ALM stands for “Augmented Linear Model”. The word “augmented” is used
to reflect that the model introduces aspects that extend beyond the
basic linear model. In some special cases `alm()`

resembles
the `glm()`

function from stats package, but with a higher
focus on forecasting rather than on hypothesis testing. You will not get
p-values anywhere from the `alm()`

function and won’t see
\(R^2\) in the outputs. The maximum
what you can count on is having confidence intervals for the parameters
or for the regression line. The other important difference from
`glm()`

is the availability of distributions that are not
supported by `glm()`

(for example, Folded Normal or Box-Cox
Normal distributions) and it allows optimising non-standard parameters
(e.g. \(\lambda\) in Asymmetric Laplace
distribution). Finally, `alm()`

supports different loss
functions via the `loss`

parameter, so you can estimate
parameters of your model via, for example, likelihood maximisation or
via minimisation of MSE / MAE, using LASSO / RIDGE or by minimising a
loss provided by user.

Although `alm()`

supports various loss functions, the core
of the function is the likelihood approach. By default the estimation of
parameters in the model is done via the maximisation of likelihood
function of a selected distribution. The calculation of the standard
errors is done based on the calculation of hessian of the distribution.
And in the centre of all of that are information criteria that can be
used for the models comparison.

This vignette contains the following sections:

All the supported distributions have specific functions which form
the following four groups for the `distribution`

parameter in
`alm()`

:

- Density functions of continuous distributions,
- Density functions for continuous non-negative data
- Density functions for continuous positive data,
- Continuous distributions on a specific interval,
- Density functions of discrete distributions,
- Cumulative functions for binary variables.

All of them rely on respective d- and p- functions in R. For example,
Log-Normal distribution uses `dlnorm()`

function from
`stats`

package.

The `alm()`

function also supports `occurrence`

parameter, which allows modelling non-zero values and the occurrence of
non-zeroes as two different models. The combination of any distribution
from (1) - (3) for the non-zero values and a distribution from (4) for
the occurrence will result in a mixture distribution
model, e.g. a mixture of Log-Normal and Cumulative Logistic or a
Hurdle Poisson (with Cumulative Normal for the occurrence part).

Every model produced using `alm()`

can be represented as:
\[\begin{equation} \label{eq:basicALM}
y_t = f(\mu_t, \epsilon_t) = f(x_t' B, \epsilon_t) ,
\end{equation}\] where \(y_t\)
is the value of the response variable, \(x_t\) is the vector of exogenous variables,
\(B\) is the vector of the parameters,
\(\mu_t\) is the conditional mean
(produced based on the exogenous variables and the parameters of the
model), \(\epsilon_t\) is the error
term on the observation \(t\) and \(f(\cdot)\) is the distribution function
that does a transformation of the inputs into the output. In case of a
mixture distribution the model becomes slightly more complicated: \[\begin{equation} \label{eq:basicALMMixture}
\begin{matrix}
y_t = o_t f(x_t' B, \epsilon_t) \\
o_t \sim \mathrm{Bernoulli}(p_t) \\
p_t = g(z_t' A)
\end{matrix},
\end{equation}\] where \(o_t\)
is the binary variable, \(p_t\) is the
probability of occurrence, \(z_t\) is
the vector of exogenous variables and \(A\) is the vector of parameters for the
\(p_t\).

In addition, the function supports scale model, i.e. the model that predicts the values of scale of distribution (for example, variance in case of normal distribution) based on the provided explanatory variables. This is discussed in some detail in a separate section.

The `alm()`

function returns, along with the set of common
for `lm()`

variables (such as `coefficient`

and
`fitted.values`

), the variable `mu`

, which
corresponds to the conditional mean used inside the distribution, and
`scale`

– the second parameter, which usually corresponds to
standard error or dispersion parameter. The values of these two
variables vary from distribution to distribution. Note, however, that
the `model`

variable returned by `lm()`

function
was renamed into `data`

in `alm()`

, and that
`alm()`

does not return `terms`

and QR
decomposition.

Given that the parameters of any model in `alm()`

are
estimated via likelihood, it can be assumed that they have
asymptotically normal distribution, thus the confidence intervals for
any model rely on the normality and are constructed based on the
unbiased estimate of variance, extracted using `sigma()`

function.

The covariance matrix of parameters almost in all the cases is calculated as an inverse of the hessian of respective distribution function. The exclusions are Normal, Log-Normal, Poisson, Cumulative Logistic and Cumulative Normal distributions, that use analytical solutions.

`alm()`

function also supports factors in the explanatory
variables, creating the set of dummies from them. In case of ordered
variables (ordinal scale, `is.ordered()`

), the ordering is
removed and the set of dummies is produced. This is done in order to
avoid the built in behaviour of R, which creates linear, squared, cubic
etc levels for ordered variables, which makes the interpretation of the
parameters difficult.

When the number of estimated parameters is calculated, in case of
`loss=="likelihood"`

the scale is considered as one of the
parameters as well, which aligns with the idea of the maximum likelihood
estimation. For all the other losses, the scale does not count (this
aligns, for example, with how the number of parameters is calculated in
OLS, which corresponds to `loss="MSE"`

).

Although the basic principles of estimation of models and predictions from them are the same for all the distributions, each of the distribution has its own features. So it makes sense to discuss them individually. We discuss the distributions in the four groups mentioned above.

This group of functions includes:

- Normal distribution,
- Laplace distribution,
- Asymmetric Laplace distribution,
- Generalised Normal distribution,
- Logistic distribution,
- S distribution,
- Student t distribution,

For all the functions in this category `resid()`

method
returns \(e_t = y_t - \mu_t\).

The density of normal distribution \(\mathcal{N}(\mu_t,\sigma)\) is: \[\begin{equation} \label{eq:Normal} f(y_t) = \frac{1}{\sqrt{2 \pi \sigma^2}} \exp \left( -\frac{\left(y_t - \mu_t \right)^2}{2 \sigma^2} \right) , \end{equation}\] where \(\sigma\) is the standard deviation of the error term. This PDF has a very well-known bell shape:

`alm()`

with Normal distribution
(`distribution="dnorm"`

) is equivalent to `lm()`

function from `stats`

package and returns roughly the same
estimates of parameters, so if you are concerned with the time of
calculation, I would recommend reverting to `lm()`

.

Maximising the likelihood of the model \(\eqref{eq:Normal}\) is equivalent to the estimation of the basic linear regression using Least Squares method: \[\begin{equation} \label{eq:linearModel} y_t = \mu_t + \epsilon_t = x_t' B + \epsilon_t, \end{equation}\] where \(\epsilon_t \sim \mathcal{N}(0, \sigma^2)\).

The variance \(\sigma^2\) is
estimated in `alm()`

based on likelihood: \[\begin{equation} \label{eq:sigmaNormal}
\hat{\sigma}^2 = \frac{1}{T} \sum_{t=1}^T \left(y_t - \mu_t
\right)^2 ,
\end{equation}\] where \(T\) is
the sample size. Its square root (standard deviation) is used in the
calculations of `dnorm()`

function, and the value is then
return via `scale`

variable. This value does not have bias
correction. However the `sigma()`

method applied to the
resulting model, returns the bias corrected version of standard
deviation. And `vcov()`

, `confint()`

,
`summary()`

and `predict()`

rely on the value
extracted by `sigma()`

.

\(\mu_t\) is returned as is in
`mu`

variable, and the fitted values are set equivalent to
`mu`

.

In order to produce confidence intervals for the mean
(`predict(model, newdata, interval="confidence")`

) the
conditional variance of the model is calculated using: \[\begin{equation} \label{eq:varianceNormalForCI}
V({\mu_t}) = x_t V(B) x_t',
\end{equation}\] where \(V(B)\)
is the covariance matrix of the parameters returned by the function
`vcov`

. This variance is then used for the construction of
the confidence intervals of a necessary level \(\alpha\) using the distribution of Student:
\[\begin{equation} \label{eq:intervalsNormal}
y_t \in \left(\mu_t \pm \tau_{df,\frac{1+\alpha}{2}} \sqrt{V(\mu_t)}
\right),
\end{equation}\] where \(\tau_{df,\frac{1+\alpha}{2}}\) is the upper
\({\frac{1+\alpha}{2}}\)-th quantile of
the Student’s distribution with \(df\)
degrees of freedom (e.g. with \(\alpha=0.95\) it will be 0.975-th quantile,
which, for example, for 100 degrees of freedom will be \(\approx 1.984\)).

Similarly for the prediction intervals
(`predict(model, newdata, interval="prediction")`

) the
conditional variance of the \(y_t\) is
calculated: \[\begin{equation}
\label{eq:varianceNormalForPI}
V(y_t) = V(\mu_t) + s^2 ,
\end{equation}\] where \(s^2\)
is the bias-corrected variance of the error term, calculated using:
\[\begin{equation}
\label{eq:varianceNormalUnbiased}
s^2 = \frac{1}{T-k} \sum_{t=1}^T \left(y_t - \mu_t \right)^2 ,
\end{equation}\] where \(k\) is
the number of estimated parameters (including the variance itself). This
value is then used for the construction of the prediction intervals of a
specify level, also using the distribution of Student, in a similar
manner as with the confidence intervals.

Laplace distribution has some similarities with the Normal one: \[\begin{equation} \label{eq:Laplace} f(y_t) = \frac{1}{2 s} \exp \left( -\frac{\left| y_t - \mu_t \right|}{s} \right) , \end{equation}\] where \(s\) is the scale parameter, which, when estimated using likelihood, is equal to the mean absolute error: \[\begin{equation} \label{eq:bLaplace} \hat{s} = \frac{1}{T} \sum_{t=1}^T \left| y_t - \mu_t \right| . \end{equation}\] So maximising the likelihood \(\eqref{eq:Laplace}\) is equivalent to estimating the linear regression \(\eqref{eq:linearModel}\) via the minimisation of \(s\) \(\eqref{eq:bLaplace}\). So when estimating a model via minimising \(s\), the assumption imposed on the error term is \(\epsilon_t \sim \mathcal{Laplace}(0, s)\). The main difference of Laplace from Normal distribution is its fatter tails, the PDF has the following shape:

`alm()`

function with `distribution="dlaplace"`

returns `mu`

equal to \(\mu_t\) and the fitted values equal to
`mu`

. \(s\) is returned in
the `scale`

variable. The prediction intervals are derived
from the quantiles of Laplace distribution after transforming the
conditional variance into the conditional scale parameter \(s\) using the connection between the two in
Laplace distribution: \[\begin{equation}
\label{eq:bLaplaceAndSigma}
s = \sqrt{\frac{\sigma^2}{2}},
\end{equation}\] where \(\sigma^2\) is substituted either by the
conditional variance of \(\mu_t\) or
\(y_t\).

The kurtosis of Laplace distribution is 6, making it suitable for modelling rarely occurring events.

Asymmetric Laplace distribution can be considered as a two Laplace
distributions with different parameters \(s\) for left and right side. There are
several ways to summarise the probability density function, the one used
in `alm()`

relies on the asymmetry parameter \(\alpha\) (Yu and
Zhang 2005): \[\begin{equation}
\label{eq:ALaplace}
f(y_t) = \frac{\alpha (1- \alpha)}{s} \exp \left( -\frac{y_t -
\mu_t}{s} (\alpha - I(y_t \leq \mu_t)) \right) ,
\end{equation}\] where \(s\) is
the scale parameter, \(\alpha\) is the
skewness parameter and \(I(y_t \leq
\mu_t)\) is the indicator function, which is equal to one, when
the condition is satisfied and to zero otherwise. The scale parameter
\(s\) estimated using likelihood is
equal to the quantile loss: \[\begin{equation} \label{eq:bALaplace}
\hat{s} = \frac{1}{T} \sum_{t=1}^T \left(y_t - \mu_t \right)(\alpha
- I(y_t \leq \mu_t)) .
\end{equation}\] Thus maximising the likelihood \(\eqref{eq:ALaplace}\) is equivalent to
estimating the linear regression \(\eqref{eq:linearModel}\) via the
minimisation of \(\alpha\) quantile,
making this equivalent to quantile regression. So quantile regression
models assume indirectly that the error term is \(\epsilon_t \sim \mathcal{ALaplace}(0, s,
\alpha)\) (Geraci and Bottai 2007).
The advantage of using `alm()`

in this case is in having the
full distribution, which allows to do all the fancy things you can do
when you have likelihood.

Graphically, the PDF of asymmetric Laplace is:

In case of \(\alpha=0.5\) the function reverts to the symmetric Laplace where \(s=\frac{1}{2}\text{MAE}\).

`alm()`

function with
`distribution="dalaplace"`

accepts an additional parameter
`alpha`

in ellipsis, which defines the quantile \(\alpha\). If it is not provided, then the
function will estimated it maximising the likelihood and return it as
the first coefficient. `alm()`

returns `mu`

equal
to \(\mu_t\) and the fitted values
equal to `mu`

. \(s\) is
returned in the `scale`

variable. The parameter \(\alpha\) is returned in the variable
`other`

of the final model. The prediction intervals are
produced using `qalaplace()`

function. In order to find the
values of \(s\) for the holdout the
following connection between the variance of the variable and the scale
in Asymmetric Laplace distribution is used: \[\begin{equation} \label{eq:bALaplaceAndSigma}
s = \sqrt{\sigma^2 \frac{\alpha^2 (1-\alpha)^2}{(1-\alpha)^2 +
\alpha^2}},
\end{equation}\] where \(\sigma^2\) is substituted either by the
conditional variance of \(\mu_t\) or
\(y_t\).

**NOTE**: in order for the Asymmetric Laplace to work
well, you might need to have large samples. This is inherited from the
pinball score of the quantile regression. If you fit the model on 40
observations with \(\alpha=0.05\), you
will only have 2 observations below the line, which does not help very
much with the fit. Similarly, the covariance matrix, produced via the
Hessian might not be adequate in this situation (because there is not
enough variability in the data due to extreme value of \(\alpha\)). The latter can be partially
addressed by using bootstrap, but do not expect miracles on small
samples.

The S distribution has the following density function: \[\begin{equation} \label{eq:S} f(y_t) = \frac{1}{4 s^2} \exp \left( -\frac{\sqrt{|y_t - \mu_t|}}{s} \right) , \end{equation}\] where \(s\) is the scale parameter. If estimated via maximum likelihood, the scale parameter is equal to: \[\begin{equation} \label{eq:bS} \hat{s} = \frac{1}{2T} \sum_{t=1}^T \sqrt{\left| y_t - \mu_t \right|} , \end{equation}\] which corresponds to the minimisation of a half of “Mean Root Absolute Error” or “Half Absolute Moment”.

S distribution has a kurtosis of 25.2, which makes it a “severe excess” distribution (thus the name). It might be useful in cases of randomly occurring incidents and extreme values (Black Swans?). Here how the PDF looks:

`alm()`

function with `distribution="ds"`

returns \(\mu_t\) in the same variables
`mu`

and `fitted.values`

, and \(s\) in the `scale`

variable.
Similarly to the previous functions, the prediction intervals are based
on the `qs()`

function from `greybox`

package and
use the connection between the scale and the variance: \[\begin{equation} \label{eq:bSAndSigma}
s = \left( \frac{\sigma^2}{120} \right) ^{\frac{1}{4}},
\end{equation}\] where once again \(\sigma^2\) is substituted either by the
conditional variance of \(\mu_t\) or
\(y_t\).

The Generalised Normal distribution is a generalisation, which has Normal, Laplace and S as special cases. It has the following density function: \[\begin{equation} \label{eq:gnormal} f(y_t) = \frac{\beta}{2s \Gamma(1/\beta)}\exp\left(-\left(\frac{|y_t - \mu|}{s}\right)^\beta\right), \end{equation}\] where \(s\) is the scale and \(\beta\) is the shape parameters. If estimated via maximum likelihood, the scale parameter is equal to: \[\begin{equation} \label{eq:gnormalScale} \hat{s} = \sqrt[^\beta]{\frac{\beta}{T} \sum_{t=1}^T \left| y_t - \mu_t \right|^{\beta}} . \end{equation}\] In the special cases, this becomes either \(\sqrt{2}\times\)RMSE (\(\beta=2\)), or MAE (\(\beta=1\)) or a half of HAM (\(\beta=0.5\)). It is important to note that although in case of \(\beta=2\), the distribution becomes equivalent to Normal, the scale of it will differ from the \(\sigma\) (this follows directly from the formula above). The relations between the two is: \(s^2 = 2 \sigma^2\).

The kurtosis of Generalised Normal distribution is determined by \(\beta\) and is equal to \(\frac{\Gamma(5/\beta)\Gamma(1/\beta)}{\Gamma(3/\beta)^2}\).

`alm()`

function with `distribution="dgnorm"`

returns \(\mu_t\) in the same variables
`mu`

and `fitted.values`

, \(s\) in the `scale`

variable and
\(\beta\) in `other$beta`

.
Note that if `beta`

is not provided in the function, then it
will estimate it. However, the estimates of \(\beta\) are known not to be consistent and
asymptotically normal if it is less than 2. **So, use with
care!** As for the intervals, they are based on the
`qgnorm()`

function from `greybox`

package and use
the connection between the scale and the variance: \[\begin{equation} \label{eq:gnormalAlphaAndSigma}
s = \left( \frac{\sigma^2 \Gamma(1/\beta)}{\Gamma(3/\beta)} \right)
^{\frac{1}{2}},
\end{equation}\] where once again \(\sigma^2\) is substituted either by the
conditional variance of \(\mu_t\) or
\(y_t\), depending on what type of
interval is needed.

The density function of Logistic distribution is: \[\begin{equation} \label{eq:Logistic}
f(y_t) = \frac{\exp \left(- \frac{y_t - \mu_t}{s} \right)} {s \left(
1 + \exp \left(- \frac{y_t - \mu_t}{s} \right) \right)^{2}},
\end{equation}\] where \(s\) is
the scale parameter, which is estimated in `alm()`

based on
the connection between the parameter and the variance in the logistic
distribution: \[\begin{equation}
\label{eq:sLogisticAndSigma}
\hat{s} = \sigma \frac{\sqrt{3}}{\pi}.
\end{equation}\] Once again the maximisation of \(\eqref{eq:Logistic}\) implies the
estimation of the linear model \(\eqref{eq:linearModel}\), where \(\epsilon_t \sim \mathcal{Logistic}(0, s)\).

Logistic is considered a fat tailed distribution, but its tails are not as fat as in Laplace. Kurtosis of standard Logistic is 4.2.

`alm()`

function with `distribution="dlogis"`

returns \(\mu_t\) in `mu`

and in `fitted.values`

variables, and \(s\) in the `scale`

variable.
Similar to Laplace distribution, the prediction intervals use the
connection between the variance and scale, and rely on the
`qlogis`

function.

The Student t distribution has a difficult density function: \[\begin{equation} \label{eq:T} f(y_t) = \frac{\Gamma\left(\frac{\nu+1}{2}\right)}{\sqrt{\nu \pi} \Gamma\left(\frac{\nu}{2}\right)} \left( 1 + \frac{x^2}{\nu} \right)^{-\frac{\nu+1}{2}} , \end{equation}\] where \(\nu\) is the number of degrees of freedom, which can also be considered as the scale parameter of the distribution. It has the following connection with the in-sample variance of the error (but only for the case, when \(\nu>2\)): \[\begin{equation} \label{eq:scaleOfT} \nu = \frac{2}{1-\sigma^{-2}}. \end{equation}\]

Kurtosis of Student t distribution depends on the value of \(\nu\), and for the cases of \(\nu>4\) is equal to \(\frac{6}{\nu-4}\). When the \(\mu \rightarrow \infty\), the distribution converges to the normal.

`alm()`

function with `distribution="dt"`

estimates the parameters of the model along with the \(\nu\) (if it is not provided by the user as
a `nu`

parameter) and returns \(\mu_t\) in the variables `mu`

and `fitted.values`

, and \(\nu\) in the `scale`

variable.
Both prediction and confidence intervals use `qt()`

function
from `stats`

package and rely on the conventional number of
degrees of freedom \(T-k\). The
intervals are constructed similarly to how it is done in Normal
distribution \(\eqref{eq:intervalsNormal}\) (based on
`qt()`

function).

In order to see how this works, we will create the following data:

```
<- cbind(rnorm(200,10,3),rnorm(200,50,5))
xreg <- cbind(500+0.5*xreg[,1]-0.75*xreg[,2]+rs(200,0,3),xreg,rnorm(200,300,10))
xreg colnames(xreg) <- c("y","x1","x2","Noise")
<- xreg[1:180,]
inSample <- xreg[-c(1:180),] outSample
```

ALM can be run either with data frame or with matrix. Here’s an example with normal distribution and several levels for the construction of prediction interval:

```
<- alm(y~x1+x2, data=inSample, distribution="dnorm")
ourModel summary(ourModel)
#> Response variable: y
#> Distribution used in the estimation: Normal
#> Loss function used in estimation: likelihood
#> Coefficients:
#> Estimate Std. Error Lower 2.5% Upper 97.5%
#> (Intercept) 485.6531 84.9830 317.9362 653.3700 *
#> x1 2.0247 2.7765 -3.4547 7.5042
#> x2 -0.8305 1.6043 -3.9966 2.3356
#>
#> Error standard deviation: 106.7513
#> Sample size: 180
#> Number of estimated parameters: 4
#> Number of degrees of freedom: 176
#> Information criteria:
#> AIC AICc BIC BICc
#> 2196.154 2196.382 2208.925 2209.519
plot(predict(ourModel,outSample,interval="p",level=c(0.9,0.95)))
```

And here’s an example with Asymmetric Laplace and predefined \(\alpha=0.95\):

```
<- alm(y~x1+x2, data=inSample, distribution="dalaplace",alpha=0.95)
ourModel summary(ourModel)
#> Warning: Choleski decomposition of hessian failed, so we had to revert to the simple inversion.
#> The estimate of the covariance matrix of parameters might be inaccurate.
#> Response variable: y
#> Distribution used in the estimation: Asymmetric Laplace with alpha=0.95
#> Loss function used in estimation: likelihood
#> Coefficients:
#> Estimate Std. Error Lower 2.5% Upper 97.5%
#> (Intercept) 663.9789 0.0000 663.9789 663.9789 *
#> x1 1.2345 0.0217 1.1915 1.2774 *
#> x2 -1.2839 90.9492 -180.7752 178.2074
#>
#> Error standard deviation: 183.319
#> Sample size: 180
#> Number of estimated parameters: 4
#> Number of degrees of freedom: 176
#> Information criteria:
#> AIC AICc BIC BICc
#> 2321.597 2321.825 2334.369 2334.962
plot(predict(ourModel,outSample))
```

There are currently three distributions in this group:

They allow the response variable to be positive or zero. Note however that the PDF of the Box-Cox Normal distribution is equal to zero in case of \(y_t=0\), which might cause some issues in the estimation.

Box-Cox Normal distribution used in the `greybox`

package
is defined as a distribution that becomes normal after the Box-Cox
transformation. This means that if \(x=\frac{y^\lambda+1}{\lambda}\) and \(x \sim \mathcal{N}(\mu, \sigma^2)\), then
\(y \sim \text{BC}\mathcal{N}(\mu,
\sigma^2)\). The density function of the Box-Cox Normal
distribution in this case is: \[\begin{equation} \label{eq:BCNormal}
f(y_t) = \frac{y_t^{\lambda-1}} {\sqrt{2 \pi \sigma^2}} \exp \left(
-\frac{\left(\frac{y_t^{\lambda}-1}{\lambda} - \mu_t \right)^2}{2
\sigma^2} \right) ,
\end{equation}\] where the variance estimated using likelihood
is: \[\begin{equation}
\label{eq:sigmaBCNormal}
\hat{\sigma}^2 = \frac{1}{T} \sum_{t=1}^T
\left(\frac{y_t^{\lambda}-1}{\lambda} - \mu_t \right)^2 .
\end{equation}\] Depending on the value of \(\lambda\), we will get different shapes of
the density function:

When \(\lambda=0\) the distribution transforms to the Log-Normal one.

Estimating the model with Box-Cox Normal distribution is equivalent to estimating the parameters of a linear model after the Box-Cox transform: \[\begin{equation} \label{eq:BCLinearModel} \frac{y_t^{\lambda}-1}{\lambda} = \mu_t + \epsilon_t, \end{equation}\] where \(\epsilon_t \sim \mathcal{N}(0, \sigma^2)\) or: \[\begin{equation} \label{eq:BCLinearModelExp} y_t = \left((\mu_t + \epsilon_t) \lambda +1 \right)^{\frac{1}{\lambda}}. \end{equation}\]

`alm()`

with `distribution="dbcnorm"`

does not
transform the provided data and estimates the density directly using
`dbcnorm()`

function from `greybox`

with the
estimated mean \(\mu_t\) and the
variance \(\eqref{eq:sigmaBCNormal}\).
The \(\mu_t\) is returned in the
variable `mu`

, the \(\sigma^2\) is in the variable
`scale`

, while the `fitted.values`

contains the
exponent of \(\mu_t\), which, given the
connection between the Normal and Box-Cox Normal distributions,
corresponds to median of distribution rather than mean. Finally,
`resid()`

method returns \(e_t =
\frac{y_t^{\lambda}-1}{\lambda} - \mu_t\). The \(lambda\) parameter can be provided by the
user via the `lambdaBC`

in ellipsis.

Folded Normal distribution is obtained when the absolute value of normally distributed variable is taken: if \(x \sim \mathcal{N}(\mu, \sigma^2)\), then \(|x| \sim \text{Fold}\mathcal{N}(\mu, \sigma^2)\). The density function is: \[\begin{equation} \label{eq:foldedNormal} f(y_t) = \frac{1}{\sqrt{2 \pi \sigma^2}} \left( \exp \left( -\frac{\left(y_t - \mu_t \right)^2}{2 \sigma^2} \right) + \exp \left( -\frac{\left(y_t + \mu_t \right)^2}{2 \sigma^2} \right) \right), \end{equation}\] which can be graphically represented as:

Conditional mean and variance of Folded Normal are estimated in
`alm()`

(with `distribution="dfnorm"`

) similarly
to how this is done for Normal distribution. They are returned in the
variables `mu`

and `scale`

respectively. In order
to produce the fitted value (which is returned in
`fitted.values`

), the following correction is done: \[\begin{equation} \label{eq:foldedNormalFitted}
\hat{y_t} = \sqrt{\frac{2}{\pi}} \sigma \exp \left(
-\frac{\mu_t^2}{2 \sigma^2} \right) + \mu_t \left(1 - 2 \Phi
\left(-\frac{\mu_t}{\sigma} \right) \right),
\end{equation}\] where \(\Phi(\cdot)\) is the CDF of Normal
distribution.

The model that is assumed in the case of Folded Normal distribution can be summarised as: \[\begin{equation} \label{eq:foldedNormalModel} y_t = \left| \mu_t + \epsilon_t \right|. \end{equation}\]

The conditional variance of the forecasts is calculated based on the
elements of `vcov()`

(as in all the other functions), the
predicted values are corrected in the same way as the fitted values
\(\eqref{eq:foldNormalFitted}\), and
the prediction intervals are generated from the `qfnorm()`

function of `greybox`

package. As for the residuals,
`resid()`

method returns \(e_t =
y_t - \mu_t\).

Rectified Normal distribution is obtained when all the negative values of normally distributed variable are set to zero: if \(x \sim \mathcal{N}(\mu, \sigma)\), then \(y = \max(0, x) \sim \text{Rect}\mathcal{N}(\mu, \sigma)\). The density function is:

\[\begin{equation} \label{eq:rectnormal} f(y_t) = I(y_t = 0) \Phi_x(0, \mu, \sigma) + I(y_t > 0) \phi_x(y_t, \mu, \sigma), \end{equation}\] where \(\Phi_x(0, \mu, \sigma)\) is the CDF and \(\phi_x(y_t, \mu, \sigma)\) is the PDF of the Normal distribution. This can be graphically represented as:

This distribution can be useful in modelling intermittent demand, when the demand sizes are not integer.

Conditional location and scale of Rectified Normal are estimated in
`alm()`

(with `distribution="drectnorm"`

)
similarly to how this is done for Normal distribution. They are returned
in the variables `mu`

and `scale`

respectively. In
order to produce the fitted value (which is returned in
`fitted.values`

), the following formula is used: \[\begin{equation} \label{eq:rectNormalFitted}
\hat{y_t} = \mu_t (1-\Phi_x(0, \mu, \sigma)) + \sigma * \phi_x(0,
\mu, \sigma) .
\end{equation}\]

The model that is assumed in the case of Rectified Normal distribution is: \[\begin{equation} \label{eq:rectifiedNormalModel} y_t = \max(\mu_t + \epsilon_t, 0). \end{equation}\]

The conditional variance of the forecasts is calculated based on the
elements of `vcov()`

(as in all the other functions), the
predicted values are corrected in the same way as the fitted values
\(\eqref{eq:foldNormalFitted}\), and
the prediction intervals are generated from the `qrectnorm()`

function of `greybox`

package. As for the residuals,
`resid()`

method returns \(e_t =
y_t - \mu_t\).

This group includes:

- Log-Normal distribution,
- Inverse Gaussian distribution,
- Gamma distribution,
- Exponential distribution,
- Log-Laplace distribution,
- Log-S distribution,
- Log-Generalised Normal distribution,

Log-Normal distribution appears when a normally distributed variable is exponentiated. This means that if \(x \sim \mathcal{N}(\mu, \sigma^2)\), then \(\exp x \sim \text{log}\mathcal{N}(\mu, \sigma^2)\). The density function of Log-Normal distribution is: \[\begin{equation} \label{eq:LogNormal} f(y_t) = \frac{1}{y_t \sqrt{2 \pi \sigma^2}} \exp \left( -\frac{\left(\log y_t - \mu_t \right)^2}{2 \sigma^2} \right) , \end{equation}\] where the variance estimated using likelihood is: \[\begin{equation} \label{eq:sigmaLogNormal} \hat{\sigma}^2 = \frac{1}{T} \sum_{t=1}^T \left(\log y_t - \mu_t \right)^2 . \end{equation}\] The PDF has the following shape:

Estimating the model with Log-Normal distribution is equivalent to estimating the parameters of log-linear model: \[\begin{equation} \label{eq:logLinearModel} \log y_t = \mu_t + \epsilon_t, \end{equation}\] where \(\epsilon_t \sim \mathcal{N}(0, \sigma^2)\) or: \[\begin{equation} \label{eq:logLinearModelExp} y_t = \exp(\mu_t + \epsilon_t). \end{equation}\]

`alm()`

with `distribution="dlnorm"`

does not
transform the provided data and estimates the density directly using
`dlnorm()`

function with the estimated mean \(\mu_t\) and the variance \(\eqref{eq:sigmaLogNormal}\). If you need a
log-log model, then you would need to take logarithms of the external
variables. The \(\mu_t\) is returned in
the variable `mu`

, the \(\sigma^2\) is in the variable
`scale`

, while the `fitted.values`

contains the
exponent of \(\mu_t\), which, given the
connection between the Normal and Log-Normal distributions, corresponds
to median of distribution rather than mean. Finally,
`resid()`

method returns \(e_t =
\log y_t - \mu_t\).

Inverse Gaussian distribution is an interesting distribution, which
is defined for positive values only and has some properties similar to
the properties of the Normal distribution. It has two parameters:
location \(\mu_t\) and scale \(\phi\) (aka “dispersion”). There are
different ways to parameterise this distribution, we use the
dispersion-based one. The important thing that distinguishes the
implementation in `alm()`

from the one in `glm()`

or in any other function is that we assume that the model has the
following form: \[\begin{equation}
\label{eq:InverseGaussianModel}
y_t = \mu_t \times \epsilon_t
\end{equation}\] and that \(\epsilon_t
\sim \mathcal{IG}(1, \phi)\). This means that \(y_t \sim \mathcal{IG}\left(\mu_t,
\frac{\phi}{\mu_t} \right)\), implying that the dispersion of the
model changes together with the conditional expectation. The density
function for the error term in this case is: \[\begin{equation} \label{eq:InverseGaussian}
f(\epsilon_t) = \frac{1}{\sqrt{2 \pi \phi \epsilon_t^3}} \exp \left(
-\frac{\left(\epsilon_t - 1 \right)^2}{2 \phi \epsilon_t} \right) ,
\end{equation}\] where the dispersion parameter is estimated via
maximising the likelihood and is calculated using: \[\begin{equation}
\label{eq:InverseGaussianDispersion}
\hat{\phi} = \frac{1}{T} \sum_{t=1}^T \frac{\left(\epsilon_t - 1
\right)^2}{\epsilon_t} .
\end{equation}\] Note that in our formulation \(\mu_t = \exp\left( x_t' B \right)\), so
that the mean is always positive. This implies that we deal with a pure
multiplicative model. In addition, we assume that \(\mu_t\) is just a scale for the
distribution, otherwise \(y_t\) would
not follow the Inverse Gaussian distribution. The density function has
following shapes depending on the values of parameters:

`alm()`

with `distribution="dinvgauss"`

estimates the density for \(y_t\) using
`dinvgauss()`

function from `statmod`

package. The
\(\mu_t\) is returned in the variables
`mu`

and `fitted.values`

, the dispersion \(\phi\) is in the variable
`scale`

. `resid()`

method returns \(e_t = \frac{y_t}{\mu_t}\). Finally, the
prediction interval for the regression model are generated using
`qinvgauss()`

function from the `statmod`

package.

Another popular distribution, defined for positive values only is called “Gamma”. It is parametrised via the shape \(k\) and scale \(\sigma^2\) and has closed forms for mean and variance: \(\mathrm{E}(x)=k \sigma^2\), \(\mathrm{V}(x)=k \sigma^4\).

The important thing that distinguishes the implementation in
`alm()`

from the one in `glm()`

or in any other
function is that we assume that the model has the following form
(similar to the Inverse Gaussian model in alm):
\[\begin{equation*}
y_t = \mu_t \times \epsilon_t
\end{equation*}\] and that \(\epsilon_t
\sim \Gamma \left(\sigma^{-2}, \sigma^2 \right)\), implying that
\(\mathrm{E}(\epsilon_t)= k \sigma^2 =
1\) and \(\mathrm{V}(\epsilon_t)=\sigma^2\). This
means that \(y_t \sim \Gamma\left(\sigma^{-2},
\sigma^2 \mu_t \right)\), meaning that the variance of the model
changes together with the conditional expectation. The density function
for the error term in this case is: \[\begin{equation} \label{eq:Gamma}
f(\epsilon_t) = \frac{1}{\Gamma(\sigma^{-2})
(\sigma^{2})^{\sigma^{-2}}} \epsilon_t^{\sigma^{-2}-1}\exp
\left(-\frac{\epsilon_t}{\sigma^2}\right),
\end{equation}\] where the scale parameter \(\sigma^2\) can be estimated via the method
of moments based on its relation to the variance: \[\begin{equation} \label{eq:GammaDispersion}
\hat{\sigma}^2 = \frac{1}{T} \sum_{t=1}^T \left(\epsilon_t - 1
\right)^2.
\end{equation}\] Note that in our formulation \(\mu_t = \exp\left( x_t' B \right)\), so
that the mean is always positive, which implies that we deal with a pure
multiplicative model. In addition, we assume that \(\mu_t\) is just a scale for the
distribution, otherwise \(y_t\) would
not follow Gamma distribution. All of this makes the model restrictive,
but arguably reasonable - otherwise the mean of the distribution might
behave uncontrollably.

The density function has following shapes depending on the values of parameters:

`alm()`

with `distribution="dgamma"`

estimates
the density for \(y_t\) using
`dgamma()`

function from `stats`

package. The
\(\mu_t\) is returned in the variables
`mu`

and `fitted.values`

, the scale \(\sigma^2\) is in the variable
`scale`

. `resid()`

method returns \(e_t = \frac{y_t}{\mu_t}\). Finally, the
prediction interval for the regression model are generated using
`qgamma()`

function from the `stats`

package.

One peculiar and very specific distribution, which can also be used in modelling is Exponential distribution. It only has one parameter, \(\lambda\), which regulates both mean and variance: \[\begin{equation*} \begin{aligned} & x \sim \mathrm{Exp}(\lambda) \\ & \mathrm{E}(x) = \frac{1}{\lambda} \\ & \mathrm{V}(x) = \frac{1}{\lambda^2} \end{aligned} . \end{equation*}\] It might be useful in cases, when one wants to model inter-arrival times.

The implementation in `alm()`

relies on the model, similar
to the Inverse Gaussian and Gamma models: \[\begin{equation*}
y_t = \mu_t \times \epsilon_t ,
\end{equation*}\] where \(\epsilon_t
\sim \mathrm{Exp} \left(1 \right)\), implying that \(\mathrm{E}(\epsilon_t) = \mathrm{V}(\epsilon_t) =
1\). This is a very restrictive model, which only works in some
special cases. If for some reason the variance and mean are not equal to
one in the empirical distribution, then the Exponential one would not be
appropriate. But in general the model formulated as above implies that
\(y_t \sim \mathrm{Exp}\left( \frac{1}{\mu_t}
\right)\), meaning that the variance of the model changes
together with the conditional expectation. The density function for the
error term in this case is: \[\begin{equation} \label{eq:Exp}
f(\epsilon_t) = \exp(-\epsilon_t).
\end{equation}\] Note that in our formulation \(\mu_t = \exp\left( x_t' B \right)\), so
that the mean is always positive, which implies that we deal with a pure
multiplicative model. In addition, we assume that \(\mu_t\) is just a scale for the
distribution, otherwise \(y_t\) would
not follow Exponential distribution.

The density function has the following shapes depending on the values of the expectation:

`alm()`

with `distribution="dexp"`

estimates
the density for \(y_t\) using
`dexp()`

function from `stats`

package. The \(\mu_t\) is returned in the variables
`mu`

and `fitted.values`

, the scale is assumed to
be equal to one. `resid()`

method returns \(e_t = \frac{y_t}{\mu_t}\). Finally, the
prediction interval for the regression model are generated using
`qexp()`

function from the `stats`

package.

**NOTE** that if the assumption of \(\mathrm{E}(\epsilon_t) = \mathrm{V}(\epsilon_t) =
1\) does not hold, the model will produce unreasonable
quantiles.

This is based on the exponent of Laplace distribution, which means that the PDF in this case is: \[\begin{equation} \label{eq:lLaplace} f(y_t) = \frac{1}{2 s y_t} \exp \left( -\frac{\left| \log y_t - \mu_t \right|}{s} \right) . \end{equation}\] The model implemented in the package has similarity with Log-Normal distribution. The MLE scale is: \[\begin{equation} \label{eq:bLogLaplace} \hat{s} = \frac{1}{T} \sum_{t=1}^T \left|\log y_t - \mu_t \right| . \end{equation}\] The density function of Log-Laplace has the following shapes:

Estimating the model with Log-Laplace distribution is equivalent to estimating the parameters of log-linear model: \[\begin{equation*} \log y_t = \mu_t + \epsilon_t, \end{equation*}\] where \(\epsilon_t \sim \mathcal{Laplace}(0, \sigma^2)\). This distribution might be useful if the data has a strong skewness (larger than in case of Log-Normal distribution).

`alm()`

with `distribution="dllaplace"`

uses
`dlaplace()`

function with the logarithm of actual values,
estimated mean \(\mu_t\) and the scale
\(\eqref{eq:sigmaLogLaplace}\). The
\(\mu_t\) is returned in the variable
`mu`

, the \(s\) is in the
variable `scale`

, while the `fitted.values`

contains the exponent of \(\mu_t\),
which corresponds to median of distribution rather than mean. Finally,
`resid()`

method returns \(e_t =
\log y_t - \mu_t\).

This is based on the exponent of S distribution, giving the PDF: \[\begin{equation} \label{eq:ls} f(y_t) = \frac{1}{4 y_t s^2} \exp \left( -\frac{\sqrt{|\log y_t - \mu_t|}}{s} \right) , \end{equation}\] The model implemented in the package has similarity with Log-Normal and Log-Laplace distributions. The MLE scale is: \[\begin{equation} \label{eq:bLogS} \hat{s} = \frac{1}{2T} \sum_{t=1}^T \sqrt{\left| \log(y_t) - \mu_t \right|} , \end{equation}\] The shape of the density function of Log-S is similar to Log-Laplace but with even more extreme values:

Estimating the model with Log-S distribution is equivalent to estimating the parameters of log-linear model: \[\begin{equation*} \log y_t = \mu_t + \epsilon_t, \end{equation*}\] where \(\epsilon_t \sim \mathcal{S}(0, \sigma^2)\). This distribution can be used for sever seldom right tail cases.

`alm()`

with `distribution="dls"`

uses
`ds()`

function with the logarithm of actual values,
estimated mean \(\mu_t\) and the scale
\(\eqref{eq:sigmaLogLaplace}\). The
\(\mu_t\) is returned in the variable
`mu`

, the \(s\) is in the
variable `scale`

, while the `fitted.values`

contains the exponent of \(\mu_t\),
which corresponds to median of distribution rather than mean. Finally,
`resid()`

method returns \(e_t =
\log y_t - \mu_t\).

This is based on the exponent of Generalised Normal distribution, giving the PDF: \[\begin{equation} \label{eq:lgnormal} f(y_t) = \frac{\beta}{2s \Gamma(1/\beta)y_t}\exp\left(-\left(\frac{|\log(y_t) - \mu|}{s}\right)^\beta\right), \end{equation}\] The model implemented in the package has similarity with Log-Normal, Log-Laplace and Log-S distributions. The MLE scale is: \[\begin{equation} \label{eq:LogAlpha} \hat{s} = \sqrt[^\beta]{\frac{\beta}{T} \sum_{t=1}^T \left| \log(y_t) - \mu_t \right|^{\beta}} . \end{equation}\] The shapes of the distribution depend on the value of parameters, giving it in some cases very long right tail:

Estimating the model with Log-Generalised Normal distribution is equivalent to estimating the parameters of log-linear model: \[\begin{equation*} \log y_t = \mu_t + \epsilon_t, \end{equation*}\] where \(\epsilon_t \sim \mathcal{GN}(0, s, \beta)\).

`alm()`

with `distribution="dlgnorm"`

uses the
`dgnorm()`

function from `greybox`

package with
the logarithm of actual values, estimated mean \(\mu_t\), the scale \(\eqref{eq:sigmaLogLaplace}\) and either
provided or estimated shape parameter \(\beta\). The \(\mu_t\) is returned in the variable
`mu`

, the \(s\) is in the
variable `scale`

and \(\beta\) is in `other$beta`

,
while the `fitted.values`

contains the exponent of \(\mu_t\), which corresponds to median of
distribution rather than mean. Finally, `resid()`

method
returns \(e_t = \log y_t - \mu_t\).

There is currently only one distribution in this group:

A random variable follows Logit-normal distribution if its logistic transform follows normal distribution: \[\begin{equation} \label{eq:logitFunction} z = \mathrm{logit}(y) = \log \left(\frac{y}{1-y}) \right), \end{equation}\] where \(y\in (0,1)\), \(y\sim \mathrm{logit}\mathcal{N}(\mu,\sigma^2)\) and \(z\sim \mathcal{N}(\mu,\sigma^2)\). The bounds are not supported, because the variable \(z\) becomes infinite. The density function of \(y\) is: \[\begin{equation} \label{eq:logitNormal} f(y_t) = \frac{1}{\sqrt{2 \pi \sigma^2} y_t (1-y_t)} \exp \left( -\frac{\left(\mathrm{logit}(y_t) - \mu_t \right)^2}{2 \sigma^2} \right) , \end{equation}\] which has the following shapes: Depending on the values of location and scale, the distribution can be either unimodal or bimodal and can be positively or negatively skewed. Because of its connection with normal distribution, the logit-normal has formulae for density, cumulative and quantile functions. However, the moment generation function does not have a closed form.

The scale of the distribution can be estimated via the maximisation of likelihood and has some similarities with the scale in Log-Normal distribution: \[\begin{equation} \label{eq:sigmaLogitNormal} \hat{\sigma}^2 = \frac{1}{T} \sum_{t=1}^T \left(\mathrm{logit}(y_t) - \mu_t \right)^2 . \end{equation}\]

Estimating the model with Log-Normal distribution is equivalent to estimating the parameters of logit-linear model: \[\begin{equation} \label{eq:logitLinearModel} \mathrm{logit}(y_t) = \mu_t + \epsilon_t, \end{equation}\] where \(\epsilon_t \sim \mathcal{N}(0, \sigma^2)\) or: \[\begin{equation} \label{eq:logitLinearModelExp} y_t = \mathrm{logit}^{-1}(\mu_t + \epsilon_t), \end{equation}\] where \(\mathrm{logit}^{-1}(z)=\frac{\exp(z)}{1+\exp(z)}\) is the inverse logistic transform.

`alm()`

with `distribution="dlogitnorm"`

does
not transform the provided data and estimates the density directly using
`dlogitnorm()`

function from `greybox`

package
with the estimated mean \(\mu_t\) and
the variance \(\eqref{eq:sigmaLogitNormal}\). The \(\mu_t\) is returned in the variable
`mu`

, the \(\sigma^2\) is in
the variable `scale`

, while the `fitted.values`

contains the inverse logistic transform of \(\mu_t\), which, given the connection
between the Normal and Logit-Normal distributions, corresponds to median
of distribution rather than mean. Finally, `resid()`

method
returns \(e_t = \mathrm{logit}(y_t) -
\mu_t\).

Beta distribution is a distribution for a continuous variable that is defined on the interval of \((0, 1)\). Note that the bounds are not included here, because the probability density function is not well defined on them. If the provided data contains either zeroes or ones, the function will modify the values using: \[\begin{equation} \label{eq:BetaWarning} y^\prime_t = y_t (1 - 2 \cdot 10^{-10}), \end{equation}\] and it will warn the user about this modification. This correction makes sure that there are no boundary values in the data, and it is quite artificial and needed for estimation purposes only.

The density function of Beta distribution has the form: \[\begin{equation} \label{eq:Beta}
f(y_t) = \frac{y_t^{\alpha_t-1}(1-y_t)^{\beta_t-1}}{B(\alpha_t,
\beta_t)} ,
\end{equation}\] where \(\alpha_t\) is the first shape parameter and
\(\beta_t\) is the second one. Note
indices for the both shape parameters. This is what makes the
`alm()`

implementation of Beta distribution different from
any other. We assume that both of them have underlying deterministic
models, so that: \[\begin{equation}
\label{eq:BetaAt}
\alpha_t = \exp(x_t' A) ,
\end{equation}\] and \[\begin{equation} \label{eq:BetaBt}
\beta_t = \exp(x_t' B),
\end{equation}\] where \(A\) and
\(B\) are the vectors of parameters for
the respective shape variables. This allows the function to model any
shapes depending on the values of exogenous variables. The conditional
expectation of the model is calculated using: \[\begin{equation} \label{eq:BetaExpectation}
\hat{y}_t = \frac{\alpha_t}{\alpha_t + \beta_t} ,
\end{equation}\] while the conditional variance is: \[\begin{equation} \label{eq:BetaVariance}
\text{V}({y}_t) = \frac{\alpha_t \beta_t}{((\alpha_t + \beta_t)^2
(\alpha_t + \beta_t + 1))} .
\end{equation}\] Beta distribution has shapes similar to the ones
of Logit-Normal one, but with shape parameters regulating respectively
the left and right tails of the distribution:

`alm()`

function with `distribution="dbeta"`

returns \(\hat{y}_t\) in the variables
`mu`

and `fitted.values`

, and \(\text{V}({y}_t)\) in the `scale`

variable. The shape parameters are returned in the respective variables
`other$shape1`

and `other$shape2`

. You will notice
that the output of the model contains twice more parameters than the
number of variables in the model. This is because of the estimation of
two models: \(\alpha_t\) \(\eqref{eq:BetaAt}\) and \(\beta_t\) \(\eqref{eq:BetaBt}\) - instead of one.

Respectively, when `predict()`

function is used for the
`alm`

model with Beta distribution, the two models are used
in order to produce predicted values for \(\alpha_t\) and \(\beta_t\). After that the conditional mean
`mu`

and conditional variance `variances`

are
produced using the formulae above. The prediction intervals are
generated using `qbeta`

function with the provided shape
parameters for the holdout. As for the confidence intervals, they are
produced assuming normality for the parameters of the model and using
the estimate of the variance of the mean based on the
`variances`

(which is weird and probably wrong).

This group includes:

These distributions should be used in cases of count data.

Poisson distribution used in ALM has the following standard probability mass function (PMF): \[\begin{equation} \label{eq:Poisson} P(X=y_t) = \frac{\lambda_t^{y_t} \exp(-\lambda_t)}{y_t!}, \end{equation}\] where \(\lambda_t = \mu_t = \sigma^2_t = \exp(x_t' B)\). As it can be noticed, here we assume that the variance of the model varies in time and depends on the values of the exogenous variables, which is a specific case of heteroscedasticity. The exponent of \(x_t' B\) is needed in order to avoid the negative values in \(\lambda_t\).

Here are several examples of the PMF of Poisson with different values of parameters \(\lambda\):

`alm()`

with `distribution="dpois"`

returns
`mu`

, `fitted.values`

and `scale`

equal
to \(\lambda_t\). The quantiles of
distribution in `predict()`

method are generated using
`qpois()`

function from `stats`

package. Finally,
the returned residuals correspond to \(y_t -
\mu_t\), which is not really helpful or meaningful…

Negative Binomial distribution implemented in `alm()`

is
parameterised in terms of mean and variance: \[\begin{equation} \label{eq:NegBin}
P(X=y_t) = \binom{y_t+\frac{\mu_t^2}{\sigma^2-\mu_t}}{y_t} \left(
\frac{\sigma^2 - \mu_t}{\sigma^2} \right)^{y_t} \left(
\frac{\mu_t}{\sigma^2} \right)^\frac{\mu_t^2}{\sigma^2 - \mu_t},
\end{equation}\] where \(\mu_t =
\exp(x_t' B)\) and \(\sigma^2\) is estimated separately in the
optimisation process. These values are then used in the
`dnbinom()`

function in order to calculate the log-likelihood
based on the distribution function.

Here are some examples of PMF of Negative Binomial distribution with different sizes and probabilities:

`alm()`

with `distribution="dnbinom"`

returns
\(\mu_t\) in `mu`

and
`fitted.values`

and \(\sigma^2\) in `scale`

. The
prediction intervals are produces using `qnbinom()`

function.
Similarly to Poisson distribution, `resid()`

method returns
\(y_t - \mu_t\). The user can also
provide `size`

parameter in ellipsis if it is reasonable to
assume that it is known.

Round up the response variable for the next example:

```
1] <- round(abs(xreg[,1]))
xreg[,<- xreg[1:180,]
inSample <- xreg[-c(1:180),] outSample
```

Negative Binomial distribution:

```
<- alm(y~x1+x2, data=inSample, distribution="dnbinom")
ourModel summary(ourModel)
#> Response variable: y
#> Distribution used in the estimation: Negative Binomial with size=25.548
#> Loss function used in estimation: likelihood
#> Coefficients:
#> Estimate Std. Error Lower 2.5% Upper 97.5%
#> (Intercept) 6.1830 0.1615 5.8644 6.5017 *
#> x1 0.0024 0.0052 -0.0079 0.0127
#> x2 -0.0011 0.0031 -0.0072 0.0049
#>
#> Error standard deviation: 86.6124
#> Sample size: 180
#> Number of estimated parameters: 5
#> Number of degrees of freedom: 175
#> Information criteria:
#> AIC AICc BIC BICc
#> 2152.028 2152.373 2167.993 2168.888
```

And an example with predefined size:

```
<- alm(y~x1+x2, data=inSample, distribution="dnbinom", size=30)
ourModel summary(ourModel)
#> Response variable: y
#> Distribution used in the estimation: Negative Binomial with size=30
#> Loss function used in estimation: likelihood
#> Coefficients:
#> Estimate Std. Error Lower 2.5% Upper 97.5%
#> (Intercept) 6.1381 0.1496 5.8429 6.4333 *
#> x1 0.0028 0.0048 -0.0068 0.0124
#> x2 -0.0003 0.0028 -0.0060 0.0053
#>
#> Error standard deviation: 86.3828
#> Sample size: 180
#> Number of estimated parameters: 4
#> Number of degrees of freedom: 176
#> Information criteria:
#> AIC AICc BIC BICc
#> 2151.235 2151.464 2164.007 2164.601
```

The final class of models includes two cases:

In both of them it is assumed that the response variable is binary
and can be either zero or one. The main idea for this class of models is
to use a transformation of the original data and link a continuous
latent variable with the binary one. As a reminder, all the models
eventually assume that: \[\begin{equation}
\label{eq:basicALMCumulative}
\begin{matrix}
o_t \sim \mathrm{Bernoulli}(p_t) \\
p_t = g(x_t' A)
\end{matrix},
\end{equation}\] where \(o_t\)
is the binary response variable and \(g(\cdot)\) is the cumulative distribution
function. Given that we work with the probability of occurrence, the
`predict()`

method produces forecasts for the probability of
occurrence rather than the binary variable itself. Finally, although
many other cumulative distribution functions can be used for this
transformation (e.g. `plaplace()`

or `plnorm()`

),
the most popular ones are logistic and normal CDFs.

Given that the binary variable has Bernoulli distribution, its log-likelihood is: \[\begin{equation} \label{eq:BernoulliLikelihood} \ell(p_t | o_t) = \sum_{o_t=1} \log p_t + \sum_{o_t=0} \log(1 - p_t), \end{equation}\] So the estimation of parameters for all the CDFs can be done maximising this likelihood.

In all the functions it is assumed that the probability \(p_t\) corresponds to some sort of unobservable `level’ \(q_t = x_t' A\), and that there is no randomness in this level. So the aim of all the functions is to estimate correctly this level and then get an estimate of probability based on it.

The error of the model is calculated using the observed occurrence variable and the estimated probability \(\hat{p}_t\). In a way, in this calculation we assume that \(o_t=1\) happens mainly when the respective estimated probability \(\hat{p}_t\) is very close to one. So, the error can be calculated as: \[\begin{equation} \label{eq:BinaryError} u_t' = o_t - \hat{p}_t . \end{equation}\] However this error is not useful and should be somehow transformed into the original scale of \(q_t\). Given that both \(o_t \in (0, 1)\) and \(\hat{p}_t \in (0, 1)\), the error will lie in \((-1, 1)\). We therefore standardise it so that it lies in the region of \((0, 1)\): \[\begin{equation} \label{eq:BinaryErrorBounded} u_t = \frac{u_t' + 1}{2} = \frac{o_t - \hat{p}_t + 1}{2}. \end{equation}\]

This transformation means that, when \(o_t=\hat{p}_t\), then the error \(u_t=0.5\), when \(o_t=1\) and \(\hat{p}_t=0\) then \(u_t=1\) and finally, in the opposite case of \(o_t=0\) and \(\hat{p}_t=1\), \(u_t=0\). After that this error is transformed using either Logistic or Normal quantile generation function into the scale of \(q_t\), making sure that the case of \(u_t=0.5\) corresponds to zero, the \(u_t>0.5\) corresponds to the positive and \(u_t<0.5\) corresponds to the negative errors. The distribution of the error term is unknown, but it is in general bimodal.

We have previously discussed the density function of logistic
distribution. The standardised cumulative distribution function used in
`alm()`

is: \[\begin{equation}
\label{eq:LogisticCDFALM}
\hat{p}_t = \frac{1}{1+\exp(-\hat{q}_t)},
\end{equation}\] where \(\hat{q}_t =
x_t' A\) is the conditional mean of the level, underlying the
probability. This value is then used in the likelihood \(\eqref{eq:BernoulliLikelihood}\) in order
to estimate the parameters of the model. The error term of the model is
calculated using the formula: \[\begin{equation} \label{eq:LogisticError}
e_t = \log \left( \frac{u_t}{1 - u_t} \right) = \log \left( \frac{1
+ o_t (1 + \exp(\hat{q}_t))}{1 + \exp(\hat{q}_t) (2 - o_t) - o_t}
\right).
\end{equation}\] This way the error varies from \(-\infty\) to \(\infty\) and is equal to zero, when \(u_t=0.5\).