On classical and other methods of discriminant analysis and estimation of log-odds

For two multinormal populations with equal covariance matrices the likelihood ratio discriminant function, an alternative allocation rule to the sample linear discriminant function when n1 ≠ n2 ,is studied analytically. With the assumption of a known covariance matrix its distribution is derived and the expectation of its actual and apparent error rates evaluated and compared with those of the sample linear discriminant function. This comparison indicates that the likelihood ratio allocation rule is robust to unequal sample sizes. The quadratic discriminant function is studied, its distribution reviewed and evaluation of its probabilities of misclassification discussed. For known covariance matrices the distribution of the sample quadratic discriminant function is derived. When the known covariance matrices are proportional exact expressions for the expectation of its actual and apparent error rates are obtained and evaluated. The effectiveness of the sample linear discriminant function for this case is also considered. Estimation of true log-odds for two multinormal populations with equal or unequal covariance matrices is studied. The estimative, Bayesian predictive and a kernel method are compared by evaluating their biases and mean square errors. Some algebraic expressions for these quantities are derived. With equal covariance matrices the predictive method is preferable. Where it derives this superiority is investigated by considering its performance for various levels of fixed true log-odds. It is also shown that the predictive method is sensitive to n1 ≠ n2. For unequal but proportional covariance matrices the unbiased estimative method is preferred. Product Normal kernel density estimates are used to give a kernel estimator of true log-odds. The effect of correlation in the variables with product kernels is considered. With equal covariance matrices the kernel and parametric estimators are compared by simulation. For moderately correlated variables and large dimension sizes the product kernel method is a good estimator of true log-odds.

Pyramidal quantum dots: single and entangled photon sources and correlation studies

Practical realisation of quantum information science is a challenge being addressed by researchers employing various technologies. One of them is based on quantum dots (QD), usually referred to as artificial atoms. Being capable to emit single and polarization entangled photons, they are attractive as sources of quantum bits (qubits) which can be relatively easily integrated into photonic circuits using conventional semiconductor technologies. However, the dominant self-assembled QD systems suffer from asymmetry related problems which modify the energetic structure. The main issue is the degeneracy lifting (the fine-structure splitting, FSS) of an optically allowed neutral exciton state which participates in a polarization-entanglement realisation scheme. The FSS complicates polarization-entanglement detection unless a particular FSS manipulation technique is utilized to reduce it to vanishing values, or a careful selection of intrinsically good candidates from the vast number of QDs is carried out, preventing the possibility of constructing vast arrays of emitters on the same sample. In this work, site-controlled InGaAs QDs grown on (111)B oriented GaAs substrates prepatterned with 7.5 μm pitch tetrahedrons were studied in order to overcome QD asymmetry related problems. By exploiting an intrinsically high rotational symmetry, pyramidal QDs were shown as polarization-entangled photon sources emitting photons with the fidelity of the expected maximally entangled state as high as 0.721. It is the first site-controlled QD system of entangled photon emitters. Moreover, the density of such emitters was found to be as high as 15% in some areas: the density much higher than in any other QD system. The associated physical phenomena (e.g., carrier dynamic, QD energetic structure) were studied, as well, by different techniques: photon correlation spectroscopy, polarization-resolved microphotoluminescence and magneto-photoluminescence.