What is PyMimic?¶
PyMimic is a pure-Python implementation of linear prediction for real-valued random variables [G62]. Linear prediction may be used for curve fitting, Kriging [C86], DACE modelling [S89], and Gaussian-process emulation [RW06]. These are the intended uses of PyMimic.
The basics¶
A linear predictor for a random variable \(Z\) based on a random process \(X := \{X_{t}\}_{t \in T}\) is a linear combination of the elements of \(X\). By denoting this predictor \(\hat{Z}\) we have that
We may form the best linear predictor (BLP) by choosing the set of coefficients, \((a_{t})_{t \in T}\), so as to minimize the mean-squared error, \(\operatorname{E}((Z - \hat{Z})^{2})\).
We may further impose the constraint that the expected values of \(Z\) and \(\hat{Z}\) are equal, i.e. that \(\operatorname{E}(\hat{Z}) = \operatorname{E}(Z)\). In order to do this, we work under the assumption that \(\operatorname{E}\) is an element of a finite-dimensional space of functions, \(F\). We then minimize the mean-squared error \(\operatorname{E}((Z - \hat{Z})^{2})\) subject to the constraint that \(\operatorname{E}_{f}(\hat{Z}) = \operatorname{E}_{f}(Z)\) for all \(\operatorname{E}_{f} \in F\). We call this the best linear unbiased predictor (BLUP).
PyMimic provides the classes Blp and Blup for the
purposes of computing realizations of these predictors.
PyMimic also provides the following submodules:
kernel, a suite of positive-definite kernels, which may be used as second-moment or covariance kernels when specifying \((Z, X)\);
basis, a suite of bases, which may be used to specify the space of mean-value functions when computing the BLUP;
testfunc, a suite of test functions, which may be used to illustrate linear prediction; and
plot, plotting routines that may be used to visualize the results of linear prediction and assist in their validation.
References¶
Cressie, N. 1986. ‘Kriging nonstationary data’ in Journal of the American Statistical Association, 81 (395): 625–34. Available at https://www.doi.org/10.1080/01621459.1986.10478315.
Goldberger, A. S. 1962. 'Best linear unbiased prediction in the generalized linear regression model' in Journal of the American Statistical Association, 57 (298): 369–75. Available at https://www.doi.org/10.1080/01621459.1962.10480665.
Rasmussen, C.E., and C.K.I. Williams. 2006. Gaussian processes for machine learning. Cambridge: MIT Press.
Sacks, J., Welch, W. J., Mitchell, T.J., and H. P. Wynn. 1989. ‘Design and analysis of computer experiments’ in Statistical science, 4 (4): 409–23. Available at https://www.doi.org/10.1214/ss/1177012413.