1. Autumn School Dynamic MSM16-18 November 2015 | L-Esch-sur-Alzette Slide 1 7. Stochastic concepts in LIAM2

2. Autumn School Dynamic MSM16-18 November 2015 | L-Esch-sur-Alzette Slide 2  Random functions  Choice  Logit_score  Align  Align_abs  Logit_regr  Other regressions Concepts

3. Autumn School Dynamic MSM16-18 November 2015 | L-Esch-sur-Alzette Slide 3  LIAM2 includes support for many random number generator functions.  For details, see LIAM2 User Guide.  Examples:  uniform(): uniform in [0, 1)  normal(): standard normal (mean=0, std=1)  randint(0, 10): random integer between 0 and 10 excluded Random functions

4. Autumn School Dynamic MSM16-18 November 2015 | L-Esch-sur-Alzette Slide 4  Stochastic simulation relies on random numbers, but true random numbers generators do not exist  Advantage of pseudo-random generators: speed and reproducibility  Fortunately, Monte-Carlo is not too picky in that department – as long as u is “random enough”  Python (and thus LIAM2) uses the Mersenne Twister (Matsumoto and Nishimura, 1997).  Use “random_seed: to fix the sequence of random numbers  In the latest version of LIAM2 (0.10.3) this can also be done in mid- simulation Random functions

5. Autumn School Dynamic MSM16-18 November 2015 | L-Esch-sur-Alzette Slide 5 Suppose event Y with two possible outcomes: 0 and 1 Pr[Y=1] = p Pr[Y=0] = 1 - p Draw a uniform random number u i If u i < p then the event is realized and Y i becomes 1 Example: Demo06 Gender (at birth) = choice([options],[probabilities]) = choice([False, True],[0.51, 0.49]) = choice([MALE, FEMALE],[0.51, 0.49]) Choice

6. Autumn School Dynamic MSM16-18 November 2015 | L-Esch-sur-Alzette Slide 6 Logit_score and logit_regr

7. Autumn School Dynamic MSM16-18 November 2015 | L-Esch-sur-Alzette Slide 7 Example: we want to simulate whether an individual is being born in or outside Belgium, and suppose that this depends on whether the individual is 50 years or over, or not. immigrant= False (Born in Belgium) True (born outside Belgium) old = False (age < 50) True (age ≥ 50) birth= -0.190136 -1.607228 oldlogistic regression on SILC Belgium for 2013 How can this be simulated? Take the inverse logit of the regression results and compare with a uniform stochast - a: -0.190136 -1.607228 * old - interm: exp(a) / (1+exp(a)) - immigrant: interm > uniform() Or use the fact that logit_score(a) = logistic(a - logit(u)) - interm2: logit_score(-0.190136 -1.607228 * old) - immigrant: interm2 > 0.5 Or use the fact that logit_regr is a combination of logit_score and the evaluation condition - immigrant: logit_regr(a) - immigrant: logit_regr(-0.190136 -1.607228 * old) Logit_score and logit_regr

8. Autumn School Dynamic MSM16-18 November 2015 | L-Esch-sur-Alzette Slide 8  Alignment is used when we want that the number of events occuring per category matches an exogenous proportion.  The methodology used for now by LIAM2 is called “alignment by sorting”, that is, for each category, the N individuals with the highest scores are selected.  The score computation has to be computed by the modeller (with logit_score() for example) Stochastic simulation using alignment

9. Autumn School Dynamic MSM16-18 November 2015 | L-Esch-sur-Alzette Slide 9  Example: we want to simulate whether an individual is being born in or outside Belgium immigrant= False (Born in Belgium) True (33.75%) old = False (age < 50) True (age ≥ 50; 36%) birth: logit_regr(-0.190136 -1.607228 * old)  But now we want the proporion of immigrants to be exactly x% - birth: logit_regr(-0.190136 -1.607228 * old, align=0.3375)  Suppose that %(old | immigrant) = 15.26% in the SILC data Stochastic simulation using alignment

10. Autumn School Dynamic MSM16-18 November 2015 | L-Esch-sur-Alzette Slide 10  score  proportions  filter (optional arguments)  take (optional arguments)  leave (optional arguments)  expressions (optional arguments)  possible_values (optional arguments)  frac_need (optional arguments) Align – structural form

11. Autumn School Dynamic MSM16-18 November 2015 | L-Esch-sur-Alzette Slide 11  It must be an expression (or a simple variable) returning a numerical value.  It will be used to rank individuals.  One will usually use logit_score() to compute the score, but it can be computed in any other way a modeller choose.  Note that the score is not modified in any way within the align() function, so if one wants a random factor, it should be added manually. Align - score

12. Autumn School Dynamic MSM16-18 November 2015 | L-Esch-sur-Alzette Slide 12  It must be an expression (or a simple variable) returning a numerical value.  It will be used to rank individuals.  One will usually use logit_score() to compute the score, but it can be computed in any other way a modeller choose.  Note that the score is not modified in any way within the align() function, so if one wants a random factor, it should be added manually.  Examples; using logit_regr -immigrant4: logit_regr(-0.190136 -1.607228 * old, align=0.3375)  is equivalent to -log_interm= logit_score(-0.190136 -1.607228 * old) -immigrant4: align(log_interm, 0.33775)  Then why is the below expression wrong? -immigrant4: align(-0.190136 -1.607228 * old, 0.33775) Align - score

13. Autumn School Dynamic MSM16-18 November 2015 | L-Esch-sur-Alzette Slide 13  take: an expression specifying individuals which should always be selected, regardless of their score.  leave: an expression specifying individuals which should never be selected, regardless of their score.  Example: suppose that we want to model that in any period, exactly 5% of the population has a cat, and nobody older than 50 has a cat. - interm: logit_score(0.0) - has_a_cat: align(interm, 0.05, leave = old) Align – take and leave

14. Autumn School Dynamic MSM16-18 November 2015 | L-Esch-sur-Alzette Slide 14  The take and leave conditions are the only conditions that give the distribution of individual characteristics priority over the alignment totals.  Example: suppose that we want to model that in any period, exactly 5% of the population has a cat, and anybody older than 50 has a cat. - interm: logit_score(0.0) - has_a_cat: align(interm, 0.05, take = old) Hence all old = 1 (about 36% of the sample) will all have a cat, and not 5%  Question: is there a better, “softer” way in modeling this? Align – take and leave

15. Autumn School Dynamic MSM16-18 November 2015 | L-Esch-sur-Alzette Slide 15  align_abs is equivalent to align(), except that it aligns to absolute numbers instead of proportions.  It also supports a few additional arguments to work on a linked entity.  Chénard’s (2001) algorithm Align_abs

16. Autumn School Dynamic MSM16-18 November 2015 | L-Esch-sur-Alzette Slide 16  score  need  filter (optional arguments)  take (optional arguments)  leave (optional arguments)  expressions (optional arguments)  possible_values (optional arguments)  frac_need (optional arguments)  link (optional arguments)  secondary_axis (optional arguments)  errors (optional arguments) Align_abs – structural form

17. Autumn School Dynamic MSM16-18 November 2015 | L-Esch-sur-Alzette Slide 17 Align_abs – specific arguments Align_abs(…, need, link=link name to entity level 2, …) e.g. align_abs(…,200, link=persons, …) called on the level of the household Select observations on entity level 1 In order to get a close as possible to alignment targets ‘need’ that pertain to entity level 2 Used on entity level 1

18. Autumn School Dynamic MSM16-18 November 2015 | L-Esch-sur-Alzette Slide 18  Continuous (expr + normal(0, 1) * mult + error_var): cont_regr(expr[, filter=None, mult=0.0, error_var=None])  Clipped continuous (always positive): clip_regr(expr[, filter=None, mult=0.0, error_var=None])  Log continuous (exponential of continuous): log_regr(expr[, filter=None, mult=0.0, error_var=None]) Other regressions

19. Autumn School Dynamic MSM16-18 November 2015 | L-Esch-sur-Alzette Slide 19 Exercises

20. Autumn School Dynamic MSM16-18 November 2015 | L-Esch-sur-Alzette Slide 20 Stochastic simulation using choice Exercise 1 Can you simulate the eye colour (eye_colour) of newborns, using the following probabilities Brown: 55% Blue: 8% Green: 2% Other: 35% (hazel, amber, …) Source: “Eye color worldwide” (https://www.quora.com/Whats-the-distribution-of-eye-colors-in-the-world) [06/11/2015]https://www.quora.com/Whats-the-distribution-of-eye-colors-in-the-world

21. Autumn School Dynamic MSM16-18 November 2015 | L-Esch-sur-Alzette Slide 21 Exercise 2 In the current setting, only the married woman decides to divorce. - divorce: logit_regr(0.6713593 * nb_children - 0.0785202 * dur_in_couple + 0.1429621 * agediff - 0.0088308 * agediff **2 - 4.546278, filter=ISFEMALE and ISMARRIED and (dur_in_couple > 0), align='al_p_divorce.csv') Suppose that you want to use two estimated logistic regression models, one for women with and without children. The two (fictitious!) models are -4.0 - 0.01 * dur_in_couple + 0.2 * agediff - 0.01 * agediff **2 for women without children - 4.0 - 0.005 * dur_in_couple + 0.2 * agediff - 0.01 * age_youngest_child for women who live with children in the household Stochastic simulation using logistic regressions

22. Autumn School Dynamic MSM16-18 November 2015 | L-Esch-sur-Alzette Slide 22 Exercise 3 Suppose that we want to implement a mandatory period of inactivity for one year for women following child birth. How would you implement this in Demo08.yml? Stochastic simulation using logistic regressions