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Multi (& Mono) Data-Driven Sparse PLS
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	*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

.. image::
	https://raw.githubusercontent.com/hlorenzo/py_ddspls/master/images/reg.png
	:width: 600

For :math:`\\lambda\\approx 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 :math:`\\lambda`.

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 .. image::
	https://raw.githubusercontent.com/hlorenzo/py_ddspls/master/images/cla.png
	:width: 600

One that figure one can see that :math:`\\lambda\\approx 0.45` can be chosen.

**Enjoy :)**