Following the thermodynamic example the Gibbs free energy is:

*G* = *U* + *P**V* − *S**Δ**T*

*Sketch of the infrastructure and customer information flow.*

Throughout this document we are going to analyze some branches of this sketch.

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`attached'' to the ticket. \begin{equation} g(x) = w_tf_t(x) + \sum_{i\in \{a\}}w_if_i(x) \end{equation} where $w$ is the margin for each sale and $f$ the frequency of sale per session. We assume that the probability, $p$, of an ancillary sale is always conditioned by the ticket sale. \begin{equation} p_a := p(a|t) = \frac{p(a)p(t|a)}{p(t)}=\frac{p(a)}{p(t)} \qquad p(t|a) = 1 \end{equation} where $p$ is the probability correspondent to the frequency $f$. The metric $m$ is the {\bf session metric} defined as the path to a {\bf conversion}. The straightes metric for the booking is: \begin{equation} ProductList - ItemSelection - CustomerDetail - Payment \end{equation} The formulation of the ticket booking frequency $f_t$ is: \begin{equation} f_t = f_{list} \cdot f_{it} \cdot f_{cd} \cdot f_{book} \end{equation} where we call $f_{st}$ the frequency of a generic step conversion.\par When the booking comes from an external campaign an additional frequency is added: \begin{equation} f_t = (1-bouncerate) \cdot f_{list} \cdot f_{it} \cdot f_{cd} \cdot f_{book} \end{equation} \section{Variant testing} If we split a page into different variants we influence the frequency of the conversion step: \begin{equation} f_{st} \to \left\{\begin{array}{c} f^a\\f^b\end{array}\right. \end{equation} We perform a time average of the two frequencies, $\mean{f^a}_t$, $\mean{f^b}_t$ and find the winning variant performing a t-test \begin{equation} t_{test} = \frac{\mean{f^a}_t - \mean{f^b}_t}{\sqrt{s_a/N_a + s_b/N_b}} \end{equation} Where $N_a$ and $N_b$ are the sample sizes and $s_a$ and $s_b$ are the standard deviation of the correspondent frequencies and follow a $\chi^2$ distribution. \par If the difference between state $a$ and state $b$ consists in more than a single change we should consider the influence of each of the $M$ element changes: \begin{equation} T^2 = \frac{N_aN_b}{N_a+N_b}(\bra{\bar a} - \bra{\bar b}) \mathrm{Cov}_{ab}^{-1}(\ket{\bar a} - \ket{\bar b}) \end{equation} where $\mathrm{Cov}_{ab}$ is the sample covariance matrix \begin{equation} \mathrm{Cov}_{ab} := \frac{1}{N_a+N_b-2} \sum_{i=1}^{N_a} \Big((\ket{a}_i - \ket{\bar a})(\bra{a}_i - \bra{\bar a}) + (\ket{b}_i - \ket{\bar b})(\bra{b}_i - \bra{\bar b})\Big) \end{equation} \section{Probability distribution} The correspondent probability of a number of conversions, $n_c$, in a day $i$, for an action $x_a$, follows the Poisson distribution: ($\Gamma(k) = (k-1)!$) \begin{equation} p_{st}(n_c,t;\lambda) = \frac{(\lambda t)^{n_c}}{n_c!}e^{-(\lambda t)} \end{equation} which is comparable to a Gaussian distribution with $\sigma = \sqrt{\lambda t}$ for $\lambda t \geq 10$. \par The probability distribution of a data set $\{n_c\}$ is: \begin{equation} P_g(\{n_c\}) = \prod_{i=1}^N \frac1{\sqrt{2\pi n_i}}e^{-\frac{(n_i-\lambda_it_i)^2}{2n_c}} \end{equation} The fitting parameter, $\lambda_i t_i$, can be determined by minimizing the $\chi^2$. \begin{equation} \chi^2 = \sum_{i=1}^{N_d}\frac{(n_i-\lambda_i t_i)^2}{n_i} \end{equation} since for Poisson distributions $\sigma_i^2 = n_i$. If we instead measure a continous quantity attached to the action (e.g. price, $x_p$) we can consider a Gaussian distribution \begin{equation} p_{st}(x_p) = \frac{1}{\sigma_p\sqrt{2\pi}} e^{-\frac{(x_p-\bar{x_p})}{2\sigma^2_p}} \end{equation} \section{Metric noise propagation} We have defined the frequency of bookings $f_t$ as the product of frequencies per conversion step. \begin{equation} f_t = f_{list} f_{it} f_{cd} f_{book} = f_1 f_2 f_3 f_4 \qquad f_t \simeq f^0_t + \sum_{i=1}^4 \partial_i f_t \cdot \Delta f_i \end{equation} The error of the total conversion propagates as follow: \begin{equation} s_t^2 = \mean{f_t - \mean{f_t}}^2 = \sum_{i=1}^4 (\partial_i f_t)^2 s_i^2 = f_t^2 \sum_{i=1}^4 (s_i^2/f_i^2) \end{equation} Where we have assumed that cross correlations $s_{12},s_{13},s_{14},\ldots$ are all zero because of statistical independence of the different pages since no test is extended on another page. That implies that the less noisy metrics are short and defined across simple pages (few biforcation).\par The typical noise generated from a Poisson process is the shot noise, which is a white noise and it's power spectrum depends on the intensity of the signal. The signal to noise ratio of a shot noise is $\mu/\sigma=\sqrt{N}$.\par To filter out the noise we have to study the autocorrelation function of the time series \begin{equation} corr_{f_t}(x) = \sum_{y=-\infty}^\infty f_t(y)f_t(y-x) \simeq f_0 e^{-cx} %\mathrm{or} f_0\left(1-\frac{x}{c}\right) \end{equation} Being a white noise we can clean the signal removing the null component of the Fourier transform: \begin{equation} f_{clean}(x) = \int \left( \hat f(k) - \frac1\pi\int_\pi^\pi f(x) dx\right) e^{-2\pi\imath xk}dk \end{equation} All the different variants of the tests (e.g. languages) should have a comparable power spectrum. \section{Learning segment performancees} During a web session the parameters, $p_i$, which can be used for segmenting the audience are: \begin{figure}[!h] \centering \includegraphics[width=.8\textwidth]{AncillarySketch} \caption{Environement parameters and ancillary offers.} \end{figure} \begin{itemize} \item {\bf behaviour}: returning, history, speed \item {\bf item}: which product from the list \item {\bf time}: time till usage, booking time frame (h/d/m/y) \item {\bf source}: referrer, searched words, social \item {\bf location}: nation, landscape, quarter, company, weather \item {\bf session}: browser, view, device, provider \item {\bf class}: quality class among similar products \item {\bf travel/specific}: type of group traveling, origin and destination \end{itemize} A learning process can be done correlating the performance of each teaser with the presence of a certain variable, e.g. via a naïve Baysian approach: \begin{equation} P(t|p_1,p_2,p_3,\ldots) = \frac{P(t)P(p_1|t)P(p_2|t)P(p_3|t)\ldots}{P(p_1,p_2,p_3,\ldots)} \end{equation} Where $P(t|p_1,p_2,p_3,\ldots)$ is the a posteriori probability of a teaser performance for a given presence of the paramters $p_i$.\par Since the number of parameters is around 1000/2000 this process is very inefficient and requires a large learning phase. Most of the environmental variables are correlated and should be hence clamped into macro environemental variables findinf the few functions that define an efficient session scoring. A big correlation matrix between all parameters should create a comprehensive hit map to guide the simplification and dimensional reduction of the parameter space. \par Second, some parameters shold be more influent in changing the teaser response and a correspondent`

learning weight’’ should be assigned to each parameter.
While the margin parameter Offer performances should be tracked over different touchpoints to understand which options is the customer considering throughout his journey. Negative offer performances are as well really important to gain information about the customer.

During a journey the customer has to take different decisions and decline an offer might be an important source of information about how he is confindent in the city he is visiting.For many biforcation points of his route we can consider a payoff matrix like: Generally *a*_{11}, *a*_{12}, *b*_{12}, *b*_{22} are negative. The determination of these parameters allow the estimation of a score about the customer/city link.

Each cluster can be separately processed.

Each external unit process the information about a single user with no communication required among different units. The central unit computes the masks and communicate the new values to each of the external unit in an asynchronous time. The external units are completely autonomous and the calculation time of the new masks do not affect the response time of the system. The masks can be time dependent and different sets of masks might be selected depending on the hour of the day or the day of the week. Given: 50M users, 10G views, 1M url, 10k pagesAccording to the performances of Neo4j there should be no large computing time required to precess 50M nodes but different levels of coarse-graining might be introduced in the graph to improve performances. That means that different databases might be used to compute different magnification of the graph.

We know the ad visibility, the page visited and the ad selection. We construct the cumulative probability of the revenue summing up each single revenue WhereThe dash represents the same categories calculated by user grouping and let the user filter the information by populating the dash.

One of the scoring parameter for a customer is the analysis of the feedbacks he writes in different channels.

Specific customer feeedbacks were used to train dictionaries for the association of some keywords to specific arguments of sentiments. The training of the dictionaries is really important for understanding the sentiment of the customer and extract the most relevant words. Those words can be connected to a particular sentiment. The customers were asked to choose a category and rate their satisfaction to each comment. We had then the chance to create topic and sentiment specific dictionaries to be able to automatic guess by user input the most probable reason of his feedback. We run an emotion recognition to understand which words are more common to different topics. We have analyzed as well the clustering and the correlation between words to see whether it was necessary to clamp words together or to assign a cluster number to the words. It is as well important to read general statistical properties of the comment to recognize whether they are formally similar.In this case we can correlate time periods together to understand which topic are more interesting in particular time of the year.