forecasting for manufacturing system
TRANSCRIPT
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Forecasting
Lecturer: Prof. Duane S. Boning
Rev 8
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Regression Review & Extensions
Single Model Coefficient: Linear Dependence
Slope and Intercept (or Offset):
Polynomial and Higher Order Models:
Multiple Parameters
Key point: linear regression can be used as long as the model islinear in the coefficients (doesnt matter the dependence in theindependent variable)
Time dependencies
Explicit
Implicit
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3
Agenda
1. Regression Polynomial regression
Example (using Excel)
2. Time Series Data & Time Series Regression
Autocorrelation
ACF
Example: white noise sequences
Example: autoregressive sequences
Example: moving average
ARIMA modeling and regression
3. Forecasting Examples
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Time Series Time as an Implicit Parameter
Data is oftencollected with a
t ime-order
An underlyingdynamic process
(e.g. due to physics
of a manufacturing
process) may create
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-5
0
5
time
x
autocorrelated
autocorrelat ion inthe data
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0
2
4
time
x
uncorrelated
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Intuition: Where Does Autocorrelation Come From?
Consider a chamber with volume V, and with gas flow in and
gas flow out at rate f. We are interested in the concentrationxatthe output, in relation to a known input concentration w.
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Key Tool: Autocorrelation Function (ACF)
Time series data: time indexi
CCF: cross-correlation function
ACF: auto-correlation function
) ACF shows the similarity of a signal
to a lagged version of same signal
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-2
0
2
4
time
x
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-0.5
0
0.5
1
lags
r(k)
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Stationary vs. Non-Stationary
Stationary series:
Process has a fixed mean
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0
5
10
time
x
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0
10
20
30
time
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Data drawn from IID gaussian
ACF: We also plot the 3 limits
values within these not significant
Note that r(0) = 1 always (a
signal is always equal to itself
with zero lag perfectlyautocorrelated at k = 0)
Sample mean
Sample variance
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-2
0
2
4
time
x
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-0.5
0
0.5
1
lags
r(k)
White Noise An Uncorrelated Series
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Autoregressive Disturbances
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0
5
10
time
x
0 5 10 15 20 25 30 35 40
-1
-0.5
0
0.5
1
lags
r(k)
Generated by:
Mean
Variance
So AR (autoregressive) behavior
increases variance of signal.
Slow drop in ACF with large
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Another Autoregressive Series
Generated by:
Slow drop in ACF with large
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0
5
10
time
x
0 5 10 15 20 25 30 35 40-1
-0.5
0
0.5
1
lags
r(k)
Slow drop in ACF with large
But now ACF alternates in sign
High negative autocorrelation:
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Random Walk Disturbances
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0
10
20
30
time
x
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0
0.5
1
lags
r(k)
Veryslow drop in ACF for = 1
Generated by:
Mean
Variance
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Moving Average Sequence
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0
2
4
time
x
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-0.5
0
0.5
1
lags
r(k)
Generated by:
Mean
Variance
So MA (moving average) behavior
also increases variance of signal.
r(1)
Jump in ACF at specific lag
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ARMA Sequence
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0
5
10
time
x
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0
0.5
1
lags
r(k)
Generated by:
Both AR & MA behavior
Slow drop in ACF with large
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0
200
400
time
x
0 5 10 15 20 25 30 35 40-1
-0.5
0
0.5
1
lags
r(k)
ARIMA Sequence
Start with ARMA sequence:
Add Integrated (I) behavior
Slow drop in ACF with large
random walk (integrative) action
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Periodic Signal with Autoregressive Noise
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0
10
20
time
x
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-0.5
0
0.5
1
lags
r(k)
Original Signal
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0
5
time
x
0 5 10 15 20 25 30 35 40-1
-0.5
0
0.5
1
lags
r(k)
After Differencing
See underlying signal with period = 5
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Cross-Correlation: A Leading Indicator
Now we have two series:
An input or explanatoryvariable x
An output variable y
CCF indicates both AR and lag:
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time
x
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0
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time
y
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-0.5
0
0.5
1
lags
rx
y(k
)
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Regression & Time Series Modeling
The ACF or CCF are helpful tools in selecting anappropriate model structure
Autoregressive terms?
xi = xi-1
Lag terms?
yi = xi-k
One can structure data and perform regressions
Estimate model coefficientvalues, significance, and
confidence intervals
Determine confidence intervals on output Check residuals
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Statistical Modeling Summary
1. Statistical Fundamentals
Sampling distributions Point and interval estimation
Hypothesis testing
2. Regression
ANOVA
Nominal data: modeling of treatment effects (mean differences)
Continuous data: least square regression
3. Time Series Data & Forecasting Autoregressive, moving average, and integrative behavior
Auto- and Cross-correlation functions
Regression and time-series modeling
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MIT OpenCourseWarehttp://ocw.mit.edu
2.854 / 2.853 Introduction to Manufacturing Systems
Fall 2010
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