Dynamic linear fashions with tfprobability

Welcome to the world of state house fashions. On this world, there’s a latent course of, hidden from our eyes; and there are observations we make concerning the issues it produces. The method evolves resulting from some hidden logic (transition mannequin); and the way in which it produces the observations follows some hidden logic (remark mannequin). There’s noise in course of evolution, and there may be noise in remark. If the transition and remark fashions each are linear, and the method in addition to remark noise are Gaussian, we’ve a linear-Gaussian state house mannequin (SSM). The duty is to deduce the latent state from the observations. Probably the most well-known method is the Kálmán filter.

In sensible purposes, two traits of linear-Gaussian SSMs are particularly engaging.

For one, they allow us to estimate dynamically altering parameters. In regression, the parameters might be seen as a hidden state; we could thus have a slope and an intercept that adjust over time. When parameters can fluctuate, we converse of dynamic linear fashions (DLMs). That is the time period we’ll use all through this put up when referring to this class of fashions.

Second, linear-Gaussian SSMs are helpful in time-series forecasting as a result of Gaussian processes might be added. A time sequence can thus be framed as, e.g. the sum of a linear pattern and a course of that varies seasonally.

Utilizing tfprobability, the R wrapper to TensorFlow Likelihood, we illustrate each points right here. Our first instance will probably be on dynamic linear regression. In an in depth walkthrough, we present on match such a mannequin, get hold of filtered, in addition to smoothed, estimates of the coefficients, and get hold of forecasts.
Our second instance then illustrates course of additivity. This instance will construct on the primary, and can also function a fast recap of the general process.

Let’s soar in.

Dynamic linear regression instance: Capital Asset Pricing Mannequin (CAPM)

Our code builds on the lately launched variations of TensorFlow and TensorFlow Likelihood: 1.14 and 0.7, respectively.

Be aware how there’s one factor we used to do currently that we’re not doing right here: We’re not enabling keen execution. We are saying why in a minute.

Our instance is taken from Petris et al.(2009)(Petris, Petrone, and Campagnoli 2009), chapter 3.2.7.
Apart from introducing the dlm bundle, this e-book offers a pleasant introduction to the concepts behind DLMs normally.

As an example dynamic linear regression, the authors characteristic a dataset, initially from Berndt(1991)(Berndt 1991) that has month-to-month returns, collected from January 1978 to December 1987, for 4 totally different shares, the 30-day Treasury Invoice – standing in for a risk-free asset –, and the value-weighted common returns for all shares listed on the New York and American Inventory Exchanges, representing the general market returns.

Let’s have a look.

# As the info doesn't appear to be obtainable on the deal with given in Petris et al. any extra,
# we put it on the weblog for obtain
# obtain from: 
# https://github.com/rstudio/tensorflow-blog/blob/grasp/docs/posts/2019-06-25-dynamic_linear_models_tfprobability/information/capm.txt"
df <- read_table(
  "capm.txt",
  col_types = record(X1 = col_date(format = "%Y.%m"))) %>%
  rename(month = X1)
df %>% glimpse()

Observations: 120
Variables: 7
$ month  <date> 1978-01-01, 1978-02-01, 1978-03-01, 1978-04-01, 1978-05-01, 19…
$ MOBIL  <dbl> -0.046, -0.017, 0.049, 0.077, -0.011, -0.043, 0.028, 0.056, 0.0…
$ IBM    <dbl> -0.029, -0.043, -0.063, 0.130, -0.018, -0.004, 0.092, 0.049, -0…
$ WEYER  <dbl> -0.116, -0.135, 0.084, 0.144, -0.031, 0.005, 0.164, 0.039, -0.0…
$ CITCRP <dbl> -0.115, -0.019, 0.059, 0.127, 0.005, 0.007, 0.032, 0.088, 0.011…
$ MARKET <dbl> -0.045, 0.010, 0.050, 0.063, 0.067, 0.007, 0.071, 0.079, 0.002,…
$ RKFREE <dbl> 0.00487, 0.00494, 0.00526, 0.00491, 0.00513, 0.00527, 0.00528, …

df %>% collect(key = "image", worth = "return", -month) %>%
  ggplot(aes(x = month, y = return, coloration = image)) +
  geom_line() +
  facet_grid(rows = vars(image), scales = "free")

Monthly returns for selected stocks; data from Berndt (1991).

Determine 1: Month-to-month returns for chosen shares; information from Berndt (1991).

The Capital Asset Pricing Mannequin then assumes a linear relationship between the surplus returns of an asset below examine and the surplus returns of the market. For each, extra returns are obtained by subtracting the returns of the chosen risk-free asset; then, the scaling coefficient between them reveals the asset to both be an “aggressive” funding (slope > 1: adjustments out there are amplified), or a conservative one (slope < 1: adjustments are damped).

Assuming this relationship doesn’t change over time, we will simply use lm for instance this. Following Petris et al. in zooming in on IBM because the asset below examine, we’ve

# extra returns of the asset below examine
ibm <- df$IBM - df$RKFREE
# market extra returns
x <- df$MARKET - df$RKFREE

match <- lm(ibm ~ x)
abstract(match)

Name:
lm(components = ibm ~ x)

Residuals:
     Min       1Q   Median       3Q      Max 
-0.11850 -0.03327 -0.00263  0.03332  0.15042 

Coefficients:
              Estimate Std. Error t worth Pr(>|t|)    
(Intercept) -0.0004896  0.0046400  -0.106    0.916    
x            0.4568208  0.0675477   6.763 5.49e-10 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual normal error: 0.05055 on 118 levels of freedom
A number of R-squared:  0.2793,    Adjusted R-squared:  0.2732 
F-statistic: 45.74 on 1 and 118 DF,  p-value: 5.489e-10

So IBM is discovered to be a conservative funding, the slope being ~ 0.5. However is that this relationship secure over time?

Let’s flip to tfprobability to research.

We wish to use this instance to display two important purposes of DLMs: acquiring smoothing and/or filtering estimates of the coefficients, in addition to forecasting future values. So in contrast to Petris et al., we divide the dataset right into a coaching and a testing half:.

# zoom in on ibm
ts <- ibm %>% matrix()
# forecast 12 months
n_forecast_steps <- 12
ts_train <- ts[1:(length(ts) - n_forecast_steps), 1, drop = FALSE]

# ensure we work with float32 right here
ts_train <- tf$solid(ts_train, tf$float32)
ts <- tf$solid(ts, tf$float32)

We now assemble the mannequin. sts_dynamic_linear_regression() does what we would like:

# outline the mannequin on the entire sequence
linreg <- ts %>%
  sts_dynamic_linear_regression(
    design_matrix = cbind(rep(1, size(x)), x) %>% tf$solid(tf$float32)
  )

We move it the column of extra market returns, plus a column of ones, following Petris et al.. Alternatively, we might middle the one predictor – this is able to work simply as effectively.

How are we going to coach this mannequin? Methodology-wise, we’ve a alternative between variational inference (VI) and Hamiltonian Monte Carlo (HMC). We are going to see each. The second query is: Are we going to make use of graph mode or keen mode? As of at the moment, for each VI and HMC, it’s most secure – and quickest – to run in graph mode, so that is the one method we present. In a couple of weeks, or months, we must always be capable to prune plenty of sess$run()s from the code!

Usually in posts, when presenting code we optimize for straightforward experimentation (that means: line-by-line executability), not modularity. This time although, with an essential variety of analysis statements concerned, it’s best to pack not simply the becoming, however the smoothing and forecasting as effectively right into a perform (which you may nonetheless step by way of in the event you needed). For VI, we’ll have a match _with_vi perform that does all of it. So after we now begin explaining what it does, don’t kind within the code simply but – it’ll all reappear properly packed into that perform, so that you can copy and execute as an entire.

Becoming a time sequence with variational inference

Becoming with VI just about appears like coaching historically used to look in graph-mode TensorFlow. You outline a loss – right here it’s accomplished utilizing sts_build_factored_variational_loss() –, an optimizer, and an operation for the optimizer to cut back that loss:

optimizer <- tf$compat$v1$practice$AdamOptimizer(0.1)

# solely practice on the coaching set!    
loss_and_dists <- ts_train %>% sts_build_factored_variational_loss(mannequin = mannequin)
variational_loss <- loss_and_dists[[1]]

train_op <- optimizer$decrease(variational_loss)

Be aware how the loss is outlined on the coaching set solely, not the entire sequence.

Now to truly practice the mannequin, we create a session and run that operation:

 with (tf$Session() %as% sess,  {
      
      sess$run(tf$compat$v1$global_variables_initializer())
   
      for (step in 1:n_iterations) {
        res <- sess$run(train_op)
        loss <- sess$run(variational_loss)
        if (step %% 10 == 0)
          cat("Loss: ", as.numeric(loss), "n")
      }
 })

Given we’ve that session, let’s make use of it and compute all of the estimates we want.
Once more, – the next snippets will find yourself within the fit_with_vi perform, so don’t run them in isolation simply but.

Acquiring forecasts

The very first thing we would like for the mannequin to present us are forecasts. With a purpose to create them, it wants samples from the posterior. Fortunately we have already got the posterior distributions, returned from sts_build_factored_variational_loss, so let’s pattern from them and move them to sts_forecast:

variational_distributions <- loss_and_dists[[2]]
posterior_samples <-
  Map(
    perform(d) d %>% tfd_sample(n_param_samples),
    variational_distributions %>% reticulate::py_to_r() %>% unname()
  )
forecast_dists <- ts_train %>% sts_forecast(mannequin, posterior_samples, n_forecast_steps)

sts_forecast() returns distributions, so we name tfd_mean() to get the posterior predictions and tfd_stddev() for the corresponding normal deviations:

fc_means <- forecast_dists %>% tfd_mean()
fc_sds <- forecast_dists %>% tfd_stddev()

By the way in which – as we’ve the total posterior distributions, we’re certainly not restricted to abstract statistics! We might simply use tfd_sample() to acquire particular person forecasts.

Smoothing and filtering (Kálmán filter)

Now, the second (and final, for this instance) factor we’ll need are the smoothed and filtered regression coefficients. The well-known Kálmán Filter is a Bayesian-in-spirit technique the place at every time step, predictions are corrected by how a lot they differ from an incoming remark. Filtering estimates are based mostly on observations we’ve seen to date; smoothing estimates are computed “in hindsight,” making use of the entire time sequence.

We first create a state house mannequin from our time sequence definition:

# solely do that on the coaching set
# returns an occasion of tfd_linear_gaussian_state_space_model()
ssm <- mannequin$make_state_space_model(size(ts_train), param_vals = posterior_samples)

tfd_linear_gaussian_state_space_model(), technically a distribution, offers the Kálmán filter functionalities of smoothing and filtering.

To acquire the smoothed estimates:

c(smoothed_means, smoothed_covs) %<-% ssm$posterior_marginals(ts_train)

And the filtered ones:

c(., filtered_means, filtered_covs, ., ., ., .) %<-% ssm$forward_filter(ts_train)

Lastly, we have to consider all these.

c(posterior_samples, fc_means, fc_sds, smoothed_means, smoothed_covs, filtered_means, filtered_covs) %<-%
  sess$run(record(posterior_samples, fc_means, fc_sds, smoothed_means, smoothed_covs, filtered_means, filtered_covs))

Placing all of it collectively (the VI version)

So right here’s the entire perform, fit_with_vi, prepared for us to name.

fit_with_vi <-
  perform(ts,
           ts_train,
           mannequin,
           n_iterations,
           n_param_samples,
           n_forecast_steps,
           n_forecast_samples) {
    
    optimizer <- tf$compat$v1$practice$AdamOptimizer(0.1)
    
    loss_and_dists <-
      ts_train %>% sts_build_factored_variational_loss(mannequin = mannequin)
    variational_loss <- loss_and_dists[[1]]
    train_op <- optimizer$decrease(variational_loss)
    
    with (tf$Session() %as% sess,  {
      
      sess$run(tf$compat$v1$global_variables_initializer())
      for (step in 1:n_iterations) {
        sess$run(train_op)
        loss <- sess$run(variational_loss)
        if (step %% 1 == 0)
          cat("Loss: ", as.numeric(loss), "n")
      }
      variational_distributions <- loss_and_dists[[2]]
      posterior_samples <-
        Map(
          perform(d)
            d %>% tfd_sample(n_param_samples),
          variational_distributions %>% reticulate::py_to_r() %>% unname()
        )
      forecast_dists <-
        ts_train %>% sts_forecast(mannequin, posterior_samples, n_forecast_steps)
      fc_means <- forecast_dists %>% tfd_mean()
      fc_sds <- forecast_dists %>% tfd_stddev()
      
      ssm <- mannequin$make_state_space_model(size(ts_train), param_vals = posterior_samples)
      c(smoothed_means, smoothed_covs) %<-% ssm$posterior_marginals(ts_train)
      c(., filtered_means, filtered_covs, ., ., ., .) %<-% ssm$forward_filter(ts_train)
   
      c(posterior_samples, fc_means, fc_sds, smoothed_means, smoothed_covs, filtered_means, filtered_covs) %<-%
        sess$run(record(posterior_samples, fc_means, fc_sds, smoothed_means, smoothed_covs, filtered_means, filtered_covs))
      
    })
    
    record(
      variational_distributions,
      posterior_samples,
      fc_means[, 1],
      fc_sds[, 1],
      smoothed_means,
      smoothed_covs,
      filtered_means,
      filtered_covs
    )
  }

And that is how we name it.

# variety of VI steps
n_iterations <- 300
# pattern dimension for posterior samples
n_param_samples <- 50
# pattern dimension to attract from the forecast distribution
n_forecast_samples <- 50

# here is the mannequin once more
mannequin <- ts %>%
  sts_dynamic_linear_regression(design_matrix = cbind(rep(1, size(x)), x) %>% tf$solid(tf$float32))

# name fit_vi outlined above
c(
  param_distributions,
  param_samples,
  fc_means,
  fc_sds,
  smoothed_means,
  smoothed_covs,
  filtered_means,
  filtered_covs
) %<-% fit_vi(
  ts,
  ts_train,
  mannequin,
  n_iterations,
  n_param_samples,
  n_forecast_steps,
  n_forecast_samples
)

Curious concerning the outcomes? We’ll see them in a second, however earlier than let’s simply shortly look on the different coaching technique: HMC.

Placing all of it collectively (the HMC version)

tfprobability offers sts_fit_with_hmc to suit a DLM utilizing Hamiltonian Monte Carlo. Latest posts (e.g., Hierarchical partial pooling, continued: Various slopes fashions with TensorFlow Likelihood) confirmed arrange HMC to suit hierarchical fashions; right here a single perform does all of it.

Right here is fit_with_hmc, wrapping sts_fit_with_hmc in addition to the (unchanged) methods for acquiring forecasts and smoothed/filtered parameters:

num_results <- 200
num_warmup_steps <- 100

fit_hmc <- perform(ts,
                    ts_train,
                    mannequin,
                    num_results,
                    num_warmup_steps,
                    n_forecast,
                    n_forecast_samples) {
  
  states_and_results <-
    ts_train %>% sts_fit_with_hmc(
      mannequin,
      num_results = num_results,
      num_warmup_steps = num_warmup_steps,
      num_variational_steps = num_results + num_warmup_steps
    )
  
  posterior_samples <- states_and_results[[1]]
  forecast_dists <-
    ts_train %>% sts_forecast(mannequin, posterior_samples, n_forecast_steps)
  fc_means <- forecast_dists %>% tfd_mean()
  fc_sds <- forecast_dists %>% tfd_stddev()
  
  ssm <-
    mannequin$make_state_space_model(size(ts_train), param_vals = posterior_samples)
  c(smoothed_means, smoothed_covs) %<-% ssm$posterior_marginals(ts_train)
  c(., filtered_means, filtered_covs, ., ., ., .) %<-% ssm$forward_filter(ts_train)
  
  with (tf$Session() %as% sess,  {
    sess$run(tf$compat$v1$global_variables_initializer())
    c(
      posterior_samples,
      fc_means,
      fc_sds,
      smoothed_means,
      smoothed_covs,
      filtered_means,
      filtered_covs
    ) %<-%
      sess$run(
        record(
          posterior_samples,
          fc_means,
          fc_sds,
          smoothed_means,
          smoothed_covs,
          filtered_means,
          filtered_covs
        )
      )
  })
  
  record(
    posterior_samples,
    fc_means[, 1],
    fc_sds[, 1],
    smoothed_means,
    smoothed_covs,
    filtered_means,
    filtered_covs
  )
}

c(
  param_samples,
  fc_means,
  fc_sds,
  smoothed_means,
  smoothed_covs,
  filtered_means,
  filtered_covs
) %<-% fit_hmc(ts,
               ts_train,
               mannequin,
               num_results,
               num_warmup_steps,
               n_forecast,
               n_forecast_samples)

Now lastly, let’s check out the forecasts and filtering resp. smoothing estimates.

Forecasts

Placing all we’d like into one dataframe, we’ve

smoothed_means_intercept <- smoothed_means[, , 1] %>% colMeans()
smoothed_means_slope <- smoothed_means[, , 2] %>% colMeans()

smoothed_sds_intercept <- smoothed_covs[, , 1, 1] %>% colMeans() %>% sqrt()
smoothed_sds_slope <- smoothed_covs[, , 2, 2] %>% colMeans() %>% sqrt()

filtered_means_intercept <- filtered_means[, , 1] %>% colMeans()
filtered_means_slope <- filtered_means[, , 2] %>% colMeans()

filtered_sds_intercept <- filtered_covs[, , 1, 1] %>% colMeans() %>% sqrt()
filtered_sds_slope <- filtered_covs[, , 2, 2] %>% colMeans() %>% sqrt()

forecast_df <- df %>%
  choose(month, IBM) %>%
  add_column(pred_mean = c(rep(NA, size(ts_train)), fc_means)) %>%
  add_column(pred_sd = c(rep(NA, size(ts_train)), fc_sds)) %>%
  add_column(smoothed_means_intercept = c(smoothed_means_intercept, rep(NA, n_forecast_steps))) %>%
  add_column(smoothed_means_slope = c(smoothed_means_slope, rep(NA, n_forecast_steps))) %>%
  add_column(smoothed_sds_intercept = c(smoothed_sds_intercept, rep(NA, n_forecast_steps))) %>%
  add_column(smoothed_sds_slope = c(smoothed_sds_slope, rep(NA, n_forecast_steps))) %>%
  add_column(filtered_means_intercept = c(filtered_means_intercept, rep(NA, n_forecast_steps))) %>%
  add_column(filtered_means_slope = c(filtered_means_slope, rep(NA, n_forecast_steps))) %>%
  add_column(filtered_sds_intercept = c(filtered_sds_intercept, rep(NA, n_forecast_steps))) %>%
  add_column(filtered_sds_slope = c(filtered_sds_slope, rep(NA, n_forecast_steps)))

So right here first are the forecasts. We’re utilizing the estimates returned from VI, however we might simply as effectively have used these from HMC – they’re almost indistinguishable. The identical goes for the filtering and smoothing estimates displayed under.

ggplot(forecast_df, aes(x = month, y = IBM)) +
  geom_line(coloration = "gray") +
  geom_line(aes(y = pred_mean), coloration = "cyan") +
  geom_ribbon(
    aes(ymin = pred_mean - 2 * pred_sd, ymax = pred_mean + 2 * pred_sd),
    alpha = 0.2,
    fill = "cyan"
  ) +
  theme(axis.title = element_blank())

12-point-ahead forecasts for IBM; posterior means +/- 2 standard deviations.

Determine 2: 12-point-ahead forecasts for IBM; posterior means +/- 2 normal deviations.

Smoothing estimates

Listed here are the smoothing estimates. The intercept (proven in orange) stays fairly secure over time, however we do see a pattern within the slope (displayed in inexperienced).

ggplot(forecast_df, aes(x = month, y = smoothed_means_intercept)) +
  geom_line(coloration = "orange") +
  geom_line(aes(y = smoothed_means_slope),
            coloration = "inexperienced") +
  geom_ribbon(
    aes(
      ymin = smoothed_means_intercept - 2 * smoothed_sds_intercept,
      ymax = smoothed_means_intercept + 2 * smoothed_sds_intercept
    ),
    alpha = 0.3,
    fill = "orange"
  ) +
  geom_ribbon(
    aes(
      ymin = smoothed_means_slope - 2 * smoothed_sds_slope,
      ymax = smoothed_means_slope + 2 * smoothed_sds_slope
    ),
    alpha = 0.1,
    fill = "inexperienced"
  ) +
  coord_cartesian(xlim = c(forecast_df$month[1], forecast_df$month[length(ts) - n_forecast_steps]))  +
  theme(axis.title = element_blank())

Smoothing estimates from the Kálmán filter. Green: coefficient for dependence on excess market returns (slope), orange: vector of ones (intercept).

Determine 3: Smoothing estimates from the Kálmán filter. Inexperienced: coefficient for dependence on extra market returns (slope), orange: vector of ones (intercept).

Filtering estimates

For comparability, listed here are the filtering estimates. Be aware that the y-axis extends additional up and down, so we will seize uncertainty higher:

ggplot(forecast_df, aes(x = month, y = filtered_means_intercept)) +
  geom_line(coloration = "orange") +
  geom_line(aes(y = filtered_means_slope),
            coloration = "inexperienced") +
  geom_ribbon(
    aes(
      ymin = filtered_means_intercept - 2 * filtered_sds_intercept,
      ymax = filtered_means_intercept + 2 * filtered_sds_intercept
    ),
    alpha = 0.3,
    fill = "orange"
  ) +
  geom_ribbon(
    aes(
      ymin = filtered_means_slope - 2 * filtered_sds_slope,
      ymax = filtered_means_slope + 2 * filtered_sds_slope
    ),
    alpha = 0.1,
    fill = "inexperienced"
  ) +
  coord_cartesian(ylim = c(-2, 2),
                  xlim = c(forecast_df$month[1], forecast_df$month[length(ts) - n_forecast_steps])) +
  theme(axis.title = element_blank())

Filtering estimates from the Kálmán filter. Green: coefficient for dependence on excess market returns (slope), orange: vector of ones (intercept).

Determine 4: Filtering estimates from the Kálmán filter. Inexperienced: coefficient for dependence on extra market returns (slope), orange: vector of ones (intercept).

Thus far, we’ve seen a full instance of time-series becoming, forecasting, and smoothing/filtering, in an thrilling setting one doesn’t encounter too usually: dynamic linear regression. What we haven’t seen as but is the additivity characteristic of DLMs, and the way it permits us to decompose a time sequence into its (theorized) constituents.
Let’s do that subsequent, in our second instance, anti-climactically making use of the iris of time sequence, AirPassengers. Any guesses what parts the mannequin may presuppose?

Determine 5: AirPassengers.

Composition instance: AirPassengers

Libraries loaded, we put together the info for tfprobability:

The mannequin is a sum – cf. sts_sum – of a linear pattern and a seasonal element:

linear_trend <- ts %>% sts_local_linear_trend()
month-to-month <- ts %>% sts_seasonal(num_seasons = 12)
mannequin <- ts %>% sts_sum(parts = record(month-to-month, linear_trend))

Once more, we might use VI in addition to MCMC to coach the mannequin. Right here’s the VI manner:

n_iterations <- 100
n_param_samples <- 50
n_forecast_samples <- 50

optimizer <- tf$compat$v1$practice$AdamOptimizer(0.1)

fit_vi <-
  perform(ts,
           ts_train,
           mannequin,
           n_iterations,
           n_param_samples,
           n_forecast_steps,
           n_forecast_samples) {
    loss_and_dists <-
      ts_train %>% sts_build_factored_variational_loss(mannequin = mannequin)
    variational_loss <- loss_and_dists[[1]]
    train_op <- optimizer$decrease(variational_loss)
    
    with (tf$Session() %as% sess,  {
      sess$run(tf$compat$v1$global_variables_initializer())
      for (step in 1:n_iterations) {
        res <- sess$run(train_op)
        loss <- sess$run(variational_loss)
        if (step %% 1 == 0)
          cat("Loss: ", as.numeric(loss), "n")
      }
      variational_distributions <- loss_and_dists[[2]]
      posterior_samples <-
        Map(
          perform(d)
            d %>% tfd_sample(n_param_samples),
          variational_distributions %>% reticulate::py_to_r() %>% unname()
        )
      forecast_dists <-
        ts_train %>% sts_forecast(mannequin, posterior_samples, n_forecast_steps)
      fc_means <- forecast_dists %>% tfd_mean()
      fc_sds <- forecast_dists %>% tfd_stddev()
      
      c(posterior_samples,
        fc_means,
        fc_sds) %<-%
        sess$run(record(posterior_samples,
                      fc_means,
                      fc_sds))
    })
    
    record(variational_distributions,
         posterior_samples,
         fc_means[, 1],
         fc_sds[, 1])
  }

c(param_distributions,
  param_samples,
  fc_means,
  fc_sds) %<-% fit_vi(
    ts,
    ts_train,
    mannequin,
    n_iterations,
    n_param_samples,
    n_forecast_steps,
    n_forecast_samples
  )

For brevity, we haven’t computed smoothed and/or filtered estimates for the general mannequin. On this instance, this being a sum mannequin, we wish to present one thing else as a substitute: the way in which it decomposes into parts.

However first, the forecasts:

forecast_df <- df %>%
  add_column(pred_mean = c(rep(NA, size(ts_train)), fc_means)) %>%
  add_column(pred_sd = c(rep(NA, size(ts_train)), fc_sds))


ggplot(forecast_df, aes(x = month, y = n)) +
  geom_line(coloration = "gray") +
  geom_line(aes(y = pred_mean), coloration = "cyan") +
  geom_ribbon(
    aes(ymin = pred_mean - 2 * pred_sd, ymax = pred_mean + 2 * pred_sd),
    alpha = 0.2,
    fill = "cyan"
  ) +
  theme(axis.title = element_blank())

Determine 6: AirPassengers, 12-months-ahead forecast.

A name to sts_decompose_by_component yields the (centered) parts, a linear pattern and a seasonal issue:

component_dists <-
  ts_train %>% sts_decompose_by_component(mannequin = mannequin, parameter_samples = param_samples)

seasonal_effect_means <- component_dists[[1]] %>% tfd_mean()
seasonal_effect_sds <- component_dists[[1]] %>% tfd_stddev()
linear_effect_means <- component_dists[[2]] %>% tfd_mean()
linear_effect_sds <- component_dists[[2]] %>% tfd_stddev()

with(tf$Session() %as% sess, {
  c(
    seasonal_effect_means,
    seasonal_effect_sds,
    linear_effect_means,
    linear_effect_sds
  ) %<-% sess$run(
    record(
      seasonal_effect_means,
      seasonal_effect_sds,
      linear_effect_means,
      linear_effect_sds
    )
  )
})

components_df <- forecast_df %>%
  add_column(seasonal_effect_means = c(seasonal_effect_means, rep(NA, n_forecast_steps))) %>%
  add_column(seasonal_effect_sds = c(seasonal_effect_sds, rep(NA, n_forecast_steps))) %>%
  add_column(linear_effect_means = c(linear_effect_means, rep(NA, n_forecast_steps))) %>%
  add_column(linear_effect_sds = c(linear_effect_sds, rep(NA, n_forecast_steps)))

ggplot(components_df, aes(x = month, y = n)) +
  geom_line(aes(y = seasonal_effect_means), coloration = "orange") +
  geom_ribbon(
    aes(
      ymin = seasonal_effect_means - 2 * seasonal_effect_sds,
      ymax = seasonal_effect_means + 2 * seasonal_effect_sds
    ),
    alpha = 0.2,
    fill = "orange"
  ) +
  theme(axis.title = element_blank()) +
  geom_line(aes(y = linear_effect_means), coloration = "inexperienced") +
  geom_ribbon(
    aes(
      ymin = linear_effect_means - 2 * linear_effect_sds,
      ymax = linear_effect_means + 2 * linear_effect_sds
    ),
    alpha = 0.2,
    fill = "inexperienced"
  ) +
  theme(axis.title = element_blank())

AirPassengers, decomposition into a linear trend and a seasonal component (both centered).

Determine 7: AirPassengers, decomposition right into a linear pattern and a seasonal element (each centered).

Wrapping up

We’ve seen how with DLMs, there’s a bunch of attention-grabbing stuff you are able to do – other than acquiring forecasts, which most likely would be the final aim in most purposes – : You possibly can examine the smoothed and the filtered estimates from the Kálmán filter, and you’ll decompose a mannequin into its posterior parts. A very engaging mannequin is dynamic linear regression, featured in our first instance, which permits us to acquire regression coefficients that adjust over time.

This put up confirmed accomplish this with tfprobability. As of at the moment, TensorFlow (and thus, TensorFlow Likelihood) is in a state of considerable inside adjustments, with wanting to turn out to be the default execution mode very quickly. Concurrently, the superior TensorFlow Likelihood improvement crew are including new and thrilling options daily. Consequently, this put up is snapshot capturing greatest accomplish these objectives now: If you happen to’re studying this a couple of months from now, chances are high that what’s work in progress now can have turn out to be a mature technique by then, and there could also be sooner methods to achieve the identical objectives. On the fee TFP is evolving, we’re excited for the issues to come back!

Berndt, R. 1991. The Observe of Econometrics. Addison-Wesley.

Murphy, Kevin. 2012. Machine Studying: A Probabilistic Perspective. MIT Press.

Petris, Giovanni, sonia Petrone, and Patrizia Campagnoli. 2009. Dynamic Linear Fashions with r. Springer.

Dynamic linear regression instance: Capital Asset Pricing Mannequin (CAPM)

Becoming a time sequence with variational inference

Acquiring forecasts

Smoothing and filtering (Kálmán filter)

Placing all of it collectively (the VI version)

Placing all of it collectively (the HMC version)

Forecasts

Smoothing estimates

Filtering estimates

Composition instance: AirPassengers

Wrapping up

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