Uncertainty Setup dialog
Use the Uncertainty Setup dialog to inspect and change the sample size, sampling method, statistics, probability bands, and samples per plot point for probability distributions. All settings are saved with your model.
To open the Uncertainty Setup dialog, select Uncertainty Options from the Result menu or Control+u. To set values for a specific variable, select the variable before opening the dialog.
The five options for viewing and changing information in the Uncertainty Setup dialog can be accessed using the Analysis option popup menu.
Uncertainty sample: To change the sample size or sampling method for the model, select the Uncertainty Sample option from the Analysis option popup menu.
The default dialog shows only a field for sample size. To view and change the sampling method, random number method, or random seed, press the More Options button.
Sample size: This number specifies how many runs or iterations Analytica performs to estimate probability distributions. Larger sample sizes take more time and memory to compute, and produce smoother distributions and more precise statistics. See Selecting the Sample Size for guidelines on selecting a sample size. The sample size must be between 2 and 32,000. You can access this number in expressions in your models as the system variable SampleSize.
Sampling method: The sampling method is used to determine how to generate a random sample of the specified sample size, m, for each uncertain quantity, X. Analytica provides three options:
Simple Monte Carlo: The simplest sampling method is known as Monte Carlo, named after the randomness prevalent in games of chance, such as at the famous casino in Monte Carlo. In this method, each of the m sample points for each uncertainty quantity, X, is generated at random from X with probability proportional to the probability density (or probability mass for discrete quantities) for X. Analytica uses the inverse cumulative method; it generates m uniform random values,
ui, for i=1,2,...m, between 0 and 1, using the specified random number method (see below). It then uses the inverse of the cumulative probability distribution to generate the corresponding values of X,
Xi where P( ) = ui for i=1,2,...m
With the simple Monte Carlo method, each value of every random variable X in the model, including those computed from other random quantities, is a sample of m independent random values from the true probability distribution for X. You can therefore use standard statistical methods to estimate the accuracy of statistics, such as the estimated mean or fractiles of the distribution, as for example described in Selecting the Sample Size.
Median Latin hypercube (the default method): With median Latin hypercube sampling, Analytica divides each uncertain quantity X into m equiprobable intervals, where m is the sample size. The sample points are the medians of the m intervals, that is, the fractiles
Xi where P( ) = (i-0.5)/m, for i=1,2,...m.
These points are then randomly shuffled so that they are no longer in ascending order, to avoid nonrandom correlations among different quantities.
Random Latin hypercube: The random Latin hypercube method is similar to the median Latin hypercube method, except that instead of using the median of each of the m equiprobable intervals, Analytica samples at random from each interval. With random Latin hypercube sampling, each sample is a true random sample from the distribution. However, the samples are not totally independent.
Choosing a sampling method: The advantage of Latin hypercube methods is that they provide more even distributions of samples for each distribution than simple Monte Carlo sampling. Median Latin hypercube is still more evenly distributed than random Latin hypercube. If you display the PDF of a variable that is defined as a single continuous distribution, or is dependent on a single continuous uncertain variable, using median Latin hypercube sampling, the distribution usually looks fairly smooth even with a small sample size (such as 20), whereas the result using simple Monte Carlo looks quite noisy.
If the variable depends on two or more uncertain quantities, the relative noise-reduction of Latin hypercube methods is reduced. If the result depends on many uncertain quantities, the performance of the Latin hypercube methods might not be discernibly better than simple Monte Carlo. Since the median Latin hypercube method is sometimes much better, and almost never worse than the others, Analytica uses it as the default method. Very rarely, median Latin hypercube can produce incorrect results, specifically when the model has a periodic function with a period similar to the size of the equiprobable intervals. For example:
X := Uniform(1, Samplesize)
Y := Sin(2*Pi*X)
This median Latin hypercube method gives very poor results. In such cases, you should use random Latin hypercube or simple Monte Carlo. If your model has no periodic function of this kind, you do not need to worry about the reliability of median Latin hypercube sampling.
Random number method: The random number method is used to determine how random numbers are generated for the probability distributions. Analytica provides three different methods for calculating a series of pseudorandom numbers.
Minimal Standard (the default method): The Minimal Standard random number generator is an implementation of Park and Miller’s Minimal Standard (based on a multiplicative congruential method) with a Bays-Durham shuffle. It gives satisfactory results for less than 100,000,000 samples.
L’Ecuyer: The L’Ecuyer random number generator is an implementation of L’Ecuyer’s algorithm, based on a multiplicative congruential method, which gives a series of random numbers with a much longer period (sequence of numbers that repeat). Thus, it provides good random numbers even with more than 100,000,000 samples. It is slightly slower than the Minimal Standard generator.
Knuth: Knuth’s algorithm is based on a subtractive method rather than a multiplicative congruential method. It is slightly faster than the Minimal Standard generator. Random seed
This value must be a number between 0 and 100,000,000 (108). The series of random numbers starts from this seed value when:
- A model is opened.
- The value in this field is changed.
- The Reset once box is checked, and the Uncertainty Setup dialog is closed by clicking the Accept or Set Default button.
Rest once: Check the Reset once box to produce the exact same series of random numbers
Statistics option: To change the statistics reported when you select Statistics as the uncertainty view for a result, select the Statistics option from the Analysis option popup menu.
Probability Bands option: To change the probability bands displayed when you select Probability Bands as the uncertainty view for a result, select the Probability Bands option from the Analysis option popup menu.
Probability density option: To change how probability density is estimated and drawn, select Probability Density from the Analysis option popup menu.
Analytica estimates the probability density function, like other uncertainty views, from the underlying array of sample values for each uncertain quantity. The probability density is highly susceptible to random sampling variation and noise. Both histogramming and kernel density smoothing techniques are available for estimating the probability density from the sample, but to ultimately reduce noise and variability it may be necessary to increase sample size (for details on selecting the sample size, see Selecting the Sample Size). The following example graphs compare the two methods on the same uncertain result:
Histogram: The histogram estimation methods partition the space of possible continuous values into bins, and then tally how many samples land in each bin. The probability density is then equal to the fraction of the Monte Carlo sample landing in a given bin divided by the bin’s width. The average number of points landing in each bin determines both the smoothness of the resulting function and the resolution of the resulting plot. With more bins, a finer resolution is obtained, but since fewer points land in each bin, the amount of random fluctuation increases resulting in a noisier plot. The Samples per PDF step interval setting sizes the bin width to match the average number of points per bin. With larger sample sizes, you can increase the Samples per PDF step interval to achieve smoother plots, since more samples will land in each bin. A number approximately equal to the square root of sample size tends to work fairly well.
You can also control how the partitioning of the space of values is performed. When Equal X axis steps is used, the range of values from the smallest to largest sample point is partitioned into equal sized bins. With this method, all bins have the same width, but the number of points falling in each bin varies. When Equal weighted probability steps is used, the bins are sized so that each bin contains approximately the same fraction of the total probability. With this method, the fraction of the sample in each bin is nearly constant, but the width of each bin varies. When Equal sample probability steps is used, the bins are partitioned so that the number of sample points in each bin is constant, with the width of each bin again varying. Equal weighted probability steps and Equal sample probability steps are exactly equivalent when standard equally-weighted Monte Carlo or Latin Hypercube sampling is being used. They differ when the Sample Weighting system variable assigns different weights to each sample along the Run index, as is sometimes employed with importance sampling, logic sampling for posterior analysis, and rare-event modeling. See Importance weighting.
Probability density plots using the histogram method default to the Step chart type, which emphasizes the histogram and reveals the bin placements. When desired, this can be changed to the standard line style from the Graph Setup, see Chart Type tab.
Smoothing: The smoothing method estimates probability density using a technique known as Kernel Density Estimation (KDE) or Kernel Density Smoothing. This technique replaces each Monte Carlo sample with a Gaussian curve, called a Kernel, and then sums the curve to obtain the final continuous curve. Unlike a histogram, the degree of smoothness and the resolution of the plot are independent. The Smoothing factor controls the smoothness or amount of detail in the estimated PDF. The more info button next to the Smoothing radio control jumps to a page on the Analytica Wiki that elaborates in more detail on how kernel density smoothing works.
Due to the randomness of Monte Carlo sampling, estimations of probability density are often quite noisy. The Smoothing method can often provide smoother and more intuitively appealing plots than Histogram methods, but the averaging effects inherent in smoothing can also introduce some minor artifacts. In particular, Smoothing tends to increase the apparent variance in your result slightly, with a greater increase when the Smoothing factor is greater. This increase in variance is also seen as a decrease in the height of peaks. Sharp cutoffs (such as is the case with a Uniform distribution, for example) become rounded with a decaying tail past the cutoff point. And when positive-only distributions begin with a very sharp rise, the density estimate may get smoothed into a plot with a tail extending into the negative value territory.
Cumulative probability option: To change how the cumulative probability values are drawn or to change their resolution, select the respective option from the Analysis option popup menu.
Analytica estimates the cumulative distribution function, like other uncertainty views, from the underlying array of sample values for each uncertain quantity. As with any simulation-based method, each estimated distribution has some noise and variability from one evaluation to the next. Cumulative probability estimates are less susceptible to noise than, for example, probability density estimates.
The Samples per CDF plot point setting controls the average number of sample values used to estimate each point on the cumulative distribution function (CDF) curve, which ultimately controls the number of points plotted on your result.
The Equal X axis steps, Equal weighted probability steps and Equal sample probability steps control which points are used in plot of the cumulative probability. Equal X axis steps spaces points equally along the X axis. Equal weighted probability steps uses the sample to estimate a set of
m+! fractiles (quantiles),
Xp, at equal probability intervals, where
p=0, q, 2q, ... 1, and
q = 1/m. The cumulative probability is plotted at each of the points
Xp, increasing in equal steps along the vertical axis. Points are plotted closer together along the horizontal axis in the regions where the density is the greatest. Equal sample probability steps plots one point each at each nth sample point, where n is the sample per CDF plot point, ignoring the weight on each sample point when they are weighted differently. The cumulative probability up to the nth point is estimated and plotted. Equal weighted probability steps and Equal sample probability steps are exactly equivalent unless unequal sample weighting is employed (see Importance weighting).