Event Counts over Time: Poisson and Pólya

The business question

A medical practice makes house calls. It expects 27 calls a week now, growing by 0.2 a week, so about 37 a week by the end of the year. Calls arrive at random, which a Poisson process describes: its variance equals its mean.

The practice's records show more variation than that, which the Pólya process allows: the expected number is itself uncertain, Gamma distributed around the trend with an extra-variation parameter h = 0.3. Where does that uncertainty sit? If it is noise from week to week, busy and quiet weeks cancel out over the year.

If it is about the year as a whole (a busy year or a quiet one), it does not. The model runs the same trend three ways and compares the year's totals.

Who it is for

Anyone forecasting counts of events: calls, claims, failures, admissions, orders.

Main outputs

The year's total in three versions at P10 / median / P90 with exact and simulated standard deviations · weekly bands for the Poisson and the shared-level versions.

How the model works

Three sheets after this one, read from left to right. Each one feeds the next.

1. 1

   Inputs sheet

   The expected calls per week (an intercept and a gradient) and the Pólya parameter h.

2. 2

   Weeks sheet

   Each week's expected calls follow the trend. The Poisson version draws VosePoisson(expected). The Pólya version of the original example draws a Gamma with mean 'expected' and a new value each week. The shared version draws one Gamma level with mean 1 for the whole year and multiplies every week's expected calls by it.

3. 3

   Results sheet

   The year's total in each version, with its exact standard deviation beside the simulated one, and the weekly bands of the Poisson and the shared versions.

ModelRisk techniques

• Poisson around a trend
• Pólya process (VosePoisson of VoseGamma)
• Exact variances
• Output ranges

Find similar models

Product

• ModelRisk

Type

• Simulation

Application

• Statistical techniques
• Forecasting

Uses

• Poisson
• Gamma

Inputs and assumptions

Every assumption in the model, with its uncertainty and where it comes from. The figures are illustrative.

Units and conventions

Calls

House calls per week.

Pólya h

The extra variation: a week's variance is mean + h × mean² instead of the Poisson's mean.

Level

The shared version's multiplier for the year: Gamma with mean 1 and variance h.

Key assumptions and limitations

Independence

Calls arrive independently within a week. In the Poisson and the weekly Pólya versions the weeks are also independent of each other.

Trend

Linear, as in the original examples.

Data

The trend (27 + 0.2 a week over 50 weeks) and h = 0.3 are those of ModelRisk's earlier Polya Time Series example. Its Poisson random walk example used the same Poisson model with 50 + 0.1 a period over 100 periods.

How to use it

1. Set the trend and h on the Inputs sheet; h = 0 makes all three versions Poisson.
2. Run a simulation from the ModelRisk ribbon (10,000 samples with a fixed seed are set).
3. Decide which version matches where your extra variation comes from, and use its total.

ModelRisk functions used

VosePoisson

Each week's calls, around the expected number.

VoseGamma

The Pólya process's uncertain rate: VoseGamma(1/h, expected × h) each week, or VoseGamma(1/h, h) once as the year's level.

VoseOutput

The 50 weekly counts of each version as an output range, and the three totals.

VoseSimMean, VoseSimStdev, VoseSimPercentile

Statistics of the simulation, and the weekly bands.

Further reading

• Monte Carlo simulation explained (Vose guide)
• Vose, D. (2008). Risk Analysis: A Quantitative Guide, 3rd edition. Wiley: the Pólya and Poisson processes.