Department Store Sales Time-Series Analysis

Independent Data Analysis Project

Author

Affiliation

John Karuitha

Karatina University, Kenya

Executive Summary

I analyze time series data from a departmental store and attempt to make forecasts. I also discuss the various terms associated with time series- trend, seasonality, and noise. I then compare the performance of the models in forecasting. The results show that a polynomial model with seasonal effects does better than a pure linear model.

Code

## Load the required packages 
###################################################
if(!require(pacman)){
    install.packages("pacman")
}

##################################################
pacman::p_load(tidyverse, janitor, skimr, corrplot, 
               Amelia, xtable, tidymodels, KableExtra, 
               rpart, rpart.plot,  arules, arulesViz, 
               forecast, fpp2, fpp3, gt, ggthemes)

##################################################
## Set theme and digits

options(digits = 2)

## Table formatting function
formatting_function <- function(data, caption = "Table 1", 
                                full_width = FALSE){
    library(kableExtra)
    data %>% 
        kbl(booktabs = TRUE, caption = caption) %>% 
        kableExtra::kable_classic(full_width = full_width,
                      latex_options = "hold_position")
}

if(!require(firatheme)){remotes::install_github("vankesteren/firatheme")}

## plots themes
theme_set(theme_clean())

1 Department Store Sales Time-Series

The file DepartmentStorSales.csv contains data on the quarterly sales for a department store over a 6-year period from 2000 to 2005

Code

## Read in the data and see first six points
sales <- read_csv("DepartmentStoreSales.csv")
head(sales) %>% 
    gt(caption = "Sample Data")

Sample Data

Quarter	Sales
1	50147
2	49325
3	57048
4	76781
5	48617
6	50898

1.1 Components of Time Series

I discuss four major components of time series (level, trend, seasonality, and noise). Discuss the meaning of these four major components. (3 points)

1.1.1 Level

In time series analysis, level refers to the average value in the series. Usually, firms may work with the level value and then adjust their stock levels for seasonal and trend components.

1.1.2 Trend

Trend in time series analysis is the sustained increase or decrease observed in a series that spans accross any seasonal patterns. In the case of the departmental store, there is an overall upward trajectory in sales. This means that, on average, overall sales are increasing even before factoring in seasonality. This trajectory is also quite predictable. The store management can then plan to stock higher levels of stock in the current quarter compared to the corresponding stock levels in the previous quarter.

1.1.3 Seasonality

Seasonality refers to the short-term recurring cycles in the series. In other words, these are the periodic ups and downs observed in the data. In the case of our departmental store sales, we can observe a periodic rise and fall in demand that occurs every 3 quarters. The seasonal component is very predictable. In this case, the department should increase the stock during the peak periods and hold relatively lower stock during the off-peak periods.

1.1.4 Noise

Noise refers to the random variation in the series. This random variation is unpredictable. Noise arises from the unknown factors that affect demand or factors that are not measurable, the known-knowns and the unknown-unknowns. This is the reason many businesses hold working capital in the form of extra cash or inventory to take care of these unpredictable ups and downs in sales.

1.2 Plotting the Data

I show the data in a time-series format and create a well-performed time plot of the data.

Code

## Create a time series data
my_sales_ts <- ts(sales, start = c(2000, 1), frequency = 4)

## View the data
print(my_sales_ts)

        Quarter  Sales
2000 Q1       1  50147
2000 Q2       2  49325
2000 Q3       3  57048
2000 Q4       4  76781
2001 Q1       5  48617
2001 Q2       6  50898
2001 Q3       7  58517
2001 Q4       8  77691
2002 Q1       9  50862
2002 Q2      10  53028
2002 Q3      11  58849
2002 Q4      12  79660
2003 Q1      13  51640
2003 Q2      14  54119
2003 Q3      15  65681
2003 Q4      16  85175
2004 Q1      17  56405
2004 Q2      18  60031
2004 Q3      19  71486
2004 Q4      20  92183
2005 Q1      21  60800
2005 Q2      22  64900
2005 Q3      23  76997
2005 Q4      24 103337

Code

## Plot the data
autoplot(my_sales_ts[, "Sales"]) + 
    labs(title = "Quarterly Sales, 2000-2005", y = "Sales", x = "Year")

I also create a seasons plot that shows the sales for each year by season. The plot shows that in spite of the seasonality, sales in each subsequent year were higher than the last.

Code

ggseasonplot(my_sales_ts[, "Sales"]) +
    labs(y =  "Sales",
         title = "Seasonal Plots")

Likewise, the sub-series plot shows the trends in sales for each quarter. For instance, the first panel is a plot of sales in \(2000Q1, 2001Q1 \cdots 2005Q1\). Again the plot shows a general upward trend.

Code

ggsubseriesplot(my_sales_ts[, "Sales"]) + 
    labs(y =  "Sales",
         title = "Subseries Plots by Quarters")

1.3 Decomposition

Decompose your time-series data into four major components (level, trend, seasonality, and noise) and plot these four components individually. Ensure to and explain your observations about different trends from your plots. (6 points)

Code

## Decompose the data into level,seasons and trend
my_dec_ts <- decompose(my_sales_ts[, "Sales"])

## Plot the decomposition
autoplot(my_dec_ts)

The decomposition above consists of 4 plots. The first plot is the observed raw data. The second plot illustrates the general upward trend of sales over the period. The trend shows that quarter after quarter, the general trend is up. The management of the store could plan future inventory on this trend among other factors.

The third plot shows the seasonal component, the short swings in sales. when stocking, the management could consider these dynamics by stocking more when the demand is high compared the the off-peak periods.

The final plot shows the random component of the sales. Here, the data appears to follow a random walk with no apparent pattern. In planning for sales, it is hard to cater for this component except for keeping some buffer stock just in case of an upward but unexpected uptick in demand.

1.4 Modelling

I run and discuss and compare different trend models in time series, including the linear trend regression model, exponential trend model, and polynomial trend model.

1.4.1 Linear trend regression model

The forecasting equation for the linear trend model is:

\(y = a + \beta t\)

Here, t is the time index. The \(\alpha\) and \(\beta\) (the “intercept” and “slope”) are come from a simple regression in with Y as the dependent variable and the time t as the independent variable.

Although useful, these models perform poorly when the trend is not linear as is the case with our data.In this case a polynomial trend model does better. Even where the trend is linear, the linear model may fail to pick seasonal or cyclical fluctuations.

1.4.2 Exponential trend model

Exponential smoothing methods use the weighted averages of past observations to make forecasts of the future. These weights reduce as the observations get older with more recent observations having a higher weight (Hyndman and Athanasopoulos 2018). The technique is easy and quick to implement and produces fairly good forecasts in several cases.

1.4.3 Polynomial trend model

Polynomial trend models model the trend in data as a smooth variation in time. This model permits us to fit a either a line or a curve to the data as opposed to the linear model that assumes a purely linear relationship over time. The polynomial trend models provide a simple, flexible forms to describe the local trend components (West and Harrison 1989). These class of models are useful for modelling data that has an increasing or decreasing trend. More specifically, The polynomial trend models can be used in those situations where the relationship between study and explanatory variables is curvilinear.

Given a polynomial regression model in one variable as follows;

\(y = \beta_{0} + \beta_{1}X + \beta_{2}X^{2} + e_{2}\)

This is a second-order model or quadratic model. \(\beta_{1}\) and \(\beta_{2}\) are called the linear effect parameter and quadratic effect parameter, respectively.This class of models can extend to include several variables and their interactions and even higher orders like cubes.

1.5 Forecasting

Partition the first 18 records as training sets and the rest of 6 records as validation sets.

1.5.1 Linear Model

Fit a linear trend model on training sets. Based on this linear trend model, make a forecast for validation sets and show the results.

Code

## Create training and testing sets 
ts_training <- window(my_sales_ts, start = c(2000, 1), 
                      end = c(2004, 2))

ts_testing <- window(my_sales_ts, start = c(2004, 2))

I then fit a linear trend model on the training data.

Code

## Fit a linear trend model
sales_model <- tslm(Sales ~ Quarter, data = ts_training)

## Summary of linear trend model
summary(sales_model)

Call:
tslm(formula = Sales ~ Quarter, data = ts_training)

Residuals:
   Min     1Q Median     3Q    Max 
-10862  -8390  -5160   1665  20742 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)    54132       5674    9.54  5.3e-08 ***
Quarter          644        524    1.23     0.24    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 11500 on 16 degrees of freedom
Multiple R-squared:  0.0861,    Adjusted R-squared:  0.029 
F-statistic: 1.51 on 1 and 16 DF,  p-value: 0.237

Code

## Forecast model with linear model
my_linear_forecast <- forecast::forecast(sales_model, newdata = ts_testing[, "Quarter"])

## Print my linear model forecast
my_linear_forecast

        Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
2004 Q3          65721 48791 82651 38872 92570
2004 Q4          66365 49175 83554 39105 93624
2005 Q1          67008 49536 84481 39299 94718
2005 Q2          67652 49873 85431 39457 95847
2005 Q3          68296 50189 86404 39580 97012
2005 Q4          68940 50483 87397 39670 98210
2006 Q1          69584 50758 88409 39729 99438

We then plot the linear model forecasts.

Code

autoplot(my_linear_forecast) + 
    labs(title = "Forecasts from Linear Trend Model")

1.5.2 Polynomial trend model with seasonal effects

I fit a polynomial trend model with seasonal effects on training sets. Based on this model, make a forecast for validation sets and show the results. Check accuracy measures of the forecasted model and show these accuracy measures. (8 points)

Code

## Fit a model with season a trend 
my_trend_model  <- forecast::tslm(Sales ~ trend + season, data = ts_training)

## Summarise the model
summary(my_trend_model)

Call:
forecast::tslm(formula = Sales ~ trend + season, data = ts_training)

Residuals:
   Min     1Q Median     3Q    Max 
 -2289   -938   -373    402   3070 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  46519.5     1065.7   43.65  1.7e-15 ***
trend          557.2       80.1    6.96  1.0e-05 ***
season2       1388.8     1112.8    1.25     0.23    
season3       8489.5     1177.3    7.21  6.8e-06 ***
season4      27735.4     1180.0   23.50  4.9e-12 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 1750 on 13 degrees of freedom
Multiple R-squared:  0.983, Adjusted R-squared:  0.978 
F-statistic:  186 on 4 and 13 DF,  p-value: 2.49e-11

Code

## A function to compute mean percent error
mape <- function(actual,pred){

  mape <- mean(abs((actual - pred)/actual))*100

  return (mape)

}

Code

## Forecasting with the model with season a trend 
my_trend_forecast <- forecast::forecast(my_trend_model, newdata = ts_testing)

my_trend_forecast

        Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
2004 Q3          65596 62734 68457 61017 70174
2004 Q4          85399 82537 88260 80820 89977
2005 Q1          58221 55318 61123 53577 62864
2005 Q2          60166 57264 63069 55523 64810
2005 Q3          67824 64773 70876 62942 72707
2005 Q4          87627 84576 90679 82745 92510
2006 Q1          60449 57330 63569 55458 65441

Code

autoplot(my_trend_forecast)

In this section I compute the error for both models. Note that the model with trend and seasons tends to do better than the linear trend model.

Code

## capturing the forecasts
forecasts <- data.frame(
    
    Quarter = (nrow(my_sales_ts) - 6):nrow(my_sales_ts),
    linear_trend = c(65721, 66365, 67008, 67652, 68296, 68940, 69584),
    trend_season = c(65596, 85399, 58221, 60166, 67824, 87627, 60449)
)
## % Error for the linear trend model
mape(ts_testing[, "Sales"], forecasts$linear_trend)

[1] 15

Code

## % Error for model with trend and seasons
mape(ts_testing[, "Sales"], forecasts$trend_season)

[1] 18

1.6 Comparison

Finally, based on the decomposition of four major components from 5.3, I compare your 5.5 and 5.6 models and discuss which model is more appropriate in this particular setting, giving reasons.

The plot below compares the comparison of both models. Note that the model with seasonality and trend (red broken line) captures the data much better than the linear model (blue line). I would choose the model with seasonality and trend given the nature of the data; It has both a trend component and a seasonal component. The linear trend model fares especially poorly in capturing seasons and cycles compared to the model with seasonality and trend.

Code

sales %>% 
    ggplot(aes(x = Quarter, y = Sales)) + 
    geom_line() + 
    geom_line(data = forecasts, aes(y = linear_trend),
              col = "blue") + 
    geom_line(data = forecasts, aes(y = trend_season),
              col = "red", linetype = "dashed") + 
    labs(title = "Comaring the Linear Trend Model and the Model with Season and Trend")

References

Hyndman, Rob J., and George Athanasopoulos. 2018. Forecasting: Principles and Practice. Melbourne: OTexts.

West, Mike, and Jeff Harrison. 1989. “Polynomial Trend Models.” In Bayesian Forecasting and Dynamic Models, 201–28. New York, NY: Springer New York. https://doi.org/10.1007/978-1-4757-9365-9_7.