What is ARIMA Forecasting and How Can it Be Used for Enterprise Analysis?
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The document provides an overview of time series forecasting using ARIMA (Autoregressive Integrated Moving Average) models. It defines the ARIMA model parameters - autoregressive (p), differencing (d), and moving average (q) - and explains how they are used to forecast future values based on past observations. Examples are given to demonstrate identifying the p, d, q values and fitting the ARIMA model to sample time series data. Limitations and use cases for ARIMA forecasting in business are also discussed.
What is ARIMA Forecasting and How Can it Be Used for Enterprise Analysis?
- 1. TIME SERIES FORECASTING AUTOREGRESSIVE INTEGRATED MOVING AVERAGE(ARIMA) A Simplistic Explainer Series For Citizen Data Scientists J o u r n e y T o w a r d s A u g m e n t e d A n a l y t i c s
- 3. Introduction • ARIMA stands for Autoregressive Integrated Moving Average model • An Auto Regressive Integrated Moving Average (ARIMA) model predicts future values of a time series by a linear combination of its past values and a series of errors. • This method is suitable for forecasting when data is stationary/non stationary, univariate and has any type of data pattern : level/trend /seasonality/cyclicity
- 4. Example Let’s take an example of month wise Nifty closing observations : The plot of these data suggests that this is non stationary data showing gradual upward trend, as shown in figure below Hence, we can choose ARIMA algorithm for forecasting because the data exhibits non stationarity and trend Actual values Months Index M1 7000 M2 7600 M3 7800 M4 8000 M5 8200 M6 8400 M7 8600 M8 8800 M9 8400 M10 8000 Forecasted values M11 8350 M12 8650 M13 8780 M14 8480 M15 8160
- 6. Standard tuning parameters Note : Refer calculations section to understand how Auto regression(p), differencing(d) and moving average(q) works Model parameters : In ARIMA, there are mainly three parameters to be set to fit the model : p: This is the component to apply autoregressive model on series d: This is the component to apply differencing on series : Basically it converts non stationary data to stationary ( stationary series : the series remains at a fairly constant level over time) q: This is the component to apply moving average model on series Model approach: In ARIMA, there are two approaches to fitting a model : Automatic and Manual When p, d and q are automatically selected by system than it’s called automatic approach and when p, d and q are manually input by user than it’s called manual approach For better forecasts, automatic approach should be chosen, as in this approach, model automatically selects and applies the right parameters based on the nature of data Forecast period : For both type of approaches , user has to input the forecast period value For example, if user wants to predict the sales value for 10 periods ahead then this value should be input as 10
- 7. Sample UI For Input/Tuning Parameters And Output
- 8. Sample UI for Selecting Inputs & Applying Tuning Parameters Select the variable you would like to Forecast Months Nifty closing If user changes the approach to Manual then this box should be displayed, showing additional provision to set p , d and q values along with forecast period Tuning parameters Approach Forecast Period Automatic Approach Forecast Period AR(p) I(d) MA(q) Manual By default this box should be displayed with default approach as Automatic. In this case parameters to fit ARIMA will be automatically detected and applied by algorithm Select the time stamp Months Nifty closing
- 9. Sample UI for output Actual values Months Index M1 7000 M2 7600 M3 7800 M4 8000 M5 8200 M6 8400 M7 8600 M8 8800 M9 8400 M10 8000 Forecasted values M11 8350 M12 8650 M13 8780 M14 8480 M15 8160 Output will contain forecasted values based on user specified time period along with line chart showing actual and forecasted series , prediction accuracy and parameters values decided by the algorithm
- 10. Limitations
- 11. Limitations Time dependent error ( decreasing with time)Time independent error ( fairly constant over time & lying within certain range) Figure 1 Figure 2 The Forecast error also known as “Residuals” should show nearly constant trend over time i.e. it should be time independent as shown in the figure 1 below in contrast to the increasing/ decreasing trend shown in figure 2 below: ARIMA model is suggested for short term forecasting There should be at least 50 observations in the input data ARIMA is only for univariate data forecasts , but there might be other variables affecting the output/dependent variable. ARIMA doesn’t take into account the influence of other predictors while forecasting, hence forecasts made might not be accurate
- 12. Business Case
- 13. Business use case Business benefit: • A pharmaceutical company can make use of these forecasts for better planning of drug production • It also helps in balancing supply and demand for the drugs Business problem : • A pharmaceutical company wants to predict the sales of drug for next two months based on past 12 months drug sales data • Data pattern : Input data exhibits non stationarity and cyclical pattern
- 14. Calculations
- 15. Calculations - Autoregression (AR) • In an autoregressive model, which is one of the components in ARIMA model, we forecast the variable of interest using a linear combination of past values of the variable • The term autoregression indicates that it is a regression of the variable against itself • An autoregressive model of order p, denoted by AR(p) model, can be written as where , c is a constant, ∅ is lag’s coefficient, 𝐞 𝒕 is an error term, 𝐩 is autoregressive model of order • This is like a multiple regression but with lagged values of yt as predictors • Order of this component (order of autoregression : AR) is given by parameter p while fitting the model : ARIMA (p,d, q) Lagged values : past values of the variable
- 16. Calculations - Integration (I) / Differencing (d) • The second component of ARIMA model i.e. I (for "integration") , is used to replace the series with the difference between their current values and the previous values (and this differencing process can be performed more than once as per the requirement ) • For example, • The equation for first order differencing is 𝒚 𝒕 = 𝒚 𝒕 − 𝒚 𝒕−𝟏 • Hence, for 𝒚 𝒕 =2 and 𝒚 𝒕−𝟏 = 1 ; 𝒚 𝒕 will be 1 • Similarly second order differencing , 𝒚 𝒕 = (𝒚 𝒕 − 𝒚 𝒕−𝟏) −(𝒚 𝒕−𝟏 − 𝒚 𝒕−𝟐) • Order of this component (order of differencing) is applied by parameter d while fitting a model : ARIMA (p,d,q)
- 17. Calculations - Moving average (MA) • A moving average model, the third component in ARIMA uses past forecast errors as a series in a model • A Moving average model of order q, denoted by MA(q) model, can be written as where , yt is predictor , c is a constant, θ is lag’s coefficient, 𝐞 𝒕 is an error term, q is moving average order • Order of this component (order of moving average : MA ) is applied by parameter d while fitting a model : ARIMA (p ,d ,q )
- 18. Identification of p,d,q values • Values of p and q are determined based on the autocorrelation(ACF) and partial auto correlation(PACF) plots and value of d depends on level of stationarity in data • In PACF plot, number of spikes indicate the order of the autoregression/AR (value of p in ARIMA(p,d,q)) • For instance, as you can see in the right figure below, there is one spike falling out of range, hence, the order of AR i.e. value of p would be 1 • In ACF plot, number of spikes indicate the order of the moving average (value of q in ARIMA(p,d,q)) • For instance , as you can see in the left figure there are five spikes falling out of range, hence, the order of MA i.e. value of q would be 5
- 19. Identification of p,d,q values Thus, p, d and q parameters in ARIMA (p , d , q) are substituted with integer values where p and q take any values between 0 to 5 and value of d is set between 0 to 2 For example, ARIMA(2,1,1) means that you have a second order autoregressive model with a first order moving average component and series has been differenced once to induce stationarity A value of 0 can be used for any of the above mentioned parameters indicating that particular component (AR/ I/ MA) should not be used. This way, the ARIMA model can be configured to perform the function of an ARMA model, and even a simple AR, I, or MA model depending on the data
- 20. Example • The automatic approach will select ideal values of Auto regression(p), differencing(d) and moving average(q) parameters based on the data pattern • For instance, if there is non stationarity in data, the algorithm will apply differencing(d) by applying d=1 in order to make it stationary • In case of manual approach, user will select optimum values of p, d and q parameters, which gives minimum value for MAPE (Mean absolute percentage error) in order to get better accuracy. This is a bit iterative process as there may be many iterations involved till the desired accuracy is achieved • After the ARIMA model is run, it will provide forecasted values of target variable(Nifty closing) for user specified periods ahead , let’s say 5 as shown in blue text in table: Forecasted values Actual index Months Index M1 7000 M2 7600 M3 7800 M4 8000 M5 8200 M6 8400 M7 8600 M8 8800 M9 8400 M10 8000 Forecasted Index M11 8350 M12 8650 M13 8780 M14 8480 M15 8160
- 21. Model Accuracy • Along with forecasted values, MAPE and prediction accuracy is displayed so user knows how much accurate the forecasts are • How MAPE and Accuracy is calculated is explained below : • Using this formula , MAPE can be calculated for 5 months ahead forecasts using recent most 5 months data (generally, actual as well as predicted values for recent most 20% data is taken to check accuracy) as shown in table below: MAPE : Where Yt is the actual, known series value for time period t, Y^t is the forecast value of the variable Y for time period t N is number of observations MAPE = 4% Hence accuracy = 100-MAPE = 96% , So model is accurate Years Actual Y Predicted Y^ Abs((Actual - predicted)/actual)*100 M12 8400 8350 0.60 M13 8600 8650 0.58 M14 8800 8780 0.23 M15 8400 8480 0.95 M16 8000 8160 2.00 MAPE = Sum(M11 to M15) =4 Accuracy =100-MAPE=96
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