Package 'otsfeatures'

AustrianWages

Description

Ordinal time series (OTS) of yearly categories of salaries for different Austrian employees

Usage

data(AustrianWages)

Format

A list with one element, which is:

data

A list with 9402 MTS.

Details

Each element in data is an ordinal time series containing 6 states (yearly categorized wages). 9402 Austrian workers considered. The series exhibit individual lengths ranging from 2 to 32 years with the median length being equal to 22. For more information, see López-Oriona et al. (2023).

References

López-Oriona Á, Weiß C, Vilar JA (2023). “Fuzzy clustering of ordinal time series based on two novel distances with financial applications.” Manuscript submitted for publication, 000-000.

---

Constructs the binarized time series associated with a given ordinal time series

Description

binarization constructs the binarized time series associated with a given ordinal time series.

Usage

binarization(series, states)

Arguments

series	An OTS (numerical vector with integers).
states	A numeric vector containing the corresponding states.

Details

Given an OTS of length $T$ with range $\mathcal{S}=\{s_0, s_1, s_2, \ldots, s_n\}$ ($s_0 < s_1 < s_2 < \ldots < s_n$), $\overline{X}_t=\{\overline{X}_1,\ldots, \overline{X}_T\}$, the function constructs the binarized time series, which is defined as $\overline{\boldsymbol Y}_t=\{\overline{\boldsymbol Y}_1, \ldots, \overline{\boldsymbol Y}_T\}$, with $\overline{\boldsymbol Y}_k=(\overline{Y}_{k,0}, \overline{Y}_{k,1},\ldots, \overline{Y}_{k,n})^\top$ such that $\overline{Y}_{k,i}=1$ if $\overline{X}_k=s_i$ ($k=1,\ldots,T, , i=0,\ldots,n$). The binarized series is constructed in the form of a matrix whose rows represent time observations and whose columns represent the states in the original series.

Value

The binarized time series.

Author(s)

Ángel López-Oriona, José A. Vilar

References

Weiß CH (2018). An introduction to discrete-valued time series. John Wiley and Sons. López-Oriona Á, Vilar JA, D’Urso P (2023). “Hard and soft clustering of categorical time series based on two novel distances with an application to biological sequences.” Information Sciences, 624, 467–492.

Examples

binarized_series <- binarization(AustrianWages$data[[100]],
states = 0 : 5) # Constructing the binarized
# time series for one OTS in dataset AustrianWages

---

Constructs the cumulative binarized time series associated with a given ordinal time series

Description

c_binarization constructs the cumulative binarized time series associated with a given ordinal time series.

Usage

c_binarization(series, states)

Arguments

series	An OTS (numerical vector with integers).
states	A numeric vector containing the corresponding states.

Details

Given an OTS of length $T$ with range $\mathcal{S}=\{s_0, s_1, s_2, \ldots, s_n\}$ ($s_0 < s_1 < s_2 < \ldots < s_n$), $\overline{X}_t=\{\overline{X}_1,\ldots, \overline{X}_T\}$, the function constructs the cumulative binarized time series, which is defined as $\overline{\boldsymbol Y}_t=\{\overline{\boldsymbol Y}_1, \ldots, \overline{\boldsymbol Y}_T\}$, with $\overline{\boldsymbol Y}_k=(\overline{Y}_{k,0}, \overline{Y}_{k,1},\ldots, \overline{Y}_{k,n-1})^\top$ such that $\overline{Y}_{k,i}=1$ if $\overline{X}_k \le s_i$ ($k=1,\ldots,T, , i=0,\ldots,n-1$). The cumulative binarized series is constructed in the form of a matrix whose rows represent time observations and whose columns represent the states in the original series.

Value

The binarized time series.

Author(s)

Ángel López-Oriona, José A. Vilar

References

Weiß CH (2018). An introduction to discrete-valued time series. John Wiley and Sons.

López-Oriona Á, Vilar JA, D’Urso P (2023). “Hard and soft clustering of categorical time series based on two novel distances with an application to biological sequences.” Information Sciences, 624, 467–492.

Examples

c_binarized_series <- c_binarization(AustrianWages$data[[100]],
states = 0 : 5) # Constructing the cumulative binarized
# time series for one OTS in dataset AustrianWages

---

Computes the cumulative conditional probabilities of an ordinal time series

Description

c_conditional_probabilities returns a matrix with the cumulative conditional probabilities of an ordinal time series

Usage

c_conditional_probabilities(series, lag = 1, states)

Arguments

series	An OTS.
lag	The considered lag (default is 1).
states	A numerical vector containing the corresponding states.

Details

Given an OTS of length $T$ with range $\mathcal{S}=\{s_0, s_1, s_2, \ldots, s_n\}$ ($s_0 < s_1 < s_2 < \ldots < s_n$), $\overline{X}_t=\{\overline{X}_1,\ldots, \overline{X}_T\}$, the function computes the matrix $\widehat{\boldsymbol F}^c(l) = \big(\widehat{f}^c_{i-1j-1}(l)\big)_{1 \le i, j \le n}$, with $\widehat{f}^c_{ij}(l)=\frac{TN_{ij}(l)}{(T-l)N_i}$, where $N_i$ is the number of elements less one or equal to $s_i$ in the realization $\overline{X}_t$ and $N_{ij}(l)$ is the number of pairs $(\overline{X}_t, \overline{X}_{t-l})$ in the realization $\overline{X}_t$ such that $\overline{X}_t \le s_i$ and $\overline{X}_{t-l} \le s_j$.

Value

A matrix with the conditional probabilities.

Author(s)

Ángel López-Oriona, José A. Vilar

References

Weiß CH (2019). “Distance-based analysis of ordinal data and ordinal time series.” Journal of the American Statistical Association.

Examples

matrix_ccp <- c_conditional_probabilities(series = AustrianWages$data[[100]],
states = 0 : 5) # Computing the matrix of
# cumulative conditional probabilities for one series in dataset AustrianWages

---

Computes the cumulative joint probabilities of an ordinal time series

Description

c_joint_probabilities returns a matrix with the cumulative joint probabilities of an ordinal time series

Usage

c_joint_probabilities(series, lag = 1, states)

Arguments

series	An OTS.
lag	The considered lag (default is 1).
states	A numerical vector containing the corresponding states.

Details

Given an OTS of length $T$ with range $\mathcal{S}=\{s_0, s_1, s_2, \ldots, s_n\}$ ($s_0 < s_1 < s_2 < \ldots < s_n$), $\overline{X}_t=\{\overline{X}_1,\ldots, \overline{X}_T\}$, the function computes the matrix $\widehat{\boldsymbol F}(l) = \big(\widehat{f}_{i-1j-1}(l)\big)_{1 \le i, j \le n}$, with $\widehat{f}_{ij}(l)=\frac{N_{ij}(l)}{T-l}$, where $N_{ij}(l)$ is the number of pairs $(\overline{X}_t, \overline{X}_{t-l})$ in the realization $\overline{X}_t$ such that $\overline{X}_t \le s_i$ and $\overline{X}_{t-l} \le s_j$.

Value

A matrix with the jcumulative oint probabilities.

Author(s)

Ángel López-Oriona, José A. Vilar

References

Weiß CH (2019). “Distance-based analysis of ordinal data and ordinal time series.” Journal of the American Statistical Association.

Examples

matrix_cjp <- c_joint_probabilities(series = AustrianWages$data[[100]],
states = 0 : 5) # Computing the matrix of
# cumulative joint probabilities for one series in dataset AustrianWages

---

Computes the cumulative marginal probabilities of an ordinal time series

Description

c_marginal_probabilities returns a vector with the cumulative marginal probabilities of an ordinal time series

Usage

c_marginal_probabilities(series, states)

Arguments

series	An OTS (numerical vector with integers).
states	A numerical vector containing the corresponding states.

Details

Given an OTS of length $T$ with range $\mathcal{S}=\{s_0, s_1, s_2, \ldots, s_n\}$ ($s_0 < s_1 < s_2 < \ldots < s_n$), $\overline{X}_t=\{\overline{X}_1,\ldots, \overline{X}_T\}$, the function computes the vector $\widehat{\boldsymbol f} =(\widehat{f}_0, \ldots, \widehat{f}_n)$, with $\widehat{f}_i=\frac{N_i}{T}$, where $N_i$ is the number of elements less than or equal to $s_i$ in the realization $\overline{X}_t$.

Value

A vector with the cumulative marginal probabilities.

Author(s)

Ángel López-Oriona, José A. Vilar

References

Weiß CH (2019). “Distance-based analysis of ordinal data and ordinal time series.” Journal of the American Statistical Association.

Examples

vector_cmp <- c_marginal_probabilities(series = AustrianWages$data[[100]],
states = 0 : 5) # Computing the vector of
# cumulative marginal probabilities for one series in dataset AustrianWages

---

Constructs a confidence interval for the ordinal asymmetry (block distance)

Description

ci_ordinal_asymmetry constructs a confidence interval for the ordinal asymmetry (block distance)

Usage

ci_ordinal_asymmetry(
  series,
  states,
  level = 0.95,
  temporal = TRUE,
  max_lag = 1
)

Arguments

series	An OTS (numerical vector with integers).
states	A numeric vector containing the corresponding states.
level	The confidence level (default is 0.95).
temporal	Logical. If temporal = TRUE (default), the interval is computed for a time series. Otherwise, the interval is computed for i.i.d. data.
max_lag	If temporal = TRUE, the maximum considered lag to compute the estimates related to the cumulative joint probabilities.

Details

If temporal = TRUE (default), the function constructs the confidence interval for the ordinal asymmetry relying on Theorem 7.1.1 in Weiß (2019). Otherwise, the interval is constructed according to Theorem 4.1 in Weiß (2019).

Value

The confidence interval.

Author(s)

Ángel López-Oriona, José A. Vilar

References

Weiß CH (2019). “Distance-based analysis of ordinal data and ordinal time series.” Journal of the American Statistical Association.

Examples

ci_asymmetry <- ci_ordinal_asymmetry(AustrianWages$data[[100]],
states = 0 : 5) # Constructing a confidence interval for the
# ordinal asymmetry for one OTS in dataset AustrianWages

---

Constructs a confidence interval for the ordinal dispersion (block distance)

Description

ci_ordinal_dispersion constructs a confidence interval for the ordinal dispersion (block distance)

Usage

ci_ordinal_dispersion(
  series,
  states,
  level = 0.95,
  temporal = TRUE,
  max_lag = 1
)

Arguments

series	An OTS (numerical vector with integers).
states	A numeric vector containing the corresponding states.
level	The confidence level (default is 0.95).
temporal	Logical. If temporal = TRUE (default), the interval is computed for a time series. Otherwise, the interval is computed for i.i.d. data.
max_lag	If temporal = TRUE, the maximum considered lag to compute the estimates related to the cumulative joint probabilities.

Details

If temporal = TRUE (default), the function constructs the confidence interval for the ordinal dispersion relying on Theorem 7.1.1 in Weiß (2019). Otherwise, the interval is constructed according to Theorem 4.1 in Weiß (2019).

Value

The confidence interval.

Author(s)

Ángel López-Oriona, José A. Vilar

References

Weiß CH (2019). “Distance-based analysis of ordinal data and ordinal time series.” Journal of the American Statistical Association.

Examples

ci_dispersion <- ci_ordinal_dispersion(AustrianWages$data[[100]],
states = 0 : 5) # Constructing a confidence interval for the
# ordinal dispersion for one OTS in dataset AustrianWages

---

Constructs a confidence interval for the ordinal skewness (block distance)

Description

ci_ordinal_skewness constructs a confidence interval for the ordinal skewness (block distance)

Usage

ci_ordinal_skewness(series, states, level = 0.95, temporal = TRUE, max_lag = 1)

Arguments

series	An OTS (numerical vector with integers).
states	A numeric vector containing the corresponding states.
level	The confidence level (default is 0.95).
temporal	Logical. If temporal = TRUE (default), the interval is computed for a time series. Otherwise, the interval is computed for i.i.d. data.
max_lag	If temporal = TRUE, the maximum considered lag to compute the estimates related to the cumulative joint probabilities.

Details

If temporal = TRUE (default), the function constructs the confidence interval for the ordinal skewness relying on Theorem 7.1.1 in Weiß (2019). Otherwise, the interval is constructed according to Theorem 4.1 in Weiß (2019).

Value

The confidence interval.

Author(s)

Ángel López-Oriona, José A. Vilar

References

Weiß CH (2019). “Distance-based analysis of ordinal data and ordinal time series.” Journal of the American Statistical Association.

Examples

ci_skewness <- ci_ordinal_skewness(AustrianWages$data[[100]],
states = 0 : 5) # Constructing a confidence interval for the
# ordinal skewness for one OTS in dataset AustrianWages

---

Computes the conditional probabilities of an ordinal time series

Description

conditional_probabilities returns a matrix with the conditional probabilities of an ordinal time series

Usage

conditional_probabilities(series, lag = 1, states)

Arguments

series	An OTS.
lag	The considered lag (default is 1).
states	A numerical vector containing the corresponding states.

Details

Given an OTS of length $T$ with range $\mathcal{S}=\{s_0, s_1, s_2, \ldots, s_n\}$ ($s_0 < s_1 < s_2 < \ldots < s_n$), $\overline{X}_t=\{\overline{X}_1,\ldots, \overline{X}_T\}$, the function computes the matrix $\widehat{\boldsymbol P}^c(l) = \big(\widehat{p}^c_{i-1j-1}(l)\big)_{1 \le i, j \le n+1}$, with $\widehat{p}^c_{ij}(l)=\frac{TN_{ij}(l)}{(T-l)N_i}$, where $N_i$ is the number of elements equal to $s_i$ in the realization $\overline{X}_t$ and $N_{ij}(l)$ is the number of pairs $(\overline{X}_t, \overline{X}_{t-l})=(s_i,s_j)$ in the realization $\overline{X}_t$.

Value

A matrix with the conditional probabilities.

Author(s)

Ángel López-Oriona, José A. Vilar

References

Weiß CH (2019). “Distance-based analysis of ordinal data and ordinal time series.” Journal of the American Statistical Association.

Examples

matrix_cp <- conditional_probabilities(series = AustrianWages$data[[100]],
states = 0 : 5) # Computing the matrix of
# conditional probabilities for one series in dataset AustrianWages

---

CreditRatings

Description

Ordinal time series (OTS) of monthly credit ratings of different European countries

Usage

data(CreditRatings)

Format

A list with one element, which is:

data

A list with 28 MTS.

Details

Each element in data is an ordinal time series containing 23 states (monthly credit ratings). The 28 countries of the European Union plus the United Kingdom are considered. The sample period spans from January 2000 to December 2017, thus resulting serial realizations of length $T=216$. For more information, see Weiß (2019).

References

Weiß CH (2019). “Distance-based analysis of ordinal data and ordinal time series.” Journal of the American Statistical Association.

---

Computes the estimated index of ordinal variation (IOV) of an ordinal time series

Description

index_ordinal_variation computes the estimated index of ordinal variation of an ordinal time series

Usage

index_ordinal_variation(series, states)

Arguments

series	An OTS.
states	A numerical vector containing the corresponding states.

Details

Given an OTS of length $T$ with range $\mathcal{S}=\{s_0, s_1, s_2, \ldots, s_n\}$ ($s_0 < s_1 < s_2 < \ldots < s_n$), $\overline{X}_t=\{\overline{X}_1,\ldots, \overline{X}_T\}$, the function computes the estimated IOV given by $\widehat{IOV}=\frac{4}{n}\sum_{k=1}^{n-1}\widehat{f}_k(1-\widehat{f}_k)$, where $\widehat{f}_k$ is the standard estimate of the cumulative marginal probability for state $s_k$ computed from the series $\overline{X}_t$.

Value

The estimated IOV.

Author(s)

Ángel López-Oriona, José A. Vilar

References

Weiß CH (2019). “Distance-based analysis of ordinal data and ordinal time series.” Journal of the American Statistical Association.

Examples

estimated_iov <- index_ordinal_variation(series = AustrianWages$data[[100]],
states = 0 : 5) # Computing the estimate of the IOV
# for one series in dataset AustrianWages

---

Computes the joint probabilities of an ordinal time series

Description

joint_probabilities returns a matrix with the joint probabilities of an ordinal time series

Usage

joint_probabilities(series, lag = 1, states)

Arguments

series	An OTS.
lag	The considered lag (default is 1).
states	A numerical vector containing the corresponding states.

Details

Given an OTS of length $T$ with range $\mathcal{S}=\{s_0, s_1, s_2, \ldots, s_n\}$ ($s_0 < s_1 < s_2 < \ldots < s_n$), $\overline{X}_t=\{\overline{X}_1,\ldots, \overline{X}_T\}$, the function computes the matrix $\widehat{\boldsymbol P}(l) = \big(\widehat{p}_{i-1j-1}(l)\big)_{1 \le i, j \le n+1}$, with $\widehat{p}_{ij}(l)=\frac{N_{ij}(l)}{T-l}$, where $N_{ij}(l)$ is the number of pairs $(\overline{X}_t, \overline{X}_{t-l})=(s_i,s_j)$ in the realization $\overline{X}_t$.

Value

A matrix with the joint probabilities.

Author(s)

Ángel López-Oriona, José A. Vilar

References

Weiß CH (2019). “Distance-based analysis of ordinal data and ordinal time series.” Journal of the American Statistical Association.

Examples

matrix_jp <- joint_probabilities(series = AustrianWages$data[[100]],
states = 0 : 5) # Computing the matrix of
# joint probabilities for one series in dataset AustrianWages

---

Computes the marginal probabilities of an ordinal time series

Description

marginal_probabilities returns a vector with the marginal probabilities of an ordinal time series

Usage

marginal_probabilities(series, states)

Arguments

series	An OTS (numerical vector with integers).
states	A numerical vector containing the corresponding states

Details

Given an OTS of length $T$ with range $\mathcal{S}=\{s_0, s_1, s_2, \ldots, s_n\}$ ($s_0 < s_1 < s_2 < \ldots < s_n$), $\overline{X}_t=\{\overline{X}_1,\ldots, \overline{X}_T\}$, the function computes the vector $\widehat{\boldsymbol p} =(\widehat{p}_0, \ldots, \widehat{p}_n)$, with $\widehat{p}_i=\frac{N_i}{T}$, where $N_i$ is the number of elements equal to $s_i$ in the realization $\overline{X}_t$.

Value

A vector with the marginal probabilities.

Author(s)

Ángel López-Oriona, José A. Vilar

References

Weiß CH (2019). “Distance-based analysis of ordinal data and ordinal time series.” Journal of the American Statistical Association.

Examples

vector_mp <- marginal_probabilities(series = AustrianWages$data[[100]],
states = 0 : 5) # Computing the vector of
# marginal probabilities for one series in dataset AustrianWages

---

Computes the estimated asymmetry of an ordinal time series

Description

ordinal_asymmetry computes the estimated asymmetry of an ordinal time series

Usage

ordinal_asymmetry(series, states, distance = "Block", normalize = FALSE)

Arguments

series	An OTS.
states	A numerical vector containing the corresponding states.
distance	A function defining the underlying distance between states. The Hamming, block and Euclidean distances are already implemented by means of the arguments "Hamming", "Block" (default) and "Euclidean". Otherwise, a function taking as input two states must be provided.
normalize	Logical. If normalize = FALSE (default), the value of the estimated asymmetry is returned. Otherwise, the function returns the normalized estimated asymmetry.

Details

Given an OTS of length $T$ with range $\mathcal{S}=\{s_0, s_1, s_2, \ldots, s_n\}$ ($s_0 < s_1 < s_2 < \ldots < s_n$), $\overline{X}_t=\{\overline{X}_1,\ldots, \overline{X}_T\}$, the function computes the estimated asymmetry given by $\widehat{asym}_{d}=\widehat{\boldsymbol p}^\top (\boldsymbol J-\boldsymbol I)\boldsymbol D\widehat{\boldsymbol p}$, where $\widehat{\boldsymbol p}=(\widehat{p}_0, \widehat{p}_1, \ldots, \widehat{p}_n)^\top$, with $\widehat{p}_k$ being the standard estimate of the marginal probability for state $s_k$, $\boldsymbol I$ and $\boldsymbol J$ are the identity and counteridentity matrices of order $n + 1$, respectively, and $\boldsymbol D$ is a pairwise distance matrix for the elements in the set $\mathcal{S}$ considering a specific distance between ordinal states, $d(\cdot, \cdot)$. If normalize = TRUE, then the normalized estimate is computed, namely $\frac{\widehat{asym}_{d}}{max_{s_i, s_j \in \mathcal{S}}d(s_i, s_j)}$.

Value

The estimated asymmetry.

Author(s)

Ángel López-Oriona, José A. Vilar

References

Weiß CH (2019). “Distance-based analysis of ordinal data and ordinal time series.” Journal of the American Statistical Association.

Examples

estimated_asymmetry <- ordinal_asymmetry(series = AustrianWages$data[[100]],
states = 0 : 5) # Computing the asymmetry estimate
# for one series in dataset AustrianWages using the block distance

---

Computes the estimated ordinal Cohen's kappa of an ordinal time series

Description

ordinal_cohens_kappa computes the estimated ordinal Cohen's kappa of an ordinal time series

Usage

ordinal_cohens_kappa(series, states, distance = "Block", lag = 1)

Arguments

series	An OTS.
states	A numerical vector containing the corresponding states.
distance	A function defining the underlying distance between states. The Hamming, block and Euclidean distances are already implemented by means of the arguments "Hamming", "Block" (default) and "Euclidean". Otherwise, a function taking as input two states must be provided.
lag	The considered lag.

Details

Given an OTS of length $T$ with range $\mathcal{S}=\{s_0, s_1, s_2, \ldots, s_n\}$ ($s_0 < s_1 < s_2 < \ldots < s_n$), $\overline{X}_t=\{\overline{X}_1,\ldots, \overline{X}_T\}$, the function computes the estimated ordinal Cohen's kappa given by $\widehat{\kappa}_d(l)=\frac{\widehat{disp}_d(X_t)-\widehat{E}[d(X_t, X_{t-l})]}{{\widehat{disp}}_d(X_t)}$, where $\widehat{disp}_{d}(X_t)=\frac{T}{T-1}\sum_{i,j=0}^nd\big(s_i, s_j\big)\widehat{p}_i\widehat{p}_j$ is the DIVC estimate of the dispersion, with $d(\cdot, \cdot)$ being a distance between ordinal states and $\widehat{p}_k$ being the standard estimate of the marginal probability for state $s_k$, and $\widehat{E}[d(X_t, X_{t-l})]=\frac{1}{T-l} \sum_{t=l+1}^T d(\overline{X}_t, \overline{X}_{t-l})$.

Value

The estimated ordinal Cohen's kappa.

Author(s)

Ángel López-Oriona, José A. Vilar

References

Weiß CH (2019). “Distance-based analysis of ordinal data and ordinal time series.” Journal of the American Statistical Association.

Examples

estimated_ock <- ordinal_cohens_kappa(series = AustrianWages$data[[100]],
states = 0 : 5) # Computing the estimated ordinal Cohen's kappa
# for one series in dataset AustrianWages using the block distance

---

Computes the standard estimated dispersion of an ordinal time series

Description

ordinal_dispersion_1 computes the standard estimated dispersion of an ordinal time series

Usage

ordinal_dispersion_1(series, states, distance = "Block", normalize = FALSE)

Arguments

series	An OTS.
states	A numerical vector containing the corresponding states.
distance	A function defining the underlying distance between states. The Hamming, block and Euclidean distances are already implemented by means of the arguments "Hamming", "Block" (default) and "Euclidean". Otherwise, a function taking as input two states must be provided.
normalize	Logical. If normalize = FALSE (default), the value of the standard estimated dispersion is returned. Otherwise, the function returns the normalized standard estimated dispersion.

Details

Given an OTS of length $T$ with range $\mathcal{S}=\{s_0, s_1, s_2, \ldots, s_n\}$ ($s_0 < s_1 < s_2 < \ldots < s_n$), $\overline{X}_t=\{\overline{X}_1,\ldots, \overline{X}_T\}$, the function computes the standard estimated dispersion given by $\widehat{disp}_{loc, d}=\frac{1}{T}\sum_{t=1}^Td\big(\overline{X}_t, \widehat{x}_{loc, d}\big)$, where $\widehat{x}_{loc, d}$ is the standard estimate of the location and $d(\cdot, \cdot)$ is a distance between ordinal states. If normalize = TRUE, then the normalized dispersion is computed, namely $\widehat{disp}_{loc, d}/$max$_{s_i, s_j \in \mathcal{S}}d(s_i, s_j)$.

Value

The standard estimated dispersion.

Author(s)

Ángel López-Oriona, José A. Vilar

References

Weiß CH (2019). “Distance-based analysis of ordinal data and ordinal time series.” Journal of the American Statistical Association.

Examples

estimated_dispersion <- ordinal_dispersion_1(series = AustrianWages$data[[100]],
states = 0 : 5) # Computing the standard dispersion estimate
# for one series in dataset AustrianWages using the block distance

---

Computes the estimated dispersion of an ordinal time series according to the approach based on the diversity coefficient (DIVC)

Description

ordinal_dispersion_2 computes the estimated dispersion of an ordinal time series according to the approach based on the diversity coefficient

Usage

ordinal_dispersion_2(series, states, distance = "Block", normalize = FALSE)

Arguments

series	An OTS.
states	A numerical vector containing the corresponding states.
distance	A function defining the underlying distance between states. The Hamming, block and Euclidean distances are already implemented by means of the arguments "Hamming", "Block" (default) and "Euclidean". Otherwise, a function taking as input two states must be provided.
normalize	Logical. If normalize = FALSE (default), the value of the estimated dispersion is returned. Otherwise, the function returns the normalized estimated dispersion.

Details

Given an OTS of length $T$ with range $\mathcal{S}=\{s_0, s_1, s_2, \ldots, s_n\}$ ($s_0 < s_1 < s_2 < \ldots < s_n$), $\overline{X}_t=\{\overline{X}_1,\ldots, \overline{X}_T\}$, the function computes the DIVC estimated dispersion given by $\widehat{disp}_{d}=\frac{T}{T-1}\sum_{i,j=0}^nd\big(s_i, s_j\big)\widehat{p}_i\widehat{p}_j$, where $d(\cdot, \cdot)$ is a distance between ordinal states and $\widehat{p}_k$ is the standard estimate of the marginal probability for state $s_k$. If normalize = TRUE, and distance = "Block" or distance = "Euclidean", then the normalized versions are computed, that is, the corresponding estimates are divided by the factors $2/m$ or $2/m^2$, respectively.

Value

The estimated dispersion according to the approach based on the diversity coefficient.

Author(s)

Ángel López-Oriona, José A. Vilar

References

Weiß CH (2019). “Distance-based analysis of ordinal data and ordinal time series.” Journal of the American Statistical Association.

Examples

estimated_dispersion <- ordinal_dispersion_2(series = AustrianWages$data[[100]],
states = 0 : 5) # Computing the DIVC dispersion estimate
# for one series in dataset AustrianWages using the block distance

---

Computes the standard estimated location of an ordinal time series

Description

ordinal_location_1 computes the standard estimated location of an ordinal time series

Usage

ordinal_location_1(series, states, distance = "Block", normalize = FALSE)

Arguments

series	An OTS.
states	A numerical vector containing the corresponding states.
distance	A function defining the underlying distance between states. The Hamming, block and Euclidean distances are already implemented by means of the arguments "Hamming", "Block" (default) and "Euclidean". Otherwise, a function taking as input two states must be provided.
normalize	Logical. If normalize = FALSE (default), the value of the standard estimated location is returned. Otherwise, the function returns the normalized standard estimated location.

Details

Given an OTS of length $T$ with range $\mathcal{S}=\{s_0, s_1, s_2, \ldots, s_n\}$ ($s_0 < s_1 < s_2 < \ldots < s_n$), $\overline{X}_t=\{\overline{X}_1,\ldots, \overline{X}_T\}$, the function computes the standard estimated location given by $\widehat{x}_{loc, d}=$argmin$_{s \in \mathcal{S}}\frac{1}{T}\sum_{t=1}^Td\big(\overline{X}_t, s\big)$, where $d(\cdot, \cdot)$ is a distance between ordinal states.

Value

The standard estimated location.

Author(s)

Ángel López-Oriona, José A. Vilar

References

Weiß CH (2019). “Distance-based analysis of ordinal data and ordinal time series.” Journal of the American Statistical Association.

Examples

estimated_location <- ordinal_location_1(series = AustrianWages$data[[100]],
states = 0 : 5) # Computing the standard location estimate
# for one series in dataset AustrianWages using the block distance

---

Computes the estimated location of an ordinal time series with respect to the lowest category

Description

ordinal_location_2 computes the estimated location of an ordinal time series with respect to the lowest category

Usage

ordinal_location_2(series, states, distance = "Block", normalize = FALSE)

Arguments

series	An OTS.
states	A numerical vector containing the corresponding states.
distance	A function defining the underlying distance between states. The Hamming, block and Euclidean distances are already implemented by means of the arguments "Hamming", "Block" (default) and "Euclidean". Otherwise, a function taking as input two states must be provided.
normalize	Logical. If normalize = FALSE (default), the value of the standard estimated location is returned. Otherwise, the function returns the normalized standard estimated location.

Details

Given an OTS of length $T$ with range $\mathcal{S}=\{s_0, s_1, s_2, \ldots, s_n\}$ ($s_0 < s_1 < s_2 < \ldots < s_n$), $\overline{X}_t=\{\overline{X}_1,\ldots, \overline{X}_T\}$, the function computes the estimated location with respect to the lowest state, that is, the state $s_j$ such that $a_j=d(s_j, s_0)$ is the closest to $\frac{1}{T}\sum_{t=1}^Td\big(\overline{X}_t, s_0\big)$ is determined, where $d(\cdot, \cdot)$ is a distance between ordinal states.

Value

The estimated location with respect to the lowest category.

Author(s)

Ángel López-Oriona, José A. Vilar

References

Weiß CH (2019). “Distance-based analysis of ordinal data and ordinal time series.” Journal of the American Statistical Association.

Examples

estimated_location <- ordinal_location_2(series = AustrianWages$data[[100]],
states = 0 : 5) # Computing the location estimate
# with respect to the lowest state for one series in dataset AustrianWages

---

Computes the estimated skewness of an ordinal time series

Description

ordinal_skewness computes the estimated skewness of an ordinal time series

Usage

ordinal_skewness(series, states, distance = "Block", normalize = FALSE)

Arguments

series	An OTS.
states	A numerical vector containing the corresponding states.
distance	A function defining the underlying distance between states. The Hamming, block and Euclidean distances are already implemented by means of the arguments "Hamming", "Block" (default) and "Euclidean". Otherwise, a function taking as input two states must be provided.
normalize	Logical. If normalize = FALSE (default), the value of the estimated skewness is returned. Otherwise, the function returns the normalized estimated skewness.

Details

Given an OTS of length $T$ with range $\mathcal{S}=\{s_0, s_1, s_2, \ldots, s_n\}$ ($s_0 < s_1 < s_2 < \ldots < s_n$), $\overline{X}_t=\{\overline{X}_1,\ldots, \overline{X}_T\}$, the function computes the estimated skewness given by $\widehat{skew}_{d}=\sum_{i=0}^n\big(d(s_i,s_n)-d(s_i,s_0)\big)\widehat{p}_i$, where $d(\cdot, \cdot)$ is a distance between ordinal states and $\widehat{p}_k$ is the standard estimate of the marginal probability for state $s_k$ computed from the realization $\overline{X}_t$.

Value

The estimated skewness.

Author(s)

Ángel López-Oriona, José A. Vilar

References

Weiß CH (2019). “Distance-based analysis of ordinal data and ordinal time series.” Journal of the American Statistical Association.

Examples

estimated_skewness <- ordinal_skewness(series = AustrianWages$data[[100]],
states = 0 : 5) # Computing the skewness estimate
# for one series in dataset AustrianWages using the block distance

---

Constructs an ordinal time series plot

Description

ots_plot constructs an ordinal time series plot

Usage

ots_plot(series, states, title = "Time series plot", labels = NULL)

Arguments

series	An OTS.
states	A numerical vector containing the corresponding states.
title	The title of the graph.
labels	The labels of the graph.

Details

Constructs an ordinal time series plot for a given OTS.

Value

The ordinal time series plot.

Author(s)

Ángel López-Oriona, José A. Vilar

References

Weiß CH (2018). An introduction to discrete-valued time series. John Wiley and Sons.

Examples

ordinal_time_series_plot <- ots_plot(series = AustrianWages$data[[100]],,
states = 0 : 5) # Constructs an ordinal
# time series plot for one series in
# dataset AustrianWages

---

Constructs a serial dependence plot based on the ordinal Cohen's kappa considering the block distance

Description

plot_ordinal_cohens_kappa constructs a serial dependence plot of an ordinal time series based on the ordinal Cohen's kappa considering the block distance

Usage

plot_ordinal_cohens_kappa(
  series,
  states,
  max_lag = 10,
  alpha = 0.05,
  plot = TRUE,
  title = "Serial dependence plot",
  bar_width = 0.12,
  ...
)

Arguments

series	An OTS.
states	A numerical vector containing the corresponding states.
max_lag	The maximum lag represented in the plot (default is 10).
alpha	The significance level for the corresponding hypothesis test (default is 0.05).
plot	Logical. If plot = TRUE (default), returns the serial dependence plot. Otherwise, returns a list with the values of the ordinal Cohens's kappa, the critical value and the corresponding p-values.
title	The title of the graph.
bar_width	The width of the corresponding bars.
...	Additional parameters for the function.

Details

Constructs a serial dependence plot based on the ordinal Cohens's kappa, $\widehat{\kappa}_d(l)$, for several lags, where $d$ is the block distance between ordinal states, that is, $d(s_i, s_j)=|i-j|$ for two states $s_i$ and $s_j$. A dashed lined is incorporated indicating the critical value of the test based on the following asymptotic approximation (under the i.i.d. assumption):

$\sqrt{\frac{T\widehat{disp}_d^2}{4\sum_{k,l=0}^{n-1}(\widehat{f}_{min\{k,l\}}-\widehat{f}_k\widehat{f}_l)^2}}\bigg(\widehat{\kappa}_d(l)+\frac{1}{T}\bigg)\sim N\big(0, 1\big),$

where $T$ is the series length, $\widehat{f_k}$ is the estimated cumulative probability for state $s_k$ and $\widehat{disp}_d$ is the DIVC estimate of the dispersion.

Value

If plot = TRUE (default), returns the serial dependence plot based on the ordinal Cohens's kappa. Otherwise, the function returns a list with the values of the ordinal Cohens's kappa, the critical value and the corresponding p-values.

Author(s)

Ángel López-Oriona, José A. Vilar

References

Weiß CH (2019). “Distance-based analysis of ordinal data and ordinal time series.” Journal of the American Statistical Association.

Examples

plot_ock <- plot_ordinal_cohens_kappa(series = AustrianWages$data[[100]],
states = 0 : 5, max_lag = 3) # Representing
# the serial dependence plot
list_ck <- plot_ordinal_cohens_kappa(series = AustrianWages$data[[100]],
states = 0 : 5, max_lag = 3, plot = FALSE) # Obtaining
# the values of the ordinal Cohens's kappa, the critical value and the p-values

---

SyntheticData1

Description

Synthetic dataset containing 80 OTS generated from four different generating processes.

Usage

data(SyntheticData1)

Format

A list with two elements, which are:

data

A list with 80 OTS.

classes

A numeric vector indicating the corresponding classes associated with the elements in data.

Details

Each element in data is a 6-state OTS of length 600. Series 1-20, 21-40, 41-60 and 61-80 were generated from binomial AR(p) processes with different coefficients (see Scenario 1 in López-Oriona et al. (2023)). Therefore, there are 4 different classes in the dataset.

References

López-Oriona Á, Weiß C, Vilar JA (2023). “Fuzzy clustering of ordinal time series based on two novel distances with financial applications.” Manuscript submitted for publication, 000-000.

---

SyntheticData2

Description

Synthetic dataset containing 80 OTS generated from four different generating processes.

Usage

data(SyntheticData2)

Format

A list with two elements, which are:

data

A list with 80 OTS.

classes

A numeric vector indicating the corresponding classes associated with the elements in data.

Details

Each element in data is a 6-state OTS of length 600. Series 1-20, 21-40, 41-60 and 61-80 were generated from binomial INARCH(p) processes with different coefficients (see Scenario 2 in López-Oriona et al. (2023)). Therefore, there are 4 different classes in the dataset.

References

López-Oriona Á, Weiß C, Vilar JA (2023). “Fuzzy clustering of ordinal time series based on two novel distances with financial applications.” Manuscript submitted for publication, 000-000.

---

SyntheticData3

Description

Synthetic dataset containing 80 OTS generated from four different generating processes.

Usage

data(SyntheticData3)

Format

A list with two elements, which are:

data

A list with 80 OTS.

classes

A numeric vector indicating the corresponding classes associated with the elements in data.

Details

Each element in data is a 6-state OTS of length 600. Series 1-20, 21-40, 41-60 and 61-80 were generated from ordinal logit AR(1) processes with different coefficients (see Scenario 3 in López-Oriona et al. (2023)). Therefore, there are 4 different classes in the dataset.

References

López-Oriona Á, Weiß C, Vilar JA (2023). “Fuzzy clustering of ordinal time series based on two novel distances with financial applications.” Manuscript submitted for publication, 000-000.

---

Performs the hypothesis test associated with the ordinal asymmetry for the block distance

Description

test_ordinal_asymmetry performs the hypothesis test associated with the ordinal asymmetry for the block distance

Usage

test_ordinal_asymmetry(
  series,
  states,
  true_asymmetry,
  alpha = 0.05,
  temporal = TRUE,
  max_lag = 1
)

Arguments

series	An OTS (numerical vector with integers).
states	A numeric vector containing the corresponding states.
true_asymmetry	The value for the true asymmetry.
alpha	The significance level (default is 0.05).
temporal	Logical. If temporal = TRUE (default), the test is performed for a time series. Otherwise, the test is performed for i.i.d. data.
max_lag	If temporal = TRUE, the maximum considered lag to compute the estimates related to the cumulative joint probabilities.

Details

If temporal = TRUE (default), the function performs the hypothesis test based on the ordinal asymmetry relying on Theorem 7.1.1 in Weiß (2019). Otherwise, the test based on Theorem 4.1 in Weiß (2019) is carried out.

Value

The results of the hypothesis test.

Author(s)

Ángel López-Oriona, José A. Vilar

References

Weiß CH (2019). “Distance-based analysis of ordinal data and ordinal time series.” Journal of the American Statistical Association.

Examples

results_test <- test_ordinal_asymmetry(AustrianWages$data[[100]],
states = 0 : 5, true_asymmetry = 2) # Performing the hypothesis test associated with the
# ordinal asymmetry for one OTS in dataset AustrianWages

---

Performs the hypothesis test associated with the ordinal dispersion for the block distance

Description

test_ordinal_dispersion performs the hypothesis test associated with the ordinal dispersion for the block distance

Usage

test_ordinal_dispersion(
  series,
  states,
  true_dispersion,
  alpha = 0.05,
  temporal = TRUE,
  max_lag = 1
)

Arguments

series	An OTS (numerical vector with integers).
states	A numeric vector containing the corresponding states.
true_dispersion	The value for the true dispersion.
alpha	The significance level (default is 0.05).
temporal	Logical. If temporal = TRUE (default), the test is performed for a time series. Otherwise, the test is performed for i.i.d. data.
max_lag	If temporal = TRUE, the maximum considered lag to compute the estimates related to the cumulative joint probabilities.

Details

If temporal = TRUE (default), the function performs the hypothesis test based on the ordinal dispersion relying on Theorem 7.1.1 in Weiß (2019). Otherwise, the test based on Theorem 4.1 in Weiß (2019) is carried out.

Value

The results of the hypothesis test.

Author(s)

Ángel López-Oriona, José A. Vilar

References

Weiß CH (2019). “Distance-based analysis of ordinal data and ordinal time series.” Journal of the American Statistical Association.

Examples

results_test <- test_ordinal_dispersion(AustrianWages$data[[100]],
states = 0 : 5, true_dispersion = 2) # Performing the hypothesis test associated with the
# ordinal dispersion for one OTS in dataset AustrianWages

---

Performs the hypothesis test associated with the ordinal skewness for the block distance

Description

test_ordinal_skewness performs the hypothesis test associated with the ordinal skewness for the block distance

Usage

test_ordinal_skewness(
  series,
  states,
  true_skewness,
  alpha = 0.05,
  temporal = TRUE,
  max_lag = 1
)

Arguments

series	An OTS (numerical vector with integers).
states	A numeric vector containing the corresponding states.
true_skewness	The value for the true skewness.
alpha	The significance level (default is 0.05).
temporal	Logical. If temporal = TRUE (default), the test is performed for a time series. Otherwise, the test is performed for i.i.d. data.
max_lag	If temporal = TRUE, the maximum considered lag to compute the estimates related to the cumulative joint probabilities.

Details

If temporal = TRUE (default), the function performs the hypothesis test based on the ordinal skewness relying on Theorem 7.1.1 in Weiß (2019). Otherwise, the test based on Theorem 4.1 in Weiß (2019) is carried out.

Value

The results of the hypothesis test.

Author(s)

Ángel López-Oriona, José A. Vilar

References

Weiß CH (2019). “Distance-based analysis of ordinal data and ordinal time series.” Journal of the American Statistical Association.

Examples

results_test <- test_ordinal_skewness(AustrianWages$data[[100]],
states = 0 : 5, true_skewness = 2) # Performing the hypothesis test associated with the
# ordinal skewness for one OTS in dataset AustrianWages

---

Computes the total cumulative correlation of an ordinal time series

Description

total_c_correlation returns the value of the total cumulative correlation for an ordinal time series

Usage

total_c_correlation(series, lag = 1, states, features = FALSE)

Arguments

series	An OTS.
lag	The considered lag (default is 1).
states	A numerical vector containing the corresponding states.
features	Logical. If features = FALSE (default), the value of the total cumulative correlation is returned. Otherwise, the function returns a matrix with the individual components of the total cumulative correlation

Details

Given an OTS of length $T$ with range $\mathcal{S}=\{s_0, s_1, \ldots, s_n\}$, $\overline{X}_t=\{\overline{X}_1,\ldots, \overline{X}_T\}$, and the cumulative binarized time series, which is defined as $\overline{\boldsymbol Y}_t=\{\overline{\boldsymbol Y}_1, \ldots, \overline{\boldsymbol Y}_T\}$, with $\overline{\boldsymbol Y}_k=(\overline{Y}_{k,0}, \ldots, \overline{Y}_{k,n-1})^\top$ such that $\overline{Y}_{k,i}=1$ if $\overline{X}_k\leq s_i$ ($k=1,\ldots,T, , i=0,\ldots,n-1$), the function computes the estimated average $\widehat{\Psi}(l)^c=\frac{1}{n^2}\sum_{i,j=0}^{n-1}\widehat{\psi}_{ij}(l)^2$, where $\widehat{\psi}_{ij}(l)$ is the estimated correlation $\widehat{Corr}(Y_{t, i}, Y_{t-l, j})$, $i,j=0, 1,\ldots,n-1$. If features = TRUE, the function returns a matrix whose components are the quantities $\widehat{\psi}_{ij}(l)$, $i,j=0,1, \ldots,n-1$.

Value

If features = FALSE (default), returns the value of the total cumulative correlation. Otherwise, the function returns a matrix of features, i.e., the matrix contains the features employed to compute the total cumulative correlation.

Author(s)

Ángel López-Oriona, José A. Vilar

Examples

tcc <- total_c_correlation(series = AustrianWages$data[[100]],
states = 0 : 5) # Computing the total cumulative correlation
# for one of the series in dataset AustrianWages
feature_matrix <- total_c_correlation(series = AustrianWages$data[[100]],
states = 0 : 5) # Computing the corresponding matrix of features

---

Computes the total mixed cumulative linear correlation (TMCLC) between an ordinal and a real-valued time series

Description

total_mixed_c_correlation_1 returns the TMCLC between an ordinal and a real-valued time series

Usage

total_mixed_c_correlation_1(
  o_series,
  n_series,
  lag = 1,
  states,
  features = FALSE
)

Arguments

o_series	An OTS.
n_series	A real-valued time series.
lag	The considered lag (default is 1).
states	A numerical vector containing the corresponding states.
features	Logical. If features = FALSE (default), the value of the TMCLC is returned. Otherwise, the function returns a vector with the individual components of the TMCLC.

Details

Given a OTS of length $T$ with range $\mathcal{S}=\{s_0, s_1, \ldots, s_n\}$, $\overline{X}_t=\{\overline{X}_1,\ldots, \overline{X}_T\}$, and the cumulative binarized time series, which is defined as $\overline{\boldsymbol Y}_t=\{\overline{\boldsymbol Y}_1, \ldots, \overline{\boldsymbol Y}_T\}$, with $\overline{\boldsymbol Y}_k=(\overline{Y}_{k,0}, \ldots, \overline{Y}_{k,n-1})^\top$ such that $\overline{Y}_{k,i}=1$ if $\overline{X}_k \leq s_i$ ($k=1,\ldots,T , i=0,\ldots,n-1$), the function computes the estimated TMCLC given by

$\widehat{\Psi}_1^m(l)=\frac{1}{n}\sum_{i=0}^{n-1}\widehat{\psi}_{i}^*(l)^2,$

where $\widehat{\psi}_{i}^*(l)=\widehat{Corr}(Y_{t,i}, Z_{t-l})$, with $\overline{Z}_t=\{\overline{Z}_1,\ldots, \overline{Z}_T\}$ being a $T$-length real-valued time series. If features = TRUE, the function returns a vector whose components are the quantities $\widehat{\psi}_{i}(l)$, $i=0,1, \ldots,n-1$.

Value

If features = FALSE (default), returns the value of the TMCLC. Otherwise, the function returns a vector of features, i.e., the vector contains the features employed to compute the TMCLC.

Author(s)

Ángel López-Oriona, José A. Vilar

Examples

tmclc <- total_mixed_c_correlation_1(o_series = SyntheticData1$data[[1]],
n_series = rnorm(600), states = 0 : 5) # Computing the TMCLC
# between the first series in dataset SyntheticData1 and white noise
feature_vector <- total_mixed_c_correlation_1(o_series = SyntheticData1$data[[1]],
n_series = rnorm(600), states = 0 : 5, features = TRUE) # Computing the corresponding
# vector of features

---

Computes the total mixed cumulative quantile correlation (TMCQC) between an ordinal and a real-valued time series

Description

total_mixed_c_correlation_2 returns the TMCQC between an ordinal and a real-valued time series

Usage

total_mixed_c_correlation_2(
  o_series,
  n_series,
  lag = 1,
  states,
  features = FALSE
)

Arguments

o_series	An OTS.
n_series	A real-valued time series.
lag	The considered lag (default is 1).
states	A numerical vector containing the corresponding states.
features	Logical. If features = FALSE (default), the value of the TMCLC is returned. Otherwise, the function returns a vector with the individual components of the TMCQC.

Details

Given a OTS of length $T$ with range $\mathcal{S}=\{s_0, s_1, \ldots, s_n\}$, $\overline{X}_t=\{\overline{X}_1,\ldots, \overline{X}_T\}$, and the cumulative binarized time series, which is defined as $\overline{\boldsymbol Y}_t=\{\overline{\boldsymbol Y}_1, \ldots, \overline{\boldsymbol Y}_T\}$, with $\overline{\boldsymbol Y}_k=(\overline{Y}_{k,0}, \ldots, \overline{Y}_{k,n-1})^\top$ such that $\overline{Y}_{k,i}=1$ if $\overline{X}_k \leq s_i$ ($k=1,\ldots,T , i=0,\ldots,n-1$), the function computes the estimated TMCQC given by

$\widehat{\Psi}_2^m(l)=\frac{1}{n}\sum_{i=0}^{n-1}\int_{0}^{1}\widehat{\psi}^\rho_{i}(l)^2d\rho,$

where $\widehat{\psi}_{i}^\rho(l)=\widehat{Corr}\big(Y_{t,i}, I(Z_{t-l}\leq q_{Z_t}(\rho)) \big)$, with $\overline{Z}_t=\{\overline{Z}_1,\ldots, \overline{Z}_T\}$ being a $T$-length real-valued time series, $\rho \in (0, 1)$ a probability level, $I(\cdot)$ the indicator function and $q_{Z_t}$ the quantile function of the corresponding real-valued process. If features = TRUE, the function returns a vector whose components are the quantities $\int_{0}^{1}\widehat{\psi}^\rho_{i}(l)^2d\rho$, $i=0,1, \ldots,n-1$.

Value

If features = FALSE (default), returns the value of the TMCQC. Otherwise, the function returns a vector of features, i.e., the vector contains the features employed to compute the TMCLC.

Author(s)

Ángel López-Oriona, José A. Vilar

Examples

tmclc <- total_mixed_c_correlation_2(o_series = SyntheticData1$data[[1]],
n_series = rnorm(600), states = 0 : 5) # Computing the TMCQC
# between the first series in dataset SyntheticData1 and white noise
feature_vector <- total_mixed_c_correlation_2(o_series = SyntheticData1$data[[1]],
n_series = rnorm(600), states = 0 : 5, features = TRUE) # Computing the corresponding
# vector of features

---