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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

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

      • 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

  • Regression-based Models

  • Predictor Selection

  • Data Quality and Clustering

  • Measuring Success

  • An Example


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

  • 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

    • 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

    • 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


    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

    • 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

    • 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

    • 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

    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

    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

    • 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

    • 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)

    • 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:


    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

    • 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)

    • 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

    Day Characterization

    - Wind direction

    - Inversion height

    - Forcing influences

    COBEL

    Forecast Weighting Function

    Local SFM

    Consensus

    Forecast

    Regional SFM

    Satellite SFM


    Measuring Success


    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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