Download Excel and Monte Carlo Applications

Monte Carlo Applications Using Excel
In deterministic modeling, we establish an outcome which corresponds to a particular set
of model inputs. River oxygen models typically input flow, temperature, and values for
rate constants and determine the dissolved oxygen concentration associated with the
value for a control variable, e.g. the BOD load. A deterministic model yields a single
value for that outcome (DO) based on the perturbation (BOD) and the set of model inputs
provided.
In probabilistic analysis, we calculate the frequency of occurrence for an outcome
associated with a particular set of model inputs. Here, a value for a control variable (e.g.
BOD) is again used, but model inputs (e.g. flow, temperature, and values for rate
constants) are now input from randomly sampled distributions representing their natural
variability (flow, temperature) or uncertainty (rate constants). Probabilistic models
generate a distribution for outcome frequencies for the value for the control variable and
the distributions of model inputs. This approach is termed Monte Carlo analysis.
Variability in the model inputs used in surface water quality analysis may be described by
one of several idealized distributions, including uniform, normal or log normal. The
uniform distribution assumes that there is an equal likelihood of occurrence over the
range specified by two bounds. The normal distribution is a symmetrical bell-shaped
curve with the most likely value in the center and diminishing likelihoods at the
extremes. The log normal distribution is similar to the normal distribution, but values are
skewed, i.e. that do not fall symmetrically about the most likely value.
The key to Monte Carlo analysis lies in functions which describe the distribution of
model inputs. Excel offers applications which utilize cumulative distribution functions to
yield values for a model input given specification of the distribution type and a mean and
standard deviation for the distribution.
Normal distribution
The cumulative distribution function identifies the probability that a data value would
exceed all of the other values in the data set. For example, in the accompanying figure, a
data value of 39.6 exceeds 40% (p = 0.4) of the data set.
1.0
0.9
0.8
Probability
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
35
40
Data Value
45
Application of Excel to Monte Carlo analysis comes at it from the other direction.
x = Application.NormInv(rnd(), mean, sd)
A random number generator is utilized to establish a probability and that probability is
applied to the cumulative distribution function to yield a data value.
Applied through many iterations, this process will yield the normal distribution of data
values described by the specified mean and standard deviation.
Frequency (0-1)
0.3
0.2
0.1
0.0
35 36 37 38 39 40 41 42 43 44 45
Data Value