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.Control¸øK NOISE macro output>PINK NOISE#:#PINK NOISE macro output>PINK NOISE>updating pink noise#fÉK NOISE macro output>PINK NOISE>gap#:#PINK NOISE macro output>PINK NOISE>scaled white noise#2#PINK NOISE macro output>PINK NOISE>white noise#¸ø¡7L¿ ‡¡7L¿ô,,,¼,¡7L¿ô¡7L¿%%d\ÜÜ,Ì\<Ìl<Ìl\Ì\l\Ì\„Á¨Ð$¨ÿÿÿÿÿÿÿÿÿTimes New Roman !ÀÀÀMacrod8¿œÅ¨ÿÿÿÿÿÿÿÿÿTimes New RomandÀÀÀStructured !7NU]cipw‚ˆ•›¢¨®¹¿ÊÐÛáèîùÿH¡7L¿ùùô&#§Éœ380,,graph63?(a¼ÿ380,,StatsVRtËQÿÿPink Noise input - created by Ed AndersonZWsDÿÿUpdated by Tom Fiddaman, Ventana Systems, 2010
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&Internally defined simulation time.
Pink Noise - Contributed by Ed Anderson, MIT/U. Texas - Austin
Description: The pink noise molecule described generates a simple random
series with autocorrelation. This is useful in representing time series,
like rainfall from day to day, in which today's value has some correlation
with what happened yesterday. This particular formulation will also have
properties such as standard deviation and mean that are insensitive both to
the time step and the correlation (smoothing) time. Finally, the output as
a whole and the difference in values between any two days is guaranteed to
be Gaussian (normal) in distribution.
Behavior: Pink noise series will have both a historical and a random
component during each period. The relative "trend-to-noise" ratio is
controlled by the length of the correlation time. As the correlation time
approaches zero, the pink noise output will become more independent of its
historical value and more "noisy." On the other hand, as the correlation
time approaches infinity, the pink noise output will approximate a
continuous time random walk or Brownian motion. Displayed above are two
time series with correlation times of 1 and 8 months. While both series
have approximately the same standard deviation, the 1-month correlation time
series is less smooth from period to period than the 8-month series, which
is characterized by "sustained" swings in a given direction. Note that this
behavior will be independent of the time-step.
The "pink" in pink noise refers to the power spectrum of the output. A time
series in which each period's observation is independent of the past is
characterized by a flat or "white" power spectrum. Smoothing a time series
attenuates the higher or "bluer" frequencies of the power spectrum, leaving
the lower or "redder" frequencies relatively stronger in the output.
Caveats: This assumes the use of Euler integration with a time step of no
more than 1/4 of the correlation time. Very long correlation times should be
avoided also as the multiplication in the scaled white noise will become
progressively less accurate.
Technical Notes: This particular form of pink noise is superior to that of
Britting presented in Richardson and Pugh (1981) because the Gaussian
(Normal) distribution of the output does not depend on the Central Limit
Theorem. (Dynamo did not have a Gaussian random number generator and hence
R&P had to invoke the CLM to get a normal distribution.) Rather, this
molecule's normal output is a result of the observations being a sum of
Gaussian draws. Hence, the series over short intervals should better
approximate normality than the macro in R&P.
MEAN: This is the desired mean for the pink noise.
STD DEVIATION: This is the desired standard deviation for the pink noise.
CORRELATION TIME: This is the smooth time for the noise, or for the more technically
minded this is the inverse of the filter's cut-off frequency in radians.
Updated by Tom Fiddaman, 2010, to include a random initial value,
correct units, and use TIME STEP$ keywordContributed by Ed Anderson, MIT/U. Texas - Austin
Description: The pink noise molecule described generates a simple random
series with autocorrelation. This is useful in representing time series,
like rainfall from day to day, in which today's value has some correlation
with what happened yesterday. This particular formulation will also have
properties such as standard deviation and mean that are insensitive both to
the time step and the correlation (smoothing) time. Finally, the output as
a whole and the difference in values between any two days is guaranteed to
be Gaussian (normal) in distribution.
Behavior: Pink noise series will have both a historical and a random
component during each period. The relative "trend-to-noise" ratio is
controlled by the length of the correlation time. As the correlation time
approaches zero, the pink noise output will become more independent of its
historical value and more "noisy." On the other hand, as the correlation
time approaches infinity, the pink noise output will approximate a
continuous time random walk or Brownian motion. Displayed above are two
time series with correlation times of 1 and 8 months. While both series
have approximately the same standard deviation, the 1-month correlation time
series is less smooth from period to period than the 8-month series, which
is characterized by "sustained" swings in a given direction. Note that this
behavior will be independent of the time-step.
The "pink" in pink noise refers to the power spectrum of the output. A time
series in which each period's observation is independent of the past is
characterized by a flat or "white" power spectrum. Smoothing a time series
attenuates the higher or "bluer" frequencies of the power spectrum, leaving
the lower or "redder" frequencies relatively stronger in the output.
Caveats: This assumes the use of Euler integration with a time step of no
more than 1/4 of the correlation time. Very long correlation times should be
avoided also as the multiplication in the scaled white noise will become
progressively less accurate.
Technical Notes: This particular form of pink noise is superior to that of
Britting presented in Richardson and Pugh (1981) because the Gaussian
(Normal) distribution of the output does not depend on the Central Limit
Theorem. (Dynamo did not have a Gaussian random number generator and hence
R&P had to invoke the CLM to get a normal distribution.) Rather, this
molecule's normal output is a result of the observations being a sum of
Gaussian draws. Hence, the series over short intervals should better
approximate normality than the macro in R&P.
MEAN: This is the desired mean for the pink noise.
STD DEVIATION: This is the desired standard deviation for the pink noise.
CORRELATION TIME: This is the smooth time for the noise, or for the more technically minded this is the inverse of the filter's cut-off frequency in radians.
Updated by Tom Fiddaman, 2010, to include a random initial value,
correct units, and use TIME STEP$ keyword²This adjusts the standard deviation of the white noise to compensate for the time step and the
correlation time to produce the appropriate pink noise std deviation.This is an independent, identically distributed random quantity drawn every time step. The distribution is gaussian with mean = 0 and variance = 1.
Note that RANDOM NORMAL is truncated +/- 6 standard deviations here.
For Vensim 1.62 syntax, remove the arguments to RANDOM NORMAL.Contributed by Ed Anderson, MIT/U. Texas - Austin
Description: The pink noise molecule described generates a simple random
series with autocorrelation. This is useful in representing time series,
like rainfall from day to day, in which today's value has some correlation
with what happened yesterday. This particular formulation will also have
properties such as standard deviation and mean that are insensitive both to
the time step and the correlation (smoothing) time. Finally, the output as
a whole and the difference in values between any two days is guaranteed to
be Gaussian (normal) in distribution.
Behavior: Pink noise series will have both a historical and a random
component during each period. The relative "trend-to-noise" ratio is
controlled by the length of the correlation time. As the correlation time
approaches zero, the pink noise output will become more independent of its
historical value and more "noisy." On the other hand, as the correlation
time approaches infinity, the pink noise output will approximate a
continuous time random walk or Brownian motion. Displayed above are two
time series with correlation times of 1 and 8 months. While both series
have approximately the same standard deviation, the 1-month correlation time
series is less smooth from period to period than the 8-month series, which
is characterized by "sustained" swings in a given direction. Note that this
behavior will be independent of the time-step.
The "pink" in pink noise refers to the power spectrum of the output. A time
series in which each period's observation is independent of the past is
characterized by a flat or "white" power spectrum. Smoothing a time series
attenuates the higher or "bluer" frequencies of the power spectrum, leaving
the lower or "redder" frequencies relatively stronger in the output.
Caveats: This assumes the use of Euler integration with a time step of no
more than 1/4 of the correlation time. Very long correlation times should be
avoided also as the multiplication in the scaled white noise will become
progressively less accurate.
Technical Notes: This particular form of pink noise is superior to that of
Britting presented in Richardson and Pugh (1981) because the Gaussian
(Normal) distribution of the output does not depend on the Central Limit
Theorem. (Dynamo did not have a Gaussian random number generator and hence
R&P had to invoke the CLM to get a normal distribution.) Rather, this
molecule's normal output is a result of the observations being a sum of
Gaussian draws. Hence, the series over short intervals should better
approximate normality than the macro in R&P.
MEAN: This is the desired mean for the pink noise.
STD DEVIATION: This is the desired standard deviation for the pink noise.
CORRELATION TIME: This is the smooth time for the noise, or for the more technically minded this is the inverse of the filter's cut-off frequency in radians.
Updated by Tom Fiddaman, 2010, to include a random initial value,
correct units, and use TIME STEP$ keyword²This adjusts the standard deviation of the white noise to compensate for the time step and the
correlation time to produce the appropriate pink noise std deviation.This is an independent, identically distributed random quantity drawn every time step. The distribution is gaussian with mean = 0 and variance = 1.
Note that RANDOM NORMAL is truncated +/- 6 standard deviations here.
For Vensim 1.62 syntax, remove the arguments to RANDOM NORMAL."
Simulation Control Parameters&The final time for the simulation.&The initial time for the simulation..The frequency with which output is stored.&The time step for the simulation.Èæ€ö€ö AzD€ö A€ö€ö€?€?€?€ö€ö¡7L¿
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