Learning Log 17
Albert Luke
April 9, 2018
Seasonal Variation and Time Series Modeling
In the previous classes we learned about how to model a time series dataset using the airpass data as an example. In the plot, we saw oscillations. Seasonal variation refers to the magnitude of these trends. Constant seasonal variation is when the magnitude of the trends remains fairly constant. If we observe constant seasonal variation, no further steps are needed, and we can begin to create a model.
library(faraway)## Warning: package 'faraway' was built under R version 3.4.4data(airpass)
attach(airpass)
plot(pass ~ year, type = "l")
Increasing seasonal variation has the magnitudes increase over time, while decreasing seasonal variation has the magnitudes decreasing over time. In the airpass example, we can see evidence of increasing seasonal variation. In cases where seasonal variation is not constant, we can use transformations or trig functions to help improve our data and make our model more accurate. Note that transformations are usually preferred as they are more effective.
We can use the square root and log transforms and check if that improves our plot. Let’s first try the log transform. Note that the log function is the natural log by default.
par(mfrow = c(2,2))
plot(pass ~ year, type = "l", main = "Original Model")
plot(log(pass) ~ year, type = "l", main = "Log of Pass Only")
plot(pass ~ log(year), type = "l", main = "Log of Year Only")
plot(log(pass) ~ log(year), type = "l", main = "Log of Both")
Looking at our plots we can see that taking the log of the year only is not very helpful. However, taking the log of the number of passengers only gives a plot that shows fairly constant seasonal variation. Taking the log of both variables gives a similar result.
Before we compare which log model is better, let’s try the square root transforms.
par(mfrow = c(2,2))
plot(pass ~ year, type = "l", main = "Original Model")
plot(sqrt(pass) ~ year, type = "l", main = "Sqrt of Pass Only")
plot(pass ~ sqrt(year), type = "l", main = "Sqrt of Year only")
plot(sqrt(pass) ~ sqrt(year), type = "l", main = "Sqrt of Both")
Looks like the square root transformations did not help much. There is still increasing seasonal variation.
Now let’s fit the models for the log transform.
LogPassOnly <- lm(log(pass) ~ year)
LogBoth <- lm(log(pass) ~ log(year))
AIC(LogPassOnly)## [1] -155.5887AIC(LogBoth)## [1] -160.6279Both models are fairly close in terms of AIC, so we go with the model that takes the log of the number of passengers since it’s easier to explain.
Modeling Seasonal Variation
In the airpass example, we are modeling the number of total airline passengers for each month from 1949 to 1951. It makes sense for the time of year to impact the number of airline travelers. For instance, we expect a larger number of travelers over the winter holidays. Thus, we need to take the season into account when creating a time series model.
Unfortunately, the data is organized in a way that requires a bit of work to clean it up before creating a model. The year variable is actually representing the month and year as a decimal. We want to be able to separate the months from the year and use them as categorical predictors.
YearOnly <- floor(airpass$year)
monthDecimal <- airpass$year - YearOnly
monthFactor <- factor (round(monthDecimal * 12))
cbind(airpass$year, monthFactor)## monthFactor
## [1,] 49.08333 2
## [2,] 49.16667 3
## [3,] 49.25000 4
## [4,] 49.33333 5
## [5,] 49.41667 6
## [6,] 49.50000 7
## [7,] 49.58333 8
## [8,] 49.66667 9
## [9,] 49.75000 10
## [10,] 49.83333 11
## [11,] 49.91667 12
## [12,] 50.00000 1
## [13,] 50.08333 2
## [14,] 50.16667 3
## [15,] 50.25000 4
## [16,] 50.33333 5
## [17,] 50.41667 6
## [18,] 50.50000 7
## [19,] 50.58333 8
## [20,] 50.66667 9
## [21,] 50.75000 10
## [22,] 50.83333 11
## [23,] 50.91667 12
## [24,] 51.00000 1
## [25,] 51.08333 2
## [26,] 51.16667 3
## [27,] 51.25000 4
## [28,] 51.33333 5
## [29,] 51.41667 6
## [30,] 51.50000 7
## [31,] 51.58333 8
## [32,] 51.66667 9
## [33,] 51.75000 10
## [34,] 51.83333 11
## [35,] 51.91667 12
## [36,] 52.00000 1
## [37,] 52.08333 2
## [38,] 52.16667 3
## [39,] 52.25000 4
## [40,] 52.33333 5
## [41,] 52.41667 6
## [42,] 52.50000 7
## [43,] 52.58333 8
## [44,] 52.66667 9
## [45,] 52.75000 10
## [46,] 52.83333 11
## [47,] 52.91667 12
## [48,] 53.00000 1
## [49,] 53.08333 2
## [50,] 53.16667 3
## [51,] 53.25000 4
## [52,] 53.33333 5
## [53,] 53.41667 6
## [54,] 53.50000 7
## [55,] 53.58333 8
## [56,] 53.66667 9
## [57,] 53.75000 10
## [58,] 53.83333 11
## [59,] 53.91667 12
## [60,] 54.00000 1
## [61,] 54.08333 2
## [62,] 54.16667 3
## [63,] 54.25000 4
## [64,] 54.33333 5
## [65,] 54.41667 6
## [66,] 54.50000 7
## [67,] 54.58333 8
## [68,] 54.66667 9
## [69,] 54.75000 10
## [70,] 54.83333 11
## [71,] 54.91667 12
## [72,] 55.00000 1
## [73,] 55.08333 2
## [74,] 55.16667 3
## [75,] 55.25000 4
## [76,] 55.33333 5
## [77,] 55.41667 6
## [78,] 55.50000 7
## [79,] 55.58333 8
## [80,] 55.66667 9
## [81,] 55.75000 10
## [82,] 55.83333 11
## [83,] 55.91667 12
## [84,] 56.00000 1
## [85,] 56.08333 2
## [86,] 56.16667 3
## [87,] 56.25000 4
## [88,] 56.33333 5
## [89,] 56.41667 6
## [90,] 56.50000 7
## [91,] 56.58333 8
## [92,] 56.66667 9
## [93,] 56.75000 10
## [94,] 56.83333 11
## [95,] 56.91667 12
## [96,] 57.00000 1
## [97,] 57.08333 2
## [98,] 57.16667 3
## [99,] 57.25000 4
## [100,] 57.33333 5
## [101,] 57.41667 6
## [102,] 57.50000 7
## [103,] 57.58333 8
## [104,] 57.66667 9
## [105,] 57.75000 10
## [106,] 57.83333 11
## [107,] 57.91667 12
## [108,] 58.00000 1
## [109,] 58.08333 2
## [110,] 58.16667 3
## [111,] 58.25000 4
## [112,] 58.33333 5
## [113,] 58.41667 6
## [114,] 58.50000 7
## [115,] 58.58333 8
## [116,] 58.66667 9
## [117,] 58.75000 10
## [118,] 58.83333 11
## [119,] 58.91667 12
## [120,] 59.00000 1
## [121,] 59.08333 2
## [122,] 59.16667 3
## [123,] 59.25000 4
## [124,] 59.33333 5
## [125,] 59.41667 6
## [126,] 59.50000 7
## [127,] 59.58333 8
## [128,] 59.66667 9
## [129,] 59.75000 10
## [130,] 59.83333 11
## [131,] 59.91667 12
## [132,] 60.00000 1
## [133,] 60.08333 2
## [134,] 60.16667 3
## [135,] 60.25000 4
## [136,] 60.33333 5
## [137,] 60.41667 6
## [138,] 60.50000 7
## [139,] 60.58333 8
## [140,] 60.66667 9
## [141,] 60.75000 10
## [142,] 60.83333 11
## [143,] 60.91667 12
## [144,] 61.00000 1The month factor is from 1 to 12, and denotes the month of the year in which the measurement was taken. Now let’s rename the numbers to the months they correspond with.
levels(monthFactor) <- c("Jan", "Feb", "Mar", "Apr", "May",
"Jun", "Jul", "Aug", "Sep", "Oct",
"Nov", "Dec")
head(monthFactor)## [1] Feb Mar Apr May Jun Jul
## Levels: Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov DecNow we can create our model.
Airpassmod <- lm(log(pass) ~ YearOnly + monthFactor)
summary(Airpassmod)##
## Call:
## lm(formula = log(pass) ~ YearOnly + monthFactor)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.156370 -0.041016 0.003677 0.044069 0.132324
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -1.214998 0.081277 -14.949 < 2e-16 ***
## YearOnly 0.120826 0.001432 84.399 < 2e-16 ***
## monthFactorFeb 0.031390 0.024253 1.294 0.198
## monthFactorMar 0.019404 0.024253 0.800 0.425
## monthFactorApr 0.159700 0.024253 6.585 1.00e-09 ***
## monthFactorMay 0.138500 0.024253 5.711 7.19e-08 ***
## monthFactorJun 0.146196 0.024253 6.028 1.58e-08 ***
## monthFactorJul 0.278411 0.024253 11.480 < 2e-16 ***
## monthFactorAug 0.392422 0.024253 16.180 < 2e-16 ***
## monthFactorSep 0.393196 0.024253 16.212 < 2e-16 ***
## monthFactorOct 0.258630 0.024253 10.664 < 2e-16 ***
## monthFactorNov 0.130541 0.024253 5.382 3.28e-07 ***
## monthFactorDec -0.003108 0.024253 -0.128 0.898
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.0593 on 131 degrees of freedom
## Multiple R-squared: 0.9835, Adjusted R-squared: 0.982
## F-statistic: 649.4 on 12 and 131 DF, p-value: < 2.2e-16In this model, we are using the month of January as a reference. Febuary, March, and December do not have significant p-values, which means the number of airline passengers in those months is not significantly different from the number of passengers in the month of January.
The slope for December is negative, which means that the month of December saw fewer airline travelers than January (although as mentioned before, this difference isn’t significant because of the large p-value). Positive slopes for the indicators mean that month saw more travelers than January.
We can plot this model onto the plot of our data. The black line is the original data, while the red line shows the model.
plot(log(pass) ~ year, type = "l")
lines(airpass$year, Airpassmod$fitted.values, col = "red")
We can see that the model fits the observed data fairly well.
Conclusion
Seasonal variation can play a big role in a time series model. In previous classes we did not account for this factor, nor did we use tranformations, which resulted in models that didn’t fit the observed data very well. We could observe that the number of airline passengers increased over time. By including seasonal variation in our model, we can see how the number of airline passengers fluctuates over the course of a given year.
Peer Review
I got some good feedback during the peer review session. The discussion portion could use some additional explanation of the significance of our results in a more general context. There were a few grammar mistakes in the introduction of the paper. The linear regression formula needs more explanation and interpretation. Some of the variable names in the dataset should also be explained in the methods section. Finally, the appendix should include the output for the R code we used to build our final model (and some of it shown in results as well).