Response modeling methodology
Response modeling methodology is a general platform for modeling monotonic convex relationships. RMM had been initially developed as a series of extensions to the original inverse Box–Cox transformation: where y is a percentile of the modeled response, Y, z is the respective percentile of a normal variate and λ is the Box–Cox parameter. As λ goes to zero, the inverse Box–Cox transformation becomes: an exponential model. Therefore, the original inverse Box-Cox transformation contains a trio of models: linear, power and exponential. This implies that on estimating λ, using sample data, the final model is not determined in advance but rather as a result of estimating. In other words, data alone determine the final model.
Extensions to the inverse Box–Cox transformation were developed by Shore and were denoted Inverse Normalizing Transformations. They had been applied to model monotone convex relationships in various engineering areas, mostly to model physical properties of chemical compounds. Once it had been realized that INT models may be perceived as special cases of a much broader general approach for modeling non-linear monotone convex relationships, the new Response Modeling Methodology had been initiated and developed.
The RMM model expresses the relationship between a response, Y, and two components that deliver variation to Y:
- The linear predictor, LP : where are regressor-variables that deliver systematic variation to the response;
- Normal errors, delivering random variation to the response.
and ε1 represents uncertainty in the explanatory variables. This is in addition to uncertainty associated with the response. Expressing ε1 and ε2 in terms of standard normal variates, Z1 and Z2, respectively, having correlation ρ, and conditioning Z2 | Z1 = z1, we may write in terms of a single error, ε:
where Z is a standard normal variate, independent of both Z1 and Z2, ε is a zero-mean error and d is a parameter. From these relationships, the associated RMM quantile function is :
or, after re-parameterization:
where y is the percentile of the response, z is the respective standard normal percentile, ε is the model’s zero-mean normal error with constant variance, σ, are parameters and MY is the response median, dependent on values of the parameters and the value of the LP, η:
where μ is an additional parameter.
If it may be assumed that cz<<η, the above model for RMM quantile function can be approximated by:
The parameter “c” cannot be “absorbed” into the parameters of the LP since “c” and LP are estimated in two separate stages.
If the response data used to estimate the model contain values that change sign, or if the lowest response value is far from zero, a location parameter, L'', may be added to the response so that the expressions for the quantile function and for the median become, respectively:
Major property of RMM – continuous monotonic convexity (CMC)
As shown earlier, the inverse Box–Cox transformation depends on a single parameter, λ, which determines the final form of the model. All three models thus constitute mere points on a continuous spectrum of monotonic convexity, spanned by λ. This property, where different known models become mere points on a continuous spectrum, spanned by the model’s parameters, is denoted the Continuous Monotonic Convexity property. The latter characterizes all RMM models, and it allows the basic “linear-power-exponential” cycle to be repeated ad infinitum, allowing for ever more convex models to be derived. Examples for such models are an exponential-power model or an exponential-exponential-power model. Since the final form of the model is determined by the values of RMM parameters, this implies that the data, used to estimate the parameters, determine the final form of the estimated RMM model. The CMC property thus grant RMM models high flexibility in accommodating the data used to estimate the parameters. References given below display published results of comparisons between RMM models and existing models. These comparisons demonstrate the effectiveness of the CMC property.Examples of RMM models
Ignoring RMM errors, we obtain the following RMM models, presented in an increasing order of monotone convexity:Adding two new parameters by introducing for η :, a new cycle of “linear-power-exponential” is iterated to produce models with stronger monotone convexity :
It is realized that this series of monotonic convex models, presented as they appear in a hierarchical order on the “Ladder of Monotonic Convex Functions”, is unlimited from above. However, all models are mere points on a continuous spectrum, spanned by RMM parameters.
Moments
The k-th non-central moment of Y is :Expanding Yk, as given on the right-hand-side, into a Taylor series around zero, in terms of powers of Z, and then taking expectation on both sides, assuming that cZ ≪ η so that η + cZ ≈ η, an approximate simple expression for the k-th non-central moment, based on the first six terms in the expansion, is:
An analogous expression may be derived without assuming cZ ≪ η. This would result in a more accurate expression.
Once cZ in the above expression is neglected, Y becomes a log-normal random variable.
RMM fitting and estimation
RMM models may be used to model random variation or to model systematic variation.In the former case, RMM quantile function is fitted to known distributions. If the underlying distribution is unknown, the RMM quantile function is estimated using available sample data. Modeling random variation with RMM is addressed and demonstrated in Shore.
In the latter case, RMM models are estimated assuming that variation in the linear predictor contribute to the overall variation of the modeled response variable. This case is addressed and demonstrated in Shore. Estimation is conducted in two stages. First the median is estimated by minimizing the sum of absolute deviations. In the second stage, the remaining two parameters, are estimated. Three estimation approaches are presented in Shore : maximum likelihood, moment matching and nonlinear quantile regression.
Literature review
Current RMM literature addresses three areas:' Developing INTs and later the RMM approach, with allied estimation methods;
' Exploring the properties of RMM and comparing RMM effectiveness to other current modelling approaches ;
Applications.
Shore developed Inverse Normalizing Transformations in the first years of the 21st century and has applied them to various engineering disciplines like statistical process control and chemical engineering. Subsequently, as the new Response Modeling Methodology had been emerging and developing into a full-fledged platform for modeling monotone convex relationships, RMM properties were explored, estimation procedures developed and the new modeling methodology compared to other approaches, for modeling random variation, and for modeling systematic variation.
Concurrently, RMM had been applied to various scientific and engineering disciplines and compared to current models and modeling approaches practiced therein. For example, chemical engineering, statistical process control, reliability engineering, forecasting, ecology, and the medical profession.