Algorithms Reference Regression

Linear Regression

LR Description

Linear Regression scripts are used to model the relationship between one numerical response variable and one or more explanatory (feature) variables. The scripts are given a dataset $(X, Y) = (x_i, y_i)_{i=1}^n$ where $x_i$ is a numerical vector of feature variables and $y_i$ is a numerical response value for each training data record. The feature vectors are provided as a matrix $X$ of size $n\,{\times}\,m$, where $n$ is the number of records and $m$ is the number of features. The observed response values are provided as a 1-column matrix $Y$, with a numerical value $y_i$ for each $x_i$ in the corresponding row of matrix $X$.

In linear regression, we predict the distribution of the response $y_i$ based on a fixed linear combination of the features in $x_i$. We assume that there exist constant regression coefficients $\beta_0, \beta_1, \ldots, \beta_m$ and a constant residual variance $\sigma^2$ such that

\[\begin{equation} y_i \sim Normal(\mu_i, \sigma^2) \,\,\,\,\textrm{where}\,\,\,\, \mu_i \,=\, \beta_0 + \beta_1 x_{i,1} + \ldots + \beta_m x_{i,m} \end{equation}\]

Distribution $y_i \sim Normal(\mu_i, \sigma^2)$ models the “unexplained” residual noise and is assumed independent across different records.

The goal is to estimate the regression coefficients and the residual variance. Once they are accurately estimated, we can make predictions about $y_i$ given $x_i$ in new records. We can also use the $\beta_j$’s to analyze the influence of individual features on the response value, and assess the quality of this model by comparing residual variance in the response, left after prediction, with its total variance.

There are two scripts in our library, both doing the same estimation, but using different computational methods. Depending on the size and the sparsity of the feature matrix $X$, one or the other script may be more efficient. The “direct solve” script LinearRegDS.dml is more efficient when the number of features $m$ is relatively small ($m \sim 1000$ or less) and matrix $X$ is either tall or fairly dense (has ${\gg}:m^2$ nonzeros); otherwise, the “conjugate gradient” script LinearRegCG.dml is more efficient. If $m > 50000$, use only LinearRegCG.dml.

Table 7

Besides $\beta$, linear regression scripts compute a few summary statistics listed below. The statistics are provided in CSV format, one comma-separated name-value pair per each line.

Name Meaning
AVG_TOT_Y Average of the response value $Y$
STDEV_TOT_Y Standard Deviation of the response value $Y$
AVG_RES_Y Average of the residual $Y - \mathop{\mathrm{pred}}(Y \mid X)$, i.e. residual bias
STDEV_RES_Y Standard Deviation of the residual $Y - \mathop{\mathrm{pred}}(Y \mid X)$
DISPERSION GLM-style dispersion, i.e. residual sum of squares / #deg. fr.
R2 $R^2$ of residual with bias included vs. total average
ADJUSTED_R2 Adjusted $R^2$ of residual with bias included vs. total average
R2_NOBIAS Plain $R^2$ of residual with bias subtracted vs. total average
ADJUSTED_R2_NOBIAS Adjusted $R^2$ of residual with bias subtracted vs. total average
R2_VS_0 * $R^2$ of residual with bias included vs. zero constant
ADJUSTED_R2_VS_0 * Adjusted $R^2$ of residual with bias included vs. zero constant

* The last two statistics are only printed if there is no intercept (icpt=0)

Table 8

The Log file for the LinearRegCG.dml script contains the above iteration variables in CSV format, each line containing a triple (Name, Iteration#, Value) with Iteration# being 0 for initial values.

Name Meaning
CG_RESIDUAL_NORM L2-norm of conjug. grad. residual, which is $A$ %*% $\beta - t(X)$ %*% $y$ where $A = t(X)$ %*% $X + diag(\lambda)$, or a similar quantity
CG_RESIDUAL_RATIO Ratio of current L2-norm of conjug. grad. residual over the initial

LR Details

To solve a linear regression problem over feature matrix $X$ and response vector $Y$, we can find coefficients $\beta_0, \beta_1, \ldots, \beta_m$ and $\sigma^2$ that maximize the joint likelihood of all $y_i$ for $i=1\ldots n$, defined by the assumed statistical model (1). Since the joint likelihood of the independent $y_i \sim Normal(\mu_i, \sigma^2)$ is proportional to the product of $\exp\big({-}\,(y_i - \mu_i)^2 / (2\sigma^2)\big)$, we can take the logarithm of this product, then multiply by $-2\sigma^2 < 0$ to obtain a least squares problem:

\[\begin{equation} \sum_{i=1}^n \, (y_i - \mu_i)^2 \,\,=\,\, \sum_{i=1}^n \Big(y_i - \beta_0 - \sum_{j=1}^m \beta_j x_{i,j}\Big)^2 \,\,\to\,\,\min \end{equation}\]

This may not be enough, however. The minimum may sometimes be attained over infinitely many $\beta$-vectors, for example if $X$ has an all-0 column, or has linearly dependent columns, or has fewer rows than columns . Even if (2) has a unique solution, other $\beta$-vectors may be just a little suboptimal1, yet give significantly different predictions for new feature vectors. This results in overfitting: prediction error for the training data ($X$ and $Y$) is much smaller than for the test data (new records).

Overfitting and degeneracy in the data is commonly mitigated by adding a regularization penalty term to the least squares function:

\[\begin{equation} \sum_{i=1}^n \Big(y_i - \beta_0 - \sum_{j=1}^m \beta_j x_{i,j}\Big)^2 \,+\,\, \lambda \sum_{j=1}^m \beta_j^2 \,\,\to\,\,\min \end{equation}\]

The choice of $\lambda>0$, the regularization constant, typically involves cross-validation where the dataset is repeatedly split into a training part (to estimate the $\beta_j$’s) and a test part (to evaluate prediction accuracy), with the goal of maximizing the test accuracy. In our scripts, $\lambda$ is provided as input parameter reg.

The solution to the least squares problem (3), through taking the derivative and setting it to 0, has the matrix linear equation form

\[\begin{equation} A\left[\textstyle\beta_{1:m}\atop\textstyle\beta_0\right] \,=\, \big[X,\,1\big]^T Y,\,\,\, \textrm{where}\,\,\, A \,=\, \big[X,\,1\big]^T \big[X,\,1\big]\,+\,\hspace{0.5pt} diag(\hspace{0.5pt} \underbrace{\lambda,\ldots, \lambda}_{\scriptstyle m}, 0) \end{equation}\]

where $[X,\,1]$ is $X$ with an extra column of 1s appended on the right, and the diagonal matrix of $\lambda$’s has a zero to keep the intercept $\beta_0$ unregularized. If the intercept is disabled by setting $icpt=0$, the equation is simply $X^T X \beta = X^T Y$.

We implemented two scripts for solving equation (4): one is a “direct solver” that computes $A$ and then solves $A\beta = [X,\,1]^T Y$ by calling an external package, the other performs linear conjugate gradient (CG) iterations without ever materializing $A$. The CG algorithm closely follows Algorithm 5.2 in Chapter 5 of [Nocedal2006]. Each step in the CG algorithm computes a matrix-vector multiplication $q = Ap$ by first computing $[X,\,1]\, p$ and then $[X,\,1]^T [X,\,1]\, p$. Usually the number of such multiplications, one per CG iteration, is much smaller than $m$. The user can put a hard bound on it with input parameter maxi, or use the default maximum of $m+1$ (or $m$ if no intercept) by having maxi=0. The CG iterations terminate when the L2-norm of vector $r = A\beta - [X,\,1]^T Y$ decreases from its initial value (for $\beta=0$) by the tolerance factor specified in input parameter tol.

The CG algorithm is more efficient if computing $[X,\,1]^T \big([X,\,1]\, p\big)$ is much faster than materializing $A$, an $(m\,{+}\,1)\times(m\,{+}\,1)$ matrix. The Direct Solver (DS) is more efficient if $X$ takes up a lot more memory than $A$ (i.e. $X$ has a lot more nonzeros than $m^2$) and if $m^2$ is small enough for the external solver ($m \lesssim 50000$). A more precise determination between CG and DS is subject to further research.

In addition to the $\beta$-vector, the scripts estimate the residual standard deviation $\sigma$ and the $R^2$, the ratio of “explained” variance to the total variance of the response variable. These statistics only make sense if the number of degrees of freedom $n\,{-}\,m\,{-}\,1$ is positive and the regularization constant $\lambda$ is negligible or zero. The formulas for $\sigma$ and $R^2$ are:

\[R^2 = 1 - \frac{\mathrm{RSS}}{\mathrm{TSS}},\quad \sigma \,=\, \sqrt{\frac{\mathrm{RSS}}{n - m - 1}},\quad R^2_{\textrm{adj.}} = 1 - \frac{\sigma^2 (n-1)}{\mathrm{TSS}}\]

where

\[\mathrm{RSS} \,=\, \sum_{i=1}^n \Big(y_i - \hat{\mu}_i - \frac{1}{n} \sum_{i'=1}^n \,(y_{i'} - \hat{\mu}_{i'})\Big)^2; \quad \mathrm{TSS} \,=\, \sum_{i=1}^n \Big(y_i - \frac{1}{n} \sum_{i'=1}^n y_{i'}\Big)^2\]

Here $\hat{\mu}_i$ are the predicted means for $y_i$ based on the estimated regression coefficients and the feature vectors. They may be biased when no intercept is present, hence the RSS formula subtracts the bias.

Lastly, note that by choosing the input option icpt=2 the user can shift and rescale the columns of $X$ to have zero average and the variance of 1. This is particularly important when using regularization over highly disbalanced features, because regularization tends to penalize small-variance columns (which need large $\beta_j$’s) more than large-variance columns (with small $\beta_j$’s). At the end, the estimated regression coefficients are shifted and rescaled to apply to the original features.

LR Returns

The estimated regression coefficients (the $\hat{\beta}_j$’s) are populated into a matrix and written to an HDFS file whose path/name was provided as the B input argument. What this matrix contains, and its size, depends on the input argument icpt, which specifies the user’s intercept and rescaling choice:

The estimated summary statistics, including residual standard deviation $\sigma$ and the $R^2$, are printed out or sent into a file (if specified) in CSV format as defined in Table 7. For conjugate gradient iterations, a log file with monitoring variables can also be made available, see Table 8.

Stepwise Linear Regression

SLR Description

Our stepwise linear regression script selects a linear model based on the Akaike information criterion (AIC): the model that gives rise to the lowest AIC is computed.

SLR Details

Stepwise linear regression iteratively selects predictive variables in an automated procedure. Currently, our implementation supports forward selection: starting from an empty model (without any variable) the algorithm examines the addition of each variable based on the AIC as a model comparison criterion. The AIC is defined as

\[\begin{equation} AIC = -2 \log{L} + 2 edf,\label{eq:AIC} \end{equation}\]

where $L$ denotes the likelihood of the fitted model and $edf$ is the equivalent degrees of freedom, i.e., the number of estimated parameters. This procedure is repeated until including no additional variable improves the model by a certain threshold specified in the input parameter thr.

For fitting a model in each iteration we use the direct solve method as in the script LinearRegDS.dml discussed in Linear Regression.

SLR Returns

Similar to the outputs from LinearRegDS.dml the stepwise linear regression script computes the estimated regression coefficients and stores them in matrix $B$ on HDFS. The format of matrix $B$ is identical to the one produced by the scripts for linear regression (see Linear Regression). Additionally, StepLinearRegDS.dml outputs the variable indices (stored in the 1-column matrix $S$) in the order they have been selected by the algorithm, i.e., $i$th entry in matrix $S$ corresponds to the variable which improves the AIC the most in $i$th iteration. If the model with the lowest AIC includes no variables matrix $S$ will be empty (contains one 0). Moreover, the estimated summary statistics as defined in Table 7 are printed out or stored in a file (if requested). In the case where an empty model achieves the best AIC these statistics will not be produced.

Generalized Linear Models

GLM Description

Generalized Linear Models [Gill2000, McCullagh1989, Nelder1972] extend the methodology of linear and logistic regression to a variety of distributions commonly assumed as noise effects in the response variable. As before, we are given a collection of records $(x_1, y_1)$, …, $(x_n, y_n)$ where $x_i$ is a numerical vector of explanatory (feature) variables of size $\dim x_i = m$, and $y_i$ is the response (dependent) variable observed for this vector. GLMs assume that some linear combination of the features in $x_i$ determines the mean $\mu_i$ of $y_i$, while the observed $y_i$ is a random outcome of a noise distribution $Prob[y\mid \mu_i]\,$2 with that mean $\mu_i$:

\[x_i \,\,\,\,\mapsto\,\,\,\, \eta_i = \beta_0 + \sum\nolimits_{j=1}^m \beta_j x_{i,j} \,\,\,\,\mapsto\,\,\,\, \mu_i \,\,\,\,\mapsto \,\,\,\, y_i \sim Prob[y\mid \mu_i]\]

In linear regression the response mean $\mu_i$ equals some linear combination over $x_i$, denoted above by $\eta_i$. In logistic regression with \(y\in\{0, 1\}\) (Bernoulli) the mean of $y$ is the same as $Prob[y=1]$ and equals $1/(1+e^{-\eta_i})$, the logistic function of $\eta_i$. In GLM, $\mu_i$ and $\eta_i$ can be related via any given smooth monotone function called the link function: $\eta_i = g(\mu_i)$. The unknown linear combination parameters $\beta_j$ are assumed to be the same for all records.

The goal of the regression is to estimate the parameters $\beta_j$ from the observed data. Once the $\beta_j$’s are accurately estimated, we can make predictions about $y$ for a new feature vector $x$. To do so, compute $\eta$ from $x$ and use the inverted link function $\mu = g^{-1}(\eta)$ to compute the mean $\mu$ of $y$; then use the distribution $Prob[y\mid \mu]$ to make predictions about $y$. Both $g(\mu)$ and $Prob[y\mid \mu]$ are user-provided. Our GLM script supports a standard set of distributions and link functions, see below for details.

Table 9

Besides $\beta$, GLM regression script computes a few summary statistics listed below. They are provided in CSV format, one comma-separated name-value pair per each line.

Name Meaning
TERMINATION_CODE A positive integer indicating success/failure as follows: 1 = Converged successfully; 2 = Maximum # of iterations reached; 3 = Input (X, Y) out of range; 4 = Distribution/link not supported
BETA_MIN Smallest beta value (regression coefficient), excluding the intercept
BETA_MIN_INDEX Column index for the smallest beta value
BETA_MAX Largest beta value (regression coefficient), excluding the intercept
BETA_MAX_INDEX Column index for the largest beta value
INTERCEPT Intercept value, or NaN if there is no intercept (if icpt=0)
DISPERSION Dispersion used to scale deviance, provided in disp input argument or estimated (same as DISPERSION_EST) if disp argument is $\leq 0$
DISPERSION_EST Dispersion estimated from the dataset
DEVIANCE_UNSCALED Deviance from the saturated model, assuming dispersion $= 1.0$
DEVIANCE_SCALED Deviance from the saturated model, scaled by DISPERSION value

Table 10

The Log file for GLM regression contains the below iteration variables in CSV format, each line containing a triple (Name, Iteration#, Value) with Iteration# being 0 for initial values.

Name Meaning
NUM_CG_ITERS Number of inner (Conj. Gradient) iterations in this outer iteration
IS_TRUST_REACHED 1 = trust region boundary was reached, 0 = otherwise
POINT_STEP_NORM L2-norm of iteration step from old point ($\beta$-vector) to new point
OBJECTIVE The loss function we minimize (negative partial log-likelihood)
OBJ_DROP_REAL Reduction in the objective during this iteration, actual value
OBJ_DROP_PRED Reduction in the objective predicted by a quadratic approximation
OBJ_DROP_RATIO Actual-to-predicted reduction ratio, used to update the trust region
GRADIENT_NORM L2-norm of the loss function gradient (omitted if point is rejected)
LINEAR_TERM_MIN The minimum value of $X$ %*% $\beta$, used to check for overflows
LINEAR_TERM_MAX The maximum value of $X$ %*% $\beta$, used to check for overflows
IS_POINT_UPDATED 1 = new point accepted; 0 = new point rejected, old point restored
TRUST_DELTA Updated trust region size, the “delta”

Table 11

Common GLM distribution families and link functions. (Here “*” stands for “any value.”)

dfam vpow link lpow Distribution Family Link Function Canonical
1 0.0 1 -1.0 Gaussian inverse  
1 0.0 1 0.0 Gaussian log  
1 0.0 1 1.0 Gaussian identity Yes
1 1.0 1 0.0 Poisson log Yes
1 1.0 1 0.5 Poisson sq.root  
1 1.0 1 1.0 Poisson identity  
1 2.0 1 -1.0 Gamma inverse Yes
1 2.0 1 0.0 Gamma log  
1 2.0 1 1.0 Gamma identity  
1 3.0 1 -2.0 Inverse Gauss $1/\mu^2$ Yes
1 3.0 1 -1.0 Inverse Gauss inverse  
1 3.0 1 0.0 Inverse Gauss log  
1 3.0 1 1.0 Inverse Gauss identity  
2 * 1 0.0 Binomial log  
2 * 1 0.5 Binomial sq.root  
2 * 2 * Binomial logit Yes
2 * 3 * Binomial probit  
2 * 4 * Binomial cloglog  
2 * 5 * Binomial cauchit  

Table 12

The supported non-power link functions for the Bernoulli and the binomial distributions. Here $\mu$ is the Bernoulli mean.

Name Link Function
Logit $\displaystyle \eta = 1 / \big(1 + e^{-\mu}\big)^{\mathstrut}$
Probit \(\displaystyle \mu = \frac{1}{\sqrt{2\pi}}\int\nolimits_{-\infty_{\mathstrut}}^{\,\eta\mathstrut} e^{-\frac{t^2}{2}} dt\)
Cloglog $\displaystyle \eta = \log \big(- \log(1 - \mu)\big)^{\mathstrut}$
Cauchit $\displaystyle \eta = \tan\pi(\mu - 1/2)$

GLM Details

In GLM, the noise distribution $Prob[y\mid \mu]$ of the response variable $y$ given its mean $\mu$ is restricted to have the exponential family form

\[\begin{equation} Y \sim\, Prob[y\mid \mu] \,=\, \exp\left(\frac{y\theta - b(\theta)}{a} + c(y, a)\right),\,\,\textrm{where}\,\,\,\mu = E(Y) = b'(\theta). \end{equation}\]

Changing the mean in such a distribution simply multiplies all $Prob[y\mid \mu]$ by $e^{\,y\hspace{0.2pt}\theta/a}$ and rescales them so that they again integrate to 1. Parameter $\theta$ is called canonical, and the function $\theta = b’^{\,-1}(\mu)$ that relates it to the mean is called the canonical link; constant $a$ is called dispersion and rescales the variance of $y$. Many common distributions can be put into this form, see Table 11. The canonical parameter $\theta$ is often chosen to coincide with $\eta$, the linear combination of the regression features; other choices for $\eta$ are possible too.

Rather than specifying the canonical link, GLM distributions are commonly defined by their variance $Var(y)$ as the function of the mean $\mu$. It can be shown from Eq. 5 that $Var(y) = a\,b’’(\theta) = a\,b’‘(b’^{\,-1}(\mu))$. For example, for the Bernoulli distribution $Var(y) = \mu(1-\mu)$, for the Poisson distribution $Var(y) = \mu$, and for the Gaussian distribution $Var(y) = a\cdot 1 = \sigma^2$. It turns out that for many common distributions $Var(y) = a\mu^q$, a power function. We support all distributions where $Var(y) = a\mu^q$, as well as the Bernoulli and the binomial distributions.

For distributions with $Var(y) = a\mu^q$ the canonical link is also a power function, namely $\theta = \mu^{1-q}/(1-q)$, except for the Poisson ($q = 1$) whose canonical link is $\theta = \log\mu$. We support all power link functions in the form $\eta = \mu^s$, dropping any constant factor, with $\eta = \log\mu$ for $s=0$. The binomial distribution has its own family of link functions, which includes logit (the canonical link), probit, cloglog, and cauchit (see Table 12); we support these only for the binomial and Bernoulli distributions. Links and distributions are specified via four input parameters: dfam, vpow, link, and lpow (see Table 11).

The observed response values are provided to the regression script as a matrix $Y$ having 1 or 2 columns. If a power distribution family is selected (dfam=1), matrix $Y$ must have 1 column that provides $y_i$ for each $x_i$ in the corresponding row of matrix $X$. When dfam=2 and $Y$ has 1 column, we assume the Bernoulli distribution for \(y_i\in\{y_{\mathrm{neg}}, 1\}\) with $y_{\mathrm{neg}}$ from the input parameter yneg. When dfam=2 and $Y$ has 2 columns, we assume the binomial distribution; for each row $i$ in $X$, cells $Y[i, 1]$ and $Y[i, 2]$ provide the positive and the negative binomial counts respectively. Internally we convert the 1-column Bernoulli into the 2-column binomial with 0-versus-1 counts.

We estimate the regression parameters via L2-regularized negative log-likelihood minimization:

\[f(\beta; X, Y) \,\,=\,\, -\sum\nolimits_{i=1}^n \big(y_i\theta_i - b(\theta_i)\big) \,+\,(\lambda/2) \sum\nolimits_{j=1}^m \beta_j^2\,\,\to\,\,\min\]

where $\theta_i$ and $b(\theta_i)$ are from (6); note that $a$ and $c(y, a)$ are constant w.r.t. $\beta$ and can be ignored here. The canonical parameter $\theta_i$ depends on both $\beta$ and $x_i$:

\[\theta_i \,\,=\,\, b'^{\,-1}(\mu_i) \,\,=\,\, b'^{\,-1}\big(g^{-1}(\eta_i)\big) \,\,=\,\, \big(b'^{\,-1}\circ g^{-1}\big)\left(\beta_0 + \sum\nolimits_{j=1}^m \beta_j x_{i,j}\right)\]

The user-provided (via reg) regularization coefficient $\lambda\geq 0$ can be used to mitigate overfitting and degeneracy in the data. Note that the intercept is never regularized.

Our iterative minimizer for $f(\beta; X, Y)$ uses the Fisher scoring approximation to the difference $\varDelta f(z; \beta) = f(\beta + z; X, Y) \,-\, f(\beta; X, Y)$, recomputed at each iteration:

\[\begin{gathered} \varDelta f(z; \beta) \,\,\,\approx\,\,\, 1/2 \cdot z^T A z \,+\, G^T z, \,\,\,\,\textrm{where}\,\,\,\, A \,=\, X^T\!diag(w) X \,+\, \lambda I\\ \textrm{and}\,\,\,\,G \,=\, - X^T u \,+\, \lambda\beta, \,\,\,\textrm{with $n\,{\times}\,1$ vectors $w$ and $u$ given by}\\ \forall\,i = 1\ldots n: \,\,\,\, w_i = \big[v(\mu_i)\,g'(\mu_i)^2\big]^{-1} \!\!\!\!\!\!,\,\,\,\,\,\,\,\,\, u_i = (y_i - \mu_i)\big[v(\mu_i)\,g'(\mu_i)\big]^{-1} \!\!\!\!\!\!.\,\,\,\,\end{gathered}\]

Here $v(\mu_i)=Var(y_i)/a$, the variance of $y_i$ as the function of the mean, and $g’(\mu_i) = d \eta_i/d \mu_i$ is the link function derivative. The Fisher scoring approximation is minimized by trust-region conjugate gradient iterations (called the inner iterations, with the Fisher scoring iterations as the outer iterations), which approximately solve the following problem:

\[1/2 \cdot z^T A z \,+\, G^T z \,\,\to\,\,\min\,\,\,\,\textrm{subject to}\,\,\,\, \|z\|_2 \leq \delta\]

The conjugate gradient algorithm closely follows Algorithm 7.2 on page 171 of [Nocedal2006]. The trust region size $\delta$ is initialized as $0.5\sqrt{m}\,/ \max\nolimits_i |x_i|_2$ and updated as described in [Nocedal2006]. The user can specify the maximum number of the outer and the inner iterations with input parameters moi and mii, respectively. The Fisher scoring algorithm terminates successfully if $2|\varDelta f(z; \beta)| < (D_1(\beta) + 0.1)\hspace{0.5pt}{\varepsilon}$ where ${\varepsilon}> 0$ is a tolerance supplied by the user via tol, and $D_1(\beta)$ is the unit-dispersion deviance estimated as

\[D_1(\beta) \,\,=\,\, 2 \cdot \big(Prob[Y \mid \! \begin{smallmatrix}\textrm{saturated}\\\textrm{model}\end{smallmatrix}, a\,{=}\,1] \,\,-\,\,Prob[Y \mid X, \beta, a\,{=}\,1]\,\big)\]

The deviance estimate is also produced as part of the output. Once the Fisher scoring algorithm terminates, if requested by the user, we estimate the dispersion $a$ from (6) using Pearson residuals

\[\begin{equation} \hat{a} \,\,=\,\, \frac{1}{n-m}\cdot \sum_{i=1}^n \frac{(y_i - \mu_i)^2}{v(\mu_i)} \end{equation}\]

and use it to adjust our deviance estimate: $D_{\hat{a}}(\beta) = D_1(\beta)/\hat{a}$. If input argument disp is 0.0 we estimate $\hat{a}$, otherwise we use its value as $a$. Note that in (7) $m$ counts the intercept ($m \leftarrow m+1$) if it is present.

GLM Returns

The estimated regression parameters (the $\hat{\beta}_j$’s) are populated into a matrix and written to an HDFS file whose path/name was provided as the B input argument. What this matrix contains, and its size, depends on the input argument icpt, which specifies the user’s intercept and rescaling choice:

Our script also estimates the dispersion $\hat{a}$ (or takes it from the user’s input) and the deviances $D_1(\hat{\beta})$ and $D_{\hat{a}}(\hat{\beta})$, see Table 9 for details. A log file with variables monitoring progress through the iterations can also be made available, see Table 10.

GLM See Also

In case of binary classification problems, consider using L2-SVM or binary logistic regression; for multiclass classification, use multiclass SVM or multinomial logistic regression. For the special cases of linear regression and logistic regression, it may be more efficient to use the corresponding specialized scripts instead of GLM.

Stepwise Generalized Linear Regression

SGLR Description

Our stepwise generalized linear regression script selects a model based on the Akaike information criterion (AIC): the model that gives rise to the lowest AIC is provided. Note that currently only the Bernoulli distribution family is supported (see below for details).

SGLR Details

Similar to StepLinearRegDS.dml our stepwise GLM script builds a model by iteratively selecting predictive variables using a forward selection strategy based on the AIC (5). Note that currently only the Bernoulli distribution family (fam=2 in Table 11) together with the following link functions are supported: log, logit, probit, and cloglog (link \(\in\{1,2,3,4\}\) in Table 11).

SGLR Returns

Similar to the outputs from GLM.dml the stepwise GLM script computes the estimated regression coefficients and stores them in matrix $B$ on HDFS; matrix $B$ follows the same format as the one produced by GLM.dml (see Generalized Linear Models). Additionally, StepGLM.dml outputs the variable indices (stored in the 1-column matrix $S$) in the order they have been selected by the algorithm, i.e., $i$th entry in matrix $S$ stores the variable which improves the AIC the most in $i$th iteration. If the model with the lowest AIC includes no variables matrix $S$ will be empty. Moreover, the estimated summary statistics as defined in Table 9 are printed out or stored in a file on HDFS (if requested); these statistics will be provided only if the selected model is nonempty, i.e., contains at least one variable.

Regression Scoring & Prediction

Reg Scoring & Predict Description

Script GLM-predict.dml is intended to cover all linear model based regressions, including linear regression, binomial and multinomial logistic regression, and GLM regressions (Poisson, gamma, binomial with probit link etc.). Having just one scoring script for all these regressions simplifies maintenance and enhancement while ensuring compatible interpretations for output statistics.

The script performs two functions, prediction and scoring. To perform prediction, the script takes two matrix inputs: a collection of records $X$ (without the response attribute) and the estimated regression parameters $B$, also known as $\beta$. To perform scoring, in addition to $X$ and $B$, the script takes the matrix of actual response values $Y$ that are compared to the predictions made with $X$ and $B$. Of course there are other, non-matrix, input arguments that specify the model and the output format, see below for the full list.

We assume that our test/scoring dataset is given by $n\,{\times}\,m$-matrix $X$ of numerical feature vectors, where each row $x_i$ represents one feature vector of one record; we have $\dim x_i = m$. Each record also includes the response variable $y_i$ that may be numerical, single-label categorical, or multi-label categorical. A single-label categorical $y_i$ is an integer category label, one label per record; a multi-label $y_i$ is a vector of integer counts, one count for each possible label, which represents multiple single-label events (observations) for the same $x_i$. Internally we convert single-label categoricals into multi-label categoricals by replacing each label $l$ with an indicator vector $(0,\ldots,0,1_l,0,\ldots,0)$. In prediction-only tasks the actual $y_i$’s are not needed to the script, but they are needed for scoring.

To perform prediction, the script matrix-multiplies $X$ and $B$, adding the intercept if available, then applies the inverse of the model’s link function. All GLMs assume that the linear combination of the features in $x_i$ and the betas in $B$ determines the means $\mu_i$ of the $y_i$’s (in numerical or multi-label categorical form) with $\dim \mu_i = \dim y_i$. The observed $y_i$ is assumed to follow a specified GLM family distribution $Prob[y\mid \mu_i]$ with mean(s) $\mu_i$:

\[x_i \,\,\,\,\mapsto\,\,\,\, \eta_i = \beta_0 + \sum\nolimits_{j=1}^m \beta_j x_{i,j} \,\,\,\,\mapsto\,\,\,\, \mu_i \,\,\,\,\mapsto \,\,\,\, y_i \sim Prob[y\mid \mu_i]\]

If $y_i$ is numerical, the predicted mean $\mu_i$ is a real number. Then our script’s output matrix $M$ is the $n\,{\times}\,1$-vector of these means $\mu_i$. Note that $\mu_i$ predicts the mean of $y_i$, not the actual $y_i$. For example, in Poisson distribution, the mean is usually fractional, but the actual $y_i$ is always integer.

If $y_i$ is categorical, i.e. a vector of label counts for record $i$, then $\mu_i$ is a vector of non-negative real numbers, one number \(\mu_{i,l}\) per each label $l$. In this case we divide the \(\mu_{i,l}\) by their sum $\sum_l \mu_{i,l}$ to obtain predicted label probabilities \(p_{i,l}\in [0, 1]\). The output matrix $M$ is the $n \times (k\,{+}\,1)$-matrix of these probabilities, where $n$ is the number of records and $k\,{+}\,1$ is the number of categories3. Note again that we do not predict the labels themselves, nor their actual counts per record, but we predict the labels’ probabilities.

Going from predicted probabilities to predicted labels, in the single-label categorical case, requires extra information such as the cost of false positive versus false negative errors. For example, if there are 5 categories and we accurately predicted their probabilities as $(0.1, 0.3, 0.15, 0.2, 0.25)$, just picking the highest-probability label would be wrong 70% of the time, whereas picking the lowest-probability label might be right if, say, it represents a diagnosis of cancer or another rare and serious outcome. Hence, we keep this step outside the scope of GLM-predict.dml for now.

Table 13

The goodness-of-fit statistics are provided in CSV format, one per each line, with four columns: (Name, CID, Disp?, Value). The columns are: “Name” is the string identifier for the statistic; “CID” is an optional integer value that specifies the $Y$-column index for per-column statistics (note that a bi/multinomial one-column Y-input is converted into multi-column); “Disp?” is an optional Boolean value ($TRUE$ or $FALSE$) that tells us whether or not scaling by the input dispersion parameter disp has been applied to this statistic; “Value” is the value of the statistic.

Name CID Disp? Meaning
LOGLHOOD_Z   + Log-likelihood $Z$-score (in st. dev.’s from the mean)
LOGLHOOD_Z_PVAL   + Log-likelihood $Z$-score p-value, two-sided
PEARSON_X2   + Pearson residual $X^2$-statistic
PEARSON_X2_BY_DF   + Pearson $X^2$ divided by degrees of freedom
PEARSON_X2_PVAL   + Pearson $X^2$ p-value
DEVIANCE_G2   + Deviance from the saturated model $G^2$-statistic
DEVIANCE_G2_BY_DF   + Deviance $G^2$ divided by degrees of freedom
DEVIANCE_G2_PVAL   + Deviance $G^2$ p-value
AVG_TOT_Y +   $Y$-column average for an individual response value
STDEV_TOT_Y +   $Y$-column st. dev. for an individual response value
AVG_RES_Y +   $Y$-column residual average of $Y - pred. mean(Y|X)$
STDEV_RES_Y +   $Y$-column residual st. dev. of $Y - pred. mean(Y|X)$
PRED_STDEV_RES + + Model-predicted $Y$-column residual st. deviation
R2 +   $R^2$ of $Y$-column residual with bias included
ADJUSTED_R2 +   Adjusted $R^2$ of $Y$-column residual w. bias included
R2_NOBIAS +   $R^2$ of $Y$-column residual, bias subtracted
ADJUSTED_R2_NOBIAS +   Adjusted $R^2$ of $Y$-column residual, bias subtracted

Reg Scoring & Predict Details

The output matrix $M$ of predicted means (or probabilities) is computed by matrix-multiplying $X$ with the first column of $B$ or with the whole $B$ in the multinomial case, adding the intercept if available (conceptually, appending an extra column of ones to $X$); then applying the inverse of the model’s link function. The difference between “means” and “probabilities” in the categorical case becomes significant when there are ${\geq}\,2$ observations per record (with the multi-label records) or when the labels such as $-1$ and $1$ are viewed and averaged as numerical response values (with the single-label records). To avoid any or information loss, we separately return the predicted probability of each category label for each record.

When the “actual” response values $Y$ are available, the summary statistics are computed and written out as described in Table 13. Below we discuss each of these statistics in detail. Note that in the categorical case (binomial and multinomial) $Y$ is internally represented as the matrix of observation counts for each label in each record, rather than just the label ID for each record. The input $Y$ may already be a matrix of counts, in which case it is used as-is. But if $Y$ is given as a vector of response labels, each response label is converted into an indicator vector $(0,\ldots,0,1_l,0,\ldots,0)$ where $l$ is the label ID for this record. All negative (e.g. $-1$) or zero label IDs are converted to the $1 +$ maximum label ID. The largest label ID is viewed as the “baseline” as explained in the section on Multinomial Logistic Regression. We assume that there are $k\geq 1$ non-baseline categories and one (last) baseline category.

We also estimate residual variances for each response value, although we do not output them, but use them only inside the summary statistics, scaled and unscaled by the input dispersion parameter disp, as described below.

LOGLHOOD_Z and LOGLHOOD_Z_PVAL statistics measure how far the log-likelihood of $Y$ deviates from its expected value according to the model. The script implements them only for the binomial and the multinomial distributions, returning NaN for all other distributions. Pearson’s $X^2$ and deviance $G^2$ often perform poorly with bi- and multinomial distributions due to low cell counts, hence we need this extra goodness-of-fit measure. To compute these statistics, we use:

We start by computing the multinomial log-likelihood of $Y$ given $P$ and $N$, as well as the expected log-likelihood given a random $Y$ and the variance of this log-likelihood if $Y$ indeed follows the proposed distribution:

\[\begin{aligned} \ell (Y) \,\,&=\,\, \log Prob[Y \,|\, P, N] \,\,=\,\, \sum_{i=1}^{n} \,\sum_{j=1}^{k+1} \,y_{i,j}\log p_{i,j} \\ E_Y \ell (Y) \,\,&=\,\, \sum_{i=1}^{n}\, \sum_{j=1}^{k+1} \,\mu_{i,j} \log p_{i,j} \,\,=\,\, \sum_{i=1}^{n}\, N_i \,\sum_{j=1}^{k+1} \,p_{i,j} \log p_{i,j} \\ Var_Y \ell (Y) \,&=\, \sum_{i=1}^{n} \,N_i \left(\sum_{j=1}^{k+1} \,p_{i,j} \big(\log p_{i,j}\big)^2 - \Bigg( \sum_{j=1}^{k+1} \,p_{i,j} \log p_{i,j}\Bigg) ^ {\!\!2\,} \right) \end{aligned}\]

Then we compute the $Z$-score as the difference between the actual and the expected log-likelihood $\ell(Y)$ divided by its expected standard deviation, and its two-sided p-value in the Normal distribution assumption ($\ell(Y)$ should approach normality due to the Central Limit Theorem):

\[Z \,=\, \frac {\ell(Y) - E_Y \ell(Y)}{\sqrt{Var_Y \ell(Y)}};\quad \mathop{\textrm{p-value}}(Z) \,=\, Prob\Big[\,\big|\mathop{\textrm{Normal}}(0,1)\big| \, > \, |Z|\,\Big]\]

A low p-value would indicate “underfitting” if $Z\ll 0$ or “overfitting” if $Z\gg 0$. Here “overfitting” means that higher-probability labels occur more often than their probabilities suggest.

We also apply the dispersion input (disp) to compute the “scaled” version of the $Z$-score and its p-value. Since $\ell(Y)$ is a linear function of $Y$, multiplying the GLM-predicted variance of $Y$ by disp results in multiplying $Var_Y \ell(Y)$ by the same disp. This, in turn, translates into dividing the $Z$-score by the square root of the dispersion:

\[Z_{\texttt{disp}} \,=\, \big(\ell(Y) \,-\, E_Y \ell(Y)\big) \,\big/\, \sqrt{\texttt{disp}\cdot Var_Y \ell(Y)} \,=\, Z / \sqrt{\texttt{disp}}\]

Finally, we recalculate the p-value with this new $Z$-score.

PEARSON_X2, PEARSON_X2_BY_DF, and PEARSON_X2_PVAL: Pearson’s residual $X^2$-statistic is a commonly used goodness-of-fit measure for linear models [McCullagh1989]. The idea is to measure how well the model-predicted means and variances match the actual behavior of response values. For each record $i$, we estimate the mean $\mu_i$ and the variance $v_i$ (or disp $\cdot v_i$) and use them to normalize the residual: $r_i = (y_i - \mu_i) / \sqrt{v_i}$. These normalized residuals are then squared, aggregated by summation, and tested against an appropriate $\chi^2$ distribution. The computation of $X^2$ is slightly different for categorical data (bi- and multinomial) than it is for numerical data, since $y_i$ has multiple correlated dimensions [McCullagh1989]:

\[X^2\,\textrm{(numer.)} \,=\, \sum_{i=1}^{n}\, \frac{(y_i - \mu_i)^2}{v_i};\quad X^2\,\textrm{(categ.)} \,=\, \sum_{i=1}^{n}\, \sum_{j=1}^{k+1} \,\frac{(y_{i,j} - N_i \hspace{0.5pt} p_{i,j})^2}{N_i \hspace{0.5pt} p_{i,j}}\]

The number of degrees of freedom #d.f. for the $\chi^2$ distribution is $n - m$ for numerical data and $(n - m)k$ for categorical data, where $k = \mathop{\texttt{ncol}}(Y) - 1$. Given the dispersion parameter disp the $X^2$ statistic is scaled by division: \(X^2_{\texttt{disp}} = X^2 / \texttt{disp}\). If the dispersion is accurate, $X^2 / \texttt{disp}$ should be close to #d.f. In fact, $X^2 / \textrm{#d.f.}$ over the training data is the dispersion estimator used in our GLM.dml script, see (7). Here we provide $X^2 / \textrm{#d.f.}$ and $X^2_{\texttt{disp}} / \textrm{#d.f.}$ as PEARSON_X2_BY_DF to enable dispersion comparison between the training data and the test data.

NOTE: For categorical data, both Pearson’s $X^2$ and the deviance $G^2$ are unreliable (i.e. do not approach the $\chi^2$ distribution) unless the predicted means of multi-label counts \(\mu_{i,j} = N_i \hspace{0.5pt} p_{i,j}\) are fairly large: all ${\geq}\,1$ and 80% are at least $5$ [Cochran1954]. They should not be used for “one label per record” categoricals.

DEVIANCE_G2, DEVIANCE_G2_BY_DF, and DEVIANCE_G2_PVAL: Deviance $G^2$ is the log of the likelihood ratio between the “saturated” model and the linear model being tested for the given dataset, multiplied by two:

\[\begin{equation} G^2 \,=\, 2 \,\log \frac{Prob[Y \mid \textrm{saturated model}\hspace{0.5pt}]}{Prob[Y \mid \textrm{tested linear model}\hspace{0.5pt}]} \end{equation}\]

The “saturated” model sets the mean $\mu_i^{\mathrm{sat}}$ to equal $y_i$ for every record (for categorical data, \(p_{i,j}^{sat} = y_{i,j} / N_i\)), which represents the “perfect fit.” For records with $y_{i,j} \in {0, N_i}$ or otherwise at a boundary, by continuity we set $0 \log 0 = 0$. The GLM likelihood functions defined in (6) become simplified in ratio (8) due to canceling out the term $c(y, a)$ since it is the same in both models.

The log of a likelihood ratio between two nested models, times two, is known to approach a $\chi^2$ distribution as $n\to\infty$ if both models have fixed parameter spaces. But this is not the case for the “saturated” model: it adds more parameters with each record. In practice, however, $\chi^2$ distributions are used to compute the p-value of $G^2$ [McCullagh1989]. The number of degrees of freedom #d.f. and the treatment of dispersion are the same as for Pearson’s $X^2$, see above.

Column-Wise Statistics

The rest of the statistics are computed separately for each column of $Y$. As explained above, $Y$ has two or more columns in bi- and multinomial case, either at input or after conversion. Moreover, each \(y_{i,j}\) in record $i$ with $N_i \geq 2$ is counted as $N_i$ separate observations \(y_{i,j,l}\) of 0 or 1 (where $l=1,\ldots,N_i$) with \(y_{i,j}\) ones and \(N_i-y_{i,j}\) zeros. For power distributions, including linear regression, $Y$ has only one column and all $N_i = 1$, so the statistics are computed for all $Y$ with each record counted once. Below we denote \(N = \sum_{i=1}^n N_i \,\geq n\). Here is the total average and the residual average (residual bias) of $y_{i,j,l}$ for each $Y$-column:

\[\texttt{AVG_TOT_Y}_j \,=\, \frac{1}{N} \sum_{i=1}^n y_{i,j}; \quad \texttt{AVG_RES_Y}_j \,=\, \frac{1}{N} \sum_{i=1}^n \, (y_{i,j} - \mu_{i,j})\]

Dividing by $N$ (rather than $n$) gives the averages for \(y_{i,j,l}\) (rather than \(y_{i,j}\)). The total variance, and the standard deviation, for individual observations \(y_{i,j,l}\) is estimated from the total variance for response values \(y_{i,j}\) using independence assumption: \(Var \,y_{i,j} = Var \sum_{l=1}^{N_i} y_{i,j,l} = \sum_{l=1}^{N_i} Var y_{i,j,l}\). This allows us to estimate the sum of squares for $y_{i,j,l}$ via the sum of squares for \(y_{i,j}\):

\[\texttt{STDEV_TOT_Y}_j \,=\, \Bigg[\frac{1}{N-1} \sum_{i=1}^n \Big( y_{i,j} - \frac{N_i}{N} \sum_{i'=1}^n y_{i'\!,j}\Big)^2\Bigg]^{1/2}\]

Analogously, we estimate the standard deviation of the residual \(y_{i,j,l} - \mu_{i,j,l}\):

\[\texttt{STDEV_RES_Y}_j \,=\, \Bigg[\frac{1}{N-m'} \,\sum_{i=1}^n \Big( y_{i,j} - \mu_{i,j} - \frac{N_i}{N} \sum_{i'=1}^n (y_{i'\!,j} - \mu_{i'\!,j})\Big)^2\Bigg]^{1/2}\]

Here $m’=m$ if $m$ includes the intercept as a feature and $m’=m+1$ if it does not. The estimated standard deviations can be compared to the model-predicted residual standard deviation computed from the predicted means by the GLM variance formula and scaled by the dispersion:

\[\texttt{PRED_STDEV_RES}_j \,=\, \Big[\frac{\texttt{disp}}{N} \, \sum_{i=1}^n \, v(\mu_{i,j})\Big]^{1/2}\]

We also compute the $R^2$ statistics for each column of $Y$, see Table 14 and Table 15 for details. We compute two versions of $R^2$: in one version the residual sum-of-squares (RSS) includes any bias in the residual that might be present (due to the lack of, or inaccuracy in, the intercept); in the other version of RSS the bias is subtracted by “centering” the residual. In both cases we subtract the bias in the total sum-of-squares (in the denominator), and $m’$ equals $m$ with the intercept or $m+1$ without the intercept.

Table 14

$R^2$ where the residual sum-of-squares includes the bias contribution.

Statistic Formula
$\texttt{R2}_j$ \(\displaystyle 1 - \frac{\sum\limits_{i=1}^n \,(y_{i,j} - \mu_{i,j})^2}{\sum\limits_{i=1}^n \Big(y_{i,j} - \frac{N_{i\mathstrut}}{N^{\mathstrut}} \sum\limits_{i'=1}^n y_{i',j} \Big)^{2}}\)
$\texttt{ADJUSTED_R2}_j$ \(\displaystyle 1 - {\textstyle\frac{N_{\mathstrut} - 1}{N^{\mathstrut} - m}} \, \frac{\sum\limits_{i=1}^n \,(y_{i,j} - \mu_{i,j})^2}{\sum\limits_{i=1}^n \Big(y_{i,j} - \frac{N_{i\mathstrut}}{N^{\mathstrut}} \sum\limits_{i'=1}^n y_{i',j} \Big)^{2}}\)

Table 15

$R^2$ where the residual sum-of-squares is centered so that the bias is subtracted.

Statistic Formula
$\texttt{R2_NOBIAS}_j$ \(\displaystyle 1 - \frac{\sum\limits_{i=1}^n \Big(y_{i,j} \,{-}\, \mu_{i,j} \,{-}\, \frac{N_{i\mathstrut}}{N^{\mathstrut}} \sum\limits_{i'=1}^n (y_{i',j} \,{-}\, \mu_{i',j}) \Big)^{2}}{\sum\limits_{i=1}^n \Big(y_{i,j} - \frac{N_{i\mathstrut}}{N^{\mathstrut}} \sum\limits_{i'=1}^n y_{i',j} \Big)^{2}}\)
$\texttt{ADJUSTED_R2_NOBIAS}_j$ \(\displaystyle 1 - {\textstyle\frac{N_{\mathstrut} - 1}{N^{\mathstrut} - m'}} \, \frac{\sum\limits_{i=1}^n \Big(y_{i,j} \,{-}\, \mu_{i,j} \,{-}\, \frac{N_{i\mathstrut}}{N^{\mathstrut}} \sum\limits_{i'=1}^n (y_{i',j} \,{-}\, \mu_{i',j}) \Big)^{2}}{\sum\limits_{i=1}^n \Big(y_{i,j} - \frac{N_{i\mathstrut}}{N^{\mathstrut}} \sum\limits_{i'=1}^n y_{i',j} \Big)^{2}}\)

Reg Scoring & Predict Returns

The matrix of predicted means (if the response is numerical) or probabilities (if the response is categorical), see Description subsection above for more information. Given Y, we return some statistics in CSV format as described in Table 13 and in the above text.

  1. Smaller likelihood difference between two models suggests less statistical evidence to pick one model over the other. ↩

  2. $Prob[y\mid \mu_i]$ is given by a density function if $y$ is continuous. ↩

  3. We use $k+1$ because there are $k$ non-baseline categories and one baseline category, with regression parameters $B$ having $k$ columns. ↩