The multi data driven sparse pls package

## Project description

mddspls is the python light package of the data-driven sparse PLS algorithm

In the high dimensional settings (large number of variables), one objective is to select the relevant variables and thus to reduce the dimension. That subspace selection is often managed with supervised tools. However, some data can be missing, compromising the validity of the sub-space selection. We propose a PLS, Partial Least Square, based method, called **dd-sPLS** for data-driven-sparse PLS, allowing jointly variable selection and subspace estimation while training and testing missing data imputation through a new algorithm called Koh-Lanta.

It contains one main class **mddspls** and one associated important method denote **predict** permitting to predict from a new dataset. The function called **perf_mddsPLS** permits to compute cross-validation.

## Data simulation

One might be interested to simulate data and test the package through **regression** and **classification**. Here a spiked model is used:

#!/usr/bin/env python import py_ddspls import numpy as np import sklearn.metrics as sklm n = 100 R_model = 10 R_X_Y = 2 T = 10 L = np.array(np.random.normal(0,1,n*R_model)).reshape((n,R_model)) p_t = 20 q = 5 Xs = {} for t in range(T): Omega_1_2 = np.diag(np.random.uniform(0,1,R_model)) u,s,vh = np.linalg.svd(np.array(np.random.normal(0,1,p_t*p_t)).reshape((p_t,p_t))) U_mod_T = vh[0:R_model,:] Xs[t] = L@Omega_1_2@U_mod_T Omega_y_1_2 = np.diag(np.concatenate((np.ones(R_X_Y),np.zeros(R_model-R_X_Y)))) u,s,vh = np.linalg.svd(np.array(np.random.normal(0,1,R_model*R_model)).reshape((R_model,R_model))) U_mod_T = vh[:,0:q] Y = L@Omega_y_1_2@U_mod_T k_groups = 2 Y_transfor = Y[:,0] lolo = np.linspace(np.min(Y_transfor),np.max(Y_transfor),k_groups+1) Y_bin = np.zeros(n) for ii in range(n): for k_i in range(k_groups): if (Y_transfor[ii]>=lolo[k_i])&(Y_transfor[ii]<lolo[k_i+1]): Y_bin[ii] = k_i if Y_transfor[ii]==lolo[k_groups]: Y_bin[ii] = k_groups-1 pos_0 = np.where(Y_bin==0)[0] pos_1 = np.where(Y_bin==1)[0] Y_classif = np.repeat("Class 2",n) Y_classif[pos_1] = "Class 1" # Missing values are introduced in blocks 1, 2 and 3 Xs[0][0,:] = None Xs[1][1:3,:] = None Xs[2][2:10,:] = None

The dd-sPLS regularization parameter is fixed to 0.6:

lambd=0.6

A train/test dataset is defined for the sack of the example:

id_train = range(30,100) id_test = range(30) Xtrain = {} Ytrain = Y[id_train,:] Xtest = {} for t in range(T): Xtrain[t] = Xs[t][id_train,:] Xtest[t] = Xs[t][id_test,:]

### Regression analysis

Let us produce *2* axes:

R=2

Start model building and tcheck results with sklearn tools:

mod_0=py_ddspls.model.ddspls(Xtrain,Ytrain,lambd=lambd,R=R,mode="reg",verbose=True) Y_est_reg = mod_0.predict(Xtest) print(sklm.mean_squared_error(Y[id_test,:],Y_est_reg))

Leave-one-out cross validation can be performed with built tools, the parameter **NCORES** permits to fix the number of cores to be used in the process

perf_model_reg = py_ddspls.model.perf_ddspls(Xs,Y,R=R,kfolds="loo",n_lambd=10,NCORES=4,mode="reg") print(perf_model_reg) fig = plt.figure() ax = fig.add_subplot(1, 1, 1) cols = ['r','g','b','y','black','brown'] for jj in range(q): ax.plot(perf_model_reg[:,1],perf_model_reg[:,jj+2],cols[jj]) ax.plot(perf_model_reg[:,1], np.sqrt((perf_model_reg[:,2:(2+q)]**2).mean(axis=1)),"brown",linewidth=2,ls="--") plt.legend(np.concatenate((1+np.arange(q),np.array("RMSE of RMSE errors").reshape((1,)))),loc='upper') plt.title('Leave-One-Out Cross-validation error against $\lambda$') plt.xlabel('$\lambda$') plt.ylabel('RMSE') plt.show()

Which returns this kind of graphics

For 0.9 one can find a minimum of the RMSE of the RMSE of each variable. This oservation can be mitigated assuming that only **Y** variables 1 and 4 are well described by the **X** dataset. In that context, a discussion with experts, might help to decide the value to give to the parameter.

### Classification analysis

Let us produce *1* axis since only one group must be discriminated:

R=1

Start model building and tcheck results with sklearn tools:

mod_0_classif=py_ddspls.model.ddspls(Xs,Y_bin,lambd=lambd,R=R,mode="clas",verbose=True) Y_est = mod_0_classif.predict(Xtest) print(sklm.classification_report(Y_est, Y_classif[id_test]=='Class 1'))

Cross validation can be performed with built tools, the parameter **NCORES** permits to use parallellization:

perf_model_class = py_ddspls.model.perf_ddspls(Xs,Y_classif,R=R,kfolds="loo",n_lambd=40,NCORES=7,mode="classif") print(perf_model_class) fig = plt.figure() ax = fig.add_subplot(1, 1, 1) ax.plot(perf_model_class[:,1], perf_model_class[:,2]) plt.title('Leave-One-Out Cross-validation error against $\lambda$') plt.xlabel('$\lambda$') plt.ylabel('Classification Error') plt.show()

Which returns this kind of graphics

One that figure one can see that a parameter approximately equal to 0.45 can be chosen.

**Enjoy :)**

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