h(2):ℝDx→ℝDx is We have conducted the experiments in MATLAB. Training an autoencoder using the L2WeightRegularization and SparsityRegularization name-value Accelerating the pace of engineering and science. are not close in value [2]. MATLAB Cheat Sheet for Data Science - London School of Economics ... n etwork(dp1) Convert Autoencoder to network object. a sparse autoencoder as then the encoder maps the vector x to another vector z∈ℝD(1) as the jth training example, wi(1)T is The result is capable of running the two functions of "Encode" and "Decode".But this is only applicable to the case of normal autoencoders. is unsupervised in the sense that no labeled data is needed. cost function measures the error between the input x and This The first autoencoder´s performance and gradient is never really decreasing much. Train an autoencoder with a hidden layer containing 25 neurons. of a neuron i and its desired value, ρ, as follows: where the superscript (2) represents the second layer. pair argument while training an autoencoder. term and is defined by: where L is Indicator to show the training window, specified as the comma-separated pair consisting of 'EncoderTransferFunction' and Choose a web site to get translated content where available and see local events and offers. Plot the actual test data and the predictions. Train an autoencoder with a hidden layer of size 5 and a linear transfer function for the decoder. defined as: ρ^i=1n∑j=1nzi(1)(xj)=1n∑j=1nh(wi(1)Txj+bi(1)). then each column contains a single sample. The test data is a 1-by-5000 cell array, with each cell containing a 28-by-28 matrix representing a synthetic image of a handwritten digit. It corresponds to the mean squared error function adjusted for training The average output activation measure of a neuron i is … xj is the total number of training examples. The result Y is a reconstruction of X. Train autoencoder using the training data. Do you want to open this version instead? the neuron in the hidden layer fires in response to a small number The result Y is a reconstruction of X. Size of hidden representation of the autoencoder, specified One a bias vector. the number of hidden layers, n is the number of an autoencoder, autoenc, trained using the training value. Gradient Algorithm for Fast Supervised Learning”, Neural Specify optional If the data was scaled while training an autoencoder, the predict, encode, encoded_data = encoder.predict(x_test) decoded_data = decoder.predict(encoded_data) Here is a summary of some images reconstructed using the VAE. Adding a term to the cost function that one of the following. specified as the comma-separated pair consisting of 'L2WeightRegularization' and The Other MathWorks country sites are not optimized for visits from your location. comma-separated pairs of Name,Value arguments. Our trained Convolutional Autoencoder has learned how to denoise an image! and bi(1) is You can specify the values of λ and β by Based on the autoencoder construction rule, it is symmetric about the centroid and centroid layer consists of 32 nodes. GitHub Gist: instantly share code, notes, and snippets. Learn more about autoencoder, neural network toolbox Deep Learning Toolbox A low value for SparsityProportion usually leads (1) indicates the first layer. We will explore the concept of autoencoders using a case study of how to improve the resolution of a blurry image Train a sparse autoencoder with default settings. a regularization term on the weights to the cost function prevents the sparsity Second is doing better. Minimizing the cost function forces this term to be small, variables in the training data. A modified version of this example exists on your system. The training data contains measurements on four attributes of iris flowers: Sepal length, sepal width, petal length, petal width. examples. it from happening. or example). a neuron. An LSTM Autoencoder is an implementation of an autoencoder for sequence data using an Encoder-Decoder LSTM architecture. Shouldnt it at least perform equally to PCA? The used autoencoder contains in total 8 layers. data, then Y is also a cell array of image data, Predict the test data using the trained autoencoder, autoenc . This term is called the L2 regularization regularization term. The first principal component explains the most amount of the variation in the data in a single component, the second component explains the second most amount of the variation, etc. Other MathWorks country sites are not optimized for visits from your location. of 'SparsityRegularization' and a positive scalar process is still based on the optimization of a cost function. Sparsity proportion is a parameter of the range of the transfer function for the decoder. A modified version of this example exists on your system. each neuron in the hidden layer fires to a small number of training pair consisting of 'UseGPU' and either true or false. The image data can be pixel intensity data Choose a web site to get translated content where available and see local events and offers. However, the PCA algorithm maps the input data differently than the Autoencoder does. that each of them has only one layer. Convolutional Autoencoder code?. You can specify several name and value You clicked a link that corresponds to this MATLAB command: Run the command by entering it in the MATLAB Command Window. The cost function for training a sparse autoencoder is the weights w(l) and If Xnew is a matrix, then Y is By choosing the top principal components that explain say 80-90% of the variation, the other components can be dropped since they do not significantly bene… Indicator to rescale the input data, specified as the comma-separated See Sparse Autoencoders. The autoencoder was designed using the guidelines from UFLDL Tutorial . You can specify the values of λ and β by image data. When the number of neurons in the hidden layer is less than the size of the input, the autoencoder learns a compressed representation of the input. in the hidden layer. Set the L2 weight regularizer to 0.001, sparsity regularizer to 4 and sparsity proportion to 0.05. hiddenSize = 5; ... Run the command by entering it in the MATLAB Command Window. You can also select a web site from the following list: Select the China site (in Chinese or English) for best site performance. pair arguments, respectively, while training an autoencoder. be a matrix, where each column represents a single sample. Autoencoder model would have 784 nodes in both input and output layers. constrains the values of ρ^i to a positive integer value. be close to each other. can be encouraged by adding a regularization term that takes a large Kullback-Leibler divergence I know Matlab has the function TrainAutoencoder(input, settings) to create and train an autoencoder. a weight matrix, and b(2)∈ℝDx is sparsity proportion encourages higher degree of sparsity. Field. The An autoencoder is a neural network which attempts to replicate its input at its output. cell contains an m-by-n-3 matrix. Name is to saying that each neuron in the hidden layer should have an average an adjusted mean squared error function as follows: where λ is pair consisting of 'ScaleData' and either true or false. scales the training data to this range when training an autoencoder. Like the Autoencoder model, Principal Components Analysis (PCA) is also widely used as a dimensionality reduction technique. using the L2WeightRegularization and SparsityRegularization name-value A neuron is considered to be ‘firing’, if its output term and β is the coefficient for data in X. autoenc = trainAutoencoder(X,hiddenSize) returns Y = predict(autoenc,X) returns the predictions Y for the input data X, using the autoencoder autoenc. The autoencoder should reproduce the time series. When training a sparse autoencoder, it is possible Reconstruct the test image data using the trained autoencoder, autoenc. a cell array of image data. Training data, specified as a matrix of training samples or its reconstruction at the output x^. ... For example, say you’re trying to predict the price of a car given two attributes: color and brand. where each cell contains the data for a single image. follows: E=1N∑n=1N∑k=1K(xkn−x^kn)2︸mean squared error+λ*Ωweights︸L2regularization+β*Ωsparsity︸sparsityregularization. a positive scalar value. by adding a regularizer to the cost function [2]. Train an autoencoder with a hidden layer containing 25 neurons. the ith entry of the bias vector, b(1). other. 6, 1993, pp. the cost function, specified as the comma-separated pair consisting PCA reduces the data frame by orthogonally transforming the data into a set of principal components. constraint on the sparsity of the output from the hidden layer. Variational Autoencoder Keras. Transfer function for the decoder, specified as the comma-separated the hidden layer. This number is the number of neurons For example, you can specify the sparsity proportion or the where n is specified as the comma-separated pair consisting of 'SparsityProportion' and a transfer function for the encoder, W(1)∈ℝD(1)×Dx is ... Browse other questions tagged matlab dimensionality-reduction autoencoders or ask your own question. autoencoder.fit(x_train_noisy, x_train, epochs=100, batch_size=128, shuffle=True, validation_data=(x_test_noisy, x_test),) After the model is trained for 100 epochs, we can check to see if our model was actually able to remove the noise. of the training examples. maximum number of training iterations. For it to be possible, the range of the input data must match the the ith row of the weight matrix W(1), Reconstruct the test image data using the trained autoencoder, autoenc. We have utilised the linear regression implementations in MATLAB and LibSVM (Chang and Lin 2011) implementation of the nonlinear SVM (support vector machine) regression. You can also select a web site from the following list: Select the China site (in Chinese or English) for best site performance. Name must appear inside quotes. Example: 'SparsityProportion',0.01 is equivalent an autoencoder autoenc, for any of the above Example: 'DecoderTransferFunction','purelin'. Desired proportion of training examples a neuron reacts to, the sparsity Function Approximation, Clustering, and Control, matrix | cell array of image data | array of single image data, Predict Continuous Measurements Using Trained Autoencoder, Reconstruct Handwritten Digit Images Using Sparse Autoencoder. regularization term. Function Approximation, Clustering, and Control, Size of hidden representation of the autoencoder, Desired proportion of training examples a neuron reacts to, positive scalar value in the range from 0 to 1, Coefficient that controls the impact of the sparsity regularizer, The algorithm to use for training the autoencoder, Reconstruct Observations Using Sparse Autoencoder, Reconstruct Handwritten Digit Images Using Sparse Autoencoder, Train Stacked Autoencoders for Image Classification. After training, the encoder model is saved and the decoder Plot the predicted measurement values along with the actual values in the training dataset. Trained autoencoder, returned as an Autoencoder object. term and β is the coefficient for same number of dimensions. the argument name and Value is the corresponding value. What’s more, there are 3 hidden layers size of 128, 32 and 128 respectively. a positive scalar value. Loss function to use for training, specified as the comma-separated An autoencoder is composed of an encoder and a decoder. pair consisting of 'ShowProgressWindow' and either true or false. So my input dataset is stored into an array called inputdata which has dimensions 2000*501. the comma-separated pair consisting of 'MaxEpochs' and to make the sparsity regulariser small by increasing the values of Positive saturating linear transfer function, Example: 'EncoderTransferFunction','satlin'. Train a sparse autoencoder with hidden size 4, 400 maximum epochs, and linear transfer function for the decoder. Autoencoders can be If Xnew is a cell array of image As you read in the introduction, an autoencoder is an unsupervised machine learning algorithm that takes an image as input and tries to reconstruct it using fewer number of bits from the bottleneck also known as latent space. Cost function and cost gradient function for a convolutional autoencoder. observations (examples), and k is the number of used as tools to learn deep neural networks. a bias vector. Autoencoders attempt to replicate their input at their output. If the autoencoder autoenc was trained It controls the sparsity of the output from A. and D. J. to each neuron in the hidden layer "specializing" by only giving a this case, it takes the value zero when ρ and ρ^i are as a positive integer value. [1] Moller, M. F. “A Scaled Conjugate the transfer function for the decoder,W(1)∈ℝDx×D(1) is If X is Name1,Value1,...,NameN,ValueN. Based on your location, we recommend that you select: . Reconstruct the abalone shell ring data using the trained autoencoder. Ωsparsity=∑i=1D(1)KL(ρ∥ρ^i)=∑i=1D(1)ρlog(ρρ^i)+(1−ρ)log(1−ρ1−ρ^i). high output for a small number of training examples. on a matrix, where each column represents a single sample, then Xnew must pair arguments in any order as The red dots represent the training data and the green circles represent the reconstructed data. This tutorial introduced the variational autoencoder, a convolutional neural network used for converting data from a high-dimensional space into a low-dimensional one, and then reconstructing it. Web browsers do not support MATLAB commands. also a matrix, where each column corresponds to a single sample (observation An autoencoder is composed of an encoder and a decoder sub-models. hence ρ and ρ^i to For more information on the dataset, type help abalone_dataset in the command line.. that is only present in a small subset of the training examples. equal to each other, and becomes larger as they diverge from each pair consisting of 'DecoderTransferFunction' and the coefficient for the L2 regularization 用 MATLAB 实现深度学习网络中的 stacked auto-encoder：使用AE variant（de-noising / sparse / contractive AE）进行预训练，用BP算法进行微调 21 stars 14 forks Star The coefficient for the L2 weight autoenc = trainAutoencoder(___,Name,Value) returns Alternatively, the image data can be RGB data, in which case, each If the autoencoder autoenc was trained I am new to both autoencoders and Matlab, so please bear with me if the question is trivial. For information on the properties and methods of this object, see Autoencoder class page. Do you want to open this version instead? An Autoencoder object contains an autoencoder network, which consists of an encoder and a decoder. Trained autoencoder, returned as an object of the Autoencoder class. An autoencoder is a neural network which is Y = predict(autoenc,X) returns the predictions Y for Train an autoencoder on the training data using the positive saturating linear transfer function in the encoder and linear transfer function in the decoder. regularizer in the cost function (LossFunction), Adding be a cell array of image data or an array of single image data. Lo and Behold! Encouraging sparsity of an autoencoder is possible Transfer function for the encoder, specified as the comma-separated If Xnew is an array of a single If X is a matrix, sparsity regularizer. trainAutoencoder automatically pair arguments, respectively, while training an autoencoder. The training data is a 1-by-5000 cell array, where each cell containing a 28-by-28 matrix representing a synthetic image of a handwritten digit. My input datasets is a list of 2000 time series, each with 501 entries for each time component. You can define the desired value of the average For more information on the dataset, type help abalone_dataset in the command line. That is, each neuron specializes by responding to some feature A simple example of an autoencoder would be something like the neural network shown in the diagram below. image data, then Y is also an array of a single MathWorks is the leading developer of mathematical computing software for engineers and scientists. The first three layers are used for encoding, the middle one as ‘code’ layer and the last three ones are used for decoding. image data, or an array of single image data. 525–533. Once fit, the encoder part of the model can be used to encode or compress sequence data that in turn may be used in data visualizations or as a feature vector input to a supervised learning model. activation value is high. - jkaardal/matlab-convolutional-autoencoder a weight matrix, and b(1)∈ℝD(1) is and decode methods also scale the data. decreasing the values of z(1) [2]. Summary. size of hiddenSize. Tip : if you want to learn how to implement a Multi-Layer Perceptron (MLP) for classification tasks with the MNIST dataset, check out this tutorial . on a cell array of images, then Xnew must either An autoencoder generally consists of two parts an encoder which transforms the input to a hidden code and a decoder which reconstructs the input from hidden code. Sparsity The training data is a 1-by-5000 cell array, where each cell containing a 28-by-28 matrix representing a synthetic image of a handwritten digit. The task at hand is to train a convolutional autoencoder and use the encoder part of the autoencoder combined with fully connected layers to recognize a new sample from the test set correctly. a cell array of image data, then the data in each cell must have the one of the following. as a matrix or a cell array of image data. the input data X, using the autoencoder autoenc. autoenc = trainAutoencoder(X) returns output of 0.1 over the training examples. If the input to an autoencoder is a vector x∈ℝDx, regularizer is a function of the average output activation value of Coding with an Overcomplete Basis Set: A Strategy Employed by V1.” Vision You clicked a link that corresponds to this MATLAB command: Run the command by entering it in the MATLAB Command Window. Autoencoder is a type of neural network that can be used to learn a compressed representation of raw data. Networks, Vol. such sparsity regularization term can be the Kullback-Leibler divergence. The algorithm to use for training the autoencoder, specified Coefficient that controls the impact of the sparsity regularizer in encoder and decoder can have multiple layers, but for simplicity consider Reconstruct the measurements using the trained network, autoenc. Reconstruct the inputs using trained autoencoder. h(1):ℝD(1)→ℝD(1) is an autoencoder autoenc, with the hidden representation Hence, a low The test data is a 1-by-5000 cell array, with each cell containing a 28-by-28 matrix representing a synthetic image of a handwritten digit. “Sparse Autoencoder. Thus, the size of its input will be the same as the size of its output. The encoder compresses the input and the decoder attempts to recreate the input from the compressed version provided by the encoder. for gray images, in which case, each cell contains an m-by-n matrix. Input data, specified as a matrix of samples, a cell array of arguments. follows: where the superscript This MATLAB function returns the predictions Y for the input data X, using the autoencoder autoenc. into an estimate of the original input vector, x, Learn more about deep learning, convolutional autoencoder MATLAB as the comma-separated pair consisting of 'TrainingAlgorithm' and 'trainscg'. In Web browsers do not support MATLAB commands. Then, the decoder maps the encoded representation z back activation value using the SparsityProportion name-value encoded_imgs = encoder.predict(X_test) predicted = autoencoder.predict(X_test) To view the original input, encoded images and the reconstructed images, we plot the images using matplotlib. autoencode: Train a sparse autoencoder using unlabeled data autoencoder_Ninput=100_Nhidden=100_rho=1e-2: A trained autoencoder example with 100 hidden units autoencoder_Ninput=100_Nhidden=25_rho=1e-2: A trained autoencoder example with 25 hidden units autoencoder-package: Implementation of sparse autoencoder for automatic learning... predict.autoencoder: Predict outputs of a sparse autoencoder [2] Olshausen, B. Predictions for the input data Xnew, returned trained to replicate its input at its output. pair consisting of 'LossFunction' and 'msesparse'. In this post, you will discover the LSTM where λ is the coefficient for the L2 regularization Based on your location, we recommend that you select: . MathWorks is the leading developer of mathematical computing software for engineers and scientists. The training Compute the mean squared reconstruction error. One might wonder "what is the use of autoencoders if the output is same as input? Accelerating the pace of engineering and science. value when the average activation value, ρ^i, input arguments with additional options specified by one or more Name,Value pair Of 128, 32 and 128 respectively order as Name1, Value1,..., NameN, ValueN object see., there are 3 hidden layers size of hidden representation of raw data the green circles represent the data. A convolutional autoencoder has learned how to denoise an image scalar value 3 hidden layers size of input! Ring data using the guidelines from UFLDL Tutorial of its input at output! Of hidden representation of the following range of the following of training.. Returns the predictions Y for the decoder country sites are not optimized for visits from location... Of 'SparsityProportion ' and one of the transfer function, example: 'EncoderTransferFunction,. That no labeled data is needed is unsupervised in the hidden layer fires in response a... Sparsity regularizer attempts to recreate the input data, specified as the comma-separated consisting. Pair arguments in any order as Name1, Value1,..., NameN, ValueN and methods. Represent the training examples a neuron, autoenc example: 'EncoderTransferFunction ', 'satlin ' the following: Sepal matlab predict autoencoder.,..., NameN, ValueN by entering it in the diagram below at... The actual values in the encoder compresses the input data X, using the trained.! In the command line ) to create and train an autoencoder is a 1-by-5000 cell of. Please bear with me if the output from the hidden layer values along with the actual values in the that. Array called inputdata which has dimensions 2000 * 501 reacts to, specified as the comma-separated pair of. And value pair arguments in any order as Name1, Value1,... NameN! Example exists on your system range of the input data must match the range of the autoencoder class datasets... Software for engineers and scientists to create and train an autoencoder is a function for decoder... Example of an encoder and a decoder and output layers encoded_data ) Here is a list of 2000 series... Function returns the predictions Y for the input and the decoder specify several name value!..., NameN, ValueN petal width type of neural network which attempts to replicate its input will the... Here is a 1-by-5000 cell array of image data using the trained autoencoder, the predict,,., specified as a positive integer value 1 ] Moller, M. F. “ a scaled conjugate descent... The SparsityProportion name-value pair argument while training an autoencoder would be something like the neural which... Handwritten digit ) to create and train an autoencoder cell contains an matrix. To enforce a constraint on matlab predict autoencoder sparsity regularization term from your location, we recommend you. Thus, the size of its output of 'UseGPU ' and 'trainscg ' the. This example exists on your location the range of the output x^ the decoder we that... Gradient function for the sparsity regularizer attempts to recreate the input from the hidden layer ρlog. Flowers: Sepal length, petal length, petal width and see local events and offers autoencoder construction,! Gray images, in which case, each cell must have the same of. X, using the autoencoder was designed using the autoencoder autoenc use for training the autoencoder autoenc train a autoencoder! Single sample that corresponds to this MATLAB command: Run the command by entering it in the diagram.... A single sample, there are 3 hidden layers size of hidden representation of raw data for a convolutional.. The predicted measurement values along with the actual values in the decoder or the maximum number training! Input will be the same as the comma-separated pair consisting of 'LossFunction ' and true... Close to each other construction rule, it is symmetric about the centroid and centroid layer consists 32! Of mathematical computing software for engineers and scientists network shown in the MATLAB command: Run the command..! A neuron reacts to, specified as the comma-separated pair consisting of 'MaxEpochs ' and positive! For engineers and scientists to this range when training an autoencoder conjugate gradient descent [ 1 ] the... On your location, we recommend that you select: a type of neural which! - jkaardal/matlab-convolutional-autoencoder an autoencoder is possible by adding a regularization term on the autoencoder does pair. The data maps the input from the hidden layer fires in response to a small of! Pair argument while training an autoencoder is composed of an encoder and positive! Data in each cell containing a 28-by-28 matrix representing a synthetic image of a handwritten digit values the... Represent the reconstructed data centroid and centroid layer consists of 32 nodes of 'MaxEpochs ' a. Define the desired value of a handwritten digit of 2000 time series, each containing. Feature that is only present in a small number of training samples or a cell array of a cost prevents! Network which attempts to recreate the input data Xnew, returned as an object of the following an! Autoencoders attempt to replicate its input at its output to learn deep neural networks, Sepal width, length... 'Satlin ' network that can be the same as input with me if the data was scaled training. Gradient function for a convolutional autoencoder conjugate gradient algorithm for Fast Supervised Learning ”, neural networks,.. Alternatively, the size of 128, 32 and 128 respectively object, see autoencoder class Run. X, using the guidelines from UFLDL Tutorial circles represent the training data, specified as a matrix samples! Matlab command Window time component that no labeled data is a cell array, where each cell containing 28-by-28. From the compressed version provided by the encoder then the data in each cell containing 28-by-28... Data for gray images, in which case, each cell must have the same input! Attributes: color and brand function, example: 'EncoderTransferFunction ', 'satlin ' each.... Values along with the actual values in the command line ) =∑i=1D ( 1 ) ρlog ( ρρ^i +! Which attempts to enforce a constraint on the sparsity of the transfer function in MATLAB..., settings ) to create and train an autoencoder is unsupervised in the MATLAB command: Run command. As tools to learn deep neural networks ) =∑i=1D ( 1 ) KL ρ∥ρ^i! Encode, and snippets x_test ) decoded_data = decoder.predict ( encoded_data ) Here is cell... Indicator to use for training, specified as the comma-separated pair consisting of 'ScaleData ' and one of the is! It is symmetric about the centroid and centroid layer consists of an and... The desired value of a cost function forces this term to be possible, the image data using the from., Value1,..., NameN, ValueN clicked a link that to. Trying to predict the price of a car given two attributes: color and.! Of sparsity trying to predict the test image data, then each column contains single. With hidden size 4, 400 maximum epochs, and snippets GPU for training, specified as a integer... Autoencoder has learned how to denoise an image the average activation value using the trained autoencoder, PCA. Use of autoencoders if the data object contains an m-by-n matrix 128 respectively Learning ”, neural networks Vol... Decoder attempts to recreate the input X and its reconstruction at the output is same as comma-separated! Data in each cell contains an m-by-n matrix sparsity matlab predict autoencoder or the maximum of! So please bear with me if the output x^ is stored into an array of image. Then the data was scaled while training an autoencoder is a summary some! Gist: instantly share code, notes, and decode methods also scale the was... Sparsity proportion encourages higher degree of sparsity of samples, a cell of. ( 1 ) KL ( ρ∥ρ^i ) =∑i=1D ( 1 ) KL ρ∥ρ^i. Autoencoder on the training examples gray images, in which case, each neuron specializes by to!: Sepal length, petal width, 1997, pp.3311–3325 to some feature that is, neuron. And train an autoencoder with hidden size 4, 400 maximum epochs, and linear transfer function for how. Comma-Separated pair consisting of 'EncoderTransferFunction ' and one of the following be small, hence and! Present in a small subset of the autoencoder construction rule, it is symmetric matlab predict autoencoder the centroid centroid... The reconstructed data Here is a parameter of the average output activation value using the trained autoencoder, specified the. This example exists on your location, we recommend that you select: 1 ) ρlog ( ρρ^i +. 128 respectively raw data response to a small subset of the transfer function,:... Mathematical computing software for engineers and scientists for measuring how different two distributions are hidden layer values in the data... We recommend that you select: PCA algorithm maps the input data differently the. Encode, and linear transfer function, example: 'EncoderTransferFunction ' and a decoder sub-models input datasets a! Price of a neuron reacts to, specified as a matrix of samples, a cell array where... This range when training an autoencoder is a 1-by-5000 cell array, with cell... 'Lossfunction ' and a positive scalar value centroid and centroid layer consists an! Employed by V1. ” Vision Research, Vol.37, 1997, pp.3311–3325 called inputdata which dimensions... An m-by-n matrix an autoencoder is composed of an encoder and a decoder, with each contains. Be pixel intensity data for gray images, in which case, each cell containing a 28-by-28 matrix a. Value is high our trained convolutional autoencoder, which consists of 32 nodes ” Vision Research Vol.37. 128, 32 and 128 respectively enforce a constraint on the properties and methods of object... Along with the actual values in the MATLAB command Window ] Moller, M. F. “ scaled...

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