U.S. Department of the Interior

U.S. Geological Survey

Techniques and Methods 4–A8

User’s Guide to the Weighted-Multiple-Linear

Regression Program (WREG version 1.0)
User’s Guide to the WeightedMultiple-Linear Regression Program

(WREG version 1.0)

By Ken Eng, Yin-Yu Chen, and Julie E. Kiang

Techniques and Methods 4–A8

U.S. Department of the Interior

U.S. Geological Survey
U.S. Department of the Interior

KEN SALAZAR, Secretary

U.S. Geological Survey

Marcia K. McNutt, Director

U.S. Geological Survey, Reston, Virginia: 2009

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reproduce any copyrighted materials contained within this report.

Suggested citation:

Eng, Ken, Chen, Yin-Yu, and Kiang, J.E., 2009, User’s guide to the weighted-multiple-linear-regression program 

(WREG version 1.0): U.S. Geological Survey Techniques and Methods, book 4, chap. A8, 21 p. (Also available at 

http://pubs.usgs.gov/tm/tm4a8.)

On the cover: Examples of plots resulting from analysis of data with the weighted-multiple-linear 

regression (WREG) program (version 1.0).
iii

Contents

Introduction.....................................................................................................................................................1

Multiple-Linear Regression..........................................................................................................................2

Independent and Dependent-Variable Transformations.........................................................................2

Estimation of Multiple-Linear-Regression Parameters ...........................................................................2

Ordinary Least Squares (OLS) ............................................................................................................3

Weighted Least Squares (WLS) .........................................................................................................3

Generalized Least Squares (GLS) ......................................................................................................4

Performance Metrics....................................................................................................................................5

Model- and Time-Sampling Errors.....................................................................................................5

Coefficient of Determination, R2

, R2

adj, and R2pseudo .................................................................................................................................

6

Leverage and Influence Statistics .....................................................................................................6

Significance of Regression Parameters ...........................................................................................7

Definition of Regions .....................................................................................................................................7

Use of the WREG Program ...........................................................................................................................8

Program Requirements.................................................................................................................................8

Installation.......................................................................................................................................................8

Input Files........................................................................................................................................................8

SiteInfo.txt..............................................................................................................................................9

FlowChar.txt .........................................................................................................................................10

LP3G.txt.................................................................................................................................................10

LP3K.txt.................................................................................................................................................11

LP3s.txt..................................................................................................................................................12

UserWLS.txt.........................................................................................................................................12

USGS########.txt................................................................................................................................13

Running the Program...................................................................................................................................13

Set Up Model................................................................................................................................................13

Select Variables ..................................................................................................................................13

Select Transformations......................................................................................................................13

Model Selection..................................................................................................................................15

GUI Outputs...................................................................................................................................................16

Regression Summary .........................................................................................................................16

Residuals Versus Estimated Flow Characteristics........................................................................17

Leverage Values Versus Observations............................................................................................17

Influence Values Versus Observations ...........................................................................................17

Output Files ...................................................................................................................................................17

ConventionalOLS.txt, ConventionalWLS.txt, and ConventionalGLS.txt......................................18

RegionofInfluenceOLS.txt, RegionofInfluenceWLS.txt, and RegionofInfluenceGLS.txt.........19

RegressionModel.txt ..........................................................................................................................20

InvXLX.txt..............................................................................................................................................20

SSres.txt and SStot.txt .......................................................................................................................20

EventLog.txt..........................................................................................................................................20
iv

Other Program Notes ..................................................................................................................................20

Acknowledgments.......................................................................................................................................20

References Cited..........................................................................................................................................20

Figures

1. Examples of input file SiteInfo.txt in text tab-delimited format .....................................................9

2. Example of input file FlowChar.txt in text tab-delimited format...................................................11

3. Example of input file LP3G.txt............................................................................................................11

4. Example of input file LP3K.txt............................................................................................................11

5. Example of input file LP3s.txt ............................................................................................................12

6. Example of input file UserWLS.txt....................................................................................................12

7. An example of a USGS########.txt file for streamflow-gaging station 09183000....................13

8. Example of WREG window used to select variables to be used in the regression .................13

9. Examples of WREG window for selecting transformations .........................................................14

10. Example of WREG window for selecting the regression..............................................................15

11. Example of WREG window for selecting parameters of the smoothing function for 

correlation as a function of distance between streamflow-gaging stations ...........................15

12. Example of WREG window for selecting option to include uncertainty in skew .....................16

13. WREG window showing the regression results for an OLS regression using the 

parameters and transformations specified in figure 9 .................................................................16

14. Examples of plots ................................................................................................................................16

15. Example of output file ConventionalOLS.txt....................................................................................17

16. Example of output file ConventionalWLS.txt ..................................................................................18

17. Example of output file ConventionalGLS.txt....................................................................................18

18. Example of output file RegionofInfluenceOLS.txt..........................................................................19

19. Example of regression model equation shown by RegressionModel.txt ..................................20

20. Example of InvXLX.txt output file......................................................................................................20

21. Examples of SSres.txt and SStot output files .................................................................................20

Tables

1. WREG input files ...........................................................................................................................8

2. WREG output files.......................................................................................................................17
v

Conversion Factors

Multiply By To obtain

Length

inch (in.) 2.54 centimeter (cm)

inch (in.) 25.4 millimeter (mm)

foot (ft) 0.3048 meter (m)

mile (mi) 1.609 kilometer (km)

mile, nautical (nmi) 1.852 kilometer (km)

yard (yd) 0.9144 meter (m)

User’s Guide to the Weighted-Multiple-Linear Regression 

Program (WREG version 1.0)

By Ken Eng, Yin-Yu Chen,1 and Julie E. Kiang

1

Former U.S. Geological Survey volunteer.

Introduction

Streamflow is not measured at every location in a stream 

network. Yet hydrologists, State and local agencies, and the 

general public still seek to know streamflow characteristics, 

such as mean annual flow or flood flows with different exceedance probabilities, at ungaged basins. The goals of this guide 

are to introduce and familiarize the user with the weightedmultiple-linear regression (WREG) program, and to also 

provide the theoretical background for program features. The 

program is intended to be used to develop a regional estimation equation for streamflow characteristics that can be applied 

at an ungaged basin, or to improve the corresponding estimate 

at continuous-record streamflow gages (henceforth referred to 

as simply gages) with short records. The regional estimation 

equation results from a multiple-linear regression that relates 

the observable basin characteristics, such as drainage area, to 

streamflow characteristics (for example, Thomas and Benson, 

1970; Giese and Mason, 1993; Ries and Friesz, 2000; Eng and 

others, 2005; Eng and others, 2007a; Eng and others, 2007b; 

Kenney and others, 2007; Funkhouser and others, 2008).

The general multiple-linear regression for estimating a 

streamflow characteristic can be given by

, (1)

where	 y is the streamflow characteristic (dependent 

variable),

xik	 are basin characteristics (independent 

variables),

i (=1, 2, 3,..., n) is the index for gage i,

k	 is the number of basin characteristics,

β0, β1, β2, and βk are the regression parameters, and

δi is the model error.

A critical issue in regional analyses is to understand 

the various sources of variability and error in the data. By 

understanding these sources, a user can select the appropriate 

approaches to estimate the regression parameters in equation 

1, and the appropriate network of gages forming the region 

used to develop an estimate. Three approaches to estimate 

regression parameters are provided in the WREG program: 

ordinary-least-squares (OLS), weighted-least-squares (WLS), 

and generalized-least-squares (GLS). All three approaches are 

based on the minimization of the sum of squares of differences between the gage values and the line or surface defined 

by the regression. The OLS approach is appropriate for many 

problems if the δi

 values are all independent of one another, 

and they have the same variance. Streamflow characteristics 

are estimated at gages using the available length of streamflow 

record. Because the length of record varies among gages, the 

precision of these estimates also varies, meaning that different 

δi values will have different variances.

A way to address the variation in the precision of estimated 

streamflow characteristics at each gage is to weight gages differently using WLS or GLS. A WLS approach reflects the precision of the estimated streamflow characteristic at that gage. An 

additional issue is that concurrent flows observed at different 

gages in a region exhibit cross correlations. If these correlations 

are not represented in a regional analysis, the regression parameters are less precise, and estimators of precision are inaccurate. 

A regional analysis that accounts for the precision of estimated 

streamflow characteristics, and the cross correlations among 

these characteristics, is known as GLS.

In addition to the variation and error in the data, the 

regression parameters in equation 1 are impacted by the choice 

of a network of gages forming a region. In a “conventional” 

regression, a region can be defined in several ways before a 

multiple-linear-regression study is initiated, such as by political 

boundaries or by physiographic boundaries. Within the context of “conventional” regressions, regions can also be defined 

during the regression study by using geographic information as 

an independent variable in the regression. Such regions can be 

defined using a variety of criteria, such as geographic grouping 

of similar residuals from an overall regression (Wandle, 1977), 

use of watershed boundaries (Neely, 1986), or physiographic 

characteristics. When performing a conventional analysis, a user 

of WREG must define the regions before using the program. 

WREG allows the user to perform conventional regressions 

using either OLS, WLS, or GLS.

An alternative to the conventional approach of pre-defining 

regions using political or physiographic boundaries is to define 

a region for each location of interest. This “region-of-influence” 

(RoI) regression approach defines a region and associated 
2 User’s Guide to the Weighted-Multiple-Linear Regression Program (WREG, v. 1.0)

multiple-linear regression for every ungaged basin (for example, 

Acreman and Wiltshire, 1987; Burns, 1990; Tasker and others, 1996; Merz and Blöschl, 2005; Eng and others, 2005; Eng 

and others, 2007a; Eng and others, 2007b). A regression is 

formed on a subset of gages for which the values of independent 

variables are, by some measure, closest to those at the ungaged 

basin of interest. While the WREG program allows testing of 

RoI regressions, the application of RoI regression to ungaged 

basins must be accomplished using other programs, such as the 

National Streamflow Statistics (NSS) Program (Ries, 2006).

The first part of this report provides an overview of the 

multiple-linear regression techniques that are employed by 

WREG. It is followed by a step-by-step guide to the actual use 

of the program, including a description of the input files, the 

use of the graphical user interface, and an explanation of the 

output files.

Multiple-Linear Regression

In practice, the dependent variable, yi, in equation 1 is an 

estimate, , often obtained from a limited sample size at each 

gage. The associated time-sampling error for the i

th gage, ηi

, is 

defined by

. (2)

Substituting equation 2 into equation 1 gives

, (3)

where εi = δi+ ηi (δi as given by equation 1). 

The ηi values from gages close together will generally be 

correlated, because the finite sample of observed streamflows 

at one gage temporally overlaps the sample from another and 

temporal variations of streamflows are spatially correlated. 

Thus, the cross correlation between ηi and ηj for gage i and j

will depend upon the cross correlation of concurrent flows at 

the two gages, and the number of concurrent years of record 

included in the dataset.

For a collection of gages with associated dependent and 

independent variables, equation 3 can be conveniently written 

in matrix notation as

, (4)

where

, (5)

and the total error, ε, is a random variable with a mean equal 

to zero and variance equal to σε

2

.

Independent and Dependent-Variable 

Transformations

The independent and dependent variables used in 

equation 5 can be transformed to obtain a linear relationship between the and X values. Common transformations 

include log (base 10), log (natural), and addition or subtraction of a constant. A user of WREG must choose appropriate 

transformations in the graphical user interface (GUI) before a 

multiple-linear regression is performed. A general transformation equation used by WREG is given as

, (6)

where V is the dependent or independent variable 

to be transformed,

Vnew is the transformed independent or 

dependent variable,

f	 is either the log (base 10), log (natural), 

or exponential function, or a 

transformation can be omitted. 

	 C1, C2, C3, and C4 are constants entered by the user.

Use of equation 6 for transforming variables is further discussed in the section of this report titled Select Transformations.

Estimation of Multiple-LinearRegression Parameters

Following transformations of the dependent and independent variables, the transformed variables are used in WREG 

to estimate multiple-linear-regression parameters. When using 

any of the least squares regression approaches (OLS, WLS, or 

GLS), the regression parameters are estimated by

. (7)

where XT is the transpose of matrix X,

Λ-1 is the inverse of the weighting matrix L (I = L-1L, 

where I is equal to the identity matrix).

The L matrix is constructed differently for OLS, WLS, 

and GLS, as described in the following sections. Once is 

determined, it can be used to estimate the regression estimate 

of at the i

th gage, iR, as

. (8)

However, a user should first check if the regression is 

adequate.

The estimators of L used in WREG for WLS and GLS 

approaches are applicable only to frequency-based streamflow 

characteristics. Alternative estimators of L to those presented 

in this manual can be explored using a user-defined option in 
Estimation of Multiple-Linear-Regression Parameters 3

WREG (see section UserWLS.txt) for non-frequency-based 

streamflow characteristics, such as flow-duration exceedences.

Ordinary Least Squares (OLS)

For the OLS approach, β is estimated by (for example, 

Montgomery and others, 2001)

, (9)

where . (10)

The OLS approach is suitable for estimating regression parameters when there is no variation in the precision of 

calculated dependent variables among gages, and the errors in 

equation 5 are independent of each other.

Weighted Least Squares (WLS)

For the WLS approach, β is estimated by (for example, 

Tasker, 1980)

, (11)

where LWLS is the covariance matrix used to determine 

weights.

The components of the LWLS matrix are a function of the 

type and source of the dependent variable. As with the OLS 

approach, the WLS approach is suitable when the errors in 

equation 5 are independent. However, for the WLS approach, 

weights in the weighting matrix are assigned so that gages that 

have more “reliable” estimates of streamflow characteristics 

have larger weights.

For streamflow characteristics calculated from a logPearson Type III frequency analysis (Bulletin 17B of the Interagency Advisory Committee on Water Data, 1982), Tasker 

(1980) provides a method for estimation of LWLS that is used 

by WREG for this option:

, (12)

where

, and (13)

, (14)

where is the model-error variance,

mi	 is the record length for the i

th gage,

is the observed mean-square error (MSE)

of estimate using ordinary-least-squares 

approach,

is the arithmetic average of the log-Pearson Type 

III deviates for all gages in the regression, and

is the arithmetic average of the skew values at all 

gages (either at-gage skew, g, or weighted 

skew, Gw ; explained below).

The log-Pearson Type III deviate values are a function of 

probability of exceedence and g (Interagency Advisory Committee on Water Data, 1982). is the arithmetic average of 

standard deviation of the annual-time series of the streamflow 

characteristic estimated by regression. This “sigma regression” 

is determined by OLS regression of the standard deviation of 

the annual-time series at each gage against basin characteristics at each gage (Tasker and Stedinger, 1989),

, (15)

where σi is the standard deviation of the annual-time 

series of the streamflow characteristic 

for the i

th gage,

xik is the kth basin characteristic for the i

th

gage,

	α0, βσ1, βσ2, and βσk are parameters, and

εσ is the model error for the sigma regression.

The σi values are a required input into the WREG program as discussed in section LP3s.txt.

The weighted skew for the i

th gage is given by (Bulletin 17B 

of the Interagency Advisory Committee on Water Data, 1982)

, (16)

where GR,i is the regional skew estimate applicable to the i

th

gage, and

, (17)

where is equal to the estimated mean square error of 

the skew value at the gage, and

is the estimated mean square error of the 

regional skew values.

A variety of methods are available to determine GR values 

(Bulletin 17B of the Interagency Advisory Committee on 

Water Data,1982). Either g or Gw values are required input to 

WREG program as discussed in section LP3G.txt.

An alternative approach to calculating is presented by 

Stedinger and Tasker (1986). Their estimator is demonstrated 

to be more precise than equation 13, but their study did not 

include a mix of approaches to compute streamflow characteristics at partial-record-stream gages (Funkhouser and others, 

2008). Use of equations 12 to 14 within the WREG program 

allows future versions to account for this mix of approaches 

for partial-record-stream gages.
4 User’s Guide to the Weighted-Multiple-Linear Regression Program (WREG, v. 1.0)

Generalized Least Squares (GLS)

For streamflow characteristics calculated from a log-Pearson Type III frequency analysis, a GLS 

approach described by Stedinger and Tasker (1985) builds on the WLS approach by accounting for 

both correlated streamflows and time-sampling errors. This GLS approach estimates the β values by

, (18)

where LGLS is a matrix containing the estimates of the covariances of εi among gages.

The main diagonal elements of LGLS thus include a part associated with the model error, δi, and 

all elements include the effect of the time-sampling error, ηi. In Tasker and Stedinger (1989), LGLS is 

estimated by

, (19)

where i and j are indices of locations of gages in the region of interest,

	 Gi and Gj are skew values equal to either g or Gw (equation 16) values for gages i and j,

	 mi and mj are record lengths for gages i and j,

	 mij is the concurrent record length for gages i and j, and

	 ρij is an estimated value for the cross-correlation of the time series of flow values used to 

calculate the streamflow characteristic at gages i and j.

Values of the cross-correlation are estimated approximately by (Tasker and Stedinger, 1989)

, (20)

where	 dij is the distance between gages i and j in miles, and

	 θ and α are dimensionless parameters estimated from data as discussed in section Model 

Selection.

The values in equation 19 and the values in equation 18 are jointly determined by iteratively 

searching for a nonnegative solution to (Stedinger and Tasker, 1985)

. (21)

Equation 19 does not account for error associated with estimating G. Depending on the actual 

magnitude of errors in estimation of skew, this additional error may unduly influence the estimation 

of β. Griffis and Stedinger (2007) proposed an approach to account for the uncertainty in the skew 

estimates in LGLS, and this approach is used as an option by WREG. As implemented in WREG, this 

option assumes that weighted skews are provided by the user, and so this option should be activated 

only when weighted skews were used. The modified LGLS matrix, LGLS,skew, is given by

, (22)
Performance Metrics 5

where and are the partial derivatives for gages i

and j calculated from the Kite (1975; 

1976) approximation for K given as

,(23)

where	 zp is the standard normal deviate corresponding to 

probability p.

The COV[gi,gj] in equation 22 is the covariance between 

the skew values at gages i and j, and is given by

, (24)

where is estimated by (Martins and Stedinger, 2002)

, and (25)

is equal to one if is positive and to minus 

one if is negative.

The Var(gi) and Var(gj) in equation 24 are approximated 

by (Griffis and Stedinger, 2009)

, (26)

where , (27)

, and (28)

. (29)

Equation 19 is a simplified version of equation 22 that 

assumes the skew is without error. Equation 19 is provided in 

the WREG program to reproduce previous studies that do not 

use equation 22.

Performance Metrics

The WREG program reports multiple performance 

metrics for multiple-linear regressions, depending upon the 

options (OLS, WLS, or GLS) selected. Specific metrics are 

reported either in the GUI or the output files of WREG.

Model- and Time-Sampling Errors

For conventional OLS and RoI regressions, the residual 

errors, ei, are computed as

, (30)

where is the estimated provided by the regression (see 

equation 8).

The residual mean square-error (MSE) is computed as

, (31)

where ei is calculated from equation 30.

The MSE metric does not distinguish the proportion of 

total error, εi that is composed of model error, δi, and timesampling error, ηi

. WLS and GLS regression provide estimates 

of the model error variance, , which is the same as the MSE

only if the time sampling error variance, , is equal to zero.

For conventional regressions using WLS and GLS, WREG 

reports the average variance of prediction, AVP, as the performance metric (Tasker and Stedinger, 1986) and is given by

, (32)

where xp is a vector containing the values of the 

independent variables of the pth gage 

augmented by a value of one.

When corresponds to the logarithm of the variable of 

interest, equation 32 can be reported as a percentage of the predicted value. When expressed in this way, the metric is known 

as the average standard error of prediction, Sp (Aitchison and 

Brown, 1957, modified for use of common logarithms), and is 

given by

. (33)

The standard model error as a percentage of the observed 

value can be calculated by substituting for AVP in equation 

33. WREG program reports both Sp and the standard model 

error for WLS and GLS regressions.

For RoI regression, a regression is developed for each 

ungaged basin of interest. An overall performance metric 

reported by WREG program for RoI regressions using OLS, 

WLS, and GLS is a root mean square error, RMSE(%). This 

metric is similar to, but not the same as the prediction error 

sum of squares, PRESS, performance metric (for example, 

Montgomery and others, 2001). Every gage is treated in turn 

as an ungaged basin and a regression is developed for that site, 

and equations 30 and 31 are used to calculate a mean-square 

error value, MSERoI, that is used in place of AVP in equation 

33 to give a root mean square error of prediction expressed as 

a percentage of the observed value, RMSE(%), given by (Eng 

and others, 2005; Eng and others, 2007a)

. (34)
6 User’s Guide to the Weighted-Multiple-Linear Regression Program (WREG, v. 1.0)

Coefficient of Determination, R2, R2

adj, and R2pseudo

A metric reported by WREG for determining the proportion of the variation in the dependent variable explained by the 

independent variables in OLS regressions, is the coefficient of 

determination, R2, (Montgomery and others, 2001) given as

, (35)

where , and (36)

, (37)

where is the arithmetic mean of all values, 

	 SST is the total sum of squares that is equal to the sum 

of the amount of variability in the observations, 

and

	 SSr is the residual sum of squares.

SST and SSr values are provided as WREG output files 

and can be used to calculate an adjusted coefficient of determination, Radj

2

, given as

. (38)

The adjusted Radj

2

 adjusts for the number of independent variables used in the regression.

For WLS and GLS regressions, a more appropriate performance metric than R2

 or Radj

2

 is the R2

pseudo described by Griffis 

and Stedinger (2007). Unlike the R2 metric in equation 35 and 

Radj

2

 in equation 38, R2

pseudo is based on the variability in the 

dependent variable explained by the regression, after removing 

the effect of the time-sampling error. The R2

pseudo is given as

, (39)

where is the model error variance from a WLS or GLS 

regression with k independent variables, and

is the model error variance from a WLS or GLS 

regression with no independent variables.

For RoI regressions using OLS, WLS, or GLS, no coefficient of determination values are reported in WREG.

Leverage and Influence Statistics

The leverage metric is used to measure how far away 

the value of one gage’s independent variables are from the 

centroid of values of the same variables at all other gages. 

This metric is reported for all gages in OLS, WLS, and GLS 

regressions and RoI regressions. For non-RoI regressions, 

leverage, h, for the i

th gage is given as

, (40)

where L is equal to either LOLS, LWLS, LGLS, or LGLS,skew.

The leverage metric for RoI regressions using either 

OLS, WLS, or GLS is given by (Eng and others, 2007b)

, (41)

where x0 is a vector of independent variables at a particular 

ungaged basin.

Unlike equation 40, equation 41 measures the leverage 

that each gage record in the RoI has on the ungaged basin 0. 

Leverage metrics associated with gages are considered large if 

these metrics exceed the criteria given by

, (42)

where 	 Ch is a constant

 For conventional regression, Ch is equal to 2 in equation 42 and reflects the observation that values twice the 

average can be considered as unusually large. For RoI 

regression, Ch is equal to 4 (Eng and others, 2007b). A 

larger Ch value for RoI regression is recommended because 

h0

T values typically exhibit greater variability, and even 

negative values, whereas hii values are always positive. 

The hlimit is reported for regressions using OLS, WLS, and 

GLS in the output files of the WREG program, and is also 

shown visually on a plot of leverage values for each gage 

(see sections Output Files and Leverage Values Versus 

Observations).

The leverage metrics in equations 40 and 41 identify 

gages whose independent variables are unusual. Such unusual 

gages may or may not have any significant impact on the 

estimated regression parameters in equations 9, 11, and 18. 

An influence metric, such as Cook’s D (Cook, 1977), indicates 

whether a gage had a large influence on the estimated regression parameter values. A generalized Cook’s D value for the i

th

gage is

, (43)

where	 v is the dimension of β,

	 Λii is the i

th main diagonal of the L covariance matrix, 

and

Lii is the i

th main diagonal of X(XT

L-1X)-1XT (Tasker 

and Stedinger, 1989).
Definition of Regions 7

For regressions, a gage that has caused large influence is 

identified if Cook’s D exceeds the limit given by

. (44)

The Cook’s D limit calculated using equation 44 is 

reported in the WREG output files (see section Output Files), 

and visually on a plot of influence values (see section Influence Values Versus Observation).

Significance of Regression Parameters

Regression parameters are tested for significance by 

WREG for regressions not using RoI regions. The null 

hypothesis is generally that the regression parameter is equal 

to zero, and the alternative hypothesis is that this parameter is 

not equal to zero. If the null hypothesis is not rejected at some 

predefined level of significance, such as 5%, then the associated independent variable is removed from the regression. For 

regressions, the test statistic that is used to evaluate the null 

hypothesis is the T value statistic given as

, (45)

where (Var βk) is the covariance value of βk, and is given as

. (46)

The T value statistic in equation 45 is assumed to follow 

a Student’s t distribution, so probabilities or “p-values” can 

be calculated. If the critical level of significance is set to 5% 

(0.05), p-values associated with the calculated T value statistic 

from equation 45 that exceed 0.05 result in acceptance of the 

null hypothesis. In this case, the regression parameter is not 

considered significant. In WREG, coefficients that are not 

significant at the 5% significance level are flagged (see section 

Regression Summary).

Definition of Regions

The region formed by a collection of gages can influence 

the regression model and the significance of the results. For 

conventional regressions, the WREG program uses all gages 

that are input by the user to develop regressions using either 

OLS, WLS, or GLS. If a subset of gages from a larger data set 

is desired by the user, input files to WREG should be modified 

to reflect the smaller subset.

For RoI regressions, a select subset from all available 

gages is formed for every ungaged basin where an estimate 

is desired. To use the RoI regression feature in WREG, a 

user would input all possible gages of interest, even though 

some gages might not be used in each RoI regression. Three 

approaches for defining hydrologic similarity among basins 

are available with WREG: independent or predictor-variable 

space RoI (PRoI) (for example, Burns, 1990), geographic 

space RoI (GRoI), and a combination of predictor-variable and 

geographic spaces called hybrid RoI (HRoI) developed by Eng 

and others (2007a).

For the GRoI option in WREG, a region of influence is 

formed using the n gages that are geographically the closest 

to the ungaged basin. In general, n is specified and is the same 

for every regression. The PRoI option in WREG is similar, but 

forms a region using the n closest gages in independent-variable 

space rather than geographic space. Thus, the region is comprised of gages whose independent variables have values that 

are the most similar to the ungaged site. Distance in independent-variable space from the ungaged basin to the i

th gage, Ri

, is 

defined in a Euclidean sense as (for example, Burns, 1990)

, (47)

where , , and are the sample standard deviations 

of x1, x2, and xk, respectively 

(computed from data from the 

entire study region).

The HRoI approach uses a region of influence of the n

closest gages in independent-variable space chosen from a subset of all gages having a geographic distance less than D from 

the ungaged basin. However, if fewer than n gages are available 

within the distance D of a given ungaged basin, then the limit 

D is ignored, HRoI reverts to GRoI and the n geographically 

closest gages are used. Thus, in the limit as D approaches zero, 

HRoI reduces to GRoI, and in the limit as D becomes arbitrarily 

large, HRoI reduces to PRoI (Eng and others, 2007a).

The user must determine the optimal D and n values. This 

determination can be accomplished by splitting the dataset 

into three equally sized subsets. Two of the three subsets are 

combined and used in an optimization step to calculate RMSE

values for various values of n and D for HRoI regionalization. 

The same subsets are used to calculate RMSE values for various values of n for both PRoI and GRoI. The lowest resulting values of RMSE and the corresponding values of n and 

D for HRoI and of just n for PRoI and GRoI are noted. The 

third subset is then used to evaluate model performance, by 

calculating the RMSE value associated with the optimal n and 

D determined in the previous step. All possible combinations 

of three subsets for this optimization-evaluation procedure are 

employed, and an overall RMSE value is then computed as the 

root mean-square value of the three individual subset values 

(Eng and others, 2005; Eng and others, 2007a).
8 User’s Guide to the Weighted-Multiple-Linear Regression Program (WREG, v. 1.0)

Use of the WREG Program

The remainder of this report provides details on how to use 

the Weighted-Multiple-Linear Regression Program (WREG). It 

can be used to set up, run, and evaluate a multiple-linear regression. As described in previous sections, the methods used by the 

program have been customized for use in the regionalization of 

streamflow characteristics. Many of the default methods may 

not be suitable for other regression problems.

The program is driven by a graphical user interface that 

leads the user through the process of setting up a regression. A 

number of input files are required, and detailed output is available in text files generated by WREG.

Notice.—MATLAB®. ©1984–2007 The MathWorks, 

Inc. was used to develop the graphical user interface and 

source code for WREG. The licensee’s rights to deployment 

of WREG are governed by the license agreement between 

licensee and MathWorks, and licensee may not modify or 

remove any license agreement file that is included with the 

MCR libraries. Installation of the application denotes acceptance of the terms of the license specified in the file named 

license.txt in the folder WREGv1 included with this manual.

Program Requirements

• Windows operating system.

• Approximately 400 MB of free disk space.

Installation

1. Unzip the distribution file (WREGv1.zip) and extract it 

to the directory of your choosing. There should be four 

files—MCRInstaller.exe, WREGv1.exe, WREGv1.ctf, 

and license.txt—and two folders—WREG_Source_Code 

and Sample_Files.

2. Run the program MCRInstaller.exe. This program installs 

MATLAB Component Runtime, software that allows 

the WREG executable program to run. The installation 

program leads you through the process with multiple 

GUI windows and may take several minutes to complete. 

Administrator privileges are required.

3. WREGv1.exe and WREGv1.ctf should be copied to 

another folder that contains input files and the program 

can be executed from that location. For example, copying 

the files into the folder Sample_Files will allow execution of the program using the sample input files. Alternatively, input files can be copied to the directory in which 

WREGv1.exe and WREGv1.ctf were installed.

4. WREGv1.exe is the program executable file and it is now 

ready to run. WREGv1.ctf is a required file that must 

be located in the same working directory as WREGv1.

exe. After WREG is run for the first time, another folder, 

WREGv1_mcr, will be created in the working directory. 

It contains additional files for running WREG. These files 

should not be changed. (If a new version of WREG is 

provided, the files WREGv1.exe and WREGv1.ctf and the 

folder WREGv1_mcr should all be deleted from the working directory. The new WREG.exe and WREG.ctf should 

be placed in the directory. A new WREG_mcr folder will 

be created the first time WREG is executed.)

5. Double-click on WREGv1.exe to start WREG.

Input Files

The input files are listed in table 1 with a brief description. 

WREG automatically reads these files to set up the regression. 

Input files that are required for a regression analysis must be 

located in the same working directory as the WREGv1.exe 

executable file. As noted in table 1, some input files are always 

required for the program to run. Others are required only for 

certain WREG options, as noted in table 1. All input files are 

text files, and all fields within them are tab-delimited. The input 

files can be created, viewed, and edited in a spreadsheet program, such as Microsoft Excel, and then saved as a tab-delimited file. They can also be created, viewed, and edited using a 

text editor. Note that some text editors may not display the input 

files in an easy-to-read format, with properly aligned columns. 

A detailed description of each file’s contents and format follows. 

Table 1. WREG input files.

File name Description WREG requirements

SiteInfo.txt Site information and basin characteristics to be used in the 

regression (the independent variables)

Always required.

FlowChar.txt Flow characteristics to be used in the regression (the dependent variables)Always required.

LP3G.txt Skew for Log-Pearson Type III distribution Always required. 

LP3K.txt K for Log-Pearson Type III distribution Always required.

LP3s.txt Standard deviation for Log-Pearson Type III distribution Always required. 

UserWLS.txt User-specified weighting matrix. Required only if the user-defined WLS option is selected.

USGS########.txt Annual time series of flow at streamflow-gaging stations Required only when using the GLS option. When needed, 

one file is required for each streamflow-gaging station 

listed in SiteInfo.txt.
Input Files 9

The examples are based on the input files in the Sample_Files 

directory distributed with WREG and use peak-flow statistics 

and basin attributes for streamflow-gaging stations in Iowa.

SiteInfo.txt

This input file contains the independent variables that 

may be used in the regression. An example file is shown in 

figure 1. All fields are tab-delimited. The first row contains 

header information describing the contents of each column. 

Information about each streamflow-gaging station is listed in 

the rows below the header.

The first eight columns are required:

1. Station ID: The USGS identifier for each streamflowgaging station.

2. Lat: Latitude of the streamflow-gaging station (or centroid 

of the watershed). The latitude and longitude are used 

to approximate the distance between streamflow-gaging 

stations in the GLS option. Latitude should be entered in 

decimal degrees (positive).

3. Long: Longitude of the streamflow-gaging station (or 

centroid of the watershed). Longitude should be entered in 

decimal degrees (either positive or negative).

4. No. Annual Series: The number of years of record available at the streamflow-gaging station, and used to estimate 

flow characteristics at the station. This information is used 

by WREG to assign weights to each streamflow gaging 

station. In general, only complete years of record should 

be used to generate flow characteristics.

5. Zero-1;NonZero-2: Flow characteristics (used for the dependent variable in a regression) will sometimes equal zero at a 

streamflow-gaging station. This field should show a 1 if the 

dependent variable for a station is equal to zero, and 2 if not. 

This field is not currently used by the WREG, but a value 

must be provided. The field is intended to facilitate addition 

of logistic regression to future versions of WREG.

6. FreqZero: The number of instances in which the annualtime series values equals zero. For example, if the regression will estimate the Q7,10, the number of years in which 

the 7-day minimum flow was equal to zero should be 

entered here.1 Note that these values may need to change 

if a different flow statistic is used as the response variable. 

This field is not currently used by WREG, but a value 

must be provided. It is included to facilitate addition of 

logistic regression to future versions of WREG.

7. Regional Skew: This field is the value of the regional 

skew used to calculate a peak-flow frequency statistic. If 

a regional skew is not applicable, then a dummy value, 

such as -99.99, needs to be entered. This field is only used 

when forming a WLS or GLS regression, but a value must 

always be supplied.

8. Cont-1:PR-2: This field should show a 1 if the station is 

a continuous streamflow-gaging station, or a 2 if it is a 

partial-record site. For peak-flow studies, crest-stage gages 

should be treated like streamflow-gaging stations. In lowflow studies, partial-record sites (or miscellaneous sites) are 

sometimes used to supplement continuous-record stations. 

At the partial-record sites, only sporadic measurements are 

made. Estimation of flow characteristics at partial-record 

sites requires specialized techniques. The resulting estimates 

of flow characteristics at partial-record sites have larger 

uncertainty than estimates at continuous-record stations.

1

The Q7,10 is the lowest stream flow for 7 consecutive days that would be 

expected to occur on average once in 10 years. It is estimated using an annualtime series of the minimum 7-day flows.

Figure 1. Example of input file SiteInfo.txt in text tab-delimited format, as shown in A, the Notepad text editor and B (next page), 

Microsoft Excel.
10 User’s Guide to the Weighted-Multiple-Linear Regression Program (WREG, v. 1.0)

header information describing the contents of each column. 

The first column, Station ID, is required and entries should 

be identical to those in column 1 of the SiteInfo.txt file. Note 

that the program requires all streamflow-gaging stations to be 

entered in the same order in both files.

Following the Station ID column, the remaining columns 

are used to define flow characteristics. The names used in the 

header should be descriptive, as they will later be used by the 

WREG to solicit user input. For display purposes, a maximum 

of 11 characters is recommended.

LP3G.txt

This file contains the skew value for each streamflow-gaging station that was used when fitting the log-Pearson Type III 

distribution. An example is shown in figure 3. All fields are tabdelimited. The first row contains header information describing 

the contents of each column. The first column, Station ID, is 

9. In the columns following these required entries, independent variables (basin characteristics) should be entered. The 

headers should be descriptive, as they will be displayed by 

WREG for selection of independent variables. For display 

purposes, a maximum of 11 characters is recommended.

Streamflow-gaging stations must be listed in ascending 

alphanumeric order (by USGS station identifier numbers) in 

this file and all others in which each row contains information 

for one station (FlowChar.txt, LP3G.txt, LP3K.txt, LP3s.txt). 

The order in which the gaging stations are listed should be the 

same as the order in which the USGS########.txt files are 

shown in Windows Explorer when sorted by name.

FlowChar.txt

This file contains the dependent variables that may be 

selected for use in the regression. An example is shown in 

figure 2. All fields are tab-delimited. The first row contains 

Figure 1. (continued)
Input Files 11

that for the annual peak flowtime series). 

Each statistic in FlowChar.txt must have 

a corresponding skew value. Note that if 

each statistic is based on the same annual 

series, the skews will be identical. If the 

statistic is not a frequency statistic and 

no skew was used in its computation, a 

dummy value of -99.99 may be used in 

the LP3G.txt file. For peak-flow studies, 

the weighted skew (equation 16) should 

be entered in LP3G.txt if it the weighted 

skew was used when fitting the log-Pearson Type III distribution. For other flow 

characteristics where a regional skew is 

not calculated, the at-site skew values, g, 

should be used. Information in this file is 

used by the WLS and GLS options.

required and entries must match those in 

column 1 of the SiteInfo.txt and FlowChar.txt files. Again, the program requires 

all streamflow-gaging stations to be 

entered in the same order in both files, in 

ascending alphanumeric gage site order.

Following the Station ID column, 

skew values for each of the frequency 

statistics listed in FlowChar.txt should be 

entered. These columns should correspond to those used in FlowChar.txt (for 

example, if column 2 of FlowChar.txt 

contains a peak for statistic, the skew 

value in column 2 of LP3G.txt should be 

LP3K.txt

This file contains the log-Pearson 

Type III distribution standard deviate, 

K, values (K values for a given exceedance probability and skew are tabulated 

in Bulletin 17B). An example is shown 

in figure 4. All fields are tab-delimited. 

The first row contains header information describing the contents of each 

column. The first column, Station ID, 

is required and must match the entries 

in column 1 of the SiteInfo.txt and 

FlowChar.txt files. Again, the program 

requires all streamflow-gaging stations 

to be entered in the same order in both 

files.

Following the Station ID column, K values for each of the frequency statistics listed in FlowChar.

txt should be entered. These columns 

Figure 2. Example of input file FlowChar.

txt in text tab-delimited format, as shown in 

Microsoft Excel.

Figure 3. Example of input file LP3G.txt 

as shown in Microsoft Excel. Note that the 

skew values entered for each streamflow 

gage are the same, since each flow 

statistic (Q2, Q50, and Q100) is based on 

the same annual-peak-flow series.

Figure 4. Example of input file LP3K.txt as 

shown in Microsoft Excel.
12 User’s Guide to the Weighted-Multiple-Linear Regression Program (WREG, v. 1.0)

should correspond to those used in 

FlowChar.txt (that is, if column 2 of 

FlowChar.txt contains the Q7,10, the K

value in column 2 of LP3K.txt should 

be that for the 7-day annual-time series). 

Each statistic in FlowChar.txt must have 

a corresponding K value. If the statistic 

is not a frequency statistic, a dummy 

value of -99.99 may be used in the 

LP3K.txt file. Information in this file is 

used by the WLS and GLS options.

LP3s.txt

This file contains the standard 

deviation, σ, of the annual-time series 

that was fit to the log-Pearson Type III 

distribution. An example is shown in 

figure 5. All fields are tab-delimited. 

The first row contains header information describing the contents of each 

column. The first column, Station ID, 

is required and must match the entries 

in column 1 of the SiteInfo.txt and 

FlowChar.txt files. Again, the program 

requires all streamflow-gaging stations 

to be entered in the same order in both 

files.

Following the Station ID column, 

σ values for each of the frequency 

statistics listed in FlowChar.txt should 

be entered. These columns should correspond to those used in FlowChar.txt 

(that is, if column 2 of FlowChar.txt 

contains the Q7,10, the σ value in column 

2 of LP3K.txt should be that for the 

7-day annual-time series). Each statistic 

in FlowChar.txt must have a corresponding value for σ. If the statistic is not a 

frequency statistic and σ is not used 

in its computation, a dummy value of 

-99.99 may be used in the LP3s.txt file. 

Information in this file is used by the 

WLS and GLS options.

UserWLS.txt

A user-defined weighting matrix 

can be specified in UserWLS.txt. This 

file contains no header information 

and simply contains the desired values 

of the L matrix (for example, equations 10, 12, and 19). The L matrix is a 

square matrix containing n rows and n

columns, where n is the number of sites 

used in the regression analysis. WREG 

inverts this matrix to assign weights to 

each station used in the analysis. When 

using WLS, only the main diagonal of 

this matrix contains nonzero values. 

In the UserWLS.txt file, observations 

that have smaller variance and are thus 

considered to be more reliable should be 

given smaller values. When the matrix is 

inverted by WREG, these observations 

will be given larger weight. Observations with large variance (for example, at 

partial-record sites) would be considered 

less reliable and should be assigned larger 

values. When the matrix is inverted by 

WREG, these observations will be given 

smaller weight. For a GLS regression, 

Figure 6. Example of input file UserWLS.txt as shown in Microsoft Excel. Because of the 

large number of columns in this file, it is difficult to read in a text editor.

Figure 5. Example of input file LP3s.txt as 

shown in Microsoft Excel.
Set Up Model 13

matrix elements off the main diagonal 

may contain nonzero values. Figure 6 

shows a portion of a UserWLS.txt file for 

a WLS regression problem.

USGS########.txt

These files contain the annual-time 

series of interest for each streamflowgaging station used in the analysis. They 

are only used when the GLS option 

is selected. The Station ID should be 

substituted in the filename for the #s. 

WREG looks for any file with “USGS” 

in the filename to use as input files. 

Consequently, no other files in the directory should have a filename containing 

USGS. WREG reads in each file sequentially in alphanumeric ascending order.2

An example file is shown in figure 7.

2

Other characters can be added to the filename, 

but they should be identical for each set of 

USGS########.txt files. For example, a set of 

peak flow files could follow the naming convention USGS########.peak.txt while a set of 

low-flow files could follow the naming convention 

USGS########.lowflow.txt. Only one set of these 

files can be in the working directory at one time.

Each file should include three 

columns and m rows, where m is the 

number of years of record available at 

the streamflow-gaging station. The first 

column contains the site identifier Station 

ID that was used in the other input files. 

The second column contains the year, 

and the third column contains the value 

of the flow statistic during that year. For 

example, for a regression to estimate the 

1% chance exceedance flood (Q100, the 

100-year return period flood), the annualflow statistic entered in this file should be 

the annual-peak flow. The year may be 

a calendar year, water year, or climatic 

year, depending on what was used to 

define the annual series.

Note that even if more than one 

type of statistic is specified in the 

FlowChar.txt file, only one annualtime series can be entered in the 

USGS########.txt file. Consequently, 

the USGS########.txt files must be 

updated if a second time series is used to 

define the dependent variable. (In other 

words, the same USGS########.txt files 

will not be valid for both peak flows and 

low flows.)

Running the Program

The WREG program uses a series 

of windows to guide the user through 

the formation, selection, and evaluation 

of a regression. The DOS window is 

displayed upon program execution, and 

will occasionally print information that 

monitors the progress of the program. 

Important information is always printed 

to a GUI window or to an output file.

Run the program by double-clicking on WREGv1.exe. If required input 

files are missing, the program will not 

run.

Set Up Model

Select Variables

The first window (fig. 8) is used 

to select variables (independent and 

dependent) for the regression. The items 

shown in the dependent variables menu 

and in the independent variables menu 

will depend on what has been included 

in the input files. One dependent variable should be selected. This is the 

flow statistic that will be estimated by 

the regression. To select more than one 

independent variable, hold down the 

CTRL key while clicking the left mouse 

button. Up to five independent variables 

can be selected.

Select Transformations

Each variable can be transformed 

by WREG (equation 5). Figure 9 shows 

the GUI window that facilitates variable 

transformation. The dependent variable and each independent variable are 

listed on the left of the window. In the 

middle part of this window, radio buttons 

are used to select whether to apply no 

transformations, to transform the variable 

using logarithms (base 10) (log10[...]), 

using natural logarithms (ln[...]), or 

using an exponential function (e[...]). By 

default, no transformations are applied. 

In the right part of the window, additional 

transformations can be applied to the 

variable. The logarithmic or exponential 

transformation will be applied to the 

entire expression specified in the right 

half of the window. The default values 

that are displayed initially will result in 

no additional changes to the variable. C1 

is used to multiply the variable by the 

constant C1. C2 is used to raise the variable to the power C2. C3 is used to add a 

constant C3 to the variable. C4 is used to 

raise the variable (including any algebraic 

transformation specified by C1, C2, or 

C3) to a power C4.

Figure 7. An example of a 

USGS########.txt file for streamflowgaging station 09183000.

Figure 8. Example of WREG window 

used to select variables to be used in the 

regression.
14 User’s Guide to the Weighted-Multiple-Linear Regression Program (WREG, v. 1.0)

Figure 9. A, Example of WREG window for selecting transformations, as it initially appears. B, After transformations have been specified.
Set Up Model 15

is the maximum distance from the ungaged location from 

which sites can be selected to form a region.

In the bottom panel of the same window (fig. 10), the 

parameter estimation method is selected using radio buttons 

as either ordinary least squares (OLS), weighted least squares 

(WLS), or generalized least squares (GLS).

If WLS or GLS is chosen, the program uses methods 

described earlier in this report to calculate the weighting 

matrix, L. The USGS########.txt files need to be supplied as 

input files if the GLS option is selected.

If conventional regression methods are used, there is also 

an option for utilizing user-specified weights in the regression. 

If UserWLS.txt was not supplied as an input file, an error message will appear if User Specified Weights is selected.

If a GLS regression is selected, a screen appears that 

allows adjustment of the parameters of a smoothing function 

(equation 18) that relates the correlation between streamflow 

time series (input in USGS########.txt files) at two sites to 

the geographic distance between them, as shown in figure 11. 

The default settings are a good starting point, but values can 

be adjusted to better fit specific datasets.

The “No. of Concurrent Years” specifies the minimum 

overlap among annual-peak flow records at streamflow-gaging 

stations before the correlation between them is used to estimate 

the smoothing function. If too few concurrent years are specified, 

the variability in correlation among streamflow-gaging stations 

may be too large to discern a relation between correlation and distance. However, if the required number of concurrent years is too 

large, there may not be enough data points to adequately define 

the relation. Generally, to fit well, the value of alpha, α, will be 

positive and near zero while theta, θ, will be slightly less than 1. 

This combination of parameters causes the correlation to be near 

1 when streamflow-gaging stations are very close, and to asymptotically approach a value near zero as the distance between them 

increases. When values are in this range, the closer θ is to one, 

the more slowly the correlation will decrease with distance. The 

closer α is to zero, the closer the asymptote will lie to zero.

After making desired changes to the parameters, click 

the “Compute” button to (re)compute and draw the smoothing 

For example, in figure 9A, the initial GUI window is shown, 

with no transformations selected. In figure 9B, selections will 

cause 1% chance exceedance flood (Q1%) and the drainage area 

(DRNAREA) to be log (base 10) transformed. No further transformations will be applied to these variables. The annual precipitation (PRECIP) will first be multiplied by 10 and then a constant 

of 1 will be added to that quantity. These are merely illustrations 

of the use of the transformations, and are not intended to suggest 

useful transformations for these variables.

Model Selection

Either conventional regression or region-of-influence 

(RoI) regression can be selected by the radio buttons in the 

upper portion of the GUI window shown in figure 10. Equations formed by conventional regression can be used to 

estimate statistics at ungaged locations. WREG forms a RoI 

regression for each streamflow-gaging station in the database, 

allowing for evaluation of RoI as a regression strategy. However, WREG does not provide a function allowing the user to 

form a RoI regression for an ungaged location.

For RoI, the method for defining the region of influence 

must also be selected—either based on geographic proximity (GRoI), proximity in predictor variable space (PRoI), or a 

hybrid approach combining geographic and predictor variable 

space (HRoI). (These methods were discussed in the section 

Definition of Regions.) If using RoI, the number of sites, n, 

to use in the region must also be specified in the field under 

the RoI radio buttons. If the GRoI or HRoI method is selected, 

then the geographic proximity, D, must also be specified; this 

Figure 10. Example of WREG window for selecting the 

regression.

Figure 11. Example of WREG window for selecting parameters 

of the smoothing function for correlation as a function of distance 

between streamflow-gaging stations.
16 User’s Guide to the Weighted-Multiple-Linear Regression Program (WREG, v. 1.0)

GUI Outputs

Regression Summary

Once parameters for the regression 

are calculated, the regression summary 

window (fig. 13) displays some basic 

function. The “Confirm” button will be 

grayed out while the program is computing the change. When satisfied with the 

fit of the smoothing function to the data, 

click the “Confirm” button.

Another window (fig. 12) will pop 

up that allows the uncertainty in skew to 

be considered in the regression calculations. [See section Generalized Least 

Squares (GLS).] This option should be 

used only if weighted skew was used in 

the calculation of the frequency statistics. The radio buttons are used to specify whether peak flows or low flows are 

being considered. Then the recurrence 

interval of the frequency statistic under 

consideration should be entered under 

T-Year, and the mean square error of the 

regional skew should be entered under 

MSE(GR). A default value of 0.302 is 

used in the program because this value 

is associated with plate I of Bulletin 17B 

of the Interagency Advisory Committee 

on Water Data (1982).

performance metrics (appropriate to 

the type of regression used) and the 

regression equation. Elements of the 

regression equation that are shown in 

red are not statistically significant at 

the 5% level. Three plots accompany 

the regression output and are shown 

in figure 14. More detailed regression 

Figure 14. A, Example of plot showing residuals versus estimated flow characteristics. B, Example of plot showing leverage values 

versus observations. C, Example of plot showing influence values versus observations.

Figure 12. Example of WREG window for 

selecting option to include uncertainty in 

skew.

Figure 13. WREG window showing the regression results for an GLS regression using 

the parameters and transformations specified in figure 9.
Output Files 17

file can also be imported into a spreadsheet program, such as 

Microsoft Excel, as a text, tab-delimited file.

results, including p-values for individual parameters, are available in the text output files.

Residuals Versus Estimated Flow Characteristics

This plot (fig. 14A) shows the regression residuals (equation 28) plotted against the estimated flow characteristics. 

Both axes will be in units of the transformed flow characteristics [for example, log (base 10)].

Leverage Values Versus Observations

This plot (figure 14B) shows the leverage values plotted by observation number. Observations are numbered in 

the order in which they appear in the input files. The red line 

shows the threshold (as calculated by equation 41) above 

which an observation may be considered to have high leverage, as calculated by the appropriate equation for the type 

of regression, discussed earlier in the section Leverage and 

Influence Statistic. Observations with high leverage should 

be checked carefully for possible errors.

Influence Values Versus Observations

This plot (fig. 14C) shows influence values plotted by 

observation number. The red line shows the threshold (as calculated by equation 45) above which an observation may be considered to have large influence, as calculated by the appropriate 

equation for the type of regression, discussed earlier in the section Leverage and Influence Statistic. Observations with high 

influence should be checked carefully for possible errors.

Output Files

The output files are listed in table 2 with a brief description. A detailed description of each file’s contents follows. 

A text editor can be used to view all of the output files. Each 

Table 2. WREG output files.

File name Description

ConventionalGLS.txt Results from a GLS regression.

ConventionalOLS.txt Results from an OLS regression.

ConventionalWLS.txt Results from a WLS regression.

RegionofInfluenceOLS.txt Results from an OLS regression using RoI.

RegionofInfluenceWLS.txt Results from a WLS regression using RoI.

RegionofInfluenceGLS.txt Results from a GLS regression using RoI.

RegressionModel.txt The regression equation (including transformations) calculated by WREG. Output only for conventional regression.

InvXLX.txt Covariance of the regression parameters (equation 46). Output only for conventional regression.

SSres.txt Residual sum of squares (equation 36). Output only for conventional regression.

SStot.txt Total sum of squares (equation 37). Output only for conventional regression.

EventLog.txt Record of the program’s execution. Helpful for diagnosing runtime errors.

NA 

 

Regression Model for Q1% 

Performance Metrics 

Mean Squared Error 0.085 

R2 77.98 

Standard Model Error

*Note: R2 is in percent.

Coefficients of Model 

Std Error T value P>|T| 

Constant 

DRNAREA 0.51 0.05 10.63 0.000

PRECIP 0.01 0.00 2.54 0.015

Transformations C1 C2 C3 C4

Q1% 

DRNAREA log10 1.00 1.00 0.00 1.00

PRECIP 1 10.00 1.00 1.00 1.00

Residuals, Leverage, and Influence of Observations 

rage Limit 0.154 

Influence Limit 0.103 

Observation Residual Leverage Influence

1 0.145 0.216 0.002

2 0.286 0.143 0.005

3 0.108 0.200 0.001

4 -0.622 0.028 0.004

5 -0.333 0.045 0.002

6 -0.435 0.053 0.004

7 -0.322 0.028 0.001

8 0.265 0.107 0.003

9 -0.020 0.110 0.000

10 0.105 0.179 0.001

11 0.346 0.183 0.011

12 -0.385 0.034 0.002

13 0.165 0.037 0.000

14 -0.207 0.037 0.001

15 -0.196 0.088 0.001

16 0.069 0.125 0.000

17 -0.269 0.056 0.002

18 -0.407 0.033 0.002

19 0.165 0.054 0.001

20 0.210 0.042 0.001

. 

. 

. 

0.97 0.70 1.39 0.173

Leve

log10 1.00 1.00 0.00 1.00

Figure 15. Example of output file ConventionalOLS.txt. Residuals, 

leverage, and influence are shown only for the first 20 observations.
18 User’s Guide to the Weighted-Multiple-Linear Regression Program (WREG, v. 1.0)

ConventionalOLS.txt, ConventionalWLS.txt, and 

ConventionalGLS.txt

These three files provide information on conventional 

regressions and are the same except for the performance 

metrics that are displayed at the head of the file. The metrics 

shown are appropriate to the parameter selection scheme: 

OLS, WLS, or GLS. Figure 15 shows an example of output in 

ConventionalOLS.txt, figure 16 shows an example of output 

in ConventionalWLS.txt, and figure 17 shows an example of 

output in ConventionalGLS.txt.

Figure 16. Example of output file ConventionalWLS.txt. Figure 17. Example of output file ConventionalGLS.txt.

Regression Model for Q1% 

 

Performance Metrics 

Avg Standard Error of Prediction 69.33 

Pseudo R2 80.92 

Standard Model Error 65.67 

*Note: Avg Standard Error of Prediction, Pseudo R2, 

and Standard Model Error are in percent. 

 

Coefficients of Model 

 Std Error T value P>|T| 

Constant 1.01 0.70 1.45 0.157

DRNAREA 0.51 0.05 10.77 0.000

PRECIP 0.01 0.00 2.51 0.017

 

Transformations C1 C2 C3 C4

Q1% log10 1.00 1.00 0.00 1.00

DRNAREA log10 1.00 1.00 0.00 1.00

PRECIP 1 10.00 1.00 1.00 1.00

 

Residuals, Leverage, and Influence of Observations 

Leverage Limit 0.154 

Influence Limit 0.103 

 Observation Residual Leverage Influence

 1 0.140 0.208 0.025

 2 0.281 0.141 0.059

 3 0.101 0.200 0.013

 4 -0.620 0.028 0.046

 5 -0.333 0.046 0.022

 6 -0.433 0.054 0.047

 7 -0.322 0.028 0.012

 8 0.262 0.115 0.043

 9 -0.021 0.107 0.000

 10 0.107 0.185 0.013

 11 0.342 0.182 0.127

 12 -0.385 0.035 0.022

 13 0.164 0.038 0.004

 14 -0.207 0.031 0.005

 15 -0.193 0.089 0.017

 16 0.073 0.135 0.004

 17 -0.269 0.056 0.018

 18 -0.406 0.033 0.023

 19 0.169 0.045 0.005

 20 0.213 0.036 0.006

 21 0.382 0.035 0.019

 22 -0.133 0.026 0.002

 23 0.089 0.027 0.001

 24 -0.056 0.140 0.002

 25 -0.148 0.053 0.005

 26 0.472 0.061 0.063

 27 0.122 0.067 0.005

 28 0.061 0.047 0.001

 29 0.183 0.060 0.009

 30 0.378 0.062 0.040

 31 0.106 0.049 0.002

 32 -0.502 0.072 0.080

 33 -0.248 0.063 0.019

 34 -0.299 0.102 0.050

 35 0.062 0.078 0.001

 36 0.018 0.057 0.000

 37 0.524 0.043 0.050

 38 0.193 0.053 0.009

 39 0.188 0.114 0.023
Output Files 19

The observations are the observed flow characteristic 

(dependent variable) at each streamflow-gaging station and 

are numbered in ascending order, as the stations appear in the 

FlowChar.txt file (and other files).

RegionofInfluenceOLS.txt, RegionofInfluenceWLS.

txt, and RegionofInfluenceGLS.txt

These three files show the same outputs but are 

produced using OLS, WLS, or GLS. The example 

shown in figure 18 is for RoI using OLS regression 

(RegionofInfluenceOLS.txt). This file provides information 

on the results of the RoI regression, and can be used to gauge 

whether an RoI regression may provide suitable results. 

However, WREG does not include a function that would 

allow users to perform a RoI regression at an ungaged location. The results shown by WREG are solely for demonstration or evaluation purposes.

WREG forms an individual RoI regression for each 

streamflow-gaging station included in the input dataset. The 

first several lines of the file show overall performance metrics for the RoI regression, when each of these individual 

regressions are considered. Next, the coefficients of the RoI 

regression built for each site in the dataset are shown. If 200 

sites are included in the dataset, 200 results will be shown—

one for each regression that was built. These individual regressions will be numbered 1 to 200, and reflect the numerically 

ascending order of the Station IDs. That is, the Station ID with 

the smallest number would be associated with regression 1 or 

observation 1 in this output file and the Station ID with the 

largest number would be associated with the regression 200 or 

observation 200 in this output file.

The output file next displays the transformations that 

were used in the regressions. WREG does not allow different transformations to be used for different RoI regressions, 

and a single transformation applies to all individual RoI 

regressions formed by WREG. Next, the PRESS-like MSE 

of the RoI residuals are shown for the regression formed 

for each site in the dataset. Finally, the output file displays 

information on leverage and influence. For each regression, 

the output shows the observations that were used to form the 

regression as well as the leverage calculated for that observation. These outputs are paired. For example, in output shown 

in figure 18, the first regression used observations 2, 8, 3, 9, 

14, 13, 12, 7, 10, and 5. Following each observation number, 

Figure 18. Example of output file RegionofInfluenceOLS.txt.

Region-of-Influence Regression Models for Q1% 

 

Performance Metrics 

Root Mean Square Error 73.45 

Pseudo R2 NA 

Standard Model Error NA 

*Note: Root Mean Square Error is in percent. 

 

Coefficients of Model 

Regression Constant Coefficients 

 DRNAREA PRECIP 

 1 4.61 0.54 -0.01 

 2 3.84 0.54 -0.00 

 3 4.10 0.29 -0.00 

 4 -2.65 0.60 0.02 

 5 2.97 0.30 -0.00 

 6 -1.28 0.70 0.01 

 . 

 . 

 . 

 

Transformations C1 C2 C3 C4 

Q1% log10 1.00 1.00 0.00 1.00 

DRNAREA log10 1.00 1.00 0.00 1.00 

PRECIP 1 10.00 1.00 1.00 1.00 

 

PRESS_RoI Residuals 

Observation Residual 

 1 0.060 

 2 0.028 

 3 0.141 

 4 0.262 

 5 0.047 

 6 0.132 

 . 

 . 

 . 

 

Leverage and Influence 

Leverage Limit 0.400 

Influence Limit NA 

Regression Observations Included in RoI 

 Model and Leverage Value 

 1 2 0.400 8 0.293 3 0.461 9 0.184 14 -0.038 13 -0.046 12 -0.075 7 -0.163 10 0.149 5 -0.165 

 2 1 0.315 8 0.169 3 0.356 13 0.050 14 0.020 12 0.035 9 0.054 7 -0.012 5 0.035 18 -0.020 

 3 2 0.385 1 0.445 8 0.112 11 0.518 13 -0.081 12 -0.113 5 -0.046 14 -0.202 17 -0.041 35 0.024 

 4 23 0.096 22 0.106 7 0.083 14 0.099 20 0.120 6 0.133 12 0.080 18 0.070 19 0.133 13 0.079 

 5 17 0.249 18 0.068 35 0.327 12 0.141 7 0.031 13 0.172 36 0.033 25 -0.080 32 0.023 14 0.035 

 6 15 0.159 19 0.098 4 0.073 20 0.077 22 0.062 14 0.086 23 0.051 16 0.174 9 0.172 7 0.047 

 

 . 

 . 

 . 
20 User’s Guide to the Weighted-Multiple-Linear Regression Program (WREG, v. 1.0)

EventLog.txt

The event log contains a record of the WREG session 

that can be useful for debugging. If the program is operating 

normally, it does not need to be considered.

Other Program Notes

Screen Capture.—As noted on many windows, the contents of that window can be captured by pressing Alt and the 

Print Screen key. This will copy an image of the window to the 

clipboard. It needs to be pasted into a document (for example, 

a WordPad document) in order to save it.

Acknowledgments

The authors would like to thank Jery R. Stedinger (Cornell University), Andrea M. Gruber-Veilleux (Cornell University), and Charles Parrett (USGS California Water Science 

Center) for their helpful comments on this report.

Future updates will be available at 

http://water.usgs.gov/software/WREG/.

References Cited

Acreman, M.C., and Wiltshire, S.E., 1987, Identification of 

regions for regional flood frequency analysis (abstract): 

EOS, v. 68, no. 44.

Aitchison, J., and Brown, J.A.C., 1957, The lognormal distribution: Cambridge University Press, 176 p.

Burn, D.H., 1990, Evaluation of regional flood frequency analysis with a region of influence approach: Water Resources 

Research, v. 26, no. 10, p. 2257–2265.

Cook, R.D., 1977, Detection of influential observation in linear regression: Technometrics, v. 19, p. 15–18.

Eng, Ken, Tasker, G.D., and Milly, P.C.D., 2005, An analysis 

of region-of-influence methods for flood regionalization in 

the Gulf-Atlantic Rolling Plains: Journal of American Water 

Resources Association, v. 41, no. 1, p. 135–143.

Eng, Ken, Milly, P.C.D., and Tasker, G.D., 2007a, Flood 

regionalization: a hybrid geographic and predictor-variable 

region-of-influence regression method: Journal of Hydrologic Engineering, ASCE, v. 12, no. 6, p. 585–591.

Eng, Ken, Stedinger, J.R., and Gruber, A.M., 2007b, Regionalization of streamflow characteristics for the Gulf-Atlantic 

Rolling Plains using leverage-guided region-of-influence 

regression, Paper 40927–3050, in Kabbes, ed., Restoring 

our natural habitat—Proceedings of the World Environthe leverage value for that observation is shown. For observation 2, the leverage was 0.400.

RegressionModel.txt

The RegressionModel.txt file gives the regression equation calculated by WREG. Transformations, variables, and 

parameters are all shown. In the example shown for conventional OLS (figs. 13 and 16), the corresponding regression equation contained in RegressionModel.txt is shown in 

figure 19. This equation can be interpreted as

.(48)

This output file is produced only for conventional 

regressions.

InvXLX.txt

This file contains the covariance matrix of the regression parameters (equation 46). Its size depends on the number 

of dependent variables used in the regression. An example 

is shown in figure 20. This matrix can be used to estimate 

the variance of individual regression estimates, which varies 

depending on the values of basin characteristics. This output 

file is created only for conventional regressions.

SSres.txt and SStot.txt

The SSres.txt file reports the residual sum of squares 

(equation 36) and the SStot.txt file reports the total sum of 

squares (equation 37). These output files are created only for 

conventional regressions. Examples are shown in figure 21.

Figure 19. Example of regression model equation shown by 

RegressionModel.txt.

log_10{[1.00*(Q1%)^(1.00)+0.00]^(1.00)} =1.15 

+0.52*log_10{[1.00*(DRNAREA)^(1.00)+0.00]^(1.00)} 

+0.00*[10.00*(PRECIP)^(1.00)+1.00]^(1.00)

 6.0173594e-001 1.4652262e-003 -1.8692077e-003

 1.4652262e-003 2.2218603e-003 -1.4027835e-005

 -1.8692077e-003 -1.4027835e-005 5.8799320e-006

Figure 20. Example of InvXLX.txt output file.

Figure 21. Example of A, SSres.txt and B, SStot.txt output files.

3.0515284e+000 1.3821829e+001
References Cited 21

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Eng, Chen, and Kiang—User’s Guide to the Weighted-Multiple-Linear Regression Program (WREG version 1.0)—Techniques and Methods 4–A8