Predictive Model Assessment with Proper Scoring Rules

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Description

Computes scores for the assessment of sharpness of a fitted model for time series of counts.

Usage

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## S3 method for class 'tsglm'
scoring(object, individual=FALSE, cutoff=1000, ...)
## Default S3 method:
scoring(response, pred, distr=c("poisson", "nbinom"), distrcoefs,
          individual=FALSE, cutoff=1000, ...)

Arguments

object

an object of class "tsglm".

individual

logical. If FALSE (the default) the average scores are returned. Otherwise a matrix with the individual scores for each observation is returned.

cutoff

positive integer. Summation over the infinite sample space {0,1,2,...} of a distribution is cut off at this value. This affects the quadratic, spherical and ranked probability score.

response

integer vector. Vector of observed values Y[1],...,Y[n].

pred

numeric vector. Vector of predicted values μ_P[1],...,μ_P[n].

distr

character giving the conditional distribution. Currently implemented are the Poisson ("poisson")and the Negative Binomial ("nbinom") distribution.

distrcoefs

numeric vector of additional coefficients specifying the conditional distribution. For distr="poisson" no additional parameters need to be provided. For distr="nbinom" the additional parameter size needs to be specified (e.g. by distrcoefs=2), see tsglm for details.

...

further arguments are currently ignored. Only for compatibility with generic function.

Details

The scoring rules are penalties that should be minimised for a better forecast, so a smaller scoring value means better sharpness. Different competing forecast models can be ranked via these scoring rules. They are computed as follows: For each score s and time t the value s(P[t],Y[t]) is computed, where P[t] is the predictive c.d.f. and Y[t] is the observation at time t. To obtain the overall score for one model the average of the score of all observations (1/n) ∑ s(P[t],Y[t]) is calculated.

For all t ≥q 1, let p[y]=P(Y[t]=y | F[t-1]) be the density function of the predictive distribution at y and ||p||^2= ∑ p[y]^2 be a quadratic sum over the whole sample space y=0,1,2,... of the predictive distribution. μ_P[t] and σ_P[t] are the mean and the standard deviation of the predictive distribution, respectively.

Then the scores are defined as follows:

Logarithmic score: logs(P[t],Y[t])= -log p[y]

Quadratic or Brier score: qs(P[t],Y[t])= -2p[y] + ||p||^2

Spherical score: sphs(P[t],Y[t])= -p[y] / ||p||

Ranked probability score: rps(P[t],Y[t])=∑ (P[t](x) - 1(Y[t]≤ x))^2 (sum over the whole sample space x=0,1,2,...)

Dawid-Sebastiani score: dss(P[t],Y[t]) = ( (Y[t]-μ_P[t]) / (σ_P[t]) )^2 + 2 log σ_P[t]

Normalized squared error score: nses(P[t],Y[t])= ( (Y[t]-μ_P[t]) \ (σ_P[t]) )^2

Squared error score: ses(P[t],Y[t])=(Y[t]-μ_P[t])^2

For more information on scoring rules see the references listed below.

Value

Returns a named vector of the mean scores (if argument individual=FALSE, the default) or a data frame of the individual scores for each observation (if argument individual=TRUE). The scoring rules are named as follows:

logarithmic

Logarithmic score

quadratic

Quadratic or Brier score

spherical

Spherical score

rankprob

Ranked probability score

dawseb

Dawid-Sebastiani score

normsq

Normalized squared error score

sqerror

Squared error score

Author(s)

Philipp Probst and Tobias Liboschik

References

Christou, V. and Fokianos, K. (2013) On count time series prediction. Journal of Statistical Computation and Simulation (published online), http://dx.doi.org/10.1080/00949655.2013.823612.

Czado, C., Gneiting, T. and Held, L. (2009) Predictive model assessment for count data. Biometrics 65, 1254–1261, http://dx.doi.org/10.1111/j.1541-0420.2009.01191.x.

Gneiting, T., Balabdaoui, F. and Raftery, A.E. (2007) Probabilistic forecasts, calibration and sharpness. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 69, 243–268, http://dx.doi.org/10.1111/j.1467-9868.2007.00587.x.

See Also

tsglm for fitting a GLM for time series of counts.

pit and marcal for other predictive model assessment tools.

permutationTest in package surveillance for the Monte Carlo permutation test for paired individual scores by Paul and Held (2011, Statistics in Medicine 30, 1118–1136, http://dx.doi.org/10.1002/sim.4177).

Examples

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###Campylobacter infections in Canada (see help("campy"))
campyfit <- tsglm(ts=campy, model=list(past_obs=1, past_mean=c(7,13)))
scoring(campyfit)

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