Basic regression: Predict fuel efficiency

In a regression problem, the aim is to predict the output of a continuous value, like a price or a probability. Contrast this with a classification problem, where the aim is to select a class from a list of classes (for example, where a picture contains an apple or an orange, recognizing which fruit is in the picture).

This tutorial uses the classic Auto MPG dataset and demonstrates how to build models to predict the fuel efficiency of the late-1970s and early 1980s automobiles. To do this, you will provide the models with a description of many automobiles from that time period. This description includes attributes like cylinders, displacement, horsepower, and weight.

This example uses the Keras API. (Visit the Keras tutorials and guides to learn more.)

# Use seaborn for pairplot.
pip install -q seaborn

import matplotlib.pyplot as plt
import numpy as np
import pandas as pd
import seaborn as sns

# Make NumPy printouts easier to read.
np.set_printoptions(precision=3, suppress=True)

import tensorflow as tf

from tensorflow import keras
from tensorflow.keras import layers

print(tf.__version__)

2022-12-27 02:21:50.841711: W tensorflow/compiler/xla/stream_executor/platform/default/dso_loader.cc:64] Could not load dynamic library 'libnvinfer.so.7'; dlerror: libnvinfer.so.7: cannot open shared object file: No such file or directory
2022-12-27 02:21:50.841824: W tensorflow/compiler/xla/stream_executor/platform/default/dso_loader.cc:64] Could not load dynamic library 'libnvinfer_plugin.so.7'; dlerror: libnvinfer_plugin.so.7: cannot open shared object file: No such file or directory
2022-12-27 02:21:50.841840: W tensorflow/compiler/tf2tensorrt/utils/py_utils.cc:38] TF-TRT Warning: Cannot dlopen some TensorRT libraries. If you would like to use Nvidia GPU with TensorRT, please make sure the missing libraries mentioned above are installed properly.
2.11.0

The Auto MPG dataset

The dataset is available from the UCI Machine Learning Repository.

Get the data

First download and import the dataset using pandas:

url = 'http://archive.ics.uci.edu/ml/machine-learning-databases/auto-mpg/auto-mpg.data'
column_names = ['MPG', 'Cylinders', 'Displacement', 'Horsepower', 'Weight',
                'Acceleration', 'Model Year', 'Origin']

raw_dataset = pd.read_csv(url, names=column_names,
                          na_values='?', comment='\t',
                          sep=' ', skipinitialspace=True)

dataset = raw_dataset.copy()
dataset.tail()

Clean the data

The dataset contains a few unknown values:

dataset.isna().sum()

MPG             0
Cylinders       0
Displacement    0
Horsepower      6
Weight          0
Acceleration    0
Model Year      0
Origin          0
dtype: int64

Drop those rows to keep this initial tutorial simple:

dataset = dataset.dropna()

The "Origin" column is categorical, not numeric. So the next step is to one-hot encode the values in the column with pd.get_dummies.

Note: You can set up the tf.keras.Model to do this kind of transformation for you but that's beyond the scope of this tutorial. Check out the Classify structured data using Keras preprocessing layers or Load CSV data tutorials for examples.

dataset['Origin'] = dataset['Origin'].map({1: 'USA', 2: 'Europe', 3: 'Japan'})

dataset = pd.get_dummies(dataset, columns=['Origin'], prefix='', prefix_sep='')
dataset.tail()

Split the data into training and test sets

Now, split the dataset into a training set and a test set. You will use the test set in the final evaluation of your models.

train_dataset = dataset.sample(frac=0.8, random_state=0)
test_dataset = dataset.drop(train_dataset.index)

Inspect the data

Review the joint distribution of a few pairs of columns from the training set.

The top row suggests that the fuel efficiency (MPG) is a function of all the other parameters. The other rows indicate they are functions of each other.

sns.pairplot(train_dataset[['MPG', 'Cylinders', 'Displacement', 'Weight']], diag_kind='kde')

<seaborn.axisgrid.PairGrid at 0x7f778177d550>

Let's also check the overall statistics. Note how each feature covers a very different range:

train_dataset.describe().transpose()

Split features from labels

Separate the target value—the "label"—from the features. This label is the value that you will train the model to predict.

train_features = train_dataset.copy()
test_features = test_dataset.copy()

train_labels = train_features.pop('MPG')
test_labels = test_features.pop('MPG')

Normalization

In the table of statistics it's easy to see how different the ranges of each feature are:

train_dataset.describe().transpose()[['mean', 'std']]

It is good practice to normalize features that use different scales and ranges.

One reason this is important is because the features are multiplied by the model weights. So, the scale of the outputs and the scale of the gradients are affected by the scale of the inputs.

Although a model might converge without feature normalization, normalization makes training much more stable.

Note: There is no advantage to normalizing the one-hot features—it is done here for simplicity. For more details on how to use the preprocessing layers, refer to the Working with preprocessing layers guide and the Classify structured data using Keras preprocessing layers tutorial.

The Normalization layer

The tf.keras.layers.Normalization is a clean and simple way to add feature normalization into your model.

The first step is to create the layer:

normalizer = tf.keras.layers.Normalization(axis=-1)

Then, fit the state of the preprocessing layer to the data by calling Normalization.adapt:

normalizer.adapt(np.array(train_features))

Calculate the mean and variance, and store them in the layer:

print(normalizer.mean.numpy())

[[   5.478  195.318  104.869 2990.252   15.559   75.898    0.178    0.197
     0.624]]

When the layer is called, it returns the input data, with each feature independently normalized:

first = np.array(train_features[:1])

with np.printoptions(precision=2, suppress=True):
  print('First example:', first)
  print()
  print('Normalized:', normalizer(first).numpy())

First example: [[   4.    90.    75.  2125.    14.5   74.     0.     0.     1. ]]

Normalized: [[-0.87 -1.01 -0.79 -1.03 -0.38 -0.52 -0.47 -0.5   0.78]]

Linear regression

Before building a deep neural network model, start with linear regression using one and several variables.

Linear regression with one variable

Begin with a single-variable linear regression to predict 'MPG' from 'Horsepower'.

Training a model with tf.keras typically starts by defining the model architecture. Use a tf.keras.Sequential model, which represents a sequence of steps.

There are two steps in your single-variable linear regression model:

• Normalize the 'Horsepower' input features using the tf.keras.layers.Normalization preprocessing layer.
• Apply a linear transformation (\(y = mx+b\)) to produce 1 output using a linear layer (tf.keras.layers.Dense).

The number of inputs can either be set by the input_shape argument, or automatically when the model is run for the first time.

First, create a NumPy array made of the 'Horsepower' features. Then, instantiate the tf.keras.layers.Normalization and fit its state to the horsepower data:

horsepower = np.array(train_features['Horsepower'])

horsepower_normalizer = layers.Normalization(input_shape=[1,], axis=None)
horsepower_normalizer.adapt(horsepower)

Build the Keras Sequential model:

horsepower_model = tf.keras.Sequential([
    horsepower_normalizer,
    layers.Dense(units=1)
])

horsepower_model.summary()

Model: "sequential"
_________________________________________________________________
 Layer (type)                Output Shape              Param #   
=================================================================
 normalization_1 (Normalizat  (None, 1)                3         
 ion)                                                            
                                                                 
 dense (Dense)               (None, 1)                 2         
                                                                 
=================================================================
Total params: 5
Trainable params: 2
Non-trainable params: 3
_________________________________________________________________

This model will predict 'MPG' from 'Horsepower'.

Run the untrained model on the first 10 'Horsepower' values. The output won't be good, but notice that it has the expected shape of (10, 1):

horsepower_model.predict(horsepower[:10])

1/1 [==============================] - 2s 2s/step
array([[ 0.101],
       [ 0.057],
       [-0.187],
       [ 0.142],
       [ 0.128],
       [ 0.05 ],
       [ 0.152],
       [ 0.128],
       [ 0.033],
       [ 0.057]], dtype=float32)

Once the model is built, configure the training procedure using the Keras Model.compile method. The most important arguments to compile are the loss and the optimizer, since these define what will be optimized (mean_absolute_error) and how (using the tf.keras.optimizers.Adam).

horsepower_model.compile(
    optimizer=tf.keras.optimizers.Adam(learning_rate=0.1),
    loss='mean_absolute_error')

Use Keras Model.fit to execute the training for 100 epochs:

%%time
history = horsepower_model.fit(
    train_features['Horsepower'],
    train_labels,
    epochs=100,
    # Suppress logging.
    verbose=0,
    # Calculate validation results on 20% of the training data.
    validation_split = 0.2)

/tmpfs/src/tf_docs_env/lib/python3.9/site-packages/keras/engine/data_adapter.py:1696: FutureWarning: The behavior of `series[i:j]` with an integer-dtype index is deprecated. In a future version, this will be treated as *label-based* indexing, consistent with e.g. `series[i]` lookups. To retain the old behavior, use `series.iloc[i:j]`. To get the future behavior, use `series.loc[i:j]`.
  return t[start:end]
CPU times: user 5.58 s, sys: 1.58 s, total: 7.16 s
Wall time: 4.84 s

Visualize the model's training progress using the stats stored in the history object:

hist = pd.DataFrame(history.history)
hist['epoch'] = history.epoch
hist.tail()

def plot_loss(history):
  plt.plot(history.history['loss'], label='loss')
  plt.plot(history.history['val_loss'], label='val_loss')
  plt.ylim([0, 10])
  plt.xlabel('Epoch')
  plt.ylabel('Error [MPG]')
  plt.legend()
  plt.grid(True)

plot_loss(history)

Collect the results on the test set for later:

test_results = {}

test_results['horsepower_model'] = horsepower_model.evaluate(
    test_features['Horsepower'],
    test_labels, verbose=0)

Since this is a single variable regression, it's easy to view the model's predictions as a function of the input:

x = tf.linspace(0.0, 250, 251)
y = horsepower_model.predict(x)

8/8 [==============================] - 0s 1ms/step

def plot_horsepower(x, y):
  plt.scatter(train_features['Horsepower'], train_labels, label='Data')
  plt.plot(x, y, color='k', label='Predictions')
  plt.xlabel('Horsepower')
  plt.ylabel('MPG')
  plt.legend()

plot_horsepower(x, y)

Linear regression with multiple inputs

You can use an almost identical setup to make predictions based on multiple inputs. This model still does the same \(y = mx+b\) except that \(m\) is a matrix and \(b\) is a vector.

Create a two-step Keras Sequential model again with the first layer being normalizer (tf.keras.layers.Normalization(axis=-1)) you defined earlier and adapted to the whole dataset:

linear_model = tf.keras.Sequential([
    normalizer,
    layers.Dense(units=1)
])

When you call Model.predict on a batch of inputs, it produces units=1 outputs for each example:

linear_model.predict(train_features[:10])

1/1 [==============================] - 0s 45ms/step
array([[ 1.68 ],
       [ 0.389],
       [ 0.388],
       [ 0.297],
       [-1.882],
       [ 0.071],
       [-1.916],
       [-0.712],
       [ 0.83 ],
       [-0.924]], dtype=float32)

When you call the model, its weight matrices will be built—check that the kernel weights (the \(m\) in \(y=mx+b\)) have a shape of (9, 1):

linear_model.layers[1].kernel

<tf.Variable 'dense_1/kernel:0' shape=(9, 1) dtype=float32, numpy=
array([[-0.453],
       [-0.035],
       [-0.519],
       [ 0.191],
       [-0.319],
       [-0.697],
       [ 0.208],
       [-0.33 ],
       [ 0.632]], dtype=float32)>

Configure the model with Keras Model.compile and train with Model.fit for 100 epochs:

linear_model.compile(
    optimizer=tf.keras.optimizers.Adam(learning_rate=0.1),
    loss='mean_absolute_error')

%%time
history = linear_model.fit(
    train_features,
    train_labels,
    epochs=100,
    # Suppress logging.
    verbose=0,
    # Calculate validation results on 20% of the training data.
    validation_split = 0.2)

/tmpfs/src/tf_docs_env/lib/python3.9/site-packages/keras/engine/data_adapter.py:1696: FutureWarning: The behavior of `series[i:j]` with an integer-dtype index is deprecated. In a future version, this will be treated as *label-based* indexing, consistent with e.g. `series[i]` lookups. To retain the old behavior, use `series.iloc[i:j]`. To get the future behavior, use `series.loc[i:j]`.
  return t[start:end]
CPU times: user 5.63 s, sys: 1.52 s, total: 7.14 s
Wall time: 4.69 s

Using all the inputs in this regression model achieves a much lower training and validation error than the horsepower_model, which had one input:

plot_loss(history)

Collect the results on the test set for later:

test_results['linear_model'] = linear_model.evaluate(
    test_features, test_labels, verbose=0)

Regression with a deep neural network (DNN)

In the previous section, you implemented two linear models for single and multiple inputs.

Here, you will implement single-input and multiple-input DNN models.

The code is basically the same except the model is expanded to include some "hidden" non-linear layers. The name "hidden" here just means not directly connected to the inputs or outputs.

These models will contain a few more layers than the linear model:

• The normalization layer, as before (with horsepower_normalizer for a single-input model and normalizer for a multiple-input model).
• Two hidden, non-linear, Dense layers with the ReLU (relu) activation function nonlinearity.
• A linear Dense single-output layer.

Both models will use the same training procedure, so the compile method is included in the build_and_compile_model function below.

def build_and_compile_model(norm):
  model = keras.Sequential([
      norm,
      layers.Dense(64, activation='relu'),
      layers.Dense(64, activation='relu'),
      layers.Dense(1)
  ])

  model.compile(loss='mean_absolute_error',
                optimizer=tf.keras.optimizers.Adam(0.001))
  return model

Regression using a DNN and a single input

Create a DNN model with only 'Horsepower' as input and horsepower_normalizer (defined earlier) as the normalization layer:

dnn_horsepower_model = build_and_compile_model(horsepower_normalizer)

This model has quite a few more trainable parameters than the linear models:

dnn_horsepower_model.summary()

Model: "sequential_2"
_________________________________________________________________
 Layer (type)                Output Shape              Param #   
=================================================================
 normalization_1 (Normalizat  (None, 1)                3         
 ion)                                                            
                                                                 
 dense_2 (Dense)             (None, 64)                128       
                                                                 
 dense_3 (Dense)             (None, 64)                4160      
                                                                 
 dense_4 (Dense)             (None, 1)                 65        
                                                                 
=================================================================
Total params: 4,356
Trainable params: 4,353
Non-trainable params: 3
_________________________________________________________________

Train the model with Keras Model.fit:

%%time
history = dnn_horsepower_model.fit(
    train_features['Horsepower'],
    train_labels,
    validation_split=0.2,
    verbose=0, epochs=100)

/tmpfs/src/tf_docs_env/lib/python3.9/site-packages/keras/engine/data_adapter.py:1696: FutureWarning: The behavior of `series[i:j]` with an integer-dtype index is deprecated. In a future version, this will be treated as *label-based* indexing, consistent with e.g. `series[i]` lookups. To retain the old behavior, use `series.iloc[i:j]`. To get the future behavior, use `series.loc[i:j]`.
  return t[start:end]
CPU times: user 6.64 s, sys: 1.48 s, total: 8.12 s
Wall time: 5.5 s

This model does slightly better than the linear single-input horsepower_model:

plot_loss(history)

If you plot the predictions as a function of 'Horsepower', you should notice how this model takes advantage of the nonlinearity provided by the hidden layers:

x = tf.linspace(0.0, 250, 251)
y = dnn_horsepower_model.predict(x)

8/8 [==============================] - 0s 1ms/step

plot_horsepower(x, y)

Collect the results on the test set for later:

test_results['dnn_horsepower_model'] = dnn_horsepower_model.evaluate(
    test_features['Horsepower'], test_labels,
    verbose=0)

Regression using a DNN and multiple inputs

Repeat the previous process using all the inputs. The model's performance slightly improves on the validation dataset.

dnn_model = build_and_compile_model(normalizer)
dnn_model.summary()

Model: "sequential_3"
_________________________________________________________________
 Layer (type)                Output Shape              Param #   
=================================================================
 normalization (Normalizatio  (None, 9)                19        
 n)                                                              
                                                                 
 dense_5 (Dense)             (None, 64)                640       
                                                                 
 dense_6 (Dense)             (None, 64)                4160      
                                                                 
 dense_7 (Dense)             (None, 1)                 65        
                                                                 
=================================================================
Total params: 4,884
Trainable params: 4,865
Non-trainable params: 19
_________________________________________________________________

%%time
history = dnn_model.fit(
    train_features,
    train_labels,
    validation_split=0.2,
    verbose=0, epochs=100)

CPU times: user 6.79 s, sys: 1.61 s, total: 8.4 s
Wall time: 5.54 s

plot_loss(history)

Collect the results on the test set:

test_results['dnn_model'] = dnn_model.evaluate(test_features, test_labels, verbose=0)

Performance

Since all models have been trained, you can review their test set performance:

pd.DataFrame(test_results, index=['Mean absolute error [MPG]']).T

These results match the validation error observed during training.

Make predictions

You can now make predictions with the dnn_model on the test set using Keras Model.predict and review the loss:

test_predictions = dnn_model.predict(test_features).flatten()

a = plt.axes(aspect='equal')
plt.scatter(test_labels, test_predictions)
plt.xlabel('True Values [MPG]')
plt.ylabel('Predictions [MPG]')
lims = [0, 50]
plt.xlim(lims)
plt.ylim(lims)
_ = plt.plot(lims, lims)

3/3 [==============================] - 0s 2ms/step

It appears that the model predicts reasonably well.

Now, check the error distribution:

error = test_predictions - test_labels
plt.hist(error, bins=25)
plt.xlabel('Prediction Error [MPG]')
_ = plt.ylabel('Count')

If you're happy with the model, save it for later use with Model.save:

dnn_model.save('dnn_model')

INFO:tensorflow:Assets written to: dnn_model/assets

If you reload the model, it gives identical output:

reloaded = tf.keras.models.load_model('dnn_model')

test_results['reloaded'] = reloaded.evaluate(
    test_features, test_labels, verbose=0)

pd.DataFrame(test_results, index=['Mean absolute error [MPG]']).T

Conclusion

This notebook introduced a few techniques to handle a regression problem. Here are a few more tips that may help:

• Mean squared error (MSE) (tf.keras.losses.MeanSquaredError) and mean absolute error (MAE) (tf.keras.losses.MeanAbsoluteError) are common loss functions used for regression problems. MAE is less sensitive to outliers. Different loss functions are used for classification problems.
• Similarly, evaluation metrics used for regression differ from classification.
• When numeric input data features have values with different ranges, each feature should be scaled independently to the same range.
• Overfitting is a common problem for DNN models, though it wasn't a problem for this tutorial. Visit the Overfit and underfit tutorial for more help with this.

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