Evaluate Threshold Transitions

This example shows how to evaluate transition functions and compute states of threshold transitions given transition variable data.

Evaluate Transition Functions

Consider defining discrete economic states based on the CPI-based Canadian inflation rate. Specifically:

• Inflation is low when its rate is below 2%.

• Inflation is moderate when its rate is at least 2% and below 8%.

• Inflation is high when its rate is at least 8%.

• Transitions are smooth and modeled using the logistic distribution.

• The transition rate from low to moderate inflation is 3.5, and the rate from moderate to high inflation is 1.5.

Create threshold transitions representing the state space based on the CPI-based Canadian inflation rate. Assign names `"Low"`, `"Med"`, and `"Hi"` to the states.

```ttL = threshold([2 8],Type="logistic",Rates=[3.5 1.5], ... StateNames=["Low" "Med" "High"]);```

`ttL` is a `threshold` object that is a model for the state space of the inflation rate.

`ttdata` computes transition function data at specified values of a threshold variable ${\mathit{z}}_{\mathit{t}}$.

Evaluate all transition functions of the inflation threshold transitions at the transition levels and 5.

```z = [2 5 8]; F = ttdata(ttL,z)```
```F = 3×2 0.5000 0.0001 1.0000 0.0110 1.0000 0.5000 ```

F is a 3-by-2 matrix of transition function values; its rows correspond to the threshold variable values `z`, and its columns correspond to the two transition functions at levels 2 and 8.

The `UseZeroLevels` name-value argument of `ttdata` evaluates transition functions relative to a zero level. This option is useful when you compare rates among the transition functions.

Illustrate comparing transition rates. Create threshold transitions, for an arbitrary threshold variable, at levels 0 and 5. Specify normal transition functions with rates 0.5 and 1.5. Create a grid of threshold data from -10 to 10.

```ttN = threshold([0 5],Type="normal",Rates=[0.5 1.5]); z = -10:0.01:10;```

Evaluate each transition function for the threshold variable data. Compute raw values and (`UseZeroLevels=false`) and values relative to level zero (`UseZeroLevels=true`).

```F0 = ttdata(ttN,z,UseZeroLevels=false); F1 = ttdata(ttN,z,UseZeroLevels=true);```

Separately plot each set of evaluated transition functions with respect to the transition function data.

```figure plot(z,F0,LineWidth=2) grid on xlabel("Level") title("Raw Threshold Function Data") legend(["Rate=0.5" "Rate=1.5"],Location="best")```

```figure plot(z,F1,LineWidth=2) grid on xlabel("Distance from Level") title("Translated Threshold Function Data") legend(["Rate=0.5" "Rate=1.5"],Location="best")```

`Use ``ttplot` to reproduce the latter plot. Adjust the plot domain by specifying the `Width` name-value argument.

```figure ttplot(ttN,Type="graph",Width=20)```

Compute States

`ttstates` computes the state path of threshold variable data relative to specified threshold levels.

Consider again the threshold transitions for the CPI-based Canadian inflation rate `ttL`. Load the Data_Canada data set, which contains the inflation rate series `INF_C`.

```load Data_Canada INF_C = DataTable.INF_C; % Inflation rate (threshold variable)```

Determine the inflation state at each sample time point in the data by passing the threshold transitions and observed inflation rate series to `ttstates`. Display the first five states.

```states = ttstates(ttL,INF_C); sp15 = ttL.StateNames(states(1:5))' % First 5 states```
```sp15 = 5x1 string "Low" "Low" "Low" "Med" "Med" ```

There is low inflation for the first 3 observations, then the process crosses the first mid-level to moderate inflation for observations 4 and 5.

Visualize the entire state path.

```figure subplot(2,1,1) ttplot(ttL,Data=INF_C) colorbar('off') subplot(2,1,2) plot(states,"mo") ylabel("State") yticks([1 2 3]) yticklabels(ttL.StateNames) axis tight ```

State transitions occur when the threshold variable crosses a transition mid-level. For TAR models, this results in an abrupt shift in the submodel computing the response. For STAR models, state changes indicate a shift in the dominant submodel in the mixture of submodel responses.