{-| Module : Gargantext.Core.Methods.Distances.Accelerate.Distributional Description : Copyright : (c) CNRS, 2017-Present License : AGPL + CECILL v3 Maintainer : team@gargantext.org Stability : experimental Portability : POSIX * Distributional Distance metric __Definition :__ Distributional metric is a relative metric which depends on the selected list, it represents structural equivalence of mutual information. __Objective :__ We want to compute with matrices processing the similarity between term $i$ and term $j$ : distr(i,j)=$\frac{\Sigma_{k \neq i,j} min(\frac{n_{ik}^2}{n_{ii}n_{kk}},\frac{n_{jk}^2}{n_{jj}n_{kk}})}{\Sigma_{k \neq i}\frac{n_{ik}^2}{ n_{ii}n_{kk}}}$ where $n_{ij}$ is the cooccurrence between term $i$ and term $j$ * For a vector V=[$x_1$ ... $x_n$], we note $|V|_1=\Sigma_ix_i$ * operator : .* and ./ cell by cell multiplication and division of the matrix * operator * is the matrix multiplication * Matrice M=[$n_{ij}$]$_{i,j}$ * opérateur : Diag(M)=[$n_{ii}$]$_i$ (vecteur) * Id= identity matrix * O=[1]$_{i,j}$ (matrice one) * D(M)=Id .* M * O * D(M) =[$n_{jj}$]$_{i,j}$ * D(M) * O =[$n_{ii}$]$_{i,j}$ * $V_i=[0~0~0~1~0~0~0]'$ en i * MI=(M ./ O * D(M)) .* (M / D(M) * O ) * distr(i,j)=$\frac{|min(V'_i * (MI-D(MI)),V'_j * (MI-D(MI)))|_1}{|V'_i.(MI-D(MI))|_1}$ [Specifications written by David Chavalarias on Garg v4 shared NodeWrite, team Pyremiel 2020] -} {-# LANGUAGE TypeFamilies #-} {-# LANGUAGE TypeOperators #-} {-# LANGUAGE ScopedTypeVariables #-} {-# LANGUAGE ViewPatterns #-} module Gargantext.Core.Methods.Distances.Accelerate.Distributional where -- import qualified Data.Foldable as P (foldl1) -- import Debug.Trace (trace) import Data.Array.Accelerate as A import Data.Array.Accelerate.Interpreter (run) import Gargantext.Core.Methods.Matrix.Accelerate.Utils import qualified Gargantext.Prelude as P -- | `distributional m` returns the distributional distance between terms each -- pair of terms as a matrix. The argument m is the matrix $[n_{ij}]_{i,j}$ -- where $n_{ij}$ is the coocccurrence between term $i$ and term $j$. -- -- ## Basic example with Matrix of size 3: -- -- >>> theMatrixInt 3 -- Matrix (Z :. 3 :. 3) -- [ 7, 4, 0, -- 4, 5, 3, -- 0, 3, 4] -- -- >>> distributional $ theMatrixInt 3 -- Matrix (Z :. 3 :. 3) -- [ 1.0, 0.0, 0.9843749999999999, -- 0.0, 1.0, 0.0, -- 1.0, 0.0, 1.0] -- -- ## Basic example with Matrix of size 4: -- -- >>> theMatrixInt 4 -- Matrix (Z :. 4 :. 4) -- [ 4, 1, 2, 1, -- 1, 4, 0, 0, -- 2, 0, 3, 3, -- 1, 0, 3, 3] -- -- >>> distributional $ theMatrixInt 4 -- Matrix (Z :. 4 :. 4) -- [ 1.0, 0.0, 0.5714285714285715, 0.8421052631578947, -- 0.0, 1.0, 1.0, 1.0, -- 8.333333333333333e-2, 4.6875e-2, 1.0, 0.25, -- 0.3333333333333333, 5.7692307692307696e-2, 1.0, 1.0] -- distributional :: Matrix Int -> Matrix Double distributional m' = run result where m = map fromIntegral $ use m' n = dim m' diag_m = diag m d_1 = replicate (constant (Z :. n :. All)) diag_m d_2 = replicate (constant (Z :. All :. n)) diag_m mi = (.*) ((./) m d_1) ((./) m d_2) -- w = (.-) mi d_mi -- The matrix permutations is taken care of below by directly replicating -- the matrix mi, making the matrix w unneccessary and saving one step. w_1 = replicate (constant (Z :. All :. n :. All)) mi w_2 = replicate (constant (Z :. n :. All :. All)) mi w' = zipWith min w_1 w_2 -- The matrix ii = [r_{i,j,k}]_{i,j,k} has r_(i,j,k) = 0 if k = i OR k = j -- and r_(i,j,k) = 1 otherwise (i.e. k /= i AND k /= j). ii = generate (constant (Z :. n :. n :. n)) (lift1 (\(Z :. i :. j :. k) -> cond ((&&) ((/=) k i) ((/=) k j)) 1 0)) z_1 = sum ((.*) w' ii) z_2 = sum ((.*) w_1 ii) result = termDivNan z_1 z_2 logDistributional :: Matrix Int -> Matrix Double logDistributional m = run $ diagNull n $ matMiniMax $ logDistributional' n m where n = dim m logDistributional' :: Int -> Matrix Int -> Acc (Matrix Double) logDistributional' n m' = result where m = map fromIntegral $ use m' -- Scalar. Sum of all elements of m. to = the $ sum (flatten m) -- Diagonal matrix with the diagonal of m. d_m = (.*) m (matrixIdentity n) -- Size n vector. s = [s_i]_i s = sum ((.-) m d_m) -- Matrix nxn. Vector s replicated as rows. s_1 = replicate (constant (Z :. All :. n)) s -- Matrix nxn. Vector s replicated as columns. s_2 = replicate (constant (Z :. n :. All)) s -- Matrix nxn. ss = [s_i * s_j]_{i,j}. Outer product of s with itself. ss = (.*) s_1 s_2 -- Matrix nxn. mi = [m_{i,j}]_{i,j} where -- m_{i,j} = 0 if n_{i,j} = 0 or i = j, -- m_{i,j} = log(to * n_{i,j} / s_{i,j}) otherwise. mi = (.*) (matrixEye n) (map (lift1 (\x -> cond (x == 0) 0 (log (x * to)))) ((./) m ss)) -- Tensor nxnxn. Matrix mi replicated along the 2nd axis. w_1 = replicate (constant (Z :. All :. n :. All)) mi -- Tensor nxnxn. Matrix mi replicated along the 1st axis. w_2 = replicate (constant (Z :. n :. All :. All)) mi -- Tensor nxnxn. w' = zipWith min w_1 w_2 -- A predicate that is true when the input (i, j, k) satisfy -- k /= i AND k /= j k_diff_i_and_j = lift1 (\(Z :. i :. j :. k) -> ((&&) ((/=) k i) ((/=) k j))) -- Matrix nxn. sumMin = sum (condOrDefault k_diff_i_and_j 0 w') -- Matrix nxn. All columns are the same. sumM = sum (condOrDefault k_diff_i_and_j 0 w_1) result = termDivNan sumMin sumM -- The distributional metric P(c) of @i@ and @j@ terms is: \[ -- S_{MI} = \frac {\sum_{k \neq i,j ; MI_{ik} >0}^{} \min(MI_{ik}, -- MI_{jk})}{\sum_{k \neq i,j ; MI_{ik}>0}^{}} \] -- -- Mutual information -- \[S_{MI}({i},{j}) = \log(\frac{C{ij}}{E{ij}})\] -- -- Number of cooccurrences of @i@ and @j@ in the same context of text -- \[C{ij}\] -- -- The expected value of the cooccurrences @i@ and @j@ (given a map list of size @n@) -- \[E_{ij}^{m} = \frac {S_{i} S_{j}} {N_{m}}\] -- -- Total cooccurrences of term @i@ given a map list of size @m@ -- \[S_{i} = \sum_{j, j \neq i}^{m} S_{ij}\] -- -- Total cooccurrences of terms given a map list of size @m@ -- \[N_{m} = \sum_{i,i \neq i}^{m} \sum_{j, j \neq j}^{m} S_{ij}\] -- distributional'' :: Matrix Int -> Matrix Double distributional'' m = -- run {- $ matMiniMax -} run $ diagNull n $ rIJ n $ filterWith 0 100 $ filter' 0 $ s_mi $ map fromIntegral {- from Int to Double -} $ use m {- push matrix in Accelerate type -} where _ri :: Acc (Matrix Double) -> Acc (Matrix Double) _ri mat = mat1 -- zipWith (/) mat1 mat2 where mat1 = matSumCol n $ zipWith min (_myMin mat) (_myMin $ filterWith 0 100 $ diagNull n $ transpose mat) _mat2 = total mat _myMin :: Acc (Matrix Double) -> Acc (Matrix Double) _myMin = replicate (constant (Z :. n :. All)) . minimum -- TODO fix NaN -- Quali TEST: OK s_mi :: Acc (Matrix Double) -> Acc (Matrix Double) s_mi m' = zipWith (\x y -> log (x / y)) (diagNull n m') $ zipWith (/) (crossProduct n m') (total m') -- crossProduct n m' total :: Acc (Matrix Double) -> Acc (Matrix Double) total = replicate (constant (Z :. n :. n)) . sum . sum n :: Dim n = dim m rIJ :: (Elt a, Ord a, P.Fractional (Exp a), P.Num a) => Dim -> Acc (Matrix a) -> Acc (Matrix a) rIJ n m = matMiniMax $ divide a b where a = sumRowMin n m b = sumColMin n m -- * For Tests (to be removed) -- | Test perfermance with this matrix -- TODO : add this in a benchmark folder distriTest :: Int -> Matrix Double distriTest n = logDistributional (theMatrixInt n)