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 1column 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 $\beta_j$â€™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 commaseparated namevalue 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  GLMstyle 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  L2norm 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 L2norm 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 all0 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 suboptimal^{1}, 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 crossvalidation where the dataset is
repeatedly split into a training part (to estimate the $\beta_j$â€™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 $\lambda$â€™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 matrixvector 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 L2norm 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:
\[R^2 = 1  \frac{\mathrm{RSS}}{\mathrm{TSS}},\quad \sigma \,=\, \sqrt{\frac{\mathrm{RSS}}{n  m  1}},\quad R^2_{\textrm{adj.}} = 1  \frac{\sigma^2 (n1)}{\mathrm{TSS}}\]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 smallvariance columns (which need large $\beta_j$â€™s)
more than largevariance columns (with small $\beta_j$â€™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 $\hat{\beta}_j$â€™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 userâ€™s intercept and rescaling choice:
 icpt=0: No intercept, matrix $B$ has size $m\,{\times}\,1$, with $B[j, 1] = \hat{\beta}_j$ for each $j$ from 1 to $m = {}$ncol$(X)$.
 icpt=1: There is intercept, but no shifting/rescaling of $X$; matrix $B$ has size $(m\,{+}\,1) \times 1$, with $B[j, 1] = \hat{\beta}_j$ for $j$ from 1 to $m$, and $B[m\,{+}\,1, 1] = \hat{\beta}_0$, the estimated intercept coefficient.
 icpt=2: There is intercept, and the features in $X$ are shifted to mean$ = 0$ and rescaled to variance$ = 1$; then there are two versions of the $\hat{\beta}_j$â€™s, one for the original features and another for the shifted/rescaled features. Now matrix $B$ has size $(m\,{+}\,1) \times 2$, with $B[\cdot, 1]$ for the original features and $B[\cdot, 2]$ for the shifted/rescaled features, in the above format. Note that $B[\cdot, 2]$ are iteratively estimated and $B[\cdot, 1]$ are obtained from $B[\cdot, 2]$ by complementary shifting and rescaling.
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 1column 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 $(x_1, y_1)$, â€¦, $(x_n, y_n)$ 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 $\beta_j$â€™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 userprovided. 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 commaseparated namevalue 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  L2norm of iteration step from old point ($\beta$vector) to new point 
OBJECTIVE  The loss function we minimize (negative partial loglikelihood) 
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  Actualtopredicted reduction ratio, used to update the trust region 
GRADIENT_NORM  L2norm 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 nonpower 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
$\theta = \mu^{1q}/(1q)$, 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
1column Bernoulli into the 2column binomial with 0versus1 counts.
We estimate the regression parameters via L2regularized negative loglikelihood 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 userprovided (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 trustregion 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 unitdispersion deviance
estimated as
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
\[\begin{equation} \hat{a} \,\,=\,\, \frac{1}{nm}\cdot \sum_{i=1}^n \frac{(y_i  \mu_i)^2}{v(\mu_i)} \end{equation}\]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 $\hat{\beta}_j$â€™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 userâ€™s intercept and rescaling choice:
 icpt=0: No intercept, matrix $B$ has size $m\,{\times}\,1$, with $B[j, 1] = \hat{\beta}_j$ for each $j$ from 1 to $m = {}$ncol$(X)$.
 icpt=1: There is intercept, but no shifting/rescaling of $X$; matrix $B$ has size $(m\,{+}\,1) \times 1$, with $B[j, 1] = \hat{\beta}_j$ for $j$ from 1 to $m$, and $B[m\,{+}\,1, 1] = \hat{\beta}_0$, the estimated intercept coefficient.
 icpt=2: There is intercept, and the features in $X$ are shifted to mean${} = 0$ and rescaled to variance${} = 1$; then there are two versions of the $\hat{\beta}_j$â€™s, one for the original features and another for the shifted/rescaled features. Now matrix $B$ has size $(m\,{+}\,1) \times 2$, with $B[\cdot, 1]$ for the original features and $B[\cdot, 2]$ for the shifted/rescaled features, in the above format. Note that $B[\cdot, 2]$ are iteratively estimated and $B[\cdot, 1]$ are obtained from $B[\cdot, 2]$ by complementary shifting and rescaling.
Our script also estimates the dispersion $\hat{a}$ (or takes it from the userâ€™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 L2SVM 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
1column 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 GLMpredict.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, nonmatrix, 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, singlelabel categorical, or multilabel categorical. A singlelabel categorical $y_i$ is an integer category label, one label per record; a multilabel $y_i$ is a vector of integer counts, one count for each possible label, which represents multiple singlelabel events (observations) for the same $x_i$. Internally we convert singlelabel categoricals into multilabel categoricals by replacing each label $l$ with an indicator vector $(0,\ldots,0,1_l,0,\ldots,0)$. In predictiononly tasks the actual $y_i$â€™s are not needed to the script, but they are needed for scoring.
To perform prediction, the script matrixmultiplies $X$ and $B$, adding the intercept if available, then applies the inverse of the modelâ€™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 $y_i$â€™s (in numerical or multilabel 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 scriptâ€™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 nonnegative 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 categories^{3}. Note again that we do not predict the labels themselves, nor their actual counts per record, but we predict the labelsâ€™ probabilities.
Going from predicted probabilities to predicted labels, in the
singlelabel 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 highestprobability
label would be wrong 70% of the time, whereas picking the
lowestprobability 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 GLMpredict.dml
for now.
Table 13
The goodnessoffit 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 percolumn statistics
(note that a bi/multinomial onecolumn Yinput is converted into multicolumn);
“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  +  Loglikelihood $Z$score (in st. dev.’s from the mean)  
LOGLHOOD_Z_PVAL  +  Loglikelihood $Z$score pvalue, twosided  
PEARSON_X2  +  Pearson residual $X^2$statistic  
PEARSON_X2_BY_DF  +  Pearson $X^2$ divided by degrees of freedom  
PEARSON_X2_PVAL  +  Pearson $X^2$ pvalue  
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 $G^2$ pvalue  
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 $Y  pred. mean(YX)$  
STDEV_RES_Y  +  $Y$column residual st. dev. of $Y  pred. mean(YX)$  
PRED_STDEV_RES  +  +  Modelpredicted $Y$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 matrixmultiplying $X$ 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 modelâ€™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 multilabel records) or when the labels such as $1$ and $1$ are viewed and averaged as numerical response values (with the singlelabel 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 asis. 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 $k\geq 1$ nonbaseline 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
loglikelihood 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.
Pearsonâ€™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 goodnessoffit measure. To compute these statistics, we use:
 the $n\times (k\,{+}\,1)$matrix $Y$ of multilabel response counts, in which $y_{i,j}$ is the number of times label $j$ was observed in record $i$
 the modelestimated probability matrix $P$ of the same dimensions that satisfies \(\sum_{j=1}^{k+1} p_{i,j} = 1\) for all $i=1,\ldots,n$ and where $p_{i,j}$ is the model probability of observing label $j$ in record $i$
 the $n\,{\times}\,1$vector $N$ where $N_i$ is the aggregated count of observations in record $i$ (all $N_i = 1$ if each record has only one response label)
We start by computing the multinomial loglikelihood of $Y$ given $P$ and $N$, as well as the expected loglikelihood given a random $Y$ and the variance of this loglikelihood 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 loglikelihood $\ell(Y)$ divided by its expected standard deviation, and its twosided pvalue in the Normal distribution assumption ($\ell(Y)$ should approach normality due to the Central Limit Theorem):
\[Z \,=\, \frac {\ell(Y)  E_Y \ell(Y)}{\sqrt{Var_Y \ell(Y)}};\quad \mathop{\textrm{pvalue}}(Z) \,=\, Prob\Big[\,\big\mathop{\textrm{Normal}}(0,1)\big \, > \, Z\,\Big]\]A low pvalue would indicate “underfitting” if $Z\ll 0$ or “overfitting” if $Z\gg 0$. Here “overfitting” means that higherprobability 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 pvalue. Since $\ell(Y)$ is a
linear function of $Y$, multiplying the GLMpredicted 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:
Finally, we recalculate the pvalue with this new $Z$score.
PEARSON_X2
, PEARSON_X2_BY_DF
, and PEARSON_X2_PVAL
:
Pearsonâ€™s residual $X^2$statistic is a commonly used goodnessoffit
measure for linear models [McCullagh1989].
The idea is to measure how
well the modelpredicted 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]:
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 Pearsonâ€™s $X^2$ and the deviance $G^2$ are unreliable (i.e. do not approach the $\chi^2$ distribution) unless the predicted means of multilabel 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:
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 pvalue of $G^2$ [McCullagh1989]. The number of degrees of freedom #d.f. and the treatment of dispersion are the same as for Pearsonâ€™s $X^2$, see above.
ColumnWise 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 \(N_iy_{i,j}\) 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 \(y_{i,j}\):
\[\texttt{STDEV_TOT_Y}_j \,=\, \Bigg[\frac{1}{N1} \sum_{i=1}^n \Big( y_{i,j}  \frac{N_i}{N} \sum_{i'=1}^n y_{i'\!,j}\Big)^2\Bigg]^{1/2}\]Analogously, we estimate the standard deviation of the residual \(y_{i,j,l}  \mu_{i,j,l}\):
\[\texttt{STDEV_RES_Y}_j \,=\, \Bigg[\frac{1}{Nm'} \,\sum_{i=1}^n \Big( y_{i,j}  \mu_{i,j}  \frac{N_i}{N} \sum_{i'=1}^n (y_{i'\!,j}  \mu_{i'\!,j})\Big)^2\Bigg]^{1/2}\]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 modelpredicted 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 sumofsquares (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 sumofsquares (in the denominator), and $m’$ equals $m$ with the intercept or $m+1$ without the intercept.
Table 14
$R^2$ where the residual sumofsquares 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 sumofsquares 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.

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

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

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