Predict fuel efficiency: regression

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In a regression problem, we aim to predict the output of a continuous value, like a price or a probability. Contrast this with a classification problem, where we aim 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 notebook uses the classic Auto MPG Dataset and builds a model to predict the fuel efficiency of late-1970s and early 1980s automobiles. To do this, we'll provide the model with a description of many automobiles from that time period. This description includes attributes like: cylinders, displacement, horsepower, and weight.

This example uses the tf.keras API, see this guide for details.

# Use seaborn for pairplot
!pip install -q seaborn
from __future__ import absolute_import, division, print_function, unicode_literals

import pathlib

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

!pip install -q tensorflow==2.0.0-alpha0
import tensorflow as tf

from tensorflow import keras
from tensorflow.keras import layers

print(tf.__version__)
2.0.0-alpha0

The Auto MPG dataset

The dataset is available from the UCI Machine Learning Repository.

Get the data

First download the dataset.

dataset_path = keras.utils.get_file("auto-mpg.data", "http://archive.ics.uci.edu/ml/machine-learning-databases/auto-mpg/auto-mpg.data")
dataset_path
Downloading data from http://archive.ics.uci.edu/ml/machine-learning-databases/auto-mpg/auto-mpg.data
32768/30286 [================================] - 0s 0us/step

'/root/.keras/datasets/auto-mpg.data'

Import it using pandas

column_names = ['MPG','Cylinders','Displacement','Horsepower','Weight',
                'Acceleration', 'Model Year', 'Origin']
raw_dataset = pd.read_csv(dataset_path, names=column_names,
                      na_values = "?", comment='\t',
                      sep=" ", skipinitialspace=True)

dataset = raw_dataset.copy()
dataset.tail()
MPG Cylinders Displacement Horsepower Weight Acceleration Model Year Origin
393 27.0 4 140.0 86.0 2790.0 15.6 82 1
394 44.0 4 97.0 52.0 2130.0 24.6 82 2
395 32.0 4 135.0 84.0 2295.0 11.6 82 1
396 28.0 4 120.0 79.0 2625.0 18.6 82 1
397 31.0 4 119.0 82.0 2720.0 19.4 82 1

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

To keep this initial tutorial simple drop those rows.

dataset = dataset.dropna()

The "Origin" column is really categorical, not numeric. So convert that to a one-hot:

origin = dataset.pop('Origin')
dataset['USA'] = (origin == 1)*1.0
dataset['Europe'] = (origin == 2)*1.0
dataset['Japan'] = (origin == 3)*1.0
dataset.tail()
MPG Cylinders Displacement Horsepower Weight Acceleration Model Year USA Europe Japan
393 27.0 4 140.0 86.0 2790.0 15.6 82 1.0 0.0 0.0
394 44.0 4 97.0 52.0 2130.0 24.6 82 0.0 1.0 0.0
395 32.0 4 135.0 84.0 2295.0 11.6 82 1.0 0.0 0.0
396 28.0 4 120.0 79.0 2625.0 18.6 82 1.0 0.0 0.0
397 31.0 4 119.0 82.0 2720.0 19.4 82 1.0 0.0 0.0

Split the data into train and test

Now split the dataset into a training set and a test set.

We will use the test set in the final evaluation of our model.

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

Inspect the data

Have a quick look at the joint distribution of a few pairs of columns from the training set.

sns.pairplot(train_dataset[["MPG", "Cylinders", "Displacement", "Weight"]], diag_kind="kde")
<seaborn.axisgrid.PairGrid at 0x7fac82d5bfd0>

png

Also look at the overall statistics:

train_stats = train_dataset.describe()
train_stats.pop("MPG")
train_stats = train_stats.transpose()
train_stats
count mean std min 25% 50% 75% max
Cylinders 314.0 5.477707 1.699788 3.0 4.00 4.0 8.00 8.0
Displacement 314.0 195.318471 104.331589 68.0 105.50 151.0 265.75 455.0
Horsepower 314.0 104.869427 38.096214 46.0 76.25 94.5 128.00 225.0
Weight 314.0 2990.251592 843.898596 1649.0 2256.50 2822.5 3608.00 5140.0
Acceleration 314.0 15.559236 2.789230 8.0 13.80 15.5 17.20 24.8
Model Year 314.0 75.898089 3.675642 70.0 73.00 76.0 79.00 82.0
USA 314.0 0.624204 0.485101 0.0 0.00 1.0 1.00 1.0
Europe 314.0 0.178344 0.383413 0.0 0.00 0.0 0.00 1.0
Japan 314.0 0.197452 0.398712 0.0 0.00 0.0 0.00 1.0

Split features from labels

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

train_labels = train_dataset.pop('MPG')
test_labels = test_dataset.pop('MPG')

Normalize the data

Look again at the train_stats block above and note how different the ranges of each feature are.

It is good practice to normalize features that use different scales and ranges. Although the model might converge without feature normalization, it makes training more difficult, and it makes the resulting model dependent on the choice of units used in the input.

def norm(x):
  return (x - train_stats['mean']) / train_stats['std']
normed_train_data = norm(train_dataset)
normed_test_data = norm(test_dataset)

This normalized data is what we will use to train the model.

The model

Build the model

Let's build our model. Here, we'll use a Sequential model with two densely connected hidden layers, and an output layer that returns a single, continuous value. The model building steps are wrapped in a function, build_model, since we'll create a second model, later on.

def build_model():
  model = keras.Sequential([
    layers.Dense(64, activation='relu', input_shape=[len(train_dataset.keys())]),
    layers.Dense(64, activation='relu'),
    layers.Dense(1)
  ])

  optimizer = tf.keras.optimizers.RMSprop(0.001)

  model.compile(loss='mse',
                optimizer=optimizer,
                metrics=['mae', 'mse'])
  return model
model = build_model()

Inspect the model

Use the .summary method to print a simple description of the model

model.summary()
Model: "sequential"
_________________________________________________________________
Layer (type)                 Output Shape              Param #   
=================================================================
dense (Dense)                (None, 64)                640       
_________________________________________________________________
dense_1 (Dense)              (None, 64)                4160      
_________________________________________________________________
dense_2 (Dense)              (None, 1)                 65        
=================================================================
Total params: 4,865
Trainable params: 4,865
Non-trainable params: 0
_________________________________________________________________

Now try out the model. Take a batch of 10 examples from the training data and call model.predict on it.

example_batch = normed_train_data[:10]
example_result = model.predict(example_batch)
example_result
array([[ 0.09340405],
       [-0.17243904],
       [-0.07171869],
       [-0.14838144],
       [ 0.5428827 ],
       [-0.16617195],
       [ 0.5833335 ],
       [ 0.56849545],
       [-0.22444534],
       [ 0.5015095 ]], dtype=float32)

It seems to be working, and it produces a result of the expected shape and type.

Train the model

Train the model for 1000 epochs, and record the training and validation accuracy in the history object.

# Display training progress by printing a single dot for each completed epoch
class PrintDot(keras.callbacks.Callback):
  def on_epoch_end(self, epoch, logs):
    if epoch % 100 == 0: print('')
    print('.', end='')

EPOCHS = 1000

history = model.fit(
  normed_train_data, train_labels,
  epochs=EPOCHS, validation_split = 0.2, verbose=0,
  callbacks=[PrintDot()])

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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()
loss mae mse val_loss val_mae val_mse epoch
995 2.482736 0.984039 2.482736 10.492274 2.388879 10.492273 995
996 2.260968 0.991715 2.260968 9.405447 2.434615 9.405448 996
997 2.443622 1.015261 2.443622 9.856659 2.371433 9.856658 997
998 2.405295 1.008966 2.405295 9.245460 2.364943 9.245461 998
999 2.471271 1.004334 2.471271 9.573922 2.304466 9.573922 999
def plot_history(history):
  hist = pd.DataFrame(history.history)
  hist['epoch'] = history.epoch

  plt.figure()
  plt.xlabel('Epoch')
  plt.ylabel('Mean Abs Error [MPG]')
  plt.plot(hist['epoch'], hist['mae'],
           label='Train Error')
  plt.plot(hist['epoch'], hist['val_mae'],
           label = 'Val Error')
  plt.ylim([0,5])
  plt.legend()

  plt.figure()
  plt.xlabel('Epoch')
  plt.ylabel('Mean Square Error [$MPG^2$]')
  plt.plot(hist['epoch'], hist['mse'],
           label='Train Error')
  plt.plot(hist['epoch'], hist['val_mse'],
           label = 'Val Error')
  plt.ylim([0,20])
  plt.legend()
  plt.show()


plot_history(history)

png

png

This graph shows little improvement, or even degradation in the validation error after about 100 epochs. Let's update the model.fit call to automatically stop training when the validation score doesn't improve. We'll use an EarlyStopping callback that tests a training condition for every epoch. If a set amount of epochs elapses without showing improvement, then automatically stop the training.

You can learn more about this callback here.

model = build_model()

# The patience parameter is the amount of epochs to check for improvement
early_stop = keras.callbacks.EarlyStopping(monitor='val_loss', patience=10)

history = model.fit(normed_train_data, train_labels, epochs=EPOCHS,
                    validation_split = 0.2, verbose=0, callbacks=[early_stop, PrintDot()])

plot_history(history)

............................................

png

png

The graph shows that on the validation set, the average error is usually around +/- 2 MPG. Is this good? We'll leave that decision up to you.

Let's see how well the model generalizes by using the test set, which we did not use when training the model. This tells us how well we can expect the model to predict when we use it in the real world.

loss, mae, mse = model.evaluate(normed_test_data, test_labels, verbose=0)

print("Testing set Mean Abs Error: {:5.2f} MPG".format(mae))
Testing set Mean Abs Error:  1.82 MPG

Make predictions

Finally, predict MPG values using data in the testing set:

test_predictions = model.predict(normed_test_data).flatten()

plt.scatter(test_labels, test_predictions)
plt.xlabel('True Values [MPG]')
plt.ylabel('Predictions [MPG]')
plt.axis('equal')
plt.axis('square')
plt.xlim([0,plt.xlim()[1]])
plt.ylim([0,plt.ylim()[1]])
_ = plt.plot([-100, 100], [-100, 100])

png

It looks like our model predicts reasonably well. Let's take a look at the error distribution.

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

png

It's not quite gaussian, but we might expect that because the number of samples is very small.

Conclusion

This notebook introduced a few techniques to handle a regression problem.

  • Mean Squared Error (MSE) is a common loss function used for regression problems (different loss functions are used for classification problems).
  • Similarly, evaluation metrics used for regression differ from classification. A common regression metric is Mean Absolute Error (MAE).
  • When numeric input data features have values with different ranges, each feature should be scaled independently to the same range.
  • If there is not much training data, one technique is to prefer a small network with few hidden layers to avoid overfitting.
  • Early stopping is a useful technique to prevent overfitting.
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