Fitting Polynomial Models with PROC ORTHOREG and GLM

The script first generates a 'Polynomial' dataset based on a polynomial function. Then, it uses PROC ORTHOREG to fit a 9th-degree polynomial model. The model results are stored and used by PROC PLM to score the zeros of the original function. Concurrently, PROC GLM is used to fit a similar model, and predictions are also calculated with PROC PLM. Finally, the predictions from both models (ORTHOREG and GLM) are merged and displayed for comparison.

Data Analysis

Type : INTERNAL_CREATION

The script creates two datasets: 'Polynomial' and 'Zeros'. The data is entirely generated via 'do' loops in DATA steps, without relying on external sources.

1 Code Block

DATA STEP Data

Explanation :
This block creates the 'Polynomial' table. It generates 101 observations by calculating 'x' and 'y' values according to a polynomial function, without using input data.

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1	title 'Polynomial Data';
2	DATA Polynomial;
3	DO i = 1 to 101;
4	x = (i-1)/(101-1);
5	y = 10**(9/2);
6	DO j = 0 to 8;
7	y = y * (x - j/8);
8	END;
9	OUTPUT;
10	END;
11	RUN;

2 Code Block

PROC ORTHOREG

Explanation :
This block uses the ORTHOREG procedure to perform a 9th-degree polynomial regression on the 'y' variable as a function of 'x'. The model is stored in the 'OStore' object and a plot of the fitted model is produced.

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1	ods graphics on;
2
3	PROC ORTHOREG DATA=Polynomial;
4	effect xMod = polynomial(x / degree=9);
5	model y = xMod;
6	effectplot fit / obs;
7	store OStore;
8	RUN;
9
10	ods graphics off;

3 Code Block

DATA STEP Data

Explanation :
This block creates the 'Zeros' table which contains the roots (zeros) of the polynomial function used to generate the initial data. These 'x' values will be used for prediction.

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1	DATA Zeros(keep=x);
2	DO j = 0 to 8;
3	x = j/8;
4	OUTPUT;
5	END;
6	RUN;

4 Code Block

PROC PLM

Explanation :
This procedure uses the 'OStore' model (created by PROC ORTHOREG) to calculate predicted values ('OPred') for each observation in the 'Zeros' table. The results are saved in the 'OZeros' table.

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1
2	PROC PLM restore=OStore noprint;
3	score
4	DATA=Zeros out=OZeros pred=OPred;
5	RUN;
6

5 Code Block

PROC PRINT

Explanation :
Displays the content of the last created table, which is 'OZeros', showing the predictions from the ORTHOREG model.

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1	PROC PRINT noobs;
2	RUN;

6 Code Block

PROC GLM

Explanation :
This block uses the GLM (General Linear Model) procedure to fit a 9th-degree polynomial model (specified by successive interactions of 'x'). The model is stored in 'GStore' for later use.

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1	PROC GLM DATA=Polynomial;
2	model y = x\|x\|x\|x\|x\|x\|x\|x\|x;
3	store GStore;
4	RUN;

7 Code Block

PROC PLM

Explanation :
Similar to the previous step, this procedure uses the 'GStore' model (created by PROC GLM) to calculate predictions ('GPred') for the 'Zeros' table. The results are stored in the 'GZeros' table.

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1
2	PROC PLM restore=GStore noprint;
3	score
4	DATA=Zeros out=GZeros pred=GPred;
5	RUN;
6

8 Code Block

DATA STEP Data

Explanation :
This block merges the 'OZeros' and 'GZeros' tables to create a new 'Zeros' table that contains the predictions from both models (ORTHOREG and GLM) side-by-side for easy comparison.

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1	DATA Zeros;
2	MERGE OZeros GZeros;
3	RUN;

9 Code Block

PROC PRINT

Explanation :
Displays the content of the final 'Zeros' table, which compares the predicted values from the ORTHOREG model ('OPred') and the GLM model ('GPred').

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1	PROC PRINT noobs;
2	RUN;

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