Moving Average (MA) Model Implementation
As we know that getting a time series data that is truly stationary in nature without differencing is difficult, since most of the data in the real world are not stationary in nature. After the brief discussion of the moving average model now we will be implementing the code example for the same. We will be using the Yahoo Finance data to collect the data of Advanced Micro Devices, Inc(AMD) stock and model it with moving average model.
Importing Libraries:
The code mentioned imports important libraries including pandas for data manipulation, numpy for mathematical computation, matplotlib for plotting the different graphs and measure relationships between variables, plot_acf and plot_pacf for plotting acf and pacf plots of the data to identify model order. ARIMA model to fit the order of the model and forecast the data provided. We will be importing yfinance which stands for yahoo finance through which we will get the access to the stock price of AMD and warnings to filter the unnecessary warnings that pop up with output. At the end we will be using matplotlib’s rcParams method which will help in setting the figure size as (15, 5), since our dataset will be spreadout.
Python3
import pandas as pdimport numpy as npimport matplotlib.pyplot as pltfrom statsmodels.graphics.tsaplots import plot_acf, plot_pacffrom statsmodels.tsa.arima.model import ARIMAimport yfinance as yfimport warningswarnings.filterwarnings("ignore")plt.rcParams['figure.figsize'] = 15, 5 |
Downloading the Stock Price of AMD and Plotting Closing Price:
Here first we will be downloading the AMD stock price and saving the values from 1st January, 2020 into a variable called AMD_values the .history method lets us go back in history and extract value from any time where the value of the stock existed. The AMD_values will act as a time series in our example, after these steps we will be plotting the closing price of the AMD_values dataframe.
Python3
AMD = yf.Ticker("AMD")# getting the historical market dataAMD_values = AMD.history(start="2020-01-01")# Plotting the close price of AMDAMD_values[['Close']].plot(); |
Output:
AMD Stock Price Closing Values (in USD)
Calculating and Plotting Rolling Average Values:
Now in the next step the we will be calculating 10-day rolling average values of closing price and adding it as a new column named rolling_av, we are calculating the rolling average value of the closing data since it helps in smoothing out the fluctuations in the time series data, such that the data could be modelled and analyzed in a better way. After this step we will be plotting the closing value and the rolling average value of the data.
Python3
AMD_values['rolling_av'] = AMD_values['Close'].rolling(10).mean()# plotting 10-day rolling average value with the closing valueAMD_values[['Close','rolling_av']].plot(); |
Output:
AMD Stock Closing Price and 10-day Rolling Value of Closing Price
ACF and PACF Plots of the Data:
Here first we will be defining a function which takes the time series data as a parameter and generate the ACF and PACF plots of it, after defining such function we will be fitting the data we are modelling. We will be plotting last 20 lags to see how the current data depends on the past noise data.
Python3
# Function to plot ACF and PACFdef plot_acf_pacf(timeseries): fig, (ax1, ax2) = plt.subplots(2, 1, figsize=(10, 7)) plot_acf(timeseries, ax=ax1, lags=75) plot_pacf(timeseries, ax=ax2, lags=75) plt.show()# Plotting ACF and PACF of the closing value time seriesplot_acf_pacf(AMD_values['Close']) |
Output:
Autocorrelations and Partial Autocorrelations
Here, we can see in the ACF plot that the first lag has a correlation of 1 since is the same error term as that of the current time series but from the second lag we can see that the ACF plot is reducing but the correlation value between the lags decreases slowly, here if we model more lag values, we’ll find out the order of q gets more than 50.
The blue region signifies where the values are no longer statistically significant. From the above plot, we can that see that the last significant lag is the 55. Therefore, our model order for the MA model will be 55. The value of q to be 55 which means the current value will depend on the last 55 error values of the data.
Fitting the MA Model and Checking for Statistical Appropriation:
In the next step we will be fitting an ARIMA model to the closing prices of AMD stock with an order of (0, 0, 55), indicating 55 lags for the moving average component or the order of the moving average model to be 55. After that we will be printing the summary of the fitted ARIMA model such that the coefficient of the past noise values could be analyzed.
Python3
#creating the modelMA_model = ARIMA(endog=AMD_values['Close'], order=(0, 0, 55))#fitting data to the modelresults = MA_model.fit()#summary of the modelprint(results.summary()) |
Output:
SARIMAX Results
==============================================================================
Dep. Variable: Close No. Observations: 1030
Model: ARIMA(0, 0, 55) Log Likelihood -2607.721
Date: Tue, 06 Feb 2024 AIC 5329.441
Time: 12:13:34 BIC 5610.868
Sample: 0 HQIC 5436.250
- 1030
Covariance Type: opg
==============================================================================
coef std err z P>|z| [0.025 0.975]
------------------------------------------------------------------------------
const 91.7676 4.863 18.872 0.000 82.237 101.298
ma.L1 0.9881 0.032 31.180 0.000 0.926 1.050
ma.L2 1.0082 0.043 23.566 0.000 0.924 1.092
ma.L3 1.0546 0.050 20.986 0.000 0.956 1.153
ma.L4 1.0883 0.060 18.148 0.000 0.971 1.206
ma.L5 1.1193 0.066 16.895 0.000 0.989 1.249
ma.L6 1.1757 0.075 15.578 0.000 1.028 1.324
ma.L7 1.2618 0.082 15.314 0.000 1.100 1.423
ma.L8 1.2647 0.088 14.362 0.000 1.092 1.437
ma.L9 1.2961 0.095 13.599 0.000 1.109 1.483
ma.L10 1.2716 0.102 12.493 0.000 1.072 1.471
ma.L11 1.3053 0.110 11.899 0.000 1.090 1.520
ma.L12 1.3568 0.118 11.528 0.000 1.126 1.588
ma.L13 1.3790 0.125 11.042 0.000 1.134 1.624
ma.L14 1.4370 0.131 10.989 0.000 1.181 1.693
ma.L15 1.4226 0.136 10.447 0.000 1.156 1.689
ma.L16 1.4259 0.141 10.105 0.000 1.149 1.703
ma.L17 1.3422 0.148 9.098 0.000 1.053 1.631
ma.L18 1.3952 0.149 9.377 0.000 1.104 1.687
ma.L19 1.3681 0.155 8.826 0.000 1.064 1.672
ma.L20 1.4254 0.160 8.897 0.000 1.111 1.739
ma.L21 1.3960 0.162 8.592 0.000 1.078 1.715
ma.L22 1.2811 0.167 7.666 0.000 0.954 1.609
ma.L23 1.2373 0.169 7.303 0.000 0.905 1.569
ma.L24 1.2439 0.172 7.214 0.000 0.906 1.582
ma.L25 1.2424 0.173 7.177 0.000 0.903 1.582
ma.L26 1.2532 0.175 7.165 0.000 0.910 1.596
ma.L27 1.2763 0.177 7.221 0.000 0.930 1.623
ma.L28 1.2407 0.176 7.046 0.000 0.896 1.586
ma.L29 1.1452 0.178 6.452 0.000 0.797 1.493
ma.L30 1.1320 0.175 6.458 0.000 0.788 1.476
ma.L31 1.1204 0.174 6.436 0.000 0.779 1.462
ma.L32 1.0165 0.175 5.798 0.000 0.673 1.360
ma.L33 0.9564 0.175 5.480 0.000 0.614 1.298
ma.L34 0.9079 0.172 5.267 0.000 0.570 1.246
ma.L35 0.8531 0.169 5.058 0.000 0.523 1.184
ma.L36 0.7645 0.166 4.605 0.000 0.439 1.090
ma.L37 0.7301 0.163 4.493 0.000 0.412 1.049
ma.L38 0.7081 0.157 4.504 0.000 0.400 1.016
ma.L39 0.6063 0.154 3.932 0.000 0.304 0.909
ma.L40 0.4872 0.148 3.293 0.001 0.197 0.777
ma.L41 0.3681 0.144 2.553 0.011 0.085 0.651
ma.L42 0.3342 0.139 2.410 0.016 0.062 0.606
ma.L43 0.3415 0.134 2.546 0.011 0.079 0.604
ma.L44 0.3485 0.129 2.710 0.007 0.096 0.601
ma.L45 0.3279 0.123 2.660 0.008 0.086 0.569
ma.L46 0.3315 0.118 2.807 0.005 0.100 0.563
ma.L47 0.3224 0.113 2.860 0.004 0.101 0.543
ma.L48 0.2858 0.105 2.734 0.006 0.081 0.491
ma.L49 0.2297 0.095 2.429 0.015 0.044 0.415
ma.L50 0.2450 0.087 2.823 0.005 0.075 0.415
ma.L51 0.1928 0.078 2.464 0.014 0.039 0.346
ma.L52 0.1343 0.070 1.923 0.054 -0.003 0.271
ma.L53 0.1699 0.059 2.885 0.004 0.054 0.285
ma.L54 0.1631 0.049 3.351 0.001 0.068 0.258
ma.L55 0.0760 0.035 2.165 0.030 0.007 0.145
sigma2 9.1346 0.333 27.418 0.000 8.482 9.788
===================================================================================
Ljung-Box (L1) (Q): 0.00 Jarque-Bera (JB): 138.76
Prob(Q): 0.96 Prob(JB): 0.00
Heteroskedasticity (H): 1.70 Skew: 0.33
Prob(H) (two-sided): 0.00 Kurtosis: 4.68
===================================================================================
From the statistical point of view we can see that coeff. of the constant is 91 which works as the mean of the closing value of the dataset and we can observe the coefficient of the noise values from ma.L1 – ma.L55 and the p values are all near 0, therefore, these observations are significant.
Predicting the values and printing some of the values:
Python3
#prediction datastart_date = '2023-12-15'end_date = '2024-02-05'AMD_values['prediction'] = results.predict(start=start_date, end=end_date)#printing last 14 values of the prediction with original and rolling avg valueprint(AMD_values[['Close','rolling_av','prediction']].tail(14)) |
Output:
Close rolling_av prediction
Date
2024-01-17 00:00:00-05:00 160.169998 146.737999 159.708700
2024-01-18 00:00:00-05:00 162.669998 149.472998 160.559394
2024-01-19 00:00:00-05:00 174.229996 153.294998 163.174427
2024-01-22 00:00:00-05:00 168.179993 156.254997 172.220122
2024-01-23 00:00:00-05:00 168.419998 158.478998 167.267264
2024-01-24 00:00:00-05:00 178.289993 161.381998 168.262082
2024-01-25 00:00:00-05:00 180.330002 164.560999 178.427955
2024-01-26 00:00:00-05:00 177.250000 167.483998 180.498107
2024-01-29 00:00:00-05:00 177.830002 170.610999 178.240948
2024-01-30 00:00:00-05:00 172.059998 171.942998 176.777833
2024-01-31 00:00:00-05:00 167.690002 172.694998 169.790058
2024-02-01 00:00:00-05:00 170.479996 173.475998 167.889296
2024-02-02 00:00:00-05:00 177.660004 173.818999 171.422463
2024-02-05 00:00:00-05:00 174.229996 174.423999 177.279664
Through the table we can conclude that the moving average model of order 15 is pretty good for the given dataset as the predictions value nearly matches the original closing value of the daily AMD stock price
Plotting Forcasted values with Original Data:
At the last step predictions are plotted with the original data set and the rolling value of the original value. This visualization helps compare the model’s forecasts against the actual data.
Python3
# Forecast future values# Forecast future closing pricesforecast_steps = 30 # Forecasting for the next 30 daysforecast_index = pd.date_range(start=AMD_values['Close'].index[-1], periods=forecast_steps+1, freq='D')[1:] # Generate datetime index for forecastforecast = results.forecast(steps=forecast_steps)# plotting the end resultsAMD_values[['Close','rolling_av','prediction']].plot()plt.plot(forecast_index, forecast, color='red', label='Forecast') |
Output:
The green line shows the predicted value by the moving average model, the blue line represents the original data that we are modelling in this code example and the orange plot is the rolling average value plot of the original data. The green line follows the trend in a pretty good way; therefore, we can conclude that the moving average model was good in analyzing the AMD stock data. The red line represents the forecasting values for next 30 days.
Understanding the Moving average (MA) in Time Series Data
Data is often collected with respect to time, whether for scientific or financial purposes. When data is collected in a chronological order, it is referred to as time series data. Analyzing time series data provides insights into how the data behaves over time, including underlying patterns that can help solve problems in various domains. Time series analysis can also aid in forecasting future values based on historical data, leading to better production, profits, policy planning, risk management, and other fields. Therefore, analysis of time series data becomes an important aspect of data science.
In this article, we will discuss Moving Average Models, which are essential for time series analysis and forecasting trends.