic and implemented for various models, e.g., linear, kernel-based, deep learning, and tree-based models. Although SHAP suffers from shortcomings, such as its TreeExplainer providing non-zero Shapley values to noise features, it is technically strong and very popular~\cite{linardatos_explainable_2020}.

The strength of the SHAP algorithm facilitates the development of new feature selection methods using Shapley values. A simple implementation would be a rank-based feature selection, which ranks the different features based on their Shapley values on which a rank cut-off value determines the final feature set. However, there are more advanced methods available. One of these more advanced techniques is borutashap~\cite{eoghan_keany_2020_4247618}. Borutashap is based on the Boruta algorithm that makes use of shadow features, i.e. features with randomly shuffled values. Boruta is built on the idea that a feature is only useful if it is doing better than the best-performing shuffled feature. To do so, Boruta compares the feature importance of the best shadow feature to all other features, selecting only features with larger feature importance than the highest shadow feature importance. Statistical interpretation is realized by repeating this algorithm for several iterations, resulting in a binomial distribution which can be used for p-value cut-off selection~\cite{boruta_base}. Borutashap improves on the underlying Boruta algorithm by using Shapley values and an optimized version of the shap TreeExplainer~\cite{eoghan_keany_2020_4247618}. As such, implementations of Borutashap are limited to tree-based models only.

Another shap-based feature selection method using statistics is shapicant~\cite{calzolari_manuel-calzolarishapicant_2022}. This feature selection method is inspired by the permutation-importance method, which first trains a model on the true dataset, and afterward, it shuffles the labels and retrains the model on the shuffled dataset. This process is repeated for a set amount of iterations, from which the average feature importances of both models are compared. If for a specific feature, the feature importance of the true dataset model is consistently larger than the importance of the shuffled dataset model, that feature is considered informative. Using a non-parametric estimation it is possible to assign a p-value to determine a wanted cut-off value~\cite{altmann_permutation_2010}. Shapicant improves on this underlying algorithm by using Shapley values. Specifically, it uses both the mean of the negative and positive Shapley values instead of Gini importances, which are only positive and frequently used for tree-based model importances. Furthermore, shapicant uses out-of-sample feature importances for more accurate estimations and an improved non-parametric estimation formula~\cite{calzolari_manuel-calzolarishapicant_2022}.

\textit{Powershap} draws inspiration from the non-parametric estimation of shapicant and the random feature usage in borutashap and improves upon all these state-of-the-art filter and wrapper algorithms resulting in at least comparable performances while being significantly faster.

\section{Powershap}\label{sec:powershap}
\textit{Powershap} builds upon the idea that a known random feature should have, on average, a lower impact on the predictions than an informative feature. To realize feature selection, the \textit{powershap} algorithm consists of two components: the \textit{Explain} component and the core \textit{powershap} component. First, in the \textit{Explain} part, multiple models are trained using different random seeds, on different subsets of the data. Each of these subsets is comprised of all the original features together with one random feature. Once the models are trained, the average impact of the features (including the random feature) is explained using Shapley values on an out-of-sample dataset. Then, in the core \textit{powershap} component, the impacts of the original features are statistically compared to the random feature, enabling the selection of all informative features.

\subsection{Powershap Algorithm}
In the \textit{Explain} component, a single known random uniform (RandomUniform) feature is added to the feature set for training a machine learning model. Unlike the Boruta algorithm, where all features are duplicated and shuffled, only a single random feature is added. In some models, such as neural networks, duplicating the complete feature set increases the scale and thereby increases the time complexity drastically. Using the Shapley values on an out-of-sample subset of the data allows for quantifying the impact on the output for each feature. The Shapley values are evaluated on unseen data to assess the true unbiased impact~\cite{breiman_random_2001}. As a final step, the absolute value of all the Shapley values is taken and then averaged ($\mu$) to get the total average impact of each feature. Compared to shapicant, only a single mean value is used here, resulting in easier statistical comparisons. Furthermore, by utilizing the absolute Shapley values, the positive values and the negative values are added to the total impact, which could result in a different distribution compared to the Gini importance. This procedure is then repeated for $I$ iterations, where every iteration retrains the model with a different random feature and uses a different subset of the data to quantify the Shapley values, resulting in an empirical distribution of average impacts that will further be used for the statistical comparison. In the codebase, the procedure explained above is referred to as the \textit{Explain} function. The pseudocode of the \textit{Explain} function is shown in Algorithm~\ref{alg:explain}.

\begin{algorithm}
    \DontPrintSemicolon

    \Fn{Explain($I\leftarrow$ Iterations, $M\leftarrow$ Model,  $\mathbf{D}^{n \times m}\leftarrow$ Data, $rs\leftarrow$ Random seed)}{
    $\mathbf{powershap}_{values}\leftarrow $size $[I,m+1]$\;
    \For{$i\leftarrow 1,2,\dots,I$}{
        $RS\leftarrow i+rs$\;
        $D_{random}^{n}\leftarrow$ RandomUniform$(RS)$ $\in [-1,1]$ size $n$\;
        $\mathbf{D}^{n\times m+1} \leftarrow \mathbf{D}^{n \times m} \cup D_{random}^{n}$\;
        $\mathbf{D}_{train}^{0.8n \times m+1},\mathbf{D}_{val}^{0.2n \times m+1}\leftarrow$ split $\mathbf{D}$\;
        $M \leftarrow$ Fit $M(\mathbf{D}_{train})$\;
        $\mathbf{S}_{values} \leftarrow$ SHAP$(M$, $\mathbf{D}_{val})$\;
        $\mathbf{S}_{values} \leftarrow |\mathbf{S}_{values}|$\;
        \For{$j\leftarrow 1,2,\dots,m+1$}{
            $\mathbf{powershap}_{values}[i][j]\leftarrow \mu(\mathbf{S}_{values}[\dots][j])$\;
        }
    }
    \textbf{return} $\mathbf{powershap}_{values}$
    }
    \caption{Powershap Explain algorithm}\label{alg:explain}
\end{algorithm}

Given the average impact of each feature for each iteration, it is then possible to compare it to the impact of the random feature in the core powershap component. This comparison is quantified using the percentile formula shown in Equation~\ref{eqn:percentile} where $\mathbf{s}$ depicts an array of average Shapley values for a single feature with the same length as the number of iterations, while $x$ represents a single value, and $\mathbb{I}$ represents the indicator function. This formula calculates the fraction of iterations where $x$ was higher than the average shap-value of that iteration and can therefore be interpreted as the p-value.
\begin{equation}\label{eqn:percentile}
    Percentile(\mathbf{s},x)=\sum_{i}^{n}\frac{\mathbb{I}(x>s_i)}{n}
\end{equation}
Note that this formula provides smaller p-values than what should be observed, the correct empirical formula is $(1+\sum_{i}^{n} \mathbb{I}(x>s_i))/(n+1)$ as explained by North et al.~\cite{north_note_2002}. This issue of smaller p-values mainly persists for lower number of iterations. However, \textit{powershap} implements Equation~\ref{eqn:percentile} as this anticonservative estimation of the p-value is desired behavior for the automatic mode (see Section~\ref{sec:automatic}). This formula enables setting a static cut-off value for the p-value instead of a varying cut-off value and results in fewer required iterations, while still providing correct results. This will be further explained at the end of Section~\ref{sec:automatic}.

As the hypothesis states that the impact of the random feature should be on average lower than any informative feature, all impacts of the random feature are again averaged, resulting in a single value that can be used in the percentile function. This results in a p-value for every original feature. This p-value represents the fraction of cases where the feature is less important, on average than a random feature. Given the hypothesis and these p-value calculations, a heuristic implementation of a one-sample one-tailed student-t smaller statistic test can be done, where the null hypothesis states that the random feature ($H_1$-distribution) is not more important than the tested feature ($H_0$-distribution)~\cite{lomax_introduction_2007}. Therefore, the positive class in this statistical test represents a true null hypothesis. This heuristic implementation does not assume a distribution on the tested feature impact scores, in contrast to a standard student-t statistic test where a standard Gaussian distribution is assumed. Then, given a threshold p-value $\alpha$, it is possible to find and output the set of informative features. The pseudocode of Algorithm~\ref{alg:base} details how the core \textit{powershap} feature selection method is realized.

\begin{algorithm}
    \DontPrintSemicolon

    \Fn{Powershap ($I\leftarrow$ Iterations, $M\leftarrow$ Model, $\mathbf{F}_{set} \leftarrow {F_{1},\dots,F_{m}}$, $\mathbf{D}\leftarrow$ Data size $[n,m]$, $\alpha \leftarrow$ required p-value)}{

    $\mathbf{powershap}_{values} \leftarrow$\textbf{Explain}($I$, $M$, $\mathbf{D}$)\;
    $S_{random}\leftarrow\mu(\mathbf{powershap}_{values}[...][m+1])$\;
    $\mathbf{P}^{m} \leftarrow$ initialize\;
    \For{$j\leftarrow 1,2,\dots,m$}{
        $\mathbf{P}[j]\leftarrow Percentile(\mathbf{powershap}_{values}[...][j]$, $S_{random})$\;
    }
    \textbf{return} $\{F_i\ |\ \forall$ $i$ : $\mathbf{P}[i]<\alpha\}$
    }
    \caption{Powershap core algorithm}\label{alg:base}
\end{algorithm}

\subsection{Automatic Mode}\label{sec:automatic}
Running the \textit{powershap} algorithm consisting of the \textit{explain} and the \textit{core} components, requires setting two hyperparameters: $\alpha$ the p-value threshold and $I$ the number of iterations. When hyperparameter tuning, one should make a trade-off between runtime and quality. On the one hand, there should be enough iterations to avoid false negatives for a given $\alpha$, especially with the anticonservative p-values. On the other hand, adding iterations increases the time complexity. To avoid the need for users to manually optimize these two hyperparameters, \textit{powershap} also has an automatic mode. This automatic mode, automatically determines and optimizes the iteration hyperparameter $I$ using statistical power calculation for $\alpha$, hence the name \textit{powershap}.

The statistical power of a test is $1-\beta$, where $\beta$ is the probability of false negatives. In this case, a false negative is a non-informative feature flagged as an informative one. If a statistical test of a tested sample outputs a p-value $\alpha$, this represents the chance that the tested sample could be flagged as \textit{significant} by chance given the current data. This is calculated using Equation~\ref{eqn:alpha}. If the data in the statistical test is small, it is possible to have a very low $\alpha$ but a large $\beta$, resulting in an output that cannot be trusted. Therefore, for a given $\alpha$, the associated power should be as close to 1 as possible to avoid any false negatives. The power of a statistical test can be calculated using the cumulative distribution function $F$ of the underlying tested distribution $H_1$ using Equation~\ref{eqn:power}. Figure~\ref{power_expl} explains this visually. In the current context, $H_0$ could represent the random feature impact distribution and $H_1$ the tested feature impact distribution.

\begin{equation}\label{eqn:alpha}
    \alpha(x)=F_{H_0}(x)
\end{equation}
\begin{equation}\label{eqn:power}
    Power(\alpha)=F_{H_1}\left(F^{-1}_{H_0}(\alpha)\right)
\end{equation}

\begin{figure}
     \centering
     \includegraphics[width=\textwidth]{figures/power_p_value_explanation.pdf}
     \caption{Visualization of p-value, effect size, and power for a standard t-test.}\label{power_expl}
\end{figure}

The power calculations require the cumulative distribution function $F$. However, the underlying distributions of the calculated feature impacts are unknown. In addition, calculating $F$ heuristically does not enable calculating the required iteration hyperparameter, which is the goal of the automatic mode. \textit{Powershap} circumvents this by mapping the underlying distributions to two standard student-t distributions as visualized in Figure \ref{power_expl}. It first calculates the pooled standard deviation, using Equation~\ref{eqn:pooledstd}, by averaging the standard deviations $\sigma$ of both distributions. It then calculates the distance $d$ between these two distributions, also called the effect size, in terms of this pooled standard deviation using the Cohen's d effect size as detailed in  Equation~\ref{eqn:effectsize}~\cite{lomax_introduction_2007}. Now, it is possible to define two standard student-t distributions with distance $\sqrt{I}\cdot d$ apart and $I-1$ degrees of freedom, where $I$ is the amount of \textit{powershap} iterations. The standard central student-t $F_{CT}$ and non-central student-t $F_{NCT}$ cumulative distribution functions are then used to calculate the power of the statistical test according to Equation~\ref{eqn:ttestpower}. This equation can in turn be used in a heuristic algorithm to solve for $I$. \textit{Powershap} uses the \textit{solve\_power} implementation of \textit{statsmodels} to determine the required $I$ from the TTestPower equation using brentq expansion for a provided required power~\cite{seabold2010statsmodels}. The \textit{powershap} pseudocode for the calculation of the effect size, power, and required iterations is shown in Algorithm~\ref{alg:analysis}.

\begin{equation}\label{eqn:pooledstd}
    PooledStd(\mathbf{s}_1,\mathbf{s}_2)=\frac{\sqrt{(\sigma^2 (\mathbf{s}_1)+\sigma^2 (\mathbf{s}_{2}))}}{2}
\end{equation}
\begin{equation}\label{eqn:effectsize}
    EffectSize(\mathbf{s}_1,\mathbf{s}_2)=\frac{\mu(\mathbf{s}_1)-\mu(\mathbf{s}_2)}{PooledStd(\mathbf{s}_1,\mathbf{s}_2)}
\end{equation}
\begin{equation}\label{eqn:ttestpower}
    TTestPower(\alpha,I,d_{calc})=F_{NCT}\left(F^{-1}_{CT}(\alpha,k=I-1),k=n-1,d=\sqrt{I}\cdot d_{calc}\right)
\end{equation}

\begin{algorithm}
    \DontPrintSemicolon
    \Fn{Analysis($\alpha \leftarrow$ required p-value, $\beta \leftarrow$ required power, $\mathbf{powershap}_{values}$)}{
    $\mathbf{S}_{random}\leftarrow \mathbf{powershap}_{values}[...][m+1]$\;
    $\mathbf{P}\leftarrow$ size $[m]$\;
    $\mathbf{N}_{required}\leftarrow$ size $[m]$\;
    \For{$j\leftarrow 1,2,\dots,m$}{
        $\mathbf{S}_i \leftarrow \mathbf{powershap}_{values}[...][j]$\;
        $\mathbf{P}[j]\leftarrow Percentile(\mathbf{S}_i$, $\mu(S_{random}))$\;
        effectsize $\leftarrow$ EffectSize($\mathbf{S}_i$, $\mathbf{S}_{random}$)\;
        $\mathbf{N}_{required}\leftarrow$ \textbf{SolveTTestPower}(effectsize, $\alpha, \beta$)\;
    }
    \textbf{return} $\mathbf{P}$, $\mathbf{N}_{required}$
    }
    \caption{Powershap analysis function}\label{alg:analysis}
\end{algorithm}

With the calculated required amount of iterations $n$, the automatic \textit{powershap} algorithm can be executed. The pseudocode to enable the automatic mode is shown in Algorithm~\ref{alg:automatic}. As can be seen, this is an expansion of the core algorithm (see Algorithm~\ref{alg:base}) and starts with an initial ten iterations to calculate the initial p-value, effect sizes, power, and required iterations for all features. Then, it searches for the largest required number of iterations $I_{max}$ of all tested features having a p-value below the threshold $\alpha$. If $I_{max}$ exceeds the already performed number of iterations $I_{old}$, automatic mode continues \textit{powershap} for the extra required iterations. This process is repeated until the performed iterations exceed the required iterations. For optimization, when the extra required iterations ($I_{max}-I_{old})$ exceed ten iterations, the automatic mode first adds ten iterations and then re-evaluates the required iterations because the required iterations are influenced by the already performed iterations. Furthermore, it is also possible to provide a stopping criterion on the re-execution of \textit{powershap} to avoid an infinite calculation.

For the automatic mode, by default, $\alpha$ is set to $0.01$ while the required power is set to $0.99$. This results in only selecting features that are more important than the random feature for all iterations. Furthermore, this also compensates for the anticonservative p-value and avoids as many false negatives as possible. Realizing the same desired behavior with the more accurate p-value estimation would require a varying $\alpha$ of $1/n$, complicating the power calculations and increasing the likelihood of false negatives.
The resulting powershap algorithm is implemented in Python as an open-source plug-and-play \textit{sklearn} compatible component. This implementation enables direct usage in conventional machine learning pipelines~\cite{scikit-learn}. The codebase  \footnote{The code and documentation can be found using the following link: \href{https://github.com/predict-idlab/PowerSHAP}{https://github.com/predict-idlab/PowerSHAP}} already supports a wide variety of models, such as linear, tree-based, and even deep learning models. To assure the quality and correctness of the implementation, we tested the functionality using unit testing.

\begin{algorithm}
    \DontPrintSemicolon

    \Fn{Powershap ($M\leftarrow$ Model, $\mathbf{F}_{set} \leftarrow {F_{1},\dots,F_{m}}$, $\mathbf{D}^{n \times m}\leftarrow$ Data, $\alpha \leftarrow$ required p-value, $\beta \leftarrow$ required power)}{

    $\mathbf{powershap}_{values} \leftarrow$\textbf{Explain}($I\leftarrow 10$, $M$, $\mathbf{D}$, $rs\leftarrow 0$)\;

    $\mathbf{P},\mathbf{N}_{required}\leftarrow$\textbf{Analysis}($\alpha$, $\beta$, $\mathbf{powershap}_{values}$)\;

    $I_{max}\leftarrow$ ceil($\mathbf{N}_{required}$[MaxArg($\mathbf{P}<\alpha$)])\;
    $I_{old}\leftarrow 10$\;

    \While{$I_{max}>I_{old}$}{
        \eIf{$I_{max}-I_{old}>10$}{
            $\mathbf{auto}_{values} \leftarrow$\textbf{Explain}($I\leftarrow 10$, $M$, $\mathbf{D}$, $rs\leftarrow 0$)\;
            $I_{old}\leftarrow I_{old}+10$\;
        }
        {
            $\mathbf{auto}_{values} \leftarrow$\textbf{Explain}($I\leftarrow I_{max}-I_{old}$, $M$, $\mathbf{D}$, $rs\leftarrow 0$)\;
            $I_{old}\leftarrow I_{max}$\;

        }
        $\mathbf{powershap}_{values} \leftarrow \mathbf{powershap}_{values} \cup \mathbf{auto}_{values}$\;

        $\mathbf{P},\mathbf{N}_{required}\leftarrow$\textbf{Analysis}($\alpha$, $\beta$, $\mathbf{powershap}_{values}$)\;

        $I_{max}\leftarrow$ ceil($Max(\mathbf{N_{required}}[i, \forall i : \mathbf{P}[i]<\alpha]))$\;

    }

    \textbf{return} $[F_i , \forall$ $i$ : $\mathbf{P}[i]<\alpha]$

    }
    \caption{Automatic Powershap algorithm version}\label{alg:automatic}
\end{algorithm}

\section{Experiments}\label{sec:experiments}
\subsection{Feature Selection Methods}
To facilitate a comparison with other feature selection techniques, we benchmark \textit{powershap} together with other frequently used techniques on both synthetic and real-world datasets. In particular, \textit{powershap} is compared with both filter and wrapper methods, and state-of-the-art shap-based wrapper methods. To provide a fair comparison, all methods, including \textit{powershap}, were used in their default out-of-the-box mode without tuning. For \textit{powershap}, this default mode is the automatic mode.
Concerning filter methods, two methods were chosen: the chi-squared and f-test feature selection from the \textit{sklearn}-library~\cite{scikit-learn}. The chi-squared test measures the dependence between a feature and the classification outcome and assigns a low p-value to features that are not independent of the outcome. As the chi-squared test only works with positive values, the values are shifted in all chi-squared experiments such that all values are positive. This has no effect on tree-estimators as they are invariant to data scaling~\cite{lomax_introduction_2007}. The F-test in \textit{sklearn} is a univariate test that calculates the F-score and p-values on the predictions of a univariate fitted linear regressor with the target~\cite{scikit-learn}. Both filter methods provide p-values that are set to the same threshold as \textit{powershap}.
As wrapper feature selection method, forward feature selection was chosen. This method is a greedy algorithm that starts with an empty set of features and trains a model with each feature separately. In every iteration, forward feature selection then adds the best feature according to a specified metric, often evaluated in cross-validation, until the metric stops improving. This is generally considered a strong method but has a very large time complexity~\cite{jovic_review_2015}.
\textit{Powershap} is also compared to shapicant~\cite{calzolari_manuel-calzolarishapicant_2022} and borutashap~\cite{eoghan_keany_2020_4247618}, two SHAP-based feature selection methods.
The default machine learning model used for all datasets and all feature selection methods, including \textit{powershap}, is a CatBoost gradient boosting tree-based estimator using 250 estimators with the overfitting detector enabled. For classification, the CatBoost model uses adjusted class weights to compensate for any potential class imbalance. The Catboost estimator often results in strong predictive performances out-of-the-box, without any hyper-parameter tuning, making it the perfect candidate for benchmarking and comparison~\cite{prokhorenkova_catboost_2019}. All experiments are performed on a laptop with a Intel(R) Core(TM) i7-9850H CPU at 2.60GHz  processor and 16 GB RAM running at 2667 MHz, with background processes to a minimum.

\subsection{Simulation Dataset}
The methods are first tested on a simulated dataset to assess their ability to discern noise features from informative features. The used simulation dataset is created using the \texttt{make\_classification} function of \textit{sklearn}. This function creates a classification dataset, however, exactly the same can be done for obtaining a regression dataset (by using \texttt{make\_regression}). The simulations are run using 20, 100, 250, and 500 total features to understand the performance on varying dimensions of feature sets. The ratio of informative features is varied as  10\%, 33\%, 50\%, and 90\% of the total feature set, allowing for assessing the quality of the selected features in terms of this ratio. The resulting simulation datasets each contain 5000 samples. Each simulation experiment was repeated five times with different random seeds. The number of redundant features, which are linear combinations of informative features, and the number of duplicate features were set to zero. Redundant features and duplicate features reduce the performance of models, but they cannot be discerned from true informative features as they are inherently informative. Therefore they are not included in the simulation dataset as the goal of \textit{powershap} is to find informative features.  The \textit{powershap} method is compared to shapicant, chi², borutashap, and the f-test for feature selection on this simulation dataset. Due to time complexity constraints, forward feature selection was not included in the simulation benchmarking.

\subsection{Benchmark Datasets}
In addition to the simulation benchmark, the different methods are also evaluated on five publicly available datasets, i.e. three classification datasets: the Madelon~\cite{vanschoren_openml_nodate}, the Gina priori~\cite{vanschoren_openml_nodate-1}, and the Scene dataset~\cite{vanschoren_openml_nodate-2}, and two regression datasets: CT location~\cite{Dua:2019} and Appliances~\cite{Dua:2019}. The details of these datasets are shown in Table~\ref{table:dataset_properties}. The Scene dataset is a multi-label dataset, however, a multi-label problem can always be reduced to a one-vs-all classification problem. Therefore only the label ``Urban" was chosen here to assess binary classification performance.

Almost all of these datasets have a large feature set, ideal for benchmarking feature selection methods. The datasets are split into a training and test set using a $75/25$ split. All methods are evaluated using both 10-fold cross-validation on the training set and 1000 bootstraps on the test set to assess the robustness of the performance. The test set is utilized to assess generalization beyond the validation set as wrapper methods tend to slightly overfit their validation set~\cite{jovic_review_2015}, while the training set is used for feature selection. The forward feature selection method was performed with 5-fold cross-validation and not 10-fold cross-validation due to the high time complexity. A validation set of 20\% of the training set is used for shapicant, using the same validation size as \textit{powershap} in Algorithm~\ref{alg:explain}. The models are evaluated with the AUC metric for classification datasets and with the $R^2$ metric for regression datasets.

\begin{table}
\begin{center}
\caption{Properties of all datasets}
\label{table:dataset_properties}
\setlength{\tabcolsep}{2pt}
\begin{tabular}{cccccc}\toprule

Dataset & Type & Source & \# features & train size & test size \\\midrule
Madelon & Classification & OpenML & 500& 1950 & 650 \\
Gina priori & Classification & OpenML & 784 & 2601 & 867  \\
Scene & Classification & OpenML & 294 & 1805 & 867  \\
CT location & Regression & UCI &  384 & 41347 & 12153  \\
Appliances & Regression & UCI & 30 & 14801 & 4934  \\\bottomrule
\end{tabular}
\end{center}
\end{table}

\section{Results}\label{sec:experiments_results}
\subsection{Simulation Dataset}

The results of the simulation benchmarking are shown in Figure~\ref{fig:total_sim_performance}. Each row of subfigures shows the duration, the percentage of informative features found, and the number of selected noise features.  These measures are shown for each feature selection method for varying feature set dimensions and varying amounts of informative features. As can be seen, the shapicant method is the slowest wrapper method while \textit{powershap} is, without doubt, the fastest wrapper method. The filter methods are substantially faster than any of the wrapper methods, as they do not train models. Furthermore, \textit{powershap} finds all informative features with a limited amount of outputted noise features up to the case with 250 total features with 50\% (125) informative features, outperforming every other method. This can be explained by the model underfitting the data. Even with higher dimensional feature sets, \textit{powershap} finds more informative features than the other methods. Interestingly, most methods do not output many noise features, except for shapicant in the experiment with 20 total and 10\% informative features.

\begin{sidewaysfigure}
    \includegraphics[width=\textwidth]{figures/total_performance_all_methods_5000.pdf}
    \caption{Simulation benchmark results using the \textit{make\_classification} \textit{sklearn} function for 5000 samples with five different \textit{make\_classification} random seeds.}
    \label{fig:total_sim_performance}
\end{sidewaysfigure}

\subsection{Benchmark Datasets}

Table~\ref{table:dataset_results_time} shows the duration of the feature selection methods and the size of the selected feature sets for each method on the different open-source datasets. Chi² does not apply to regression problems and is therefore not included in the results of the CT location and Appliances datasets. The table shows that \textit{powershap} is again the fastest wrapper method, while the number of selected features is in line with the other methods. The filter methods tend to output more features, while forward feature selection outputs a more conservative set of features.

The performance of the selected feature sets for each classification benchmark dataset is shown in Figure~\ref{fig:classification_benchmark} and in Figure~\ref{fig:regression_benchmark} for the regression benchmarks. These figures show that \textit{powershap} provides a steady performance on all datasets, consistently achieving the best or equal performance on both the cross-validation and test sets. However, even in cases with equal performance, \textit{powershap} achieves these performances considerably quicker, especially compared to shapicant and forward feature selection. The CT location dataset performances show that forward feature selection tends to overfit on the cross-validation dataset while \textit{powershap} is more robust.

\begin{table}
\begin{center}
\caption{Benchmarks results for duration and selected features. "default" indicates no feature selection or all features.}
\setlength{\tabcolsep}{4pt}
\label{table:dataset_results_time}
\begin{tabular}{cccccccc}\toprule
 \multicolumn{8}{c}{\textbf{duration (s)}} \\\cmidrule(lr){1-8}
 \textbf{Dataset} & \multicolumn{1}{c}{powershap }  & \multicolumn{1}{c}{borutashap } & \multicolumn{1}{c}{shapicant} & \multicolumn{1}{c}{forward } & \multicolumn{1}{c}{chi²} & \multicolumn{1}{c}{f test}& \multicolumn{1}{c}{default}\\\midrule
\textbf{Madelon} & 132s & 186s & 632s & 10483s & $<1s$ & $<1s$ & N/A \\
\textbf{Gina priori} & 184s & 299s & 812s & 68845s & $<1s$ & $<1s$ & N/A \\
\textbf{Scene} & 115s & 220s & 749s & 12496s & $<1s$ & $<1s$ & N/A \\
\textbf{CT location} & 459s & 543s & 1553s & 56879s &  N/A  & $<1s$ & N/A \\
\textbf{Appliances} & 34s & 48s &134s & 1913s &  N/A & $<1s$ & N/A \\\midrule
\multicolumn{8}{c}{\textbf{selected features}} \\\cmidrule(lr){1-8}

\textbf{Madelon} & 22 & 10 & 30 & 8 & 43 & 18 & 500\\
\textbf{Gina priori} & 105 & 37 & 106 & 26 & 328 & 405 & 784 \\
\textbf{Scene} & 36 & 14 & 56 & 15 & 93 & 220 & 294 \\
\textbf{CT location} & 123 & 162 & 74 & 75 & N/A & 350 & 384 \\
\textbf{Appliances} & 24 & 24 & 10 & 13 & N/A & 20 & 30\\\bottomrule
\end{tabular}
\end{center}
\end{table}

\section{Discussion}\label{sec:discussion}

For the above test results, we used the default automatic \textit{powershap} implementation. However, similar to many other feature selection methods, \textit{powershap} can be further optimized or tuned. One of these optimizations is the use of a \textit{convergence} mode to extract as many informative features as possible. In this mode, \textit{powershap} continues recursively in automatic mode where in every recursive iteration, \textit{powershap} re-executes but with any previously found and selected features excluded from the considered feature set. This process continues, until no more informative features can be found. The convergence mode is especially useful in use-cases with high dimensional feature sets or datasets with a large risk of underfitting as it reduces the feature set dimension each recursive iteration to facilitate finding new informative features. As a basic experiment on the simulation benchmark, using the convergence mode for 500 features and 90\% (450) informative features, the percentage of found features increases from around 38\% (170) to 73\% (330) without adding noise features. However, the duration also increases to the same duration as shapicant.

\begin{figure}
     \centering
     \begin{subfigure}{\textwidth}
         \centering
         \includegraphics[width=\textwidth]{figures/benchmark_performance_all_methods_class.pdf}
         \caption{Classification benchmark dataset performances.}
         \label{fig:classification_benchmark}
     \end{subfigure}

     \begin{subfigure}{\textwidth}
         \centering
         \includegraphics[width=0.66\textwidth]{figures/benchmark_performance_all_methods_regres.pdf}
         \caption{Regression benchmark dataset performances.}
         \label{fig:regression_benchmark}
     \end{subfigure}
    \caption{Benchmark performances. The error bars represent the standard deviation. }
    \label{fig:benchmark_bar_performance}
\end{figure}

Other possible optimizations are also applicable to other feature selection methods, such as applying backward feature selection after \textit{powershap} to eliminate any noise features, redundant, or duplicate features. Another possibility is optimizing the used machine learning model to better match the dataset and rerun \textit{powershap}, e.g. by using more CatBoost estimators for datasets with large sample sizes and high dimensional feature sets.

In the benchmarking results, there are datasets where including all features perform equally well or even better, such as in the case of the Gina prior test set. In these cases, the filter methods perform well but output large feature sets, while the forward feature selection performs the worst. Alternatively, \textit{powershap} can be used here as a fast wrapper-based dimensionality reduction method to retain approximately the same performance with a much smaller feature set. As such, there will still be a trade-off for each use-case between filter and wrapper methods based on time and performance.

We are aware that the current design of the benchmarks has some limitations. For the simulation benchmark, the \texttt{make\_classification} function uses by default a hypercube to create its classification problem, resulting in a linear classification problem, which is inherently easier to classify~\cite{scikit-learn}. Furthermore, wrapper methods, and thus also \textit{powershap}, are highly dependent on the used model, as the feature selection quality suffers from modeling issues such as for example overfitting and underfitting. Therefore, the true potential achievable performances on the benchmark datasets may differ since every use-case and dataset requires its own tuned model to achieve optimal performance. Additionally, the cut-off values and hyperparameters of none of the methods were optimized and are either set to the same value as in \textit{powershap} or used with their default values. This might impact the performance and could have skewed the benchmark results in both directions. However, choosing the same model and the same values for hyperparameters (if possible) in all experiments, reduces potential performance differences and facilitates a fair enough comparison.

\section{Conclusion}\label{sec:conclusion}
We propose \textit{powershap}, a wrapper feature selection method using Shapley values and statistical tests to determine the significance of features. \textit{powershap} uses power calculations to optimize the number of required iterations in an automatic mode to realize fast, strong, and reliable feature selection. Benchmarks indicate that \textit{powershap}'s performance is significantly faster and more reliable than comparable state-of-the-art shap-based wrapper methods. \textit{Powershap} is implemented as an open-source plug-and-play \textit{sklearn} component, increasing its accessibility and ease of use, making it a power-full Shapley feature selection method, ready for your next feature set.

\paragraph{Acknowledgements.}
Jarne Verhaeghe is funded by the Research Foundation Flanders (FWO, Ref. 1S59522N) and designed \textit{powershap}. Jeroen Van Der Donckt implemented \textit{powershap} as an \textit{sklearn} component. Sofie Van Hoecke and Femke Ongenae supervised. A special thanks goes to Gilles Vandewiele for proof-reading the manuscript.

\paragraph{Code.}
The code and documentation can be found using the following link:\\ \href{https://github.com/predict-idlab/PowerSHAP}{https://github.com/predict-idlab/PowerSHAP}

\bibliographystyle{splncs04}

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\cite{ref_article1,ref_book1,ref_proc1,ref_url1}.

redpajama_set_name: RedPajamaArXiv


This reversible Hawaiian print cobbler apron, is far the most beautiful aprons so far. Set against gorgeous blues, the boat is sailing on the gorgeous blue waters of Hawaii.Tropical palm trees can be seen throughout this print, along with the gorgeous Hibiscus flowers.
The other side of this handmade, reversible cobbler apron, is made from a striped tropical print. There are rows of pink Hibiscus flowers, schools of tropical fish, yellow Hibiscus flowers, and dolphins playing in the nearby water.
Pockets on both sides are double stitched in center, so that the pockets don't pull away-adds strength.


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Simpson's Diversity Index: The Diversity Metric You Aren't Tracking Yet
By Eric Knudsen, Ph.D.
What Does Workplace Diversity Look Like in 2018?
How do you currently measure employee diversity in your organization? Like many others, your business probably measures the breakdown of gender, ethnicity, and other employee demographics. Such breakdowns (e.g., 45 percent female / 55 percent male) are very common for organizations to monitor and action as "outcomes" of diversity initiatives (e.g., "Did the introduction of a structured interview process increase diversity in our workplace?"). But how do we know if our diversity efforts have succeeded?
Within Namely's 2018 Workforce Diversity Report, one key table refers to a metric called Simpson's Diversity Index (SDI), a metric that offers organizations a more robust way to "quantify" diversity. Simply put, this index distills the measurement of diversity into a single, trackable metric. Here's how you can get started measuring it today.
What is Simpson's Diversity Index?
Simpson's Diversity Index (SDI) originated as a tool for measuring the diversity of species in an ecosystem––in our case, we'll use it to measure employees in an organization. The metric was designed to capture two critical elements of diversity: richness and evenness. Richness refers to the number of different groups represented (e.g., how many ethnicities are present), while evenness refers to the spread across those groups (e.g., whether employees are spread evenly).
Examples of the Simpson Index for Sample Group Breakdowns:
Simpson Index
Least Diverse 500 0 0 0 0.00
v 500 50 50 50 0.39
Most Diverse 50 50 50 50 0.75
Note: Data is illustrative
The philosophy behind the Simpson Index is that both of these criteria matter. For example, you are probably not a diverse community if only two groups are represented compared with ten groups (i.e., low in richness), and if you have 90 members in one group and one member in each of ten other groups (i.e., low in evenness). The index incorporates both of these criteria in a single, clean snapshot of diversity. The metric ranges in score from zero to one, where zero represents a complete lack of diversity, and one representing (get ready for it…) infinite diversity!
Why use Simpson's Diversity Index?
Although infinite diversity would be fantastic, in organizations we typically work with demographics that have a limited number of groups, such as ethnicity. However, this metric is still useful to HR professionals and their stakeholders because it captures the essence of common diversity measures in a single, reportable number.
It's also readily explained to those who want more detail on what the metric actually represents: the probability that two randomly-selected employees are from different groups. By monitoring this metric as a supplement to traditional breakdowns, you can now more objectively determine if shifts in representation across your company could be considered increases or decreases in diversity.
How do you calculate Simpson's Diversity Index?
The formula for Simpson's Diversity Index is:
D = 1 - ( ( Σ n(n-1) ) / ( N (N-1) ) )
n = number of individuals of each ethnicity
N = total number of individuals of all ethnicities
The value of D ranges between 0 and 1
To make calculating this metric even easier for you, download this free spreadsheet or use our Diversity Index Calculator to simply enter employee counts for each group of the demographic you're interested in studying. You can't change what you don't measure, so get started now!
Calculating Simpson's Diversity Index for your organization can help you gauge just how diverse your organization is, but it won't help you understand how you compare to your competitors and peers. For that added context, you might want to consider using Namely's quarterly benchmarking reports to see just how your business stacks up. In addition to comparing your SDI score to Namely's database of over 1,300 companies, Namely benchmarking reports gives you an in-depth look at your company's diversity, pay equity, and more. Learn more about Namely benchmarking reports here.
Get Your Own Workplace Diversity Reporting Template
Topics: HR, Diversity & Inclusion, HR Data
Eric Knudsen, Ph.D.
Eric Knudsen is a Manager of People Analytics at Namely, the HR, payroll, and benefits platform built for today's employees. Connect with Eric and the Namely team on Twitter, Facebook, and LinkedIn.

redpajama_set_name: RedPajamaCommonCrawl


Annita Georgaca - Vainanidis, Attorney-at-Law, member of the Athens Bar, Supreme Court, Partner.
For eighteen years (1980-1998), Annita was the Chief Legal Counsel of the biggest department stores of Athens with more than 650 employees and several trading departments with diversified legal needs. Resigning from her post, Annita joined the Firm's daily practice in 1998.
Annita has valuable experience in Commercial and Corporate Law, Property Law, Law of Employment, Law of International and Retail Trade, Debt Collection, Litigation and she also deals with Adoptions having an international element.
Annita graduated from the University of Athens with honors.


Description: Deepa, Taapsee, Vijayachander and Raghu are ghosts staying in their ancestral house. They are shown to be waiting for the wife of Vijayachander to come home. They don't know how they were killed. A family visits their house for tenancy but they scare the family away. The broker who deals with the house is also scared as he spots the ghost of Taapsee and informs their presence to the house owner Ramu (Rajiv Kanakala), an NRI and his cop friend Vamshi (Raja Ravindra). Vamshi is apprehensive while Vamshi dismisses it as a concoction while in real, Raja connives with Yadagiri to force Ramu to sell his house at a throwaway price by fabricating and peddling the ghost story.


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Brownfield kan syfta på:

Brownfield, Maine, stad i USA
Brownfield, Texas, stad i USA

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