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+autoplot(ent_hosp_tbl, Employed) +``` +Intentaremos usar varios modelos, incluso algunos que claramente +no son correctos: + +1. Nivel local +2. Nivel local y tendencia +3. Nivel local con estacionalidad (12 meses) +4. Nivel local con tendencia y estacionalidad (12 meses) +5. Nivel semilocal con tendencia y estacionalidad +6. Nivel semilocal con tendencia, estacionalidad, y una componente +autoregresiva. + +En este ejemplo usaremos el paquete *bsts* que implementa modelos de espacio +de estados usuales de manera relativamente fácil y rápida (aunque es necesario hacer +diagnósticos aparte), y tiene algunas funciones convenientes para evaluar pronósticos. + +```{r} +#| message: false +#| warning: false +library(bsts) +set.seed(998) +#inicial_nivel <- NormalPrior(11000, 1500, initial.value = 11000) +mod_spec_1 <- AddLocalLevel(list(), y = ent_hosp_ts) +mod_spec_2 <- AddLocalLinearTrend(list(), y = ent_hosp_ts) +mod_spec_3 <- AddLocalLevel(list(), y = ent_hosp_ts) |> + AddSeasonal(nseasons = 12, y = ent_hosp_ts) +mod_spec_4 <- AddLocalLinearTrend(list(), y = ent_hosp_ts) |> + AddSeasonal(nseasons = 12, y = ent_hosp_ts) +mod_spec_5 <- AddSemilocalLinearTrend(list(), y = ent_hosp_ts) |> + AddSeasonal(nseasons = 12, y = ent_hosp_ts) +mod_spec_6 <- AddSemilocalLinearTrend(list(), y = ent_hosp_ts) |> + AddSeasonal(nseasons = 12, y = ent_hosp_ts) |> + AddAr(lags = 2, y = ent_hosp_ts) +specs <- list(mod_spec_1, mod_spec_2, mod_spec_3, mod_spec_4, mod_spec_5, mod_spec_6) +ajustes <- map(specs, function(spec){ + bsts(ent_hosp_ts, spec, niter = 40000, ping = 10000) +}) +``` + +En primer lugar, podemos comparar las predicciones a un paso de cada modelo. +En este caso, los modelos 3 y 4 son claramente superiores a 1 y 2 en desempeño +a un paso. El mejor modelo en desempeño a un paso es el modelo 4, por la tendencia +lineal de la serie. + +```{r} +#| fig-height: 6 +#| fig-width: 7 +CompareBstsModels(ajustes, burn = 20000) +``` + +```{r} +ajuste <- ajustes[[6]] +plot(ajuste, "components") +``` + + +```{r} +#| fig-height: 3 +#| fig-width: 7 +pred_errors_tbl <- + bsts.prediction.errors(ajuste, burn = 20000)$in.sample |> + t() |> as_tibble() |> + mutate(t = 1: length(ent_hosp_ts)) |> + pivot_longer(-c(t), names_to = "sim", values_to = "valor") |> + group_by(t) |> + summarise(valor = mean(valor)) |> + as_tsibble(index = t) +# filtramos a partir de 12 meses (mínimo requerido para +# estimar estacionalidad): +ACF(pred_errors_tbl |> filter(t > 12), valor, lag_max = 20) |> + autoplot() + ylim(c(-1,1)) +``` +Ahora podemos hacer pronósticos: + + + + +```{r} +#| warning: false +#| fig-height: 3 +#| fig-width: 7 +ajuste <- ajustes[[6]] +pred <- predict(ajuste, horizon = 36, burn = 30000) + plot(pred) +``` + +Podemos hacer también pronósticos agregados. Como tenemos simulaciones de los +pronósticos, esto es fácil, incluyendo intervalos predictivos. En el siguiente +ejemplo, hacemos un intervalo para la suma de los 36 meses que pronosticamos: + +```{r} +#| fig-width: 4 +#| fig.height: 3 +pred$distribution |> dim() +apply(pred$distribution, 1, sum) |> qplot() +print("Intervalo de 90% para el agregado:") +apply(pred$distribution, 1, sum) |> quantile(c(0.05, 0.50, 0.95)) + +``` + +## Análisis de datos diarios + +Consideremos por ejemplo la siguiente serie de nacimientos por día +en México (ver la sección de introducción de series de tiempo): + +```{r} +natalidad_tbl <- read_rds("../datos/natalidad.rds") |> + ungroup() |> + arrange(fecha) |> + filter(fecha >= ymd("2008-01-01")) |> + mutate(nacimientos = log(n)) |> + select(-fecha_str, -n) |> + as_tsibble(index = fecha) +``` + + +```{r} +y <- natalidad_tbl |> as.ts(frequency = 365.25) +y <- zoo(natalidad_tbl$nacimientos, natalidad_tbl$fecha) +natalidad_tbl <- natalidad_tbl |> + mutate(feb_29 = as.numeric(month(fecha)==2 & day(fecha) == 29)) +feb_29 <- natalidad_tbl$feb_29 +mar_1_feb_29 <- lag(feb_29, default = 0) +``` + +En este caso, es necesario: + +- Agregar estacionalidad con periodo de 7 días +- Agregar estacionalidad con periodo de 1 año +- Agregar efectos de días de asueto, incluyendo posibles efectos que +no ocurren solamente el día en cuestión, sino en los "hombros" del día +(unos días antes o después). + +La estacionalidad anual la modelamos con armónicos de periodo 365.25, +es decir, con $\lambda_j = 2\pi j /365.25$. Cada armónico agrega dos +coeficientes al espacio de estados. + +### Estacionalidad trigonométrica {-} + +Supongamos que queremos modelar estacionalidad anual para datos mensuales +usando armónicos. +El primer armónico tiene un ciclo de 12 meses, así +que ponemos +$$\lambda_1 = 2\pi /12 = \pi/6$$ +Consideramos las funciones + +$$f_1(t) = a\cos(\lambda_1 t) + b \sin(\lambda_1 t)$$ + +Con el tamaño de a y b se define la amplitud, y la fase +depende de los tamaños relativos de $a$ y $b$. Esto lo podemos +ver, por ejemplo, notando que $a\sin x + b\cos x = \sqrt{a^2 + b^2}\sin(x + \alpha)$. + + +```{r} +#| fig-height: 3 +#| fig-width: 7 +arm_1 <- tibble(t=1:25) |> + mutate(f_1 = -2 * cos(t * pi / 6) + 4 * sin(t * pi / 6)) |> + mutate(n = 1) |> +bind_rows( + tibble(t=1:25) |> + mutate(f_1 = 0.5 * cos(t * pi / 6) + 0.1 * sin(t * pi / 6)) |> + mutate(n = 2) +) +ggplot(arm_1, aes(x = t, y = f_1, colour = factor(n))) + geom_line() + + geom_point() + + geom_vline(xintercept = c(1, 13, 25)) +``` + +Podemos considerar armónicos más altos. El siguiente es + +$$f_2(t) = f_1(t) + a_2\cos(2\lambda_1 t) + b_2 \sin(2\lambda_1 t)$$ +para obtener una funciones como las que siguen: + +```{r} +#| fig-height: 3 +#| fig-width: 7 +arm_2 <- arm_1 |> + mutate(f_2 = f_1 + -3 * cos(2 * t * pi / 6) + 2 * sin(2 * t * pi / 6)) +ggplot(arm_2, aes(x = t, y = f_2, colour = factor(n))) + geom_line() + + geom_point() + + geom_vline(xintercept = c(1, 13, 25)) +``` +Y así sucesivamente. Cualquier función periodica puede aproximarse bien +mediante una suma de cosenos y senos, siempre y cuando utilicemos +suficientes armónicos. + +Nota que en los ejemplos de arriba necesitamos dos coeficientes para cada +armónico. La componente estacional del modelo podemos escribirla como +(para estacionalidad constante): + +$$\gamma_t = \sum_{j=1}^L (a_j \cos(\lambda_j t) + a^*_j \sin(\lambda_jt)).$$ +Y hacemos estocásticas estas componentes sustituyendo +$$a_{j,t+1} = a_{j,t} + \omega_{j,t}, a^*_{j,t+1} = a^*_{j,t} + \omega^*_{j,t}$$ +donde las $\omega,\omega^*$ son normales independientes con +varianza $\sigma_\omega^2$. + +Esta es la forma más simple de agregar estacionalidad trigonométrica, +pero tiene la desventaja de que los errores que agregamos se multiplican +después por senos y cosenos, lo cual quiere decir que su efecto no es +igual en diferentes partes del periodo. La otra forma más usual +puedes verla en la ayuda de *AddTrig* de bsts o en la sección 3.2.1 de +@durbinkoopman. + +--- + +Nuestra primera parte del modelo es entonces (donde podemos +ajustar el número de frecuencias o armónicos que incluímos +comparando los modelos producidos): + +```{r} +modelo_natalidad_1 <- AddLocalLevel(list(), y = y) |> + AddSeasonal(nseasons = 7, y = y) |> + AddTrig(period = 365.25, frequencies = 1:3, y = y) +``` + +### Días de asueto + +Definiremos componentes para algunos de los días de asueto/especiales más +importantes, considerando que su efecto puede ir más allá del día particular. +En la versión estática, tenemos un coeficiente distinto para cada uno +de esos días en la ventana que hayamos elegido: + + +```{r} +año_nuevo <- NamedHoliday(holiday.name = "NewYearsDay", days.before = 4, + days.after = 4) +navidad <- NamedHoliday(holiday.name = "Christmas", days.before = 4, + days.after = 4) +pascua <- NamedHoliday(holiday.name = "EasterSunday", + days.after = 5, days.before = 5) +feb_14 <- NamedHoliday(holiday.name = "ValentinesDay", + days.after = 3, days.before = 3) +halloween <- NamedHoliday(holiday.name = "Halloween", + days.after = 5, days.before = 5) +sept_16 <- FixedDateHoliday(holiday.name = "Independencia", + month = "September", day = 16, days.before = 2, + days.after = 2) +mayo_1 <- FixedDateHoliday(holiday.name = "Trabajo", + month = "May", day = 1, days.before = 2, + days.after = 2) +``` + + +Y nuestro modelo es entonces, agregando también un coeficiente +para febrero 29 y marzo 1 cuando sigue de 29 de febrero: + +```{r} +modelo_natalidad <- AddLocalLevel(list(), y = y) |> + AddSeasonal(nseasons = 7, y = y) |> + AddTrig(period = 365.25, frequencies = 1:3, y = y) |> + AddRandomWalkHoliday(holiday = año_nuevo, y = y) |> + AddRandomWalkHoliday(holiday = navidad, y = y,) |> + AddRandomWalkHoliday(holiday = pascua, y = y) |> + AddRandomWalkHoliday(holiday = feb_14, y = y) |> + AddRandomWalkHoliday(holiday = halloween, y = y) |> + AddRandomWalkHoliday(holiday = sept_16, y = y) |> + AddRandomWalkHoliday(holiday = mayo_1, y = y) |> + AddDynamicRegression(y ~ feb_29 + mar_1_feb_29) +``` + +```{r} +#ajuste_nat <- bsts(y, modelo_natalidad, niter = 1000) +ajuste_nat <- read_rds("cache/ajuste_nat.rds") +``` + +```{r} +#| fig-height: 6 +#| fig-width: 7 +mean_contrib <- ajuste_nat$state.contributions |> + apply(c(2, 3), mean) +mean_contrib_tbl <- mean_contrib |> t() |> as_tibble() |> + mutate(fecha = natalidad_tbl$fecha) |> + mutate(trend_c = trend - mean(trend)) |> + select(-trend) |> + pivot_longer(cols = c(seasonal.7.1:dynamic, trend_c), + names_to = "componente", values_to = "valor") +ggplot(mean_contrib_tbl, aes(x = fecha, y = valor)) + + facet_wrap(~ componente) + geom_point() +``` + +Podemos ver con más detalle cómo es el efecto de los días de +asueto y sus "hombros", tomando por ejemplo una parte 2008-2009 +vemos la estructura: + +```{r} +#| fig-height: 6 +#| fig-width: 7 +ggplot(mean_contrib_tbl |> + filter(year(fecha) == 2009 & month(fecha) < 5 | + year(fecha) == 2008 & month(fecha) >= 12), + aes(x = fecha, y = valor)) + + facet_wrap(~ componente) + geom_line() + geom_point(size = 0.5) +``` + +**Nota**: este modelo todavía requiere considerar qué otros efectos +adicionales están actuando. La distribución de errores a un paso +tiene colas largas, y todavía hay cierta información que podríamos +usar para mejorar las predicción si consideramos la ACF de las innovaciones: + +```{r} +#| fig-height: 5 +#| fig-width: 6 +error <- ajuste_nat$one.step.prediction.errors |> apply(2, mean) +error_sin_2008 <- error[-c(1:365)] +qqnorm(error_sin_2008) +qqline(error_sin_2008) +natalidad_tbl$error <- error +ACF(natalidad_tbl |> filter(year(fecha) > 2008), + error) |> autoplot() + ylim(c(-1,1)) +``` +```{r} +natalidad_tbl |> + filter(year(fecha) > 2008) |> + arrange(desc(abs(error))) |> + mutate(dia_sem = weekdays(fecha)) +``` + +- En la tabla de arriba vemos que falta incluir algunos otros días de asueto, los que feriados +que se mueven a los lunes. +- También días feriados con fecha fija que caen en Domingo tienen +un efecto distinto (es posible construir manualmente la serie +de días de asueto excluyendo los feriados que caen en domingo, por ejemplo). + + + +## Nowcasting y uso de covariables + +::: callout-note +# Nowcasting + +En nowcasting buscamos predecir los valores contemporáneos de una +serie de tiempo que, por retrasos en reporte, no tenemos disponibles. +Usualmente utilizamos variables asociadas contemporáneas que sí +están disponibles, y se trata de pronósticos a corto plazo + +::: + +Un ejemplo es el indicador oportuno de actividad económica que se publica + mensualmente. Este indicador busca estimar el IGAE (indicador global de + actividad económica), y utiliza variables económicas, financieras y otras, + disponibles al momento (mientras que el IGAE se publica con unos dos meses + de retraso). Ver aquí: https://www.inegi.org.mx/investigacion/ioae/. + + +Una estrategia puede ser incluír variables relevantes cómo en regresión +dinámica. Sin embargo, si estamos buscando entre una cantidad relativamente +grande de predictores, es posible obtener mejores resultados haciendo + selección de variables. En el caso de *bsts* se utiliza una +inicial de *spike-slab*, que es una mezcla de una masa de probabilidad mayor a 0 +para un valor del coeficiente igual a 0 (es decir, la variable está excluída +del modelo), y una distribución normal centrada en 0 (el coeficiente puede +tomar un valor positivo o negativo). + + +En este ejemplo consideraremos (@scottvarian2015) el pronóstico oportuno de +los números semanales +reclamaciones pagos de seguro de desempleo en Estados Unidos, que se reportan +varios días después de que cierra cada semana. En el artículo citado, +utilizan medidas relativa de popularidad de búsquedas relacionadas +con desempleo en Google, que se pueden tener de manera completa en cuanto +la semana cierra. Ver también [aquí](https://www.nber.org/system/files/working_papers/w19567/w19567.pdf) o [aquí](https://www.unofficialgoogledatascience.com/2017/07/fitting-bayesian-structural-time-series.html). + +Este código está adaptado de [aqui](https://www.unofficialgoogledatascience.com/2017/07/fitting-bayesian-structural-time-series.html), ver también [este reporte](https://storage.googleapis.com/pub-tools-public-publication-data/pdf/41335.pdf). + + +Primero construiremos un modelo con tendencia lineal y estacionalidad. +En este caso, como consideramos una serie relativamente +corta (no más de 10 años), y la estacionalidad es dinámica, podemos +usar un periodo de 52 semanas, aún cuando sabemos que algunos años +tienen 53 semanas. + +```{r} +data(iclaims) +original.claims <- initial.claims +initial.claims <- original.claims[1:436, ] +new.data <- original.claims[437:456, ] +ss <- AddLocalLinearTrend(list(), initial.claims$iclaimsNSA) +ss <- AddSeasonal(ss, initial.claims$iclaimsNSA, nseasons = 52) +model1 <- bsts(initial.claims$iclaimsNSA, + state.specification = ss, + niter = 1000) +``` + +```{r} +plot(model1) +plot(model1, "components") +``` + +Y podemos construir predicciones: + +```{r} +pred1 <- predict(model1, horizon = 12) +plot(pred1, plot.original = 156) +``` + +Entre las variables de términos de búsqueda están + +```{r} +names(initial.claims)[-1] +``` + +Seleccionamos una inicial para el tamaño esperado del modelo (y después +seleccionamos según desempeño predictivo): + +```{r} +model2 <- bsts(iclaimsNSA ~ ., + state.specification = ss, + niter = 1000, + data = initial.claims, + expected.model.size = 2) +model3 <- bsts(iclaimsNSA ~ ., + state.specification = ss, + niter = 1000, + data = initial.claims, + expected.model.size = 6) # Passed to SpikeSlabPrior. +``` + +Comparamos los modelos con el error de pronóstico a un paso, y el +modelo 2 es el de mejor desempeño: + +```{r} +CompareBstsModels(list("Modelo s.vars" = model1, + "Modelo esp2" = model2, + "Modelo esp6" = model3), + colors = c("black", "red", "blue")) +``` +```{r} +pred2 <- predict(model2, horizon = 12, newdata = new.data[1:12, ]) +plot(pred2, plot.original = 156) +``` +Podemos ver qué variables fueron seleccionadas: + +```{r} +plot(model2, "coefficients") +``` + + +## Agregación de encuestas + +Para cada elección, se publica en México un gran número de encuestas sobre las preferencias partidistas. Con el fin de obtener una estimación de las preferencias subyacentes, las encuestas pueden combinarse en una encuesta de encuestas. Mostramos un modelo dinámico bayesiano para combinar encuestas y producir una única estimación. + +```{r} +#| warning: false +library(lubridate) +mex_2018_tbl <- + read_csv("../datos/wiki_encuestas_2018_en_campana.csv") |> + mutate(id_encuesta = row_number()) |> + select(id_encuesta, everything()) |> + mutate(Muestra = as.numeric(Muestra)) |> + mutate(margen_error = str_remove(`Margen de error`, "±" )) |> + mutate(margen_error = as.numeric(margen_error)) |> + rowwise() |> + mutate(no_contesto = + sum(`Ninguno Nulo/Blanco`, `NC/NS`, na.rm = TRUE)) |> + ungroup() |> + select(-c(`Margen de error`, `Ninguno Nulo/Blanco`, `NC/NS`)) |> + mutate(Zavala = ifelse(is.na(Zavala), 0, Zavala)) |> + mutate(Rodríguez = ifelse(is.na(Rodríguez ), 0, Rodríguez )) |> + mutate(Otros = Zavala + Rodríguez) |> + mutate(publicacion = dmy(paste(`Publicación (2018)`, 2018))) |> + select(id_encuesta, Encuestadora, publicacion, Muestra, + Obrador:Meade, Otros, no_contesto, margen_error) +``` + +Generalmente las casas encuestadores usan una fórmula simple +para el error, como si fuera muestreo aleatorio simple, que rara +vez se usa en la práctica (para encuestas casa por casa, por ejemplo), +y no toma el cuenta de que muchas veces se usan reponderaciones de los +datos. Ambas cosas generalmente producen varianzas considerablemente +más altas. + +```{r} +ggplot(mex_2018_tbl, aes( + x = margen_error, y = round(100 / sqrt(Muestra),1))) + + geom_point() + geom_abline() +``` + +Supondremos margen de error de 6 para aquellos que no reportan +tamaño de muestra o error, y usamos un factor de diseño igual a 2. +Adicionalmente, como buscamos preferencias efectivas, descontamos +aquellos que no reportaron preferencia: + +```{r} +# suponemos margen de error 6 para las que no reportan +# generalmente se reporta el error para un porcentaje de 50% +# como si fuera muestreo aleatorio simple +# Usamos un factor de diseño igual a 2 +mex_2018_tbl <- mex_2018_tbl |> + mutate(margen_error = ifelse(is.na(margen_error), 6, margen_error)) |> + mutate(n_inf = (100 / (sqrt(2) * margen_error))^2) |> + mutate(n_efectiva = n_inf * (Obrador + Anaya + Meade + Otros)/100) +``` + +Finalmente, normalizamos a preferencia efectiva: + +```{r} +pref_tbl <- mex_2018_tbl |> + pivot_longer(Obrador:Otros, names_to = "candidato", + values_to = "pref_cruda") |> + group_by(id_encuesta) |> + mutate(p = pref_cruda / sum(pref_cruda)) |> + mutate(n_cand = p * n_efectiva) |> + select(-c(p, pref_cruda)) |> + pivot_wider(names_from = candidato, values_from = n_cand) +``` + +Ahora consideramos el **modelo de observaciones** que utilizaremos. En primer lugar, +las observaciones son conteos, que suponemos multinomiales: + +$$n_i \sim \textrm{Multinom}(\pi_i, N_{i}^{eff})$$ +donde $n_i$ es el vector de conteos observado, $N_i^{eff}$ es la muestra +efectiva, y $\pi_i$ es el vector de preferencias de los candidatos +para la encuesta $i$. Ahora definimos +$\mu_i$ tal que +$$\textrm{softmax}(\mu_i) = \pi_i$$ +Ahora consideramos $\mu_i$ para cada encuesta. Esta cantidad dependerá de: + +- La preferencia subyacente $p(t_i)$ de los votantes al tiempo $t_i$ que se tomó +la encuesta $i$. +- Sesgos propios de la casa encuestadora $c(i)$, que hizo la encuesta $i$, que denotamos como $d_{c(i)}$. Estos ocurren porque +diversas metodologías alcanzan a distintos sectores de posibles votantes, o +consideran diferentes poblaciones objetivo. Suponemos +que en promedio sobre las casas encuestadores, este valor es cercano +a cero. +- Sesgos general $b$ en el que incurren todas las casas encuestadoras. Este error +sistemático ocurre por ejemplo, cuando un sector particular no responde a encuestas, y también tiene preferencias distintas a las de la población general. + +El modelo entonces es + +$$\mu_i = \alpha(t_i) + b + d_{c(i)}$$ +Nuestro interés es estimar $\alpha(t)$ para cada $t$, y en consecuencia +nuestra estimación de la preferencia será + +$$p (t) = \textrm{softmax}(\alpha(t)),$$ +es decir, la preferencia quitando el sesgo de casa encuestadora y el sesgo general de todas las encuestas. Buscaremos hacer una proyección +$p(T)$ si $T$ es el día de la elección. + +Ahora construímos la parte dinámica del modelo. Supondremos los +sesgos generales y por casas encuestadoras como fijos en el tiempo +(esto puede cambiarse si existen cambios de metodología en el tiempo por +ejemplo). + +En este ejemplo usaremos el modelo más simple de nivel local, +como en [en este modelo de elecciones en Alemania](https://www.sowi.uni-mannheim.de/media/Lehrstuehle/sowi/Gschwend/Articel/forecasting_elections_in_multiparty_systems_a_bayesian_approach_combining_polls_and_fundamentals.pdf), o [el modelo del Economist para la elección presidencial de 2020](https://github.com/TheEconomist/us-potus-model): + +- Modelo de nivel local, donde los choques ocurren cada día y +cambian las preferencias. Las predicciones al día de la elección +son las últimas estimaciones junto con la incertidumbre asociada +a la evolución de las preferencias. +- Otra alternativa es también modelar tendencia (usual o autorregresiva), +si consideramos que el "momento" de los candidatos es importante. Las +predicciones al día de elección incluyen esa tendencia. + +En este caso, tenemos usaremos (recuerda que la longitud de $p$ es +el número de candidatos). En primer lugar, ponemos + + +$$\alpha(t+1) = \alpha(t) + \epsilon(t)$$ +con $\epsilon(t) \sim NMV(0, W)$, donde $W$ es una matriz de varianzas y covarianzas para los choques, que +pueden tener patrones de correlación entre candidatos. + +```{r} +primer_dia <- pref_tbl$publicacion |> min() +ultimo_dia <- pref_tbl$publicacion |> max() +fecha_eleccion <- lubridate::ymd("2018-07-01") +#dias_tbl <- tibble(dia = seq(primer_dia, fecha_eleccion, by = "day")) |> +# mutate(id_dia = row_number()) +dias_tbl <- read_csv("../datos/dias.csv") +# identificar dias +pref_tbl <- pref_tbl |> + left_join(dias_tbl |> rename(publicacion = dia)) |> + ungroup() +# numerar casas +pref_tbl <- pref_tbl |> ungroup() |> + mutate(id_casa = as.integer(factor(Encuestadora))) +cands_tbl <- pref_tbl |> pivot_longer(Obrador:Otros, names_to = "candidato", + values_to = "n_pref") |> + group_by(id_encuesta) |> + mutate(porcentaje_pref = 100 * n_pref / sum(n_pref)) +``` + +Y podemos usar un modelo como el que sigue: + +```{r} +#| message: false +#| warning: false +library(cmdstanr) +mod_elecciones <- cmdstan_model("./src/series-de-tiempo/modelo-eleccion.stan") +print(mod_elecciones) +``` + +```{r} +n_obs <- pref_tbl |> + ungroup() |> + select(Obrador:Otros) |> + as.matrix() |> + apply(c(1,2), as.integer) + +casas <- pref_tbl$id_casa +dia_enc <- pref_tbl$id_dia +datos_lst <- list(K = ncol(n_obs), N = nrow(n_obs), n_obs = n_obs + 1, + S = max(casas), num_casa_enc = casas, + dia_enc = dia_enc, T = max(dias_tbl$id_dia)) +ajuste <- mod_elecciones$sample(data = datos_lst, + parallel_chains = 4, max_treedepth = 12, init = 0.01, + step_size = 0.01, refresh = 1000, + iter_sampling = 500, iter_warmup = 500) +``` + +Extraemos la componente de interés, la resumimos y la transformamos +a porcentaje de preferencia: + +```{r} +#| warning: false +alpha_tbl <- ajuste$draws("alpha", format = "df") |> + pivot_longer(contains("alpha"), + names_to = "variable", values_to = "valor") |> + separate(variable, into = c("var", "id_dia", "cand"), + convert = TRUE, extra = "drop") +``` + + +```{r} +prop_tbl <- alpha_tbl |> group_by(.draw, id_dia) |> + mutate(valor = exp(valor)/sum(exp(valor))) |> + group_by(id_dia, cand) |> + summarise(media = mean(valor), q5 = quantile(valor, 0.05), + q95 = quantile(valor, 0.95), .groups = "drop") +``` + +Y podemos var ahora las estimaciones suavizadas y los pronósticos +desde el momento que terminaron las encuestas hasta el día de la elección: + +```{r} +#| fig-width: 7 +#| fig-height: 3 +#| message: false +cands_tbl <- cands_tbl |> + left_join(tibble(cand = 1:4, candidato = colnames(n_obs))) +prop_tbl <- prop_tbl |> + left_join(dias_tbl) +ggplot(prop_tbl, aes( y = 100 * media, group = factor(cand), + colour = factor(cand))) + + geom_line(aes(x = dia)) + + geom_ribbon( + aes(x = dia,ymax = 100 * q95, ymin = 100 * q5, fill = factor(cand)), + alpha = 0.1, colour = NA) + + geom_point(data = cands_tbl, aes(x = publicacion, y = porcentaje_pref), alpha = 0.4) + ylab("Preferencia") +``` +Por ejemplo la matriz de correlaciones estimada (para los +choque del nivel) es: + +```{r} +ajuste$summary("Omega") |> select(variable, median, sd) |> + mutate(across(where(is.numeric), ~ round(.x, 3))) +``` + +Este modelo tiene varias ventajas sobre hacer promedios simples +o suavizamientos locales (como loess), por ejemplo: + +- Considera el error de estimación de cada encuesta. +- Cuando distintas casas publican a frecuencias distintas, es importante +considerar que esos datos informan tanto la preferencia como *el sesgo de la casa encuestadora*. Esto implica, por ejemplo, que los encuestadores con encuestas más grandes o con más encuestas no dominan la estimación. +- Contiene parámetros más apropiados para calibrar las predicciones contra datos de elecciones anteriores (los intervalos deben ser consistentes con +los resultados finales de la elección), incluyendo efectos de casa, sesgos +generales, y rapidez en la evolución del sistema. + + + +## Contrafactuales de impacto causal + +Los modelos de espacio de estados también presentan una herramienta +útil para hacer inferencia causal, específicamente cuando nos interesa +el efecto de una intervención en una métrica que se mide en el tiempo. + +En un ejemplo anterior, vimos cómo utilizar estos modelos, y variables +dummy, para hacer análisis de efecto de intervenciones (efecto +de ley de cinturones de seguridad). La hipótesis básica es que +no existen puertas traseras una vez que condicionamos a las covariables +incluidas. Esto sucedería +si alguna variable $U_t$ afecta a la variable de intervención $X_t$ +al mismo tiempo que afecta el estado del sistema $\theta_t$ o sus +valores futuros. En este caso, +el coeficiente de la variable $X_t$ puede estar contaminado con +una correlación no +causal. Por ejemplo, un camino de puerta trasera podría ser que hay una +decisión de implica poner la ley de cinturones y un reglamento de +velocidad máxima en autopistas en fechas cercanas. + +Igualmente, el conjunto al que condicionamos no debe contener colisionadores +que formen una ruta no causal entre la variable intervenida $X_t$ y +el estado del sistema $\theta_t$. En el caso de los cinturones, +por ejemplo, usar como control $R_t$, el número de personas con +heridas menores por accidentes es mala idea, pues es afectado +tanto por la ley como por el estado del sistema (o un antecesor del +estado). + +En ese ejemplo, recaemos en nuestros supuestos +de que no existes puertas traseras y en la forma funcional del +modelo para hacer la estimación del contrafactual. + + +### Modelos predictivos del contrafactual + +Otra forma de abordar el problema es el siguiente (@causalimpact) , +ver también [Google Causal Impact](https://google.github.io/CausalImpact/)): + +- Construimos un modelo predictivo para la serie que vamos a intervenir, +utilizando solamente variables que **sabemos que no serán afectadas por la intervención**. +- Para constrruir este modelo sólo usamos los **datos anteriores a la intervención** +que estamos considerando. +- Una alternativa es utilizar **modelos de espacio de estados**, lo cual +más adelante nos permite estimar adecuadamente errores de estimación en +efectos causales (toman en cuenta la autocorrelación, y la incertidumbre +que aumenta conforme el tiempo pasa después de la intervención). +- Usando este modelo pronosticamos los valores de la serie de interés +después de que ocurre la intervención con nuestro modelo +pre-intervención. Estos pronósticos, por el +supuesto en el primer inciso, constituyen nuestro **contrafactual** si la +intervención no se hubiera aplicado. +- **Comparamos** los valores observados de la serie de interés, después +de la intervención, con los pronósticos para estimar el efecto causal +de la intervención. + +Nótese que no es necesario suponer una forma parámétrica para +la intervención, podemos estimarla como la diferencia señalada arriba +(individualmente en cada periodo o de forma agregada sobre cierto +tiempo después de la intervención), y es posible estimar efectos +rezagados de la intervención. + + +### Ejemplo 1 + +Tomamos el siguiente ejemplo del paquete *CausalImpact*: + +```{r} +#| warning: false +library(CausalImpact) +library(bsts) +set.seed(1241) +x <- 100 + arima.sim(model = list(ar = 0.999), n = 100) +x_int <- rep(0, 100) +y <- 1.2 * x + rnorm(100) +``` + +Suponemos que la intervencion se hace en el tiempo $t=71$, y que +tiene el efecto de aumentar la $y$ en 10 unidades: + +```{r} +y[71:100] <- y[71:100] + 10 +x_int[71:100] <- 1 +datos_tbl <- tibble(y, x, x_int) +``` + +Los dos series, antes de la intervención, muestra correlación. +Construiremos un + +```{r} +datos_tbl |> mutate(t = row_number()) |> + filter(t < 71) |> + ggplot(aes(x = t)) + + geom_line(aes(y = y)) + + geom_line(aes(y = x), colour = "red") +``` + +- Si sabemos que la serie $x$ roja no puede ser afectada por nuestra +intervención sobre $y$, y suponemos que +la relación entre $y$ y $x$ se mantendría después del tiempo de la +intervención, entonces podemos contruir un modelo predictivo de $y$ en +función de $x$ para construir contrafactuales. + +Un ejemplo es: si imaginamos que una promoción se hace +exclusivamente en una región $A$ +del país, podemos usar otras regiones del país como predictores de las +ventas en la región $A$. Muchas veces estos modelos tienden a ser más +precisos y simples pues no es necesario incluir +factores como estacionalidad, pues están sujetos a fuentes de +variación similares. + +--- + +En términos de contrafactuales: una vez que construimos este modelo, hacemos pronósticos con nuestro +modelo para $t>t_{int}$, después de la intervención, que denotamos como $\hat{y}_t^0$, que son estimaciones de $y_t^0$ (el contrafactual sin +la intervención). Después de la intervención observamos +$y_t^1$, así que calculamos, para cada $t$: + +$$y_t^1 - \hat{y}_t^0$$ +Esta cantidad tiene una distribución posterior, que representa lo que sabemos +del efecto causal individual sobre la región tratada $A$. + +:::callout-note +# Método de Impacto causal + +Requerimos los siguientes supuestos fuertes: + +- La serie de interés $y$ puede ser explicada +en términos de otras series de tiempo que **no son afectadas** por la intervención. +- La relación entre la serie de interés $y$ y la serie de control se supone +**estable** durante el periodo post-intervención. + +::: + + + + +### Ejemplo {-} + +Usamos la idea de construir un contrafactual con el método +de impacto causal: + +En primer lugar, excluímos los datos post-tratamiento: + +```{r} +post_periodo <- c(71, 100) +y_mod <- y +post_y <- y[post_periodo[1] : post_periodo[2]] +y_mod[post_periodo[1] : post_periodo[2]] <- NA +``` + +Construimos un modelo. En este caso usamos regresión estática pero +agregamos una componente de nivel local: + +```{r} +#| fig-width: 6 +#| fig-height: 4 +set.seed(886) +mod_esp_1 <- AddLocalLevel(list(), y_mod) +mod_esp_2 <- AddLocalLinearTrend(list(), y_mod) +ajuste_pre_1 <- bsts(y_mod ~ x, mod_esp_1, niter = 5000, ping = 5000) +ajuste_pre_2 <- bsts(y_mod ~ x, mod_esp_2, niter = 5000, ping = 5000) +CompareBstsModels(list(ajuste_pre_1, ajuste_pre_2)) +``` + +Seleccionamos el primer modelo: + +```{r} +#| fig-width: 6 +#| fig-height: 4 +plot(ajuste_pre_1) +plot(ajuste_pre_1, "components") +``` + +Extraemos predicciones que automáticamente hace *bsts* + +```{r} +# quitar burn-in +estado_contrib <- ajuste_pre_1$state.contributions[-seq_len(500), , ,drop = FALSE] +# sumar contribuciones de componentes +pred_cf <- rowSums(aperm(estado_contrib, c(1, 3, 2)), dims = 2) +# extraer period post-intervencion +pred_cf <- pred_cf[, 71:100] +dim(pred_cf) +``` + +Y estimamos el efecto promedio en estos 30 días utilizando +el valor observado de $y$ menos las simulaciones correspondientes +del contrafactual: + +```{r} +media_dia_post <- apply(pred_cf, 1, mean) +diferencia <- mean(y[71:100]) - media_dia_post +mean(diferencia) +quantile(diferencia, c(0.025, 0.975)) |> round(3) +``` + +O más simplemente: + +```{r} +CausalImpact(bsts.model = ajuste_pre_1, + post.period.response = y[71:100]) +``` + +Podemos hacer esto automáticamente usando funciones del +paquete: + +```{r} +pre_periodo <- c(1, 70) +post_periodo <- c(71, 100) +impact <- CausalImpact(datos_tbl |> select(y, x), + pre_periodo, post_periodo, model.args = list(niter = 5000)) +plot(impact) +``` +```{r} +summary(impact) +``` + +### Ejemplo 2 {-} + + +Nótese adicionalmente esta estimación es no-paramétrica, y no establecemos +un modelo explícito para el efecto de la intervención. + +```{r} +#| warning: false +library(CausalImpact) +library(bsts) +set.seed(461) +x <- 100 + arima.sim(model = list(ar = 0.999), n = 100) +x_int <- rep(0, 100) +y <- 1.2 * x + rnorm(100) +y[71:100] <- y[71:100] + 5 * (71:100 - 71) * exp(-25 * (71:100 -71) /100) +x_int[71:100] <- 1 +datos_tbl <- tibble(y, x, x_int) +``` + + +```{r} +pre_period <- c(1, 70) +post_period <- c(71, 100) +impact <- CausalImpact(datos_tbl |> select(y, x), pre_period, post_period) +plot(impact) +``` + +En casos como este, muchas veces tiene más sentido considerar el +efecto acumulado en lugar del promedio (por ejemplo si la variable $y$ +son clicks o compras): + +```{r} +summary(impact) +``` + + +## Ejemplo: cinturones de seguridad + +En el ejemplo de cinturones de seguridad, es necesario encontrar +series que se razonable que no son afectadas por la introducción +de la ley de cinturones de seguridad. + +Supondremos (aunque puede ser discutible), que la ley no tendrá un +efecto considerable en kilómetros recorridos o precio de la gasolina, +y tampoco tendrá un efecto sobre pasajeros que viajan en los asientos +traseros. Estas variables son candidatas para construir un modelo +predictivo antes de la introducción de la ley. + + +```{r} +uk_drivers <- Seatbelts |> as_tibble() |> + mutate(ind = row_number()) +pre_periodo <- c(1, which(uk_drivers$law == 0) |> max()) +pre_periodo +post_periodo <- c(pre_periodo[2] + 1, nrow(uk_drivers)) +datos_mod_tbl <- uk_drivers |> select(drivers, kms, rear, PetrolPrice,ind) +``` + +Podemos construir nuestro modelo manualmente (usualmente, varios +modelo que comparamos en capacidad predictiva y calibración de intervalos): + +```{r} +datos_ent_tbl <- datos_mod_tbl +datos_ent_tbl <- datos_mod_tbl |> filter(ind <= 140) +modelo_ent <- AddLocalLevel(list(), + sigma.prior = SdPrior(5, 10), + y = datos_ent_tbl$drivers) |> + AddSeasonal(nseasons = 12, datos_ent_tbl$drivers) +ajuste_ent <- bsts(drivers ~ kms + rear + PetrolPrice, + modelo_ent, data = datos_ent_tbl, + niter = 5000, ping = 5000) +``` +```{r} +plot(ajuste_ent) +``` +Checamos predicciones validando un periodo de validación hasta +antes de la intervención: + +```{r} +preds_val <- predict(ajuste_ent, newdata = datos_mod_tbl |> + filter(ind > 140, ind < 169)) +drivers_val <- datos_mod_tbl |> + filter(ind > 140, ind < 169) |> pull(drivers) +tibble(pred = preds_val$mean, + pred_inf = preds_val$interval[1, ], + pred_sup = preds_val$interval[2, ], + drivers = drivers_val) |> + ggplot(aes(x = drivers, y = pred, ymin = pred_inf,ymax = pred_sup )) + geom_point() + geom_linerange(colour = "red") + + geom_abline() + coord_fixed() +# MAPE +mean(abs(preds_val$mean - drivers_val)/drivers_val) +``` +Estas son las variables que utilizamos en el modelo (el color indica +el signo cuando entra en el modelo): + +```{r} +plot(ajuste_ent, "coefficients") +``` + + +Y ahora ajustamos el modelo a todos los datos pre intervención: + +```{r} +datos_mod_tbl$y <- datos_mod_tbl$drivers +datos_mod_tbl$y[post_periodo[1]:post_periodo[2]] <- NA +modelo_1 <- AddLocalLevel(list(), + sigma.prior = SdPrior(5, 10), + datos_mod_tbl$y) |> + AddSeasonal(nseasons = 12, datos_mod_tbl$y) +ajuste_1 <- bsts(y ~ kms + rear + PetrolPrice, modelo_1, data = datos_mod_tbl, + niter = 5000, ping = 5000) +``` + +```{r} +impacto <- CausalImpact(bsts.model = ajuste_1, + post.period.response = + datos_mod_tbl$drivers[post_periodo[1]:post_periodo[2]]) +plot(impacto) +``` + + +```{r} +impacto +``` + +Finalmente, podemos checar cómo cambiaron las variables del modelo +predictivo post-entrenamiento. Por ejemplo, para las muertes +y heridas graves de pasajeros +de los asientos traseros: + +```{r} +uk_drivers |> + ggplot(aes(x = ind)) + + geom_line(aes(y = drivers)) + + geom_line(aes(y = rear), colour = "red") +``` +O más simplemente, podemos usar los defaults de CausalImpact: + +```{r} +impacto_auto <- CausalImpact(datos_mod_tbl |> select(-y), pre_periodo, post_periodo) +impacto_auto +``` + +*Nota*: los valores de las iniciales que usa por default *CausalImpact* +son diferentes a los de *bsts*. + + +