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Building Statistical Forecast Models. Wes Wilson MIT Lincoln Laboratory April, 2001. Experiential Forecasting. Idea: Base Forecast on observed outcomes in previous similar situations (training data) Possible ways to evaluate and condense the training data Categorization

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building statistical forecast models

Building Statistical Forecast Models

Wes Wilson

MIT Lincoln Laboratory

April, 2001

experiential forecasting
Experiential Forecasting
  • Idea: Base Forecast on observed outcomes in previous similar situations (training data)
  • Possible ways to evaluate and condense the training data
    • Categorization
      • Seek comparable cases, usually expert-based
    • Statistical
      • Correlation and significance analysis
    • Fuzzy Logic
      • Combines Expert and Statistical analysis
  • Belief: Incremental changes in predictors relate to incremental changes in the predictand
  • Issues
    • Requirements on the Training Data
    • Development Methodology
    • Automation
outline
Outline
  • Regression-based Models
  • Predictor Selection
  • Data Quality and Clustering
  • Measuring Success
  • An Example
statistical forecast models
Statistical Forecast Models
  • Multi-Linear Regression

F = w0 + S wi Pi

wi = Predictor Weighting

w0 = Conditional Climatology  Mean Predictor Values

  • GAM: Generalized Additive Models

F = w0 + S wi fi(Pi)

fi = Structure Function, determined during regression

  • PGAM: Pre-scaled Generalized Additive Models

F = w0 + S wi fi(Pi)

fi = Structure Function, determined prior to regression

  • The constant term w0 is conditional climatology less the weighted mean bias of the scaled predictors
models based on regression
Models Based on Regression
  • Training Data for one predictor
    • P vector of predictor values
    • E vector of observed events
  • Residual
      • R2 = || FP – E ||2
  • Regression solutions are obtained by adjusting the parametric description of the forecast model (parameters w) until the objective function J(w) = R2 is minimized
  • Multi-Linear Regression (MLR)
      • J(w) = || Aw – E ||2
  • MLR is solved by matrix algebra; the most stable solution is provided by the SVD decomposition of A
regression and correlation
Regression and Correlation
  • Training Data for one predictor
    • P vector of predictor values
    • E vector of observed events
    • Error Residual: R2 = || FP – E ||2
  • Correlation Coefficient r(P, E) = DP •DE / sDPsDE
  • Fundamental Relationship. Let F0 be a forecast equation with error residuals E0 (||E0||=R0). Let W0 + W1 P be a BLUE correction for E0, and let F = F0 + E0 . The error residual RF of F satisfies
      • RF2 = R02  [ 1 - r(P, E0)2 ]
model training considerations
Model Training Considerations
  • Assumption: The training data are representative of what is expected during the implementation period
  • Simple models are less likely to capture undesirable (non-stationary) short-term fluctuations in the training data
  • The climatology of the training period should match that expected in the intended implementation period (decade scale)
  • It is irrational to expect that short training periods can lead to models with long-term skill
    • Plan for repeated model tuning
    • Design self-tuning into the system
  • It is desirable to have many more training cases than model parameters

The only way to prepare for the future is to prepare to be surprised;

that doesn’t mean we have to be flabbergasted. Kenneth Boulding

slide8
GAM
  • An established statistical technique, which uses the training data to define nonlinear scaling of the predictors
  • Standard implementation represents the structure functions as B-splines with many knots, which requires the use of a large set of training data
  • The forecast equations are determined by linear regression including the nonlinear scaling of the predictors

F = w0 + Siwi fi(Pi)

  • The objective is to minimize the error residual
  • The structure functions are influence by all of the predictors, and may change if the predictor mix is altered
  • If a GAM model has p predictors and k knots per structure function, then the regression model has np+1 (linear) regression parameters
pgam pre scaled gam
PGAM: Pre-scaled GAM
  • A new statistical technique, which permits the use of training sets that are decidedly smaller than those for GAM
  • Once the structure functions are selected, the forecast equations are determined by linear regression of the pre-scaled predictors

F = w0 + S wi fi(Pi)

  • Determination of the structure functions is based on enhancing the correlation of the (scaled) predictor with the error residual of conditional climatology
      • Maximize r( fi(Pi), DE )
  • The structure function is determined for each predictor separately
  • Composite predictors should be scaled as composites
  • The structure functions often have interpretations in terms of scientific principles and forecasting techniques
predictors
Predictors
  • Every Method Involves a Choice of Predictors
  • The Great Predictor Set: Everything relevant and available
  • Possible Reduction based on Correlation Analysis
  • Predictor Selection Strategies
    • Sequential Addition
    • Sequential Deletion
    • Ensemble Decision ( SVD )
  • Changing the predictor list changes the model weights; for GAM, it also changes the structure functions
computing solutions for the basic regression problem
Computing Solutions for the Basic Regression Problem
  • Setting: Predictor List { Pi }n and observed outcomes b over the m trials of the training set
  • Basic Linear Regression Problem

A w = b

where the columns of the m by n matrix A are the lists of observed predictor values over the trials

  • Normal Equations: ATA w = ATb
  • Linear Algebra: w = (ATA)-1 Atb
  • Optimization: Find x to minimize R2 = | Aw – b |2
svd singular value decomposition
SVD – Singular Value Decomposition

S

0

[ S | 0 ] T =

      • A = U S VT where U and V are orthogonal matrices
          • and S = [ S | 0 ]T where S is diagonal with positive diagonal entries
      • UT A w = S VT w = UT b
      • Set w = VTw, b = [UTb]n
  • Restatement of the Basic Problem
          • S VT w = b or S w = b
      • (original problem space) (VT-transformed problem space)
  • Since U is orthogonal, the error residual is not altered by this restatement of the problem

CAUTION: Analysis of Residuals can be misleading unless the

dynamic ranges of the predictor values have been standardized

structure of the error residual vector
Structure of the Error Residual Vector

0

0

  • Truncated Problem: For i > k , . set wi = 0. This increases the . error residual to
  • Rk2 = Sk+1mbi2= R*2+ Sk+1nbi2
  • si’s are usually decreasing
  • sn > 0, or reduce predictor list
  • For i < n, wi = bi / si
  • For i > n, there is no solution. This is the portion of the problem that is not resolved by these predictors
  • Magnitude of the unresolved portion of the problem: .R*2 = Sn+1mbi2

Sw = b

s1

s2

s3

*

sn

w1

w2

w3

*

wn

b1

b2

b3

*

bn

bn+1

*

*

*

*

bm

=

controlling predictor selection
Controlling Predictor Selection
  • SVD / PC analysis provides guidance
  • Truncation in w space reduces the degrees of freedom
  • Truncation does not provide nulling of predictors: . since 0 components of w. do not lead to 0 components of w = V w
  • Seek a linear forecast model of the form
      • F( a ) = aT w = S wi ai , a is a vector of predictor values
  • Predictor Nulling:
    • The ith predictor is eliminated from the problem if wi = 0
  • Benefits of predictor nulling
    • Provides simple models
    • Eliminate designated predictors (missing data problem)
    • Quantifies the incremental benefit provided by essential predictors (sensor benefit problem)
predictor selection process
Predictor Selection Process
  • Gross Predictor Selection (availability & correlation)
  • SVD for problem sizing an gross error estimation
  • Truncation and Predictor Nulling maximal model(s)

( there may be more than one good solution)

  • Successive Elimination in the Original Problem Space

 minimal model (until SD starts to grow rapidly)

  • Successive Augmentation in the Original Problem Space
  • At this point, the good solutions are bracketed between the maximal and the minimal models; exhaustive searches are probably feasible, cross validation is wise.
creating 15z satellite forecast models 1
Creating 15z Satellite Forecast Models (1)
  • 149 marine stratus days from 1996 to 2000
  • 51 sectors and 3 potential predictors per sector (153)
  • Compute the correlation for each predictor with the residual from conditional climatology
  • Retain only predictors, which have correlation greater than .25, reduces the predictor list to 45 predictors
  • Separate analysis for two data sets, Raw and PGAM
  • Truncate each when SD reduction drops below 1.5 %

RAW:

PGAM:

creating 15z satellite forecast models 2

Raw Data

SVD Raw 6

PGAM Data

SVD PGAM 6

Creating 15z Satellite Forecast Models (2)

Sigma PC 6

1.134

Sigma PC 6

0.999

Sigma

1.148

Sigma

0.999

  • SVD Truncate 6  Pred.Nulling
  • In the Truncation space:

Null to 7 predictors with acceptable error growth

  • Maximal Problems (R-8,P-7)
  • Minimal Problems (R-5,P-4)
  • Neither problem would accept augmentation according to the strict cross-validation test
  • Different predictors were selected
data quality and clustering
Data Quality and Clustering
  • DQA is similar to NWP
    • need to do the training set
    • probably need to work to tighter standards
  • Data Clustering
    • During training - manual ++
    • For implementation - fully automated
  • Conditional Climatology based on Clustering
satellite statistical model mit ll
Satellite Statistical Model (MIT/LL)
  • 1-km visible channel (brightness)
  • Data pre-processing
    • re-mapping to 2 km grid
    • 3x3 median smoother
    • normalized for sun angle
    • calibrated for lens graying
  • Grid points grouped into sectors
    • topography
    • physical forcing
    • operational areas
  • Sector statistics
    • Brightness
    • Coverage
    • Texture
  • 4 year data archive, 153 predictors
  • PGAM Regression Analysis

SECTORIZATION

consensus forecast
Consensus Forecast

Day Characterization

- Wind direction

- Inversion height

- Forcing influences

COBEL

Forecast Weighting Function

Local SFM

Consensus

Forecast

Regional SFM

Satellite SFM

conclusions
Conclusions
  • PGAM, SVD/PC, and Predictor Nulling provides a systematic way to approach the development of Linear Forecast models via Regression
  • This methodology provides a way to investigate the elimination of specific predictors, which could be useful in the development of contingency models
  • We are investigating full automation
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