Transform Time Series

Multiple time series can be selected and each selected time series is standardized or transformed.

For standardization the user can use any of the following four standardization types:

1. Gaussian: The rank ordering function from the sample marginal distribution to marginal Gaussian distribution is applied.
2. Uniform: The rank ordering function from the sample marginal distribution to uniform Gaussian distribution is applied.
3. Linear [0,1]: The samples of the time series are linearly transformed so that the minimum is transformed to 0 and the maximum to 1.
4. Normalized (or z-score): The samples of the time series are transformed linearly so that they have mean zero and standard deviation 1. 

For transformation, there are four types:
1. Box Cox: For the Box Cox transform, a parameter lambda (or a range of parameters to produce equivalently many transformed time series) determines the strength of the power Box-Cox transform that aims at normalizing the histogram of the time series. For lambda=0 the transform reduces to the natural logarithm.
2. Differencing: The transform of first differences (y(i)=x(i)-x(i-1)) is supposed to remove slow trends from the time series. This transform can be further extended to differences of a higher order t, that is at a given lag t, y(i)=x(i)-x(i-t).  The order (lag) t (or an array of orders) can be given as input parameter.
3. Log-difference: In financial applications, the analysis is on the time series of returns rather than the financial indices. The returns are defined as the first differences of the logarithms of the index values, and this transform is implemented here.
4. Detrend: For this transform first a polynomial of a given degree is fitted to the time series and then subtracted from the time series. 

The first of the four transforms attempts to make the magnitudes of the time series Gaussian while the other three transforms aim to reduce non-stationarity of the time series. First differences or logarithm of first differences, as well as polynomial detrending, aims at removing the slow drifts in the time series. Lag differences may be used to reduce the effect of periodicity if the selected lag matches the period of the cycle in the data.

 

Select one or more time series ...

Multiple time series names can be selected from this list. This can be done by standard use of the mouse: a click on the first name and Shift^click on the last name selects all names between the first and the last name; a click on the first name and subsequent Ctrl^clicks on other names select these names including the first name; Ctrl^a selects the whole list.
 

Standardization types

Any of the four standardization types, or even all standardization types (along with selected transformation types) can be selected. The standardization types are briefly explained below.

Gaussian : A rank ordering function from the sample marginal distribution to marginal Gaussian distribution is applied. Let x(i), i = 1,...,N, the selected time series. The position plotting transform estimates the marginal cumulative density function (cdf) as F(x(i)) = ((i)-0.326)/(N+0.348), where (i) is the order of x(i) in the N sample of the time series. Then the Gaussian standardization transform reads  y(i) = Ö-1(F(x(i)), where Ö-1 is the inverse marginal Gaussian cdf.

Uniform : A rank ordering function from the sample marginal distribution to marginal Uniform distribution is applied. Let x(i), i = 1,...,N, the selected time series. The position plotting transform estimates the marginal cumulative density function (cdf) as F(x(i)) = ((i)-0.326)/(N+0.348), where (i) is the order of x(i) in the N sample of the time series. Then the Unifrom standardization transform reads  y(i) = F(x(i).

Linear [0,1] : The samples of the time series are linearly transformed so that the minimum is transformed to 0 and the maximum to 1. Let xmin and xmax the minimum and maximum of the time series x(i), i = 1,...,N. Then the Linear standardization transform reads  y(i) = (x(i) - xmin) / (xmax - xmin).

Normalized (or z-score) : The samples of the time series are transformed linearly so that they have mean zero and standard deviation 1.  Let m and SD the mean and standard deviation of the samples of the time series x(i), i = 1,...,N. Then the Normalized standardization transform reads  y(i) = (x(i) - m) / SD.

 

Transformation types

Any of the four transformation types, or even all transformation types (along with selected standardization types) can be selected. The transformation types are briefly explained below.

Log-difference: This transform takes the first differences of the natural logarithms of the time series. If the time series contains negative value(s) then the logarithm transform is applied as y(i)=log(x(i)-xmin+1), for each value x(i) of a given time series, where xmin is the minimum value of the time series. Then the transform reads y(i)-y(i-1).

Box Cox : This is the Box Cox power transform and requires a parameter lambda. The transform is (x(i)^lambda-1)/lambda if lambda is not zero, and log(x(i)-xmin+1) if lambda is zero, where xmin is the minimum value of the time series. For the lambda parameter the user can give multiple values (according to the standard matlab format for forming arrays) and then the time series is transformed with the Box Cox formula for each lambda value.

Differencing : The transform of differencing for a given order (lag) t is defined as x(i)-x(i-t) (for t=1 this is the common first difference). For the order parameter the user can give multiple values (according to the standard matlab format for forming arrays) and then equally many differenced time series are obtained for each order.

Detrend : This is the detrending of the time series with a polynomial. First a polynomial of a given degree is fitted to the time series and then the fitted values are subtracted from the corresponding values of the time series. For the polynomial degree parameter the user can give multiple values (according to the standard matlab format for forming arrays) and then equally many detrended time series are obtained for each polynomial degree.

 

OK

By pressing this button, the standardizations of the selected time series starts and when it is completed the window is closed.
 

Cancel

The window is closed without segmenting any time series.
 

Help

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