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. If required, install MRCE or remMap from GitHub (<https://github.com/cran/MRCE>, <https://github.com/cran/remMap>).
<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>
<p>The function <code>joinet</code> fits univariate and multivariate regression. Set the argument <code>alpha.base</code> to 0 (ridge) or 1 (lasso).</p>
<p>Standard methods are available. The function <code>predict</code> returns the predicted values from the univariate (<code>base</code>) and multivariate (<code>meta</code>) models. The function <code>coef</code> returns the estimated intercepts (<code>alpha</code>) and slopes (<code>beta</code>) from the multivariate model (input-output effects). And the function <code>weights</code> returns the weights from stacking (output-output effects).</p>
<p>The function <code>cv.joinet</code> compares the predictive performance of univariate (<code>base</code>) and multivariate (<code>meta</code>) regression by nested cross-validation. The argument <code>type.measure</code> determines the loss function.</p>
