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        "author": "Hadrien Lorenzo",
        "author_email": "hadrien.lorenzo.2015@gmail.com",
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        "description": "=====================================\nMulti (& Mono) Data-Driven Sparse PLS\n=====================================\n\n\t*mddspls is the python light package of the data-driven sparse PLS algorithm*\n\nIn 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.\n\nIt 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.\n\nData simulation\n===============\nOne might be interested to simulate data and test the package through **regression** and **classification**. Here a spiked model is used::\n\n\t#!/usr/bin/env python\n\n\timport py_ddspls\n\timport numpy as np\n\timport sklearn.metrics as sklm\n\t\n\tn = 100\n\tR_model = 10\n\tR_X_Y = 2\n\tT = 10\n\tL = np.array(np.random.normal(0,1,n*R_model)).reshape((n,R_model))\n\n\tp_t = 20\n\tq = 5\n\n\tXs = {}\n\n\tfor t in range(T):\n\t\tOmega_1_2 = np.diag(np.random.uniform(0,1,R_model))\n\t\tu,s,vh = np.linalg.svd(np.array(np.random.normal(0,1,p_t*p_t)).reshape((p_t,p_t)))\n\t\tU_mod_T = vh[0:R_model,:]\n\t\tXs[t] = L@Omega_1_2@U_mod_T\n\n\tOmega_y_1_2 = np.diag(np.concatenate((np.ones(R_X_Y),np.zeros(R_model-R_X_Y))))\n\tu,s,vh = np.linalg.svd(np.array(np.random.normal(0,1,R_model*R_model)).reshape((R_model,R_model)))\n\tU_mod_T = vh[:,0:q]\n\tY = L@Omega_y_1_2@U_mod_T\n\n\tk_groups = 2\n\tY_transfor = Y[:,0]\n\tlolo = np.linspace(np.min(Y_transfor),np.max(Y_transfor),k_groups+1)\n\tY_bin = np.zeros(n)\n\tfor ii in range(n):\n\t\tfor k_i in range(k_groups):\n\t\t\tif (Y_transfor[ii]>=lolo[k_i])&(Y_transfor[ii]<lolo[k_i+1]):\n\t\t\t\tY_bin[ii] = k_i\n\t\t\tif Y_transfor[ii]==lolo[k_groups]:\n\t\t\t\tY_bin[ii] = k_groups-1\n\n\tpos_0 = np.where(Y_bin==0)[0]\n\tpos_1 = np.where(Y_bin==1)[0]\n\tY_classif = np.repeat(\"Class 2\",n)\n\tY_classif[pos_1] = \"Class 1\"\n\n\t# Missing values are introduced in blocks 1, 2 and 3\n\tXs[0][0,:] = None\n\tXs[1][1:3,:] = None\n\tXs[2][2:10,:] = None\n\n\nThe dd-sPLS regularization parameter is fixed to 0.6::\n\n\tlambd=0.6\n\nA train/test dataset is defined for the sack of the example::\n\n\tid_train = range(30,100)\n\tid_test = range(30)\n\tXtrain = {}\n\tYtrain = Y[id_train,:]\n\tXtest = {}\n\tfor t in range(T):\n\t\tXtrain[t] = Xs[t][id_train,:]\n\t\tXtest[t] = Xs[t][id_test,:]\n\nRegression analysis\n-------------------\n\nLet us produce *2* axes::\n\n\tR=2\n\nStart model building and tcheck results with sklearn tools::\n\n\tmod_0=py_ddspls.model.ddspls(Xtrain,Ytrain,lambd=lambd,R=R,mode=\"reg\",verbose=True)\n\tY_est_reg = mod_0.predict(Xtest)\t\t\n\tprint(sklm.mean_squared_error(Y[id_test,:],Y_est_reg))\n\nLeave-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 ::\n\n\tperf_model_reg = py_ddspls.model.perf_ddspls(Xs,Y,R=R,kfolds=\"loo\",n_lambd=10,NCORES=4,mode=\"reg\")\n\tprint(perf_model_reg)\n\t\n\tfig = plt.figure()\n\tax = fig.add_subplot(1, 1, 1)\n\tcols = ['r','g','b','y','black','brown']\n\tfor jj in range(q):\n\t\tax.plot(perf_model_reg[:,1],perf_model_reg[:,jj+2],cols[jj])\n\n\tax.plot(perf_model_reg[:,1],\n\t\tnp.sqrt((perf_model_reg[:,2:(2+q)]**2).mean(axis=1)),\"brown\",linewidth=2,ls=\"--\")\n\tplt.legend(np.concatenate((1+np.arange(q),np.array(\"RMSE of RMSE errors\").reshape((1,)))),loc='upper')\n\tplt.title('Leave-One-Out Cross-validation error against $\\lambda$')\n\tplt.xlabel('$\\lambda$')\n\tplt.ylabel('RMSE')\n\tplt.show()\n\nWhich returns this kind of graphics\n\n.. image::\n\thttps://raw.githubusercontent.com/hlorenzo/py_ddspls/master/images/reg.png\n\t:width: 600\n\nFor 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.\n\nClassification analysis\n-----------------------\n\nLet us produce *1* axis since only one group must be discriminated::\n\n\tR=1\n\nStart model building and tcheck results with sklearn tools::\n\n\tmod_0_classif=py_ddspls.model.ddspls(Xs,Y_bin,lambd=lambd,R=R,mode=\"clas\",verbose=True)\n\tY_est = mod_0_classif.predict(Xtest)\n\tprint(sklm.classification_report(Y_est, Y_classif[id_test]=='Class 1'))\n\nCross validation can be performed with built tools, the parameter **NCORES** permits to use parallellization::\n\n\tperf_model_class = py_ddspls.model.perf_ddspls(Xs,Y_classif,R=R,kfolds=\"loo\",n_lambd=40,NCORES=7,mode=\"classif\")\n\tprint(perf_model_class)\n\n\tfig = plt.figure()\n\tax = fig.add_subplot(1, 1, 1)\n\tax.plot(perf_model_class[:,1], perf_model_class[:,2])\n\tplt.title('Leave-One-Out Cross-validation error against $\\lambda$')\n\tplt.xlabel('$\\lambda$')\n\tplt.ylabel('Classification Error')\n\tplt.show()\n\nWhich returns this kind of graphics\n\n.. image::\n\thttps://raw.githubusercontent.com/hlorenzo/py_ddspls/master/images/cla.png\n\t:width: 600\n\nOne that figure one can see that a parameter approximately equal to 0.45 can be chosen.\n\n**Enjoy :)**",
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