WO2017151113A1

WO2017151113A1 - Estimating the impact of weather in electricity bills

Info

Publication number: WO2017151113A1
Application number: PCT/US2016/020202
Authority: WO
Inventors: Ioannis Akrotirianakis; Amit Chakraborty
Original assignee: Siemens Corporation
Priority date: 2016-03-01
Filing date: 2016-03-01
Publication date: 2017-09-08

Abstract

A method for determining the impact of weather on electricity costs includes executing (21) at least one regression model to calculate (22, 23) regression coefficients for one or more regressor variables from a set of weather and energy usage data; and splitting (24) energy usage associated with weather from energy usage not associated with weather based on the regression coefficients and confidence intervals calculated for the one or more regression models, wherein the one or more regressor variables include temperature, heating degree days (HDDs), and cooling degree days (CDDs), HDD_i = max{ T_b T_i, 0}, CDD_i = max{ T_i T_b, 0}, T_i is an average temperature for day i, and T_b is a baseline temperature selected to separate an amount of energy used for heating and from an amount of energy used for cooling, and the energy usage not associated with weather is a total energy usage minus the energy usage associated with weather.

Description

ESTIMATING THE IMPACT OF WEATHER IN ELECTRICITY BILLS

Technical Field

Embodiments of the present disclosure are directed to identifying the portion of an electricity bills that is related to energy used for heating or cooling a residential building.

Discussion of the Related Art

Electricity consumption depends in part on the weather conditions of the region where it is distributed. Higher temperatures during summer increase consumption due to the longer and more intense use of air-conditioning systems. In winter, lower temperatures increase the use of electric heating systems and lighting due to longer night periods which results in higher electricity demand. Electricity bills tend to increase substantially when there are periods of extreme weather conditions, namely periods where the temperature is abnormally high or low due to the extensive use of HVAC systems.

There is a significant research effort and interest in the relationship that exists between energy consumption and weather. Electricity consumption increases as the temperatures increase mainly due to the extensive use of air conditioning. As a result the electricity bills increase abnormally during those summer months when there is heat wave that lasts for several days. It would be beneficial for the consumers to know what portion of their electricity cost was spent for cooling their home. If that amount is high and the remaining (non-weather related) amount was kept at normal levels, then the consumers would have enough evidence to realize that the increase in their bill was caused by the unusual weather conditions during that month. Identifying the impact of weather factors in electricity bills has many uses. Consumers can use this information to take specific actions to reduce their consumption and thereby their bill. For example, by reducing the target temperature of a normal A/C by two degrees, consumers can reduce electricity consumption by 10%, without sacrificing much of their comfort. Other uses include the installation of double-glazing windows, better thermal insulation of their entire home, and use of energy efficient light bulbs.

Every household needs a certain basic amount of energy to function properly throughout the year. That energy, usually called the baseline load, mostly includes use of electricity for cooking, lighting, washing and drying machines. Cooling equipment, such as air conditioning and electric fans, and heating equipment, such as electric heaters, are used at certain seasons of the year and depend on the weather conditions of the region where the household exists. Therefore the influence/impact of the weather on an electricity bill can be quantified if the amount of energy used by heating and cooling equipment in a household can be estimated.

Baseline electric loads needed by a household are usually determined during the months of May and September-October. This is because the outside temperatures during those months are generally between 65-70°F and therefore there is no need to use any heating or cooling equipment in a house. The temperatures between 65-70°F are called base temperatures because they are used to identify the baseline energy loads needed in a household. Different building types may have different base temperatures depending on the insulation material used. The most frequently used base temperature is 65 °F, since it can capture the heating and cooling requirements of a large portion of buildings. In addition a large portion of humans feel comfortable within a building when the outside temperature is 65°F. When the outside temperature is higher than the base temperature, cooling equipment may be used so that the inside temperature of a building is comfortable for its residents. Similarly, when the outside temperature is lower than 65°F, heating equipment may be used so that the temperature inside the building remains at comfortable levels for its inhabitants. This extra amount of energy used for heating and cooling the building when the temperature shifts above and below the base temperature is responsible for fluctuations in electricity bills during different seasons.

Summary

Exemplary embodiments of the disclosure as described herein generally include methods and systems for a parametric multiple regression model that can incorporate different weather factors and identify their impact on an electricity bill. Exemplary, non-limiting computer implementation of a model according to an embodiment of the invention uses Java and was tested using electricity and weather data from three different cities in the United States. The correlation between the temperature and the electricity consumption are identified, and linear and quadratic models are built that describe the relationship between them. The results show that a model according to an embodiment of the invention can accurately identify the portion of an electricity bill spent on heating or cooling purposes. An implementation according to an embodiment of the invention is highly scalable and can handle tens of thousands of households in different geographic locations.

According to an embodiment of the disclosure, there is provided a method for determining the impact of weather on electricity costs, including executing at least one regression model to calculate regression coefficients for one or more regressor variables from a set of weather and energy usage data, and splitting energy usage associated with weather from energy usage not associated with weather based on the regression coefficients and confidence intervals calculated for the one or more regression models. The at least one regression model is expressed as y = Χβ + ε, wherein y is an ^-dimensional vector of energy usage samples wherein n is a number of samples, β is a (k + l)-vector of regression coefficients wherein β = (βο, βι, ... , y¾), is an n x (k + 1) matrix of samples of one of the regressor variables, and ε = (si, 82, ... , ε„) is an ^-dimensional vector of random errors. The one or more regressor variables include temperature, heating degree days (HDDs), and cooling degree days (CDDs), wherein HDDj = max{ Tb - Tj, 0}, CDDj = max{ Z - Tb, 0}, is an average temperature for day /^', and Tb is a baseline temperature selected to separate an amount of energy used for heating and from an amount of energy used for cooling, and the energy usage not associated with weather is a total energy usage minus the energy usage associated with weather.

According to a further embodiment of the disclosure, the at least one regression models is a linear model, for which k=\, wherein energy usage associated with weather is a total amount of energy used for heating and cooling over N days, expressed as E_weather = X ^1=1 HDDi. + β X ∑i=l CDDi, wherein is a first order regression coefficient for the HDDs, and β is a first order regression coefficient for the CDDs.

According to a further embodiment of the disclosure, the at least one regression models is a quadratic model, for which k=2, for each of the temperature, HDDs, and CDDs.

According to a further embodiment of the disclosure, the method includes preparing weather and energy usage data for statistical calculations by obtaining daily temperature averages from hourly temperature data, and calculating HDDs and CDDs from temperature data, and reporting the energy usage associated with weather to a user.

According to a further embodiment of the disclosure, X is defined as

X = and a solution vector β= {X^TX)^'lX^Ty minimizes (y - XP (y -

Χβ).

According to a further embodiment of the disclosure, the method includes calculating a covariance matrix and errors associated with the regression coefficients from the at least one regression model, wherein errors are calculated as differences between actual sample values y and estimated sample values y = Χβ, and the covariance is calculated as οον(β) = c²(X^TX}^'1 wherein where σ² is the variance of the coefficients, estimated as d² = ^^1-1^¹—

n- Qc+ l ^) ~.

According to a further embodiment of the disclosure, the method includes calculating confidence intervals for true values of the regression coefficients, wherein a \00(\-a)% confidence interval for the regression coefficient ¾ is calculated from j - ta/2;n-(fc+ l)

wherein is a (j, j)^th of the covariance matrix, and t represents the Student's i-distribution.

According to a further embodiment of the disclosure, the at least one regression model is expressed as

+ ^ε, wherein ^χ _ί-; is a sample of one of the regressor variables at j time points before a current value of x, wherein the j time points are measured each hour.

According to a further embodiment of the disclosure, the one or more regressor variables include temperature include humidity and wind chill factor.

According to a another embodiment of the disclosure, there is provided a method for determining the impact of weather on electricity costs, including executing at least one regression model to calculate regression coefficients for one or more regressor variables from a set of weather and energy usage data, wherein the at least one regression model is expressed as y = Χβ + ε, wherein y is an ^-dimensional vector of energy usage samples wherein n is a number of samples, β is a (k + l)-vector of regression coefficients wherein β = (βο, βι, ... , y¾), is an n x (k + 1) matrix of samples of one of the regressor variables defined as

X = vector of random

errors, wherein a solution vector β= {X X) Xy minimizes (y - Χβ) (y - Χβ). The one or more regressor variables include temperature, heating degree days (HDDs), and cooling degree days (CDDs), HDDj = max{ T_b— T 0}, CDDj = max{ Z - 7¾, 0}, Γ, is an average temperature for day /^', and T_b is a baseline temperature selected to separate an amount of energy used for heating and from an amount of energy used for cooling, and the energy usage not associated with weather is a total energy usage minus the energy usage associated with weather.

According to a further embodiment of the disclosure, the at least one regression models is a linear model y = βο + βι%ι + ε, for which k=\, for each of the temperature, HDDs, and CDDs, wherein energy usage associated with weather is a total amount of energy used for heating and cooling over N days, expressed as E_weath_er ⁼

+ βΐ CDDi, wherein is a first order regression coefficient for the HDDs, and is a first order regression coefficient for the CDDs.

According to a further embodiment of the disclosure, the method includes splitting energy usage associated with weather from energy usage not associated with weather based on the regression coefficients calculated for the one or more regression models.

According to a further embodiment of the disclosure, the at least one regression models is a linear model y = βο + βιΧι + β2*2+ ε, for which k=2 and = x\ , for each of the temperature, HDDs, and CDDs.

According to another embodiment of the disclosure, there is provided a non-transitory program storage device readable by a computer, tangibly embodying a program of instructions executed by the computer to perform the method steps for determining the impact of weather on electricity costs.

Brief Description of the Drawings

FIG. 1 is a flowchart of the main components of an energy usage analysis engine according to an embodiment of the invention.

FIG. 2 is a flowchart of an analytics engine according to an embodiment of the invention. FIGS. 3A-C are plots of the daily data between the temperature and the electricity usage for Bastrop, Texas, including fitting a linear model and a quadratic model, according to an embodiment of the invention.

FIGS. 4A-C are plots of the daily data between the heating degree days (HDD) and the electricity usage for Bastrop, Texas, including fitting a linear model and a quadratic model, according to an embodiment of the invention.

FIGS. 5A-C are plots of the daily data between the cooling degree days (cDD) and the electricity usage for Bastrop, Texas, including fitting a linear model and a quadratic model, according to an embodiment of the invention.

FIGS. 6A-C are plots of the daily data between the temperature and the electricity usage for Washington D.C., including fitting a linear model and a quadratic model, according to an embodiment of the invention.

FIGS. 7A-C are plots of the daily data between the heating degree days (HDD) and the electricity usage for Washington D.C., including fitting a linear model and a quadratic model, according to an embodiment of the invention.

FIGS. 8A-C are plots of the daily data between the cooling degree days (cDD) and the electricity usage for Washington D.C., including fitting a linear model and a quadratic model, according to an embodiment of the invention. FIGS. 9A-C are plots of the daily data between the temperature and the electricity usage for the first dataset from Lawrence, Kansas, including fitting a linear model and a quadratic model, according to an embodiment of the invention.

FIGS. lOA-C are plots of the daily data between the heating degree days (HDD) and the electricity usage for the first dataset from Lawrence, Kansas, including fitting a linear model and a quadratic model, according to an embodiment of the invention.

FIGS. 11 A-C are plots of the daily data between the cooling degree days (cDD) and the electricity usage for the first dataset from Lawrence, Kansas, including fitting a linear model and a quadratic model, according to an embodiment of the invention.

FIGS. 12A-C are plots of the daily data between the temperature and the electricity usage for the second dataset from Lawrence, Kansas, including fitting a linear model and a quadratic model, according to an embodiment of the invention.

FIGS. 13 A-C are plots of the daily data between the heating degree days (HDD) and the electricity usage for the second dataset from Lawrence, Kansas, including fitting a linear model and a quadratic model, according to an embodiment of the invention.

FIGS. 14A-C are plots of the daily data between the cooling degree days (cDD) and the electricity usage for the second dataset from Lawrence, Kansas, including fitting a linear model and a quadratic model, according to an embodiment of the invention. FIGS. 15A-C are plots of the daily data between the temperature and the electricity usage for the third dataset from Lawrence, Kansas, including fitting a linear model and a quadratic model, according to an embodiment of the invention.

FIGS. 16A-C are plots of the daily data between the heating degree days (HDD) and the electricity usage for the third dataset from Lawrence, Kansas, including fitting a linear model and a quadratic model, according to an embodiment of the invention.

FIGS. 17A-C are plots of the daily data between the cooling degree days (cDD) and the electricity usage for the third dataset from Lawrence, Kansas, including fitting a linear model and a quadratic model, according to an embodiment of the invention.

FIG. 18 is a block diagram of an exemplary computer system for implementing a method and user interface (UI) for effective video surveillance, according to an embodiment of the disclosure.

Detailed Description of Exemplary Embodiments

Exemplary embodiments of the disclosure as described herein generally include methods and user interfaces (UI) for effective video surveillance. Accordingly, while the disclosure is susceptible to various modifications and alternative forms, specific embodiments thereof are shown by way of example in the drawings and will herein be described in detail. It should be understood, however, that there is no intent to limit the disclosure to the particular forms disclosed, but on the contrary, the disclosure is to cover all modifications, equivalents, and alternatives falling within the spirit and scope of the disclosure. Statistical Background

In this section are presented the main statistical notions used in an analysis according to an embodiment of the invention. The correlation coefficient, denoted by R, is a measure of the strength of a linear relationship between two variables x and y, such as temperature and electricity consumption. The correlation coefficient is calculated from n pairs of observations on the two variables x and y. That is, if there is a sample of observations

(xi,yi), (x₂,y2), · · · , (xn,y_n), then the correlation coefficient R of the variables x and y is given by the formula

where σ_χ and o_y represent the standard deviations of the x and y samples, respectively, and are defined as follows:

The correlation coefficient has some important properties:

• The value of R is always in the interval [-1, 1].

• The magnitude of R provides information about the strength of the linear relationship between x and y. In addition its sign indicates the direction or trend followed by the observations. In particular: if R > 0, then the pattern of the data (x_;, yi) has an increasing trend; if R < 0, then the pattern of the data (x_;, y,) has a decreasing trend; if R = +1 , then all sample points (x_;, y,) lie exactly on a straight line with positive slope; and if R = -1, then all sample points (x_;, y,) lie exactly on a straight line with negative slope.

• A value of R relatively close to 0 means that the linear relationship between the variables x and y is weak.

The correlation coefficient is close to 0 when there is no clear pattern of any relationship, that is, the values of the y variable do not change in a specific direction as the values of x change. In practice, a safe conclusion cannot be drawn from any relationship between x and y when R6(-0.5, 0.5).

If the relationship between variables x and y is linear, then the variables may be connected by the formula:

where βο is called the intercept of the line with the -axis, and βι represents the slope of the line. The slope βι can also be thought of as the change in y per unit change in x. Estimates of the values of βο and βι may be calculated by using sample data. More specifically βι may be calculated by the formula: where

$xx ∑i=i(-^i is the sum of squared deviations from the mean x, and

is the sum of the cross products of the deviations. In addition, an estimate of the intercept β is determined by the formula βο = y - βιΧ, where x and y are the sample means of the x and y variables, respectively.

The estimates β₀ and β can then be used to locate the best linear fit, defined as: 9 = βο + βιΧ-

The line defined above provides the best fit to the data in the sense that the error defined by the sum of squares of the deviations: e = ∑?=_(y< - yd² = ∑ {vi - to - xi)². is minimized. The magnitude of the error reveals information on the divergence of the observed data from the linear model. A small error means that most of the observations are scattered close to the linear model.

Data Description

The data used in a statistical analysis according to an embodiment of the invention can be classified as primitive, which includes temperature, relative humidity, electricity usage, etc., and derived, which includes heating and cooling degree Days. eMeter, a software company that specializes in software for managing smart grid and smart meter technologies that is now owned by Siemens AG, has provided premitive data from three different geograpical areas of the United states: Bastrop, Texas; Washington, D.C.; and Lawrence, Kansas. These datasets provide access to hourly/daily electricity demand as well as hourly figures of weather parameters such as temperature, relative humidity, etc.

A statistical analysis according to an embodiment of the invention uses daily and monthly demand. The daily electricity demand, DE_t, can be calculated by summing the hourly electricity loads for each day, whereas the monthly demand, ME_t, can be estimated as the sum of the daily consumption. The daily temperature, DT_t, can be calculated as the sum of the hourly temperatures divided by 24, and the monthly temperature, MT_t, can be calculated as the sum of the daily temperatures divided by the number of days in the month.

Similar figures can be obtained for daily and monthly relative humidity, denoted by DRH_t and MRH_t, respectively. It has been well documented in the literature that relative humidity incorporates both absolute humidity and temperature, and as a result, relative humidity and temperature are not strictly independent. There are many models that select relative humidity as an independent variable based on the fact that consumers react to perceived comfort which is better described by relative humidity rather than absolute humidity.

In addition, an analysis according to an embodiment of the invention also uses heating and cooling Degree Days, denoted by HDD_t and CDD_t, respectively. HDD_t and CDD_t are defined as

HDA = max{ T_b - T, 0}, (1)

CDD_t = max{ T_t - T_b, 0}, (2) where T_t is the average temperature for the day t, and Tb represents the base temperature that is selected in such a way that separates the amount of energy spent for heating and cooling. The most commonly used values for the base temperature T_b are selected in the range (63 °F, 68°F), depending on the region where the measurements are obtained. Several values of Tb have been in the above mentioned range, and the results are relatively insensitive to the particular choice. The base temperature can be thought of as the range where no significant amounts of energy are consumed for heating and cooling activities.

If the HDD_t value is positive, then energy is used for heating purposes during the time period t. Similarly, if the CDD_t is positive, energy is used for cooling purposes. The magnitude of HDD_t can also be used to relate the cost of heating with the temperature. That is, the larger the value of HDD_t the colder the outside temperature is, and as a result, more space heating is used. In addition, analyses according to embodiments of the invention may also use monthly HDD and CDD, which are defined as

MHDD^ max^^ HDDt i - Y^₌₁ CDD_{t il} 0}, (3) MCDD_t =

0}, (4) where m is the number of days in a given month.

It is rather common to have hourly data for weather factors, such as temperature, relative humidity, etc., and electricity usage. To perform an analysis data preparation, the hourly values may be converted into daily values. This can be performed by calculating an average temperature of each day from the hourly values:

Daily Temp = —

24- W_ t_\ Hourly _ Tem.pi.

Similarly calculations may be performed for the relative humidity, and other weather related factors. Once the daily temperatures have been determined, the heating degree days {HDD) and cooling degree days (CDD), for every day, may be calculated from EQS. (1) and (2), respectively.

If there is information about the billing cycle, then the weather factors and usage data can be assigned to those cycles. This can be used to analyze billings that occur in the same month or the same season and identify patterns in the consumption of electricity and the influence of weather factors. For simplicity, embodiments of the invention assume that the bills coincide with months, however, this billing cycle choice is exemplary and non-limiting. Energy Usage Analysis Engine

FIG. 1 is a flowchart of the main components of an energy usage analysis engine according to an embodiment of the invention. Referring now to the figure, weather and energy data read from databases or files is prepared at step 11 so that it can be used in specific statistical calculations. The data may be recorded in different levels of granularity. For example energy usage values may be recorded on an hourly or daily basis, but temperatures may have daily values. For example, if energy is provided for every hour, the average daily energy value is calculated. In addition, the heating and cooling degree days are calculated as described above. Once all the data have been formed, the data is provided at step 12 as input to an analytics engine, which calculates appropriate statistical models. These models are used to determine the amount of energy used for weather related activities. More details about the analytics engine are described in the next section. The results are may be reported to the users at step 13.

According to an embodiment of the invention, an analytics engine performs a basic statistical analysis of the data based on a selection of regressor variables. According to an embodiment of the invention, temperature and heating/cooling degree days, i.e., HDD and CDD, will be used to determine the amount of energy consumed for heating and cooling. Note that this choice of regressor variables is exemplary and non-limiting, and other choices may be used in other embodiments of the invention. A flowchart of an analytics engine according to an embodiment of the invention is depicted in FIG. 2. Referring now to the figure, a method begins at step 21 by calculating two sets of regression models: a set of linear models relating usage to temperature, heating and HDD, and cooling and CDD, respectively:

Usage¹ = /¾ + β[ Temp , (5a) Usage^h = β% + β£ HDD , (5b)

Usage^c = βξ + β{ CDD; (5c) and a set of quadratic models relating usage to temperature, heating and HDD, and cooling and CDD, respectively:

Usage¹ = βο + β Temp + β Temp² , (6a) Usage^h = β% + βζ HDD + β£ HDD² , (6b) sage⁰ = βξ + β{ CDD + β CDD²; (6c)

Usage^h contains electricity usage data that correspond to non-zero HDDs, i.e.,

Usage^h = { Usage, : HDD, > 0 } , (7) and similarly, Usage^c contains electricity usage data that correspond to non-zero CDDs, i.e., Usage^c = {Usage, : CDD, > 0} . (8)

A regression analysis according to an embodiment of the invention performed in step 21 can calculate values for the coefficients /¾, β , β , β , βξ, and β , and optionally β , β , and β The computational details for determining the regression coefficients are described below.

The covariance matrix and errors associated with the regression models are determined in step 22, and details are provided below. Since the values specified by the regression analysis are only estimates of the true values for the coefficients /¾, β , β , β^, /¾ , β{, β , β , and βξ according to embodiments of the invention, intervals and confidence levels may be specified at step 23 within which the true values of these parameters exists. Details of the computational procedures used to determine the lower and upper bounds of these intervals are provided below.

Finally, at step 24, the electricity usage associated with weather is split from the usage that is not related to weather. The coefficients βζ and βΊ play a role in determining the amount of energy consumed for heating and cooling. More specifically, β provides the daily amount of energy that is used for heating, when there is a unit of decrease in the temperature below the base temperature, e.g., below 65°F. Similarly, β{ provides the amount of energy that is used for cooling, when there is a unit of increase in the temperature above the base temperature, e.g., above 65°F.

To better understand the value of the coefficients

and βζ, consider an example. Assume that βζ = 1.5, βΊ = 2.1, and the base temperature is 65°F. On Monday, the outside temperature was 60°F and as a result HDDuonday = 5. During that day an electric heater was used to heat a building and maintain its inside temperature at 65 °F. The energy that was consumed was βζ HDD_Monday = 1.5 x 5 = 7.5 kWh. On Tuesday, the average outside temperature increased and it became 68°F and as a result CDD uesday = 3. During Tuesday, air conditioning was used to cool the building to maintain its inside temperature to 65°F. The energy that was consumed was βΊ x CDD uesday = 2.1 x 3 = 6.3 kWh.

The total amount of energy used for heating during a specified number of days, N, is given by: = ?i X ∑i_{= 1} HDD_t. (9)

Similarly, the total amount of energy used for cooling during a specified number of days, N, is given by:

E_C00lmg = i x ∑f_{= 1} CDDi. (io)

The sum of the heating and cooling energy provides the energy consumption related to the weather and is given by:

Eweather -^Aeaizng + E_c00n_ng. (1 1)

The amount of energy that is not attributed to weather conditions is given by:

Enon-weather Efotal E_weather_> ( 12) where E_totai is the total amount of electricity consumed by the building during N days, that is:

Etotal - ∑JLi ^i - (^{1 3})

Regression Coefficients

This section provides details on determining the coefficients of regression models according to embodiments of the invention. A regression process according to an embodiment of the invention is presented in its most general form. Suppose there are k regressors

( xi, x₂, x_k), from which the variable y is to be predicted. In addition, suppose there are n sets of observations ( Xi,i, Xi,2, ... , Xi.k, yd, for /^' = 1 , 2, ... , n.

A model according to an embodiment of the invention that relates the regressors and the response is y_t = βο + βιΧι,ι + βϊΧι,ι + ... + fikXi,k + ¾ for /^' = 1, 2, ... , n. (14)

EQ. (14) is a system of n equations with k+ 1 unknowns, the coefficients βο, βι,ι, y¾, - β^ The quantities ¾ represent random errors, such as noise. EQ. (14) can be written in a compact form as γ = Χβ + ε, where y and ε are ^-dimensional vectors y = ( yi, y2, y_n), ε = (¾ ¾ ··· , ¾), β is a (k + l)-vector β = (βο, βι, ... , βί), and is an n x (k + 1) matrix defined as

Embodiments of the invention can find a vector β, that satisfies the following optimization program: τηπ Κβ) = (γ - Χβ)^τ(γ - Χβ). (16) In other words, this is a least-squares estimators, β, of the unknown coefficients β. A solution of the optimization program (16) is β = {Χ^τΧ) Χ^τγ. (17)

Note that X^TX is a k + l)x k + 1) non-singular matrix of small dimension, since k is usually a small number. Hence finding an inverse matrix (X^TX)^'1 does not pose a serious computational burden. Exemplary, non-limiting values of k are 1, for a linear least squares, and 2, for a quadratic least squares.

Once β is determined, the regression model can be fit as

9 = Χβ. (18)

The difference between the actual observations, y, and the fitted/estimated values, y, is called the residual or the error, e, of the regression, i.e., e = y - y . (19)

According to other embodiments of the invention, the relationship between the regressors and the response variable may follow a nonlinear relationship. One way to address nonlinear relationships is through the use of polynomial regression models, defined as

Y = β₀ + β_{ι Χ} + β₂χ2 + ... _qXq + _e where q is called the degree of the polynomial. The coefficients βο, βι, β₂, ... , P_q can be determined similar to those defined in equation (14). In particular the matrix X now becomes:

When q=2, the above model is called quadratic and it is defined as

Y = β₀ + β χ + β₂χ² + e

Note that equations (6a)-(6c) are quadratic polynomials, and their coefficients can be determined by using the new matrix X defined by EQ. (20). Since the derivative of a quadratic model is non-constant, a quadratic model is useful for using hourly weather data, which can vary non- linearly over the course of a day, to predict energy usage on an hourly basis during the day.

Covariance Matrix and Errors

The variance of the estimated coefficients, β, can be expressed in terms of the elements of the of the matrix {X^TX)^~l. More specifically, according to embodiments of the disclosure, the covariance matrix of the regression coefficients can be defined as οον{β) = ^QfXy¹ = c C, (21) where σ² is the variance of the coefficients, and C = ( ¾)^_1. The covariance matrix, cov(/?), is a symmetric matrix whose j-t diagonal element represents the variance of /?_;, that is var(fij) = G²C_jj. In addition the (i, j)-t element of the matrix οον(β) represents the covariance between and j, that is

The variance σ² is estimated in terms of the sum of squares of the residuals:

SSE = ∑ ₌₁(y, - yd² = ∑?₌₁ ^². (22)

Hence, according to an embodiment of the invention, an estimate of the variance σ² can be defined by

SSE

σ^Λ = (23) n- (k+ l)

Confidence Intervals

According to an embodiment of the invention, it is useful to construct confidence interval estimates of the regression coefficients y¾, for i = 0, 1, ... , k+l . A 100(1 -a) % confidence interval for the regression coefficient ¾ is given by j - ta/2;n-(k+l) Cj ≤ j ≤ j + ^a/2;n-(k+l) fa Cj ' A) where t represents the Student's /-distribution. Statistical Analysis of Provided Data

Embodiments of the invention seek answers to the following questions:

• Is there any relationship between energy consumption and temperature in the datasets provided?

• If there is, how strong is that relationship?

According to an embodiment of the invention, several scatter plots were created of the temperature and electricity usage for the 3 cities, after which regression analysis was performed and models were indentified that can be used to predict future electricity usage by taking into account the temperatures in the corresponding cities.

1. Bastrop, Texas

1.1 Temperature vs Usage

FIG. 3A is a plot of the daily data between the temperature and the electricity usage. The correlation coefficient is l R^{l B}Temp-vs-Usage — — fi

The correlation coefficient measures the strength of the linear relationship that may exist between the temperature and the usage. A correlation coefficient with value close to 1 reveals a strong linear relationship with positive slope. On the other hand, a correlation value close to -1, indicates a strong linear relationship with negative slope. In the case of Bastrop dataset, the value of Rremp-vs-usage ^{= ~}0.4239 reveals that there is not a strong linear relationship between the temperature and electricity usage. This can be seen clearly in FIG. 3 A, where the plotted data seem to follow a nonlinear pattern.

FIG. 3B depicts the application of a linear regression. The error of the residuals is

B(I )

eTemp-vs-usage ⁼ 13.8984. The linear model has the following form:

Usage = 0.0255 Temp + 0.6066. Here, the positive sign indicates a general upward trend of the data. FIG. 3C depicts application of a quadratic model:

Usage = 0.0033 Temp² -0.4195 Temp + 14.81.

By fitting a quadratic, the error of the residuals decreases to S_jemp-vs- sage ⁼ 8.4477. 1.2 HDD vs Usage

A statistical analysis was also performed using heating and cooling degree (HDD) days. FIG.4 A is a plot of the daily data between HDD and usage. The correlation coefficient is

HDD-vs-Usage = 0.8065.

The fact that the value of R HDD-vs-Usage is close to 1 reveals a positive linear correlation of the

HDD and usage.

FIG. 4B depicts a fit of a linear regression model, defined as Usage = 0.0914 HDD + 0.8440. The error of the residuals is ^HDD-vs-usage ^~ 4.7521.

A quadratic model was also fitted to the data, as illustrated in FIG. 4C: Usage = 0.0015 HDD² - 0.0514 HDD + 1.0365.

B(2 )

In this case the residual error was slightly reduced to ^ DD-vs-usage ⁼ 4.6627. 1.3 CDD vs Usage

A similar analysis was performed analysis for the cooling degree days (CDD). FIG. 5A we have plotted the daily data between CDD and usage. The correlation coefficient is

^^■CDD-vs-Usage ⁼ 0.9358.

The fact that the value of Rc_DD-vs-usage is ^verY close to 1, reveals a strong linear relationship between CDD and electricity usage. In fact this is evident from the scatter diagram shown in FIG. 5A.

FIG. 5B depicts a fit of a linear regression model, defined as Usage = 0.14 CDD + 0.2596.

The error of the residuals is s_CDD-vs-usage ⁼ 8 886.

A quadratic model was also fitted to the data, as illustrated in FIG. 5C: Usage = -0.0033 CDD² + 0.2149 CDD + 0.1545. In this case the residual error was slightly reduced to s_CDD-vs-usage ⁼ 8.3141.

2 Washington, DC

2.1 Temperature vs Usage

FIG. 6A is a plot of the daily data between the temperature and the electricity usage. The correlation coefficient is nw

^Temp-vs- Usage 0.4239.

The value of RtEMP-vs-usage reveals that there is not a strong linear relationship between the temperature and electricity usage in this dataset. This can be seen clearly in FIG. 6A, where the plotted data seem to follow a nonlinear pattern.

FIG. 6B illustrates the application of a linear regression. The error of the residuals is

⁼ 7-65- The linear model has the following form:

Usage = -0.0109 Temp + 2.3317.

Here, the negative sign indicates a general negative trend of the data. By fitting a quadratic the error of the residuals decreases to e * _vs__{a e} = 5.65. The quadratic model is given by the following formula:

Usage = 0.0001 Temp² - 0.1251 Temp + 5.1509. 2.2 HDD vs Usage

A statistical analysis was also performed using the heating and cooling degree (HDD) days. FIG. 7A is a plot of the daily data between HDD and usage. The correlation coefficient is

^HDD-vs-Usage ⁼ 0.7435.

The fact that the value of R oD-vs-usage *^s close to 1 reveals a positive linear correlation of the HDD and usage.

FIG. 7B depicts a fit of a linear regression model, defined as

Usage = 0.0343 HDD + 1.0698.

The error of the residuals is e^oo-vs-usage ⁼ 5- 14.

A quadratic model was fit to the data in FIG. 7C, which is defined as Usage = 0.001 HDD² - 0.0074HD + 1.3757.

In this case the residual error was slightly reduced to e^^__vs__Usage = 4.71

2.3 CDD vs Usage

A similar analysis was also performed analysis for the Cooling Degree Days (CDD). Fig. 8A is a plot of the daily data between CDD and usage. The correlation coefficient is

RcDD-vs-Usage 0.4168. The fact that the value of RcDD-vs-usage is positive reveals an upwards tread. However, the fact that the value of RcDD-vs-usage is ^not close enough to 1 indicates that there is no strong linear relationship between CDD and electricity usage. In fact, the scatter diagram in FIG. 8A shows that there is little relation between the CDD and the usage. In this case the use of different regression models is not likely to reduce the error substantially.

FIG. 8B depicts a fit of a linear regression model, defined as

Usage = 0.0248 CDD + 1.4038.

The error of the residuals is pp_^> _vs__Usage = 3.0645.

As illustrated in FIG. 8C, a quadratic model was fit to the data, defined as Usage = -0.0006 CDD² + 0.0352 CDD + 1.3688.

In this case the residual error was slightly reduced to s^^__vs__Usage= 3.0592, as expected. Higher order models (up to degree 5) were tried, which did not substantially reduce the error.

3. Lawrence, Kansas

This section analyzes the dataset from Lawrence Kansas. Three different datasets were provided, with names:

• wsagel-l_w7u0.csv,

• wsagel-l_w7rg.csv, • usagel— I _wdxk.csv.

An analysis of each of those datasets is presented in the following subsections.

3.1 Dataset: usagel-l_w7u0.csv

3.1.1. Temperature vs Usage

FIG. 9A is a plot of the daily data between the temperature and the electricity usage. The correlation coefficient is

^Temp-vs-Usage

The small value of the correlation coefficient in this dataset reveals that there is no strong linear relationship between the temperature and the electricity usage. This can be seen clearly in FIG. 9A, where the plotted data fail to reveal any specific pattern. The positive sign, however, indicates a general upward trend of the data.

FIG. 9B depicts the application of a linear regression. The error of the residuals is ^eT^mp°-vs-usage⁼ 685.1331. The linear model has the following form:

Usage = 0.3409 Temp + 142.6864

By fitting a quadratic, shown in FIG. 9C, the error of the residuals decreases to ^T^mp°-vs-usage

650.6. The quadratic model is given by the following formula:

Usage = 0.0.219 Temp² - 2.0151 Temp + 197.155. The large values of the errors of the residuals show a significant deviation from a linear or quadratic relationship between the temperature and the electricity usage. Large residual errors were observed even when fitting polynomials of degree up to 5.

3.1.2. HDD vs Usage

A statistical analysis was performed using the heating and cooling degree (HDD) days. FIG. 1 OA is a plot of the daily data between HDD and usage. The correlation coefficient is

which indicates that it is not possible to identify any specific relationship between HDD and electricity usage.

FIG. 1 OB illustrates a fit of a linear regression model, defined as

Usage = 0.3946 HDD + 144.4209.

The error of the residuals is ^e DD-vs-usage = 439.

FIG. 10C illustrates the fit of a quadratic model, defined as

Usage = -0.0163 HDD² + 1.2194HDD + 137.0344.

In this case the residual error was slightly reduced to SnoD^s-usage = 435 3.1.3. CDD vs Usage

A similar analysis was performed for the cooling degree days (CDD). FIG. 11 A is a plot of the daily data between CDD and usage. The correlation coefficient is uLw7uO _ r₎ fi on

nCDD -vs- Usage U. OZO / .

Note that in this case the correlation coefficient is closer to 1 than in the case of the Temperature and HDD. As a result the CDD seems to have a noticeable linear relationship with the electricity usage. This can be seen in FIG. 11 A.

FIG. 1 IB illustrates the fit of a linear regression model, defined as

Usage = 2.8849 CDD + 132.7424.

The error of the residuals is ^ecDD- s-usage

FIG. l lC illustrates the fit of a quadratic model, defined as Usage = 0.0577 CDD2 + 1.2601 CDD + 140.8451.

The error of the residuals is &cDD-vs-usage^~ 439.315, which is the same as the linear regression error.

3.2. Dataset: usagel-1 w7rg.csv

3.2.1. Temperature vs Usage

FIG. 12A is a plot of the daily data between the temperature and the electricity The correlation coefficient is

_RLw7rg _{= Q ηη}^

l^lTemp-vs-Usage

The value of the correlation coefficient in this dataset reveals a noticeable linear relationship between the temperature and the electricity usage, which can be seen in FIG. 12A. The positive sign indicates a general upward trend in the data. FIG. 12B depicts the application of a linear regression. The error of the residuals is ^Temp-vs-usage ⁼ 1957.7. The linear model has the following form:

Usage = 4.9833 Temp - 133.5788.

By fitting a quadratic, the error of the residuals decreases to ^Temp-vs-usage ⁼ 1361.9. The quadratic model is given by the following formula:

Usage = 0.1433 Temp² - 10.4434 Temp + 223.0613.

Note here that the large values of the residual errors indicate show a deviation from a linear and a quadratic relationship between the temperature and the electricity usage. Large residual errors were observed even when fitting polynomials of degree up to 5. 3.2.2. HDD vs Usage

A statistical analysis was performed using the heating and cooling degree (HDD) days. FIG. 13A is a plot of the daily data between HDD and usage. The correlation coefficient is n lw7rg _{= n 7} „

^HDD-vs-Usage which indicates that it is not possible to identify any specific relationship between HDD and electricity usage. The negative sign of the correlation coefficient reveals a downwards trend in the data.

FIG. 13B illustrates a fit of a linear regression model, defined as Usage = -0.7236 HDD + 79.5135.

The error of the residuals is

FIG. 13C illustrates the fit of a quadratic model, defined as

Usage = 0.0654 HDD² - 4.0364 HDD + 109.1802.

In this case the residual error was slightly reduced to e_HDD__vs__Usage = 530.51 3.2.3. CDD vs Usage

A similar analysis was performed for the cooling degree days (CDD). FIG. 14A is a plot of the daily data between CDD and usage. The correlation coefficient is n lw7rg = 0 71 78

^CDD-vs-Usage u. ^/ ± ^/ o. As a result the CDD seems to have a noticeable linear relationship with the electricity usage, which can be seen in FIG. 14A.

FIG. 14B illustrates a fit of a linear regression model, defined as

Usage = 12.7654 CDD + 131.2998.

The error of the residuals is ^cDD-vs-usage ⁼ 579.3239.

FIG. 14C illustrates the fit of a quadratic model, defined as

Usage = -0.1777 CDD² + 17.7737CDD + 106.3233.

The error of the residuals is e_CDD__v ⁽ _s__Usage = 579.3239, which is the same as the linear regression error.

3.3. Dataset: usagel-1 wdxk.csv

3.3.1. Temperature vs Usage

FIG. 15A is a plot of the daily data between the temperature and the electricity usage. The correlation coefficient is Lwdxk —

nTemp-vs-Usage ou ^/ o.

The value of the correlation coefficient in this dataset reveals that there seems to be a noticeable linear relationship between the temperature and the electricity usage, which can be seen in FIG. 15 A. The positive sign indicates a general upwards trend in the data. FIG. 15B illustrates a fit of a linear regression model. The error of the residuals is ^eT mp^vs-usage ⁼ 75.78. The linear model has the following form:

Usage = 0.2935 Temp - 0.0221.

By fitting a quadratic, the error of the residuals decreases to Sremp^vs-usage ⁼ 61.613. The quadratic model is given by the following formula:

Usage = 0.0068 Temp² - 0.5097 Temp + 21.33.

3.3.2. HDD vs Usage

A statistical analysis was performed using the heating and cooling degree (HDD) days. FIG. 16A is a plot of the daily data between HDD and usage. The correlation coefficient is u Lwdxk _ 0 1 4

nHDD-vs-Usage ^"υ· which indicates that there is a linear relationship between HDD and electricity usage. The negative sign of the correlation coefficient reveals a downwards trend in the data.

FIG. 16B illustrates a fit of a linear regression model, defined as

Usage = -0Λ57 HDD + 16.6287.

The error of the residuals is e^D^X-vs-usage ⁼ 24.397.

FIG. 16C illustrates the fit of a quadratic model, defined as

Usage = -0.0009 HDD² -0.1196 HDD + 16.35. In this case the residual error was slightly reduced to e_HDD_^_s__Usage= 24.33. 3.3.3. CDD vs Usage

A similar analysis was performed for the cooling degree days (CDD). FIG. 17A is a plot of the daily data between CDD and usage. The correlation coefficient is n Lwdxk

nCDD-vs-Usage

As a result the CDD seems to have a noticeable linear relationship with the electricity usage, which can be seen in FIG. 17A.

FIG. 17B illustrates a fit of a linear regression model, defined as

Usage = 0.8141 CDD + 11.89. error of the residuals is e_CDD__vs__Usage

FIG. 17C illustrates the fit of a quadratic model, defined as

Usage = 0.0356 CDD² - 0.2013 CDD + 17.122.

The error of the residuals is e DD-VS-U sage ^~ 24.3194, substantially the same as the linear case.. System Implementations

It is to be understood that embodiments of the present invention can be implemented in various forms of hardware, software, firmware, special purpose processes, or a combination thereof. In one embodiment, the present invention can be implemented in software as an application program tangible embodied on a computer readable program storage device. The application program can be uploaded to, and executed by, a machine comprising any suitable architecture.

FIG. 18 is a block diagram of an exemplary computer system for implementing a XXXX according to an embodiment of the invention. Referring now to FIG. 18, a computer system 181 for implementing the present invention can comprise, inter alia, a central processing unit (CPU) 182, a memory 183 and an input/output (I/O) interface 184. The computer system 181 is generally coupled through the I/O interface 184 to a display 185 and various input devices 186 such as a mouse and a keyboard. The support circuits can include circuits such as cache, power supplies, clock circuits, and a communication bus. The memory 183 can include random access memory (RAM), read only memory (ROM), disk drive, tape drive, etc., or a combinations thereof. The present invention can be implemented as a routine 187 that is stored in memory 183 and executed by the CPU 182 to process the signal from the signal source 188. As such, the computer system 181 is a general purpose computer system that becomes a specific purpose computer system when executing the routine 187 of the present invention.

The computer system 181 also includes an operating system and micro instruction code. The various processes and functions described herein can either be part of the micro instruction code or part of the application program (or combination thereof) which is executed via the operating system. In addition, various other peripheral devices can be connected to the computer platform such as an additional data storage device and a printing device.

It is to be further understood that, because some of the constituent system components and method steps depicted in the accompanying figures can be implemented in software, the actual connections between the systems components (or the process steps) may differ depending upon the manner in which the present invention is programmed. Given the teachings of the present invention provided herein, one of ordinary skill in the related art will be able to contemplate these and similar implementations or configurations of the present invention.

While the present invention has been described in detail with reference to exemplary embodiments, those skilled in the art will appreciate that various modifications and substitutions can be made thereto without departing from the spirit and scope of the invention as set forth in the appended claims.

Claims

WHAT IS CLAIM IS;

1. A computer- implemented method for determining the impact of weather on electricity costs, the method implemented by the computer comprising the steps of:

executing at least one regression model to calculate regression coefficients for one or more regressor variables from a set of weather and energy usage data; and

splitting energy usage associated with weather from energy usage not associated with weather based on the regression coefficients and confidence intervals calculated for the one or more regression models,

wherein the at least one regression model is expressed as y = Χβ + ε, wherein

y is an ^-dimensional vector of energy usage samples wherein n is a number of samples, β is a (k + 1)- vector of regression coefficients wherein β = (βο, βι, ... , y¾),

is an n ^χ (k + 1) matrix of samples of one of the regressor variables, and

ε = (ει, ¾, ... , ε„) is an ^-dimensional vector of random errors,

wherein the one or more regressor variables include temperature, heating degree days (HDDs), and cooling degree days (CDDs),

wherein HDDj = max{ T_b - T, 0} , CDDj = max{ T - T_b, 0} ,

T is an average temperature for day /^', and

T_b is a baseline temperature selected to separate an amount of energy used for heating and from an amount of energy used for cooling, and the energy usage not associated with weather is a total energy usage minus the energy usage associated with weather.

2. The method of claim 1, wherein the at least one regression models is a linear model, for which k=\ , wherein energy usage associated with weather is a total amount of energy used for heating and cooling over N days, expressed as E_weath_er = X

HDDi. + βΐ X CDDi, wherein β oΐh is a first order regression coefficient for the HDDs, and is a first order regression coefficient for the CDDs.

3. The method of claim 1 , wherein the at least one regression models is a quadratic model, for which k=2, for each of the temperature, HDDs, and CDDs.

4. The method of claim 1 , further comprising preparing weather and energy usage data for statistical calculations by obtaining daily temperature averages from hourly temperature data, and calculating HDDs and CDDs from temperature data, and reporting the energy usage associated with weather to a user.

5. The method of claim 1 , wherein is defined as X =

and a solution vector β= (X X)^' Xy minimizes (y - Χβ) (y - Χβ).

6. The method of claim 1 , further comprising calculating a covariance matrix and errors associated with the regression coefficients from the at least one regression model, wherein errors are calculated as differences between actual sample values y and estimated sample values y = Χβ, and the covariance is calculated as οον(β) = σ²( ¾)^_1 wherein where σ² is the variance of the coefficients, estimated as d² = ^^1-1^¹— ~- n-ik+l)

7. The method of claim 1 , further comprising calculating confidence intervals for true values of the regression coefficients, wherein a \00(\-a)% confidence interval for the regression coefficient ¾ is calculated from ft - ta/2;n-(fc+l) _^J ^{2 C}j_>j ^≤ ft ^≤ ft ⁺ ¾n-(fc+l) J^²^-, wherein is a (J, j)^th of the covariance matrix, and t represents the Student's i-distribution.

8. The method of claim 1 , wherein the at least one regression model is expressed as y_t = βο + fiiXt-i + /¾-2 + ... + βρΧι-ρ + ε,

wherein x_t._j is a sample of one of the regressor variables at j time points before a current value of x, wherein the j time points are measured each hour.

9. The method of claim 1, wherein the one or more regressor variables include temperature include humidity and wind chill factor.

10. A computer- implemented method for determining the impact of weather on electricity costs, the method implemented by the computer comprising the steps of:

executing at least one regression model to calculate regression coefficients for one or more regressor variables from a set of weather and energy usage data, wherein the at least one regression model is expressed as y = Χβ + ε, wherein

y is an ^-dimensional vector of energy usage samples wherein n is a number of samples, β is a (k + l)-vector of regression coefficients wherein β = (βο, βι, ... , y¾),

is an n ^χ (k + 1) matrix of samples of one of the regressor variables defined as

ε = (si, 82, ... , ε„) is an ^-dimensional vector of random errors, wherein a solution vector β= (X^TX)^'lX^Ty minimizes (y - Χβ)^τ(γ - Χβ),

wherein the one or more regressor variables include temperature, heating degree days (HDDs), and cooling degree days (CDDs), HDD, = max{ T_b - T 0} , CDD, = max{ T_t - T_b, 0} ,

Tj is an average temperature for day /^', and

11. The method of claim 10, wherein the at least one regression models is a linear model y = βο + βι%ι + ε, for which k=l, for each of the temperature, HDDs, and CDDs,

wherein energy usage associated with weather is a total amount of energy used for heating and cooling over N days, expressed as E_weather = Pi X ∑_{i= 1} HDDi. + β X ∑_{j= 1} DDi, wherein

oh

βΐ is a first order regression coefficient for the HDDs, and is a first order regression coefficient for the CDDs.

12. The method of claim 10, further comprising splitting energy usage associated with weather from energy usage not associated with weather based on the regression coefficients calculated for the one or more regression models.

13. The method of claim 10, wherein the at least one regression models is a linear model y = βο + βιΧι + β2*2+ ε, for which k=2 and = x\ , for each of the temperature, HDDs, and CDDs.

14. A non-transitory program storage device readable by a computer, tangibly embodying a program of instructions executed by the computer to perform the method steps for determining the impact of weather on electricity costs, the method comprising the steps of: executing at least one regression model to calculate regression coefficients for one or more regressor variables from a set of weather and energy usage data; and

y is an ^-dimensional vector of energy usage samples wherein n is a number of samples, β is a (k + l)-vector of regression coefficients wherein β = (βο, βι, ... , y¾), is an n ^χ k + 1) matrix of samples of one of the regressor variables, and ε = (ει, 82, ... , ε„) is an ^-dimensional vector of random errors,

wherein HDDj = max{ T_b - T, 0}, CDDj = max{ T - T_b, 0},

Tj is an average temperature for day /^', and

15. The computer readable program storage device of claim 14, wherein the at least one regression models is a linear model, for which k=\, wherein energy usage associated with weather is a total amount of energy used for heating and cooling over N days, expressed as

E_Weather = β_\ X H D D(. + β£ X ∑Li C j, wherein β oΐh is a first order regression coefficient for the HDDs, and is a first order regression coefficient for the CDDs.

16. The computer readable program storage device of claim 14, wherein the at least one regression models is a quadratic model, for which k=2, for each of the temperature, HDDs, and CDDs.

17. The computer readable program storage device of claim 14, the method further comprising preparing weather and energy usage data for statistical calculations by obtaining daily temperature averages from hourly temperature data, and calculating HDDs and CDDs from temperature data, and reporting the energy usage associated with weather to a user.

18. The computer readable program storage device of claim 14, wherein is defined as X = -

Χβ).

19. The computer readable program storage device of claim 14, the method further comprising calculating a covariance matrix and errors associated with the regression coefficients from the at least one regression model,

wherein errors are calculated as differences between actual sample values y and estimated sample values y = Χβ, and the covariance is calculated as οον(β) = σ²(Ι¾)^_1 wherein where σ² is the variance of the coefficients, estimated as d² = ^'^" n¹-^Qc—+ l ^) ~.

20. The computer readable program storage device of claim 14, the method further comprising calculating confidence intervals for true values of the regression coefficients, wherein a 100(1 -a) % confidence interval for the regression coefficient ¾ is calculated from j - t_a/2;n-(k+l) J ² Cj,j ≤ j ≤ j + t_a/_2;n-(k+l) J ²Cj,j; wherein j j is a (/^', 7) of the covariance matrix, and t represents the Student's i-distribution.

21. The computer readable program storage device of claim 14, wherein the at least one regression model is expressed as

+ ^ε,

wherein χ_ί-; is a sample of one of the regressor variables at j time points before a current value of x, wherein the j time points are measured each hour.

22. The computer readable program storage device of claim 14, wherein the one or more regressor variables include temperature include humidity and wind chill factor.