g13ba filters a time series by an ARIMA model.

# Syntax

C# |
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public static void g13ba( double[] y, int ny, int[] mr, int nmr, double[] par, double cy, double[] b, out int ifail ) |

Visual Basic |
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Public Shared Sub g13ba ( _ y As Double(), _ ny As Integer, _ mr As Integer(), _ nmr As Integer, _ par As Double(), _ cy As Double, _ b As Double(), _ <OutAttribute> ByRef ifail As Integer _ ) |

Visual C++ |
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public: static void g13ba( array<double>^ y, int ny, array<int>^ mr, int nmr, array<double>^ par, double cy, array<double>^ b, [OutAttribute] int% ifail ) |

F# |
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static member g13ba : y : float[] * ny : int * mr : int[] * nmr : int * par : float[] * cy : float * b : float[] * ifail : int byref -> unit |

#### Parameters

- y
- Type: array<System..::..Double>[]()[][]An array of size [ny]
*On entry*: the ${Q}_{y}^{\prime}$ backforecasts, starting with backforecast at time $1-{Q}_{y}^{\prime}$ to backforecast at time $0$, followed by the time series starting at time $1$, where ${Q}_{y}^{\prime}={\mathbf{mr}}\left[9\right]+{\mathbf{mr}}\left[12\right]\times {\mathbf{mr}}\left[13\right]$. If there are no backforecasts, either because the ARIMA model for the time series is not known, or because it is known but has no moving average terms, then the time series starts at the beginning of y.

- ny
- Type: System..::..Int32
*On entry*: the total number of backforecasts and time series data points in array y.*Constraint*: ${\mathbf{ny}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1+{Q}_{y}^{\prime},\mathbf{\_npar}\right)$.

- mr
- Type: array<System..::..Int32>[]()[][]An array of size [nmr]
*On entry*: the orders vector for the filtering model, followed by the orders vector for the ARIMA model for the time series if the latter is known. The orders appear in the standard sequence $\left(p,d,q,P,D,Q,s\right)$ as given in the**G13**class. If the ARIMA model for the time series is supplied, then the method will assume that the first ${Q}_{y}^{\prime}$ values of the array y are backforecasts.*Constraints*:the filtering model is restricted in the following ways:- ${\mathbf{mr}}\left[0\right]+{\mathbf{mr}}\left[2\right]+{\mathbf{mr}}\left[3\right]+{\mathbf{mr}}\left[5\right]>0$, i.e., filtering by a model which contains only differencing terms is not permitted;
- ${\mathbf{mr}}\left[\mathit{k}\right]\ge 0$, for $\mathit{k}=0,1,\dots ,6$;
- if ${\mathbf{mr}}\left[6\right]=0$, ${\mathbf{mr}}\left[3\right]+{\mathbf{mr}}\left[4\right]+{\mathbf{mr}}\left[5\right]=0$;
- if ${\mathbf{mr}}\left[6\right]\ne 0$, ${\mathbf{mr}}\left[3\right]+{\mathbf{mr}}\left[4\right]+{\mathbf{mr}}\left[5\right]\ne 0$;
- ${\mathbf{mr}}\left[6\right]\ne 1$.

the ARIMA model for the time series is restricted in the following ways:- ${\mathbf{mr}}\left[\mathit{k}\right]\ge 0$, for $\mathit{k}=7,8,\dots ,13$;
- if ${\mathbf{mr}}\left[13\right]=0$, ${\mathbf{mr}}\left[10\right]+{\mathbf{mr}}\left[11\right]+{\mathbf{mr}}\left[12\right]=0$;
- if ${\mathbf{mr}}\left[13\right]\ne 0$, ${\mathbf{mr}}\left[10\right]+{\mathbf{mr}}\left[11\right]+{\mathbf{mr}}\left[12\right]\ne 0$;
- ${\mathbf{mr}}\left[13\right]\ne 1$.

- nmr
- Type: System..::..Int32
*On entry*: the number of values specified in the array mr. It takes the value $7$ if no ARIMA model for the time series is supplied but otherwise it takes the value $14$. Thus nmr acts as an indicator as to whether backforecasting can be carried out.*Constraint*: ${\mathbf{nmr}}=7$ or $14$.

- par
- Type: array<System..::..Double>[]()[][]An array of size [dim1]
**Note:**dim1 must satisfy the constraint:- if ${\mathbf{nmr}}=7$, $\mathbf{\_npar}={\mathbf{mr}}\left[0\right]+{\mathbf{mr}}\left[2\right]+{\mathbf{mr}}\left[3\right]+{\mathbf{mr}}\left[5\right]$;
- if ${\mathbf{nmr}}=14$, $\mathbf{\_npar}={\mathbf{mr}}\left[0\right]+{\mathbf{mr}}\left[2\right]+{\mathbf{mr}}\left[3\right]+{\mathbf{mr}}\left[5\right]+\phantom{\rule{0ex}{0ex}}{\mathbf{mr}}\left[7\right]+{\mathbf{mr}}\left[9\right]+{\mathbf{mr}}\left[10\right]+{\mathbf{mr}}\left[12\right]$.

**Note:**the first constraint (i.e., ${\mathbf{mr}}\left[0\right]+{\mathbf{mr}}\left[2\right]+{\mathbf{mr}}\left[3\right]+{\mathbf{mr}}\left[5\right]>0$) on the orders of the filtering model, in parameter mr, ensures that $\mathbf{\_npar}>0$.*On entry*: the parameters of the filtering model, followed by the parameters of the ARIMA model for the time series, if supplied. Within each model the parameters are in the standard order of non-seasonal AR and MA followed by seasonal AR and MA.

- cy
- Type: System..::..Double

- b
- Type: array<System..::..Double>[]()[][]An array of size [nb]
*On exit*: the filtered output series. If the ARIMA model for the time series was known, and hence ${Q}_{y}^{\prime}$ backforecasts were supplied in y, then b contains ${Q}_{y}^{\prime}$ ‘filtered’ backforecasts followed by the filtered series. Otherwise, the filtered series begins at the start of b just as the original series began at the start of y. In either case, if the value of the series at time $t$ is held in ${\mathbf{y}}\left[t-1\right]$, then the filtered value at time $t$ is held in ${\mathbf{b}}\left[t-1\right]$.

- ifail
- Type: System..::..Int32%
*On exit*: ${\mathbf{ifail}}={0}$ unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).

# Description

From a given series ${y}_{1},{y}_{2},\dots ,{y}_{n}$, a new series ${b}_{1},{b}_{2},\dots ,{b}_{n}$ is calculated using a supplied (filtering) ARIMA model. This model will be one which has previously been fitted to a series ${x}_{t}$ with residuals ${a}_{t}$. The equations defining ${b}_{t}$ in terms of ${y}_{t}$ are very similar to those by which ${a}_{t}$ is obtained from ${x}_{t}$. The only dissimilarity is that no constant correction is applied after differencing. This is because the series ${y}_{t}$ is generally distinct from the series ${x}_{t}$ with which the model is associated, though ${y}_{t}$ may be related to ${x}_{t}$. Whilst it is appropriate to apply the ARIMA model to ${y}_{t}$ so as to preserve the same relationship between ${b}_{t}$ and ${a}_{t}$ as exists between ${y}_{t}$ and ${x}_{t}$, the constant term in the ARIMA model is inappropriate for ${y}_{t}$. The consequence is that ${b}_{t}$ will not necessarily have zero mean.

The equations are precisely:

the appropriate differencing of ${y}_{t}$; both the seasonal and non-seasonal inverted autoregressive operations are then applied,

followed by the inverted moving average operations

Because the filtered series value ${b}_{t}$ depends on present and past values ${y}_{t},{y}_{t-1},\dots \text{}$, there is a problem arising from ignorance of ${y}_{0},{y}_{-1},\dots \text{}$ which particularly affects calculation of the early values ${b}_{1},{b}_{2},\dots \text{}$, causing ‘transient errors’. The method allows two possibilities.

$${w}_{t}={\nabla}^{d}{\nabla}_{s}^{D}{y}_{t}\text{,}$$ | (1) |

$${u}_{t}={w}_{t}-{\Phi}_{1}{w}_{t-s}-\cdots -{\Phi}_{P}{w}_{t-s\times P}$$ | (2) |

$${v}_{t}={u}_{t}-{\varphi}_{1}{u}_{t-1}-\cdots -{\varphi}_{p}{u}_{t-p}$$ | (3) |

$${z}_{t}={v}_{t}+{\Theta}_{1}{z}_{t-s}+\cdots +{\Theta}_{Q}{z}_{t-s\times Q}$$ | (4) |

$${b}_{t}={z}_{t}+{\theta}_{1}{b}_{t-1}+\cdots +{\theta}_{q}{b}_{t-q}\text{.}$$ | (5) |

(i) | The equations (1), (2) and (3) are applied from successively later time points so that all terms on their right-hand sides are known, with ${v}_{t}$ being defined for $t=\left(1+d+s\times D+s\times P\right),\dots ,n$. Equations (4) and (5) are then applied over the same range, taking any values on the right-hand side associated with previous time points to be zero.
This procedure may still however result in unacceptably large transient errors in early values of ${b}_{t}$. |

(ii) | The unknown values ${y}_{0},{y}_{-1},\dots \text{}$ are estimated by backforecasting. This requires that an ARIMA model distinct from that which has been supplied for filtering, should have been previously fitted to ${y}_{t}$. |

For efficiency, you are asked to supply both this ARIMA model for ${y}_{t}$ and a limited number of backforecasts which are prefixed to the known values of ${y}_{t}$. Within the method further backforecasts of ${y}_{t}$, and the series ${w}_{t}$, ${u}_{t}$, ${v}_{t}$ in (1), (2) and (3) are then easily calculated, and a set of linear equations solved for backforecasts of ${z}_{t},{b}_{t}$ for use in (4) and (5) in the case that $q+Q>0$.

Even if the best model for ${y}_{t}$ is not available, a very approximate guess such as

or

can help to reduce the transients substantially.

$${y}_{t}=c+{e}_{t}$$ |

$$\nabla {y}_{t}={e}_{t}$$ |

The backforecasts which need to be prefixed to ${y}_{t}$ are of length ${Q}_{y}^{\prime}={q}_{y}+{s}_{y}\times {Q}_{y}$, where ${q}_{y}$ and ${Q}_{y}$ are the non-seasonal and seasonal moving average orders and ${s}_{y}$ the seasonal period for the ARIMA model of ${y}_{t}$. Thus you need not carry out the backforecasting exercise if ${Q}_{y}^{\prime}=0$. Otherwise, the series ${y}_{1},{y}_{2},\dots ,{y}_{n}$ should be reversed to obtain ${y}_{n},{y}_{n-1},\dots ,{y}_{1}$ and
g13aj
should be used to forecast ${Q}_{y}^{\prime}$ values, ${\hat{y}}_{0},\dots ,{\hat{y}}_{1-{Q}_{y}^{\prime}}$. The ARIMA model used is that fitted to ${y}_{t}$ (as a forward series) except that, if ${d}_{y}+{D}_{y}$ is odd, the constant should be changed in sign (to allow, for example, for the fact that a forward upward trend is a reversed downward trend). The ARIMA model for ${y}_{t}$ supplied to the filtering method must however have the appropriate constant for the forward series.

The series ${\hat{y}}_{1-{Q}_{y}^{\prime}},\dots ,{\hat{y}}_{0},{y}_{1},\dots ,{y}_{n}$ is then supplied to the method, and a corresponding set of values returned for ${b}_{t}$.

# References

Box G E P and Jenkins G M (1976)

*Time Series Analysis: Forecasting and Control*(Revised Edition) Holden–Day# Error Indicators and Warnings

Errors or warnings detected by the method:

Some error messages may refer to parameters that are dropped from this interface
(NPAR) In these
cases, an error in another parameter has usually caused an incorrect value to be inferred.

- ${\mathbf{ifail}}=1$
On entry, ${\mathbf{nmr}}\ne 7$ and ${\mathbf{nmr}}\ne 14$.

- ${\mathbf{ifail}}=2$
On entry, the orders vector mr does not satisfy the constraints given in [Parameters].

- ${\mathbf{ifail}}=4$
On entry, ny is too small to successfully carry out the requested filtering, (see [Parameters]).

- ${\mathbf{ifail}}=5$
On entry, the work array wa is too small.

- ${\mathbf{ifail}}=6$
On entry, the array b is too small.

- ${\mathbf{ifail}}=7$
- The orders vector for the filtering model is invalid.

- ${\mathbf{ifail}}=8$
- The orders vector for the ARIMA model is invalid. (Only occurs if ${\mathbf{nmr}}=14$.)

- ${\mathbf{ifail}}=9$
- The initial values of the filtered series are indeterminate for the given models.

- ${\mathbf{ifail}}=-999$
- Internal memory allocation failed.

# Accuracy

Accuracy and stability are high except when the MA parameters are close to the invertibility boundary.

# Parallelism and Performance

None.

# Further Comments

If an ARIMA model is supplied,
a local workspace array
of fixed
length is
allocated internally by g13ba. The total size of
this array
amounts to $K$ integer
elements, where
$K$ is the expression defined in the description of the parameter wa.

The time taken by g13ba is approximately proportional to

with an appreciable fixed increase if an ARIMA model is supplied for the time series.

$${\mathbf{ny}}\times \left({\mathbf{mr}}\left[0\right]+{\mathbf{mr}}\left[2\right]+{\mathbf{mr}}\left[3\right]+{\mathbf{mr}}\left[5\right]\right)\text{,}$$ |

# Example

This example reads a time series of length $296$. It reads the univariate ARIMA $\left(4,0,2,0,0,0,0\right)$ model and the ARIMA filtering $\left(3,0,0,0,0,0,0\right)$ model for the series. Two initial backforecasts are required and these are calculated by a call to
g13aj
.
The backforecasts are inserted at the start of the series and g13ba is called to perform the calculations.

Example program (C#): g13bae.cs