# Time-series Analysis! ARIMA or Prophet?

## A very brief comparison between Auto ARIMA and Prophet by Facebook

Time Series is a class of data science problems where the primary values of interest are a series of data points measured over some time. This notebook aims to provide the basic building blocks of some of the more modern algorithms/techniques (and data!) for solving these types of problems.

Is ARIMA the first thing you think of when you hear about time series? It might be time to explore other ventures and methodologies. There is a lot of innovation and modern techniques being actively developed, and some of them are outperforming the traditional ARIMA models. We’ll look at some of these models and try to apply them to stock market data to predict the price.

Models explored in this notebook:

Auto ARIMAX

Let’s get some data!

`!pip install pmdarimaimport lightgbm as lgbimport numpy as npimport pandas as pdfrom fbprophet import Prophetfrom matplotlib import pyplot as pltfrom pmdarima import auto_arimafrom sklearn.metrics import mean_absolute_error, mean_squared_errormyfavouritenumber = 23seed = myfavouritenumbernp.random.seed(seed)`

# Nifty-50 Stock Market Data

The dataset used is stock market data of the Nifty-50 index from NSE (National Stock Exchange) India over the last 20 years (2000–2019)

The historic VWAP (Volume Weighted Average Price) is the target variable to predict. VWAP is a trading benchmark used by traders that gives the average price the stock has traded throughout the day, based on both volume and price.

The stock used is BAJAJFINSV.

# Data Preparation

Reading the market data of BAJAJFINSV stock and preparing a training dataset and validation dataset.

`df = pd.read_csv("/kaggle/input/nifty50-stock-market-data/BAJAJFINSV.csv")df.set_index("Date", drop=False, inplace=True)df.head()`

Plotting the target variable VWAP over time

`df.VWAP.plot(figsize=(14, 7))`

# Feature Engineering

Almost every time series problem will have some external features or some internal feature engineering to help the model.

Let’s add some basic features like lag values of available numeric features that are widely used for time series problems. Since we need to predict the price of the stock for a day, we cannot use the feature values of the same day since they will be unavailable at actual inference time. We need to use statistics like mean, the standard deviation of their lagged values.

We will use three sets of lagged values, one previous day, one looking back 7 days, and another looking back 30 days as a proxy for last week and last month metrics.

`df.reset_index(drop=True, inplace=True)lag_features = ["High", "Low", "Volume", "Turnover", "Trades"]window1 = 3window2 = 7window3 = 30df_rolled_3d = df[lag_features].rolling(window=window1, min_periods=0)df_rolled_7d = df[lag_features].rolling(window=window2, min_periods=0)df_rolled_30d = df[lag_features].rolling(window=window3, min_periods=0)df_mean_3d = df_rolled_3d.mean().shift(1).reset_index().astype(np.float32)df_mean_7d = df_rolled_7d.mean().shift(1).reset_index().astype(np.float32)df_mean_30d = df_rolled_30d.mean().shift(1).reset_index().astype(np.float32)df_std_3d = df_rolled_3d.std().shift(1).reset_index().astype(np.float32)df_std_7d = df_rolled_7d.std().shift(1).reset_index().astype(np.float32)df_std_30d = df_rolled_30d.std().shift(1).reset_index().astype(np.float32)for feature in lag_features:    df[f"{feature}_mean_lag{window1}"] = df_mean_3d[feature]    df[f"{feature}_mean_lag{window2}"] = df_mean_7d[feature]    df[f"{feature}_mean_lag{window3}"] = df_mean_30d[feature]        df[f"{feature}_std_lag{window1}"] = df_std_3d[feature]    df[f"{feature}_std_lag{window2}"] = df_std_7d[feature]    df[f"{feature}_std_lag{window3}"] = df_std_30d[feature]df.fillna(df.mean(), inplace=True)df.set_index("Date", drop=False, inplace=True)df.head()`

For boosting models, it is beneficial to add DateTime features like an hour, day, month, as applicable to provide the model information about the time component in the data. For time series models, it is not explicitly required to pass this information, but we could do so, and we will see in this notebook so that all models are compared on the same set of features.

`df.Date = pd.to_datetime(df.Date, format="%Y-%m-%d")df["month"] = df.Date.dt.monthdf["week"] = df.Date.dt.weekdf["day"] = df.Date.dt.daydf["day_of_week"] = df.Date.dt.dayofweek`

Splitting the data into train and validation along with features.

• Train: Data from 26th May 2008 to 31st December 2018.
• valid: Data from 1st January 2019 to 31st December 2019.
`df_train = df[df.Date < "2019"]df_valid = df[df.Date >= "2019"]exogenous_features = ["High_mean_lag3", "High_std_lag3",Low_mean_lag3", "Low_std_lag3","Volume_mean_lag3","Volume_std_lag3","Turnover_mean_lag3"Turnover_std_lag3", "Trades_mean_lag3", "Trades_std_lag3","High_mean_lag7", "High_std_lag7", "Low_mean_lag7", "Low_std_lag7","Volume_mean_lag7", "Volume_std_lag7", "Turnover_mean_lag7","Turnover_std_lag7", "Trades_mean_lag7", "Trades_std_lag7","High_mean_lag30", "High_std_lag30", "Low_mean_lag30", "Low_std_lag30","Volume_mean_lag30", "Volume_std_lag30", "Turnover_mean_lag30","Turnover_std_lag30", "Trades_mean_lag30", "Trades_std_lag30","month", "week", "day", "day_of_week"]`

The additional features supplied to time series problems are called exogenous regressors.

# Auto ARIMAX

ARIMA (Auto-Regressive Integrated Moving Average) models explain a given time series based on its past values, that is, its lags and the lagged forecast errors, so that equation can be used to forecast future values.

ARIMA models require specific input parameters: p for the AR(p) part, q for the MA(q) part, and d for the I(d) part. Thankfully, there is an automatic process by which these parameters can be chosen, which is called Auto ARIMA.

When exogenous regressors are used with ARIMA, it is commonly called ARIMAX.

https://en.wikipedia.org/wiki/Autoregressive_integrated_moving_average

`model = auto_arima(df_train.VWAP, exogenous=df_train[exogenous_features], trace=True, error_action="ignore", suppress_warnings=True)model.fit(df_train.VWAP, exogenous=df_train[exogenous_features])forecast = model.predict(n_periods=len(df_valid), exogenous=df_valid[exogenous_features])df_valid["Forecast_ARIMAX"] = forecast`

The best ARIMA model is ARIMA(2, 0, 1), which has the lowest AIC.

`df_valid[["VWAP", "Forecast_ARIMAX"]].plot(figsize=(14, 7))`
`print("RMSE of Auto ARIMAX:", np.sqrt(mean_squared_error(df_valid.VWAP, df_valid.Forecast_ARIMAX)))print("\nMAE of Auto ARIMAX:", mean_absolute_error(df_valid.VWAP, df_valid.Forecast_ARIMAX))`

The Auto ARIMAX model seems to do a reasonably good job in predicting the stock price given data until the previous day. Can other models beat this benchmark?

Prophet is an open-source time series model developed by Facebook. It was released in early 2017. An excerpt from the homepage:

Prophet is a procedure for forecasting time series data based on an additive model where non-linear trends are fit with yearly, weekly, and daily seasonality, plus holiday effects. It works best with time series that have strong seasonal effects and several seasons of historical data. Prophet is robust to missing data and shifts in the trend, and typically handles outliers well.

Note that the default parameters are used for Prophet. They can be tuned to improve the results.

`model_fbp = Prophet()for feature in exogenous_features:    model_fbp.add_regressor(feature)model_fbp.fit(df_train[["Date", "VWAP"] + exogenous_features].rename(columns={"Date": "ds", "VWAP": "y"}))forecast = model_fbp.predict(df_valid[["Date", "VWAP"] + exogenous_features].rename(columns={"Date": "ds"}))df_valid["Forecast_Prophet"] = forecast.yhat.valuesmodel_fbp.plot_components(forecast)`
`df_valid[["VWAP", "Forecast_ARIMAX", "Forecast_Prophet"]].plot(figsize=(14, 7))`
`print("RMSE of Auto ARIMAX:", np.sqrt(mean_squared_error(df_valid.VWAP, df_valid.Forecast_ARIMAX)))print("RMSE of Prophet:", np.sqrt(mean_squared_error(df_valid.VWAP, df_valid.Forecast_Prophet)))print("\nMAE of Auto ARIMAX:", mean_absolute_error(df_valid.VWAP, df_valid.Forecast_ARIMAX))print("MAE of Prophet:", mean_absolute_error(df_valid.VWAP, df_valid.Forecast_Prophet))`

# Conclusions and Tips

• Auto ARIMAX is a great baseline model, but newer algorithms like Facebook’s Prophet are potent and are getting smarter by the day. Don’t feel afraid to try out new techniques.
• Setting up an appropriate validation framework is extremely important. It enables you to try and experiment with various models and objectively compare them.
• Lag-based features are handy in providing trend information about the time series data. Rolling statistics are a common way of generating these.
• Exogenous regressors help in providing external information about the time series. They tend to be very important in most models.
• Converting a time series to stationary and then modeling is a common approach for building solutions and can significantly improve results.

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