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Regression on Principal Components - Stochastic Hydrology - Lecture Notes, Study notes of Mathematical Statistics

Avinashilingam University Mathematical Statistics

The main points which I found very interesting are: Regression on Principal Components, Independent Variables, Multiple Regression Analysis, Principal Components, Stochastic Model, Standard Deviation, Matrix of Transformed Data, Regression Model, Prediction Model

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Regression on Principal components:

• In the development of a stochastic model for a

dependent variable Y, the first step usually is to do

PCA on the independent variables.

• The derived principal components are used as

independent variables in a multiple regression

analysis with the dependent variable Y.

Regression on Principal Components

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Partial preview of the text

Download Regression on Principal Components - Stochastic Hydrology - Lecture Notes and more Study notes Mathematical Statistics in PDF only on Docsity!

Regression on Principal components:

In the development of a stochastic model for a

dependent variable Y, the first step usually is to do

PCA on the independent variables.

The derived principal components are used as

independent variables in a multiple regression

analysis with the dependent variable Y.

Regression on Principal Components 3

Procedure:

Independent variables are standardized.

where X

i,j

is the i

observation on the j

variable,

and s

are the mean and standard deviation of the j

variable.

Dependant variables are centered.

where Y

is the i

observation on y, is the mean of

y.

Regression on Principal Components 4 ( (^) ij j ) ij j X x x s − = i i y = Y − y y j x

The regression model is

Where

Y is nx1 vector of n observations of the centered

dependent variable,

Z is nxp matrix of n values for transformed data of p

variables, and

Β is a px1 vector of unknown parameters.

Regression on Principal Components 6 Y = Z Β 1 p i j ij j y β z = = or ∑

The matrix Β is estimated as Regression on Principal Components 7 ( ) 1 ˆ '^ ' Z Z Z Y − Β =

The annual rainfall in mm for 10 stations and observed basin annual yield (Y) in mm 9

Year x

x

y

A regression equation is obtained using all the 10

stations annual rainfall data is as follows

The multiple linear regression equation is as follows.

10 Example – 1 (Contd.) (19 1 )^ (19 10 ) (10 1 ) Y X

× × ×

= Β ( ) 1 ˆ '^ ' X X X Y − Β =

Y = 782.4 + 0.1861 x

+ 0.0484 x

0.0198 x 3

0.0019 x

+ 0.1196 x

+ 0.1555 x

+ 0.0232 x

0.1948 x

+ 0.0799 x

0.0041 x 10

12 Example – 1 (Contd.) ( (^) ij j ) ij j X x x s − = i i y = Y − y Station Mean Std.dev.

x

1 1873.2 434.

x

2 3839.9 927.

x

3 5925.8 1250.

x

4 4436.0 1846.

x

5 6068.9 914.

x

6 4441.0 1091.

x

7 3254.0 608.

x

8 3307.3 599.

x

9 2969.7 621.

x

10 2859.6 462.

y^3569 -

Standardized annual rainfall and centered observed basin annual yield (Y). 13

Year x

x

y

The eigenvalues and eigenvectors for the covariance

matrix

15 Example – 1 (Contd.) Eigenvalues 4.945 2.631 1.047 0.364 0.307 0.257 0.205 0.140 0.063 0. Eigenvectors 0.390 -0.165 0.211 -0.191 0.451 -0.304 0.149 -0.043 -0.644 -0. 0.381 -0.188 -0.053 -0.543 -0.127 0.215 -0.265 -0.574 0.189 0. 0.393 0.029 0.382 0.235 0.074 -0.128 -0.328 0.319 0.227 0. 0.298 -0.321 -0.390 -0.111 0.246 0.400 0.425 0.437 0.210 0. 0.404 -0.065 0.179 -0.121 -0.589 -0.056 -0.093 0.393 -0.013 -0. 0.371 -0.161 -0.229 0.546 -0.116 -0.394 0.301 -0.402 0.241 -0. 0.122 0.462 0.521 -0.117 0.237 0.148 0.428 -0.136 0.393 -0. 0.317 0.338 -0.122 0.444 0.069 0.603 -0.241 -0.134 -0.333 -0. 0.136 0.529 -0.237 -0.201 -0.412 -0.110 0.388 0.031 -0.275 0. 0.160 0.443 -0.477 -0.192 0.351 -0.358 -0.358 0.155 0.235 -0.

S − λ I = 0

( S^ −^ λ I^ ) X =^0

The eigenvalues and % variance explained:

16 Example – 1 (Contd.) Eigenvalues % variance explained 4.945 49. 2.631 26. 1.047 10. 0.364 3. 0.307 3. 0.257 2. 0.205 2. 0.140 1. 0.063 0. 0.042 0.

> 95% variance

explained by first

6 principal

components

( ) j Trace S λ

18 Example – 1 (Contd.)

 - 0.390 -0.165 0.211 -0.191 0.451 -0. Z = X A - 0.381 -0.188 -0.053 -0.543 -0.127 0. - 0.393 0.029 0.382 0.235 0.074 -0. - 0.298 -0.321 -0.390 -0.111 0.246 0. - 0.404 -0.065 0.179 -0.121 -0.589 -0. - 0.371 -0.161 -0.229 0.546 -0.116 -0. - 0.122 0.462 0.521 -0.117 0.237 0. - 0.317 0.338 -0.122 0.444 0.069 0. - 0.136 0.529 -0.237 -0.201 -0.412 -0. - 0.160 0.443 -0.477 -0.192 0.351 -0.

0.17 0.36 -0.34 -0.82 -0.39 -0.73 -0.08 -0.95 -1.29 -1.
0.89 -0.18 1.50 -0.60 0.95 -0.02 0.72 0.24 -0.17 -0.
0.27 0.55 1.17 -0.87 0.23 0.35 2.01 1.83 0.96 0.
0.29 -0.57 0.91 -0.54 0.63 0.34 1.64 1.37 0.77 0.
0.49 0.42 0.51 -0.78 0.66 -0.44 1.06 0.42 1.78 1.
-0.36 -1.21 -0.01 -0.56 -0.51 -0.73 -0.28 -1.45 -0.54 -0.
-1.13 -1.60 -0.70 -0.80 -1.01 -1.22 1.31 -0.50 0.13 0.
-0.93 -0.90 -1.58 -0.95 -1.04 -0.81 -0.79 -2.33 -1.64 -2.
-1.20 -1.50 -1.32 -1.10 -1.47 -0.81 -0.72 -0.61 -0.06 -0.
-0.62 0.38 -0.01 -0.44 0.87 0.64 -0.11 0.59 1.60 0.
-1.05 -0.23 -0.62 -0.63 -0.92 -0.59 -0.97 -0.48 -0.63 1.
-0.49 0.91 0.37 1.05 1.36 0.22 -0.36 0.60 0.28 -0.
- 0.19 0.77 -0.20 0.92 0.00 -0.72 -1.07 -0.86 -1.32 -0.
- 0.23 0.00 0.08 0.66 -0.21 -0.38 -0.36 1.02 -0.62 -0.
- 0.51 0.37 0.65 1.24 -0.03 1.95 -1.30 0.16 -0.90 -0.
- 2.94 2.47 1.78 2.50 2.53 2.77 -0.60 0.36 -0.76 0.
-0.93 -0.96 -1.42 0.29 -1.01 0.27 -1.07 0.31 0.69 1.
-0.42 -0.14 -1.38 0.28 -0.78 -0.60 -0.11 -0.35 0.66 0.
- 1.15 1.07 0.62 1.15 0.14 0.51 1.06 0.63 1.07 1. - 10 x Z =
  - 19 x
-‐1.283 -‐1.278 1.259 -‐0.492 0.139 0. Z = X A =
- 1.133 0.192 1.776 0.367 -‐0.068 -‐0.
- 1.825 2.512 0.974 0.410 0.245 0.
- 1.273 1.972 0.998 0.826 0.040 0.
- 1.176 2.403 0.134 -‐1.034 -‐0.232 -‐0.
-‐1.907 -‐0.547 0.724 0.067 0.087 -‐0.
-‐2.395 1.519 0.673 0.048 0.429 0.
-‐3.779 -‐2.327 1.050 -‐0.058 -‐0.483 -‐0.
-‐3.115 0.333 -‐0.480 0.475 -‐0.042 -‐0.
- 0.830 1.247 -‐0.735 0.055 -‐1.484 -‐0.
-‐1.700 0.094 -‐1.054 -‐0.154 0.348 -‐0.
- 1.345 -‐0.585 -‐0.305 -‐0.126 -‐1.214 1.
-‐0.430 -‐2.241 0.012 -‐0.817 0.203 0.
- 0.243 -‐0.433 -‐0.169 0.391 0.593 1.
- 1.336 -‐2.171 -‐0.751 1.326 0.157 -‐0.
- 5.527 -‐2.835 -‐0.199 -‐0.183 0.169 -‐0.
-‐1.202 0.857 -‐2.516 0.433 0.246 -‐0.
-‐1.219 0.366 -‐0.975 -‐0.743 0.059 0.
- 2.341 0.921 -‐0.416 -‐0.792 0.808 0.081 19 x

Regression analysis is performed on these

components

19 Example – 1 (Contd.) Y = Z Β ( ) 1 ˆ '^ ' Z Z Z Y − Β = -314.

-463. -500. -629. -554.

-397.

-224.

-70. -328.

Y =

The regression equation is

21 Example – 1 (Contd.)

y = 192.1569 P

+ 13.29536 P

33.1304 P c

73.92323 P

64.0569 P c - 15.6921 P c 2 R = 0.

The eigenvalues and % variance explained is

22 Example – 1 (Contd.) Eigenvalues % variance explained 4.945 49. 2.631 26. 1.047 10. 0.364 3. 0.307 3. 0.257 2. 0.205 2. 0.140 1. 0.063 0. 0.042 0.

Regression on Principal Components - Stochastic Hydrology - Lecture Notes, Study notes of Mathematical Statistics

Related documents

Partial preview of the text

Download Regression on Principal Components - Stochastic Hydrology - Lecture Notes and more Study notes Mathematical Statistics in PDF only on Docsity!

Regression on Principal components:

dependent variable Y, the first step usually is to do

PCA on the independent variables.

independent variables in a multiple regression

analysis with the dependent variable Y.

Procedure:

where X

is the i

observation on the j

variable,

and s

are the mean and standard deviation of the j

variable.

where Y

is the i

observation on y, is the mean of

y.

Where

Y is nx1 vector of n observations of the centered

dependent variable,

Z is nxp matrix of n values for transformed data of p

variables, and

Β is a px1 vector of unknown parameters.

Year x

x

x

x

x

x

x

x

x

x

y

A regression equation is obtained using all the 10

stations annual rainfall data is as follows

The multiple linear regression equation is as follows.

× × ×

Y = 782.4 + 0.1861 x

+ 0.0484 x

0.0019 x

+ 0.1196 x

+ 0.1555 x

+ 0.0232 x

0.1948 x

+ 0.0799 x

x

x

x

x

x

x

x

x

x

x

y^3569 -

Year x

x

x

x

x

x

x

x

x

x

y

The eigenvalues and eigenvectors for the covariance

matrix

S − λ I = 0

The eigenvalues and % variance explained:

> 95% variance

explained by first

6 principal

components