Plot threshold transitions
ttplot plots transition functions of threshold
transitions. To evaluate the transition function for observations of the threshold variable,
plots on the axes specified by
ax instead of the current axes (
gca) using any of the input argument combinations in the previous syntaxes.
returns a handle
h = ttplot(___)
h to the threshold transitions plot. Use
h to modify properties of the plot after
you create it.
Plot Discrete Threshold Transitions
Create discrete threshold transitions at 0 and 2.
t = [0 2]; tt = threshold(t)
tt = threshold with properties: Type: 'discrete' Levels: [0 2] Rates:  StateNames: ["1" "2" "3"] NumStates: 3
tt is a
threshold object. The specified thresholds split the space into three states.
Plot the threshold transitions.
ttplot graphs a gradient plot by default. The -axis represents the value of the threshold variable (currently undefined) and the state-space:
The system is in state 1 when .
The system is in state 2 when .
The system is in state 3 when .
Because the transitions are discrete,
ttplot graphs the levels as lines—the regime switches abruptly when crosses a threshold variable.
Because is undefined, the -axis is arbitrary. When you specify threshold variable data by using the
Data name-value argument, the -axis is the sample index.
Plot Smooth Threshold Transitions
This example shows how to create two logistic threshold transitions with different transition rates, and then display a gradient plot of the transitions.
Load the yearly Canadian inflation and interest rates data set. Extract the inflation rate based on consumer price index (
INF_C) from the table, and plot the series.
load Data_Canada INF_C = DataTable.INF_C; plot(dates,INF_C); axis tight
Assume the following characteristics of the inflation rate series:
Rates below 2% are low.
Rates at least 2% and below 8% are medium.
Rates at least 8% are high.
A logistic transition function describes the transition between states well.
Transition between low and medium rates are faster than transitions between medium and high.
Create threshold transitions to describe the Canadian inflation rates.
t = [2 8]; % Thresholds r = [3.5 1.5]; % Transition rates statenames = ["Low" "Med" "High"]; tt = threshold(t,Type="logistic",Rates=r,StateNames=statenames)
tt = threshold with properties: Type: 'logistic' Levels: [2 8] Rates: [3.5000 1.5000] StateNames: ["Low" "Med" "High"] NumStates: 3
Plot the threshold transitions; show the gradient of the transition function between the states, and overlay the data.
Visually Compare Transition Functions
Create normal cdf threshold transitions at levels 0 and 5, with rates 0.5 and 1.5.
t = [0 5]; r = [0.5 1.5]; tt = threshold(t,Type="normal",Rates=r)
tt = threshold with properties: Type: 'normal' Levels: [0 5] Rates: [0.5000 1.5000] StateNames: ["1" "2" "3"] NumStates: 3
To compare the behavior of the transition functions, plot their graphs at the same level.
Plot the transition functions at their levels. Evaluate the transition function over a 1-D grid of values by using
ttdata, and then plot the results.
lower = tt.Levels(1) - 3/min(tt.Rates); upper = tt.Levels(end) + 3/min(tt.Rates); z = lower:0.1:upper; F = ttdata(tt,z,UseZeroLevels=false); figure plot(z,F,LineWidth=2) grid on xlabel("Level") legend(["Level 0, Rate 0.5" "Level 5, Rate 1.5"],Location="NorthWest")
Plot Exponential Threshold Transitions
Create smooth threshold transitions for the Australian to US dollar exchange rate to model price parity.
Load the Australia/US purchasing power and interest rates data set. Extract the log of the exchange rate
EXCH from the table.
load Data_JAustralian EXCH = DataTable.EXCH;
Consider a two-state system where:
State 1 occurs when the Australian dollar buys more than the US dollar (
State 2 occurs when the US dollar buys more than the Australian dollar (
States are weighed more highly as the system deviates from parity (
Create threshold transitions representing the system. To attribute a greater amount of mixing away from the threshold, specify an exponential transition function. Set the transition rate to 2.5.
tt = threshold(0,Type="exponential",Rates=2.5)
tt = threshold with properties: Type: 'exponential' Levels: 0 Rates: 2.5000 StateNames: ["1" "2"] NumStates: 2
Plot the threshold transitions with the threshold data.
Try improving the display by experimenting with the transition band width (
Width name-value argument).
Plot the transition function.
tt — Threshold transitions
Threshold transitions, with
NumStates states, specified as a
tt must be fully specified (no
Specify optional pairs of arguments as
the argument name and
Value is the corresponding value.
Name-value arguments must appear after other arguments, but the order of the
pairs does not matter.
Before R2021a, use commas to separate each name and value, and enclose
Name in quotes.
Type="graph" specifies plotting a line plot of the transition
function at each threshold level on the same axes.
Type — Plot type
"gradient" (default) |
Plot type, specified as a value in this table.
|Gradient shading of the mixing level in each transition band.|
Graphs of transition functions at each level.
Data — Data on threshold variable zt to include in plot
 (empty array) (default) | numeric vector
Data on a threshold variable zt to include in the plot, specified as a numeric vector.
Data with gradient
shading of transition bands (
Width — Width of transition bands
positive numeric scalar
Width of transition bands, specified as a positive numeric scalar.
For gradient plots (
ttplottruncates transition function data outside of the bands.
For transition function graphs (
ttplotsets the x-axis limits to
ttplot selects the band width
h — Plot handle
Plot handle, returned as a graphics object.
h contains a unique
plot identifier, which you can use to query or modify properties of the plot.
The mixing level is the degree to which adjacent states contribute to a response.
Transition functions F vary between 0 and 1; adjacent states are assigned weights F and 1 – F. The mixing level between adjacent states is the minimum weight min(F, 1 – F).
The following characteristics define the mixing behavior of each transition type:
Discrete transitions have no mixing.
Normal and logistic transitions achieve maximum mixing at threshold levels.
Exponential transitions achieve maximum mixing on either side of threshold levels.
For more details, see
Widthname-value argument to adjust the display of transition function graph (
Type="graph") plots with varying rates. In multilevel gradient plots (
Type="gradient"), a large enough width results in overlapping transition bands that can misrepresent data. By default,
ttplotchooses an appropriate width for displaying all transitions.