Sparse Regression Incorporating Graphical Structure Among Predictors

<p>With the abundance of high-dimensional data in various disciplines, sparse regularized techniques are very popular these days. In this article, we make use of the structure information among predictors to improve sparse regression models. Typically, such structure information can be modeled by the connectivity of an undirected graph using all predictors as nodes of the graph. Most existing methods use this undirected graph edge-by-edge to encourage the regression coefficients of corresponding connected predictors to be similar. However, such methods do not directly use the neighborhood information of the graph. Furthermore, if there are more edges in the predictor graph, the corresponding regularization term will be more complicated. In this article, we incorporate the graph information node-by-node, instead of edge-by-edge as used in most existing methods. Our proposed method is very general and it includes adaptive Lasso, group Lasso, and ridge regression as special cases. Both theoretical and numerical studies demonstrate the effectiveness of the proposed method for simultaneous estimation, prediction, and model selection. Supplementary materials for this article are available online.</p

Chaotic behavior of four typical fractional-order chaotic systems.

<p>(a) Chaotic behavior of the fractional-order Chen system when a = 35, b = 3, c = 28, q₁ = 0.93, q₂ = 0.9, and q₃ = 0.88; (b) Chaotic behavior of the fractional-order Lorzen system when a = 10, b = 28, c = 8/3, q₁ = 0.993, q₂ = 0.993, and q₃ = 0.993; (c) Chaotic behavior of the fractional-order Rössler system when a = 0.5, b = 0.2, c = 10, q₁ = 0.9, q₂ = 0.85, and q₃ = 0.95; (d) Chaotic behavior of the fractional-order Lü system when a = 36, b = 3, c = 20, q₁ = 0.985, q₂ = 0.99, and q₃ = 0.98.</p

Schematic of fractional-order chaotic system parameter estimation.

<p>The parameter estimation for the fractional-order chaotic system can be considered a multidimensional continuous optimization problem, where <i>q</i> and <i>θ</i> are the decision variables.</p

Division of the solution space for <i>x_j</i>.

<p>An arbitrary <i>x_j</i> is set a value between (<i>a</i>_j, <i>b</i>_j), and it is divided into 2^m subset by quantum parallelism.</p

Flowchart of the QPPSO algorithm.

<p>The procedure of the QPPSO algorithm.</p

Convergence graph of the objective function.

<p>(a) Convergence graph of the fractional-order Chen system, (b) Convergence graph of the fractional-order Lorzen system, (c) Convergence graph of the fractional-order Rössler system, (d) Convergence graph of the fractional-order Lü system.</p

Quantum circuit to simultaneously evaluate <i>f</i>(<i>x</i>).

<p><i>U_f</i> is the quantum circuit that takes inputs, such as </p><p></p><p></p><p><mo>|</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>〉</mo></p><p></p><p></p>, to <p></p><p></p><p></p><p><mo>|</mo></p><p><mi>x</mi><mo>,</mo><mi>y</mi><mo>⊕</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></p><mo>〉</mo><p></p><p></p><p></p><p></p>.<p></p

2D-Metal–Organic-Framework-Nanozyme Sensor Arrays for Probing Phosphates and Their Enzymatic Hydrolysis

The detection of phosphates and their enzymatic hydrolysis is of great importance because of their essential roles in various biological processes and numerous diseases. Compared with individual sensors for detecting one given phosphate at a time, sensor arrays are able to discriminate multiple phosphates simultaneously. Although nanomaterial-based sensor arrays have shown great promise for the discrimination of phosphates, very few of them have been explored for probing phosphates involved enzymatic hydrolysis. To fill this gap, herein we fabricated two-dimensional-metal–organic-framework (2D-MOF)-nanozyme-based sensor arrays by modulating their peroxidase-mimicking activity with various phosphates, including AMP, ADP, ATP, pyrophosphate (PPi), and phosphate (Pi). The sensor arrays were used to successfully discriminate the five phosphates not only in aqueous solutions but also in biological samples. The practical application of the sensor arrays was then validated with blind samples, where 30 unknown samples containing phosphates were accurately identified. Moreover, the sensor arrays were successfully applied to probing hydrolytic processes involving ATP and PPi that are catalyzed by apyrase and PPase, respectively. This work demonstrates a nanozyme-based sensor array as a convenient and reliable analytical platform for probing phosphates and their related enzymatic processes, which could be applied to other analytes and enzymatic reactions

Quantum encoding of QPPSO.

<p>For an <i>n</i> dimensional space, each basis of this space can be divided into 2^<i>m</i> states with <i>m</i> gates only.</p

Tuning trajectories of the parameters of fractional-order chaotic systems by the QPPSO method.

<p>(a) Tuning trajectory of the parameters of the fractional-order Chen system, (b) Tuning trajectory of the parameters of the fractional-order Lorzen system, (c) Tuning trajectory of the parameters of the fractional-order Rössler system, (d) Tuning trajectory of the parameters of the fractional-order Lü system.</p