How to determine the type of regression to be used? I am relatively new into the machine learning field and I came up with the following problem that is giving me some headaches, so any help on it would be greatly appreciated for my inner peace. 
Consider the following example. For each day of every week of 2017 we have a set of tickets that are classified in 3 main categories (P1, P2 and P3) and we came of with the following results:
Year 2017

Category P1

| Week number | Day of week | Number of tickets|
| ----------- | ------------| ---------------- |
|     30      |   Monday    | 9                |
|     30      |   Tuesday   | 4                |
|     30      |   Wednesday | 9                |
|     30      |   Thursday  | 12               |
|     30      |   Friday    | 9                |
|     30      |   Saturday  | 1                |
|     30      |   Sunday    | 1                |
|     31      |   Monday    | 3                |
|     31      |   Tuesday   | 2                |
|     31      |   Wednesday | 13               |
|     31      |   Thursday  | 10               |
|     31      |   Friday    | 5                |
|     31      |   Saturday  | 0                |
|     31      |   Sunday    | 2                |
   ......

Category P2
| Week number | Day of week | Number of tickets|
| ----------- | ------------| ---------------- |
|     30      |   Monday    | 51               |
|     30      |   Tuesday   | 59               |
|     30      |   Wednesday | 59               |
|     30      |   Thursday  | 94               |
|     30      |   Friday    | 43               |
|     30      |   Saturday  | 16               |
|     30      |   Sunday    | 29               |
|     31      |   Monday    | 48               |
|     31      |   Tuesday   | 59               |
|     31      |   Wednesday | 41               |
|     31      |   Thursday  | 47               |
|     31      |   Friday    | 32               |
|     31      |   Saturday  | 18               |
|     31      |   Sunday    | 38               |
   .......  

Category P3
| Week number | Day of week | Number of tickets|
| ----------- | ------------| ---------------- |
|     30      |   Monday    | 40               |
|     30      |   Tuesday   | 41               |
|     30      |   Wednesday | 44               |
|     30      |   Thursday  | 34               |
|     30      |   Friday    | 32               |
|     30      |   Saturday  | 13               |
|     30      |   Sunday    | 4                |
|     31      |   Monday    | 41               |
|     31      |   Tuesday   | 56               |
|     31      |   Wednesday | 44               |
|     31      |   Thursday  | 46               |
|     31      |   Friday    | 46               |
|     31      |   Saturday  | 17               |
|     31      |   Sunday    | 4                |
   ......

Basically I want to predict the number of tickets for each category (P1, P2, and P3) that might be generated on the next day and I use the number of weeks as a reference to lock up this problem, that is, I start on week 1 and every time I finish with Sunday then I go on the next week and I continue with my prediction of tickets based on the data of previous days, besides, once I have reached week 53 then I should stop with the prediction because that is the beginning of another year. 
I am not sure what type of regression I should use for this problem but my guess is that I could use a simple linear regression because I am predicting the data of the following day based on the previous number of tickets that were opened and week number is just like a boundary for setting up a start and end point. On the other hand, if I were to predict the number of tickets that might be opened in this 2018, then I should use multivariate multiple regression on data of years 2015, 2016 and  2017 because I should use week number and day of the week as my dependent variables to predict the number of tickets that might be opened.
If I am wrong, please correct me because I am in this site to learn as much as I can in this exciting field of machine learning. 
Note: Please feel free to download my data to see if you can find any more insights https://github.com/alexbr9007/DataSets 
 A: If I understand your problem correctly, you essentially have a time-series of three output values at times $t=1, ..., 365$, corresponding to days in a year.  Since you presently only have data for one year, this means that you cannot really hypothesise and test annual seasonality in your data.  Realistically, you are not going to be able to look at a single year of data and also create a decent predictive model for that same year.  (At best you could split your data into train-test split via random sampling, but with only a single year of data in total, this would not give you much to work with.)  Once you can get multiple years of data then it may be possible to create a predictive model for outcomes in a future year.
Your suggested model for this data is to use day and week as explanatory variables for the time effect, and presumably factor(category) to separate the three categories of tickets.  This means that you assume there is a possible main effect for each of these terms, but no interaction.  That is one way to model the data, but it is not the only way.  Some other options include using seasonal terms (i.e., sinusoidal terms), including a term for public holidays, or categorising day into only weekdays and weekends.
Rather than hypothesising a model form at the outset, and trying to create a predictive model immediately, it would be better to undertake some basic exploratory analysis using some simple graphs of the data, to see if you can get any sense of the nature of your data.  This would be useful for a broader project of modelling if you can get additional years of data, but it gives you a starting point where you have a basic understanding of what are plausible things to add into your model.  A few obvious exploratory plots you should look at are:


*

*Time-series plot of your data over the year, for the three ticket categories.   Try making one plot with all 365 daily values for each ticket category, and if this is too noisy to interpret easily, make a smoothed plot with the 52 weekly averages for each ticket category.  This should give you a basic sense of whether there is some seasonality in the data (though you can't explore and test with the same data set).  Is there a seasonal pattern in the data?  Are there days where the tickets spike?  (E.g., Saturdays, public holidays, Superbowl Sunday, etc.)

*Frequency plot showing the discrete Fourier transform of your data in the frequency domain.  This is another plot that will help you to see seasonality in the data.  With only a year of data it will not be able to detect anything larger than a six-month seasonality.  If you can get multiple years of data then you should be able to detect an annual seasonal effect.  In any case, it is most useful for seeing higher frequency fluctuations that might escape your eye in a standard time-series plot.

*Day-of-the-week plot showing the distribution of outputs for the three categories on each day-of-the-week (e.g., box and whiskers plot or violin plot).  Are tickets more likely to be sold on any particular day of the week?  (E.g., more tickets sold on weekend days.)  You might also want to separate public holidays into another category and treat these distinctly.
Once you have done some basic exploration, you could try fitting a regression model to the data.  Since your output is a count variable (non-negative integer), the obvious starting point for modelling would be the negative-binomial GLM.  Once you have fit a basic model with appropriate explanatory variables you can produce the residuals for the model fit.  This would allow you to also create an auto-correlation plot of the residuals to see if there is evidence of auto-correlated errors in your data (after filtering out explanatory variables like day-of-the-week).  This could lead to a more complex time-series model.

Note: If you are able to post your full data set on here then I'm sure there would be users who would happily play with it for you and give some insights.
