Last Updated on August 28, 2020

Multi-output regression involves predicting two or more numerical variables.

Unlike normal regression where a single value is predicted for each sample, multi-output regression requires specialized machine learning algorithms that support outputting multiple variables for each prediction.

Deep learning neural networks are an example of an algorithm that natively supports multi-output regression problems. Neural network models for multi-output regression tasks can be easily defined and evaluated using the Keras deep learning library.

In this tutorial, you will discover how to develop deep learning models for multi-output regression.

After completing this tutorial, you will know:

- Multi-output regression is a predictive modeling task that involves two or more numerical output variables.
- Neural network models can be configured for multi-output regression tasks.
- How to evaluate a neural network for multi-output regression and make a prediction for new data.

Letâ€™s get started.

## Tutorial Overview

This tutorial is divided into three parts; they are:

- Multi-Output Regression
- Neural Networks for Multi-Outputs
- Neural Network for Multi-Output Regression

## Multi-Output Regression

Regression is a predictive modeling task that involves predicting a numerical output given some input.

It is different from classification tasks that involve predicting a class label.

Typically, a regression task involves predicting a single numeric value. Although, some tasks require predicting more than one numeric value. These tasks are referred to as **multiple-output regression**, or multi-output regression for short.

In multi-output regression, two or more outputs are required for each input sample, and the outputs are required simultaneously. The assumption is that the outputs are a function of the inputs.

We can create a synthetic multi-output regression dataset using the make_regression() function in the scikit-learn library.

Our dataset will have 1,000 samples with 10 input features, five of which will be relevant to the output and five of which will be redundant. The dataset will have three numeric outputs for each sample.

The complete example of creating and summarizing the synthetic multi-output regression dataset is listed below.

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# example of a multi-output regression problem from sklearn.datasets import make_regression # create dataset X, y = make_regression(n_samples=1000, n_features=10, n_informative=5, n_targets=3, random_state=2) # summarize shape print(X.shape, y.shape) |

Running the example creates the dataset and summarizes the shape of the input and output elements.

We can see that, as expected, there are 1,000 samples, each with 10 input features and three output features.

1 |
(1000, 10) (1000, 3) |

Next, let’s look at how we can develop neural network models for multiple-output regression tasks.

## Neural Networks for Multi-Outputs

Many machine learning algorithms support multi-output regression natively.

Popular examples are decision trees and ensembles of decision trees. A limitation of decision trees for multi-output regression is that the relationships between inputs and outputs can be blocky or highly structured based on the training data.

Neural network models also support multi-output regression and have the benefit of learning a continuous function that can model a more graceful relationship between changes in input and output.

Multi-output regression can be supported directly by neural networks simply by specifying the number of target variables there are in the problem as the number of nodes in the output layer. For example, a task that has three output variables will require a neural network output layer with three nodes in the output layer, each with the linear (default) activation function.

We can demonstrate this using the Keras deep learning library.

We will define a multilayer perceptron (MLP) model for the multi-output regression task defined in the previous section.

Each sample has 10 inputs and three outputs, therefore, the network requires an input layer that expects 10 inputs specified via the “*input_dim*” argument in the first hidden layer and three nodes in the output layer.

We will use the popular ReLU activation function in the hidden layer. The hidden layer has 20 nodes, which were chosen after some trial and error. We will fit the model using mean absolute error (MAE) loss and the Adam version of stochastic gradient descent.

The definition of the network for the multi-output regression task is listed below.

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... # define the model model = Sequential() model.add(Dense(20, input_dim=10, kernel_initializer='he_uniform', activation='relu')) model.add(Dense(3)) model.compile(loss='mae', optimizer='adam') |

You may want to adapt this model for your own multi-output regression task, therefore, we can create a function to define and return the model where the number of input and number of output variables are provided as arguments.

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# get the model def get_model(n_inputs, n_outputs): model = Sequential() model.add(Dense(20, input_dim=n_inputs, kernel_initializer='he_uniform', activation='relu')) model.add(Dense(n_outputs)) model.compile(loss='mae', optimizer='adam') return model |

Now that we are familiar with how to define an MLP for multi-output regression, let’s explore how this model can be evaluated.

## Neural Network for Multi-Output Regression

If the dataset is small, it is good practice to evaluate neural network models repeatedly on the same dataset and report the mean performance across the repeats.

This is because of the stochastic nature of the learning algorithm.

Additionally, it is good practice to use k-fold cross-validation instead of train/test splits of a dataset to get an unbiased estimate of model performance when making predictions on new data. Again, only if there is not too much data and the process can be completed in a reasonable time.

Taking this into account, we will evaluate the MLP model on the multi-output regression task using repeated k-fold cross-validation with 10 folds and three repeats.

Each fold the model is defined, fit, and evaluated. The scores are collected and can be summarized by reporting the mean and standard deviation.

The *evaluate_model()* function below takes the dataset, evaluates the model, and returns a list of evaluation scores, in this case, MAE scores.

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# evaluate a model using repeated k-fold cross-validation def evaluate_model(X, y): results = list() n_inputs, n_outputs = X.shape[1], y.shape[1] # define evaluation procedure cv = RepeatedKFold(n_splits=10, n_repeats=3, random_state=1) # enumerate folds for train_ix, test_ix in cv.split(X): # prepare data X_train, X_test = X[train_ix], X[test_ix] y_train, y_test = y[train_ix], y[test_ix] # define model model = get_model(n_inputs, n_outputs) # fit model model.fit(X_train, y_train, verbose=0, epochs=100) # evaluate model on test set mae = model.evaluate(X_test, y_test, verbose=0) # store result print('>%.3f' % mae) results.append(mae) return results |

We can then load our dataset and evaluate the model and report the mean performance.

Tying this together, the complete example is listed below.

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# mlp for multi-output regression from numpy import mean from numpy import std from sklearn.datasets import make_regression from sklearn.model_selection import RepeatedKFold from keras.models import Sequential from keras.layers import Dense # get the dataset def get_dataset(): X, y = make_regression(n_samples=1000, n_features=10, n_informative=5, n_targets=3, random_state=2) return X, y # get the model def get_model(n_inputs, n_outputs): model = Sequential() model.add(Dense(20, input_dim=n_inputs, kernel_initializer='he_uniform', activation='relu')) model.add(Dense(n_outputs)) model.compile(loss='mae', optimizer='adam') return model # evaluate a model using repeated k-fold cross-validation def evaluate_model(X, y): results = list() n_inputs, n_outputs = X.shape[1], y.shape[1] # define evaluation procedure cv = RepeatedKFold(n_splits=10, n_repeats=3, random_state=1) # enumerate folds for train_ix, test_ix in cv.split(X): # prepare data X_train, X_test = X[train_ix], X[test_ix] y_train, y_test = y[train_ix], y[test_ix] # define model model = get_model(n_inputs, n_outputs) # fit model model.fit(X_train, y_train, verbose=0, epochs=100) # evaluate model on test set mae = model.evaluate(X_test, y_test, verbose=0) # store result print('>%.3f' % mae) results.append(mae) return results # load dataset X, y = get_dataset() # evaluate model results = evaluate_model(X, y) # summarize performance print('MAE: %.3f (%.3f)' % (mean(results), std(results))) |

Running the example reports the MAE for each fold and each repeat, to give an idea of the evaluation progress.

**Note**: Your results may vary given the stochastic nature of the algorithm or evaluation procedure, or differences in numerical precision. Consider running the example a few times and compare the average outcome.

At the end, the mean and standard deviation MAE is reported. In this case, the model is shown to achieve a MAE of about 8.184.

You can use this code as a template for evaluating MLP models on your own multi-output regression tasks. The number of nodes and layers in the model can easily be adapted and tailored to the complexity of your dataset.

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... >8.054 >7.562 >9.026 >8.541 >6.744 MAE: 8.184 (1.032) |

Once a model configuration is chosen, we can use it to fit a final model on all available data and make a prediction for new data.

The example below demonstrates this by first fitting the MLP model on the entire multi-output regression dataset, then calling the *predict()* function on the saved model in order to make a prediction for a new row of data.

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# use mlp for prediction on multi-output regression from numpy import asarray from sklearn.datasets import make_regression from keras.models import Sequential from keras.layers import Dense # get the dataset def get_dataset(): X, y = make_regression(n_samples=1000, n_features=10, n_informative=5, n_targets=3, random_state=2) return X, y # get the model def get_model(n_inputs, n_outputs): model = Sequential() model.add(Dense(20, input_dim=n_inputs, kernel_initializer='he_uniform', activation='relu')) model.add(Dense(n_outputs, kernel_initializer='he_uniform')) model.compile(loss='mae', optimizer='adam') return model # load dataset X, y = get_dataset() n_inputs, n_outputs = X.shape[1], y.shape[1] # get model model = get_model(n_inputs, n_outputs) # fit the model on all data model.fit(X, y, verbose=0, epochs=100) # make a prediction for new data row = [-0.99859353,2.19284309,-0.42632569,-0.21043258,-1.13655612,-0.55671602,-0.63169045,-0.87625098,-0.99445578,-0.3677487] newX = asarray([row]) yhat = model.predict(newX) print('Predicted: %s' % yhat[0]) |

Running the example fits the model and makes a prediction for a new row.

**Note**: Your results may vary given the stochastic nature of the algorithm or evaluation procedure, or differences in numerical precision. Consider running the example a few times and compare the average outcome.

As expected, the prediction contains three output variables required for the multi-output regression task.

1 |
Predicted: [-152.22713 -78.04891 -91.97194] |

## Further Reading

This section provides more resources on the topic if you are looking to go deeper.

## Summary

In this tutorial, you discovered how to develop deep learning models for multi-output regression.

Specifically, you learned:

- Multi-output regression is a predictive modeling task that involves two or more numerical output variables.
- Neural network models can be configured for multi-output regression tasks.
- How to evaluate a neural network for multi-output regression and make a prediction for new data.

**Do you have any questions?**

Ask your questions in the comments below and I will do my best to answer.

Lors de lâ€™exÃ©cution est gÃ©nÃ©rÃ©e l’erreur suivante:

File “C:\ProgramData\Anaconda3\lib\site-packages\keras\backend\tensorflow_backend.py”, line 703, in is_tensor

return isinstance(x, tf_ops._TensorLike) or tf_ops.is_dense_tensor_like(x)

AttributeError: module ‘tensorflow.python.framework.ops’ has no attribute ‘_TensorLike’

Comment rÃ©soudre?

Sorry to hear that, perhaps confirm that your version of Keras and TensorFlow are up to date:

https://machinelearningmastery.com/setup-python-environment-machine-learning-deep-learning-anaconda/

The tips here may also help:

https://machinelearningmastery.com/faq/single-faq/why-does-the-code-in-the-tutorial-not-work-for-me

Hy

Running this example I get the following error:

AttributeError: module ‘tensorflow.python.framework.ops’ has no attribute ‘_TensorLike’

A solution?

What version of Keras and TensorFlow are you using?

You must use the follow versions or higher:

Hi Jason,

could you tell me why a model is compiled for each cross validation fold?

Each model requires training a new model from scratch in order to establish an unbiased estimate of model performance when making predictions on out of sample instances.

If you are new to k-fold cross-validation, you can get started here:

https://machinelearningmastery.com/k-fold-cross-validation/

I thought a model had to be instantiated once and then was passed to the cross validation loop. I missed something I guess…

In the example below taken from one of your excellent article ( Add Binary Flags for Missing Values ) I understand that the model is instantiated and that this instance of the model is evaluated on each cross validation training folds :

[ step 1 ] model = RandomForest()

[ step 2 ] cv = RepeatedStratifiedKFold( n_splits = 10 , n_repeats = 3 )

[ step3 ] scores = cross_val_score( model , X , y , scoring = ‘accuracy’ , cv = cv , n_jobs = -1 )

Here it seems to me that for each training fold a model is instanciated and then evaluated on the corresponding testing folds.

Both logic seems different… or I missed something.

I tested passing the model to the evaluation function :

def evaluate_model( model , X , y ) :

results = list()

n_inputs = X.shape[ 1 ]

n_outputs = y.shape[ 1 ]

cv = RepeatedKFold( n_splits = 10 , n_repeats = 1 , random_state = 999 )

for train_ix, test_ix in cv.split( X ) :

X_train, X_test = X[train_ix], X[test_ix]

y_train, y_test = y[train_ix], y[test_ix]

model.fit( X_train , y_train , verbose = 0 , epochs = 100 )

mae = model.evaluate( X_test , y_test , verbose = 0 )

results.append( mae )

print( f’mae : {mae:.3f}’ )

return results

mae keeps decreasing… I do not understand why…

It is invalid as you continue to train the same model each loop.

The model must be re-defined and re-fit each cross-validation loop otherwise the evaluation is optimistic.

It does the same thing.

In that case, the model is created and fit a new for each cross-validation fold. You just don’t see it as it happens internally to the function.

OK. I thought the model was created once and then fitted on each cross validation fold.

I have been doing a little experiment.

cv = RepeatedStratifiedKFold( n_splits = 10 , n_repeats = 1 , random_state = 777 )

and then [ A ] :

> scores_A = []

> for train_ix , test_ix in cv.split( X , y ) :

> X_train , X_test = X[ train_ix , : ] , X[ test_ix , : ]

> y_train , y_test = y[ train_ix ] , y[ test_ix ]

> model = AdaBoostClassifier()

> model.fit( X_train , y_train )

> y_test_pred = model.predict( X_test )

> score = roc_auc_score( y_test , y_test_pred )

> print( f’score : {score}’ )

> scores_A.append( score )

After that [ B ] :

> scores_B = []

> model = AdaBoostClassifier()

> for train_ix , test_ix in cv.split( X , y ) :

> X_train , X_test = X[ train_ix , : ] , X[ test_ix , : ]

> y_train , y_test = y[ train_ix ] , y[ test_ix ]

> #model = AdaBoostClassifier()

> model.fit( X_train , y_train )

> y_test_pred = model.predict( X_test )

> score = roc_auc_score( y_test , y_test_pred )

> print( f’score : {score}’ )

> scores_B.append( score )

The difference between [ A ] and [ B ] is that with [ B ] the model is instantiated outside the loop once whereas with [ A ] it is created anew for each fold.

scores_A and _B are the same ( even when I change the random state parameter for RepeatedStratifiedKFold and/or the model used : RandomForest…).

This experiment makes me think the model has to be created once and be fitted on each cross validation fold.

For me, the sole difference between model_A = AdaBoostClassifier() an model_B = AdaBoostClassifier() is there memory location this is why I do not understand why the get_model function is called in the evaluate_model function…

PS :

I was not able to replicate the scores given by

> cross_val_score( model , X , y , scoring = ‘roc_auc’ , cv = cv , n_jobs = -1 )

The cross-validation procedure requires the model be re-fit each evaluation.

Internally cross_val_score will clone the model and refit from scratch each iteration. This is functionality equilivient to re-defining and re-fitting each iteration.

HI Jason,

Just out of curiosity, why would you build a multi-output model instead of multiple models of single outputs? When would it be better/worse?

Try both and see what works best for your specific dataset.

Hi Jason. How can I apply those codes into specific datasets? Since you were using random values.

This will help you to load your dataset:

https://machinelearningmastery.com/load-machine-learning-data-python/

Hi Jason,

First of all, congratulations on your website and for your detailed explanations and examples. I really appreciate it!

I have a question for you, as I’ve already spent a considerable amount of time searching online, without significant success. How can we handle a situation where we have partial ground truth for our targets? (e.g. output vectors with some NaN values)

So far, I understand that if you provide a y_train with NaNs, then the loss function won’t behave properly. If we provided a loss function for each output variable (I think keras allows that with loss=[‘mse’, ‘mse’, …]), then in theory, we could dismiss a group of NaNs within a batch of an output variable, by filtering them out (practically making both y_true and y_pred = 0).

The problem is that if we create a custom loss function and try to replace NaNs with 0s in both vectors, then keras throws an error (I think it relates to the use of non-tensorflow functions to filter nans, thus making keras unable to compute the derivative of the loss function)

Could you think of an elegant solution on this problem? Or maybe a completely different approach? (i assume that this way is sensible from the optimization perspective)

Thanks in advance for your time!

Thanks!

It is common to replace missing inputs with an imputed value using statistics or a model.

For some models, you can mark missing values with a special value and allow the model to treat missing as just another value.

Also, you can mark missing values for some models and configure them to ignore them, e.g. a masking layer in neural nets.

I hope that gives you some ideas.

Hi Jason,

Thanks for your reply.

In this case, it’s not missing inputs, and I can’t really use any statistics to learn the missing targets.

Regarding the special values, I haven’t seen anything related in Keras. Are you aware of such symbol?

Also, I checked the masking layers, but that’s again only for the inputs. This wouldn’t affect the y_true in the loss function.

In general, it seems like a trivial problem which doesn’t seem to have a trivial solution…

Maybe I need to check how people handle missing labels on multi-label classification.

Hopefully I’ll figure it out soon!

You can use zero padding in the putput (yhat) then manually ignore zero padded values when using the output or evaluating predoctions.

This is very common in seq2seq problems in NLP.

This is a great nn-model for regression. I tried using mean sq log error for the loss, so I can interpret the reslt a bit better.

Now, when you evaluate or use the evaluation, is there a difference if you setup the repeat times as desired? Or just keep calling the evaluation function with just 2 or 3 repeats?

Just wandering with this algorithm as I never use tried RepeatedKFolds before.

Thanks for sharing J.

Perhaps you can keep the loss as mse, and use sq log error as a metric?

Repeated k-fold cross-validation is a good practice for stochastic models, if they are not too expensive to fit. 3 repeats is conservative, 10 or 30 is better.

More here:

https://machinelearningmastery.com/repeated-k-fold-cross-validation-with-python/

can you please explain how the mean absolute error in Keras loss function is calculated on multi-output vectors? Is it the average of corresponding values in vectors averaged over dimensions of the output and samples?

I believe it is error averaged across variables and samples.

Hello

I have a question

My data is not time dependent

Not image and video

Which Deep Learning model is more suitable for predicting my data?

Thanks for guiding me

An MLP.

Also, this will help:

https://machinelearningmastery.com/when-to-use-mlp-cnn-and-rnn-neural-networks/

Hi Jason,

Can you be my teacher

I’m happy to answer questions about machine learning.

Hi, thanks for the tutorial.

If the goal is simply prediction, what are the benefits of fitting a multi-output NN instead of multiple single-output NN? What are the gains in this case? Is there a paper or reference you can recommend?

The problem will require a single or multiple outputs. You must use the model that achieves the goals of the project.

Hi, thank you for this article!

I didn’t get once you go through the Cross validation step how you choose one configuration with respect to the others in order to make prediction on new data. I mean, once all the loop is finished what has to be done?

See this:

https://machinelearningmastery.com/make-predictions-scikit-learn/

Ok I think I’ve understood.

Based on what you have written in this article, I evaluate the model using k-fold-cross validation and based on what the score is I change the Layers and their parameters in my model accordingly. Then I make my network learning on all the data already used during validation and finally I make the predictions.

To conclude, Validation is used just to measure the performance of the model I have built, is that right?

The train set is used to fit the model.

The test set is used to evaluate the model.

The validation set can be used to tune the model or stop training the model at the right time.

More here:

https://machinelearningmastery.com/difference-test-validation-datasets/

Wondering about this: With “model.compile(loss=’mae’, optimizer=’adam’)”, I basically instruct keras to minimize the combined loss of both output values together. This averages out the two errors stemming from the two separate target values, which could lead to a large positive error on target 1 which is compensated by a large negative error on target 2.

I imagine to avoid this, I would have to create 2 separate submodels that optimize each loss individually – probably using the functional API?

Yes.

Another approach, a model with two output and two losses (so-called multi-output model):

https://machinelearningmastery.com/keras-functional-api-deep-learning/

Hi, I really appreciate your content!

I am starting with deep learning models and I have a project on mind.

What I want to do, giving certain input values (for example: 10 features), I want to predict a curve. I mean, I want to obtain a 2D curve (where Y axis will be force and X axis will be time). Can “Multi-output regression” be a solution? I will predict time steps (t0=5, t1=16, t2=26…).

What other solution can I use for this case?

thanks in advance

If you want a curve, then perhaps use curve fitting directly:

https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.curve_fit.html

I tested this on my data set and found that it worked if iloc was used. ex:

X_train, X_test = X.iloc[train_ix], X.iloc[test_ix]

y_train, y_test = y.iloc[train_ix], y.iloc[test_ix]

Nice work!

Can I do this to forecast t+1, t+2 and so on?

Yes.

Hi Jason, how would it be for multi-output binary classification?

For example, forecast if t+1 is 0 or 1 and t+2 is 0 or 1 and t+3 is 0 or 1.

1. Output layer: model.add(Dense(n_outputs, activation=’sigmoid’))?

2. Loss function: ‘binary_crossentropy’?

Thanks in advance.

Great tutorial!!!

Yes, I believe so. Try it and see.

Hi Jason, first of all, thanks for your great content! I am also using MLP for doing some multi-output regression, but I found that when I tested the model, the output would always be the same regardless of the input (this also happened at training stage). I have normalized my input data and the dimension of my input and output are 450 and 120, respectively, also I used tanh activation function to bound my output within range [-1,1]. Do you have any suggestions on this? Thanks in advance ðŸ™‚

It may suggestion your model require further tuning for your dataset.

The suggestions here will help:

https://machinelearningmastery.com/start-here/#better

Hey Jason, I really love your works here. I’m working on a dataset_1 where the output classes of this dataset_1 has some other features dataset, say dataset_2, that the model could also learn from. The challenge I’m facing is that dataset_1 and dataset_2 are totally different, so there’s no way I could merge them on some common features.

I’d like to know if there’s a way to train a model that would be able to learn on dataset_1 then subsequently learn on dataset_2?

I am considering using dimensionality reduction to reduce the features of the dataset_2 to a single value and then use this single value as an output to dataset_1 in a multi-output model. Do you think this is a good approach?

Thanks!

Try it and compare to other approaches.

Perhaps try to ensemble their predictions?

Thanks a lot for this amazing article. I love your works. I actually need some advice on a multioutput regression problem. In the problem, the outputs are percentage. and all the outputs need to sum up to 1.

For example, i have to predict 3 outputs A, B, C. each of them can be any value between 0 and 1. let’s say,

A = 0.6,B = 0.3,C = 0.1 or A = 1.0, B = 0.0, C = 0.0

so, A+B+C is 1. how do i approach this problem.

Thanks in advance.

Thanks!

Use a softmax activation:

https://machinelearningmastery.com/softmax-activation-function-with-python/

Hi

I appreciate your post. I have a question, how to determine the number of hidden layers and their dense? Is there any relation between them and the shape of input/output?

Thanks.

Good question, I answer it here:

https://machinelearningmastery.com/faq/single-faq/how-many-layers-and-nodes-do-i-need-in-my-neural-network

Hey Jason,

Absolutely enjoy reading these concise tutorials. What are the available metrics for multi-output regression models when using grid search? When compiling the model, any other metric besides [“Accuracy”] raises an error. Is accuracy the MAE or R^2 value in this case?

Thanks.

You can use the same old regression metrics either for overall error or per output variable error.

Accuracy is inappropriate for regression, you can use MSE, RMSE or MAE as common error metrics.

Hi

If I want to build deep model , as output will be vector of 20 elements and each element will take a value from on two possible values …..what kind of system i can use??,, any hep ..to solve this problem

I recommend testing a suite of different models in order to discover what works well or best for your dataset.

Perhaps start with a simple MLP.

thanks for reply

if any link to MLP example to fellow??

Yes, the above tutorial is exactly this, an MLP (multilayer perceptron) for multi-output regression.

Hi there, first of all thank you for the tutorial! This website has been immensely helpful for me in recent months as I’ve started to incorporate machine learning into my research. My main question for this post is why you chose he_uniform for your kernel initialization on the output layer. I know that it’s the best choice for relu activations, but I can’t find anything anywhere saying that it should also be used for linear activation. Maybe you have a good source/reason for this choice here? To be honest, I can’t find much about what is the best canned initializer for linear activation so I’m really curious to hear your thoughts.

You’re welcome!

It probably should not be used for linear activation. Probably won’t make a difference though.

Hi, how can i get multiple text output after training model for multiple text input features,

Hi Ramesh…You may find the following resource of interest:

https://blog.paperspace.com/combining-multiple-features-outputs-keras/

Hi

If there is example based on dividing data set into training and testing Multi-Output Regression

It would be the same for single output regression:

https://machinelearningmastery.com/train-test-split-for-evaluating-machine-learning-algorithms/

Unless, you have a time series, then this will help:

https://machinelearningmastery.com/time-series-forecasting-supervised-learning/

Hi,

you mentioned that if the data set is small , can use k folds for evaluation …what about if the data set is large?? , for example ..instead of 1000 , maybe 5000 or 10000

Those are still small datasets, and k-fold cross validation.

Nevertheless, for very large data (millions/billions of samples) or for data where small samples are stable/stationary, you can use a train/test split.

okay..Thanks alots Jason

Hi ,

May I know how to import x y data which i have as excel instead of auto generated x y data as per above code.

I want to predict multiple outputs and above concept is nice to start with, but since i am new to coding would like to get help on how to import my excel data of multiple xs and multiple ys

Thank you

Your help will be much appreciated

Save your excel to CVS, then load your CSV as a numpy array:

https://machinelearningmastery.com/load-machine-learning-data-python/

which deep learning will be used for prediction of crop production ,humidity ,temperature and pH and also rain fall ?

Hi, just wondering, how can this neural network use multiple inputs and outputs if it is using Keras’s sequential API? Everything I’ve read online says that a Functional API is required for multiple inputs and outputs? thanks!

Yes, the functional API is required if you want to use separate input models or output models.

To have a vector input or vector output the sequential model can be used directly.

Thank you for your reply, I really appreciate it. Would it be possible to explain the difference between using “seperate input models” and simply using “vector input or vector output” if it wouldn’t be too much trouble? For example, I have data with 5 numerical inputs and 9 numerical outputs. The inputs represent the performance characteristics of a op-amp, e.g gain and slew rate, the outputs are the widths of transistors which give those performance characteristics. Would these be vector inputs and outputs or separate models? Sorry if this is a foolish question, this is all quite new to me. Thanks again for your help and this fantastic article.

Separate input models would be multiple disjoint vectors as input to the model for example.

You can see models with separate input models here:

https://machinelearningmastery.com/keras-functional-api-deep-learning/

Yes, you’re describing a single model that takes a vector in and predicts a vector out. You could also model it with a separate “model” for output of each element in the output vector if you want – this might make sense if the elements were unrelated/uncorrelated.

Hi Jason,

I have a question about activation functions in multi-output regression neural networks. I notice you choose the default linear function for the output and ReLU for the hidden layer.

In my case, my outputs can be very small numbers, and often the predictions return negative values which are physically meaningless. How can I restrict the outputs to be positive only? Is a linear output required for regression models, or can I change the output transfer function to tansig, logsig, etc.?

Thanks!

Perhaps you can use a sigmoid activation then scale the output to the desired range.

Hi! Nice work

Please help me here:

yhat = model.predict(newX)

model is not defined. Rest works fine I think

Why this error

Perhaps you have skipped some lines of code in the example, this will help:

https://machinelearningmastery.com/faq/single-faq/why-does-the-code-in-the-tutorial-not-work-for-me

Hi Jason,

Great!! thank you so much for your nice tutorial. However, can I know how to develop an explainer for developing an interpretable model such as Permutation Feature Importance in a multi-output regression?

Thank you

Sorry, I cannot help you with model interpretation – I don’t have tutorials on this topic.

Hi Jason,

Would you please help me that how I can develop multi output classification neural network?

Thanks

See this example:

https://machinelearningmastery.com/multi-class-classification-tutorial-keras-deep-learning-library/

Hi Jason,

Thanks for this tutorial, helped me on a real project.

I have a related question please.

I have a set of image matrix data set (lets say red, green, blue, white etc.) and the output colour is decided by a set of three input (x, y, z) (numerical values). Therefore the data looks like this (everything numerical, nothing categorical):

Input

X1,Y1,Z1 –> Blue Matrix (flattened 1-D arrays)

X2,Y2,Z2 –> Red Matrix

‘

‘

Xn,Yn,Zn –> White Matrix

Now I want build a NN regression model, like you showed here, where if [X,Y,Z] are input, it predicts the entire colour matrix (/ flattened 1-D array)

now, I understand this output layer will have a shape = shape(1-D colour matrix) (similar to 2 in your example). I also understand that the output layer’s shape is >> input layer’s size.

Can you please give some comments ? is this possible (technically I dont see why not)? Which algorithm you suggest is ideal.

thanks

Perhaps the model can predict a class label and you can write code to then create the matrix – if it is the same for each class label.

Hi,

First of all, thank you for this amazing explanation, it was really helpful.

My question is, can we use RNN or LSTM to predict many output variables because I’ve been searching and found LSTM may be the best option but didn’t found any implementation to start with.

Thank you.

You’re welcome.

Yes, see this:

https://machinelearningmastery.com/faq/single-faq/how-do-you-use-lstms-for-multi-step-time-series-forecasting

Hello

I want to know how MAE for multi regression output is calculated numerically , if you can mention the mathematical equation for better understanding?

Thanks in advance

Mean of the absolute differences, more here:

https://scikit-learn.org/stable/modules/model_evaluation.html#mean-absolute-error

Dear Jason,

Thank you for this nice tutorial.

I have a complex multi-output regression task, in which, the input elements and output elements are not independent. I have designed a network with multi-layers of 1D convolutions by trial and error. I was wondering if you could recommend a well-known network architecture for multi-output regression problems.

I guess it makes more sense to optimize the hyperparameters of a well-known model to my problem rather than designing it from the scratch.

Thank you in advance,

Samaneh

We cannot know what architecture will work well or best for a dataset, I recommend testing a suite of models and model configurations in order to discover what works best for your specific dataset.

Hi,

I run this code: I have two features(inputs) and 13 output, but I have this error:

—————————————————————————

TypeError Traceback (most recent call last)

in

16 mae = model.evaluate(x_test, y_test, verbose=0)

17 # store result

—> 18 print(‘>%.3f’ % mae)

19 results.append(mae)

20 return results

TypeError: must be real number, not list

######################################

def evaluate_model(x, y):

results = list()

n_inputs, n_outputs = x.shape[1], y.shape[1]

# define evaluation procedure

cv = RepeatedKFold(n_splits=10, n_repeats=3, random_state=1)

# enumerate folds

for train_ix, test_ix in cv.split(x):

# prepare data

x_train, x_test = x[train_ix], x[test_ix]

y_train, y_test = y[train_ix], y[test_ix]

# define model

model = build_regression(n_inputs, n_outputs)

# fit model

model.fit(x_train, y_train, verbose=0, epochs=100)

# evaluate model on test set

mae = model.evaluate(x_test, y_test, verbose=0)

# store result

print(‘>%.3f’ % mae)

results.append(mae)

return results

Sorry to hear that, these tips may help:

https://machinelearningmastery.com/faq/single-faq/why-does-the-code-in-the-tutorial-not-work-for-me

Hi Jason Brownlee,

Thank you for this pedagogically explained model.

I would like to know how I can save this code in a file. Do you have a code to save it?

You’re welcome.

This will help you to save the code:

https://machinelearningmastery.com/faq/single-faq/how-do-i-copy-code-from-a-tutorial

Thanks Jason Brownlee, but sorry that was a stupid question.

No problem.

Hi Jason, Just I want to Train a model with a set of inputs in numbers and set of outputs in numbers without knowing relationship between them(input and output).

Yes, that is the definition of regression problems.

Hi, first of all, thank you for the tutorial! my question is: I have students’ id as a sample and I want to predict 3 grades in the future to evaluate their performance but the problem is that the student id is repeated because they have many homework and many quizzes and so on. how can I predict the grades for each student separately? Then I want to save these predicted grades in a new file with a unique student-id to decide the student performance. Note: the repeated for student-id are different from student to other. any suggestion? Thanks in advance

Student id is a nominal data. This should not be input to the model. I would suggest you to create a vector for each student id, holding the grades of homework and quiz, and predict the final grade. No student id should be in this vector.

Hi, I’ve bought some of your books and they are great. BTW, for multi outputs regression, are there different weights and bias for each output? Or all outputs share the same w and b? In that case, how are the performance of multi outputs regression compared to separate single output regression?

Likely different because there is nothing in the model to enforce that they should be the same (but you can construct a model that way if you insist). For performance comparison, I doubt if that is possible because it sounds like you are comparing apples to oranges.

Hi, there is a shared metrics between the classification and regression? I want to compare the performance of regression NN with classification NN?

You’re comparing apples to oranges. Note, regression is about predicting continuous values that the numbers have its literal meaning. Classification, however, are predicting nominal values that the numbers at the output are just names. This vast difference causes one metric can make no sense in the wrong problem.

First of all, congratulations on your website and for your detailed explanations and examples. I really enjoy it!

Multiple output regression involves predicting two or more numerical variables.

In the end (# evaluate a model using repeated k-fold cross-validation), on the part of loading dataset, this error apears

ValueError: too many values to unpack (expected 2)

Can you tell me what can be?

Appreciated.

Can’t tell what exactly it is but this is how to trigger this error:

In python you can do multiple assignments in one line using tuple notation as above. But if the number of elements on either side of the assignment do not match, you have this “too many values to unpack” error.

Can we develop a multi-output regression model for RNN and CNN as well? If yes, do you have any tutorials for that?

Hi Jason, thank you for your useful explanation. I have a problem that is Multi-Output Regression, I think. I want to reconstruct images with MLP. I have 12000 data and create a fully connected layer NN. My loss (MSE) in training and validation is low but the prediction is awful. When I normalized my input data performance became better but the prediction is not good again and not on the scale of ground truth. Could give me some advice, please?

Hi Alice…It is possible to overfit the training data.

This means that the model is learning the specific random variations in the training dataset at the cost of poor generalization of the model to new data.

This can be seen by improved skill of the model at making predictions on the training dataset and worse skill of the model at making predictions on the test dataset.

A good general approach to reducing the likelihood of overfitting the training dataset is to use k-fold cross-validation to estimate the skill of the model when making predictions on new data.

A second good general approach in addition to using k-fold cross-validation is to have a hold-out test set that is only used once at the end of your project to help choose between finalized models.

Thanks for the reply but unfortunately prediction on training data is not good as well.

Thanks for the reply but prediction on the training dataset is not good as well

How can we develop a multi-output regression model for RNN and CNN? If yes, do you have any tutorials for that?

How add another hidden layer?

How to add two more hidden layers?

Hi Shan…the following resource explains how to add additional layers in multi-layer perceptron models:

https://machinelearningmastery.com/build-multi-layer-perceptron-neural-network-models-keras/

Hi there!

Great tutorial! Quick question, currently we are getting a single MAE value for all 3 of the predicted values. How can I get 3 MAE values, one for each of the predicted values?

Thanks in advance!

Hi David…You could capture the MAE each time in a variable before “appending” it

mae = model.evaluate(X_test, y_test, verbose=0) # store result

print(‘>%.3f’ % mae)

Hi. I would like to ask you the following.

If we have a regression problem with 2 targets, should we flatten the vector before computing L1Loss (or MAE in keras)?

Hi Marianna…You may find the following beneficial:

https://moviecultists.com/why-flattening-is-used

Thank you for answering.

I found it quite beneficial, but in my 2 target predictions, I want to know i.e. that the prediction p1 is a pair with the prediction p2, in a y.shape [1,2]. So, I think that flattening is not the way to go.

Another question that I would like to ask you though is for this particular example and considering that the normalized targets p1, p2 follow different distributions (normal, power-law), after training my model, I found out that p1 has good predictions, but p2 hasn’t. It seems that these different targets have different losses, and p2 has a bigger loss.

What my next actions should be in order to improve the predictions for the p2 target with the larger loss?

Thank you for your time and consideration

Hi. i wanna ask question.

i want to predict some variable with the 3 input factors that effect to this variable, how to make the input data from 3 factors?

Hi Jason,

I want to ask something related to the multi outputs model, so I have build a model with multi output and after the training when I compare single and multi output model, the loss produce from the multi output is very large. What should I do to make the model with multi output to have less loss? is it possible to run the optimizer and loss individually on multi output? because it seems like the loss is decreasing very slow compare with single output.

Hi Indra,

The following may help clarify:

https://machinelearningmastery.com/feature-selection-to-improve-accuracy-and-decrease-training-time/

https://machinelearningmastery.com/classification-accuracy-is-not-enough-more-performance-measures-you-can-use/

Hi Jason,

Great article, thank you for detailed explanation on the process.

i am looking to build a multi- output regression model for 4 target variables using tabular data as input. What kind of neural network model do you recommend for such a problem?

Additionally, apart from NNs, what other models do you recommend for such as scenario? i wanted to use XGB but according to my understanding it doesnâ€™t support multi target regression models.

Thanks again!

Your comment is awaiting moderation.

Hello KayD…the following may be of interest to you:

https://machinelearningmastery.com/deep-learning-models-for-multi-output-regression/

https://machinelearningmastery.com/multi-output-regression-models-with-python/

Hi Dr. Jason,

Is it possible to develop a machine learning model where there are multiple outputs (say 10 to 20) and those outputs are mutually dependent. I mean to say that for a specific set of inputs one one combination of outputs are expected.

I have training data set where inputs are images and each image corresponds to a specific output vector. Now if I try to use such 1000 images for training set to predict output vector for similar and unknown images.

can you please guide me if there is any way to tackle this kind of problems.

Hi Shirsendu…The following may be of interest to you:

https://machinelearningmastery.com/multi-output-regression-models-with-python/

Hello I am trying to develop a N Network that predicts handwriting letters. I have created my dataset and now I want to fit a model.

My perspective is that I have to use multi output regression but not with numerical values. Instead I must use sigmoid activation function in the output layer. Am I correct?

How can I do this ? Can you please help me?

PS I have to congratulate you for this great work you have dove some far. You have probably the best site I’ve ever seen (and I have seen too many).

Hi Elias…You may wish to consider CNNs for this task:

https://machinelearningmastery.com/handwritten-digit-recognition-using-convolutional-neural-networks-python-keras/

Hello,

I tried to plot two learning curves for the k-fold cross-validation. These are intended to represent the test data and the validation data. Unfortunately I didn’t succeed for def evaluate_model(X, y). Is there a solution for this?

Thank you in advance!

Hi Adrian…Please clarify what issues you are encountering so that we may better assist you.

Hi Dr. Jason Brownlee,

I am trying to develop a multi-output regression model (4 inputs, 4 outputs). I have been successful so far with the Neural Network algorithm (4-5-5-4 architecture); it is feedforward. I am using time series data I collected from a physical experiment for the input and the resulting output.

I am now trying to basically cover my tracks and compare results between models developed with other machine learning algorithms that may be applicable and can produce multiple outputs.

Do you have any suggestions for algorithms I could explore?

I see you mention in this post “decision trees and ensembles of decision trees” (random forest?). Are there any others?

Best regards,

OLB

Hi Oscar…You may wish to investigate sequence to sequence prediction models:

https://machinelearningmastery.com/develop-encoder-decoder-model-sequence-sequence-prediction-keras/

Hi Dr. Jason Brownlee,

Thank you for the helpful lessons.

When I run your code, it occurred the error: “IndentationError: expected an indented block”, as below:

def evaluate_model(X, y):

results = list()

n_inputs, n_outputs = X.shape[1], y.shape[1]

def evaluate_model(X, y):

Input In [31]

def evaluate_model(X, y):

^

IndentationError: expected an indented block

Please tell me why and how to fix it.

Thank you.

Hi, I have a training dataset whose training data is a curve (x-axis is the frequency) which includes 50,000 points on it and for each curve i have more than 100 conresponding parameters/labels. I would like to train a model so that for curve it can regress those hundreds of parameters.

The first thing I thought of was to use mlp…. Do you have any suggestions?

Thank you

Hi WDDouble…An MLP would be a great starting point for regression as you described. The following resource may help clarify when to use various model types:

https://machinelearningmastery.com/when-to-use-mlp-cnn-and-rnn-neural-networks/

Dear Jason,

Thanks for sharing such useful information with a clear description. It is very helpful to me.

I have an important question that is related to my project. I have to define 10 weights for every 10 input values so that each input feature has a specified weight, and the loss function should minimize the cost by considering these features’ weights. It means features with the highest weight have more priority for minimizing the error.

Let’s describe more:

Consider we define wights range from 1 to 5 for 10 input features.

input feature weight

1 5

2 3

3 4

4 4

5 3

6 1

7 5

8 2

9 1

10 2

How can I define a custom weighted loss function for weighing each element?

Hi Aref…The following resource may be of interest:

https://stackoverflow.com/questions/62059060/how-to-set-custom-weights-in-layers

Dear Jason,

I would be thankful if you put code for defining a custom weighted loss function here.

Thank you for your recommendation Aref!

guys,

I don’t understand why my question disappeared after a few minutes?! It is so annoying.

Sorry, I could not find my question, but I found it now. It was a bit wired.

Hi there. I was wondering how we can formulate the mean squared error for a multi-output regression over a training data set. The mean squared error for a single output regression would be 1/2m(sum(y_p-y)^2) and the sum is over the training data set. Now, how would it be for multi-output regression?

Hi Nasim…The following resource may be of interest to you:

https://stackoverflow.com/questions/66707594/rmse-for-multidimensional-data

Hi Jason,

Can you tell me, once modeled, how can we find the values of input variables to optimize (minimize one, maximize another etc.) multiple output variables?

Cheers,

Med

Hi Med…The following resource may be of interest to you:

https://machinelearningmastery.com/tour-of-optimization-algorithms/

Hello! I’m currently working on a dataset that involves 2 dependent variables however those dependent variables are also dependent on one another. Will the algorithm still work in this case? Thanks!

Hi JJ Dela Cruz…I would recommend that you try the model on your scenrio. Please let us know what you find.