Mean Square Error Measure Research Articles

Nearest-Neighbor Forecasts of U.S Interest Rates

We employ a nonlinear, nonparametric method to model the stochastic behavior of changes in several short and long term U.S interest rates. We apply a nonlinear autoregression to the series using the locally weighted regression (LWR) estimation method, a nearest-neighbor method, and evaluate the forecasting performance with a measure of root mean square error (RMSE). We compare the forecasting performance of the nonparametric fit to the performance of two benchmark linear model: an autoregressive model and a random-walk-with-drift model. The nonparametric model exhibits greater out-of-sample forecast accuracy that of the linear predictors for most U.S interest rate series. The improvements in forecast accuracy are statistically significant and robust. This evidence establishes the presence of significant nonlinear mean predictability in U.S interest rates, as well as the usefulness of the LWR method as modeling strategy for these benchmark series

Global and local statistical properties of fixed-length nonparametric smoothers

The main purpose of this study is to analyze the global and local statistical properties of nonparametric smoothers subject to a priori fixed length restriction. In order to do so, we introduce a set of local statistical measures based on their weighting system shapes and weight values. In this way, the local statistical measures of bias, variance and mean square error are intrinsic to the smoothers and independent of the data to which they will be applied on. One major advantage of the statistical measures relative to the classical spectral ones is their easiness of calculation. However, in this paper we use both in a complementary manner. The smoothers studied are based on two broad classes of weighting generating functions, local polynomials and probability distributions. We consider within the first class, the locally weighted regression smoother (loess) of degree 1 and 2 (L1 and L2), the cubic smoothing spline (CSS), and the Henderson smoothing linear filter (H); and in the second class, the Gaussian kernel (GK). The weighting systems of these estimators depend on a smoothing parameter that traditionally, is estimated by means of data dependent optimization criteria. However, by imposing to all of them the condition of an equal number of weights, it will be shown that some of their optimal statistical properties are no longer valid. Without any loss of generality, the analysis is carried out for 13- and 9-term lengths because these are the most often selected for the Henderson filters in the context of monthly time series decomposition.

Uso de modelos não-lineares para o ajuste de curvas de crescimento de cavalos pantaneiros

Dados de altura da cernelha e idade de 26 cavalos Pantaneiros, obtidos, na maioria, do nascimento até 36 meses, foram ajustados aos modelos de respostas não-lineares de Brody, Richards, Gompertz, Logístico, Weibull e Morgan-Mercer-Flodin. Estes seis modelos matemáticos foram comparados com o uso de uma medida de curvatura média e do erro médio quadrático combinado. O modelo de Weibull foi escolhido. A assíntota desta curva representa a altura esperada na maturidade. Os machos apresentaram um valor maior desta quantidade do que as fêmeas. O índice de maturidade, contudo, é maior para as fêmeas. Observou-se uma indicação de associação negativa entre altura na maturidade e índice de maturidade somente nas fêmeas. Tais resultados indicam que as fêmeas amadurecem mais cedo. Após testes de normalidade e homogeneidade de variância, as diferenças entre sexos foram analisadas com o uso do teste-t. Somente a altura da cernelha ao nascimento apresentou diferença significativa.

Optimal state estimation for stochastic systems: an information theoretic approach

In this paper, we examine the problem of optimal state estimation or filtering in stochastic systems using an approach based on information theoretic measures. In this setting, the traditional minimum mean-square measure is compared with information theoretic measures, Kalman filtering theory is reexamined, and some new interpretations are offered. We show that for a linear Gaussian system, the Kalman filter is the optimal filter not only for the mean-square error measure, but for several information theoretic measures which are introduced in this work. For nonlinear systems, these same measures generally are in conflict with each other, and the feedback control policy has a dual role with regard to regulation and estimation. For linear stochastic systems with general noise processes, a lower bound on the achievable mutual information between the estimation error and the observation are derived. The properties of an optimal (probing) control law and the associated optimal filter, which achieve this lower bound, and their relationships are investigated. It is shown that for a linear stochastic system with an affine linear filter for the homogeneous system, under some reachability and observability conditions, zero mutual information between estimation error and observations can be achieved only when the system is Gaussian.

General Decompositions of MSE-Based Skill Scores: Measures of Some Basic Aspects of Forecast Quality

Abstract Skill scores defined as measures of relative mean square error—and based on standards of reference representing climatology, persistence, or a linear combination of climatology and persistence—are decomposed. Two decompositions of each skill score are formulated: 1) a decomposition derived by conditioning on the forecasts and 2) a decomposition derived by conditioning on the observations. These general decompositions contain terms consisting of measures of statistical characteristics of the forecasts and/or observations and terms consisting of measures of basic aspects of forecast quality. Properties of the terms in the respective decompositions are examined, and relationships among the various skill scores—and the terms in the respective decompositions—are described. Hypothetical samples of binary forecasts and observations are used to illustrate the application and interpretation of these decompositions. Limitations on the inferences that can be drawn from comparative verification based on skil...

Early Expression of Smooth-Pursuit Eye Movement Abnormalities in Children of Schizophrenic Parents

Disordered smooth-pursuit eye movements (SPEM) and, specifically, small anticipatory saccades that disrupt SPEM have been hypothesized to be a marker of genetic vulnerability to schizophrenia. This study compares SPEM in children of schizophrenic parents with normally developing control children to assess whether SPEM abnormalities are also present in a subset of at-risk children. With infrared oculography, SPEM was examined in 13 children of schizophrenic parents and 19 normally developing controls (aged 6 to 15 years). Measures of smooth-pursuit gain and root mean square error were used in addition to more specific measures of catch-up saccades and anticipatory saccades. Children of schizophrenic parents differed from normally developing controls on gain and root mean square error, but not on catch-up saccades. Small anticipatory saccades were significantly more frequent in the at-risk group. The percentage of total eye movements due to anticipatory saccades identified 54% of the at-risk group (compared with none of the control group) as performing more than two standard deviations above (worse than) the control mean. The presence of increased anticipatory saccades is evidence for an oculomotor dysfunction that may be a phenotype of the genetic risk for schizophrenia, expressed years prior to the possible development of clinical illness.

Theoretical analysis of the high-rate vector quantization of LPC parameters

The paper presents a theoretical analysis of high-rate vector quantization (VQ) systems that use suboptimal, mismatched distortion measures, and describes the application of the analysis to the problem of quantizing the linear predictive coding (LPC) parameters in speech coding systems. First, it is shown that in many high-rate VQ systems the quantization distortion approaches a simple quadratically weighted error measure, where the weighting matrix is a "sensitivity matrix" that is an extension of the concept of the scalar sensitivity. The approximate performance of VQ systems that train and quantize using mismatched distortion measures is derived, and is used to construct better distortion measures. Second, these results are used to determine the performance of LPC vector quantizers, as measured by the log spectral distortion (LSD) measure, which have been trained using other error measures, such as mean-squared (MSE) or weighted mean-squared error (WMSE) measures of LEPC parameters, reflection coefficients and transforms thereof, and line spectral pair (LSP) frequencies. Computationally efficient algorithms for computing the sensitivity matrices of these parameters are described. In particular, it is shown that the sensitivity matrix for the LSP frequencies is diagonal, implying that a WMSE measured LSP frequencies converges to the LSD measure in high-rate VQ systems. Experimental results to support the theoretical performance estimates are provided.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Existence and uniqueness results for neural network approximations

Some approximation theoretic questions concerning a certain class of neural networks are considered. The networks considered are single input, single output, single hidden layer, feedforward neural networks with continuous sigmoidal activation functions, no input weights but with hidden layer thresholds and output layer weights. Specifically, questions of existence and uniqueness of best approximations on a closed interval of the real line under mean-square and uniform approximation error measures are studied. A by-product of this study is a reparametrization of the class of networks considered in terms of rational functions of a single variable. This rational reparametrization is used to apply the theory of Pade approximation to the class of networks considered. In addition, a question related to the number of local minima arising in gradient algorithms for learning is examined.

Predicting protein secondary structure using neural net and statistical methods

A comparison of neural network methods and Bayesian statistical methods is presented for prediction of the secondary structure of proteins given their primary sequence. The Bayesian method makes the unphysical assumption that the probability of an amino acid occurring in each position in the protein is independent of the amino acids occurring elsewhere. However, we find the predictive accuracy of the Bayesian method to be only minimally less than the accuracy of the most sophisticated methods used to date. We present the relationship of neural network methods to Bayesian statistical methods and show that, in principle, neural methods offer considerable power, although apparently they are not particularly useful for this problem. In the process, we derive a neural formalism in which the output neurons directly represent the conditional probabilities of structure class. The probabilistic formalism allows introduction of a new objective function, the mutual information, which translates the notion of correlation as a measure of predictive accuracy into a useful training measure. Although a similar accuracy to other approaches (utilizing a mean-square error) is achieved using this new measure, the accuracy on the training set is significantly and tantalizingly higher, even though the number of adjustable parameters remains the same. The mutual information measure predicts a greater fraction of helix and sheet structures correctly than the mean-square error measure, at the expense of coil accuracy, precisely as it was designed to do. By combining the two objective functions, we obtain a marginally improved accuracy of 64.4%, with Matthews coefficients C α, C β and C coil of 0.40, 0.32 and 0.42, respectively. However, since all methods to date perform only slightly better than the Bayes algorithm. which entails the drastic assumption of independence of amino acids, one is forced to conclude that little progress has been made on this problem, despite the application of a variety of sophisticated algorithms such as neural networks, and that further advances will require a better understanding of the relevant biophysics.

Optimal quantization by matrix searching

Optimal quantization, a fundamental problem in source coding and information theory, can be formulated as a discrete optimization problem. In 1964 Bruce (“Optimum Quantization,” Sc.D. thesis, MIT, May 1964) devised a dynamic programming algorithm for discrete optimal quantization. For the mean-square error measure and when the amplitude density function of the quantized signal is represented by a histogram of N points, Bruce's algorithm can compute the optimal K-level quantizer in O( KN 2) time. This paper shows that the same problem can be solved in O( KN) time. The improvement is made by a better understanding of the objective function in this particular non-linear programming problem and the use of Aggarwal et al.'s matrix-searching technique.

Estimating a probability using finite memory

Let <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\{X_{i}\}_{i=1}^{\infty}</tex> be a sequence of independent Bernoulli random variables with probability <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</tex> that <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X_{i} = 1</tex> and probability <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q=1-p</tex> that <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X_{i} = 0</tex> for all <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i \geq 1</tex> . Time-invariant finite-memory (i.e., finite-state) estimation procedures for the parameter p are considered which take <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X_{1}, \cdots</tex> as an input sequence. In particular, an n-state deterministic estimation procedure is described which can estimate p with mean-square error <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O(\log n/n)</tex> and an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -state probabilistic estimation procedure which can estimate <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</tex> with mean-square error <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O(1/n)</tex> . It is proved that the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O(1/n)</tex> bound is optimal to within a constant factor. In addition, it is shown that linear estimation procedures are just as powerful (up to the measure of mean-square error) as arbitrary estimation procedures. The proofs are based on an analog of the well-known matrix tree theorem that is called the Markov chain tree theorem.

Forecasting temporally aggregated vector ARMA processes

AbstractIf interest centres on forecasting a temporally aggregated multiple time series and the generation process of the disaggregate series is a known vector ARMA (autoregressive moving average) process then forecasting the disaggregate series and temporally aggregating the forecasts is at least as efficient, under a mean squared error measure, as forecasting the aggregated series directly. Necessary and sufficient conditions for equality of the two forecasts are given. In practice the data generation process is usually unknown and has to be determined from the available data. Using asymptotic theory it is shown that also in this case aggregated forecasts from the disaggregate process will usually be superior to forecasts obtained from the aggregated process.

Forecasting Contemporaneously Aggregated Vector ARMA Processes

Given a multiple time series that is generated by a multivariate ARMA process and assuming the objective is to forecast a weighted sum of the individual variables, then under a mean squared error measure of forecasting precision, it is preferable to forecast the disaggregated multiple time series and aggregate the forecasts, rather than forecast the aggregated series directly, if the involved processes are known. This result fails to hold if the processes used for forecasting are estimated from a given set of time series data. The implications of these results for empirical research are investigated using different sets of economic data.

Modeling Multipath Fading Responses Using Multitone Probing Signals and Polynomial Approximation

We show in quite a general way that highly accurate modeling of multipath fading responses is possible using low-order complex polynormals. This applies to all terrestrial radio systems in the channelized common carrier bands below 15 GHz, where channel widths are 40 MHz or less. The context of the study is a new multipath experiment being conducted in New Jersey over a 23-mile path at 11 GHz. The transmitted signal consists of up to nine tones in a 40-MHz bandwidth. These tones are coherently processed, sampled, and digitized in the receiver and recorded, during fading events, for later off-line reductions. Simple routines can be used to determine polynomial coefficients from these recorded data. This paper describes the signal processing and data reduction methods and analyzes them to assess the accuracy of polynomial fitting. The analysis uses a mean-square error measure and assumes a representative form for the underlying response function. Our results predict that the vast majority of multipath fading responses can be accurately approximated over bandwidths of 40 (62) MHz using first- (second-) order complex polynomials.

A Numerical Method for the Solution of the Junction of Cylindrical Waveguides (Short Papers)

In order to solve the waveguide junction problem numerically, we express the fields in the guides by truncated modal expansions and construct an error function which is a measure of the mean-square error in the matching of the boundary conditions at the junction. The minimum of this error leads to a set of linear equations for the modal amplitudes. Offset rectangular waveguides with amplitude and current excitations are studied. A weighting factor which multiplies the error contribution due to the magnetic field is studied, and a criterion for its selection given.