This tutorial shows how to adjust prediction intervals in time series forecasting using Keras recurrent neural networks and Python. We build on a previous article on stock market forecasting, in which we created a forecast for the S&P500 stock market index. The prediction interval used in this last article was a single day. Other time-series prediction problems may require us to make predictions that range further ahead, say, several days, weeks, or months. We can modify the prediction interval by slightly changing the data preparation and model architecture. This article shows how this works. We will adjust the prediction intervals to create a single-step forecast for a longer time frame.

The remainder of this article proceeds as follows. We begin with a brief overview of different methods we can use to adjust the time series prediction interval. Once we are familiar with these concepts, we turn to the practical part in Python. We will train a simple neural network on stock market data. After validating the performance of our model, we will prepare the data in a way that allows us to forecast a single but more extended step into the future.

## Ways of Adjusting Prediction Intervals

When forecasting one step, the prediction interval is the point in time for which a prediction model will simulate the next value. There are three different ways to change the prediction interval:

**Multi-Step Rolling Forecasting:**Another way is to train the model on its output. We do this by maintaining the predictions and reusing them as input in the subsequent training run. In this way, the projections range one-time step further ahead with each iteration. After seven iterations, based on daily input time steps, the model will have provided the output for a weekly prediction. We have covered this rolling forecasting approach in a separate tutorial.**Deep Multi-Output Forecasting:**A third option is to create a multi-output model that provides an entire series of predictions with multiple timesteps. We have covered this multi-output forecasting approach in a separate tutorial.**Single-step forecasting with bigger timesteps:**In a single-stage forecasting approach, the input data defines the length of a time step. Changing the size of the input steps will change the output steps to the same extent. For example, a model that uses daily prices as input data will also provide day-to-day forecasts. We will cover this forecasting approach in the following section.

## Predicting the Price of the S&P500 One Week Ahead

Let’s begin with the hands-on part. In this part, we will use Python to create a single-step forecasting model with more extended timesteps. Our model will make projections that reach one week ahead. For this purpose, we reuse most of the code from the previous article on univariate single-step daily forecasting. So we won’t go into all the details and will only speak about the areas in which we must adjust the code. Changes are necessary to data preparation and model architecture.

In the following, we develop a single-variate neural network model that forecasts the S&P500 stock market index. The code is available on the GitHub repository.

### Prerequisites

Before starting the coding part, make sure that you have set up your Python 3 environment and required packages. If you don’t have an environment, you can follow these steps to set up the Anaconda environment.

Also, make sure you install all required packages. In this tutorial, we will be working with the following standard packages:

In addition, we will be using *Keras *(2.0 or higher) with *Tensorflow* backend, the machine learning library sci-kit-learn, and pandas DataReader to interact with the yahoo finance API.

You can install packages using console commands:

*pip install <package name>**conda install <package name>*(if you are using the anaconda packet manager)

### Step #1 Load the Data

In the following, we will modify the prediction interval of the neural network model we developed in a previous post. As a result, the model will generate predictions for the market price of the S&P500 Index that range one week ahead.

As before, we start loading the stock market data via an API.

# A tutorial for this file is available at www.relataly.com import math # Fundamental package for scientific computing with Python import numpy as np # Additional functions for analysing and manipulating data import pandas as pd # Date Functions from datetime import date, timedelta # This function adds plotting functions for calender dates from pandas.plotting import register_matplotlib_converters # Important package for visualization - we use this to plot the market data import matplotlib.pyplot as plt # Formatting dates import matplotlib.dates as mdates # Packages for measuring model performance / errors from sklearn.metrics import mean_absolute_error, mean_squared_error # Tools for predictive data analysis. We will use the MinMaxScaler to normalize the price data from sklearn.preprocessing import MinMaxScaler # Deep learning library, used for neural networks from tensorflow.keras.models import Sequential # Deep learning classes for recurrent and regular densely-connected layers from tensorflow.keras.layers import LSTM, Dense from tensorflow.keras.callbacks import EarlyStopping import tensorflow as tf import seaborn as sns sns.set_style('white', { 'axes.spines.right': False, 'axes.spines.top': False}) # check the tensorflow version and the number of available GPUs print('Tensorflow Version: ' + tf.__version__) physical_devices = tf.config.list_physical_devices('GPU') print("Num GPUs:", len(physical_devices)) # Setting the timeframe for the data extraction end_date = date.today().strftime("%Y-%m-%d") start_date = '2010-01-01' # Getting S&P500 quotes stockname = 'S&P500' symbol = '^GSPC' # You can either use webreader or yfinance to load the data from yahoo finance # import pandas_datareader as webreader # df = webreader.DataReader(symbol, start=start_date, end=end_date, data_source="yahoo") import yfinance as yf #Alternative package if webreader does not work: pip install yfinance df = yf.download(symbol, start=start_date, end=end_date) df.head(5)

### Step #2 Adjusting the Shape of the Input Data and Exploration

We have a DataFrame that contains the daily price quotes for the S&P500. Next, we prepare the data to include the weekly price quotes. If we want our model to provide weekly price predictions, we need to change the data so that the input contains weekly price quotes. A simple way to achieve this is to iterate through the rows and only keep every 7th row.

# Changing the data structure to a dataframe with weekly price quotes df["index1"] = range(1, len(df) + 1) rownumber = df.shape[0] lst = list(range(rownumber)) list_of_relevant_numbers = lst[0::7] df_weekly = df[df["index1"].isin(list_of_relevant_numbers)] df_weekly.head(5)

After this, we quickly create a line plot to validate that everything looks as expected.

# Creating a Lineplot years = mdates.YearLocator() fig, ax1 = plt.subplots(figsize=(16, 6)) ax1.xaxis.set_major_locator(years) ax1.legend([stockname], fontsize=12) plt.title(stockname + ' from '+ start_date + ' to ' + end_date) sns.lineplot(data=df['Close'], label=stockname, linewidth=1.0) plt.ylabel('S&P500 Points') plt.show()

### Step #3 Preprocess the Data

Before we can train the neural network, we first need to define the shape of the training data. We use weekly price quotes and define an input_sequence_length of 50 weeks.

# Feature Selection - Only Close Data train_df = df.filter(['Close']) data_unscaled = df.values # Transform features by scaling each feature to a range between 0 and 1 mmscaler = MinMaxScaler(feature_range=(0, 1)) np_data = mmscaler.fit_transform(data_unscaled) # Creating a separate scaler that works on a single column for scaling predictions scaler_pred = MinMaxScaler() df_Close = pd.DataFrame(df['Close']) np_Close_scaled = scaler_pred.fit_transform(df_Close) # Set the sequence length - this is the timeframe used to make a single prediction sequence_length = 25 # Prediction Index index_Close = train_df.columns.get_loc("Close") # Split the training data into train and train data sets # As a first step, we get the number of rows to train the model on 80% of the data train_data_length = math.ceil(np_data.shape[0] * 0.8) # Create the training and test data train_data = np_data[0:train_data_length, :] test_data = np_data[train_data_length - sequence_length:, :] # The RNN needs data with the format of [samples, time steps, features] # Here, we create N samples, sequence_length time steps per sample, and 6 features def partition_dataset(sequence_length, train_df): x, y = [], [] data_len = train_df.shape[0] for i in range(sequence_length, data_len): x.append(train_df[i-sequence_length:i,:]) #contains sequence_length values 0-sequence_length * columsn y.append(train_df[i, index_Close]) #contains the prediction values for validation (3rd column = Close), for single-step prediction # Convert the x and y to numpy arrays x = np.array(x) y = np.array(y) return x, y # Generate training data and test data x_train, y_train = partition_dataset(sequence_length, train_data) x_test, y_test = partition_dataset(sequence_length, test_data) # Print the shapes: the result is: (rows, training_sequence, features) (prediction value, ) print(x_train.shape, y_train.shape) print(x_test.shape, y_test.shape) # Validate that the prediction value and the input match up # The last close price of the second input sample should equal the first prediction value print(x_test[1][sequence_length-1][index_Close]) print(y_test[0])

** Out:** (245, 50, 1)

### Step #4 Building a Time Series Prediction Model

The first layer of neurons in our neural network needs to fit the input values from the data. Therefore, we need 50 neurons – one for each input price quote.

We use the following input arguments for the model fit:

**x_train:**Vector, matrix, or array of training data. It can also be a list (as in our case) if the model has multiple inputs.**y_train**: Vector, matrix, or array of target data. This is the labeled data the model tries to predict; in other words, these are the results of x_train.- Epochs: The integer value defines how often the model goes through the training set.
**Batch size:**Integer value that defines the number of samples that will be propagated through the network. After each propagation, the network adjusts the weights of the nodes in each layer.

# Configure the neural network model model = Sequential() # Model with n_neurons Neurons n_neurons = x_train.shape[1] * x_train.shape[2] print(n_neurons, x_train.shape[1], x_train.shape[2]) model.add(LSTM(n_neurons, return_sequences=True, input_shape=(x_train.shape[1], x_train.shape[2]))) model.add(LSTM(n_neurons, return_sequences=False)) model.add(Dense(25, activation="relu")) model.add(Dense(1)) # Compile the model model.compile(optimizer="adam", loss="mean_squared_error") # Training the model epochs = 10 early_stop = EarlyStopping(monitor='loss', patience=2, verbose=1) history = model.fit(x_train, y_train, batch_size=16, epochs=epochs, callbacks=[early_stop])

### Step #5 Evaluate Model Performance

Next, we validate the model by calculating our predictions’ mean-squared and root-mean-squared errors. However, in time series forecasting metrics can be misleading. It is good to double-check model results using illustrations. Therefore, we plot the input sequences and the forecast to see if our model can continue the time series in a plausible way.

# Get the predicted values y_pred_scaled = model.predict(x_test) y_pred = scaler_pred.inverse_transform(y_pred_scaled) y_test_unscaled = scaler_pred.inverse_transform(y_test.reshape(-1, 1)) # Mean Absolute Error (MAE) MAE = mean_absolute_error(y_test_unscaled, y_pred) print(f'Median Absolute Error (MAE): {np.round(MAE, 2)}') # Mean Absolute Percentage Error (MAPE) MAPE = np.mean((np.abs(np.subtract(y_test_unscaled, y_pred)/ y_test_unscaled))) * 100 print(f'Mean Absolute Percentage Error (MAPE): {np.round(MAPE, 2)} %') # Median Absolute Percentage Error (MDAPE) MDAPE = np.median((np.abs(np.subtract(y_test_unscaled, y_pred)/ y_test_unscaled)) ) * 100 print(f'Median Absolute Percentage Error (MDAPE): {np.round(MDAPE, 2)} %') # The date from which on the date is displayed display_start_date = "2018-01-01" # Add the difference between the valid and predicted prices train = pd.DataFrame(train_df[:train_data_length + 1]).rename(columns={'Close': 'x_train'}) valid = pd.DataFrame(train_df[train_data_length:]).rename(columns={'Close': 'y_test'}) valid.insert(1, "y_pred", y_pred, True) valid.insert(1, "residuals", valid["y_pred"] - valid["y_test"], True) df_union = pd.concat([train, valid]) # Zoom in to a closer timeframe df_union_zoom = df_union[df_union.index > display_start_date] # Create the lineplot fig, ax1 = plt.subplots(figsize=(16, 8)) plt.title("Predictions vs Ground Truth") plt.ylabel(stockname, fontsize=18) sns.despine(); sns.lineplot(data=df_union_zoom, linewidth=1.0, palette='CMRmap', ax=ax1) plt.show()

** Out: **114.9 (RMSE) /

**Out:**155.9 (ME)

At the bottom, we can see the differences between predictions and valid data. Positive values signal that the projections were too optimistic. Negative values mean that the predictions were too pessimistic and that the actual value turned out to be higher than the prediction.

### Step #6 Predicting for the Next Week

What can be more satisfying than to see a newly trained model at work? Let’s use our new model to predict next week’s price for the S&P500. We will create a fresh input_sequence with prices from the past N days. Then we scale this input sequence and include it as input in our call to the model.predict() function.

# Get fresh data until today and create a new dataframe with only the price data new_df = df N = sequence_length # Get the last N steps closing price values and scale the data to be values between 0 and 1 last_N_steps = new_df[-sequence_length:].values last_N_steps_scaled = mmscaler.transform(last_N_steps) # Create an empty list and Append past N steps X_test_new = [] X_test_new.append(last_N_steps_scaled) # Convert the X_test data set to a numpy array and reshape the data pred_price_scaled = model.predict(np.array(X_test_new)) pred_price_unscaled = scaler_pred.inverse_transform(pred_price_scaled.reshape(-1, 1)) # Print last price and predicted price for the next week price_today = np.round(new_df['Close'][-1], 2) predicted_price = np.round(pred_price_unscaled.ravel()[0], 2) change_percent = np.round(100 - (price_today * 100)/predicted_price, 2) plus = '+'; minus = '' print(f'The close price for {stockname} at {end_date} was {price_today}') print(f'The predicted close price is {predicted_price} ({plus if change_percent > 0 else minus}{change_percent}%)')

**Out:** 2652.3

So for the 9th of April 2020, the model predicts that the S&P500 will close at:

**2652.3**

Considering today’s (2nd of April 2020) price is 2528 points, our model expects the S&P to gain roughly 124 points in the coming seven days. Of course, this is by no means financial advice. As we have seen before, our model is often wrong.

## Summary

This article has shown how to adjust the prediction intervals for a time series forecasting model. We have created a neural network that predicts the price of the S&P500 one week in advance. Finally, we trained and validated the model and made a forecast for the next week.

Varying the input shape is a quick approach to changing the forecasting time steps. However, increasing the length of the time steps also reduces the amount of data we can use for training and testing. In our case, we still have enough data available. But in other cases, where less information is available, this can become a problem. The preferred method is to use a rolling forecast approach or create a multi-output forecast in such a case.

I hope this article was helpful. Should you have questions or remarks, let me know in the comments.

## Sources and Further Readings

- Charu C. Aggarwal (2016) Recommender Systems: The Textbook
- Kin Falk (2019) Practical Recommender Systems
- Andriy Burkov (2020) Machine Learning Engineering
- Oliver Theobald (2020) Machine Learning For Absolute Beginners: A Plain English Introduction
- Aurélien Géron (2019) Hands-On Machine Learning with Scikit-Learn, Keras, and TensorFlow: Concepts, Tools, and Techniques to Build Intelligent Systems
- David Forsyth (2019) Applied Machine Learning Springer

*The links above to Amazon are affiliate links. By buying through these links, you support the Relataly.com blog and help to cover the hosting costs. Using the links does not affect the price.*

hi there,

i did not get the code. first you tried to convert daily data into week data. but in neural model, you utilized daily data. Also did prediction on daily x_test data. then how you predicted next week data?

in the daily timeframe you could also forecast a price 7 days (or 7 bars ) in the future by adjusting the samples. For instance, see below. I tested it and it works. So it just forecasts 1 bar but 7 bars ahead instead of just 1 bar ahead. Of course the length of the array pred_price_unscaled will also be nfut bars shorter.

Let me know what you think

nfut = 7

def partition_dataset(sequence_length, train_df):

x, y = [], []

data_len = train_df.shape[0]

for i in range(sequence_length, data_len – nfut + 1):

x.append(train_df[i-sequence_length:i,:]) #contains sequence_length values 0-sequence_length * columsn

y.append(train_df[i + nfut – 1, index_Close]) #contains the prediction values for validation (3rd column = Close), for single-step prediction

# Convert the x and y to numpy arrays

x = np.array(x)

y = np.array(y)

return x, y