Filtered Model Output Statistics (FMOS)

A computer-implemented method for estimating a set of regression coefficients for use in estimating a model forecast variable, comprising:

• receiving, at a computer programmed with appropriate software, data of an n-vector aMOS and an n-vector bMOS, the elements of aMOS and bMOS comprising a set of modeled output statistics (MOS) estimates of the true values of forecast correction coefficients a and b;
• receiving, at the computer, data of an n-vector aprior and an n-vector bprior, the elements of aprior and bprior comprising a set of prior estimates of forecast correction coefficients a and b;
• estimating a spatial covariance matrix Ga for aprior spatial covariance matrix Gb for bprior;
• estimating a spatial error covariance matrix Ra for aMOS and a spatial error covariance matrix Rb for bMOS;
• and
• transforming the data of aMOS, bMOS, aprior, bprior, Ga, Gb, Ra, and Rb, into a set of estimated regression coefficients aFMOS and bFMOS, wherein
• aFMOS=aprior=Ga(Ga=Ra)−1(aMOS−aprior);
• and
• bFMOS=bprior=Gb(Gb=Rb)−1(bMOS−bprior).

A computer-implemented method for estimating a set of regression coefficients for use in estimating a model forecast variable, comprising:

• receiving, at a computer programmed with appropriate software, data of values of a plurality of analysis/observation variables yik, each value of yik representing a value of an ith variable yi during a kth event;
• receiving, at the computer, data of values of a plurality of corresponding forecast variables fik, each value of fik being related to a corresponding value of yik by the relation fik=aiyik+bi+εik, where ai and bi are regression coefficients and εik is a random number associated with the kth forecast;
• receiving, at the computer, data of values of a plurality of mean variables yi and fi, each of yi and fi comprising average values of yi and fik over a plurality of realizations;
• transforming the data of the yik, fik, yi, and fi in to a first set of estimated regression coefficients aiMOS and biMOS, the first set of estimated regression coefficients comprising a set of modeled output statistics (MOS) estimates of the true values of regression coefficients ai and bi;
• receiving, at the computer, data of a first set of second set of estimated regression coefficients aiprior and biprior, the second set of estimated regression coefficients representing a prior estimate of regression coefficients ai and bi;
• transforming the values of aiMOS into the n-vector aMOS and the values of biMOS into the n-vector bMOS;
• transforming the values of aiprior into the n-vector aprior a prior and the values of biprior into an n-vector bprior;
• estimating a spatial covariance matrix Ga for aprior and spatial covariance matrix Gb for bprior;
• estimating a spatial error covariance matrix Ra for aMOS and a spatial error covariance matrix Rb for bMOS;
• and
• transforming the data of aMOS, bMOS, aprior, Ga, Gb, Ra, and Rb into a set of estimated regression coefficients aFMOSand bFMOS, wherein
• aFMOS=aprior=Ga(Ga=Ra)−1(aMOS−aprior);
• and
• bFMOS=bprior=Gb(Gb=Rb)−1(bMOS−bprior).

A computer-implemented method of forecasting weather from a small data set, comprising:

• receiving, at a computer programmed with appropriate software, data of values of a set of observation variables yik, each of yik representing a value of an observation variable yi at a kth event;
• receiving, at the computer, data of n-vectors of filtered model output coefficients aFMOS and bFMOS, wherein
• aFMOS=aprior=Ga(Ga=Ra)−1(aMOS−aprior);
• and
• bFMOS=bprior=Gb(Gb=Rb)−1(bMOS−bprior);
• and wherein aMOS and b bMOS comprise n-vectors of a set of MOS estimated regression coefficients, aprior and bPrior are n-vectors of prior estimated coefficients, Ga is the spatial covariance matrix for aprior, Gb is the spatial covariance matrix for bprior, Ra is the spatial error covariance matrix for aMOS, and Rb is the spatial error covariance matrix for bMOS;
• and
• transforming each observation variable yik into a corresponding forecast variable fik, where fik=aiFMOSyik=biFMOSyikεik, aiFMOS and biFMOS are the ith elements of the n-vectors aFMOS and bFMOS, respectively, and εik is a random number associated with the kth forecast;
• wherein the transformation of yik into fik represents an accurate description of relationship between yik and fik.