模拟此模型中的10000个观测值

rng(1)% For reproducibilityn = 1e4;d = 1e3;NZ = 0.1;X = Sprandn（n，d，nz）;y = x（：，100） + 2*x（：，200） + 0.3*randn（n，1）;

Train a linear regression model. Reserve 30% of the observations as a holdout sample.

CVMdl = fitrlinear(X,Y,'Holdout'，，，，0。3); Mdl = CVMdl.Trained{1}

MDL=RegressionLinear ResponseName: 'Y' ResponseTransform: 'none' Beta: [1000x1 double] Bias: -0.0066 Lambda: 1.4286e-04 Learner: 'svm' Properties, Methods

CVMdl是a回归专业模型。It contains the propertyTrained，，，，which is a 1-by-1 cell array holding aRegressionLinearmodel that the software trained using the training set.

Extract the training and test data from the partition definition.

trainIdx = training(CVMdl.Partition); testIdx = test(CVMdl.Partition);

Predict the training- and test-sample responses.

yHatTrain = predict(Mdl,X(trainIdx,:)); yHatTest = predict(Mdl,X(testIdx,:));

因为有一个正规化强度MDL，，，，yHatTrainandYhattestare numeric vectors.

Predict responses from the best-performing, linear regression model that uses a lasso-penalty and least squares.

Simulate 10000 observations as in预测测试样本响应。

rng(1)% For reproducibilityn = 1e4;d = 1e3;NZ = 0.1;X = Sprandn（n，d，nz）;y = x（：，100） + 2*x（：，200） + 0.3*randn（n，1）;

从 $$10^{-5}$$through $$10^{-1}$$。

lambda = logspace（-5，-1,15）;

Cross-validate the models. To increase execution speed, transpose the predictor data and specify that the observations are in columns. Optimize the objective function using SpaRSA.

X=X'; CVMdl = fitrlinear(X,Y,'ObservationsIn'，，，，'列'，，，，'KFold'，，，，5，，，，'lambda'，，，，lambda,。。。'学习者'，，，，“最少”，，，，'Solver'，，，，'sparsa'，，，，“正则化”，，，，'套索'）；numCLModels = numel(CVMdl.Trained)

numCLModels = 5

CVMdl是a回归专业模型。因为fitrlinear一世mplements 5-fold cross-validation,CVMdlcontains 5RegressionLinear该软件在每个折叠上训练的模型。

Display the first trained linear regression model.

MDL1=CVMdl.Trained{1}

MDL1=RegressionLinear ResponseName: 'Y' ResponseTransform: 'none' Beta: [1000x15 double] Bias: [-0.0049 -0.0049 -0.0049 -0.0049 -0.0049 -0.0048 ... ] Lambda: [1.0000e-05 1.9307e-05 3.7276e-05 7.1969e-05 ... ] Learner: 'leastsquares' Properties, Methods

MDL1是aRegressionLinear模型对象。fitrlinearconstructedMDL1通过在前四倍上进行训练。因为兰姆达是a sequence of regularization strengths, you can think ofMDL1as 11 models, one for each regularization strength in兰姆达。

估计交叉验证的MSE。

mse = kfoldloss（cvmdl）;

更高的值兰姆达lead to predictor variable sparsity, which is a good quality of a regression model. For each regularization strength, train a linear regression model using the entire data set and the same options as when you cross-validated the models. Determine the number of nonzero coefficients per model.

MDL=fitrlinear(X,Y,'ObservationsIn'，，，，'列'，，，，'lambda'，，，，lambda,。。。'学习者'，，，，“最少”，，，，'Solver'，，，，'sparsa'，，，，“正则化”，，，，'套索'）；numNZCoeff = sum(Mdl.Beta~=0);

在同一图中，为每个正则化强度绘制跨验证的MSE和非零系数的频率。在日志刻度上绘制所有变量。

数字;[H，HL1，HL2] = Plotyy（Log10（Lambda），Log10（MSE），。。。log10(Lambda),log10(numNZCoeff)); hL1.Marker ='o';hl2.marker ='o';Ylabel（h（1），'log_ {10} mse'）Ylabel（H（2），'log_{10} nonzero-coefficient frequency'）Xlabel('log_ {10} lambda'）hold离开

选择平衡预测变量稀疏性和低MSE的正规化强度的索引（例如兰姆达(10)）。

一世dxFinal = 10;

提取与最小MSE相对应的模型。

MDLFinal = selectModels(Mdl,idxFinal)

MDLFinal = RegressionLinear ResponseName: 'Y' ResponseTransform: 'none' Beta: [1000x1 double] Bias: -0.0050 Lambda: 0.0037 Learner: 'leastsquares' Properties, Methods

idxnzcoeff =find(MdlFinal.Beta~=0)

idxnzcoeff =2×1100200

EstCoeff = Mdl.Beta(idxNZCoeff)

EstCoeff =2×11。00511。9965

MDLFinal是aRegressionLinear具有一个正规化强度的模型。非零系数EstCoeff接近模拟数据的系数。

模拟10个新观察结果，并使用表现最佳模型预测相应的响应。

Xnew = Sprandn（D，10，NZ）;yhat =预测（mdlfinal，Xnew，'ObservationsIn'，，，，'列'）；