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 in regards to the issues it produces. The method evolves because of some hidden logic (transition mannequin); and the way in which it produces the observations follows some hidden logic (remark mannequin). There may be noise in course of evolution, and there’s noise in remark. If the transition and remark fashions each are linear, and the method in addition to remark noise are Gaussian, we now have a linear-Gaussian state house mannequin (SSM). The duty is to deduce the latent state from the observations. Probably the most well-known approach 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 considered as a hidden state; we could thus have a slope and an intercept that fluctuate over time. When parameters can fluctuate, we converse of dynamic linear fashions (DLMs). That is the time period we’ll use all through this submit 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 Chance, we illustrate each elements right here. Our first instance will probably be on dynamic linear regression. In an in depth walkthrough, we present on how one can match such a mannequin, how one can get hold of filtered, in addition to smoothed, estimates of the coefficients, and how one can get hold of forecasts.
Our second instance then illustrates course of additivity. This instance will construct on the primary, and may additionally function a fast recap of the general process.

Let’s bounce in.

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

Our code builds on the not too long ago launched variations of TensorFlow and TensorFlow Chance: 1.14 and 0.7, respectively.

Word how there’s one factor we used to do recently 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.
Moreover introducing the dlm bundle, this e-book supplies a pleasant introduction to the concepts behind DLMs on the whole.

For 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 information doesn't appear to be accessible 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 = listing(X1 = col_date(format = "%Y.%m"))) %>%
  rename(month = X1)
df %>% glimpse()

Observations: 120
Variables: 7
$ month   1978-01-01, 1978-02-01, 1978-03-01, 1978-04-01, 1978-05-01, 19…
$ MOBIL   -0.046, -0.017, 0.049, 0.077, -0.011, -0.043, 0.028, 0.056, 0.0…
$ IBM     -0.029, -0.043, -0.063, 0.130, -0.018, -0.004, 0.092, 0.049, -0…
$ WEYER   -0.116, -0.135, 0.084, 0.144, -0.031, 0.005, 0.164, 0.039, -0.0…
$ CITCRP  -0.115, -0.019, 0.059, 0.127, 0.005, 0.007, 0.032, 0.088, 0.011…
$ MARKET  -0.045, 0.010, 0.050, 0.063, 0.067, 0.007, 0.071, 0.079, 0.002,…
$ RKFREE  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 underneath research 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 are able to simply use lm as an example this. Following Petris et al. in zooming in on IBM because the asset underneath research, we now have

# extra returns of the asset underneath research
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 need 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 not like 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]

# be certain 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 whole 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 may heart the one predictor – this is able to work simply as properly.

How are we going to coach this mannequin? Methodology-wise, we now have 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 in the present day, for each VI and HMC, it’s most secure – and quickest – to run in graph mode, so that is the one approach we present. In a number of weeks, or months, we should always be capable to prune plenty of sess$run()s from the code!

Usually in posts, when presenting code we optimize for simple experimentation (which 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 properly right into a operate (which you could possibly nonetheless step by in case you wished). For VI, we’ll have a match _with_vi operate 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 operate, so that you can copy and execute as a complete.

Becoming a time sequence with variational inference

Becoming with VI just about seems like coaching historically used to look in graph-mode TensorFlow. You outline a loss – right here it’s completed 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)

Word how the loss is outlined on the coaching set solely, not the whole 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 now have that session, let’s make use of it and compute all of the estimates we need.
Once more, – the next snippets will find yourself within the fit_with_vi operate, 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 the intention 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(
    operate(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 now have the total posterior distributions, we’re not at all restricted to abstract statistics! We may simply use tfd_sample() to acquire particular person forecasts.

Smoothing and filtering (Kálmán filter)

Now, the second (and final, for this instance) factor we are going to need are the smoothed and filtered regression coefficients. The well-known Kálmán Filter is a Bayesian-in-spirit methodology 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 up to now; smoothing estimates are computed “in hindsight,” making use of the whole 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, supplies 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(listing(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 whole operate, fit_with_vi, prepared for us to name.

fit_with_vi <-
  operate(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(
          operate(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(listing(posterior_samples, fc_means, fc_sds, smoothed_means, smoothed_covs, filtered_means, filtered_covs))
      
    })
    
    listing(
      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 measurement for posterior samples
n_param_samples <- 50
# pattern measurement to attract from the forecast distribution
n_forecast_samples <- 50

# this 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 in regards to the outcomes? We’ll see them in a second, however earlier than let’s simply rapidly look on the various coaching methodology: HMC.

Placing all of it collectively (the HMC version)

tfprobability supplies 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 Chance) confirmed how one can arrange HMC to suit hierarchical fashions; right here a single operate 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 <- operate(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(
        listing(
          posterior_samples,
          fc_means,
          fc_sds,
          smoothed_means,
          smoothed_covs,
          filtered_means,
          filtered_covs
        )
      )
  })
  
  listing(
    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 now have

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 may simply as properly 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 below are the filtering estimates. Word that the y-axis extends additional up and down, so we are able to 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).

To date, 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 elements the mannequin may presuppose?

Determine 5: AirPassengers.

Composition instance: AirPassengers

Libraries loaded, we put together the information for tfprobability:

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

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

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

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

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

fit_vi <-
  operate(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(
          operate(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(listing(posterior_samples,
                      fc_means,
                      fc_sds))
    })
    
    listing(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 need to present one thing else as a substitute: the way in which it decomposes into elements.

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) elements, 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(
    listing(
      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 part (each centered).

Wrapping up

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

This submit confirmed how one can accomplish this with tfprobability. As of in the present day, TensorFlow (and thus, TensorFlow Chance) is in a state of considerable inner adjustments, with desperate to grow to be the default execution mode very quickly. Concurrently, the superior TensorFlow Chance growth crew are including new and thrilling options day-after-day. Consequently, this submit is snapshot capturing how one can finest accomplish these targets now: Should you’re studying this a number of months from now, likelihood is that what’s work in progress now can have grow to be a mature methodology by then, and there could also be sooner methods to realize the identical targets. On the price TFP is evolving, we’re excited for the issues to return!

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 fashions with tfprobability

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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