# State decoding and prediction

Source:`vignettes/v05_state_decoding_and_prediction.Rmd`

`v05_state_decoding_and_prediction.Rmd`

This vignette^{1} introduces the Viterbi algorithm for state
decoding. The decoded state sequence in combination with the estimated
model parameters can be used for forecasting.

## State decoding using the Viterbi algorithm

For financial markets, it is of special interest to infer the
underlying (hidden) states in order to gain insight about the actual
market situation. Decoding a full time series \(S_1, \ldots, S_T\) is called *global
decoding*. Hereby, we aim to find the most likely trajectory of
hidden states under the estimated model. Global decoding can be
accomplished by using the so-called Viterbi algorithm which is a
recursive scheme enabling to find the global maximum without being
confronted with huge computational costs. To this end, we follow Zucchini, MacDonald, and Langrock (2016) and define \[\zeta_{1i} = Pr(S_1 = i, X_1 = x_1) = \delta_i
p_i(x_1)\] for \(i = 1, \ldots,
N\) and for the following \(t = 2,
\ldots, T\) \[\zeta_{ti} =
\operatorname*{max}_{s_1, \ldots, s_{t-1}} Pr(S_{t-1} = s_{t-1}, S_t =
i, X_t = x_t).\] Then, the trajectory of most likely states \(i_1, \ldots, i_T\) can be calculated
recursively from \[i_T =
\operatorname*{argmax}_{i = 1, \ldots, N} \zeta_{Ti}\] and for
the following \(t = T-1, \ldots, 1\)
from \[i_t = \operatorname*{argmax}_{i = 1,
\ldots, N} (\zeta_{ti} \gamma_{i, i_{t+1}}).\] Transferring the
state decoding to HHMMs is straightforward: at first the coarse-scale
state process must be decoded. Afterwards, by using this information the
fine-scale state process can be decoded, see Adam
et al. (2019).

## The `decode_states()`

function

We revisit the DAX model of the vignette on model estimation:

`data(dax_model_3t)`

The underlying states can be decoded via the
`decode_states()`

function:

```
dax_model_3t <- decode_states(dax_model_3t)
#> Decoded states
```

We now can visualize the decoded time series:

`plot(dax_model_3t)`

## Invariance towards state labelling

Mind that the model is invariant to permutations of the state labels.
Therefore, fHMM provides the option to switch labels
after decoding via the `reorder_states()`

function, for
example:

`dax_model_3t <- reorder_states(dax_model_3t, 3:1)`

## Prediction

Having decoded the underlying states, it is possible to compute the state probabilities of next observations. Based on these probabilities and in combination with the estimated state-dependent distributions, next observations can be predicted, compare Zucchini, MacDonald, and Langrock (2016):

```
predict(dax_model_3t, ahead = 10)
#> state_1 state_2 state_3 lb estimate ub
#> 1 0.00000 0.02446 0.97554 -0.01065 0.00123 0.01311
#> 2 0.00012 0.04773 0.95215 -0.01092 0.00120 0.01332
#> 3 0.00036 0.06988 0.92976 -0.01119 0.00116 0.01352
#> 4 0.00070 0.09095 0.90835 -0.01145 0.00113 0.01371
#> 5 0.00115 0.11099 0.88786 -0.01170 0.00110 0.01390
#> 6 0.00169 0.13007 0.86825 -0.01195 0.00107 0.01408
#> 7 0.00231 0.14821 0.84948 -0.01218 0.00104 0.01426
#> 8 0.00301 0.16547 0.83152 -0.01241 0.00101 0.01443
#> 9 0.00379 0.18189 0.81432 -0.01263 0.00098 0.01459
#> 10 0.00463 0.19752 0.79785 -0.01285 0.00095 0.01476
```

## References

*Methods in Ecology and Evolution*10 (9): 1536–50.

*Chapman and Hall/CRC*.