Forecasting

Time series data

https://www.google.com/finance

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What is different?

• Data are dependent over time
• Specific pattern types
• Trends - long term increase or decrease
• Seasonal patterns - patterns related to time of week, month, year, etc.
• Cycles - patterns that rise and fall periodically
• Subsampling into training/test is more complicated
• Similar issues arise in spatial data
• Dependency between nearby observations
• Location specific effects
• Typically goal is to predict one or more observations into the future.
• All standard predictions can be used (with caution!)

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Beware spurious correlations!

http://www.google.com/trends/correlate

http://www.newscientist.com/blogs/onepercent/2011/05/google-correlate-passes-our-we.html

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Also common in geographic analyses

http://xkcd.com/1138/

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Beware extrapolation!

http://www.nature.com/nature/journal/v431/n7008/full/431525a.html

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Google data

library(quantmod)
from.dat <- as.Date("01/01/08", format="%m/%d/%y")
to.dat <- as.Date("12/31/13", format="%m/%d/%y")
getSymbols("GOOG", src="google", from = from.dat, to = to.dat)

[1] "GOOG"

head(GOOG)

           GOOG.Open GOOG.High GOOG.Low GOOG.Close GOOG.Volume
2008-01-02     692.9     697.4    677.7      685.2     4306848
2008-01-03     685.3     686.9    676.5      685.3     3252846
2008-01-04     679.7     681.0    655.0      657.0     5359834
2008-01-07     653.9     662.3    637.4      649.2     6404945
2008-01-08     653.0     660.0    631.0      631.7     5341949
2008-01-09     630.0     653.3    622.5      653.2     6744242

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Summarize monthly and store as time series

mGoog <- to.monthly(GOOG)
googOpen <- Op(mGoog)
ts1 <- ts(googOpen,frequency=12)
plot(ts1,xlab="Years+1", ylab="GOOG")

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Example time series decomposition

• Trend - Consistently increasing pattern over time
• Seasonal - When there is a pattern over a fixed period of time that recurs.
• Cyclic - When data rises and falls over non fixed periods

https://www.otexts.org/fpp/6/1

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Decompose a time series into parts

plot(decompose(ts1),xlab="Years+1")

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Training and test sets

ts1Train <- window(ts1,start=1,end=5)
ts1Test <- window(ts1,start=5,end=(7-0.01))
ts1Train

    Jan   Feb   Mar   Apr   May   Jun   Jul   Aug   Sep   Oct   Nov   Dec
1 692.9 528.7 471.5 447.7 578.3 582.5 519.6 472.5 476.8 412.1 357.6 286.7
2 308.6 334.3 333.3 343.8 395.0 418.7 424.2 448.7 459.7 493.0 537.1 588.1
3 627.0 534.6 529.2 571.4 526.5 480.4 445.3 489.0 455.0 530.0 615.7 563.0
4 596.5 604.5 617.8 588.8 545.7 528.0 506.7 611.2 540.8 509.9 580.1 600.0
5 652.9                                                                  

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Simple moving average

\[ Y_{t}=\frac{1}{2*k+1}\sum_{j=-k}^k {y_{t+j}}\]

plot(ts1Train)
lines(ma(ts1Train,order=3),col="red")

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Exponential smoothing

Example - simple exponential smoothing \[\hat{y}_{t+1} = \alpha y_t + (1-\alpha)\hat{y}_{t-1}\]

https://www.otexts.org/fpp/7/6

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Exponential smoothing

ets1 <- ets(ts1Train,model="MMM")
fcast <- forecast(ets1)
plot(fcast); lines(ts1Test,col="red")

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Get the accuracy

accuracy(fcast,ts1Test)

                   ME   RMSE    MAE     MPE   MAPE   MASE    ACF1 Theil's U
Training set   0.9464  48.78  39.35 -0.3297  7.932 0.3733 0.07298        NA
Test set     156.1890 205.76 160.78 18.1819 18.971 1.5254 0.77025     3.745

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Notes and further resources

• Forecasting and timeseries prediction is an entire field
• Rob Hyndman’s Forecasting: principles and practice is a good place to start
• Cautions
• Be wary of spurious correlations
• Be careful how far you predict (extrapolation)
• Be wary of dependencies over time
• See quantmod or quandl packages for finance-related problems.