Description: Implements high-dimensional multivariate regression by stacked generalisation (Wolpert 1992 <doi:10.1016/S0893-6080(05)80023-1>). For positively correlated outcomes, a single multivariate regression is typically more predictive than multiple univariate regressions. Includes functions for model fitting, extracting coefficients, outcome prediction, and performance measurement.
<p>We simulate the <code>p</code> inputs from a multivariate normal distribution. For the mean, we use the <code>p</code>-dimensional vector <code>mu</code>, where all elements equal zero. For the covariance, we use the <code>p</code><spanclass="math inline">\(\times\)</span><code>p</code> matrix <code>Sigma</code>, where the entry in row <spanclass="math inline">\(i\)</span> and column <spanclass="math inline">\(j\)</span> equals <code>rho</code><spanclass="math inline">\(^{|i-j|}\)</span>. The parameter <code>rho</code> determines the strength of the correlation among the inputs, with small <code>rho</code> leading weak correlations, and large <code>rho</code> leading to strong correlations (0 <<code>rho</code>< 1). The input matrix <code>X</code> has <code>n</code> rows and <code>p</code> columns.</p>
<p>We simulate the input-output effects from independent Bernoulli distributions. The parameter <code>pi</code> determines the number of effects, with small <code>pi</code> leading to few effects, and large <code>pi</code> leading to many effects (0 <<code>pi</code>< 1). The scalar <code>alpha</code> represents the intercept, and the <code>p</code>-dimensional vector <code>beta</code> represents the slopes.</p>
<p>From the intercept <code>alpha</code>, the slopes <code>beta</code> and the inputs <code>X</code>, we calculate the linear predictor, the <code>n</code>-dimensional vector <code>eta</code>. Rescale the linear predictor to make the effects weaker or stronger. Set the argument <code>family</code> to <code>"gaussian"</code>, <code>"binomial"</code>, or <code>"poisson"</code> to define the distribution. The <code>n</code> times <code>p</code> matrix <code>Y</code> represents the outputs. We assume the outcomes are <em>positively</em> correlated.</p>