Sangram Keshari Patro

Business Report - 8 PG Program in Data Science and Business Analytics submitted by Sangram Keshari Patro BATCH:PGPDSBA.O.AUG24.B Contents 1 Objective 4 2 Data Description 4 3 Data Overview 3.1 Importing necessary libraries and the dataset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Structure and type of data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Statistical summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 4 4 4 4 Exploratory Data Analysis 4.1 Univariate Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 ’Rose’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 ’Sparkling’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Bivariate Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Rose sales vs Year . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Sparkling sales vs Year . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Rose sales vs Month . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 Sparkling sales vs Month . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Plotting the Empirical Cumulative Distribution . . . . . . . . . . . . . . . . . . 4.3.1 Rose sales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Sparkling sales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Plotting the average RetailSales per month and the month on month percentage 4.4.1 Rose sales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Sparkling sales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . change of RetailSales. . . . . . . . . . . . . . . . . . . . . . . . . - 5 Data Pre-processing 5.1 Additive Model . . . . . . . . 5.1.1 Rose sales . . . . . . . 5.1.2 Sparkling sales . . . . 5.2 Multiplicative Model . . . . . 5.2.1 Rose sales . . . . . . . 5.2.2 Sparkling sales . . . . 5.3 Model Comparison Summary 6 Build forecasting models 6.1 Linear Regression . . . . . . 6.1.1 Model Building . . . 6.1.2 Model Evaluation . . 6.2 Simple Average . . . . . . 6.2.1 Model Building . . . 6.2.2 Model Evaluation . . 6.3 Moving Average . . . . . . 6.3.1 Model Building . . . 6.3.2 Model Evaluation . . 6.4 Exponential Models (Single, 6.4.1 Model Building . . . 6.4.2 Model Evaluation . . 6.4.3 Model Building . . . 6.4.4 Model Evaluation . . 6.4.5 Model Building . . . 6.4.6 Model Evaluation . . 6.4.7 Model Building . . . 6.4.8 Model Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Double, Triple) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Check for Stationarity 28 7.1 Rose Sales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 7.2 Sparkling Sales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 8 Model building of stationary data 8.1 Plotting the Autocorrelation function plots . . . . . . . . . . . 8.1.1 Plotting PACF . . . . . . . . . . . . . . . . . . . . . . . 8.1.2 Plotting ACF . . . . . . . . . . . . . . . . . . . . . . . . 8.1.3 Plotting PACF . . . . . . . . . . . . . . . . . . . . . . . 8.1.4 Plotting ACF . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Manual ARIMA Model . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 (Rose Sales)ARIMA model with p=0,d=1 and q=2 . . 8.2.2 (Rose Sales)ARIMA model with p=1,d=1 and q=2 . . . 8.2.3 (Sparkling Sales)ARIMA model with p=2,d=1 and q=2 8.2.4 (Sparkling Sales)ARIMA model with p=2,d=1 and q=1 8.3 Auto ARIMA Model . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Manual SARIMA Model . . . . . . . . . . . . . . . . . . . . . . 8.4.1 (Rose Sales)SARIMA model ((0,1,2)(2, 1, 2, 12)) . . . 8.4.2 (Sparkling Sales)SARIMA model ((1,1,2)(0, 1, 2, 12)) . 8.5 Auto SARIMA Model . . . . . . . . . . . . . . . . . . . . . . . 8.5.1 (Sparkling Sales)SARIMA(0,0,1)(0,1,1)[12] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - 9 Comparison of all different models 43 10 Final Forecast 44 11 Actionable Insights and Business Recommendations 45 List of Figures- Table depicting the datatype for both types of wine. . . . . . . . . . . . . . . . . . . . Statistical summary of the data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Timeseries plot of ’Rose’ wine sales . . . . . . . . . . . . . . . . . . . . . . . . . . . . Timeseries plot of ’Sparkling’ wine sales . . . . . . . . . . . . . . . . . . . . . . . . . Rose sales vs Year . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sparkling sales vs Year . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rose sales vs Month . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sparkling sales vs Month . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ECDF plot of Rose sales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ECDF plot of Sparkling sales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rose sales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sparkling sales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rose sales(Additive Model) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sparkling sales(Additive Model) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rose sales(Multiplicative Model) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sparkling sales(Multiplicative Model) . . . . . . . . . . . . . . . . . . . . . . . . . . Rose sales(Linear Regression) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sparkling sales(Linear Regression) . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rose sales(Simple Average) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sparkling sales(Simple Average) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rose sales(Moving Average) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sparkling sales(Moving Average ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rose sales(Single Exponential Model and its parameters) . . . . . . . . . . . . Sparkling sales(Single Exponential Model and its parameters) . . . . . . . . . Rose sales(Double Exponential Model and its parameters) . . . . . . . . . . . Sparkling sales(Double Exponential Model and its parameters) . . . . . . . . Rose sales(Triple Exponential Model and its parameters) . . . . . . . . . . . . Sparkling sales(Triple Exponential Model and its parameters) . . . . . . . . . Rose sales(Triple Exponential Model) and its parameters . . . . . . . . . . . . Sparkling sales(Triple Exponential Model and its parameters) . . . . . . . . . Dickey-Fuller test results for Rose dataset with and without differencing the data . . . Dickey-Fuller test results for Sparkling dataset with and without differencing the data Rose sales(PACF plot) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rose sales(ACF plot) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sparkling sales(PACF plot) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sparkling sales(ACF plot) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (Rose Sales)ARIMA model with p=0,d=1 and q=2 . . . . . . . . . . . . . . . . . . . (Rose Sales)ARIMA model with p=1,d=1 and q=2 . . . . . . . . . . . . . . . . . . . . (Sparkling Sales)ARIMA model with p=2,d=1 and q=2 . . . . . . . . . . . . . . . . . (Sparkling Sales)ARIMA model with p=2,d=1 and q=1 . . . . . . . . . . . . . . . . . Comparison of all the ARIMA models . . . . . . . . . . . . . . . . . . . . . . . . . . . Rose sales(ARIMA model with p=1,d=1 and q=2) . . . . . . . . . . . . . . . . . . . Sparkling sales(ARIMA model with p=2,d=1 and q=2 ) . . . . . . . . . . . . . . . (Rose Sales)SARIMA model ((0,1,2)(2, 1, 2, 12)) . . . . . . . . . . . . . . . . . . . . . (Sparkling Sales)SARIMA model ((1,1,2)(0, 1, 2, 12)) . . . . . . . . . . . . . . . . . . Comparison of all the SARIMA models . . . . . . . . . . . . . . . . . . . . . . . . . . (Rose Sales)SARIMA model ((0,1,2)(2, 1, 2, 12)) and residuals plot . . . . . . . . . . (Sparkling Sales)SARIMA model ((1,1,2)(0, 1, 2, 12)) and residuals plot . . . . . . . (Rose Sales)SARIMA(5,1,1)(1,0,1)[12] . . . . . . . . . . . . . . . . . . . . . . . . . . . (Sparkling Sales)SARIMA(0,0,1)(0,1,1)[12] . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of all the AUTO SARIMA models . . . . . . . . . . . . . . . . . . . . . . Comparison of all different models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Final forecast plot for Rose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Final forecast plot for Sparkling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - 1 Objective The primary objective of this project is to analyze and forecast wine sales trends for the 20th century based on historical data provided by ABC Estate Wines. We aim to equip ABC Estate Wines with the necessary insights and foresight to enhance sales performance, capitalize on emerging market opportunities, and maintain a competitive edge in the wine industry. 2 Data Description The data provided is of sales of two types of wine Rose and Sparkling from the period of 1980 to 1995. 3 3.1 Data Overview Importing necessary libraries and the dataset Datasets for both types of wine are printed. Both have 187 rows & 2 columns. 3.2 Structure and type of data Data is explored further. The dataset is free from duplicate rows and contains some null values. Figure 1: Table depicting the datatype for both types of wine. 3.3 Statistical summary Figure 2: Statistical summary of the data 4 4 Exploratory Data Analysis 4.1 4.1.1 Univariate Analysis ’Rose’ Figure 3: Timeseries plot of ’Rose’ wine sales Observations Sales Trend: There is a noticeable overall downward trend in the sales of rosé wine from 1980 to 1995. Seasonality: Spikes in sales occur regularly, indicating a possible seasonal pattern. Volatility: Early years -) show high fluctuation in sales, which stabilizes over time. Peak Period: The highest sales were recorded around-, after which a gradual decline is observed. Business Recommendations Seasonal Marketing: Utilize peak seasons to promote rosé wine through targeted campaigns and discounts. Revitalization Strategy: Investigate causes of declining demand and consider rebranding or new packaging to rejuvenate consumer interest. Diversify Portfolio: Consider expanding into other wine categories or beverages to offset the decline in rosé sales. Customer Analysis: Conduct market research to understand changing preferences and tailor offerings accordingly. 5 4.1.2 ’Sparkling’ Figure 4: Timeseries plot of ’Sparkling’ wine sales Observations Sales Trend: Overall, there is a clear upward trend in sparkling wine sales over the years. Seasonality: Sales show consistent yearly spikes, likely during festive seasons. Volatility: Peaks are sharp and regular, indicating predictable consumer behavior patterns. Growth: From 1980 to 1995, both the baseline and peak sales values have increased steadily. Business Recommendations Seasonal Promotions: Intensify marketing during known high-demand months (e.g., holidays) to maximize sales. Capacity Planning: Scale production and distribution in advance of seasonal peaks. Product Positioning: Leverage the premium and celebratory image of sparkling wine to appeal to aspirational consumers. International Markets: Given strong domestic growth, explore export opportunities during peak global festive seasons. 4.2 4.2.1 Bivariate Analysis Rose sales vs Year Figure 5: Rose sales vs Year Observations Box Plot: Sales of rosé wine declined over the years. Variability and median decreased, with more outliers in early years indicating occasional high sales. Monthly Trends: November and December show consistently higher sales, suggesting seasonality. All months show a general downward trend. 7 Business Recommendations Seasonal Targeting: Focus promotions around November–December due to peak sales in these months. Revival Strategy: Revamp marketing for rosé wine to counter steady decline over time. Outlier Utilization: Analyze high-sales months in early years to replicate successful strategies. 4.2.2 Sparkling sales vs Year Figure 6: Sparkling sales vs Year Observations Box Plot (Sparkling Wine): Sales increased slightly from 1980 to 1995. Distribution remained steady, but outliers rose after 1985, showing sporadic sales spikes. 8 Monthly Trends (Sparkling Wine): Clear seasonality with high sales in November and December. Other months showed stable but lower sales over time. Business Recommendations Seasonal Focus: Intensify campaigns in November–December to leverage strong seasonal demand. Peak Analysis: Investigate reasons behind outlier months with high sales to replicate success. Stable Base: Maintain consistent supply across months while optimizing for year-end spikes. 4.2.3 Rose sales vs Month 9 Figure 7: Rose sales vs Month Observations Trend Plot: Clear upward trend from January to December; highest spikes in November–December. Yearly Comparison: Sales peaked in the 1980s; gradual decline post-1990. Seasonality remains strong across years. Business Recommendations Seasonal Focus: Boost marketing in late Q4 to leverage peak sales. Sales Recovery: Study 1980s peak years to extract replicable growth strategies. Product Strategy: Consider bundling or festive packaging for holiday months. 4.2.4 Sparkling sales vs Month 10 Figure 8: Sparkling sales vs Month 11 Insights Trend Plot: Sales remain flat in the first half, then rise sharply from July to December, with steep peaks in November–December. Yearly Comparison: Consistent year-end spikes across all years; 1995 shows strongest finish. Overall upward trend post-June. Business Recommendations Holiday Strategy: Prioritize inventory and campaigns for Q4 to align with demand surges. Growth Momentum: Scale up operations mid-year to prepare for strong second-half sales. Year-End Promotions: Launch targeted festive offers and high-value bundles to boost conversions. 4.3 Plotting the Empirical Cumulative Distribution This particular graph tells us what percentage of data points refer to what number of Sales. 4.3.1 Rose sales Figure 9: ECDF plot of Rose sales 4.3.2 Sparkling sales Figure 10: ECDF plot of Sparkling sales 12 4.4 4.4.1 Plotting the average RetailSales per month and the month on month percentage change of RetailSales. Rose sales Figure 11: Rose sales 4.4.2 Sparkling sales Figure 12: Sparkling sales 13 5 Data Pre-processing I have forward filled the 2 null values encountered in Rose dataset and no null value is found is Sparkling dataset. After removing null values data is decomposed into trend,seasonality and residuals and then visualized by plotting. Perform Decomposition 5.1 5.1.1 Additive Model Rose sales Figure 13: Rose sales(Additive Model) Insights Trend Plot: Long-term decline observed in overall values post-1981, indicating a steady downward trend. Seasonality: Clear recurring peaks annually, with strong cyclical behavior suggesting seasonal effects. Residuals: Randomly scattered, indicating minimal pattern left; model captures most structure. 14 Business Recommendations Seasonal Planning: Leverage predictable seasonal peaks for campaign timing and resource allocation. Decline Strategy: Investigate causes for long-term decline and strategize for revival. Model Reliability: Proceed with forecasting, as residuals show no significant patterns. 5.1.2 Sparkling sales Figure 14: Sparkling sales(Additive Model) Insights Trend Plot: Mild dip till 1983, followed by a steady rise peaking around-, then slight decline. Seasonality: Strong, consistent seasonal spikes each year, showing high demand fluctuations. Residuals: Randomly scattered around zero with minor variance; model fits the data well. Business Recommendations Peak Timing: Focus marketing and logistics around predictable annual spikes. Capacity Planning: Align operations with the rising trend observed mid-80s to early-90s. Monitoring Needed: Investigate post-1990 plateau or decline to inform future strategy. 15 5.2 5.2.1 Multiplicative Model Rose sales Figure 15: Rose sales(Multiplicative Model) Insights Trend Plot: Noticeable decline throughout the period, indicating weakening growth over time. Seasonality: Clear yearly patterns with relative seasonal effects remaining proportional over time. Residuals: Fluctuations appear stable and centered, suggesting a good model fit. Business Recommendations Stabilize Decline: Identify and mitigate drivers of downward trend. Seasonal Focus: Seasonal proportionality suggests timing remains key – optimize around peak months. Model Validity: Continue using multiplicative models for accurate forecasting and planning. 16 5.2.2 Sparkling sales Figure 16: Sparkling sales(Multiplicative Model) Insights Trend Plot: Gradual decline in long-term trend, indicating reduced popularity or demand. Seasonality: Strong multiplicative seasonality – peaks maintain proportional relationship to trend. Residuals: Stable and random around 1, supporting model appropriateness. Business Recommendations Address Downtrend: Explore marketing or innovation strategies to reverse trend. Leverage Seasonality: Plan production and promotions around predictable seasonal peaks. Model Reliability: Multiplicative model effectively captures structure – use for forecasting. 5.3 Model Comparison Summary Rose Data: Multiplicative model shows clearer seasonal variation with stable residuals. Best fit. Sparkling Data: Multiplicative model better captures proportional seasonality with a declining trend. Best fit. Conclusion Multiplicative models are more suitable for both Rose and Sparkling datasets due to their ability to model varying seasonal effects relative to trend levels. Then the data is split into train data and test data. 17 6 6.1 6.1.1 Build forecasting models Linear Regression Model Building Linear regression model is built on train data and fitted on test data to get the following plot. Rose Sales Figure 17: Rose sales(Linear Regression) Sparking Sales Figure 18: Sparkling sales(Linear Regression) 18 6.1.2 Model Evaluation The RMSE for the forecast is given below. 6.2 6.2.1 Simple Average Model Building Simple Average model is built on train data and fitted on test data to get the following plot. Rose Sales Figure 19: Rose sales(Simple Average) 19 Sparking Sales Figure 20: Sparkling sales(Simple Average) 6.2.2 Model Evaluation The RMSE for the forecast is given below. 6.3 6.3.1 Moving Average Model Building Moving Average model is built on the whole data to get the following plot. The model is built with different window sizes i.e. different point moving averages and best fit is found to be 2-point moving average by looking at the least RMSE. 20 Rose Sales Figure 21: Rose sales(Moving Average) Sparking Sales Figure 22: Sparkling sales(Moving Average ) 6.3.2 Model Evaluation The RMSE for the forecast is given below. 21 The best models for both dataset are 2-point averages. 6.4 Exponential Models (Single, Double, Triple) Exponential smoothing methods consist of flattening time series data. Exponential smoothing averages or exponentially weighted moving averages consist of forecast based on previous periods data with exponentially declining influence on the older observations. Exponential smoothing methods consist of special case exponential moving with notation ETS (Error, Trend, Seasonality) where each can be none(N), additive (N), additive damped (Ad), Multiplicative (M) or multiplicative damped (Md). One or more parameters control how fast the weights decay. These parameters have values between 0 and 1. Single Exponential Model The simplest of the exponentially smoothing methods is naturally called simple exponential smoothing (SES). This method is suitable for forecasting data with no clear trend or seasonal pattern. In Single ES, the forecast at time (t + 1) is given by Winters,1960 Ft+1 = αYt + (1 − α)Ft Parameter is called the smoothing constant and its value lies between 0 and 1. Since the model uses only one smoothing constant, it is called Single Exponential Smoothing. Note: Here, there is both trend and seasonality in the data. So, we should have directly gone for the Triple Exponential Smoothing but Simple Exponential Smoothing and the Double Exponential Smoothing models are built over here to get an idea of how the three types of models compare in this case. 6.4.1 Model Building Single Exponential Model model is built on train data and fitted on test data to get the following plot. 22 Rose Sales Figure 23: Rose sales(Single Exponential Model and its parameters) Sparking Sales Figure 24: Sparkling sales(Single Exponential Model and its parameters) 6.4.2 Model Evaluation The RMSE for the forecast is given below. 23 Double Exponential Model One of the drawbacks of the simple exponential smoothing is that the model does not do well in the presence of the trend. This model is an extension of SES known as Double Exponential model which estimates two smoothing parameters. Applicable when data has Trend but no seasonality. Two separate components are considered: Level and Trend. Level is the local mean. One smoothing parameter α corresponds to the level series A second smoothing parameter β corresponds to the trend series. 6.4.3 Model Building Double exponential model is built on train data and fitted on test data to get the following plot. Rose Sales Figure 25: Rose sales(Double Exponential Model and its parameters) Sparking Sales Figure 26: Sparkling sales(Double Exponential Model and its parameters) 24 6.4.4 Model Evaluation The RMSE for the forecast is given below. Here, we see that the Double Exponential Smoothing has actually done well when compared to the Simple Exponential Smoothing. This is because of the fact that the Double Exponential Smoothing model has picked up the trend component as well. Triple Exponential Model 6.4.5 Model Building Holt-Winters ETS(A,A,A) Model (Additive Error, Additive Trend, Additive Seasonality) model is built on train data and fitted on test data to get the following plot. Rose Sales Figure 27: Rose sales(Triple Exponential Model and its parameters) 25 Sparking Sales Figure 28: Sparkling sales(Triple Exponential Model and its parameters) 6.4.6 Model Evaluation The RMSE for the forecast is given below. Triple Exponential Smoothing has performed the best on the test as expected since the data had both trend and seasonality. But we see that our triple exponential smoothing is under forecasting. Let us try to tweak some of the parameters in order to get a better forecast on the test set. Now I have built another model i.e. Holt-Winters ETS(A,A,M) Model (Additive Error, Additive Trend, Multiplicative Seasonality) 6.4.7 Model Building Holt-Winters ETS(A,A,M) model is built on train data and fitted on test data to get the following plot. 26 Rose Sales Figure 29: Rose sales(Triple Exponential Model) and its parameters 27 Sparking Sales Figure 30: Sparkling sales(Triple Exponential Model and its parameters) 6.4.8 Model Evaluation The RMSE for the forecast is given below. Triple Exponential Smoothing has performed the best on the test as expected since the data had both trend and seasonality. 7 Check for Stationarity Stationarity in Time Series A time series is considered to be stationary when its statistical properties such as the mean, variance, and autocorrelation remain constant over time. Stationarity allows us to model and forecast time series using historical data because the behavior of the series does not change over time. In a stationary series, the autocorrelation at lag k depends only on k, and not on the specific time t. Let Xt denote the value of the time series at time t. The autocorrelation at lag k is the correlation between Xt and Xt−k . 28 Checking for Stationarity To test for stationarity, we use the Dickey-Fuller test (specifically, the Augmented Dickey-Fuller or ADF test). Null Hypothesis (H0 ): The time series is non-stationary. Alternative Hypothesis (H1 ): The time series is stationary. Interpretation: If the p-value < 0.05: Reject the null hypothesis ⇒ The time series is stationary. If the p-value ≥ 0.05: Fail to reject the null hypothesis ⇒ The time series is non-stationary. The test is depicted below. The data is also differentiated once and then checked for stationarity and we find that on single order differencing we are able to get a stationary data. 7.1 Rose Sales Figure 31: Dickey-Fuller test results for Rose dataset with and without differencing the data 29 7.2 Sparkling Sales Figure 32: Dickey-Fuller test results for Sparkling dataset with and without differencing the data 8 Model building of stationary data 8.1 Plotting the Autocorrelation function plots Using ACF and PACF in Time Series Forecasting ACF (Autocorrelation Function) and PACF (Partial Autocorrelation Function) plots help identify the order of ARIMA models. ACF Plot: Shows correlation between the time series and its lags. Useful for identifying the MA (Moving Average) order q. PACF Plot: Shows correlation between the series and its lags after removing intermediate effects. Helps determine the AR (AutoRegressive) order p. General Guidelines: If ACF cuts off after lag q and PACF tails off ⇒ MA(q) model. If PACF cuts off after lag p and ACF tails off ⇒ AR(p) model. 30 If both tail off slowly ⇒ Consider ARMA or ARIMA model with differencing. Note: Bars that lie outside the blue confidence region (typically at 95%) are considered statistically significant. Rose Sales 8.1.1 Plotting PACF Figure 33: Rose sales(PACF plot) 8.1.2 Plotting ACF Figure 34: Rose sales(ACF plot) 31 Sparkling Sales 8.1.3 Plotting PACF Figure 35: Sparkling sales(PACF plot) 32 8.1.4 Plotting ACF Figure 36: Sparkling sales(ACF plot) We can clearly observe the seasonality from the ACF plot. Note: The data has some seasonality so ideally we should build a SARIMA model. But for demonstration purposes we are building an ARIMA model both by looking at the minimum AIC criterion and by looking at the ACF and the PACF plots. 8.2 Manual ARIMA Model p,d,q values are changed and all permutations are tried to get the least AIC value. I have built the best 2 ARIMA models to compare. 8.2.1 (Rose Sales)ARIMA model with p=0,d=1 and q=2 The model result is as follows. 33 Figure 37: (Rose Sales)ARIMA model with p=0,d=1 and q=2 All the coefficients are statistically significant. 8.2.2 (Rose Sales)ARIMA model with p=1,d=1 and q=2 The model result is as follows. Figure 38: (Rose Sales)ARIMA model with p=1,d=1 and q=2 Two of the coefficients are not statistically significant considering 95% confidence. 34 8.2.3 (Sparkling Sales)ARIMA model with p=2,d=1 and q=2 The model result is as follows. Figure 39: (Sparkling Sales)ARIMA model with p=2,d=1 and q=2 All of the coefficients are statistically significant considering 95% confidence. 8.2.4 (Sparkling Sales)ARIMA model with p=2,d=1 and q=1 The model result is as follows. 35 Figure 40: (Sparkling Sales)ARIMA model with p=2,d=1 and q=1 one of the coefficients is not statistically significant considering 95% confidence. Figure 41: Comparison of all the ARIMA models The best ARIMA model for Rose and Sparkling data are shown below. The plots are obtained by forecasting the test data using the best models are attached below. 36 Rose Sales Figure 42: Rose sales(ARIMA model with p=1,d=1 and q=2) Sparking Sales Figure 43: Sparkling sales(ARIMA model with p=2,d=1 and q=2 ) 8.3 Auto ARIMA Model Auto ARIMA Model is built by using pmdarima library. For Rose data we get the best model to be ARIMA model with p=0,d=1 and q=2 which I have already built. And for Sparkling data the best model comes out to be ARIMA model with p=0,d=0 and q=1. RMSE is more than the best model that I built using ARIMA model with p=2,d=1 and q=2. Hence, I can safely ignore this model. 37 8.4 Manual SARIMA Model p,d,q parameters and Seasonal parameters (P,D,Q)S values are changed and all permutations are tried to get the least AIC value. I have built the best SARIMA model to compare. 8.4.1 (Rose Sales)SARIMA model ((0,1,2)(2, 1, 2, 12)) The model result is as follows. Figure 44: (Rose Sales)SARIMA model ((0,1,2)(2, 1, 2, 12)) Some of the coefficients are statistically insignificant. 38 8.4.2 (Sparkling Sales)SARIMA model ((1,1,2)(0, 1, 2, 12)) The model result is as follows. Figure 45: (Sparkling Sales)SARIMA model ((1,1,2)(0, 1, 2, 12)) All of the coefficients are statistically significant considering 95% confidence. Figure 46: Comparison of all the SARIMA models 39 The plots are obtained by forecasting the test data using the best models are attached below. Rose Sales Figure 47: (Rose Sales)SARIMA model ((0,1,2)(2, 1, 2, 12)) and residuals plot 40 Sparking Sales Figure 48: (Sparkling Sales)SARIMA model ((1,1,2)(0, 1, 2, 12)) and residuals plot 8.5 Auto SARIMA Model Auto SARIMA Model is built by using pmdarima library. For Rose data we get the best model to be SARIMA(5,1,1)(1,0,1)[12] which I have built. (Rose Sales)SARIMA(5,1,1)(1,0,1)[12] The model result is as follows. 41 Figure 49: (Rose Sales)SARIMA(5,1,1)(1,0,1)[12] Some of the coefficients are statistically insignificant. 8.5.1 (Sparkling Sales)SARIMA(0,0,1)(0,1,1)[12] The model result is as follows. Figure 50: (Sparkling Sales)SARIMA(0,0,1)(0,1,1)[12] All of the coefficients are statistically significant considering 94% confidence. 42 Figure 51: Comparison of all the AUTO SARIMA models 9 Comparison of all different models Figure 52: Comparison of all different models Out of all the models the best model is found to be Best model for Rose - Triple Exponential Smoothing (with α=0.0995, β=1.3e-09 and γ=1.2e-07) Best model for Sparkling - Triple Exponential Smoothing (with α=0.07569, β=0.0324 and γ=0.479) Now, we will take our best model and forecast 12 months into the future with appropriate confidence intervals to see how the predictions look. We have to build our model on the full data for this. 10 Final Forecast Figure 53: Final forecast plot for Rose Figure 54: Final forecast plot for Sparkling 44 11 Actionable Insights and Business Recommendations Rosé Wine Analysis Trend: A noticeable decline in overall sales is observed after 1982. Initial years show higher volatility, followed by more stabilized patterns in later years. Seasonality: Sales display consistent seasonal peaks, with significant surges around November and December, aligning with festive demand. Model Suitability: The multiplicative model provides a better fit due to the seasonal peaks maintaining a proportional relationship with the overall trend. Business Recommendations: – Launch well-timed seasonal campaigns and festive bundles, especially targeting Q4. – Address the long-term downward trend through innovative marketing, rebranding, or introducing limitededition variants. – Investigate patterns during peak early years to extract strategies that can be revived or adapted. – Consider diversifying into related beverage segments to counteract declining core demand. Sparkling Wine Analysis Trend: The sales trend shows steady growth up to 1995, with some slowdown visible after 1990, though year-end performance remains strong. Seasonality: Strong and predictable seasonal effects occur yearly, with the highest sales typically in November and December, reflecting strong holiday influence. Model Suitability: Both additive and multiplicative models fit reasonably well, but the multiplicative model better captures the proportional nature of seasonal fluctuations. Business Recommendations: – Increase inventory and promotional efforts in the second half of the year, especially from July onwards. – Capitalize on holiday momentum by introducing special sparkling wine editions or high-margin bundles. – Examine strong-performing periods to replicate success factors across markets. – Explore export opportunities and upscale positioning in premium product lines.

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