Regression analysis

Regression analysis

General

The regression analysis option in hymosincludes:

• computation of correlation matrix, and
• fitting of following type of functions: polynomial, simple linear, exponential, power, logarithmic, hyperbolic, multiple linear and stepwise.
  Relation curves^image010.gif!
  The multiple linear functions can be fitted by means of multiple or step-wise regression techniques. The algebraic forms of the functions are presented in next section.
  

  Series codes

  Data of series with the same time interval, read from the database, can enter the regression analysis. The series can be selected by selecting a series in the series list box and pressing the '>>' button. To deselect one of the series, selected the series in the spreadsheet and press '<<'. For regression analysis between series a minimum of two series must be selected. When only one series is selected, a regression against time will be calculated. When selecting a series, the minimum and maximum values of the series for the defined time period will be calculated. Each series must be given a code.
  

  Coding of variables

  0: free variable available for regression
  1: forced variable, which will enter regression
  2: variable not entering regression
  3: dependent variable (only one per selection )
  

  Minimum value / Maximum value

  The lower and upper boundaries for each series must be set; data outside these boundaries are eliminated; hymosshows default the minimum and maximum values.
  

  Regression function

  Choose a regression option from the list box by clicking the function.If you select <Polynomial> the degree of the polynomial has to be entered as well.
  If you select <Stepwise> following data have to be entered:
  
• maximum number of steps in the stepwise regression analysis;
• whether or not the results of each step in the stepwise analysis have to be printed; if you enter <N>, only the results of the last step will be presented.
• F-values to enter and delete variables from the regression analysis, which only applies for the free variables .
  

  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. The coefficients of the regression equation will be stored in the data base. There are, however, some limitations to the number of coefficients that can be stored:
  
• in case of polynomial regression, the degree of the polynomial should be £4;
• in case of multiple/stepwise regression the number of independent variables should be £4.
  

  Regression equations

  The following types of regression equations are available, with Y the dependent variable and X~j~ 's the independent variables:
  

  Type	Equation
  Polynomial	Relation curves^image012.gif! with: n = degree of polynomial: n £4, C~j~ = coefficient
  Simple linear	Y~i~ = A + B*.X~i~ with: A,B = coefficients
  Exponential 1	Y~i~ = A exp(B*X~i~ ) with: A,B = coefficients
  Exponential 2	Y~i~ = A exp(B/X~i~ ) with: A,B = coefficients
  Power	Y~i~ = A.X~i~ B with: A,B = coefficients
  Logarithmic	Y~i~ = A + B.ln(X~i~ ) with: A,B = coefficients
  Hyperbolic	Y~i~ = A + B/X~i~ with: A,B = coefficients
  Multiple linear	Relation curves^image014.gif! with: n = number of series on independent variables, n£4 C~j~ = coefficient

  
  

  Computation procedure

  All coefficients are determined by the least squares method.
  The Multiple Linear equation may be established by multiple or by stepwise regression:
  
• In the multiple regression situation all independent variables enter the regression equation at one go.
• In the stepwise regression situation the independent variables enter the regression equation one by one. The order of entry may be:
• free: the variables to which this option applies are called free variables ; their entry is determined by statistical properties; the variable added is the one which makes the greatest improvement in goodness of fit. Among the free varia­bles only the significant variables are included in the final regression equation.
• predetermined: the variables to which this option applies are called forced variables ; i.e. their entry is not based on correlation but solely requested by the user.
  

  Confidence limits

  For the linear regression methods, confidence intervals can be computed. Based on the sampling distributions of the regression parameters, the following estimates and confidence limits hold (see e.g. Kottegoda and Rosso, 1998).
  If the linear regression function is as follows:
  Relation curves^image016.gif!
  where:
  Y = dependent variable, also called response variable,
  X = independent variable or explanatory variable,
  a,b= regression coefficients,
  e= residual because of imperfect match of regression function through measurement points.
  An unbiased estimate of the error variance is given by:
  Relation curves^image018.gif!
  where:
  Relation curves^image020.gif!
  The parameter n is the total number of values for making the regression function and_x_ _m ,y _m are the mean values of the two series. Note that n-2 appears in the denominator to reflect the fact that two degrees of freedom have been lost in estimating (a, b)
  

  Relation curves^image022.gif!

  A (100-a) percent confidence interval for the mean response to some input value x~i~ of X is given by:
  Relation curves^image024.gif!
  where:
  t~n-2,1-~ _{a /2} = Student's t distribution with n-2 degrees of freedom.
  Note that the farther away x~i~ is from its mean the wider the confidence interval will be because the last term under the root sign expands in that way.
  HYMOS uses by default a 95% confidence interval of the regression line.
  Example
  In the underneath table some 17 years of annual rainfall (2) and runoff (3) data of a basin are presented. Regression analysis will be applied to validate the runoff series. No changes took place in the drainage characteristics of the basin.
  From the HYMOSfunctions list select the Regression function. Add the two series to the spreadsheet and give the rainfall series code 1, and the discharge series code 3. Select the linear regression function and press <Execute>.
  Table: Example computation of confidence limits for regression analysis

  Year	X=Rainfall	Y=Runoff	(X-X~m~ )2	(Y-Y~m~ )2	(X-X~m~ )(Y-Y~m~ )	Yest	UC1	LC1
  1	2	3	4	5	6	7	8	9
  1961	1130	740	44100	36100	39900	737	801	673
  1962	1280	1040	3600	12100	-6600	875	922	827
  1963	1270	960	4900	900	-2100	866	914	818
  1964	1040	610	90000	102400	96000	654	732	576
  1965	1080	590	67600	115600	88400	691	762	619
  1966	1150	820	36100	12100	20900	755	816	694
  1967	1670	1090	108900	25600	52800	1234	1317	1150
  1968	1540	1020	40000	8100	18000	1114	1176	1052
  1969	990	570	122500	129600	126000	608	695	521
  1970	1190	780	22500	22500	22500	792	848	736
  1971	1520	1090	32400	25600	28800	1096	1155	1036
  1972	1370	960	900	900	900	958	1004	911
  1973	1650	1240	96100	96100	96100	1215	1295	1135
  1974	1510	1030	28900	10000	17000	1086	1145	1028
  1975	1600	1340	67600	168100	106600	1169	1241	1098
  1976	1300	870	1600	3600	2400	893	940	847
  1977	1490	1060	22500	16900	19500	1068	1124	1012
  x~m~ ,S~xx~	1340	930	790200	786200	727100

  
  From this table the values for S~xx~ , S~yy~ and S~xy~ can be computed (last value of columns 4,5 and 6). From these values the error variance and confidence limits (8+9) can be computed together with the Student's t value. Note that t~n-2,1-~ _{a /2} = 2.131 and s~e~ = 88.3 mm. HYMOS will show the following graph.