In this notebook, we will use LSTM layers to develop time series forecasting models. The dataset used for the examples of this notebook is on air pollution measured by concentration of particulate matter (PM) of diameter less than or equal to 2.5 micrometers. There are other variables such as air pressure, air temparature, dewpoint and so on. Two time series models are developed - one on air pressure and the other on pm2.5. The dataset has been downloaded from UCI Machine Learning Repository. https://archive.ics.uci.edu/ml/datasets/Beijing+PM2.5+Data
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from __future__ import print_function
import os
import sys
import pandas as pd
import numpy as np
%matplotlib inline
from matplotlib import pyplot as plt
import seaborn as sns
import datetime
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#set current working directory
os.chdir('D:/Practical Time Series')
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#Read the dataset into a pandas.DataFrame
df = pd.read_csv('datasets/PRSA_data_2010.1.1-2014.12.31.csv')
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print('Shape of the dataframe:', df.shape)
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#Let's see the first five rows of the DataFrame
df.head()
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"""
Rows having NaN values in column pm2.5 are dropped.
"""
df.dropna(subset=['pm2.5'], axis=0, inplace=True)
df.reset_index(drop=True, inplace=True)
To make sure that the rows are in the right order of date and time of observations, a new column datetime is created from the date and time related columns of the DataFrame. The new column consists of Python's datetime.datetime objects. The DataFrame is sorted in ascending order over this column.
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df['datetime'] = df[['year', 'month', 'day', 'hour']].apply(lambda row: datetime.datetime(year=row['year'], month=row['month'], day=row['day'],
hour=row['hour']), axis=1)
df.sort_values('datetime', ascending=True, inplace=True)
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#Let us draw a box plot to visualize the central tendency and dispersion of PRES
plt.figure(figsize=(5.5, 5.5))
g = sns.boxplot(df['pm2.5'])
g.set_title('Box plot of pm2.5')
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In [9]:
plt.figure(figsize=(5.5, 5.5))
g = sns.tsplot(df['pm2.5'])
g.set_title('Time series of pm2.5')
g.set_xlabel('Index')
g.set_ylabel('pm2.5 readings')
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In [10]:
#Let's plot the series for six months to check if any pattern apparently exists.
plt.figure(figsize=(5.5, 5.5))
g = sns.tsplot(df['pm2.5'].loc[df['datetime']<=datetime.datetime(year=2010,month=6,day=30)], color='g')
g.set_title('pm2.5 during 2010')
g.set_xlabel('Index')
g.set_ylabel('pm2.5 readings')
#Let's zoom in on one month.
plt.figure(figsize=(5.5, 5.5))
g = sns.tsplot(df['pm2.5'].loc[df['datetime']<=datetime.datetime(year=2010,month=1,day=31)], color='g')
g.set_title('pm2.5 during Jan 2010')
g.set_xlabel('Index')
g.set_ylabel('pm2.5 readings')
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Gradient descent algorithms perform better (for example converge faster) if the variables are wihtin range [-1, 1]. Many sources relax the boundary to even [-3, 3]. The pm2.5 variable is mixmax scaled to bound the tranformed variable within [0,1].
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from sklearn.preprocessing import MinMaxScaler
scaler = MinMaxScaler(feature_range=(0, 1))
df['scaled_pm2.5'] = scaler.fit_transform(np.array(df['pm2.5']).reshape(-1, 1))
Before training the model, the dataset is split in two parts - train set and validation set. The neural network is trained on the train set. This means computation of the loss function, back propagation and weights updated by a gradient descent algorithm is done on the train set. The validation set is used to evaluate the model and to determine the number of epochs in model training. Increasing the number of epochs will further decrease the loss function on the train set but might not neccesarily have the same effect for the validation set due to overfitting on the train set.Hence, the number of epochs is controlled by keeping a tap on the loss function computed for the validation set. We use Keras with Tensorflow backend to define and train the model. All the steps involved in model training and validation is done by calling appropriate functions of the Keras API.
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"""
Let's start by splitting the dataset into train and validation. The dataset's time period if from
Jan 1st, 2010 to Dec 31st, 2014. The first fours years - 2010 to 2013 is used as train and
2014 is kept for validation.
"""
split_date = datetime.datetime(year=2014, month=1, day=1, hour=0)
df_train = df.loc[df['datetime']<split_date]
df_val = df.loc[df['datetime']>=split_date]
print('Shape of train:', df_train.shape)
print('Shape of test:', df_val.shape)
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#First five rows of train
df_train.head()
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In [14]:
#First five rows of validation
df_val.head()
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#Reset the indices of the validation set
df_val.reset_index(drop=True, inplace=True)
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"""
The train and validation time series of scaled_pm2.5 is also plotted.
"""
plt.figure(figsize=(5.5, 5.5))
g = sns.tsplot(df_train['scaled_pm2.5'], color='b')
g.set_title('Time series of scaled pm2.5 in train set')
g.set_xlabel('Index')
g.set_ylabel('Scaled pm2.5 readings')
plt.figure(figsize=(5.5, 5.5))
g = sns.tsplot(df_val['scaled_pm2.5'], color='r')
g.set_title('Time series of scaled pm2.5 in validation set')
g.set_xlabel('Index')
g.set_ylabel('Scaled pm2.5 readings')
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Now we need to generate regressors (X) and target variable (y) for train and validation. 2-D array of regressor and 1-D array of target is created from the original 1-D array of columm log_PRES in the DataFrames. For the time series forecasting model, Past seven days of observations are used to predict for the next day. This is equivalent to a AR(7) model. We define a function which takes the original time series and the number of timesteps in regressors as input to generate the arrays of X and y.
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def makeXy(ts, nb_timesteps):
"""
Input:
ts: original time series
nb_timesteps: number of time steps in the regressors
Output:
X: 2-D array of regressors
y: 1-D array of target
"""
X = []
y = []
for i in range(nb_timesteps, ts.shape[0]):
X.append(list(ts.loc[i-nb_timesteps:i-1]))
y.append(ts.loc[i])
X, y = np.array(X), np.array(y)
return X, y
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X_train, y_train = makeXy(df_train['scaled_pm2.5'], 7)
print('Shape of train arrays:', X_train.shape, y_train.shape)
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X_val, y_val = makeXy(df_val['scaled_pm2.5'], 7)
print('Shape of validation arrays:', X_val.shape, y_val.shape)
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#X_train and X_val are reshaped to 3D arrays
X_train, X_val = X_train.reshape((X_train.shape[0], X_train.shape[1], 1)), X_val.reshape((X_val.shape[0], X_val.shape[1], 1))
print('Shape of arrays after reshaping:', X_train.shape, X_val.shape)
Now we define the MLP using the Keras Functional API. In this approach a layer can be declared as the input of the following layer at the time of defining the next layer.
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from keras.layers import Dense, Input, Dropout
from keras.layers.recurrent import LSTM
from keras.optimizers import SGD
from keras.models import Model
from keras.models import load_model
from keras.callbacks import ModelCheckpoint
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#Define input layer which has shape (None, 7) and of type float32. None indicates the number of instances
input_layer = Input(shape=(7,1), dtype='float32')
The LSTM layers are defined for seven timesteps. In this example, two LSTM layers are stacked. The first LSTM returns the output from each all seven timesteps. This output is a sequence and is fed to the second LSTM which returns output only from the last step. The first LSTM has sixty four hidden neurons in each timestep. Hence the sequence returned by the first LSTM has sixty four features.
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lstm_layer1 = LSTM(64, input_shape=(7,1), return_sequences=True)(input_layer)
lstm_layer2 = LSTM(32, input_shape=(7,64), return_sequences=False)(lstm_layer1)
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dropout_layer = Dropout(0.2)(lstm_layer2)
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#Finally the output layer gives prediction for the next day's air pressure.
output_layer = Dense(1, activation='linear')(dropout_layer)
The input, dense and output layers will now be packed inside a Model, which is wrapper class for training and making predictions. The box plot of pm2.5 shows the presence of outliers. Hence, mean absolute error (MAE) is used as absolute deviations suffer less fluctuations compared to squared deviations.
The network's weights are optimized by the Adam algorithm. Adam stands for adaptive moment estimation and has been a popular choice for training deep neural networks. Unlike, stochastic gradient descent, adam uses different learning rates for each weight and separately updates the same as the training progresses. The learning rate of a weight is updated based on exponentially weighted moving averages of the weight's gradients and the squared gradients.
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ts_model = Model(inputs=input_layer, outputs=output_layer)
ts_model.compile(loss='mean_absolute_error', optimizer='adam')#SGD(lr=0.001, decay=1e-5))
ts_model.summary()
The model is trained by calling the fit function on the model object and passing the X_train and y_train. The training is done for a predefined number of epochs. Additionally, batch_size defines the number of samples of train set to be used for a instance of back propagation.The validation dataset is also passed to evaluate the model after every epoch completes. A ModelCheckpoint object tracks the loss function on the validation set and saves the model for the epoch, at which the loss function has been minimum.
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save_weights_at = os.path.join('keras_models', 'PRSA_data_PM2.5_LSTM_weights.{epoch:02d}-{val_loss:.4f}.hdf5')
save_best = ModelCheckpoint(save_weights_at, monitor='val_loss', verbose=0,
save_best_only=True, save_weights_only=False, mode='min',
period=1)
ts_model.fit(x=X_train, y=y_train, batch_size=16, epochs=20,
verbose=1, callbacks=[save_best], validation_data=(X_val, y_val),
shuffle=True)
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Prediction are made for the pm2.5 from the best saved model. The model's predictions, which are on the standardized pm2.5, are inverse transformed to get predictions of original pm2.5.
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best_model = load_model(os.path.join('keras_models', 'PRSA_data_PM2.5_LSTM_weights.09-0.0117.hdf5'))
preds = best_model.predict(X_val)
pred_pm25 = scaler.inverse_transform(preds)
pred_pm25 = np.squeeze(pred_pm25)
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from sklearn.metrics import mean_absolute_error
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mae = mean_absolute_error(df_val['pm2.5'].loc[7:], pred_pm25)
print('MAE for the validation set:', round(mae, 4))
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#Let's plot the first 50 actual and predicted values of pm2.5.
plt.figure(figsize=(5.5, 5.5))
plt.plot(range(50), df_val['pm2.5'].loc[7:56], linestyle='-', marker='*', color='r')
plt.plot(range(50), pred_pm25[:50], linestyle='-', marker='.', color='b')
plt.legend(['Actual','Predicted'], loc=2)
plt.title('Actual vs Predicted pm2.5')
plt.ylabel('pm2.5')
plt.xlabel('Index')
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