Time Series Analysis

Topics in Machine LearningFall 2011

School of Electrical Engineeringand Computer Science

Time Series Discussions

• Overview• Basic definitions• Time domain• Forecasting• Frequency domain• State space

Why Time Series Analysis?

• Sometimes the concept we want to learn is the relationship between points in time

Time series: a sequence of

measurements over time

A sequence of random variablesx1, x2, x3, …

What is a time series?

Time Series ExamplesDefinition: A sequence of measurements over time

Finance Social science Epidemiology Medicine Meterology Speech Geophysics Seismology Robotics

Three Approaches

• Time domain approach– Analyze dependence of current value on past values

• Frequency domain approach– Analyze periodic sinusoidal variation

• State space models– Represent state as collection of variable values– Model transition between states

Sample Time Series Data

Johnson & Johnson quarterly earnings/share, 1960-1980

Sample Time Series Data

Yearly average global temperature deviations

Sample Time Series Data

Speech recording of “aaa…hhh”, 10k pps

Sample Time Series Data

NYSE daily weighted market returns

Not all time data will exhibit strong patterns…

LA annual rainfall

…and others will be apparent

Canadian Hare counts

Time Series Discussions

• Overview• Basic definitions• Time domain• Forecasting• Frequency domain• State space

Definitions

• Mean

• Variance

mean

variance

Definitions

• Covariance

• Correlation

N

i

yixi

Nyx

YXCov1

))((),(

YX

YXCovrYXCor

),(),(

Y

X

Y

X

Y

Xr = -1 r = -.6 r = 0

Y

X

Y

Xr = +.3r = +1

Y

Xr = 0

Correlation

Redefined for Time

•

•

•

,...2,1,0)()( tforXEt tXErgodic?

Mean function

),()0()()( tht

X

XX XXCorhh

Autocorrelation

),()( thtX XXCovh lag

Autocovariance

Autocorrelation Examples

Positivelag

Negative

lag

Stationarity – When there is no relationship

• {Xt} is stationary if– X(t) is independent of t– X(t+h,t) is independent of t for each h

• In other words, properties of each section are the same

• Special case: white noise

Time Series Discussions

• Overview• Basic definitions• Time domain• Forecasting• Frequency domain• State space

Linear Regression

• Fit a line to the data• Ordinary least squares

– Minimize sum of squareddistances between pointsand line

• Try this out at http://hspm.sph.sc.edu/courses/J716/demos/LeastSquares/LeastSquaresDemo.html

y = x +

R2: Evaluating Goodness of Fit

• Least squares minimizesthe combined residual

• Explained sum of squaresis difference between lineand mean

• Total sum of squares is the total of these two

y = x +

i iYYRSS 2)(

i

YYESS 2)(

ii i YYYYRSSESSTSS 22 )()(

R2: Evaluating Goodness of Fit

• R2, the coefficientof determination

• 0 R2 1• Regression minimizes RSS and so

maximizes R2

y = x + TSSRSS

TSSESSR 12

R2: Evaluating Goodness of Fit

TSSRSS

TSSESSR 12

R2: Evaluating Goodness of Fit

TSSRSS

TSSESSR 12

R2: Evaluating Goodness of Fit

TSSRSS

TSSESSR 12

Linear Regression

• Can report:– Direction of trend (>0, <0, 0)– Steepness of trend (slope)– Goodness of fit to trend (R2)

Examples

What if a linear trend does not fit my data well?

• Could be no relationship• Could be too much local variation

– Want to look at longer-term trend– Smooth the data

• Could have periodic or seasonality effects– Add seasonal components

• Could be a nonlinear relationship453423121 QbQbQbQbtbaX t

Moving Average

• Compute an average of the last m consecutive data points

• 4-point moving average is

• Smooths white noise4

)( 321)4(

tttt

MAxxxxx

k

kjjtjt xam

• Can apply higher-order MA

• Exponential smoothing

• Kernel smoothing

Power Load Data

5 week

53 week

Piecewise Aggregate Approximation

• Segment the data into linear pieces

Interesting paper

Nonlinear Trend Examples

Nonlinear Regression

Fit Known Distributions

ARIMA: Putting the pieces together

• Autoregressive model of order p: AR(p)• Moving average model of order q: MA(q)• ARMA(p,q)

ARIMA: Putting the pieces together

• Autoregressive model of order p: AR(p)

• Moving average model of order q: MA(q)• ARMA(p,q)

tptpttt wxxxx ..2211

AR(1),

0 20 40 60 80 100

-20

24

9.0

AR(1),

0 20 40 60 80 100

-4-2

02

49.0

ARIMA: Putting the pieces together

• Autoregressive model of order p: AR(p)

• Moving average model of order q: MA(q)

• ARMA(p,q)

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tqtqttt wwwwx ..2211

ARIMA: Putting the pieces together

• Autoregressive model of order p: AR(p)

• Moving average model of order q: MA(q)

• ARMA(p,q)– A time series is ARMA(p,q) if it is stationary and

tptpttt wxxxx ..2211

tqtqttt wwwwx ..2211

qtqtt

tptpttt

www

wxxxx

..

..

2211

2211

ARIMA (AutoRegressive Integrated Moving Average)

• ARMA only applies to stationary process• Apply differencing to obtain stationarity

– Replace its value by incremental change from last value

• A process xt is ARIMA(p,d,q) if– AR(p)– MA(q)– Differenced d times

• Also known as Box Jenkins

Differenced x1 x2 x3 x41 time x2-x1’ x3’-x2’ x4’-x3’

2 times x3’-2x2’+x1’ x4’-2x3’+x2’

Time Series Discussions

• Overview• Basic definitions• Time domain• Forecasting• Frequency domain• State space

Express Data as Fourier Frequencies

• Time domain– Express present as function of the past

• Frequency domain– Express present as function of oscillations, or

sinusoids

Time Series Definitions

• Frequency, , measured at cycles per time point• J&J data

– 1 cycle each year– 4 data points (time points) each cycle– 0.25 cycles per data point

• Period of a time series, T = 1/– J&J, T = 1/.25 = 4– 4 data points per cycle– Note: Need at least 2

Fourier Series

• Time series is a mixture of oscillations– Can describe each by amplitude, frequency and

phase– Can also describe as a sum of amplitudes at all

time points– (or magnitudes at all frequencies)

– If we allow for mixtures of periodic series then

)2sin()2cos( ttxt

Take a look

q

iiiiit ttx

1

)]2sin()2cos([

Example

)100/62sin(3)100/62cos(21

ttxt

)100/402sin(7)100/402cos(63

ttxt

)100/102sin(5)100/102cos(42

ttxt

3214 tttt xxxx

How Compute Parameters?

• Regression• Discrete Fourier Transform

• DFTs represent amplitude and phase of series components

• Can use redundancies to speed it up (FFT)

2/

1

)]/2sin()/()/2cos()/([n

jiit ntjnjntjnjx

n

t

ntjitexnnjdd

1

/221

)/()(

Breaking down a DFT

• Amplitude

• Phase

22 ))/(())/((|)/(|)/( njdInjdRnjdnjA

)))/((/))/(((tan)/( 1 njdRnjdInj

Example

-1

0

1

2

GB

P

-1

0

1

2

GB

P

-1

0

1

2

GB

P

-1

0

1

2

GB

P

-1

0

1

2

GB

P

-1

0

1

2

GB

P

1 frequency

2 frequencies

3 frequencies

5 frequencies

10 frequencies

20 frequencies

Periodogram

• Measure of squared correlation between– Data and– Sinusoids oscillating at frequency of j/n

– Compute quickly using FFT

n

tt

n

tt ntjx

nntjx

nnjP

1

2

1

2 ))/2sin(2())/2cos(2()/(

Example

P(6/100) = 13, P(10/100) = 41, P(40/100) = 85

Wavelets

• Can break series up into segments– Called wavelets– Analyze a window of time separately– Variable-sized windows

Time Series Discussions

• Overview• Basic definitions• Time domain• Forecasting• Frequency domain• State space

State Space Models

• Current situation represented as a state– Estimate state variables from noisy observations

over time– Estimate transitions between states

• Kalman Filters– Similar to HMMs

• HMM models discrete variables• Kalman filters models continuous variables

Conceptual Overview

• Lost on a 1-dimensional line• Receive occasional sextant position readings

– Possibly incorrect• Position x(t), Velocity x’(t)

x

Conceptual Overview

0 10 20 30 40 50 60 70 80 90 1000

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

• Current location distribution is Gaussian• Transition model is linear Gaussian• The sensor model is linear Gaussian• Sextant Measurement at ti: Mean = i and Variance = 2

i

• Measured Velocity at ti: Mean = ’i and Variance = ’2i

Noisy information

Kalman Filter Algorithm

• Start with current location (Gaussian)• Predict next location

– Use current location– Use transition function (linear Gaussian)– Result is Gaussian

• Get next sensor measurement (Gaussian)• Correct prediction

– Weighted mean of previous prediction and measurement

0 10 20 30 40 50 60 70 80 90 1000

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Conceptual Overview

• We generate the prediction for time i+, prediction is Gaussian• GPS Measurement: Mean = i+ and Variance = 2

i +

• They do not match

prediction Measurement at i

0 10 20 30 40 50 60 70 80 90 1000

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Conceptual Overview

• Corrected mean is the new optimal estimate of position• New variance is smaller than either of the previous two variances

measurement at i+

corrected estimate

prediction

Updating Gaussian Distributions

• One-step predicted distribution is Gaussian

• After new (linear Gaussian) evidence, updated distribution is Gaussian

tx ttttttt dxexPxXPeXP )|()|()|( :11:11

)|()|()|( :11111:11 tttttt eXPXePeXP

PriorTransition

Previous step

New measurement

Why Is Kalman Great?

• The method, that is…• Representation of state-based series with

general continuous variables grows without bound

Why Is Time Series Important?

• Time is an important component of many processes

• Do not ignore time in learning problems• ML can benefit from, and in turn benefit,

these techniques– Dimensionality reduction of series– Rule discovery– Cluster series– Classify series– Forecast data points– Anomaly detection