4.10. Predicting House Prices on Kaggle
Open the notebook in Colab

Now that we have introduced some basic tools for building and training deep networks and regularizing them with techniques including dimensionality reduction, weight decay, and dropout, we are ready to put all this knowledge into practice by participating in a Kaggle competition. Predicting house prices is a great place to start. The data is fairly generic and does not exhibit exotic structure that might require specialized models (as audio or video might). This dataset, collected by Bart de Cock in 2011 [DeCock, 2011], covers house prices in Ames, IA from the period of 2006-2010. It is considerably larger than the famous Boston housing dataset of Harrison and Rubinfeld (1978), boasting both more examples and more features.

In this section, we will walk you through details of data preprocessing, model design, and hyperparameter selection. We hope that through a hands-on approach, you will gain some intuitions that will guide you in your career as a data scientist.

4.10.1. Downloading and Caching Datasets

Throughout the book, we will train and test models on various downloaded datasets. Here, we implement several utility functions to facilitate data downloading. First, we maintain a dictionary DATA_HUB that maps a string (the name of the dataset) to a tuple containing both a URL to locate the dataset and a SHA-1 key which we will use to verify the integrity of the file. All of our datasets are hosted on site whose address is assigned to DATA_URL below.

import os
from mxnet import gluon
import zipfile
import tarfile

# Saved in the d2l package for later use
DATA_HUB = dict()

# Saved in the d2l package for later use
DATA_URL = 'http://d2l-data.s3-accelerate.amazonaws.com/'

The following download function downloads the dataset, caching it in a local directory (in ../data by default) and returns the name of the downloaded file. If a file corrsponding to this dataset already exists in the cache directory and its SHA-1 matches the one stored in DATA_HUB, our code will use the cached file to avoid clogging up your internet with redundant downloads.

# Saved in the d2l package for later use
def download(name, cache_dir=os.path.join('..', 'data')):
    """Download a file inserted into DATA_HUB, return the local filename."""
    assert name in DATA_HUB, "%s does not exist" % name
    url, sha1 = DATA_HUB[name]
    return gluon.utils.download(url, cache_dir, sha1_hash=sha1)

We also implement two additional functions: one is to download and extract a zip/tar file and the other to download all the files from DATA_HUB (most of the datasets used in this book) into the cache directory.

# Saved in the d2l package for later use
def download_extract(name, folder=None):
    """Download and extract a zip/tar file."""
    fname = download(name)
    base_dir = os.path.dirname(fname)
    data_dir, ext = os.path.splitext(fname)
    if ext == '.zip':
        fp = zipfile.ZipFile(fname, 'r')
    elif ext in ('.tar', '.gz'):
        fp = tarfile.open(fname, 'r')
        assert False, 'Only zip/tar files can be extracted'
    if folder:
        return os.path.join(base_dir, folder)
        return data_dir

# Saved in the d2l package for later use
def download_all():
    """Download all files in the DATA_HUB"""
    for name in DATA_HUB:

4.10.2. Kaggle

Kaggle is a popular platform that hosts machine learning competitions. Each competition centers on a dataset and many are sponsored by stakeholders who offer prizes to the winning solutions. The platform helps users to share interact via forums and shared code, fostering both collaboration and competition. While leaderboard chasing often spirals out of control, with researchers focusing myopically on pre-processing steps rather than asking fundamental questions, there is also tremendous value in the objectivity of a platform that facillitates direct quantitative comparisons between competing approaches as well as code sharing so that everyone can learn what did and did not work. If you want to participate in a Kaggle competitions, you will first need to register for an account (see Fig. 4.10.1).


Fig. 4.10.1 Kaggle website

On the House Prices Prediction page, as illustrated in Fig. 4.10.2, you can find the dataset (under the “Data” tab), submit predictions, see your ranking, etc., The URL is right here:


Fig. 4.10.2 House Price Prediction

4.10.3. Accessing and Reading the Dataset

Note that the competition data is separated into training and test sets. Each record includes the property value of the house and attributes such as street type, year of construction, roof type, basement condition, etc. The features consist of various data types. For example, the year of construction is represented by an integer, the roof type by discrete categorical assignments, and other features by floating point numbers. And here is where reality complicates things: for some examples, some data is altogether missing with the missing value marked simply as na. The price of each house is included for the training set only (it is a competition after all). We will want to partition the training set to create a validation set, but we only get to evaluate our models on the official test set after uploading predictions to Kaggle. The “Data” tab on the competition tab has links to download the data.

To get started, we will read in and process the data using pandas, an efficient data analysis toolkit, so you will want to make sure that you have pandas installed before proceeding further. Fortunately, if you are reading in Jupyter, we can install pandas without even leaving the notebook.

# If pandas is not installed, please uncomment the following line:
# !pip install pandas

%matplotlib inline
import d2l
from mxnet import autograd, init, np, npx
from mxnet.gluon import nn
import pandas as pd

For convenience, we can download and cache the Kaggle housing dataset using the script we defined above.

# Saved in the d2l package for later use
DATA_HUB['kaggle_house_train'] = (
    DATA_URL + 'kaggle_house_pred_train.csv',

# Saved in the d2l package for later use
DATA_HUB['kaggle_house_test'] = (
    DATA_URL + 'kaggle_house_pred_test.csv',

To load the two csv files containing training and test data respectively we use Pandas.

train_data = pd.read_csv(download('kaggle_house_train'))
test_data = pd.read_csv(download('kaggle_house_test'))
Downloading ../data/kaggle_house_pred_train.csv from http://d2l-data.s3-accelerate.amazonaws.com/kaggle_house_pred_train.csv...
Downloading ../data/kaggle_house_pred_test.csv from http://d2l-data.s3-accelerate.amazonaws.com/kaggle_house_pred_test.csv...

The training dataset includes \(1460\) examples, \(80\) features, and \(1\) label, while the test data contains \(1459\) examples and \(80\) features.

(1460, 81)
(1459, 80)

Let’s take a look at the first \(4\) and last \(2\) features as well as the label (SalePrice) from the first \(4\) examples:

print(train_data.iloc[0:4, [0, 1, 2, 3, -3, -2, -1]])
   Id  MSSubClass MSZoning  LotFrontage SaleType SaleCondition  SalePrice
0   1          60       RL         65.0       WD        Normal     208500
1   2          20       RL         80.0       WD        Normal     181500
2   3          60       RL         68.0       WD        Normal     223500
3   4          70       RL         60.0       WD       Abnorml     140000

We can see that in each example, the first feature is the ID. This helps the model identify each training example. While this is convenient, it does not carry any information for prediction purposes. Hence, we remove it from the dataset before feeding the data into the network.

all_features = pd.concat((train_data.iloc[:, 1:-1], test_data.iloc[:, 1:]))

4.10.4. Data Preprocessing

As stated above, we have a wide variety of data types. We will need to process the data before we can start modeling. Let us start with the numerical features. First, we apply a heuristic, replacing all missing values by the corresponding variable’s mean. Then, to put all variables on a common scale, we rescale them to zero mean and unit variance:

(4.10.1)\[x \leftarrow \frac{x - \mu}{\sigma}.\]

To verify that this indeed transforms our variable such that it has zero mean and unit variance, note that \(E[(x-\mu)/\sigma] = (\mu - \mu)/\sigma = 0\) and that \(E[(x-\mu)^2] = \sigma^2\). Intuitively, we normalize the data for two reasons. First, it proves convenient for optimization. Second, because we do not know a priori which features will be relevant, we do not want to penalize coefficients assigned to one variable more than on any other.

numeric_features = all_features.dtypes[all_features.dtypes != 'object'].index
all_features[numeric_features] = all_features[numeric_features].apply(
    lambda x: (x - x.mean()) / (x.std()))
# After standardizing the data all means vanish, hence we can set missing
# values to 0
all_features[numeric_features] = all_features[numeric_features].fillna(0)

Next we deal with discrete values. This includes variables such as ‘MSZoning’. We replace them by a one-hot encoding in the same way that we previously transformed multiclass labels into vectors. For instance, ‘MSZoning’ assumes the values ‘RL’ and ‘RM’. These map onto vectors \((1, 0)\) and \((0, 1)\) respectively. Pandas does this automatically for us.

# Dummy_na=True refers to a missing value being a legal eigenvalue, and
# creates an indicative feature for it
all_features = pd.get_dummies(all_features, dummy_na=True)
(2919, 331)

You can see that this conversion increases the number of features from 79 to 331. Finally, via the values attribute, we can extract the NumPy format from the Pandas dataframe and convert it into MXNet’s native ndarray representation for training.

n_train = train_data.shape[0]
train_features = np.array(all_features[:n_train].values, dtype=np.float32)
test_features = np.array(all_features[n_train:].values, dtype=np.float32)
train_labels = np.array(train_data.SalePrice.values,
                        dtype=np.float32).reshape(-1, 1)

4.10.5. Training

To get started we train a linear model with squared loss. Not surprisingly, our linear model will not lead to a competition-winning submission but it provides a sanity check to see whether there is meaningful information in the data. If we cannot do better than random guessing here, then there might be a good chance that we have a data processing bug. And if things work, the linear model will serve as a baseline giving us some intuition about how close the simple model gets to the best reported models, giving us a sense of how much gain we should expect from fancier models.

loss = gluon.loss.L2Loss()

def get_net():
    net = nn.Sequential()
    return net

With house prices, as with stock prices, we care about relative quantities more than absolute quantities. Thus we tend to care more about the relative error \(\frac{y - \hat{y}}{y}\) than about the absolute error \(y - \hat{y}\). For instance, if our prediction is off by USD 100,000 when estimating the price of a house in Rural Ohio, where the value of a typical house is 125,000 USD, then we are probably doing a horrible job. On the other hand, if we err by this amount in Los Altos Hills, California, this might represent a stunningly accurate prediction (there, the median house price exceeds 4 million USD).

One way to address this problem is to measure the discrepancy in the logarithm of the price estimates. In fact, this is also the official error metric used by the competition to measure the quality of submissions. After all, a small value \(\delta\) of \(\log y - \log \hat{y}\) translates into \(e^{-\delta} \leq \frac{\hat{y}}{y} \leq e^\delta\). This leads to the following loss function:

(4.10.2)\[L = \sqrt{\frac{1}{n}\sum_{i=1}^n\left(\log y_i -\log \hat{y}_i\right)^2}.\]
def log_rmse(net, features, labels):
    # To further stabilize the value when the logarithm is taken, set the
    # value less than 1 as 1
    clipped_preds = np.clip(net(features), 1, float('inf'))
    return np.sqrt(2 * loss(np.log(clipped_preds), np.log(labels)).mean())

Unlike in previous sections, our training functions will rely on the Adam optimizer (a slight variant on SGD that we will describe in greater detail later). The main appeal of Adam vs vanilla SGD is that the Adam optimizer, despite doing no better (and sometimes worse) given unlimited resources for hyperparameter optimization, people tend to find that it is significantly less sensitive to the initial learning rate. This will be covered in further detail later on when we discuss the details in Section 11.

def train(net, train_features, train_labels, test_features, test_labels,
          num_epochs, learning_rate, weight_decay, batch_size):
    train_ls, test_ls = [], []
    train_iter = d2l.load_array((train_features, train_labels), batch_size)
    # The Adam optimization algorithm is used here
    trainer = gluon.Trainer(net.collect_params(), 'adam', {
        'learning_rate': learning_rate, 'wd': weight_decay})
    for epoch in range(num_epochs):
        for X, y in train_iter:
            with autograd.record():
                l = loss(net(X), y)
        train_ls.append(log_rmse(net, train_features, train_labels))
        if test_labels is not None:
            test_ls.append(log_rmse(net, test_features, test_labels))
    return train_ls, test_ls

4.10.6. k-Fold Cross-Validation

If you are reading in a linear fashion, you might recall that we introduced k-fold cross-validation in the section where we discussed how to deal with model section (Section 4.4). We will put this to good use to select the model design and to adjust the hyperparameters. We first need a function that returns the \(i^\mathrm{th}\) fold of the data in a k-fold cross-validation procedure. It proceeds by slicing out the \(i^\mathrm{th}\) segment as validation data and returning the rest as training data. Note that this is not the most efficient way of handling data and we would definitely do something much smarter if our dataset was considerably larger. But this added complexity might obfuscate our code unnecessarily so we can safely omit here owing to the simplicity of our problem.

def get_k_fold_data(k, i, X, y):
    assert k > 1
    fold_size = X.shape[0] // k
    X_train, y_train = None, None
    for j in range(k):
        idx = slice(j * fold_size, (j + 1) * fold_size)
        X_part, y_part = X[idx, :], y[idx]
        if j == i:
            X_valid, y_valid = X_part, y_part
        elif X_train is None:
            X_train, y_train = X_part, y_part
            X_train = np.concatenate((X_train, X_part), axis=0)
            y_train = np.concatenate((y_train, y_part), axis=0)
    return X_train, y_train, X_valid, y_valid

The training and verification error averages are returned when we train \(k\) times in the k-fold cross-validation.

def k_fold(k, X_train, y_train, num_epochs,
           learning_rate, weight_decay, batch_size):
    train_l_sum, valid_l_sum = 0, 0
    for i in range(k):
        data = get_k_fold_data(k, i, X_train, y_train)
        net = get_net()
        train_ls, valid_ls = train(net, *data, num_epochs, learning_rate,
                                   weight_decay, batch_size)
        train_l_sum += train_ls[-1]
        valid_l_sum += valid_ls[-1]
        if i == 0:
            d2l.plot(list(range(1, num_epochs+1)), [train_ls, valid_ls],
                     xlabel='epoch', ylabel='rmse',
                     legend=['train', 'valid'], yscale='log')
        print('fold %d, train rmse: %f, valid rmse: %f' % (
            i, train_ls[-1], valid_ls[-1]))
    return train_l_sum / k, valid_l_sum / k

4.10.7. Model Selection

In this example, we pick an untuned set of hyperparameters and leave it up to the reader to improve the model. Finding a good choice can take time, depending on how many variables one optimizes over. With a large enough dataset, and the normal sorts of hyperparameters reason, k-fold cross-validation tends to be reasonably resilient against multiple testing. However, if we try an unreasonably large number of options we might just get lucky and find that our validation performance is no longer representative of the true error.

k, num_epochs, lr, weight_decay, batch_size = 5, 100, 5, 0, 64
train_l, valid_l = k_fold(k, train_features, train_labels, num_epochs, lr,
                          weight_decay, batch_size)
print('%d-fold validation: avg train rmse: %f, avg valid rmse: %f'
      % (k, train_l, valid_l))
fold 0, train rmse: 0.169738, valid rmse: 0.157274
fold 1, train rmse: 0.162218, valid rmse: 0.189319
fold 2, train rmse: 0.163707, valid rmse: 0.167919
fold 3, train rmse: 0.167962, valid rmse: 0.154935
fold 4, train rmse: 0.162838, valid rmse: 0.182808
5-fold validation: avg train rmse: 0.165293, avg valid rmse: 0.170451

Notice that someimes the number of training errors for a set of hyperparameters can be very low, even as the number of errors on \(k\)-fold cross-validation is considerably higher. This indicates that we are overfitting. Throughout training you will want to monitor both numbers. No overfitting might indicate that our data can support a more powerful model. Massive overfitting might suggest that we can gain by incorporating regularization techniques.

4.10.8. Predict and Submit

Now that we know what a good choice of hyperparameters should be, we might as well use all the data to train on it (rather than just \(1-1/k\) of the data that is used in the cross-validation slices). The model that we obtain in this way can then be applied to the test set. Saving the estimates in a CSV file will simplify uploading the results to Kaggle.

def train_and_pred(train_features, test_feature, train_labels, test_data,
                   num_epochs, lr, weight_decay, batch_size):
    net = get_net()
    train_ls, _ = train(net, train_features, train_labels, None, None,
                        num_epochs, lr, weight_decay, batch_size)
    d2l.plot(np.arange(1, num_epochs + 1), [train_ls], xlabel='epoch',
             ylabel='rmse', yscale='log')
    print('train rmse %f' % train_ls[-1])
    # Apply the network to the test set
    preds = net(test_features).asnumpy()
    # Reformat it for export to Kaggle
    test_data['SalePrice'] = pd.Series(preds.reshape(1, -1)[0])
    submission = pd.concat([test_data['Id'], test_data['SalePrice']], axis=1)
    submission.to_csv('submission.csv', index=False)

One nice sanity check is to see whether the predictions on the test set resemble those of the k-fold cross-validation process. If they do, it is time to upload them to Kaggle. The following code will generate a file called submission.csv (CSV is one of the file formats accepted by Kaggle):

train_and_pred(train_features, test_features, train_labels, test_data,
               num_epochs, lr, weight_decay, batch_size)
train rmse 0.162449

Next, as demonstrated in Fig. 4.10.3, we can submit our predictions on Kaggle and see how they compare to the actual house prices (labels) on the test set. The steps are quite simple:

  • Log in to the Kaggle website and visit the House Price Prediction Competition page.

  • Click the “Submit Predictions” or “Late Submission” button (as of this writing, the button is located on the right).

  • Click the “Upload Submission File” button in the dashed box at the bottom of the page and select the prediction file you wish to upload.

  • Click the “Make Submission” button at the bottom of the page to view your results.


Fig. 4.10.3 Submitting data to Kaggle

4.10.9. Summary

  • Real data often contains a mix of different data types and needs to be preprocessed.

  • Rescaling real-valued data to zero mean and unit variance is a good default. So is replacing missing values with their mean.

  • Transforming categorical variables into indicator variables allows us to treat them like vectors.

  • We can use k-fold cross validation to select the model and adjust the hyper-parameters.

  • Logarithms are useful for relative loss.

4.10.10. Exercises

  1. Submit your predictions for this tutorial to Kaggle. How good are your predictions?

  2. Can you improve your model by minimizing the log-price directly? What happens if you try to predict the log price rather than the price?

  3. Is it always a good idea to replace missing values by their mean? Hint: can you construct a situation where the values are not missing at random?

  4. Find a better representation to deal with missing values. Hint: what happens if you add an indicator variable?

  5. Improve the score on Kaggle by tuning the hyperparameters through k-fold cross-validation.

  6. Improve the score by improving the model (layers, regularization, dropout).

  7. What happens if we do not standardize the continuous numerical features like we have done in this section?

4.10.11. Discussions