Mean quadratic error

To create Regression Trees collect your known input data into a matrix X. Collect the responses to X in a response variable Y and perform following steps.

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t = classregtree(X, Y, ‘names, ′{‘x1′, ‘x2′, …})

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treetype = type(t) % tells that it is regression or classification tree

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view(t) % returns a text description of the tree view(t, ‘mode, ′‘graph′) % returns a graphic description of the tree.

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qetoler—Defines tolerance on quadratic error per node for regression trees. Splitting nodes stops when quadratic error per node drops below qetoler * qed, where qed is the quadratic error for the entire data computed before the decision tree is grown: qed = norm(y − ybar) with ybar estimated as the average of the input array Y. Default value is 1e − 6.

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The tree predicts the response values at the circular leaf (terminal) nodes based on a series of questions about the response at the nonterminal branching nodes. A true answer to any question follows the branch to the left; a false follows the branch to the right. For example, we can use the tree to predict the HIV prevalence when the mortality and progression has some fixed value.

predict(tree, [2200, 4, …]) % Predict the response variable when x1 = 2200, x2 = 4, ….

That is, suppose the points (x_i, y_i), (x₂;y₂), …, (x_c;y_c) are all the samples belonging to the leaf-node l. Then our model for l is just ŷ=1c∑i=1cyi, the sample mean of the response variable in that cell. This is a piecewise-constant model.

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We can use a variety of other methods of the classregtree class, such as cutvar, cuttype, and cutcategories, to get more information about the split at the node 3 (e.g., cutvar(t,3) % what variable determines the split?).

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cost = test(t, ′resubstitution′) computes the cost of the tree t using a resubstitution method. The cost of the tree is the sum over all terminal nodes of the estimated probability of a node times the cost of a node. If t is a regression tree, the cost of a node is the average squared error over the observations in that node. Cost is a vector of cost values for each subtree in the optimal pruning sequence for t. The resubstitution cost is based on the same sample that was used to create the original tree, so it under estimates the likely cost of applying the tree to new data.

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To find predictor importance, we use imp = predictorImportance(t)

(provides vector (of length as X) of values with higher value indicates that corresponding independent variable has most impact on the response variable)

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yfit = eval(t, X) takes a classification or regression tree t and a matrix X of predictors, and produces a vector yfit of predicted response values. For a regression tree, yfit(i) is the fitted response value for a point having the predictor values X(i, :).

The principle of the wavelet denoising can be summarized in the following procedure:

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Decomposition. Compute the wavelet coefficients s˜λ using the FWT.

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Thresholding. Apply the thresholding function ρ_ɛ to the wavelet coefficients s˜λ, thus reducing the relative importance of the coefficients with small absolute value.

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Reconstruction. Reconstruct a denoised version s_C from the thresholded wavelet coefficients using the fast inverse wavelet transform.

The thresholding parameter ɛ depends on the variance of the noise and on the sample size N. The thresholding function ρ we consider corresponds to hard thresholding:

[66]ρɛ(a)={aif|a|>ɛ0if|a|≤ɛ

Donoho and Johnstone (1994) have shown that there exists an optimal ɛ for which the relative quadratic error between the signal s and its estimator s_C is close to the minimax error for all signals s∈H, where H belongs to a wide class of function spaces, including Hölder and Besov spaces. They showed using the threshold

[67]ɛD=σn2lnN

yields an error which is close to the minimum error. The threshold ɛ_D depends only on the sampling N and on the variance of the noise σ_n; hence, it is called universal threshold. However, in many applications, σ_n is unknown and has to be estimated from the available noisy data s. For this, the present authors have developed an iterative algorithm (see Azzolini et al. (2005)), which is sketched in the following:

1.

Initialization

(a)

given sk,k=0,…,N−1. Set i=0 and compute the FWT of s to obtain s˜λ

(b)

compute the variance σ02 of s as a rough estimate of the variance of n and compute the corresponding threshold ɛ0=2lnNσ021/2;

(c)

set the number of coefficients considered as noise Nnoise=N.

2.

Main loop repeat

(a)

set Nnoise′=Nnoise and count the wavelet coefficients N_noise with modulus smaller than ɛ_i;

(b)

compute the new variance σi+12 from the wavelet coefficients whose modulus is smaller than ɛ_i and the new threshold ɛi+1=2lnNσi+121/2;

(c)

set i=i+1 until (Nnoise′==Nnoise).

3.

Final step

(a)

compute s_C from the coefficients with modulus larger than ɛ_i using the inverse FWT.

The procedure for examination and construction of a linear regression model consists of following steps [71,72].

(1)

Proposal of a model

The procedure should always start from the simplest model, with individual independent controllable variables not raised to powers other than the first, and with no interaction terms of the type x_jx_k included. Only in cases when it is known that the model contains functions of the controllable variables is an exception made.

(2)

Exploratory data analysis in regression

The scatter of individual variables and all possible pair combinations are examined. The scatter plots of x_j vs. x_k or the index plots x_j vs. j are often used here. In this step of a regression analysis, the significance of individual variables with reference to scatter and the presence of multicollinearity is examined. An approximately linear relationship between variables in scatter plots of x_j vs. x_k indicates multicollinearity. Also, in this step the influential points causing multicollinearity are detected.

(3)

Parameter estimation

The parameters of the proposed regression model and the corresponding basic statistical characteristics of this model are determined by the classical least-squares method (LS). Individual parameters are tested for significance by using the Student t-test, the determination coefficient R^2 and the predicted determination coefficient R^P2. Other statistical characteristics calculated are the total F-test, the model significance test, the model complexity test, the mean quadratic error of prediction MEP and the Akaike information criterion AIC, to examine the linearity of model.

(4)

Analysis of regression diagnostics

The statistical analysis of classical residuals leads to estimates of residual variance σ^2e^, residual standard deviation Se^, residual skewness g^1e^, residual kurtosis g^2e^, the Pearson χ²-test of residual normality and the Jarque-Berra normality test. Different diagnostic graphs are used to examine the regression diagnostics for identification of influential points, and to test the conditions for the least-squares method, namely homoscedasticity, absence of autocorrelation, and normality of error distribution. If influential points are found, it has to be decided whether these points should be eliminated from data. If points are eliminated, the whole data treatment must be repeated. When there are several controllable variables, the significance of each variable and its function is examined by partial-regression graphs and by the partial-residual graph.

(5)

Construction of a more accurate model

According to the test for fulfilment of the conditions for the least-squares method, and the result of regression diagnostics, a more accurate regression model is constructed as follows.

(a)

When heteroscedasticity is found in the data, the weighted least-squares method (WLS) is used.

(b)

When autocorrelation is found in the data, the generalized least-squares method (GLS) is used.

(c)

When some restrictions apply to the parameters, the conditioned least-squares method (CLS) is used.

(d)

When multicollinearity is found in the data, the method of rational ranks (MRV) is used.

(e)

When all variables are subject to random errors, the extended least-squares method (ELS) is used.

(f)

When the data have an error distribution other than normal, or the data contain outliers or high leverage points, some robust methods are used.

(6)

Evaluation of the quality of the model proposed

With the use of classical tests, regression diagnostics and some supplementary information about the “model + data + method”, the quality of the proposed linear regression model is evaluated.

(7)

Analysis of calibration models

For a calibration model proposed for the given signal value y
*,the quality of the independent variable x
* together with its confidence interval is estimated. Before application of the calibration model, the detection limit and the determination limit should be estimated. These limits determine the allowable lower limit of the calibration model.

(8)

Statistical hypothesis testing

In some cases, to compare several straight lines, statistical hypothesis testing is performed.

These criteria use colors coded in the RGB acquisition color space to estimate the fidelity of the demosaiced image compared with the original image.

1

Mean Absolute Error

This criterion evaluates the mean absolute error between the original image I and the demosaiced image Î. Denoted by MAE, it is expressed as follows (Chen et al., 2008; Li and Randhawa, 2005):

(82)MAE(I,Iˆ)=13XY∑k=R,G,B∑x=0X−1∑y=0Y−1|Ix,yk−Iˆx,yk|,

where Ix,yk is the level of the color component k at the pixel whose spatial coordinates are (x,y) in the image I. X and Y are, respectively, the number of columns and rows of the image.

The MAE criterion can be used to measure the estimation errors of a specific color component. For example, this criterion is evaluated on the red color plane as

(83)MAER(I,Iˆ)=1XY∑x=0X−1∑y=0Y−1|Ix,yR−Iˆx,yR|.

MAE values range from 0 to 255, and the lower MAE value, the better demosaicing quality.

2

Mean Square Error

This criterion measures the mean quadratic error between the original image and the demosaiced image. Denoted by MSE, it is defined as follows (Alleysson et al., 2005):

(84)MSE(I,Iˆ)=13XY∑k=R,G,B∑x=0X−1∑y=0Y−1(Ix,yk−Iˆx,yk)2.

The MSE criterion can also measure the error on each color plane, as in Eq. (23). The optimal quality of demosaicing is reached when MSE is equal to 0, whereas the worst is measured when MSE is close to 255².

3

Peak Signal-to-Noise Ratio

The PSNR criterion is a widely used distortion measurement to estimate the quality of image compression. Many authors (e.g., Alleysson et al., 2005; Hirakawa and Parks, 2005; Lian et al., 2007; Wu and Zhang, 2004) use this criterion to quantify the performance reached by demosaicing schemes. The PSNR is expressed in decibels as

(85)PSNR(I,Iˆ)=10⋅log10(d2MSE(I,Iˆ)),

where d is the maximum color component level. When the color components are quantized with 8 bits, d is set to 255.

Like the preceding criteria, PSNR can be applied to a specific color plane. For the red color component, it is defined as

(86)PSNRR(I,Iˆ)=10⋅log10(d2MSER(I,Iˆ)).

The higher the PSNR value, the better the demosaicing quality. The PSNR measured on demosaiced images generally ranges from 30 to 40 dB (i.e., MSE ranges from 65.03 to 6.50).

4

Correlation

A correlation measurement between the original image and the demosaiced image is used by Su and Willis (2003) to quantify the demosaicing performance. The correlation criterion between two grey-level images I and Î is expressed as

(87)C(I,Iˆ)=|(∑x=0X−1∑y=0Y−1Ix,yIˆx,y)−XYμμˆ[(∑x=0X−1∑y=0Y−1Ix,y2)−XYμ2]1/2[(∑x=0X−1∑y=0Y−1Iˆx,y2)−XYμˆ2]1/2|,

where μ and μˆ are the mean grey levels in the two images.

When a color demosaiced image is considered, the correlation level C^k (I^k,Î^k), k ∈ {R,G,B} between the original and demosaiced color planes is estimated. The mean of the three correlation levels is used to measure the quality of demosaicing. The correlation levels C range between 0 and 1, and a measurement close to 1 can be considered a satisfying demosaicing quality.

Theoretically, the permeability should vary with pressure according to [11]

(1)d=d∞1+μp.

(2)μ=1R32η3π2πR0TM,

(3)R=43Rsρ1–ρ

(c) In Maple I used the following code for the default example f(x)=ex, a=0:

(c) In Maple we can use the following modification of the above code:

Bonus question: We can use the formula for summing up geometric series:

1R+h=1R11−(−hR)=1R⋅(1+(−hR)+(−hR)2+(−hR)3+⋯).

Now we see why we can ignore higher powers when h is negligible compared to R, since then the higher powers of the very small number −rR go to zero fast.

Suppose we continuously monitor the output of a quantum system whose evolution is described by Eq. (A.11) via homodyne detection, yielding a measurement of the field quadrature Y(t)=Aout(t)+Aout†(t) [where A_out is defined in (A.25)]. We denote by Yt={Y(t′):0≤t′≤t} the algebra generated by the observation process Y and by Yt′ its commutant, Yt′={A:[A,Y]=0,∀Y∈Yt}. For 0 ≤ t′≤ t we then have the nondemolition property (Bouten et al., 2007)

(B.22)[jt(X),Y(t′)]=0

[such that jt(X)∈Yt] and the self-nondemolition property

(B.23)[Y(t),Y(t′)]=0.

This allows us to map the quantum filtering problem to a classical filtering problem which we can tackle by methods from classical nonlinear filtering theory.

We define the quantum state E[⋅]=tr(ρ0⊗ρvac⋅) and seek to find an estimate πt(X)∈Yt for the Heisenberg operator X(t) = j_t(X) which minimizes the quadratic error E[(πt(X)−jt(X))2] based on the observations Y (t). As this requirement is equivalent to Eq. (B.21) we directly find

(B.24)πt(X)=E[jt(X)|Yt].

We will now show how to calculate π_t(X) explicitly using the characteristic function technique (Gough and Kostler, 2010), for the case of a single vacuum input. Due to the orthogonality property (B.20) of the conditional expectation we have the identity E[(πt(X)−jt(X))Ct]=0 for all Ct∈Yt. We can thus make the following Ansatz

(B.25a)dCt=gtCtdY(t),

(B.25b)dπt(X)=αtdt+βtdY(t),

for adapted stochastic processes αt,βt∈Yt. This implies Ct,πt(X)∈Yt [while on the other hand jt(X)∉Yt]. We can then use dE[(πt(X)−jt(X))Ct]=0, which, due to Itō-rules consists of three terms. We find for the first term

I=E[(dπt(X)−djt(X))Ct]=E[αtCt]dt+E[βtdY(t)Ct]−E[djt(X)Ct]=E[αtCt]dt+E[βtjt(s+s†)Ct]dt−E[jt(L*X)Ct]dt=E[αtCt]dt+E[βtπt(s+s†)Ct]dt−E[πt(L*X)Ct]dt,

where to go from the second to the third line we used (i) the fact that Y is generated by a Wiener process and therefore the increments dY (t) are independent of Y (t) at equal times, i.e., E[dY(t)f(Y(t))]=E[E[dY(t)]f(Y(t))] for a measureable function f, (ii) E[dY(t)]=E[jt(s+s†)]dt, and (iii) for the field initially in vacuum we have E[dA(t)]=0 and E[dA†(t)]=0. In particular this means that E[βtdA(t)Ct]=E[βtdA†(t)Ct]=0. To go to the fourth line we used the orthogonality property, thus E[jt(X)Ct]=E[πt(X)Ct]. For the second term we find

II=E[(πt(X)−jt(X))dCt]=E[(πt(X)−jt(X))gtCtjt(s+s†)]dt=E[πt(X)πt(s+s†)gtCt]dt−E[πt(X(s+s†))gtCt]dt

by the same reasoning as above. Going from the second to the third line we used (B.19c), which implies π_t(π_t(X)j_t(s)) = π_t(X)π_t(s), and j_t(XY ) = j_t(X)j_t(Y ). For the third term we obtain by using the Itō table (A.14)

III=E[(dπt(X)−djt(X))dCt]=E[(βt(X)dY(t)−djt(X))gtCtdY(t)]=E[βt(X)gtCt]dt−E[jt([s†,X])gtCt]dt=E[βt(X)gtCt]dt−E[πt([s†,X])gtCt]dt.

As g_t is an arbitrary function we can deduce from I + II + III = 0 by equating the coefficients of g_t and g_tC_t that

0=αt+βtπt(s+s†)−πt(L*X),0=πt(X)πt(s+s†)−πt(X(s+s†))+βt−πt([s†,X])=πt(X)πt(s+s†)−πt(Xs+s†X)+βt,

where both equalities hold almost surely with respect to E. Solving this for α_t and β_t, and using the ansatz (B.25b) we find for the quantum filter

(B.26)dπt(X)=πt(L*X)dt+[πt(Xs+s†X)−πt(X)πt(s+s†)]dW,

where

(B.27)dW(t)=dY(t)−πt(s+s†)dt

is the innovations process. W(t) is a Wiener process with E[dW]=0, dW² = dt and therefore E[πs(X)dW(t)]=0 for 0 ≤ s ≤ t. The innovations process can be interpreted as the difference between the observed changed dY (t) and the expected change π_t(s + s^†)dt of the field. We can obtain the adjoint equation for the conditional density matrix ρ_c by identifying π_t(X) = tr(ρ_c(t)X). This leads to the SME (B.8).

In finance, continuous time models, particularly diffusion processes, are frequently used to characterise the dynamics of major economic variables such as exchange rates, asset valuations and interest rates. One model frequently used to characterise the dynamics of interest rates is the diffusion model in continuous time, also known as the Itô process, which is given by the stochastic differential equation,

(1)drt=μ(rt)dt+σ(rt)dWt,

where Wt is standard Brownian motion, and μ(rt) and σ(rt) are called the drift function and the diffusion (or volatility function), respectively. Parametric models are often useful to finance professionals because observations can be interpreted in terms of parameters. A parametric representation of Model (1) is given by

(2)drt=μ(rt,θ)dt+σ(rt,θ)dWt,

where θ is a d-dimensional unknown parameter within a parameter space Θ⊂Rd for some positive integer d. In the market, models are continuously refined, and their ability to fit current prices is continuously tested. Models are less frequently tested for their goodness of fit to historical data. Research in this context has exhibited notable growth and development, but no definitive model has yet been produced. For this reason, one of the main goals of this work is to apply our goodness of fit approach to the diffusion processes that are studied in the literature for modelling interest rates.

In this paper, we consider a goodness-of-fit test based on empirical processes such as model checks. The aim is to introduce a test that compares the goodness of fit for parametric forms of the drift and volatility functions in interest rate models based on empirical processes. The hypotheses being studied are

(3)H0:μ∈{μ(⋅,θ):θ∈Θ}

for the parametric form of the drift, and

(4)H0:σ∈{σ(⋅,θ):θ∈Θ}

for the parametric form of the volatility function. An alternative way to address the problem would be to use the proposed tests for the integrated regression models. In regression models, this approach is discussed in Stute (1997), and a similar approach is later considered for time series in Koul and Stute (1999). The literature on the goodness of fit of diffusion models based on empirical processes, is scarce, with the exception of the some recent works: Lee and Wee (2008), who propose a test based on the empirical process of the residues of the diffusion model; and Negri and Nishiyama (2009), Negri and Nishiyama (2010) and Masuda et al. (2010), who propose a goodness-of-fit test based on a score-marked empirical process for both continuous and discrete time observations of ergodic diffusion processes. The above works only consider tests of a simple null hypothesis. In this sense, our proposal addresses a more general case: the composite hypothesis for a parametric form of the drift function. We also propose a test for the volatility function based on empirical processes that have not been considered in previous works.