Change Point Analysis of the Chlorophyll-a and LDI Data from the IWRM 

STATUS Network Cycle 1 Lakes Study 

Xu-Feng Niu 

Professor of Statistics 

Department of Statistics 

Florida State University 

Tallahassee, FL 32306-4330 

niu@stat.fsu.edu; (850)644-4008 

August 2004 
Change Point Analysis 

a. The detection procedure. 

Niu et al. (2000) introduced an iterative procedure for detecting and modeling level-shift change

points. The procedure is similar to that suggested by Chang (1982) and further developed by Chang, 

Tiao, and Chen (1988) for detecting outliers and level shifts in time series analysis. Statistical details

of this procedure can also be found in Pankratz (1991, Chapter 8). 

For simplicity, let us consider a biological response Y, after an appropriate transformation. Suppose 

that observations { ( , ) Xi Yi , } are available where n is the sample size and X is an

independent variable such as LDI. Moreover, we assume that the observation are arranged in the 

following manner: 

i = 1, 2,L, n

• The values { , Xi } are distinct. If several are corresponding to a single X

value, median of the is taken to be the response value for the X value. 

i = 1, 2,L, n ' Yi s

' Yi s

• { ( , ) Xi Yi , i = 1, 2,L, n } are sorted according to the values of X from least to greatest. 

For each integer l > 1, define the step variable ( ) 0 i S l = for i < l and ( ) 1 for . i S l = i l ≥

Step 1. Fit the linear regression model: 

0 1 ( ) ( ) ( ) ( ) Y l i i i = + β β ε l S l + l , i = 1, 2,L, n , (1) 

where for a fixed l, the ( )' i ε l s are assumed to be independent and identically distributed normal

random variables with mean zero and variance . 2 σ (l)

Step 2. Calculate the values { 1 1 L l( ) = β β ( )l /se( ( )l ) ) ) , l = 2, 3,L, (n −1) } where 1 se( ( β l)) ) is the

estimated standard error of 1 β (l) ) . 

Step 3. Let and compare with the critical value C=3.0 (or 

C=3.5). The critical value C=3.0 (or C=3.5) corresponds roughly to 

1 L l( ) = max{L(2), L(3),L, L(n −1)} 1 L l( )

α = 0.10 (or α = 0.05), or the

10% (or the 5%) significance level, based on the simulation results of Chang, Tiao, and Chen (1988). 

If is significant, we conclude that the response Y has a change point at 1 L l( ) 1 Xl with a level-shift 

1 β (l) ) . 

Step 4. Let . Repeat Steps 1-3 on the new response variable for detecting a

possible second change point. Continue the process until no further change point can be identified. 

*

1 1 1 ( ) ( ) Y Y i i i = − β l S l * Yi
3

Step 5. Suppose that k change points are detected in the response variable Y and the corresponding X

values are 1 2 { , , , k l l l X X L X }. Fit the model 

0 1 1 2 2 ( ) ( ) ( ) Y S i i i k i k i = + β β β l + S l +L + β S l + ε , i = 1, 2,L, n . (2) 

Then the estimated coefficients 1 2 { , , , } β β βk

) ) ) L will be the k estimated level-shift values.

b. An illustration example. 

 In this section, we present an example of change point analysis for chlorophyll a vs LDI at 547 lakes. 

If multiple samples of chlorophyll a were collected at a lake, median of the chlorophyll a values was 

calculated. 

 First, we put the LDI values into nine classes, with LDI 1.0-1.4 as Class 1, LDI 1.5-2.4 as Class 2, 

and so on. Figure 1 shows the boxplots of chlorophyll a values corresponding to the nine LDI classes. 

The chlorophyll a values have quite large variation with the max around 500 ug/L. 

 The median of chlorophyll a values at each LDI level were calculated and used for the change point 

analysis. For example, 17 lakes with LDI=1.0 and the median of chlorophyll a values at the 17 lakes 

is 0.5. The medians of chlorophyll a values at the 76 LDI levels (from 1.0 to 8.6 with 0.1 increment, 

no lakes with LDI=8.1) are presented in Table 2. 

 The natural logarithm transformation was performed on the chlorophyll a data. After the 

transformation, change point analysis of chlorophyll a vs LDI was conducted. 

 Two change points are detected in the chlorophyll a data. The largest change point was at LDI=6.3, 

which is significant at the 5% level (95% confidence). The second change point occurred at LDI=1.9, 

which is significant at the 20% level (80% confidence). Figures 2 and 3 present the fitted regression

model and the histogram of the residuals from the model, respectively. The R-square of the regression 

is only 0.31, due to the large variation among the chlorophyll a observations. 
4

0 100 200 300 400 500

Chl-a

123456 7 8 9

LDI

Figure 1. Boxplot of Chlorophyll-a vs LDI

Note: LDI 1.0 – 1.4 = Class 1; LDI 1.5 – 2.4 = Class 2; ..., so on 

 LDI 8.5 – 9.4 = Class 9 (only one point with LDI=8.6) 
5

Table 2. Medians of Chlorophyll a Values and Corresponding LDI Values 

LDI Median of 

Chlorophyll a 

Log of Median of 

Chllorophyll a LDI Median of 

Chlorophyll a

Log of Median of 

Chllorophyll a 

1.0 0.5 -0.69 4.8 13 2.56 

1.1 5.45 1.7 4.9 2.5 0.92 

1.2 2.6 0.96 5 21 3.04 

1.3 0.52 -0.64 5.1 3.2 1.16 

1.4 3.75 1.32 5.2 0.5 -0.69 

1.5 0.62 -0.47 5.3 0.42 -0.86 

1.6 0.6 -0.51 5.4 11.42 2.44 

1.7 4.3 1.46 5.5 0.63 -0.46 

1.8 1.2 0.18 5.6 41.72 3.73 

1.9 0.5 -0.69 5.7 29 3.37 

2 5.3 1.67 5.8 0.46 -0.77 

2.1 1.2 0.18 5.9 0.46 -0.77 

2.2 11.85 2.47 6 8.15 2.1 

2.3 6.2 1.82 6.1 1.3 0.26 

2.4 10.6 2.36 6.2 10 2.3 

2.5 2.2 0.79 6.3 1.63 0.49 

2.6 15.5 2.74 6.4 150 5.01 

2.7 0.5 -0.69 6.5 33 3.5 

2.8 5.3 1.67 6.6 4.62 1.53 

2.9 0.55 -0.6 6.7 23 3.14 

3 6.25 1.83 6.8 0.5 -0.69 

3.1 0.5 -0.69 6.9 63 4.14 

3.2 5.75 1.75 7 2.6 0.96 

3.3 1.4 0.34 7.1 1.7 0.53 

3.4 4.15 1.42 7.2 5.9 1.77 

3.5 1.2 0.18 7.3 38 3.64 

3.6 6.08 1.8 7.4 22 3.09 

3.7 7.2 1.97 7.5 19.65 2.98 

3.8 4.5 1.5 7.6 5.1 1.63 

3.9 8.71 2.16 7.7 62 4.13 

4 0.95 -0.05 7.8 135 4.91 

4.1 0.5 -0.69 7.9 38 3.64 

4.2 13.02 2.57 8 13 2.56 

4.3 15.5 2.74 8.2 4.8 1.57 

4.4 41.75 3.73 8.3 49 3.89 

4.5 1.95 0.67 8.4 48 3.87 

4.6 2.02 0.71 8.5 47 3.85 

4.7 4.73 1.55 8.6 88 4.48 

Change Points: LDI=1.9 with about 80% confidence; LDI=6.3 with about 95% confidence. 

Bold-case numbers are the values just after the change points.
6

LDI

Log of Median of Chl-a

2 4 6 8

-1 0 1

2

3 4 5

Figure 2. Log of Median of Chlorophyll-a vs LDI

Note: 1) Two change points were detected. The first one at LDI=1.9 with about 80% 

 confidence, and the second one at LDI=6.3 with about 95% confidence. 

 2) If using the original scale of Chl-a, no change points are around LDI=2; 

 Using the log-scale of Chl-a, LDI=1.9 is the second largest change point. 
7

-4 -2 0 2

0

5

10

15

20

Histogram

Figure 3. Histogram of Residuals
8

Fitted Regression Model (Natural Logarithm of Chlorophyll a): 

In the model, the response variable is Chlorophyll a after the natural logarithm 

transformation. The two independent variables are: 

x1=0 for LDI 1-1.9; x1=1 for LDI 2-8.6; 

x2=0 for LDI 1-6.3; x2=1 for LDI 6.4-8.6; 

Coefficients: 

 Value Std. Error t value Pr(>|t|) 

(Intercept) 0.2602 0.4294 0.6061 0.5464 

 x1 0.9837 0.4757 2.0681 0.0422 (LDI 1 – 1.9) 

 x2 1.6703 0.3546 4.7111 0.0000 (LDI 2 – 6.2) 

Residual standard error: 1.358 on 73 degrees of freedom 

Multiple R-Squared: 0.3137 

F-statistic: 16.69 on 2 and 73 degrees of freedom, 

 the p-value is 1.076e-006 

Means of Chlorophyll a: LDI 1-1.9: 2; LDI 2-6.3: 7.5; 

 LDI 6.4-8.6: 38.8. 

Medians of Chlorophyll a: LDI 1-1.9: 0.9; LDI 2-6.3: 4.6; 

 LDI 6.4-8.6: 28. 

Potential Change Points Based on Medians: 

 LDI: 6.3 1.9 6.5 7.2 8.2 

Test Statistics: 5.28 2.01 1.63 1.57 1.52 

 (The last three are not significant) 
9

References: 

Chang, I. (1982) “Outliers in Time Series," Unpublished Ph.D. dissertation, Department of Statistics, 

University of Wisconsin, Madison. 

Chang, I., Tiao, G. C., and Chen, C. (1988) “Estimation of Time Series Parameters in the Presence of 

 Outliers," Technometrics, 30, 193-204. 

Niu, X., Lin, P., and Meeter, D. (2000).“Detecting Change Points in the Species Composition and Water 

Quality Data of WCA2A”. Department of Statistics, Florida State University, Tallahassee. 

Technical Report Submitted to the Florida Department of Environmental Protection. 

 Pankratz A. (1991) Forecasting with Dynamic Regression Models. John Wiley & Sons Inc. New York.