01 Compute relation curves

Compute relation curves

General

A relation curve gives a functional relationship between two series of the following form Y_t = F(X_t+t1 ). The curves can be used for:
1. Detection of random errors,
2. Detection of systematic errors,
3. Filling in of missing data, and
4. Forecasting purposes.
If there is a distinct one to one relationship between two series random errors will be shown in a relation curve plot as outliers. To arrive at a one to one relationship (i.e. elimination of looping) the introduction of a time shift (t1) between the two series may be necessary.
By comparing two relation curves or data of one period with the curve of another period, shifts in relationships, e.g. in water level series due to changes in the gauge zero, can be detected.
The relation curve fitted to the data of two series can be used to fill-in missing data in the dependent variable of the relation (Y), see Section Interpolation.

If the series in the relation are mutually shifted in time, with sufficient lead-time for the independent variable X (t1 negative), the relation curve may be used to forecast the dependent variable in the relation Y from observations on X.
The parameters of the established relationships for a period of time can be stored in the data base for e.g. later comparison, filling-in missing data.
The main options under <Relation curves> include:

• optimisation of time shift t1,
• plotting of time series data Y_t versus X_t+t1 ,
• fitting a polynomial to Y_t , X_t+t1 ,
• validation of relation curve,
• display and comparison of relation curves.
  

  Series codes

  X- and Y-Series can be selected by selecting a series in the series list box and pressing the Select X or Select Y button. If an X-Series is selected HYMOS will calculate the data range of this series for the specified period.
  

  Time Shift

  A time shift can be entered manually or calculated by HYMOS . If an optimised time shift is chosen the user can divide the data range into three intervals, the lower and upper limits of each data interval must be entered. By pressing execute, HYMOS will calculate for each interval the optimised time shift.
  

  Parameter Computation

  By checking the compute relation checkbox in the parameter computation you are authorised to enter values for parameter computation. Choose the number of intervals, and for each interval the lower and upper limit and the polynomial degree.
  

  Draw data

  Plot the time series data Y versus the shifted X, see plotting of X-Y data.
  

  Save Relation

  Save the calculated relation for the time period defined in the Relation Validity Period. The start and end-time of this period can be changed by double-clicking the time-label.
  

  Optimisation of time shift

  Optimisation of time shifts may be considered if loopings are apparent in the relation curve caused by a translation phenomenon, like the passage of a flood wave. For this consider a relation curve of daily water levels of stations X and Y which are spaced at some distance s km from each other along the same river, X upstream of Y. Let the average flow velocity be u m/s, then the propagation velocity or celerity of the flood wave c is approximately 1.5(B_r /B_s )u, where B_r = river width and B_s = storage width (river + flood plain). So, if the river is in-bank, c = 1.5u, and the time shift to be applied between X and Y for a one to one relationship amounts: time = distance/celerity = -s*1000/(1.5u) sec. or t1 = -s*1000/(1.5u*86,400) days. If one wants to work with full day shifts t1 should be rounded to the nearest integer, (in HYMOS this rounding is not necessary). Note that when the river is in flood the celerity may differ from 1.5u and a different time shift may be necessary.
  In case the required time shift is known beforehand, the shift can be implemented by means of entering different start dates for the series X and Y.
  In the previous example a physical reasoning for the time shift has been given and field data were assumed to be available to compute the time shift. hymosprovides also a computational procedure to derive a time shift between the series based on cross-correlation analysis. The time-lag of maximum correlation is computed. Next a parabola is fitted to the maximum value in the cross-correlogram and its surrounding. The maximum of this parabola will be the maximum displayed on your screen (shown with two decimals). The user still has the freedom to implement the computed time shift or to use another shift. If the required time shift is not an integer, hymoswill apply a weighted interpolation between surrounding series elements of X.
  

  Plotting of X-Y data

  Prior to fitting a relation curve to the observations, a plot of the X-Y data is advised to:
  
• Detect outliers,
• Inspect the absence of looping,
• Determine existence of break-points in the relation, and
• Estimation of degree of polynomial to fit the data.

Fitting of relation curve

The X-Y data can be fitted by a relation equation of polynomial type:

Y_t = c₀ + c₁ .X_i + c₂ .X_i ² + c₃ .X_i ³ + .... i=t+t1

where: c_j = coefficient
The order of the polynomial is input. The maximum order of the polynomial is 9. However, if the parameters are to be stored in the data base the maximum order is 4. The least squares
principle is applied to estimate the coefficients.
At maximum 3 intervals of X may be applied for which a set of coefficients is estimated. To obtain an intersection between the relation equations of successive intervals an overlap between the intervals should be applied to force an intersection in the preset overlap. If in this overlap no intersecti­on is found, then that value of X will be taken as boundary for which the difference between the two equations is minimum.

Error analysis

During the calculation, a report file is made with error analysis. The error analysis gives following information:

• boundaries and coefficients of the polynomials,
• time of observation,
• observed Y and estimated Y data,
• difference between observed and estimated Y,
• relative estimation error in percent,
• standard error of estimate per interval of X,
• overall standard error of estimate.
  The standard error of estimate is estimated by:
  
  

  Validation of relation curve

  To investigate shifts in the relationship between two series caused by natural or human factors, like changes in cross-sections, or shifts in gauge zero's, changes in gauge locations, etc. the relation curve developed for a particular period of time is compared with data of a following period.
  hymosprovides for this type of analysis the following options:
• The X-Y data and relation curve parameters are both read from the data base;
• The X-Y data are read from the database and the relation curve parameters are entered manually.
  Next by visual inspection the validity of the relation equation is judged.