Recommended
More Related Content
Similar to SalesyearQ1Q2Q3Q43.569199510004.1521995.25000000000001003.9981995.50.docx
Similar to SalesyearQ1Q2Q3Q43.569199510004.1521995.25000000000001003.9981995.50.docx (20)
More from jeffsrosalyn
More from jeffsrosalyn (20)
Recently uploaded
Recently uploaded (20)
SalesyearQ1Q2Q3Q43.569199510004.1521995.25000000000001003.9981995.50.docx
- 2. 0000319011413152.51001151315311011613155.512011713156. 300003113011813148.599990814012213150.399993915012313 146.550003116012413148.30000311701251314918012813149.8 00003119012913147.699996920013013146.9499969210131131 47.949996922020113144.349990823020413144.0500031240205 13142.099990825020613140.300003126020713141.0500031270 20813140.149993928021113140.649993929021213138.7530021 31313831021413136.949996932021513136.5330219131413402 2013141.7535022113143.099990836022213142.5999908370225 13142.899993938022613142.399993939022713142.6499939400 22813142.899993941030113145.949996942030413140.5430305 13140.599990844030613142.449996945030713143.3499908460 30813143.050003147031113141.649993948031213140.0500031 49031313138.899993950031413136.551031513133.1499939520 31813133.099990853031913133.599990854032013133.7555032 113135.300003156032213135.599990857032513137.599990858 032613136.599990859032713137.149993960032813138.399993 961040113136.149993962040213139.449996963040313139.564 040413140.149993965040513135.899993966040813135.399993 967040913136.050003168041013136.800003169041113135.257 0041213134.449996971041513135.849990872041613136.09999 0873041713 R Homework for Chapter 20 Coffee Prices 2013 Coffee is the world’s second largest legal export commodity (after oil) and is the second largest source of foreign exchange for developing nations.The Unite States consumes about one-fifth of the world’s coffee. The International Coffee Organization (ICO) computes a coffee price index using Colombian, Brazilian, and a mixture of other coffee data. Data are provided for the daily
- 3. ICO price index (in US dollars) from January 2013 to April 2013. You can find the data file on Blackboard. Download it and put it in the same folder as your R program file. Then, use the following command to read the data coffee <- read.table('Coffee_prices_2013.txt', sep = 't', header = TRUE) 1. Make a time series plot (not scatterplot) of price against time. Use “Coffee Price” and “Time” as label for y-axis and x-axis, respectively. Which time series components are evident from the plot? 2. Smooth the coffee price series using simple moving averages (SMA) of length 2 and 8. Add the two smoothed curves (one in red and one in green) to the plot made in (a) and compare them. 3. Apply single exponential smoothing (SES) to the coffee price series with weights α = 0.8 and α = 0.2, respectively. Add the two smoothed curves (one in orange and one in purple) to the plot made in (a) and compare them. 4. Find autocorrelation between the original time series and each of the first 5 lags. Then, fit an autoregressive model with the lags whose autocorrelations are greater than 0.8. Write down the fitted model and add the smoothed curve in blue to the plot made in (a). Which lag does the model depend on most? Why? 5. Suppose we know that the next value in the series was, in fact, 138.90. Compute the corresponding absolute percentage error (APE) for each of the models you have fitted before. Which model gives us the best
- 4. prediction? 1 Coffee Prices 2013 Chapter 20: Time Series Analysis Home Depot Sales The Home Depot chain of home improvement stores grew in the 1980s and 1990s faster than any other retailer in history. By 2005, it was the second largest retailer in the United States. But its extraordinary record of growth was slowed by the financial crisis of 2008. How do different methods of modeling time series compare for understanding these data? Let’s first read the dataset into R HD <- read.table('Home_Depot_2012_GE19.txt', sep = 't', header = TRUE) and look at its structure str(HD) ## 'data.frame': 72 obs. of 6 variables: ## $ Sales: num 3.57 4.15 4 3.75 4.36 ... ## $ year : num 1995 1995 1996 1996 1996 ... ## $ Q1 : int 1 0 0 0 1 0 0 0 1 0 ... ## $ Q2 : int 0 1 0 0 0 1 0 0 0 1 ... ## $ Q3 : int 0 0 1 0 0 0 1 0 0 0 ... ## $ Q4 : int 0 0 0 1 0 0 0 1 0 0 ... As we can see, there are 72 observations for each of 6 variables. Both Sales and year are numerical variables,
- 5. while Q1, Q2, Q3 and Q4 are the indicator variables for the quarters. Let’s plot the time series of Home Depot quarterly sales: plot(HD$year, HD$Sales, xlab = 'Year', ylab = 'Sales', main = 'Home Depot Quarterly Sales 1995 - 2012') lines(HD$year, HD$Sales) 1995 2000 2005 2010 5 1 0 1 5 2 0 2 5 Home Depot Quarterly Sales 1995 − 2012 Year S a le s As we can see, there was a consistent increasing trend until the
- 6. end of 2006. After that, sales fell sharply. 1 They appear to have been recovering. Throughout this period, however, there are fluctuations around the trend that appear to be seasonal because they repeat every four quarters. Smoothing Methods First, Let’s try a simple moving average (SMA). For data with a strong seasonal component, it’s a good idea to choose a moving average length based on the period. The period of Home Depot sales is 4. We therefore apply a SMA of length 2 and 4, respectively. We use SMA() function in TTR package to do this: library(TTR) ## Registered S3 method overwritten by 'xts': ## method from ## as.zoo.xts zoo sma1 <- SMA(HD$Sales, n = 2) sma2 <- SMA(HD$Sales, n = 4) The sma1 and sma2 return the smoothed series, which are plotted below: par(mfrow = c(1, 2)) plot(HD$year, HD$Sales, xlab = 'Year', ylab = 'Sales', main = 'SMA of Length 2') lines(HD$year, HD$Sales) lines(HD$year, sma1, col = 'red')
- 7. plot(HD$year, HD$Sales, xlab = 'Year', ylab = 'Sales', main = 'SMA of Length 4') lines(HD$year, HD$Sales) lines(HD$year, sma2, col = 'red') 1995 2000 2005 2010 5 1 0 1 5 2 0 2 5 SMA of Length 2 Year S a le s 1995 2000 2005 2010 5 1
- 8. 0 1 5 2 0 2 5 SMA of Length 4 Year S a le s As expected, the SMA of longer length shows a smoother series. The SMA of length 4 smooths out most of the seasonal effects, and has trouble modeling the sudden change in 2007. But, it provides a good description of the trend. The SMA of length 2 is more wiggly, capturing the seasonal effects and the sudden change better. Second, let’s try an exponential moving average (EMA). We need to choose a smoothing weight. We will use α = 0.5, which weights the current data value equally all the rest in the past. For comparison reasons, we 2
- 9. also try α = 0.2, which makes the smoothing heavily depend on the data in the past. The initial value is chosen as the first observed data point. We use EMA() function in TTR package to do this: ema1 <- EMA(HD$Sales, ratio = 0.5, n = 1) ema2 <- EMA(HD$Sales, ratio = 0.2, n = 1) par(mfrow = c(1, 2)) plot(HD$year, HD$Sales, xlab = 'Year', ylab = 'Sales', main = 'EMA (alpha = 0.5)') lines(HD$year, HD$Sales) lines(HD$year, ema1, col = 'red') plot(HD$year, HD$Sales, xlab = 'Year', ylab = 'Sales', main = 'EMA (alpha = 0.2)') lines(HD$year, HD$Sales) lines(HD$year, ema2, col = 'red') 1995 2000 2005 2010 5 1 0 1 5 2 0 2 5
- 10. EMA (alpha = 0.5) Year S a le s 1995 2000 2005 2010 5 1 0 1 5 2 0 2 5 EMA (alpha = 0.2) Year S a le s
- 11. As we can see, the EMA with a bigger α provides a less smooth series. The EMA of α = 0.5 follows the seasonal pattern more closely and models the sudden change better. The EMA of α = 0.2 totally fails, because it is too smooth to capture the seasonal effects, and responds too slowly to the trend (under-estimate before 2007 and over-estimate after). Autoregressive Method Let’s fit an autoregressive model. Because we know that the seasonal pattern is of 4 quarters long, we use 4 lagged variables as the predictors. The autocorrelations regarding those lags are given by acf(HD$Sales, lag.max = 4, plot = FALSE) ## ## Autocorrelations of series 'HD$Sales', by lag ## ## 0 1 2 3 4 ## 1.000 0.912 0.838 0.831 0.838 It’s interesting to see that the autocorrelation of lag4 is 0.838, the second largest among all. It is also the evidence of the seasonal effects. 3 Now let’s use ar() function to fit the autoregressive model: ar1 <- ar(HD$Sales, aic = FALSE, order.max = 4, demean = FALSE, intercept = TRUE, method = 'ols')
- 12. ar1 ## ## Call: ## ar(x = HD$Sales, aic = FALSE, order.max = 4, method = "ols", demean = FALSE, intercept = TRUE) ## ## Coefficients: ## 1 2 3 4 ## 0.4695 -0.2551 0.2233 0.4871 ## ## Intercept: 1.694 (0.4544) ## ## Order selected 4 sigma^2 estimated as 1.612 The fitted AR(4) model is ŷt = 1.694 + 0.4695yt−1 − 0.255yt−2 + 0.223yt−3 + 0.487yt−4 Unfortunately, the ar1 does not provide the fitted values. But, we can use the residuals e to compute them using ŷ = y − e: fitted.ar1 <- HD$Sales - ar1$resid Let’s plot the fitted series and the residuals: par(mfrow = c(1, 2)) plot(HD$year, HD$Sales, xlab = 'Year', ylab = 'Sales', main = 'AR(4)') lines(HD$year, HD$Sales) lines(HD$year, fitted.ar1, col = 'red') plot(HD$year, ar1$resid, xlab = 'Year', ylab = 'Residuals') abline(0, 0) 1995 2000 2005 2010
- 13. 5 1 0 1 5 2 0 2 5 AR(4) Year S a le s 1995 2000 2005 2010 − 4 − 2 0 2
- 14. 4 Year R e si d u a ls As we can see, the AR(4) model follows the seasonal pattern well, but seems to have great difficulty capturing the sudden change in 2007. The residual plot shows some disturbance around the time of the financial 4 crisis. Because sales at that time stopped resembling previous behavior with respect to growth, the 4 lagged variables are less successful predictors. Multiple Regression-based Model Let’s fit a multiple regression model to Sales using year and the dummy variables. We use year to model trend and the dummy variables seasonal component. Since there is a linear trend until the end of 2006, we will fit the model to the time series before 2007 only. Note that we will use the last quarter Q4 as baseline, so
- 15. only put the first three dummy variables in the model. HD.new <- HD[HD$year<2007,] imod <- lm(Sales ~ year + Q1 + Q2 + Q3, data = HD.new) summary(imod) ## ## Call: ## lm(formula = Sales ~ year + Q1 + Q2 + Q3, data = HD.new) ## ## Residuals: ## Min 1Q Median 3Q Max ## -1.9176 -0.6004 -0.1046 0.3649 4.7269 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) -3.515e+03 1.084e+02 -32.414 < 2e-16 *** ## year 1.762e+00 5.414e-02 32.554 < 2e-16 *** ## Q1 5.885e-01 4.967e-01 1.185 0.242652 ## Q2 1.755e+00 4.921e-01 3.567 0.000901 *** ## Q3 4.554e-01 4.878e-01 0.934 0.355686 ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 1.178 on 43 degrees of freedom ## Multiple R-squared: 0.966, Adjusted R-squared: 0.9629 ## F-statistic: 305.7 on 4 and 43 DF, p-value: < 2.2e-16 The coefficient of year is positive and highly significant, showing a positive linear trend in the time series of Sales . The significant coefficient of Q2 indicates that the sales in second quarter is on average significantly higher than the last quarter (baseline). The whole model looks great with high R2 and significant F-test. Comparison
- 16. So far, we have applied a few different methods to the Home Depot series. Suppose we want to forecast the sales for the first quarter of 2013. (Note: the original series ends at the fourth quarter of 2012.) In the smoothing methods, we use the last average in the series to forecast: (yhat.sma1 <- sma1[length(sma1)]) ## [1] 18.627 (yhat.sma2 <- sma2[length(sma2)]) ## [1] 18.908 5 (yhat.ema1 <- ema1[length(ema1)]) ## [1] 18.89295 (yhat.ema2 <- ema2[length(ema2)]) ## [1] 18.38313 In the autoregressive model, we may use predict() function to forecast: (yhat.ar1 <- predict(ar1, n.ahead = 1, se.fit = FALSE)) ## Time Series: ## Start = 73 ## End = 73 ## Frequency = 1 ## [1] 19.31656 In the multiple regression, we also need predict() function to forecast:
- 17. data.new <- data.frame(year = 2013, Q1 = 1, Q2 = 0, Q3 = 0) yhat <- predict(imod, newdata = data.new) We know that the actual sales in the first quarter of 2013 were $19.124$B. Which model does provide the best prediction? We can compute absolute percentage error (APE) for each prediction: y.true <- 19.124 abs(y.true - yhat.sma1)/abs(y.true)*100 ## [1] 2.598829 abs(y.true - yhat.sma2)/abs(y.true)*100 ## [1] 1.129471 abs(y.true - yhat.ema1)/abs(y.true)*100 ## [1] 1.208153 abs(y.true - yhat.ema2)/abs(y.true)*100 ## [1] 3.87401 abs(y.true - yhat.ar1)/abs(y.true)*100 ## Time Series: ## Start = 73 ## End = 73 ## Frequency = 1 ## [1] 1.006905 abs(y.true - yhat)/abs(y.true)*100 ## 1 ## 76.65737 The AR(4) model seems to offer the best prediction, because it yields the smallest APE (1.01%). The multiple regression model gives the worst forecast, because the financial crisis changed the trend of the time series
- 18. after 2007, but the model assumes the trend remains the same. 6 Home Depot SalesSmoothing MethodsAutoregressive MethodMultiple Regression-based ModelComparison