lecture2

1. Social Forecasting
Lecture 2: Performance Evaluation and Validation
Thomas Chadefaux
1

2. Fundamentals of Forecasting

3. Time series data
• Daily IBM stock prices
• Monthly rainfall
• Annual Google profits
• Quarterly beer production
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300
400
500
600
1960 1970 1980 1990 2000 2010
Time
.
2

4. Time series vs. cross-sectional data
3

5. Beer production
Forecasting is estimating how the sequence of observations
will continue into the future.
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300
400
500
600
1960 1980 2000
Time
.
Forecasts from ETS(M,A,M)
4

6. Beer production
350
400
450
500
1995 2000 2005 2010
Time
.
Forecasts from ETS(M,A,M)
5

7. Defining your data as Time series
#yearly data: one observation per year
y <- ts(c(123, 39, 78, 52, 110), start = 2012, frequency =
y
## Time Series:
## Start = 2012
## End = 2016
## Frequency = 1
## [1] 123 39 78 52 110
6

8. Monthly data
# Monthly data
y <- ts(y, start = 2003, frequency = 12)
y
## Jan Feb Mar Apr May
## 2003 123 39 78 52 110
Note that quarterly data would require “frequency = 4”, weekly
data frequency = 52, etc.
7

9. Time plots
autoplot(LAprices) + ggtitle('House Prices in LA')
4e+05
5e+05
6e+05
7e+05
2008 2010 2012 2014 2016 2018 2020
Time
LAprices
House Prices in LA
8

10. Time plots: seasonal
autoplot(a10)+
ggtitle("Antidiabetic drug sales") +
ylab("$ million") +
xlab("Year")
10
20
30
1995 2000 2005
Year
$
million
Antidiabetic drug sales
9

11. Seasonal plots
1991
1991
1992
1992
1993
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1995
1995
1996 1996
1997
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2003 2003
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2006 2006
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2008
2008
10
20
30
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
Month
$
million
Seasonal plot: antidiabetic drug sales
10

12. SubSeasonal plots
10
20
30
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
Month
$
million
Seasonal subseries plot: antidiabetic drug sales
11

13. Multiple time series
Chicago Los Angeles New York
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2e+05
4e+05
6e+05
$ million
value
Seasonal subseries plot: antidiabetic drug sales
12

14. Simple Forecasting Methods

15. How to forecast. . .
400
450
500
1995 2000 2005 2010
Year
megalitres
Quarterly beer production
How would you forecast these data?
13

16. How to forecast. . .
80
90
100
110
1990 1991 1992 1993 1994 1995
Year
thousands
Number of pigs slaughtered
How would you forecast these data?
14

17. How to forecast. . .
3600
3700
3800
3900
4000
0 50 100 150 200 250 300
Day
Dow−Jones index
How would you forecast these data?
15

18. Average method
• Forecast of all future values is equal to mean of historical data
{y1, . . . , yT }.
• Forecasts: ŷT+h|T = ȳ = (y1 + · · · + yT )/T
meanf(beer2, h=10, level = 95)
## Point Forecast Lo 95 Hi 95
## 2010 Q3 433.5135 346.801 520.2261
## 2010 Q4 433.5135 346.801 520.2261
## 2011 Q1 433.5135 346.801 520.2261
## 2011 Q2 433.5135 346.801 520.2261
## 2011 Q3 433.5135 346.801 520.2261
## 2011 Q4 433.5135 346.801 520.2261
## 2012 Q1 433.5135 346.801 520.2261
## 2012 Q2 433.5135 346.801 520.2261
## 2012 Q3 433.5135 346.801 520.2261 16

19. Plotting mean
beer2 <- window(ausbeer,start=1992,end=c(2007,4))
autoplot(beer2) +
autolayer(meanf(beer2, h=11), PI=TRUE, series="Mean") +
ggtitle("Forecasts for quarterly beer production") +
xlab("Year") + ylab("Megalitres") +
guides(colour=guide_legend(title="Forecast"))
350
400
450
500
1995 2000 2005 2010
Megalitres
Forecast
Mean
Forecasts for quarterly beer production
17

20. Naïve method
• Forecasts equal to last observed value.
• Forecasts: ŷT+h|T = yT .
naive(beer2, h=10, level = 95)
## Point Forecast Lo 95 Hi 95
## 2008 Q1 473 344.98474 601.0153
## 2008 Q2 473 291.95908 654.0409
## 2008 Q3 473 251.27106 694.7289
## 2008 Q4 473 216.96948 729.0305
## 2009 Q1 473 186.74917 759.2508
## 2009 Q2 473 159.42793 786.5721
## 2009 Q3 473 134.30345 811.6965
## 2009 Q4 473 110.91816 835.0818
## 2010 Q1 473 88.95421 857.0458
## 2010 Q2 473 68.18020 877.8198 18

21. Plotting naïve
beer2 <- window(ausbeer,start=1992,end=c(2007,4))
autoplot(beer2) +
autolayer(naive(beer2, h=11), PI=TRUE, series="Mean") +
ggtitle("Forecasts for quarterly beer production") +
xlab("Year") + ylab("Megalitres") +
guides(colour=guide_legend(title="Forecast"))
250
500
750
1995 2000 2005 2010
Megalitres
Forecast
Mean
Forecasts for quarterly beer production
19

22. Seasonal naïve method
• Forecasts equal to last value from same season.
• Forecasts: ŷT+h|T = yT+h−m(k+1), where m = seasonal period
and k is the integer part of (h − 1)/m (i.e., the number of
complete years in the forecast period prior to time T + h )
• E.g., h = 1 and m = 12 (i.e. monthly data) → k = 0, so
ŷT+h|T = yT+1−12(0+1) = yT−11).
• E.g. we are in January, so we predict February using January
-11 months = Feb of the previous year.
20

23. Seasonal naïve method
snaive(beer2, h=10, level = 95)
## Point Forecast Lo 95 Hi 95
## 2008 Q1 427 394.1080 459.8920
## 2008 Q2 383 350.1080 415.8920
## 2008 Q3 394 361.1080 426.8920
## 2008 Q4 473 440.1080 505.8920
## 2009 Q1 427 380.4837 473.5163
## 2009 Q2 383 336.4837 429.5163
## 2009 Q3 394 347.4837 440.5163
## 2009 Q4 473 426.4837 519.5163
## 2010 Q1 427 370.0294 483.9706
## 2010 Q2 383 326.0294 439.9706
21

24. Plotting seasonal naive
beer2 <- window(ausbeer,start=1992,end=c(2007,4))
autoplot(beer2) +
autolayer(snaive(beer2, h=11), PI=TRUE, series="Seasonal
ggtitle("Forecasts for quarterly beer production") +
xlab("Year") + ylab("Megalitres") +
guides(colour=guide_legend(title="Forecast"))
350
400
450
500
1995 2000 2005 2010
Megalitres
Forecast
Seasonal naïve
Forecasts for quarterly beer production
22

25. Drift method
• Forecasts equal to last value plus average change.
• Forecasts:
ŷT+h|T = yT +
h
T − 1
T
X
t=2
(yt − yt−1) (1)
= yT +
h
T − 1
(yT − y1). (2)
• Equivalent to extrapolating a line drawn between first and last
observations.
23

26. Drift method
rwf(beer2, h=10, level = 95)
## Point Forecast Lo 95 Hi 95
## 2008 Q1 473 344.98474 601.0153
## 2008 Q2 473 291.95908 654.0409
## 2008 Q3 473 251.27106 694.7289
## 2008 Q4 473 216.96948 729.0305
## 2009 Q1 473 186.74917 759.2508
## 2009 Q2 473 159.42793 786.5721
## 2009 Q3 473 134.30345 811.6965
## 2009 Q4 473 110.91816 835.0818
## 2010 Q1 473 88.95421 857.0458
## 2010 Q2 473 68.18020 877.8198
24

27. Plotting Drift
dj2 <- window(dj,end=250)
autoplot(dj2) +
autolayer(rwf(dj2, drift=TRUE, h=42), PI=TRUE, series="Dr
ggtitle("Dow Jones Index (daily ending 15 Jul 94)") +
xlab("Day") + ylab("") +
guides(colour=guide_legend(title="Forecast"))
3600
3800
4000
0 50 100 150 200 250 300
Day
Forecast
Drift
Dow Jones Index (daily ending 15 Jul 94)
25

28. Simple forecasting methods
All together
beer2 <- window(ausbeer,start=1992,end=c(2007,4))
autoplot(beer2) +
autolayer(meanf(beer2, h=11), PI=FALSE, series="Mean") +
autolayer(naive(beer2, h=11), PI=FALSE, series="Naïve") +
autolayer(snaive(beer2, h=11), PI=FALSE, series="Seasonal naïv
ggtitle("Forecasts for quarterly beer production") +
xlab("Year") + ylab("Megalitres") +
guides(colour=guide_legend(title="Forecast"))
400
450
500
1995 2000 2005 2010
Megalitres
Forecast
Mean
Naïve
Seasonal naïve
Forecasts for quarterly beer production
26

29. Simple forecasting methods
All together
dj2 <- window(dj,end=250)
autoplot(dj2) +
autolayer(meanf(dj2, h=42), PI=FALSE, series="Mean") +
autolayer(rwf(dj2, h=42), PI=FALSE, series="Naïve") +
autolayer(rwf(dj2, drift=TRUE, h=42), PI=FALSE, series="Drift"
ggtitle("Dow Jones Index (daily ending 15 Jul 94)") +
xlab("Day") + ylab("") +
guides(colour=guide_legend(title="Forecast"))
3700
3800
3900
4000
Forecast
Drift
Mean
Naïve
Dow Jones Index (daily ending 15 Jul 94)
27

30. Simple forecasting methods
Summary of R functions
• Mean: meanf(y, h=20)
• Naïve: naive(y, h=20)
• Seasonal naïve: snaive(y, h=20)
• Drift: rwf(y, drift=TRUE, h=20)
28

31. Performance evaluation

32. The problem of overfitting
A model which fits the data well does not necessarily forecast well.
A perfect fit can always be obtained by using a model with enough
parameters.
Over-fitting a model to data is as bad as failing to identify the
systematic pattern in the data
29

33. The problem of overfitting: an example
0.0 0.2 0.4 0.6 0.8 1.0
−50
0
50
100
150
200
x
y
30

34. three models
#model fitting
linearmodel = lm(y~x)
#prediction on test data set
predict_linear = predict(linearmodel,
list(x = testx))
z = xˆ2
# fitting
quadraticmodel<- lm(y~ x + z)
# prediction on test data set
predict_quadratic = predict(quadraticmodel,
list(x = testx, z = testxˆ2))
#fitting
smoothspline = smooth.spline(x,y,df = 20) 31

35. Plots
0.0 0.2 0.4 0.6 0.8 1.0
−50
0
50
100
150
200
Example of Overfitting, Normal Fitting and Underfitting.
X
Y
32

36. MSE
library(MLmetrics)
MSE(predict_linear,testy)
## [1] 14449.75
MSE(predict_quadratic,testy)
## [1] 2054.563
MSE(predict_spline,testy)
## [1] 587.3641
33

37. Data partitioning
Time
Ridership
1400
1600
1800
2000
2200
2400
2600
1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006
Training Validation Future
34

38. When to use which partition?
Fit the model only to training period
Assess performance on validation period
Deploy model by joining training+validation; forecast the future
35

39. How to choose a validation period?
Depends on:
• Forecast horizon
• Seasonality
• Length of series
• Underlying conditions affecting series
36

40. Partitioning time series in R
Time
Ridership
1400
1600
1800
2000
2200
2400
2600
1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006
Training Validation Future
37

41. Which model to choose?
yt+h = trend + trend
tslm(train.ts ~ trend )
Time
Ridership
1400
1600
1800
2000
2200
2400
2600
1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006
Training Validation Future
38

42. Which model to choose?
yt+h = trend + trend2
tslm(train.ts ~ trend + I(trendˆ2))
Time
Ridership
1400
1600
1800
2000
2200
2400
2600
1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006
Training Validation Future
39

43. Which model to choose?
yt+h = trend + trend2 + trend3
In R:
tslm(train.ts ~ trend + I(trendˆ2) + I(trendˆ3))
Ridership
1400
1600
1800
2000
2200
2400
2600
Training Validation Future
40

44. Which model to choose?
yt+h = trend + trend2 + season
In R:
tslm(train.ts ~ trend + I(trendˆ2) + season)
Ridership
1400
1600
1800
2000
2200
2400
2600
Training Validation Future
41

45. Choosing the model: compare errors
head(ridership.lm.pred$mean )
## Apr May Jun Jul
## 2001 2004.271 2045.419 2008.675 2128.560
## Aug Sep
## 2001 2187.911 1875.032
head(valid.ts)
## Apr May Jun Jul
## 2001 2023.792 2047.008 2072.913 2126.717
## Aug Sep
## 2001 2202.638 1707.693
42

46. MAE: Mean Absolute Error
Gives the magnitude of the absolute error
1
v
v
X
t=1
| ˆ
yt − yt|
ridership.lm <- tslm(train.ts ~ trend)
ridership.lm.pred <- forecast(ridership.lm, h = stepsAhead, leve
sum(abs(ridership.lm.pred$mean - valid.ts))
## [1] 7539.736
ridership.lm <- tslm(train.ts ~ trend + I(trendˆ2))
ridership.lm.pred <- forecast(ridership.lm, h = stepsAhead, leve
sum(abs(ridership.lm.pred$mean - valid.ts))
## [1] 4814.579
ridership.lm <- tslm(train.ts ~ trend + I(trendˆ2)+ season)
ridership.lm.pred <- forecast(ridership.lm, h = stepsAhead, leve
43

47. MAPE: Mean Absolute Percentage Error
Percentage deviation. Useful to compare across series
1
v
v
X
t=1
|
ˆ
yt − yt
yt
| × 100
ridership.lm <- tslm(train.ts ~ trend + I(trendˆ2))
ridership.lm.pred <- forecast(ridership.lm, h = stepsAhead,
sum(abs((ridership.lm.pred$mean - valid.ts) /valid.ts ))
## [1] 2.547263
ridership.lm <- tslm(train.ts ~ trend + I(trendˆ2)+ season
ridership.lm.pred <- forecast(ridership.lm, h = stepsAhead,
sum(abs((ridership.lm.pred$mean - valid.ts) /valid.ts ))
## [1] 2.411532
44

48. Mean Squared Error and Root Mean Squared Error
MSE =
1
v
v
X
t=1
( ˆ
yt − yt)2
RMSE =
1
v
v
X
t=1
( ˆ
yt − yt)2
45

49. Mean Squared Error and Root Mean Squared Error
ridership.lm <- tslm(train.ts ~ trend + I(trendˆ2))
ridership.lm.pred <- forecast(ridership.lm, h = stepsAhead,
sum(sqrt((ridership.lm.pred$mean - valid.ts)ˆ2 ))
## [1] 4814.579
ridership.lm <- tslm(train.ts ~ trend + I(trendˆ2)+ season
ridership.lm.pred <- forecast(ridership.lm, h = stepsAhead,
sum(sqrt((ridership.lm.pred$mean - valid.ts)ˆ2 ))
## [1] 4742.101
46

50. Time series cross-validation
Traditional evaluation
time
Training data Test data
47

51. Time series cross-validation
Traditional evaluation
time
Training data Test data
Time series cross-validation
time
48

52. Time series cross-validation
Traditional evaluation
time
Training data Test data
Time series cross-validation
time
• Forecast accuracy averaged over test sets.
• Also known as “evaluation on a rolling forecasting origin”
48

53. tsCV function
set.seed(0)
s1 <- (rnorm(100, mean=0.1))
s2 <- (rnorm(100, mean=-0.1))
s3 <- cumsum(c(s1, s2))
ecv <- tsCV(s3, rwf, drift=TRUE, h=1, initial =100)
plot(s3, type='l', ylim=c(-20,20))
lines(c(s3 + ecv), type='l', col=2)
pred <- (rwf(s3[1:100], drift=TRUE, h=100 ))$mean
lines(pred, type='l', col=3)
A good way to choose the best forecasting model is to find the model with
the smallest RMSE computed using time series cross-validation.
49

54. tsCV function
0 50 100 150 200
−20
−10
0
10
20
Index
s3
50

55. Prediction intervals

56. Prediction intervals
• A forecast ŷT+h|T is (usually) the mean of the conditional
distribution yT+h | y1, . . . , yT .
• A prediction interval gives a region within which we expect
yT+h to lie with a specified probability.
• Assuming forecast errors are normally distributed, then a 95%
PI is
ŷT+h|T ± 1.96σ̂h
where σ̂h is the st dev of the h-step distribution.
• When h = 1, σ̂h can be estimated from the residuals.
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57. Prediction intervals
Naive forecast with prediction interval:
res_sd <- sqrt(mean(resˆ2, na.rm=TRUE))
c(tail(goog200,1)) + 1.96 * res_sd * c(-1,1)
## [1] 519.3103 543.6462
naive(goog200, level=95, bootstrap=T)
## Point Forecast Lo 95 Hi 95
## 201 531.4783 522.8631 541.2396
## 202 531.4783 519.5798 546.3474
## 203 531.4783 516.6695 550.4248
## 204 531.4783 514.0899 554.9091
## 205 531.4783 511.7058 573.2582
## 206 531.4783 509.4558 580.5680
## 207 531.4783 507.7254 581.1676
## 208 531.4783 505.8039 584.2847
## 209 531.4783 504.1997 586.8647
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58. Easiest way to generate prediction intervals: bootstrap
We can simulate the next observation of a time series using
yT+1 = ŷT+1|T + eT+1
we can replace eT+1 by sampling from the collection of errors we
have seen in the past (i.e., the residuals). Adding the new simulated
observation to our data set, we can repeat the process to obtain
yT+2 = ŷT+2|T + eT+2
Doing this repeatedly, we obtain many possible futures. Then we
can compute prediction intervals by calculating percentiles for each
forecast horizon
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59. Prediction intervals
• Computed automatically using: naive(), snaive(), rwf(),
meanf(), etc.
• Use level argument to control coverage.
• Check residual assumptions before believing them.
• Usually too narrow due to unaccounted uncertainty.
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