# Time Series Analysis

## FOURIER

FOURIER Fourier or inverse Fourier transform

### Synopsis

`FOURIER(Sequence,Inverse,Separate)`

### Arguments

Sequence: the data sequence to be transformed

Inverse: if true, the inverse Fourier transform is calculated, defaults to false

Separate: if true, the real and imaginary parts are given separately, defaults to false

### Description

This array function returns the Fourier or inverse Fourier transform of the given data sequence.

The output consists of one column of complex numbers if Separate is false and of two columns of real numbers if Separate is true.

If Separate is true the first output column contains the real parts and the second column the imaginary parts.

### Note

If Sequence is neither an n by 1 nor 1 by n array, this function returns #VALUE!

## HPFILTER

HPFILTER Hodrick Prescott Filter

### Synopsis

`HPFILTER(Sequence,λ)`

### Arguments

Sequence: the data sequence to be transformed

λ: filter parameter λ, defaults to 1600

### Description

This array function returns the trend and cyclical components obtained by applying the Hodrick Prescott Filter with parameter λ to the given data sequence.

The output consists of two columns of numbers, the first containing the trend component, the second the cyclical component.

### Note

If Sequence is neither an n by 1 nor 1 by n array, this function returns #VALUE! If Sequence contains less than 6 numerical values, this function returns #VALUE!

## INTERPOLATION

INTERPOLATION interpolated values corresponding to the given abscissa targets

### Synopsis

`INTERPOLATION(abscissae,ordinates,targets,interpolation)`

### Arguments

abscissae: abscissae of the given data points

ordinates: ordinates of the given data points

targets: abscissae of the interpolated data

interpolation: method of interpolation, defaults to 0 ('linear')

### Description

The output consists always of one column of numbers.

Possible interpolation methods are:

0: linear;

1: linear with averaging;

2: staircase;

3: staircase with averaging;

4: natural cubic spline;

5: natural cubic spline with averaging.

### Note

The abscissae should be given in increasing order. If the abscissae is not in increasing order the INTERPOLATION function is significantly slower. If any two abscissae values are equal an error is returned. If any of interpolation methods 1 ('linear with averaging'), 3 ('staircase with averaging'), and 5 ('natural cubic spline with averaging') is used, the number of returned values is one less than the number of targets and the target values must be given in increasing order. The values returned are the average heights of the interpolation function on the intervals determined by consecutive target values. Strings and empty cells in abscissae and ordinates are ignored. If several target data are provided they must be in the same column in consecutive cells.

## PERIODOGRAM

PERIODOGRAM periodogram of the given data

### Synopsis

`PERIODOGRAM(ordinates,filter,abscissae,interpolation,number)`

### Arguments

ordinates: ordinates of the given data

filter: windowing function to be used, defaults to no filter

abscissae: abscissae of the given data, defaults to regularly spaced abscissae

interpolation: method of interpolation, defaults to none

number: number of interpolated data points

### Description

If an interpolation method is used, the number of returned values is one less than the number of targets and the targets values must be given in increasing order.

The output consists always of one column of numbers.

Possible interpolation methods are:

0: linear;

1: linear with averaging;

2: staircase;

3: staircase with averaging;

4: natural cubic spline;

5: natural cubic spline with averaging.

Possible window functions are:

0: no filter (rectangular window)

1: Bartlett (triangular window)

2: Hahn (cosine window)

3: Welch (parabolic window)

### Note

Strings and empty cells in abscissae and ordinates are ignored. If several target data are provided they must be in the same column in consecutive cells.