Many indicators are constructed as functions of several variables not necessarily strongly correlated. The ultimate validity condition is their subject domain meaning. Frequently, they are used to measure a certain distance to an ideal modelling condition. Correlational methods, such as factor or principal components analyses, are very good exploratory techniques but they are restricted to linear relationships between the variables. So, you have a lot of work to do on your own.

Factor analysis is what you look for.

Statistics can't solve your problem. That's a subject-matter question.

Considering you example (Speed=Distance/Time) I agree with Jochen Wilhelm. In reality you are creating a new variable (speed) with meaning totally different of the distance and the time. In this case the validity of the third variable is not a statistical problem. It is important to compare the velocities in your research? Maybe if you are testing different routes, or different transportation ways. I don't know any kind of test to validate the existence of a new variable.

In other hand, exists the problem of different but correlated, variables. In this cases, Factor Analysis (as said by Tom Broekel) or Principal Component Analysis (in my opinion) is interesting because, in this case, you can change two or more variables by one or more principal component(s) whithout a significative loss of the variance. In this case you are, in fact, combining two variables (in example) to one principal component and preservating the amount of variance explained in two variables. The percent of variance explained by the principal component is the key. If you have 2 variables and the first principal component can explain, say 99% of the variance, it is coherent in some analysis, to use these principal components instead 2 variables. In other hand you can have two variables and the first principal component can explain only 60% of the variance of the initial variables, the substitution is not appropriate.

I expecte you can use princibal component.

You do not need a statistical analysis to decide whether Speed=Distance/Time. What you should do is to calculate the new variable, calculate descriptive statistics, plot a histogram, plot the new variable against other available variables and calculate correlation coefficients with other variables in order to check whether the new variable behaves in an expected way.

However, in general, if you want to combine items by taking averages or other linear combinations, it may be necessary to calculate a Cronbach Alpha on the items which you want to combine, in order to check whether the items are consistent/compatible and whether some inverted items need to be re-coded before combining them.

Plot distance vs time. If they are close to a line (or curve) there is one combined variable.

Regards

Be aware that constructing speed as the ratio of distance over time builds in a positive correlation of the resulting new variable with distance (and therefore with other variables that are correlated with distance), and a negative correlation of the new variable with time (and with other variables correlated with time). That's OK, as long as you take that into account in interpreting results. An alternative approach (in a multiple regression framework) would be to use distance (or its log) as the dependent variable, with time (or its log) as a control variable, and then examine the effects on distance of other predictor variables, controlling for time. Generally speaking, results should be similar to those obtained by predicting the speed-ratio variable from the predictor variables.

Also, I doubt whether Principal Components Analysis would be useful in this case. As others in this thread have noted, the issue is largely theoretical/substantive, not statistical. Distance and time will, in general, be positively correlated -- long journeys tend to take more time than short journeys, regardless of speed, unless the journeys are all relatively short. So (in most data sets, with a full range of journey lengths), even if a PCA shows distance and time loading on a single factor, they will likely both load positively. That would provide no justification for you to form an index of the two as a ratio, with distance in the numerator and time in the denominator. Such results would, instead, justify an additive indicator as the sum of time and distance (perhaps weighted by their factor loadings or factor score coefficients). That indicator would not represent the theoretical construct, "velocity; rather, it would just represent "journey duration" as an amalgam of distance and time. It does not sound like that is the construct of theoretical interest to you.

If S=D/T then log(S)=log(D)-log(T). So if you fit a linear regression of the form log(S) = β0 + β1 log(D) + β2 log(T) then you may test H0( β0=0; β1=1; β2=-1). Alternatively, in SPSS you have under the Regression techniques, "Curve Fitting" which offers a number of curves including the hyperbola. If you fit a hyperbola in this case, you obtain R-squared as a goodness-of-fit measure.

In your example Distance is constant. There will then be a direct relationship between average speed and time of travel. If you measure speed and time independently (for example, you directly measure speed at half travel distance and you also clock the total travel time) and plot them, there will only be a perdect linear relationship between the variables if speed is constant throughout the journey, or if speed at the measurement point is a constant offset from the average speed. If speed is erratic at the measurement point then there may be no strong statistical relationship between the two variables. It really depends on the physical system and the hypothesis you are seeking to prove.

In situations where a third variable can be mathematically derived from the first two it is definitely superior to do that

We should take a step back and ask why would we want to do that. I will make a guess that there is one or more target variables that we want to estimate based on the initial ones (like Subburaj suggested). In the example let's say it is the trip cost. In that case, we would create a model that takes the initial variables as inputs, and another that takes the combined variables as inputs. We would then compute performance metrics (e.g., r-squared) for each model and assess whether if there is a loss of performance with the simplified model, and if the benefits of the simplification (e.g., in terms of data collection, computation time) are worth this loss. As with any model calibration process, we should take the adequate steps to validate the models and mitigate the risk of overfitting.

Dear Mayur,

When:

dx: is the distance travelled and dt: is the time needed to travel the distance dx, then speed is mathematically defined as the first derivative of dx in function of the time interval dt, needed to travel dx,....at least in physical mechanics.

It is my opinion that a physical variable like a first derivative (like speed) is by a mathematical analytical expression (speed = dx/dt) and hence not stochastically.

Hence what do you want to learn with a stochastical approach here?

The physical definition of speed in physics was already applied by the Egyptians?

I don't see your problem!

But nobody will stop you from testing whether / =

Cheers

Roy

Hayashi, C. (1952). On the prediction of phenomena from qualitative data and the quantification of qualitative data from the mathematico-statistical point of view. Annals of the Institute of Statistical Mathematics, 69-98.

Hayashi, C. (1954a). Multidimensional quantification I. Proceedings of the Japan Academy, 30, 61-65.

Hayashi, C. (1954b). Multidimensional quantification II. Proceedings of the Japan Academy, 30, 165-169.

Hi Mayur,

As has been written already, the problem you described is one of dimensional analyses and it is domain-specific (in your described case it is the physics of motion and not statistics). If you were to use statistics (e.g. correlation) then you would not be addressing the physical relationships, because correlation does not imply causation.

If you are looking at dimension reduction, then the underlying physics must be applied by using domain knowledge. For dimensional reduction may I suggest reading material related to Dimensional Analyses and Buckingham's Pi Theorem?

Incidentally, if in your example, the mass of the vehicle remained invariant, then you can also derive an average "Specific Power" variable (0.5V^2/t). Alternatively, if you had a third variable (mass) as part of your experiment, then of course you could reduce the three variables to just one (Power). This example illustrats the need to apply the underlying physics!

I hope this helps.

Sorry, I have no idea.

With best wishes,

Yoichi Ando

"Speed" is a well defined concept and is defined to be "Distance/Time" . Therefore, the combination of the two variables namely "distance" and "time" into one variable namely "speed" is not a valid one.

If we have 100 pairs of observations on distance traveled and time taken to travel same distance then automatically we have 100 observations on speed.

I am assuming that "Distance", "Speed" and "Time" were used as (unfortunate) examples, what Mayur wanted to know (I surmise) was how to combine variables in general. So, if you have observations of Y, X1 and X2 and you want to find out how Y depends on X1 and X2, the first step would be to plot scatter plots and then try non-linear regression (or curve fitting) to model Y as a function of X1 and X2. If you have a prior idea of what the functions should be, say Y=f(X1, X2) then you fit the function to the data, calculate f(X1, X2) for the fitted function, call it Y(hat), then plot a scatter plot of Y(hat) against Y(observed); the plot should be a straight line if the formula f(X1,X2) is the correct one, or the points should be scattered around a straight line if the observations contain random errors.

Dear Mayur,

Is your investigation based on experimental data or work ? If so, you perhaps should combine techniques of error analysis (accuracy of measurements; analysis/identification of possible systematic uncertainties) and statistical investigation in order to develop/evaluate modelling techniques : the way and the rigourness you do the job makes the difference towards success or failure.

Most of the time, you have different experimental variables (or conceptual ones if some test methods do not exist yet...) that you want to relate to each other, and you want to see what kind of relationship possibly exist between them. So if I use the writings of François Steffens here above, you could fail with Y = f(X1, X2) if you did not chose a relevant shape for the function f; or if you neglected another important parameter X3 or X4; or if you have a systematic deviation or uncertainty in your problem you did not identify. If your experimental data are too rough (not statistically relevant or not representative), you can look in the direction of "Fuzzy logic" techniques. "Domain knowledge" (Sanjiv Sharma) is the result of this kind of research process; intermediate steps are individual researcher/engineer contributions...

I remember of an interesting application work of this kind conducted by Freek Bos (TU Delft; now TU Eindhoven) in the field of structural engineering/mechanical analysis, but I could not find the right reference of the corresponding paper/conference paper back. You should search here if you think it can help : https://www.researchgate.net/profile/Freek_Bos/research

It is difficult and probably not relevant to give more detailled advice without talking about your specific investigation areas. In a lot of practical engineering problems (e.g. quality control systems), one often use more or less empirical relationships, as developping "complete" models is often out of reach (at least on short term...), because of side effects of different natures... Iterative work with progressive refinements is then often an interesting strategy.

Good luck !

I hope you will be able to an answer:

Hayashi, C. (1952). On the prediction of phenomena from qualitative data and the quantification of qualitative data from the mathematico-statistical point of view. Annals of the Institute of Statistical Mathematics, 69-98.

Hayashi, C. (1954a). Multidimensional quantification I. Proceedings of the Japan Academy, 30, 61-65.

Hayashi, C. (1954b). Multidimensional quantification II. Proceedings of the Japan Academy, 30, 165-169.

See an example:

Ando, Y. (1991). Architectural Acoustics, in McGraw-Hill Yearbook of Science &

Technology 1991, McGraw-Hill, New York.

With best wishes,

Yoichi Ando

Why not try different combinations of your two veraibles into one and see how your correlation with an «y-value» comes out? Statistics in the early stages is much a game of finding patterns and trying out things. You could also try a multiple regression and see if both parameters survives entering the regression formula. This is usually step 2 in the process. If the parameters are statistically independant, that is no correlations or patterns seen in a x-y plot, then they should both survive a multiple regreession and live side by side, or in a combined one. Statistics is different from analytical formulas though. Linear regression formulas cant be combined.

PCA is what I can think of. I would look at the correlation and see if they are linear correlated.

Please read carefully Appendix I and Section 4.5 in the book,

Ando, Y. (1998). Architectural Acoustics, Blending Sound Sources, Sound Fields,

and Listeners. AIP Press/Springer-Verlag, New York.

With best wishes,

Yoichi Ando

A problem like this can be approached with numerous different ways. Only your imagiation (and knowledge) can limit the methodes.

Biplot. any matrix of rank two can be displayed as a biplot which consists of a vector for each row and a vector for each column, chosen so that any element of the matrix is exactly the inner product of the vector corresponding to its row and to its column. If a matrix is of higher rank, one may display it approximately by a biplot of a matrix of rank two which approximates the original matrix. The biplot provides a useful tool of data analysis and allows the visual appraisal of the structure of large data matrices. It is especially revealing in principal component analysis, where the biplot can show inter-unit distances and indicate clustering of units as well as display variances and correlations of the variables. (Gabriel 1971)

Hi Mayur,

Although it has been quite a while since you posted the question, and others have provided very interesting statistical and physical suggestions, may I also suggest you look into the PCA and the active subspace method (ASM) approach?

The latter essentially allows you to create superpositions of the different variables in your problem. As such, you can reduce the dimensionality of your problem. Say you have an output Y which depends on X1, X2 as Y=f(X1,X2). By applying this method you can identify whether there is a spacific pattern, that is, whether Y is actually mostly dependent in a -say- linear superposition of X1,X2 like Y=f(aX1+bX2). Thus, now -within reasonable accuracy- you can reduce the dimensionality from 2 to 1.

Hope this is a bit helpful,

Spyros

Thanks, Spyros - an interesting point about studying the applicability of the Active Subspace Method for this example.

Although, I am still a little confused about the original question! Statistically, one can only say that the *correlation* between effects and factors may be made stronger by some combination of two or more factors. It does not imply that the combination of the factors will have any scientific (e.g. physical) meaning. Certainly, by considering the interactions amongst the different factors could results in a new narrative. This new narrative would be informed by, for example, Spyros's linear superposition approach, Y = f(aX1 + bX2).

Analysis of two variables can give u some idea

https://www.ess.uci.edu/~yu/class/ess210b/lecture.3.regression.all.pdf

This article mainly focuses on correlation, regression and significance test

Hi,

It is not clear your aim, but I advise you to create a variable as combination of others if this makes sense to you. The reason is that you are reducing the number of parameters, so you generate a simpler problem. In statistics, for instance in predictive modelling, the reduction of the parameter space is a very relevant matter (https://en.wikipedia.org/wiki/Dimensionality_reduction). In mechanics, in Fluid Mechanics for instance, makes no sense to study velocity, density and viscosity as separated parameters, since you can create the Reynolds number (https://en.wikipedia.org/wiki/Reynolds_number). As aforementioned, I am not sure about your objective, but surely to reduce the number of parameters will help you. Good luck!

The problem is required to be defined perfectly.

Many indicators are constructed as functions of several variables not necessarily strongly correlated. The ultimate validity condition is their subject domain meaning. Frequently, they are used to measure a certain distance to an ideal modelling condition. Correlational methods, such as factor or principal components analyses, are very good exploratory techniques but they are restricted to linear relationships between the variables. So, you have a lot of work to do on your own.

I like Jorge's suggestions. The point of factor analysis or principle component analysis is to come up with a single theoretical factor calculated from weightings of measurable variables.

I recommend you Handbook on Constructing Composite Indicators, https://www.oecd.org/sdd/42495745.pdf

Regards,

Hanna

I agree with the view of Jorge Ortiz Pinilla .

Look for main axis transformation for the variables.

It is mathematics using the potential energy surface.

Very enjoyable discussion

I agree Dr. Jorge Ortiz Pinilla

@Mohammad A. Fawzy, if you agree, then why don't you click the "recommend" link? It's a better way to announce your support that's integrated into the ResearchGate system (similar to likes, up-votes and thumbs up).

yes @ Tom Arjannikov

https://www.ess.uci.edu/~yu/class/ess210b/lecture.3.regression.all.pdf

Article Statistics review 7: Correlation and Regression

Great discussion. Following

I agree with Augustine Senanu Kukah.

I like to add that the the discussion is meaningful and fruitful also.

I hope that you can carefully read these material and find your method of analyses.

Hayashi, C. (1952). On the prediction of phenomena from qualitative data and the quantification of qualitative data from the mathematico-statistical point of view. Annals of the Institute of Statistical Mathematics, 69-98.

Hayashi, C. (1954a). Multidimensional quantification I. Proceedings of the Japan Academy, 30, 61-65.

Hayashi, C. (1954b). Multidimensional quantification II. Proceedings of the Japan Academy, 30, 165-169.

With best wishes,

Yoichi Ando

Yoichi Ando mentioned some important and valuable materials.

These will be very much helpful in selecting the method of analysis to be applied in the problem.

I think there is no need for any significance test. One can define the ratio “distance/time to travel the same distance” as a genuine statistical sample element, suitable for all common statistical operations.

In any case, I think it needs for significance test.

With best wishes,

Yoichi Ando

The need for significance tests depends on the purpose of the exercise. There would be no purpose to test for significance in calculating a rate such as mph, cf/s, etc. These are just a away to obtain a single value from several measurements. In other cases, such as factor analysis or principle component analysis, variables with lower correlation are preferred. Test for correlation among predictor variables would be used to eliminate highly correlated variables. The variables to use or drop are up to the desecration of the researcher.

Sorry for me it is hard to understand.

If your paper is not scientific, then it may be.

Yoichi Ando

No further comment

Forming speed as new single derived variable from distance and time needs no statistical test.

Speed is a well defined concept in dynamics.

Thus, I thank that no test is necessary here.

follow

follow

Mohammad A. Fawzy , you could follow a question by clicking the "Follow" link that is located just below the original question, right between "Recommend" and "Share" links. You could un-follow by clicking the "Following" link in the same location.

Sorry I cannot get your question what do you want to discuss with me.

Looking forward to hearing from you.

best

Yoichi Ando