Maritally Unattached

Starting from a 2p atomic orbital aligned along the molecular axis, we solve the three-dimensional TDSE in single-active-electron approximation with the split-operator method on a Cartesian grid with 512 points in each dimension, a grid spacing of 0.25 a.u. and a time step of 0.02 a.u. While propagating up to a final time T = 1500 a.u., outgoing parts of the wave function are projected onto Volkov states^{44 (link)}. The potential for a single neon atom is chosen as in ref. ^{45 (link)} but with the singularity removed using a pseudopotential^{46 (link)} for angular momentum l = 1. The clockwise circularly polarized pulse has a 12-cycle sin² envelope and a peak field strength of 0.096 a.u. To obtain the momentum distribution for the dimer we multiply two copies of the atomic distribution by e^±ik·R/2, respectively (|R|/2 = 2.93 a.u.) and then add them coherently with an additional factor of ± 1 depending on the type of interference, gerade, or ungerade. To account for different possible orientations of the dimer with respect to the polarization plane, we vary the angle between them in 8 steps to cover a range from 0 to 45°, project the molecular photoelectron momentum distribution (PMD) onto the polarization plane and add these projections together with their geometrical weights. The PMDs are then averaged over the ATI peaks to obtain the final distributions shown in Fig. 2.

Given a pedigree consisting of K parent–child relationships. Consider the set $\mathcal{A}$ consisting of vectors r of length K where each component is either 0 or 1, with 0 indicating that the child in the corresponding relationship has inherited the parent’s maternal allele, while 1 indicates inheritance of the paternal allele. We call such a vector an inheritance pattern. Each value of r organizes the alleles of the persons in the pedigree into subsets of alleles that must be identical as long as we disregard mutations, which is reasonable to do for SNPs. Restricting ourselves to the typed persons, we represent such a partition as a vector of length 2T of subset indices, with each of the T pairs representing the maternal and paternal alleles of a person. We enumerate the subsets using consecutive integers starting from zero. For any two subsets, if there exists one or more persons in which alleles from exactly one of the subsets occur, consider the first person, in a fixed ordering of the persons, in which this happens, and the subset with an allele in this person. This subset will then be indexed with a lower integer compared to the other subset. Finally, for each pair of integers representing the alleles of a single person, if the first integer is larger than the second, we switch the two integers. This creates a unique code for each partitioning of unordered pairs of alleles: We call this an IBD code.
As an example, consider a nephew and his paternal uncle. We get two possible IBD codes (0, 1, 2, 3) and (0, 1, 0, 2). As the pedigree may be defined by $K=5$ relationships, we have that the corresponding r vector has $2^5=32$ possible values. Each of these values map to one of the IBD codes above. If we assign equal probability to each of the possible r vectors, the induced probabilities on the two IBD codes above are both 0.5.
Let us now write $r_i$ for the inheritance pattern at locus i. Let h(r) denote the IBD code for an $r\in \mathcal{A}$ . Given the IBD code for a locus and the observational and population models above, we may compute the probability of the observed data at the locus by conditioning on and summing over all possible combinations of alleles for the subsets indicated by the IBD code. Write $L_i(h(r_i))$ for the probability2 of the data at locus i. The functions $L_i$ are determined by our observational and population models. The complete model probability can now be written $$\begin{array}{c}\hfill \text{Pr}(\text{data}\mid \text{pedigree})=\sum _{(r_1,\cdots ,r_N)}\text{Pr}(r_1,\cdots ,r_N)\prod _{i=1}^N{L}_i(h(r_i))\end{array}$$ where we sum over all possible inheritance patterns for all loci.
It remains to specify the joint probability model for the vectors $r_1,r_2,\cdots ,r_N$ . To simplify we assume a Markov model so that each $r_{i+1}$ is independent of $r_1,\cdots ,r_{i-1}$ given $r_i$ . Specifically, we assume there is a given probability $p_i$ for an odd number of crossovers between locus i and $i+1$ independently for all relationships defining the pedigree. This yields the conditional probabilities $$\begin{array}{c}\hfill T_i(r_i,r_{i+1})\stackrel{\text{def}}{=}\text{Pr}(r_{i+1}\mid r_i)=\prod _{j=1}^K{p}_i^{I(r_{\mathit{ij}}\ne r_{i+1,j})}{(1-p_i)}^{I(r_{\mathit{ij}}=r_{i+1,j})},\end{array}$$ where we write $r_i=(r_{i1},r_{i2},\cdots ,r_{\mathit{iK}})$ and $r_{i+1}=(r_{i+1,1},r_{i+1,2},\cdots ,r_{i+1,K})$ . The Markov assumption together with a uniform probability on $r_1$ now yields a joint probability model for $r_1,r_2,\cdots ,r_N$ .

We attempted the classification of dementia status in 146 samples after removing missing values from the 177 that were used in the feature selection process. The 146 samples had a slight class imbalance, with 89 demented versus 57 non-demented patients. Before training our models, we randomly selected 57 patients from the demented group using the sample() function from the random module in Python3. Then, the rows were shuffled using sklearn.utils version 0.22.2.post1. As a result, 114 samples were utilized after balancing the class label. The 32 samples were held out for final assessment. The hippocampal tau stage feature, which had 50% missing values, was dropped during the training process. Age and brain weight were removed before training the models, ending up with 22 features and 114 samples for classification. The dataset was split into a training set of 70% (80 samples) and a testing set of 30% (34 samples).
Seven classification algorithms were trained to classify individuals’ dementia status from the 22 top-ranked features. Scikit-learn version 0.22.2.post1 was used to implement and train the ML classifiers, and then measure their classification performance. Logistic regression was implemented using the sklearn.linear_model package where penalty was set to 12, the regularization parameter C was set to 1, the maximum number of iterations taken for the solvers to converge was set to 2000, and other parameters were set to default values. A decision tree classifier was implemented using the sklearn.tree package. K-nearest neighbors classifier was implemented using the sklearn.neighbors with the number of neighbors set to 5, the function “uniform weights” used for prediction, the “Minkowski” distance metric utilized for the tree, and with other parameters were set to default values. The linear discriminant analysis classifier was implemented using the sklearn.discriminant_analysis package with singular value decomposition for solver hyperparameter and other parameters were set to default values. The Gaussian naïve Bayes classifier was implemented using sklearn.naive_bayes. The support vector machine with a radial basis function kernel (SVM-RBF) was implemented using sklearn.svm with the regularization parameter C set to 1, the kernel coefficient gamma = “scale” and other parameters were set to default values. The support vector machine with a linear kernel (SVM-LINEAR) was implemented using the sklearn.svm package with regularization parameter C set to 1, with a “linear” kernel, gamma coefficient “scale” and other parameters were set to default. The sklearn.metrics package was used to report classification performance. Training and performance evaluation were performed 500 times, from which the average performance measure was calculated as overall performance. Accuracy, balanced accuracy, F1-score, precision, sensitivity and specificity utilizing regression plots were measures used for performance. ML models and feature selection libraries were built using Python 3.7.3.

We analysed data from people with clinical diagnoses of Alzheimer’s disease (n = 18),^{28 (link)} behavioural variant of FTD (bvFTD; n = 16),^{29 (link)} CBS (n = 17),^{30 (link)} PSP (n = 36),^{31 (link)} nfvPPA (n = 26) and svPPA (n = 26),^{32 (link),33 (link)} as well as healthy controls (n = 33). Patients were recruited from specialist memory and movement disorders clinics at Cambridge University Hospitals NHS Trust in observational studies (REC references 07/Q0102/3, 10/H0308/34, 12/EE/0475, 14/LO/2045 and 16/LO/1735).
Verbal fluency tests were administered during the baseline visit. Participants were asked to name as many words as they could that (i) began with the letter ‘P’ (excluding people and place names) and (ii) that belonged to the category of ‘animals’, to assess phonemic/letter and semantic/category fluency, respectively. Words were recorded over 60 s for each task and transcribed by the examiner. Features (e.g. frequency and concreteness) were extracted for bigrams when available; otherwise, the bigram was included as a single entry. Where ratings for pluralized words were unavailable, the word properties for the singular version were extracted. In case of homonyms, when the intended word was unclear, the most frequent word was transcribed then analysed. We indicated ‘NA’ for features for which a rating was not available. Where a feature was not available for a given word (i.e. outside of the psycholinguistic databases), the word was excluded from the PCA. The total word count, excluding errors and repetitions, was calculated.
For each of the words, we obtained ratings for psycholinguistic properties from the MRC Psycholinguistic Database³⁴ and the English Lexicon Project^{35 (link)} as listed in Table 1.