Product announcement: Actuarial Forecast Software version 5.0 October 7, 2013
An actuarial forecast is an estimate of average future demand due to failures. It is more accurate (unbiased) and precise (less variance) than alternatives. The Actuarial Forecast
Software computes an actuarial forecast for a specified (future) calendar interval and confidence [prediction or tolerance limit]. It also computes the distribution of future demands for estimating prediction intervals. The Software makes forecasts for more than one future calendar interval, for variable (random) operating hours, for dead-forever and repairable systems, products, and service parts.
Quality and service departments should forecast demand and estimate its distribution, for all service parts not just those tracked by serial number. They need confidence limits for risk analysis and prediction limits for spares recommendations (fill rate or service level). Accountants should estimate warranty reserves to cover the risk of more-than-expected warranty claims. Insurance companies should forecast claims and their distributions. Health providers should forecast epidemic cases and compare country-by-country differences in survivor functions and cast counts (www.fieldreliability.com/SARSStat.ppt). Auto companies that extend warranties should reimburse customers who previously paid for repairs covered by the warranty extension (e.g., GM and Saturn CVT transmissions).
Some forecasting software says, “Please enter forecast ________.” Some forecasting software does time series extrapolation. Such forecasts lag parts demands, because they don't account for age-specific field reliability. Such forecasts result in parts shortages during the early portion of a product's life cycle and excess spares due to obsolescence at the ends of product life cycles. Why not account for age-specific failure rate estimates and their joint distribution, as well as installed base, and lead-time? Why not quantify sample and extrapolation uncertainty?
1. Obtain vehicle registration counts by year, make, model, and engine, by zip code, http://www.polk.com. Identify zip codes that service your clients’ and their market shares of those zip codes, http://en.wikipedia.org/wiki/Voronoi_diagram. This provides the car installed base by age in your clients’ neighborhoods.
2. Obtain automotive catalogs that tell which parts and how many go into which cars, http://en.wikipedia.org/wiki/Andrew_V%C3%A1zsonyi#CITEREFV.C3.A1zsonyi2002. This provides gozintos.
3. Obtain parts sales from as many sources and potential clients as possible. Do not use a modem; that method is patented, https://www.google.com/patents/US5765143. This provides part demand samples.
4. Compute parts installed base V(I-N)-1 where V is a vector of the vehicle counts corresponding to data obtained in step 3, and N is the gozinto matrix {N(car, part)}.
5. Estimate parts’ actuarial demand rates from date from steps 3 and 4 and the estimates’ variance-covariance matrix. Extrapolate actuarial rates to age of oldest installed base and estimate standard errors of extrapolations.
6. Make actuarial forecasts using each client’s installed base and estimate confidence limits on average demand per part number.
7. Recommend stock levels for each client and each part according to each client’s desired service level or fill rate using the distribution of client’s part demands and prediction limits.
8. Recommend demand driven part prices and whether to stock parts based on vehicle installed base, opportunity costs, and scarcity, (ask for PandQ.ppt).
Triad Systems Corporation made millions of $$$ selling actuarial forecasts and stock level recommendations in the late 1990s. The forecasts were evidently better than alternatives, especially for cars coming out of warranty and old cars. Triad was bought in a leveraged buyout by a competitor.
I learned actuarial methods while working for the (former) US Air Force Logistics Command (AFLC) on engine reliability and maintenance. The AFLC actuarial method assumes constant failure rates within age intervals, Poisson demands, and ignores variance induced by variable flying hours per aircraft in the flying hour program. While working for Apple Computer, my boss Mike Johanns suggested a literature search on self-service systems and statistics, because he regarded time to computer failure as service time in a self-service system. Apple had ships and returns counts for all service parts, as if observing inputs and outputs of a self-service system. The only hit was my 1973 NLRQ paper that gave the nonparametric maximum likelihood reliability estimator from ships and returns counts of dead-forever parts (ask if you want a copy). Later, for the automotive aftermarket, I extended the estimators to deal with repairable systems using stores’ part sales, gozinto catalogs, and vehicle registrations near stores. That means actuarial methods can be applied to all products and service parts, without having to track them by serial number.
Today most companies aren’t even as advanced as the AFLC circa 1960. The naïve forecast, installed base times annual failure rate, accounts for installed base but ignores age dependence due to:
· Infant mortality, warranty expiration, wearout, and periodic inspection
· Periodic maintenance and seasonal usage
· Retirement
· Environment, operating hours per calendar interval, and other factors
The Actuarial Forecast discretizes the naïve forecast to account for age-specific failure or demand rates; that is,
SUM [a(j)n(j)], SUM [f(j)n(j)’], or SUM [d(j)n(j)],
where a(j) is the actuarial failure rate per product of age j and n(j) counts surviving products of age j during the forecast interval; f(j) is the discrete probability density function for age j and n(j)’ counts original sales that would have age j in the forecast calendar interval, or d(j) is the age-specific demand rate per repairable product of age j. (The rate a(j) is the age-specific failure rate for nonrepairable products.) The Actuarial Forecast is unbiased and has greater precision than time series extrapolation or the naïve forecast.
The Actuarial Forecast typically uses discrete nonparametric, estimates of failure rates. The estimates could be either maximum likelihood or least squares estimates. Actuarial failure rates can be estimated from the nonparametric maximum likelihood Nelson-Aalen failure rate estimator corresponding to the Kaplan-Meier reliability estimator, for dead-forever products or parts. For repairable systems or parts, they can be estimated from the Nelson-Aalen estimator if ages and first failures and times between failures are known. Ask for alternatives.
Some people assume a failure rate function and plug that into the actuarial forecast formula. Assuming a failure rate function incurs loss of information and assumes that the future resembles the past. The Actuarial Forecast requires no unwarranted distribution assumptions, preserves all information in data, and has no sample uncertainty, except that in estimating the actuarial rates. It has no other uncertainty, except for extrapolation of a failure rate for the oldest installed base in the forecast calendar interval. It also facilitates estimation with missing data. It accommodates variable operating hours per calendar interval, given actuarial rates per operating hour and the operating hour distribution per calendar time interval (version 3.0). The Actuarial Forecast Software also facilitates what-if studies, because it allows you to change the extrapolation to suit past observations and future scenarios. The Actuarial Forecast Software uses functional principal component analysis to extrapolate the covariance matrix to quantify extrapolation uncertainty too.
Actuarial Forecast Software Versions 1-5:
1. Computes the actuarial forecast, SUM [â(j)n(j)]. This is for nonrepairable systems with discrete failure rate function a(j) also known as actuarial rates.
2. Computes the actuarial forecast, SUM [d(j)n(j)], where the d(j) are estimates of age-specific demand rates, whether from first or subsequent failures, of repairable systems. The demand rates should incorporate the joint distribution of times to first failures and times between subsequent failures. They may be neither independent nor identically distributed.
3. Incorporates usage, operating hours per calendar hour, in case it differs in the forecast calendar interval from the usage in prior calendar time. This modification is necessary, because, if usage is great, units may age through several age intervals in the forecast calendar interval. Contact me if you want to incorporate random usage in the forecast interval.
4. Uses the covariance matrix of the actuarial rate estimates to estimate confidence intervals on the actuarial forecast and confidence and prediction intervals. It also extrapolates the actuarial rate for the oldest installed base and the covariance matrix. Version 4 simulates failures in the forecast interval, with dependent binomial distributions to represent correlation among the failure rate estimates. Version 5 computes the distribution of failures in the forecast interval, using the Cramer-Rao asymptotic variance-covariance matrix. This estimates the distribution of demand in the forecast interval, not just its expected value, the actuarial forecast. This obviates assumptions such as Poisson, negative binomial, and normal.
Variance of Actuarial Forecast
The variance is needed for confidence bounds and prediction intervals. The variance of the actuarial forecast is
Var[SUM [â(j)*n(j)]] = SUM n(j)2Var[â(j)] + 2*SUM [n(i)n(j)Covar[â(i), â(j)]].
The Actuarial Forecast Software offers: empirical covariance estimates from broom chart actuarial rates and asymptotic inverse of the Fisher information matrix. Sorry, the covariances of actuarial rate estimates are not zero! If estimators are unbiased, the Cramer-Rao asymptotic lower bound is the inverse of the observed Fisher information (matrix). If the estimators are maximum likelihood estimators, then under some conditions, they are asymptotically unbiased, normally distributed, and achieve the minimum covariance matrix, the inverse of the observed Fisher information matrix. The Kaplan-Meier and Nelson-Aalen estimators are maximum likelihood estimators (almost, depending on which version you use). One of the estimators, from ships and returns counts, is also a maximum likelihood estimator, although with greater variance than that of the Kaplan-Meier estimator. (Please ask forNwsRev2.doc page 4 “What Price Required Data?” for a comparison.)
Variance of Demand is not Variance of Forecast
The variance formula above is the variance of an estimate of expected demand, not the variance of demand itself. Demand is a random variable equal to the sum of demands from each age interval,
Demand = SUM R(i)
where R(i) denotes random demands from the n(i) survivors in the forecast calendar interval, i = 1,2,...,k. The R(i) are independent but not identically distributed binomial random variables with probabilities a(i). The variance of the demand random variable is
Var[Demand] = SUM [Var[R(i)]] + 2*SUM Covar[R(i), R(j)] = SUM [n(i)a(i)(1-a(i))].
Fortunately, the covariances of R(i) and R(j) are zero, because the n(i) are independent cohorts, and the variances of R(i) are those of binomial random variables.
However, if the a(i) failure rates are estimates, Var[Demand] should include the sample uncertainty in estimates of the actuarial rates â(i). This is represented in the variance-covariance matrix of the â(i) failure rate estimates. There are several alternative estimates for the variance-covariance matrix:
1. Empirical: Compute different estimators of the â(i) failure rates, preferably from independent and identically distributed samples, and compute their variance-covariance matrix
2. Asymptotic: Use the asymptotic variance-covariance matrix of â(i) maximum likelihood estimators, the inverse of the Fisher information matrix
3. Please let me know of other methods or suggestions...
Greenwood’s formula approximates the variance of the Kaplan-Meier nonparametric reliability estimator, but neither the variances of the â(i) estimators nor their covariances. They are not independent!
Distribution of demand
The US Air Force Logistics Command actuarial method assumes the demand distribution is Poisson. This assumption is correct if the installed base was accumulated as a Poisson process and if all failure rates are constant regardless of age. In reality, demand has the distribution of the sum of independent, non-identically distributed binomial random demands, a convolution. This distribution is estimable and computable by simulation or analytically. Estimates of the exact demand distribution parameters have been incorporated in the Actuarial Forecast Software. The Actuarial Forecast Software forecasts for specified future periods, the variance and standard deviation of future demands, and the empirical distribution of simulated demands. If the user inputs variable operating hours per calendar interval and actuarial rates per hour, the Actuarial Forecast is computed for specified future operating hour intervals.
Inputs
The Actuarial Forecast Software input consists of the current installed base n(i) values, estimates of the actuarial (age-specific) failure or demand rates â(i) (alternatively f(i) and n(i)’, the discrete probability density of age at failure and original installed base by shipping calendar interval), and an estimate of the covariance matrix of the actuarial or demand rates. The covariance matrix could be that of maximum likelihood â(i) estimates, either from grouped ages at failures and survivors’ ages, from ships and returns counts, or even subjective estimates. The Software requires actuarial rate(s) and their variances and covariances for the oldest units in the forecast calendar interval. Versions 4 and 5 make a linear extrapolation of these rates, variances, and covariances. Other alternatives are available on request.
Format
The Actuarial Forecast Software consists of an Excel workbook and VBA functions and subroutines. The workbook consists of two spreadsheets: one for forecast computations and the second for Cholesky decomposition of the variance-covariance matrix for simulating demands. An alternative estimates the variance-covariance matrix of actuarial rate estimates and standard errors of extrapolations for use in computing the distribution of future demands. Your inputs can be linked to tables 1 and 2 of the forecast computation spreadsheet. The inputs and forecasts computations are done in tables:
1. Input actuarial failure or demand rate estimates, length of forecast interval, and installed base by age, including future installed base if any in the forecast interval. The table uses stepwise, piecewise linear regression to extrapolate the actuarial rates to cover oldest installed base in the forecast interval. The piecewise failure rate function is constant followed by a linear increasing (to represent wearout) or decreasing (to represent retirement) line segment. The extrapolation can be altered if necessary to exclude infant mortality, warranty expiration anticipations, and other phenomena by simply changing the input failure rates.
2. Input the variance-covariance matrix of the estimates. They too will be extrapolated, using the same subset of ages that yielded the best stepwise, piecewise linear regression. The variance-covariance matrix could be estimated empirically from broom chart actuarial rates, from the asymptotic variance-covariance of the Nelson-Aalen failure rate estimates, or other estimates. Please contact me for help with estimation of the variance-covariance matrix; some are unpublished statistics.
3. Actuarial hindcasts and forecasts, for multiple periods in the future if the forecast interval (table 1) is longer than one actuarial age interval. If the usage rate is not 1:1 in the forecast interval, here is the place to enter it.
4. Confidence interval estimates on the actuarial forecast(s). This will include a single confidence interval estimate if the forecast interval equals the actuarial age interval (and usage rate remains 1:1) and simultaneous confidence intervals if multiple forecasts are requested. Please note that these are confidence intervals on mean demand in the forecast interval; the are not prediction or tolerance intervals.
5. Sensitivity of actuarial forecasts to the parameters used to make the linear extrapolation of the failure rate function.
6. Simulation of demand in the forecast interval, using the (multi)variate normal approximation to the binomial distribution of failures from each part n(i) of the installed base. Percentiles represent prediction limits on demand. Demand distribution mean, standard deviation, skew, and kurtosis are computed by spreadsheet formulas.
Methods
The forecast computation spreadsheet constructs the actuarial forecast, confidence limits on it, for the calendar age interval that you specify, and it estimates demand distributions for prediction and tolerance intervals on future demands.
Software is available for Macintosh and Windows, as Excel workbooks with VBA programming. Depending on available data, you might also need reliability estimation software. It is also useful to work together to customize the actuarial forecast software for your needs, data, and appropriate extrapolation(s). For more information, specifications, free samples, or software, please contact Larry George at Problem Solving Tools, pstlarry@yahoo.com. Thank you and may your forecasts be accurate and precise.