Regression model
- Multiple linear regression modeling is used for statistical prediction based on past forecasts for the statistical period.
- In such modeling, the predictand Y is related to the N predictors Xi. The predictand is estimated from a linear combination of predictors.
![](equation/eq2.png)
Here, ai represents the regression coefficients, b is the regression constant and ε is the error term.
- The coefficients ai and the constant b are determined such that the sum of the squares of estimation errors is minimized.
- The analysis procedure is detailed below.
- Calculation of the factors ai and b is based on past observation data variables such as temperature and precipitation and on past forecast (i.e., hindcast) elements for the same period from the verification period (30-year statistical period (1991 - 2020) by default).
- Prediction of objective variables from real-time forecast elements multiplied by these factors is conducted using the relevant simultaneous equation.
- Mapping from the objective variable to three categorized forecasts based on the ranking is conducted.
- In the guidance tool, the probability density function (PDF) is assumed to have normal distribution.
Here the mean (xs) is a prediction value from the regression model and the standard deviation (σn) is the error of the model, assumed to be its RMSE based on hindcast data.
- Threshold values for the three categories are determined from past observation for the verification period (from 1991 to 2020 by default).
- Probability for each tercile category (below-, near- and above-normal) is calculated with reference to the PDF of guidance and the threshold values for the three categories.
- The cross-validation technique (Bishop 2006) is not used to create the regression model.
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![](./fig/fig1.png)
Conceptual diagram for a linear regression model of the predictand y and two predictors (x1 and x2).
![](./fig/fig2.png)
Sample of predicted PDF with normal distribution. xs and σn denote the mean forecast and the standard deviation, respectively.
![](./fig/fig3.png)
Sample climatorogical and predicted anomalous PDF for guidance forecasting.
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