Shapley Decomposition

This package consists of two applications of shapley values in descriptive analysis: 1) a generalized module for decomposing change over time, using shapley values¹ (initially influenced by the World Bank's Job Structure tool²) and 2) shapley and owen values based decomposition of R^2 (contribution of independent variables to a goodness of fit metric -R^2 in this case-) for linear regression models³.

Notes

Identities or functions with independently moving variables have independent contributions to the result as well. Therefore this module is better useful for functions or identities with dependently moving variables (though it works as well for the independent movements). It should be noted that being able to decompose the contribution of variables doesn't mean that the results are always clearly interpretable. Many features of variables like; scale, dependency mode, change dynamics (slow paced/fast paced, instant/lagged), etc. deserves attention when interpreting their individual contribution to the change or result.

Both for the first and second application, the computation time increases exponentially as the number of variables increase. This is the result of powersets and so 2^n calculations.

Shapley value:

$v(i) = \sum \limits _{S \subseteq M \setminus i} \phi(s) \cdot [V(S \cup {i})-V(S)]$

$\phi(s) = (m-1-s)! \cdot s!/m!$

where $i \in M$ and M is the main set of variables and $m=|M|, s=|S|$. For shapley change decomposition, $[V(S \cup {i_{t_1} })-V(S\cup {i_{t_0} })]$ and s is the number of variables with $t_1$ instance.

Owen value:

$o(i) = \sum \limits _{R \subseteq N \setminus k} \sum \limits _{T \subseteq B_k \setminus i} \phi(r) \cdot \omega(t) \cdot [V(Q \cup T \cup {i})-V(Q \cup T)]$

$\phi(r) = (n-1-r)! \cdot r!/n!$

$\phi(t) = (b_k-1-t)! \cdot t!/b_k!$

where $i \in M$ and M is the main set of variables. N is the powerset of coalition/group set composed of i individuals. $Q = \bigcup_{r \in R}B_r$ and $n=|N|, r=|R|, b_k=|B_k|, t=|T|$.

Installation

Run the following to install

pip install shapley_decomposition

Workings

shapley_decomposition.shapley_change module consists of three functions: samples(), shapley_values() and decomposition(). shapley_change.samples(dataframe) returns powerset pairs that model uses. shapley_change.shapley_values(dataframe, "your function") returns weighted differences for each variable, sum of which gives the shapley value. shapley_change.decomposition(dataframe, "your function") returns decomposed total change by variable contributions. These functions of shapley_change module accepts either or both of the data and function inputs:

1. The structure of input data is important. Module accepts pandas dataframes or 2d arrays:

• If pandas dataframe is used as input, both the dependent variable and the independent variables should be presented in the given format (variable names as index and years as columns):

  	year1	year2
  y	y_value	y_value
  x1	x1_value	x1_value
  x2	x2_value	x2_value
  ...	...	...
  xn	xn_value	xn_value

• If an array is preferred, note that module will convert it to a pandas dataframe format and expects y and xs in the following order:

  [[y_value,y_value],
    [x1_value,x1_value],
    [x2_value,x2_value]]
    ...
  

2. Identity or function defines the relation between xs and y. Due to the characteristic of shapley decomposition the sum of xs' contributions should be equal to y, i.e. exact (with plus minus 0.0001 freedom in this module due to the residue of arithmetic operations), therefore no place for residuals, or an input relation that fails to create the given y will shoot a specific error. Function input is expected in text format. It is evaluated by a custom syntax parser (as the eval() function has its security risks). Expected format for the function input is the right hand side of the equation:

   • "x1+x2*(x3/x4)**x5"
   • "(x1+x2)*x3+x4"
   • "x1*x2**2"

   All arithmetic operators and paranthesis operations are usable:

   • "+" , "-" , "*" , "/" or "÷", "**" or "^"

3. If shapley_change.decomposition(df,"your function", cagr=True) is called, a yearly_growth (using compound annual growth rate - cagr) column will be added, which will index the decomposition to cagr of the y. Default is cagr=False.

The shapley_decomposition.shapley_r2 module consists of three functions as well: samples(), shapley_decomposition() and owen_decomposition. shapley_r2.samples(dataframe) returns powerset variable pairs that model uses. shapley_r2.shapley_decomposition(dataframe) returns the decomposition of model r^2 to the contributions of variables. shapley_r2.owen_decomposition(dataframe, [["x1","x2"],[..]]) returns the owen decomposition of model r^2 to the contributions of variables and groups/coalitions. Input features expected by shapley_r2 functions are as:

1. The expected format for the input dataframe or array is:

	x1	x2	...	xn	y
0	x1_value	x2_value	...	xn_value	y_value
1	x1_value	x2_value	...	xn_value	y_value
2	x1_value	x2_value	...	xn_value	y_value
...	...	...	...	...	...
n	x1_value	x2_value	...	xn_value	y_value

2. shapley_r2.owen_decomposition expects the group/coalition structure as the second input. This input should be a list of list showing the variables grouped within coalition/group lists. For example a model of 8 variables, x1,x2,...,x8 has three groups/coalitions which are formed as group1:(x1,x2,x3), group2:(x4) and group3:(x5,x6,x7,x8). Then the second input of owen_decomposition should be [["x1","x2","x3"],["x4"],["x5","x6","x7","x8"]]. Even if it is a singleton like group2 which has only x4, variable name should be in a list. If every group is a singleton, then the owen values will be equal to shapley values.

3. As the computation time increases exponentially with the number of variables. For the shapley_decomposition function a default upper variable limit of 10 variables has been set. Same limit applies for owen_decomposition but as the number of groups, not individual variables. However in users' own discretion more variables can be forced by calling the function as shapley_r2.shapley_decomposition(df, force=True) or shapley_r2.owen_decomposition(df, [groups], force=True).

Examples

1. As the first influence for the model was from WB's Job Structure, accordingly first example is decomposition of change in value added per capita of Turkey from 2000 to 2018 according to "x1*x2*x3*x4" where x1 is value added per worker, x2 is employment rate, x3 is participation rate, x4 is share of 15-64 population in total population. This is an identity.

import pandas
from shapley_decomposition import shapley_change

df=pandas.DataFrame([[8237.599210,15026.707520],[27017.637990,43770.525560],[0.935050,0.891050],[0.515090,0.57619],[0.633046,0.668674]],index=["val_ad_pc","val_ad_pw","emp_rate","part_rate","working_age"], columns=[2000,2018])
print(df)

	2000	2018
val_ad_pc	8237.599210	15026.707520
val_ad_pw	27017.637990	43770.525560
emp_rate	0.935050	0.891050
part_rate	0.515090	0.57619
part_rate	0.633046	0.668674

shapley_change.decomposition(df,"x1*x2*x3*x4")

	2000	2018	dif	shapley	contribution
val_ad_pc	8237.599210	15026.707520	6789.108310	6789.108310	1.000000
val_ad_pw	27017.637990	43770.525560	16752.887570	5431.365538	0.800012
emp_rate	0.935050	0.891050	-0.044000	-556.985657	-0.082041
part_rate	0.515090	0.576190	0.061100	1285.200011	0.189303
working_age	0.633046	0.668674	0.035628	629.528410	0.092726

2. Second example is the decomposition of change in non-parametric skewness of a normally distributed sample, after the sample is altered with additional data. We are trying to understand how the change in mean, median and standard deviation contributed to the change in skewness parameter. Non parametric skewness is calculated by "(x1-x2)/x3", (mean-median)/standard deviation.

import numpy as np
import pandas
from shapley_decomposition import shapley_change

np.random.seed(210)

data = np.random.normal(loc=0, scale=1, size=100)

add = [np.random.uniform(min(data), max(data)) for m in range(5,10)]

altered_data = np.concatenate([data,add])

med1, med2 = np.median(data), np.median(altered_data)
mean1, mean2 = np.mean(data), np.mean(altered_data)
std1, std2 = np.std(data, ddof=1), np.std(altered_data, ddof=1)
sk1 = (np.mean(data)-np.median(data))/np.std(data, ddof=1)
sk2 = (np.mean(altered_data)-np.median(altered_data))/np.std(altered_data, ddof=1)

df=pandas.DataFrame([[sk1,sk2],[mean1,mean2],[med1,med2],[std1,std2]], columns=["0","1"], index=["non_par_skew","mean","median","std"])

shapley_change.decomposition(df,"(x1-x2)/x3")

	0	1	dif	shapley	contribution
non_par_skew	0.065803	0.044443	-0.021359	-0.021359	1.000000
mean	-0.247181	-0.285440	-0.038259	-0.036146	1.692288
median	-0.315957	-0.333088	-0.017131	0.016184	-0.757719
std	1.045188	1.072090	0.026902	-0.001398	0.065432

3. Third example uses shapley_r2 decomposition with the fish market database from kaggle⁴:

import numpy as np
import pandas
from shapley_decomposition import shapley_r2

df=pandas.read_csv("Fish.csv")
#ignoring the species column
shapley_r2.shapley_decomposition(df.iloc[:,1:])

	shapley_values	contribution
Length1	0.194879	0.220131
Length2	0.195497	0.220829
Length3	0.198097	0.223766
Height	0.116893	0.132040
Width	0.179920	0.203233

#using the same dataframe

groups = [["Length1","Length2","Length3"],["Height","Width"]]

shapley_r2.owen_decomposition(df.iloc[:,1:], groups)

	owen_values	contribution	group_owen
Length1	0.157523	0.177934	b1
Length2	0.158178	0.178674	b1
Length3	0.160276	0.181045	b1
Height	0.141092	0.159374	b2
Width	0.268218	0.302972	b2

	owen_values	contribution
b1	0.475977	0.537653
b2	0.409309	0.462347

# if the species are included as categorical variables

from sklearn.preprocessing import OneHotEncoder

enc = OneHotEncoder(handle_unknown="ignore")
encti = enc.fit_transform(df[["Species"]]).toarray()

df2 = pandas.concat([pandas.DataFrame(encti,columns=["Species"+str(n) for n in range(len(encti[0]+1))]),df.iloc[:,1:]],axis=1)
shapley_r2.shapley_decomposition(df2, force=True) # as the number of variables bigger than 10, force=True

groups=[["Species0","Species1","Species2","Species3","Species4","Species5","Species6"],["Length1","Length2","Length3"],["Height","Width"]]

shapley_r2.owen_decomposition(df2, groups) #no need for force as the number of groups does not exceed 10

Footnotes

1. https://www.rand.org/content/dam/rand/pubs/papers/2021/P295.pdf ↩

2. https://datatopics.worldbank.org/jobsdiagnostics/jobs-tools.html ↩

3. https://www.scitepress.org/papers/2017/61137/ ↩

4. https://www.kaggle.com/datasets/aungpyaeap/fish-market?resource=download ↩