Abstract:
A system and method of accurately predicting and reporting a value of a property based on a weighted average of values predicted by at least two prediction models. The system and method include the steps of accessing predicted values, determined by the prediction models, for the property; determining property-specific proportional prediction error distribution information for each predicted value determined by each prediction model; assigning a weight to the predicted value determined by each prediction model by using the property-specific proportional prediction error distribution information; and generating a property-specific weighted average value based on combination of the weight and the predicted value determined by each prediction model and reporting the property-specific weighted average value to minimize prediction error during prediction of the property value.

Description:
This application claims the benefit of U.S. Provisional Application No. 60/342,842, filed on Dec. 28, 2001, hereby incorporated by reference. 
    
    
     FIELD OF THE INVENTION 
     The present invention relates to a method and apparatus for predicting the value of real estate based on a weighted average of predicted values from at least two different value prediction methods. 
     BACKGROUND OF THE INVENTION 
     Determining an accurate estimate of the value of a property is important to entities such as buyers, sellers, real estate agents, lenders, and financial institutions that participate in a real estate market. For example, accurate estimates enable sellers to formulate realistic expectations as to the values of the properties being sold and to set appropriate sales prices for those properties. Buyers, on the other hand, rely on accurate estimates to formulate offers for the purchase of properties. Lenders may use predicted property values to apply loan-to-value guidelines and lenders and financial institutions that buy and sell mortgages may depend on these estimates to determine the value of a particular mortgage or portfolio of mortgages and the risk associated with holding such mortgages. 
     There are several known methods for estimating the value of a property at a specified time, four of which are the hedonic or property characteristic methods, the repeat sales methods, the tax assessment methods, and the Neural Network methods. Hedonic methods estimate the value of a property by assigning values to attributes of the property and totaling the values. For example, a hedonic method may estimate the value of a home by assigning a value to the home style, the neighborhood, each bedroom, each bath room, etc. 
     Repeat sales methods of estimation use actual sales prices and time data points to estimate a market-level price index, wherein the present value of a property may be estimated using the prior sale information. For example, a repeat sales index model proposed by Bailey, Muth, and Nourse (the BMN model) specifies that the change in the logarithm price of a property over a known period of time is equal to a logarithmic price index plus and error term. Another repeat sales model by Case and Shiller (the Case-Shiller model) improves on the BMN method. Unlike the BMN model that assumes that the error term is independent, the Case-Shiller model assumes that the error term is a linear function of the time between sales. 
     Tax assessment methods rely upon historical tax assessment data to predict the value of a property or estimate a price index for a collection of properties. The tax assessment methods use the proportional relationship between assessed values in any given geographic area, such as metropolitan statistical areas (MSA&#39;s), counties, or zip code areas, and sale values to predict property values. 
     Neural Network methods predict property values using the same data and underlying principles as in the hedonic model. Neural Network methods use a network training process that forms multiple linear combinations of the property characteristic variables, passes these variables through “activation functions”, then forms a linear combination of these results that is then compared with the desired output, i.e., the observed property value. Thereafter, the coefficients of the linear combinations are iteratively adjusted in an attempt to make the output of the network mimic the observed property value. 
     Each of the four methods referenced above for predicting property values do not predict the value of a property with absolute accuracy. Each method is considered to operate within acceptable bounds of accuracy and each method is expected to predict the value of a property within a few percent, for example 5%, of an actual sale price and within a certain percentage of the time, for example 15% of the time. To improve the overall prediction accuracy from currently available prediction methods, the values predicted by these methods may be averaged. However, some methods have been determined to be more accurate than others. Accordingly, there is a need to improve the accuracy of a prediction based on an average value. 
     SUMMARY OF THE INVENTION 
     The invention relates to a system and method of accurately predicting and reporting a value of a property based on a weighted average of values predicted by at least two prediction models. The method includes the steps of accessing predicted values, determined by the prediction models, for the property; determining property-specific proportional prediction error distribution information for each predicted value determined by each prediction model; assigning a weight to the predicted value determined by each prediction model by using the property-specific proportional prediction error distribution information; generating a property-specific weighted average value based on combination of the weight and the predicted value determined by each prediction model and reporting the property-specific weighted average value to minimize prediction error during prediction of the property value. 
     Additional features and advantages of the invention will be set forth in the description that follows, and in part will be apparent from the description, or may be learned; by practice of the invention. The objectives and advantages of the invention will be realized and attained by the system and method particularly pointed out in the written description and claims hereof as well as the appended drawings. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       In order to assist the understanding of this invention, reference will now be made to the appended drawings, in which like reference numerals refer to like elements. The drawings are exemplary only, and should not be construed as limiting the invention. 
         FIG. 1  is a diagram of a computing system in accordance with an embodiment of the present invention; 
         FIG. 2  is a graph of PPE variance versus predicted property value; 
         FIG. 3  illustrates a correlation coefficients matrix for the PPE for four models; 
         FIG. 4  illustrates the average estimated mean PPE assigned to the 4, 3, 2, and 1 model categories; and 
         FIG. 5  illustrates a summary of the reconciliation weights by different model combinations. 
     
    
    
     DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     Reference will now be made in detail to the preferred embodiments of the present invention, examples of which are illustrated in the accompanying drawing. The present invention described below extends the functionality of the inventive system and method for predicting the value of a real estate based on a weighted average of predicted values from multiple value prediction methods. 
       FIG. 1  illustrates a computer system  100  that is configured to predict the value of a real estate based on a weighted average of predicted values from multiple methods. Computer system  100  comprises a processor  102 , a memory component  104  and a display component  106 . Processor  102  may be programmed to implement a method for predicting and reporting real estate prices. Memory component  104  stores data used by processor  102  and display component  106  enables user interactions with computer system  100 . 
     According to the invention, accuracy of the values predicted by a repeat sales (RS) model, a tax assessment (T) model, a hedonic or property characteristic (PC) model, and a Neural Network (NN) model may be improved by combining the predictions made by any combination of two or more of the methods and by weighting the predictions attributable to the individual methods prior to combining them, wherein greater weight is given to the more accurate methods of prediction relative to the less accurate methods. 
     Specifically, each of the prediction methods is associated with a proportional prediction error PPE. The PPE is the difference between the actual sale price and the predicted sale price for a property divided by the actual sale price. The PPE for, a particular method may vary in relation to variables, such as sale price magnitude or sale-resale interval. The appropriate weight to be assigned to a particular method may vary in relation to the variance of the PPE for that method. Therefore, accuracy of a prediction made using a weighted combination of methods may be further improved by modifying the weight of a prediction attributed to a particular method based on the variance of the PPE for the method. 
     In the inventive system, the predicted values from any number of prediction methods/models may be combined to produce a weighted average value. In an embodiment of the invention, the values predicted by the (RS) model, the (I) model, the (PC) model, and the (NN) model are combined using weights obtained from a regression model. 
     The following steps demonstrate a method for determining the weighted average of four values predicted by four different models. It is apparent that one of ordinary skill in the art could routinely determine a weighted average for any number of models based on the concepts underlying the four model weighted average. 
     Specifically, if P i  denotes the sale price, i.e. known value of property i, and let P iR , P iT , P iPC , P iNN  denote the values predicted by the RS model, the T model, the PC model, and the NN model, respectively, an unbiased weight for combining the four predicted values can be estimated using an ordinary least squares (OLS) regression. The weight can be estimated by using the following equation wherein:
 
 P   i   =W   RS   P   iRS   +W   T   P   iT   +W   PC   P   iPC +(1− W   RS   −W   T   −W   PC ) P   iNN +ε i   Equation 1 
 
where W xx , ∀xx=RS, T, PC is the model specific weight. In Equation 1, the condition is imposed that the sum of weights for all models is equal to one. Note that the Neural Network model reconciliation weight is chosen as the constrained weight, but any one of the model weights could have been chosen. Therefore, Equation 1 may be rewritten as:
 
 P   i   −P   iNN   =W   RS ( P   iRS   −P   iNN )+ W   T ( P   iT   −P   iNN )+ W   PC ( P   iPC   −P   iNN )−ε i   Equation 2 
 
     The error term in Equation 2 is likely to be heteroskedastic with the variance of the error positively correlated with the value of the properties. In other words, it is expected that the error associated with a predicted value will be related to the magnitude of the predicted value. To correct for the heteroskedasticity, both sides of Equation 2 may be divided by P i . After this division and the rearrangement of the terms of the equation, the following relationship is obtained.
 
 PPE   iNN   =W   RS ( PPE   iNN   −PPE   iRS )+ W   T ( PPE   iNN   −PPE   iT )+ W   PC ( PPE   iNN   −PPE   iPC )+ε  Equation 3 
 
where PPE ixx =(1−P ixx /P i ), ∀ xx=RS, T, PC, NN, and is the proportional prediction error for the RS, A, PC, and NN models. Based on the foregoing, a column matrix of PPE values may be calculated for each model, wherein each individual PPE value is that for a particular property i. The model weights produced from OLS estimation of Equation 3 are: 
         Equation   ⁢           ⁢   4   ⁢       :     ⁢     
     [           W   RS               W   T               W   PC           ]       =         [           E   ⁡     [       (       PPE   NN     -     PPE   RS       )     2     ]             E   ⁡     [       (       PPE   NN     -     PPE   RS       )     ⁢     (       PPE   NN     -     PPE   T       )       ]             E   ⁡     [       (       PPE   NN     -     PPE   RS       )     ⁢     (       PPE   NN     -     PPE   PC       )       ]                 E   ⁡     [       (       PPE   NN     -     PPE   RS       )     ⁢     (       PPE   NN     -     PPE   T       )       ]             E   ⁡     [       (       PPE   NN     -     PPE   T       )     2     ]             E   ⁡     [       (       PPE   NN     -     PPE   T       )     ⁢     (       PPE   NN     -     PPE   PC       )       ]                 E   ⁡     [       (       PPE   NN     -     PPE   RS       )     ⁢     (       PPE   NN     -     PPE   PC       )       ]             E   ⁡     [       (       PPE   NN     -     PPE   T       )     ⁢     (       PPE   NN     -     PPE   PC       )       ]             E   ⁡     [       (       PPE   NN     -     PPE   PC       )     2     ]             ]       -   1       ·               [           ⁢           E   ⁡     [       (       PPE   NN     -     PPE   RS       )     ⁢     PPE   NN       ]                 E   ⁡     [       (       PPE   NN     -   PPE     )     ⁢     PPE   NN       ]                 E   ⁡     [       (       PPE   NN     -     PPE   PC       )     ⁢     PPE   NN       ]             ]     ⁢     
     ⁢     W   NN       =     1   -     W   RS     -     W   T     -     W   PC                 
 
where E signifies a mean or “expected” value. Based on this formulation of the reconciliation scheme, the models with lower prediction error will have higher weights
 
     The inverse of the moment matrix in Equation 4 (which is the first matrix to the right of the equal sign) can be solved analytically. To simplify the notation of Equation 4 and solve the inverse of the moment matrix, the model can be rewritten as: 
         Equation   ⁢           ⁢   5   ⁢       :     ⁢     
     [           W   RS               W   T               W   PC           ]       =         [           a   11           a   12           a   13               a   21           a   22           a   23               a   31           a   32           a   33           ]       -   1       ⁡     [           c   1               c   2               c   3           ]           
 
or in matrix form as W=A −1 C., where the elements of A are denoted by a ij  and the elements of C by c i . The diagonal elements in A are described in terms of variances and covariances in:
 
Equation 5.1
 
 a   kk   =E {( PPE   NN   −PPE   xx ) 2 }=σ 2   NN +σ 2   XX −2ρ NNxx σ NN σ xx   +E {( PPE   NN   −PPE   xx )} 2  
 
where kk is 11, 22, and 33 and xx is RS, T, and PC, respectively.
 
the off diagonal terms of A are described in:
 
Equation 5.2
 
 a   ij   =E {( PPE   NN   −PPE   xx )( PPE   NN   −PPE   yy )}
 
−σ NN .ρ NNxx σ NN σ xx −ρ NNyy σ NN σ yy +ρ xxyy σ xx σ yy +½({ E ( PPE   NN   −PPE   xx )} 2 
 
+{E( PPE   NN   −PPE   yy )} 2   − 
 
{E( PPE   xx   −PPE   yy )} 2 ) 
 
where i,j=1, 2, and 3 and xx, yy=RS, T, and PC, respectively.
 
and the elements of C are described in:
 
     Equation 5.3
 
 c   k   =E {( PPE   NN   −PPE   xx ) PPE   NN }=σ 2   NN −ρ NNxx σ NN σ xx   +{E ( PPE   NN )} 2   −E ( PPE   NN ) E ( PPE   xx ) 
 
where k is 11, 22, and 33 and xx is RS, T, and PC, respectively.
 
     In Equations 5.1-5.3, ρ the correlation coefficient between any two different models. The ρ xxyy (∀ xx=RS, T, PC, NN and yy=RS, T, PC, NN, where xx≠yy) values may be determined by performing a simple correlation of the PPE values for any pair of two different models.  FIG. 2  illustrates the relationship of PPE variance and property value for the physical characteristic model. 
     The values for σ (standard deviation or variance) for each model are provided by determining the standard deviation for the set of PPE&#39;s that exist for the model. The relationship of the standard deviation and the correlation coefficient for any two models is provided by the equation ρ xxyy =σ xxyy /[(σ xx )(σ yy )], where σ xxyy  is the covariance between any two models xx, and yy. 
     With renewed reference to Equation 5, let B=A −1 =1/|A|adj(A), where the elements of B are denoted by b ij , so that W=B*C. The inverse solution is a symmetrical matrix that takes the following form element by element: 
       Equation   ⁢           ⁢   6   ⁢     :         
               b   11     =       ⁢       (     1   /        A          )     *     (         a   22     ⁢     a   33       -       a   32     ⁢     a   23         )                     b   12     =       ⁢       b   21     =       -     (     1   /        A          )       *     (         a   21     ⁢     a   33       -       a   31     ⁢     a   23         )                       b   13     =       ⁢       b   31     =       (     1   /        A          )     *     (         a   21     ⁢     a   32       -       a   31     ⁢     a   22         )                       b   22     =       ⁢       (     1   /        A          )     *     (         a   11     ⁢     a   33       -       a   31     ⁢     a   13         )                     b   23     =       ⁢       b   32     =       -     (     1   /        A          )       *     (         a   11     ⁢     a   32       -       a   31     ⁢     a   12         )                       b   33     =       ⁢       (     1   /        A          )     *     (         a   11     ⁢     a   22       -       a   12     ⁢     a   21         )                        A        =       ⁢         a   11     ⁢     a   22     ⁢     a   33       +       a   21     ⁢     a   32     ⁢     a   13       +       a   31     ⁢     a   12     ⁢     a   23       -       a   31     ⁢     a   22     ⁢     a   13       -                     ⁢         a   11     ⁢     a   32     ⁢     a   23       -       a   33     ⁢     a   21     ⁢     a   12                   
 
     Based on this solution of the inverse matrix, the final form of the reconciliation weights can be written as, 
         Equation   ⁢           ⁢   7   ⁢       :     ⁢     
     [           W   RS               W   T               W   PC           ]       =       [           b   11           b   12           b   13               b   21           b   22           b   23               b   31           b   32           b   33           ]     *     [           c   1               c   2               c   3           ]           
         W   NN     =     1   -     W   RS     -     W   T     -     W   PC           
 
     Equation 7 can be solved for each of the reconciliation weights. For example, the solution for the repeat sales reconciliation weight is provided by the equation:
 
 W   RS   =b   11   *c   1   +b   12   *c   2   +b   13   *c   3 . 
 
     The OLS regression model described in Equations 1-3 may not actually be run due to the segmentation of the data based on correlation between PPE variance and time intervals for the RS and T models, or between PPE, variance and value magnitude for the PC and NN models. As a result of this segmentation, the amount of data available for regression for each possible combination of models may be too small and difficult to implement. By analytically solving the OLS regression, Equation 7 may be used to determine the weights using property level PPE variances/means (σ) for all of the models, and county level correlation coefficients (ρ) of PPE between any two models. As a result, all unknowns may be solved by substituting known PPE&#39;s, σ&#39;s, and ρ&#39;s in Equation 7 based on the relationships established by Equations 5.1-5.3 and 6. 
       FIG. 3  illustrates a correlation coefficients matrix for the PPE for each of the four models discussed above. For each county, the correlation coefficients of PPE are estimated using sample correlation coefficients of in-sample PPE&#39;s. Based on the in-sample PPEs, all of the sample correlation coefficients in  FIG. 2  are positive. This means that, in general, if one model predicts the value of a property well, the other models will do the same and vice versa. According to  FIG. 2 , the hedonic and Neural Network models have the highest correlation, followed by the tax model and hedonic model, and next by the tax model and Neural Network model. The repeat sales model has the weakest correlation with the other three models. 
     In addition to the four model case, three and two model case reconciliation models also may be derived from the inventive system. Given two, three, and four model weighting schemes, all possible combinations of model predictions can be categorized and assigned weights. In an embodiment of the invention, to create property specific PPE variance measures, non-linear empirical models are developed on a county level for each prediction model.  FIG. 4  illustrates the average estimated mean PPE assigned to the 4, 3, 2, and 1 model categories. According to the invention, the closer to zero the mean PPE, the more accurate the model is. According to  FIG. 4 , the repeat sales model  402  is most accurate, followed by the tax  404 , hedonic  406  and Neural Network  408  models, in that order. The reconciliation scheme produces a weighted model  410 , which is more accurate than any other model, thereby indicating the reconciliation scheme is working effectively. The frequency  412  column of  FIG. 4  indicates the number of properties for which there was usable data. 
       FIG. 5  illustrates a summary of the reconciliation weights by different model combinations. Higher weights indicate models with greater influence in the final weighted values. In the four model case, the general model hierarchy can be seen as repeat sales, tax, hedonic and Neural Network models. 
     In the foregoing example, the weights for the averaging of multiple valuation models are determined using actual standard deviations (σ) that are based on actual PPE&#39;s. Use of actual PPE&#39;s, however, requires knowledge of an actual sale price as well as a predicted price. In some instances, only a predicted value or price is known. It has been determined that the foregoing method may still be used in these instances by substituting a predicted standard deviation (a) for the actual standard deviation in Equations 5.1-5.3. 
     Predicted standard deviations may be determined as a result of the existence of a correlation between PPE variance and certain variables, depending upon the model under consideration. For the repeat sales and tax assessment models, a correlation exists between PPE variance and the interval of time since the last sale or last tax assessment. For the repeat sales model, the longer the prediction time interval, the higher the dispersion of PPE. As a result, weights for the repeat sales model should be lower for longer prediction intervals relative to other models. Similarly, a positive correlation has been observed between the tax assessment model prediction interval, i.e., the interval from assessment time to prediction reference time, and the PPE variance of the tax assessment model. As a result, the tax model may also be expected to have a relatively lower weight when the tax model prediction interval is longer. 
     The property characteristic and Neural Network models do not demonstrate a correlation between PPE variance and time interval as do the repeat sales and tax assessment models. Both the physical characteristic and the Neural Network models do demonstrate, however, a correlation between the PPE variance and the magnitude of the property value. Small and large predicted prices in terms of dollars have higher PPE variance relative to mid-ranged predicted values. According to  FIG. 2 , a sharp drop off in PPE variance is shown for low value predictions to moderate value predictions, followed by a more gradual increase in PPE variance as the predicted value increases. 
     The relationship between PPE variance and another variable such as value or time interval may be modeled as follows. First, it is assumed that the difference between the log price and its predicted value for model xx (tax, property characteristic, and Neural Network models), 
         ln   ⁡     (         P   ^     ixx       P   i       )       ,       
 
is a normal distribution with mean zero, then the mean and variance of the model xx prediction error Mean(PPE ixx ) and Var(PPE ixx ) can be written as: 
       Equation   ⁢           ⁢   8   ⁢     :         
         Mean   ⁡     (     PPE   ixx     )       =         Mean   ⁡     (         P   ^     ixx       P   i       )       -   1     =         Mean   ⁡     [     exp   ⁡     (     ln   ⁡     (         P   ^     ixx       P   i       )       )       ]       -   1     =       exp   ⁡     [       Var   ⁡     (     ln   ⁡     (         P   ^     ixx       P   i       )       )       /   2     ]       -   1.             
 
and, 
       Equation   ⁢           ⁢   9   ⁢     :         
         Var   ⁡     (     PPE   ixx     )       =       Var   ⁡     (         P   ^     ixx       P   i       )       =       Var   ⁡     [     exp   ⁡     (     ln   ⁡     (         P   ^     ixx       P   i       )       )       ]       =       exp   ⁡     [     Var   ⁡     (     ln   ⁡     (         P   ^     ixx       P   i       )       )       ]       ⁢       {       exp   ⁡     [     Var   ⁡     (     ln   ⁡     (         P   ^     ixx       P   i       )       )       ]       -   1     }     .               
 
     Assuming the relationship established by Equation 9, in the tax, property characteristic, and Neural Network models, an error variance model may be used to regress the squared model residuals on a quadratic function of the prediction interval (tax model) or prediction size (property characteristic and Neural Network models) to estimate the variance of the error. As a result, the following relationship is established: 
       Equation   ⁢           ⁢   10   ⁢     :         
           (       σ   ^     xx     )     2     =       Var   ⁡     [     ln   ⁡     (         P   ^     ixx       P   i       )       ]       -       α   1     ⁢     y   2       +       α   2     ⁢   y     +   ɛ         
 
where y and y 2  are the time interval or the value magnitude and the time interval or the value magnitude squared, respectively. Using an OLS method and substituting known values for ({circumflex over (σ)} xx ) 2  and y, the values for α 1  and α 2  can be determined. Once these coefficients are known, ({circumflex over (σ)} xx ) 2  can be determined for any property, even though actual sales data is unavailable for that property.
 
     For the repeat sale model, the 
       Var   ⁡     [     ln   ⁡     (       P   iR       P   i       )       ]         
 
can be obtained from the second stage regression of the Case-Shiller three stage estimation procedures.
 
     The new framework for determining the prediction model variances and the model weights incorporate the concept of prediction interval (tax and repeat sales models) and prediction size (property characteristic and Neural Network models). The prediction interval and prediction size determine the size of the prediction error through the estimated variance functions, which in turn determines the predicted variance. The use of prediction interval and prediction size to calculate the predicted variance in the tax, property characteristic, and Neural Network models makes their predicted variances consistent with the repeat sales model predicted variance. The prediction intervals and prediction sizes are the major factors in determining the model weights because the weights are inversely related to the variance of the prediction error. Thus, a property with a larger variance will have lower weight and vice versa. 
     In addition to the reconciliation and weighting of multiple predictions, an exclusion filter may be applied to the property characteristic and Neural Network models prior to application of the reconciliation scheme. This may be done to remove obvious prediction outliers from the models. Using the log variance model, the probability that the predicted value is within +/−5% PPE is calculated. Then the five percent of the properties that have the lowest probability of being within +/−5% PPE are excluded for each model. 
     It will be apparent to those skilled in the art that variations and modifications of the present invention can be made without departing from the scope or spirit of the invention. For example, the program language and code selected to implement the aspects of the present invention may be varied without departing from the intended scope of the claims. Further, the computer hardware selected to implement the aspects of the present invention may also be varied without departing from the intended scope of the claims. Thus, it is intended that the present invention cover all such modifications and variations of the invention, provided they come within the scope of the appended claims and their equivalents.