First steps in time series

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First steps in time series
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Time series look different
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But they are not
Xt systematic pattern noise
Obscures the pattern
Often made with two basic components
trend seasonality
periodic
Underlying linear/non-linear, time changing,
uncanny model
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Monthly car production in Spain
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• Trend analysis
• Smooth data using an underlying model
• Moving average
• Exponential moving average
• Fit any function
• linear, splines, log, exp, your guess

Good for one-period ahead forecasting (weather)
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linear
Detect and model trend Substract trend
ma(12)
Stationarity (mean and variance)
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• Seasonality
• Look for autocorrelations
• absolute values
• increments
• other combination of variables!
  (intuitionexpertise)

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lag 1 npat 58 corr -0.0397427758 lag 2
npat 57 corr -0.274627552 lag 3 npat 56
corr -0.265544135 lag 4 npat 55 corr
0.20316958 lag 5 npat 54 corr -0.0689740424
lag 6 npat 53 corr -0.0261346967 lag 7
npat 52 corr -0.106164043 lag 8 npat 51
corr 0.242284075 lag 9 npat 50 corr
-0.245981648 lag 10 npat 49 corr -0.315532046
lag 11 npat 48 corr -0.0585207867 lag 12 npat
47 corr 0.94479132 lag 13 npat 46 corr
-0.0744984938 lag 14 npat 45 corr
-0.255095906 lag 15 npat 44 corr -0.272366416
lag 16 npat 43 corr 0.197473796 lag 17 npat
42 corr 2.4789972E-05 lag 18 npat 41 corr
-0.0635903421 lag 19 npat 40 corr
-0.138785333 lag 20 npat 39 corr 0.311166354
lag 21 npat 38 corr -0.224766545 lag 22 npat
37 corr -0.337246239 lag 23 npat 36 corr
-0.056239266 lag 24 npat 35 corr 0.916841357
xt vs xt -lag
Structure seasonality
anticorrelations
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vs
lag 1 inc 1 npat 57 corr -0.390023157 lag
2 inc 1 npat 56 corr -0.101618385 lag 3
inc 1 npat 55 corr -0.184521702 lag 4 inc 1
npat 54 corr 0.275796168 lag 5 inc 1 npat
53 corr -0.11691902 lag 6 inc 1 npat 52
corr 0.0605651902 lag 7 inc 1 npat 51 corr
-0.146978369 lag 8 inc 1 npat 50 corr
0.309853501 lag 9 inc 1 npat 49 corr
-0.18309118 lag 10 inc 1 npat 48 corr
-0.127671412 lag 11 inc 1 npat 47 corr
-0.35079422 lag 12 inc 1 npat 46 corr
0.944415972 lag 13 inc 1 npat 45 corr
-0.390599355
12-month correlation!
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vs
lag 1 inc 2 npat 56 corr 0.0635841158 lag
2 inc 2 npat 55 corr -0.661845284 lag 3
inc 2 npat 54 corr -0.14900299 lag 4 inc 2
npat 53 corr 0.236847899 lag 5 inc 2 npat
52 corr 0.0775265432 lag 6 inc 2 npat 51
corr -0.1446445 lag 7 inc 2 npat 50 corr
0.0427922726 lag 8 inc 2 npat 49 corr
0.268605346 lag 9 inc 2 npat 48 corr
-0.13164323 lag 10 inc 2 npat 47 corr
-0.660207622 lag 11 inc 2 npat 46 corr
0.0676404933 lag 12 inc 2 npat 45 corr
0.956921988 lag 13 inc 2 npat 44 corr
0.0444132255
2-month anticorrelation!
Danger correlations vs error
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Fourier analysis uncovers periodicity
const
2?/48.131
Single tick 2?
More sophisticated analysis is possible but
brings little further information
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Reasonable bets
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Best linear fit neural fit to 12 based on the
last 12 months
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Magnification of the last 12 months (8 train
patterns 4 predictions)
prediction
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Linear fit heavily depends on one variable Neural
net finds non-linear relations that enhance
correlations
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Simple is good

• If in doubt, start with a simple dependency
• Ex xt xt-lag
• lag 1 day
• Weather forecast
• Donuts
• lag 7 days
• Donuts
• Electricity load curve
• lag 1 year
• Electricity load curve
• Sales

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ARIMA
Define Backward shift Backward
difference Polynomials of degree p and q
Auto-Regressive Moving Average time series models
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AR(p)
I(d)
MA(q)
Zeros in polynomials must lie outside unit circle
Example ARIMA(1,0,0)
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NN enhancement

• Rather than using recursive NN,
• carry out linear analysis
• preprocess data to ARIMA like
• perform a linear forecast
• feed a NN with all the linear analysis
• preprocessed data
• linear prediction

The NN will learn (if any) the underlying law
controlling the departures of real data from
linear analysis
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Leave-one-out NN
When the number of data are very small,
leave out
Vars Goal Pattern 1 Vars Goal Pattern
2 Vars Goal Pattern 3 Vars Goal Pattern
4 Vars Goal Pattern 5 Vars Goal Pattern 6
train
train
leave out
step 1
step n
Collect statistics
Predict 1
Predict n
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About noise
We have discarded noise Be careful asset1
trend1 noise1
finances asset2 trend2
noise2
correlations
Cov(i,j) 30003000 Huge CPU time
Brownian motion N(0,?t)
VaR
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Summary

• Time series are often structured
• Analyze trend seasonality noise
• Build a linear model with preprocessed data
• Build NN on top of previous analysis