Decomposing Data Matrix into Factors Factor Network based segmentation Vertex-level attributes
Use Case 2
Stefan Lyócsa
Masaryk University
<FINTECH MANAGEMENT www.fintedt-ho2020.eu
Štefan Lyócsa FinTech
Decomposing Data Matrix into Factors Factor Network based segmentation Vertex-level attributes
Goal
The goal is to improve credit model by using network variables. Instead of the dissimilarity matrix, we create a matrix using a Factor Network model approach.
• We create an adjacency matrix.
• We calculate vertex level network variables.
• We augment credit models with new variables.
• We compare the forecasting accuracy.
Stefan Lyöcsa FinTech
	Decomposing Data Matrix into Factors
	Factor Network based segmentation
	Vertex-level attributes
We assume that loan characteristics (size, duration, ...) are driven by some unobserved (latent) factors. We can think of these factors are variables that more accurately (with less noise) loan characteristics. Let n by the number of loans, p number of loan
characteristics. X be a n x p matrix, and k an arbitrary number of factors. The factor representation of X is:
X = FW + e
Here F is a n x k matrix of factors, W a kxp matrix of coefficients and € a nxp matrix of residuals. Factors and residuals are mutually independent and normally distributes.
Stefan Lyöcsa FinTech
Decomposing Data Matrix into Factors Factor Network based segmentation Vertex-level attributes
Example
Let's have three loan characteristics, loan size, loan duration, interest rate (p = 3), and two underlying (unobserved) factors (k = 2). A loan with values of 1000 (size), 36 (duration), 15.5 (interest rate) could be decomposed into:
1000 = + /i,2^2,i +
36 = /i,iWi,2 + /l,2^2,2 + ^1,2 15.5 = /i,iWi,3 + /l,2^2,3 + ^1,3
Stefan Lyöcsa FinTech
Decomposing Data Matrix into Factors Factor Network based segmentation Vertex-level attributes
Example
1000 = + /i,2^2,i +
36 = /i,iWi,2 + /l,2^2,2 + ^1,2 15.5 = /l,l^l,3 + /l,2^2,3 + ^1,3
Instead of three variables, we have two (Factor 1 and 2). The factors are unique to each loan, while coefficients are common to each loan. We use the two factors to re-construct the three values of loan characteristics. The more we can re-construct with as few factors as possible, the better. Factors are estimated using singular value decomposition approach.
Stefan Lyöcsa FinTech
	Decomposing Data Matrix into Factors
	Factor Network based segmentation
	Vertex-level attributes
How to create a network? We are going to use previously estimated factors - instead of variables. Only if two loans have similar factor values they should be connected. Again, we hope that in the network bad loans are going to be clustered together.
Let's have a binary adjacency matrix G e {0, l}nxn7 where:
> 7) and gij = 0 otherwise
where $ is the cumulative distribution function of the normal distribution. We follow the previous literature and choose 7 = 0.10 and fl = <l>-i(_L_).
9i.
3
= 1 if
= $ e + {FFT)hJ
Stefan Lyöcsa FinTech
Decomposing Data Matrix into Factors Factor Network based segmentation Vertex-level attributes
Create network in R
Import data again (but do not delete anything from previous Use Case). We arbitrarily select variables that we think might identify a bad loan:
• X = DT[,c(;int;,;durm;,;linctot;,;noliab;)] Run the following function:
• AM = FN_SVD(X,p=0.75,gam=0.10)
• g = graph.from_adjacencyjiiatrix(AM, mode = ;undirected; , weighted = TRUE)
We can visualize the Network Factor Model:
• plot(g, graph = ;NFM;, vertex.label=NA,vertex.size
= 3, main = ;Network factor model of the P2P applicants networks;)
Stefan Lyöcsa FinTech
Decomposing Data Matrix into Factors Factor Network based segmentation Vertex-level attributes
Create the network in R
Network factor model of the P2P applicants networks
Stefan Lyöcsa FinTech
Decomposing Data Matrix into Factors Factor Network based segmentation Vertex-level attributes
• vertex degree,
• harmonic centrality,
• Community detection - Louvain method.
To address the issue of isolated vertices, one can assume that the shortest distance between vertex i and an isolated vertex j is oc, while conveniently assuming that 1/oc = 0. Harmonic centrality is therefore:
where d(i,j) is the shortest path from vertex i to vertex j in the network.
Stefan Lyöcsa FinTech
Decomposing Data Matrix into Factors Factor Network based segmentation Vertex-level attributes
Estimating vertex level attributes in R
The following function calculates centrality and community:
• NetDscr=BVC(g)
Now add variable into the model:
• DT$Deg = NetDscr$VCentrality[,l]
• DT$Hac = NetDscr$VCentrality[,2]
• DT = data.frajne(DT,NetDscr$Community)
Stefan Lyöcsa FinTech
Preparing data
Decomposing Data Matrix into Factors Factor Network based segmentation Vertex-level attributes
Define the matrix of input and output variables:
• indep = as.matrix(DT[1:(N-NF),c(;new;,;ver3;,;ver4;,;lfi; ;undG;,;female;,;lamt;,;int;,;durm;,;educprim;,;educbasic;. ;educvocat;,;educsec;,;msmar;,;msco;,;mssi;,;msdi;,;nrodep: ;espem;,;esfue;,;essem;,;esent;,;esret;,;dures;,;exper;, ;linctot;,;noliab;,;lliatot;,;norli;,;noplo;,;lamountplo;,
;lamntplr;,;lamteprl;,;nopearlyrep;,;Deg;,;Hac;,paste(;g;J
• dep = DT[1:(N-NF),;RR2;]
Stefan Lyöcsa FinTech
Preparing data
Decomposing Data Matrix into Factors Factor Network based segmentation Vertex-level attributes
• pred = as.matrix(DT[(N-NF+1):N,c(;new;,;ver3;,;ver4;,;lfj ;undG;,;female;,;lamt;,;int;,;durm;,;educprim;,;educbasic;. ;educvocat;,;educsec;,;msmar;,;msco;,;mssi;,;msdi;,;nrodep: ;espem;,;esfue;,;essem;,;esent;,;esret;,;dures;,;exper;, ;linctot;,;noliab;,;lliatot;,;norli;,;noplo;,;lamountplo;,
;lamntplr;,;lamteprl;,;nopearlyrep;,;Deg;,;Hac;,paste(;g;J
• ytrue = DT[(N-NF+1):N,;RR2;]
Stefan Lyöcsa FinTech
LASSO model
Decomposing Data Matrix into Factors Factor Network based segmentation Vertex-level attributes
Model estimation:
• m3_L = cv.glmnet(x=indep,y=dep,nfolds=30,alpha=l)
• coef (m3_L,s=;lambda.lse;)
Forecast loan returns:
• yhat = predict (m3_L,newx=pred,s=m3_L$lambda.lse)
Calculate mean squared error:
• LASSCLFN = mean((yhat-ytrue)2)
Stefan Lyöcsa FinTech
RIDGE model
Decomposing Data Matrix into Factors Factor Network based segmentation Vertex-level attributes
Model estimation:
• m3_R = cv.glmnet(x=indep,y=dep,nfolds=30,alpha=0)
• coef (m3_R,s=;lambda.lse;)
Forecast loan returns:
• yhat = predict (m3_R,newx=pred,s=m3_R$lambda.lse)
Calculate mean squared error:
• RIDGE_FN = mean((yhat-ytrue)2)
Stefan Lyöcsa FinTech
Elastic net model
Decomposing Data Matrix into Factors Factor Network based segmentation Vertex-level attributes
• m3_E25 = cv.glmnet(x=indep,y=dep,nfolds=30,alpha=0.25)
• m3_E50 = cv.glmnet(x=indep,y=dep,nfolds=30,alpha=0.50)
• m3_E75 = cv.glmnet(x=indep,y=dep,nfolds=30,alpha=0.75)
• yhat = predict(m3_E25 ,newx=pred,s=m3_E25$lambda.lse)
• yhat = predict(m3_E50 ,newx=pred,s=m3_E50$lambda.lse)
• yhat = predict(m3_E75 ,newx=pred,s=m3_E75$lambda.lse)
• EN25FN = mean((yhat-ytrue)2)
• EN50FN = mean((yhat-ytrue)2)
• EN75FN = mean((yhat-ytrue)2)
Stefan Lyöcsa FinTech
Decomposing Data Matrix into Factors Factor Network based segmentation Vertex-level attributes
Comparing forecast accuracy
Is network approach worth the strugle?
• MSEs = c(0LS, LASSO, RIDGE, EN25, EN50, EN75, LASSCLN, RIDGE_N, EN25N, EN50N, EN75N, LASSCLFN, RIDGE_FN, EN25FN, EN50FN, EN75FN)
• names(MSEs) =c('0LS',  'LASSO',  'RIDGE', 'EN25',
'EN50',  'EN75',  'LASS0_N' ,  'RIDGE_N' ,  'EN25N',  'EN50N', 'EN75N',  'LASS0_FN','RIDGE_FN' ,  'EN25FN',  'EN50FN', 'EN75FN')
• MSEs = sort(MSEs)
• cbind(MSEs)
Stefan Lyöcsa FinTech
Decomposing Data Matrix into Factors Factor Network based segmentation Vertex-level attributes
Comparing forecast accuracy
the factor network approach worth the struggle?
MSEs
EN75FN 848.6278 EN50FN 849.5642 RIDGE_FN 850.8609 LA550_N 856.3854 EN25N 858.0379 LASSO_FN 859.8096 RIDGE_N 860.3076 EN50N 862.0838 EN75N 864.3380 EN25FN 864.3384 EN50 870.3685 LASSO 871.5219 EN75 874.5317 EN25 874.8249 RIDGE 929.6943 OLS 995.6180
Stefan Lyöcsa FinTech
Decomposing Data Matrix into Factors Factor Network based segmentation Vertex-level attributes
Use Case 2
Stefan Lyócsa
Masaryk University
<FINTECH MANAGEMENT www.fintedt-ho2020.eu
Štefan Lyócsa FinTech