Researched and Presented by:
Christopher Thompson
Which forecast error measuring tool is the best?
Of the four error tracking tools we have learned about today (MAD, MSE, MAPE and seasonality), which method provides the best way to track forecast error? Give examples.
Mean Absolute Percent Error (MAPE)
A problem with both the MAD and MSE is that their values depend on the magnitude of the item being forecast. If the forecast item is measured in thousands, the MAD and MSE values can be exceptionally large. To avoid this problem, we can use the mean absolute percent error (MAPE). This is computed as the average of the absolute difference between the forecasted and actual values, expressed as a percentage of the actual values. That is, if we have forecasted and actual values for n periods, the MAPE is calculated as:
Mean absolute percent error (MAPE)
The average of the absolute differences between the forecast and actual values, expressed as a percent of actual values.
“Three commonly used measures for summarizing historical errors are the mean absolute deviation (MAD), the mean squared error (MSE), and the mean absolute percent error (MAPE). MAD is the average absolute error, MSE is the average of squared errors, and MAPE is the average absolute percent error.
MAD is the easiest to compute, but weights errors linearly. MSE squares errors, thereby giving more weight to larger errors, which typically cause more problems. MAPE should be used when there is a need to put errors in perspective.”
I think it is a judgment call when to use those method because each method is good for certain situation. However, I do prefer to use MAPE method to calculate forecast errors.
Retrieved from:
http://highered.mcgraw-hill.com/sites/dl/free/0073525251/886181/stevenson11e_sample_ch03.pdf
The mean squared error is arguably the most important criterion used to evaluate the performance of a predictor or an estimator. (The subtle distinction between predictors and estimators is that random variables are predicted, and constants are estimated.) The mean squared error is also useful to relay the concepts of bias, precision, and accuracy in statistical estimation. In order to examine a mean squared error, you need a target of estimation or prediction, and a predictor or estimator that is a function of the data. Suppose that the target, whether a constant or a random variable, is denoted as . The mean squared error of the estimator or predictor for is
One use for these measures is to compare the accuracy of alternative forecasting methods. For instance, a manager could compare the results to determine one which yields the lowest MAD, MSE, or MAPE for a given set of data. Another use is to track error performance over time to decide if attention is needed. Overall, the operations manager must settle on the relative importance of historical performance versus responsiveness and whether to use MAD, MSE, or MAPE to measure historical performance. MAD is the easiest to compute, but weights errors linearly. MSE squares errors, thereby giving more weight to larger errors, which typically cause more problems. MAPE should be used when there is a need to put errors in perspective.
The first measure of the overall forecast error for a model is the mean absolute deviation (MAD). This value is computed by taking the sum of the absolute values of the individual forecast errors (deviations) and dividing by the number of periods of data. The mean squared error (MSE) is a second way of measuring overall forecast error. MSE is the average of the squared differences between the forecasted and observed values. A drawback of using the MSE is that it tends to accentuate large deviations due to the squared term. For example, if the forecast error for period 1 is thrice as large as the error for period 2, the squared error in period 1 is nine times as large as that for period 2. Hence, using MSE as the measure of forecast error typically indicates that we prefer to have several smaller deviations rather than even one large deviation. A problem with both the MAD and MSE is that their values depend on the magnitude of the item being forecast. If the forecast item is measured in thousands, the MAD and MSE values can be exceptionally large. To avoid this problem, we can use the mean absolute percent error (MAPE). This is computed as the average of the absolute difference between the forecasted and actual values, expressed as a percentage of the actual values. The MAPE is perhaps the easiest measure to interpret. For example, a result that the MAPE is 6% is a clear statement that it is not dependent on issues such as the magnitude of the input data.
References: Heizer, Jay H. Operations management – 10th Edition Textbook
The MAPE is the easiest measurement to interpret because using percentages give a straight forth picture of a forecast. And we all know that when you are in meetings all managers throw out percentages to make their point.
According to the text on page 116, the MAPE is perhaps the easiest measure to interpret. For example, a result that the MAPE is 6% is a clear statement that is not dependent on issues such as the magnitude of the input data.
A problem with both the MAD and MSE is that their values depend on the magnitude of the item being forecast. If the forecast item is measured in thousands, the MAD and MSE values can be very large. To avoid this problem, we can use the mean absolute percent error (MAPE). This is computed as the average of the absolute difference between the forecasted and actual values, expressed as a percentage of the actual values. That is, if we have forecasted and actual values for n periods, the MAPE is calculated as:
The average of the absolute differences between the forecast and actual values, expressed as a percent of actual values.
Ref: Operations Management, tenth Edition
What other forecast measuring tool(s) can we use besides MAD, MSE, and MAPE to better show the direction of the forecast error?
There are several very efficient tools used to find errors in forecasts. One of those tools, are using a tracking signal. A Tracking signal is defined in our textbook as:
A tracking signal is a measurement of how well a forecast is predicting actual values. As forecasts are updated every week, month, or quarter, the newly available demand data are compared to the forecast values. The tracking signal is computed as the cumulative error divided by the mean absolute deviation (MAD).
Our textbook further explains the meaning of positive and negative tracking signals:
Positive tracking signals indicate that demand is greater than forecast. Negative signals mean that demand is less than forecast. A good tracking signal—that is, one with a low cumulative error—has about as much positive error as it has negative error. In other words, small deviations are okay, but positive and negative errors should balance one another so that the tracking signal centres closely around zero. A consistent tendency for forecasts to be greater or less than the actual values (that is, for a high absolute cumulative error) is called a bias error. Bias can occur if, for example, the wrong variables or trend line are used or if a seasonal index is misapplied.
This information is clear and concise. I would like to see this application performed in excel.
Render, Jay Heizer and Barry (). Operations Management [10] (VitalSource Bookshelf), Retrieved from
http://devry.vitalsource.com/books/9781256081487/id/ch4box38
Forecast bias is a tendency for a forecast to be consistently higher or lower than the actual value. Forecast bias is distinct from forecast error in that a forecast can have any level of error and be completely unbiased.
Bias= the sum of error/the number of forecast errors
Running sum of forecast errors (RSFE)- provides a measure of forecast bias
RSFE= the sum of forecast error for the number of periods
Information retrieved from
http://www.scmfocus.com/demandplanning/2012/02/forecastbias/
What about Trend Projections and the Least Squares Method?
Trend projection
A time-series forecasting method that fits a trend line to a series of historical data points and then projects the line into the future for forecasts. (Render 119)
If we decide to develop a linear trend line by a precise statistical method, we can apply the least-squares method. This approach results in a straight line that minimizes the sum of the squares of the vertical differences or deviations from the line to each of the actual observations. (Render 119)
A least-squares line is described in terms of its y-intercept (the height at which it intercepts the y-axis) and its expected change (slope). If we can compute the y-intercept and slope, we can express the line with the following equation:
y(hat) = a + b x
(Render 119)
Render, Jay Heizer and Barry. Operations Management, 10th Edition. Pearson Learning Solutions. <vbk:9781256081487#outline (8.8.8)>.
Another forecast measuring tool is Trend Projections which is a time-series forecasting method that fits a trend line to a series of historical data points and then projects the line into the future for forecasting the technique fits a trend line to a series of historical data points and then projects the line into the future for medium to long-range forecasts.
Trend projection: This technique fits a trend line to a series of historical data points and then projects the line into the future for medium to long-range forecasts. Several mathematical trend equations can be developed (for example, exponential and quadratic), but in this section, we will look at linear (straight-line) trends only.
http://devry.vitalsource.com/#/books/9781256081487/pages/55879738
Tracking signal is one such tool that shows the direction of the forecast and compare with the actual values. A tracking signal is a measurement of how well a forecast is predicting actual values. As forecasts are updated every week, month, or quarter, the newly available demand data are compared to the forecast values. As used in forecasting, the tracking signal is the number of mean absolute deviations that the forecast value is above or below the actual occurrence. A tracking signal (TS) can be calculated using the arithmetic sum of forecast deviations divided by the mean absolute deviation:
TS = RSFE/MAD
Where RSFE = The running sum of forecast errors, considering the nature of the error. (For example, negative errors cancel positive errors and vice versa.
MAD = The average of all the forecast errors (disregarding whether the deviations are positive or negative). It is the average of the absolute deviations.
Once tracking signals are calculated, they are compared with predetermined control limits. When a tracking signal exceeds an upper or lower limit, there is a problem with the forecasting method, and management may want to re-evaluate the way it forecasts demand.
How does a bias and trend adjustment affect MAD, MSE, and MAPE?
Forecast accuracy can be determined by computing the bias, mean absolute deviation (MAD), mean square error
(MSE) or mean absolute percent error (MAPE) for the forecast using different values for alpha. Bias is the sum of the forecast errors [Σ(FE)]. For the exponential smoothing example above, the computed bias would be:
(60 – 41.5) + (72 – 54.45) + (58 – 66.74) + (40 – 60.62) = 6.69
If one assumes that a low bias indicates an overall low forecast error, one could compute the bias for a few potential values of alpha and assume that the one with the lowest bias would be the most accurate. However, caution must be observed in that wildly inaccurate forecasts may yield a low bias if they tend to be both over forecast and under forecast (negative and positive). For example, over three periods a firm may use a particular value of alpha to over forecast by 75,000 units (-75,000), under forecast by 100,000 units (+100,000), and then over forecast by 25,000 units (-25,000), yielding a bias of zero (-75,000 + 100,000 25,000 = 0). By comparison, another alpha yielding over forecasts of 2,000 units, 1,000 units, and 3,000 units would result in a bias of 5,000 units. If normal demand was 100,000 units per period, the first alpha would yield forecasts that were off by as much as 100 percent while the second alpha would be off by a maximum of only 3 percent, even though the bias in the first forecast was zero.
A safer measure of forecast accuracy is the mean absolute deviation (MAD). To compute the MAD, the forecaster sums the absolute value of the forecast errors and then divides by the number of forecasts (Σ |FE| ÷N). By taking the absolute value of the forecast errors, the offsetting of positive and negative values is avoided. This means that both an over forecast of 50 and an under forecast of 50 are off by 50. Using the data from the exponential smoothing example, MAD can be computed as follows:
(|60 – 41.5| + |72 – 54.45| + |58 – 66.74| + |40 – 60.62|) ÷ 4 = 16.35
Therefore, the forecaster is off an average of 16.35 units per forecast. When compared to the result of other alphas, the forecaster will know that the alpha with the lowest MAD is yielding the most accurate forecast.
Mean square error (MSE) can also be utilized in the same fashion. MSE is the sum of the forecast errors squared divided by N-1 [Σ(P(FE)) ÷ (N-1)]. Squaring the forecast errors eliminates the possibility of offsetting negative numbers, since none of the results can be negative. Utilizing the same data as above, the MSE would be:
[(18.5) + (17.55) + (–8.74) + (–20.62)] ÷ 3 = 383.94
As with MAD, the forecaster may compare the MSE of forecasts derived using various values of alpha and assume the alpha with the lowest MSE is yielding the most accurate forecast.
The mean absolute percent error (MAPE) is the average absolute percent error. To arrive at the MAPE one must take the sum of the ratios between forecast error and actual demand times 100 (to get the percentage) and divide by N[(Σ | Actual demand —forecast | ÷Actual demand) × 100 ÷ N]. Using the data from the exponential smoothing example, MAPE can be computed as follows:
[(18.5/60) + 17.55/72 + 8.74/58 + 20.62/48) × 100] ÷ 4 = 28.33%
As with MAD and MSE, the lower the relative error the more accurate the forecast.
It should be noted that in some cases the ability of the forecast to change quickly to respond to changes in data patterns is more important than accuracy. Therefore, one’s choice of forecasting method should reflect the relative balance of importance between accuracy and responsiveness, as determined by the forecaster.
http://www.encyclopedia.com/topic/Forecasting.aspx
Bias is something that is often introduced into the raw data by the way the data is obtained. It is something that you can discover and remove. The problem is sometimes it is introduced without knowing, often from not having clearly defined procedures as to how to obtain something.
Post Problem 4.13b Here
B. Use a three-year moving average to forecast demand in years 4,5,6 Year A Moving Average Error
1 45
2 50
3 52
4 56 45+50+52/3=49 7
5 58 50+52+56/3=52.7 5.3
6 52+56+58/3=55.3
Made=12.3/2=6.2
Question: Post Problem 4.13a Here
4.13 (a) Exponential smoothing, a = 0.6:
Absolute
Year Demand Exponential Smoothing a = 0.6 Deviation
1 45 41 4.0
2 50 41.0 + 0.6(45–41) = 43.4 6.6
3 52 43.4 + 0.6(50–43.4) = 47.4 4.6
4 56 47.4 + 0.6(52–47.4) = 50.2 5.8
5 58 50.2 + 0.6(56–50.2) = 53.7 4.3
6 ? 53.7 + 0.6(58–53.7) = 56.3
{= 25.3
MAD = 5.06
(b) Exponential smoothing, a = 0.9:
Absolute
Year Demand Exponential Smoothing a = 0.9 Deviation
1 45 41 4.0
2 50 41.0 + 0.9(45–41) = 44.6 5.4
3 52 44.6 + 0.9(50–44.6 ) = 49.5 2.5
4 56 9.5 + 0.9(52–49.5) = 51.8 4.2
5 58 51.8 + 0.9(56–51.8) = 55.6 2.4
6 ? 55.6 + 0.9(58–55.6) = 57.8
{= 18.5
MAD = 3.7
As you can see in the following table, demand for heart transplant surgery at Washington General Hospital has increased steadily in the past few years: The director of medical services predicted 6 years ago that demand in year 1 would be 41 surgeries. a) Use exponential smoothing, first with a smoothing constant of .6 and then with one of .9, to develop forecasts for years 2 through 6.
Year Heart Transplants Forecast Using alpha = 0.6 Forecast Using alpha = 0.9
1 45 41.000 41.000
2 50 43.400 44.600
3 52 47.360 49.460
4 56 50.144 51.746
5 58 53.658 55.575
6 56.263 57.757
My calculations came out the same as yours in the forecast table. The MAD calculated by Willian of 5.06 and 3.7 were also the same.
While MAD, MSE and MAPE are great ways to track forecasts, nothing to me does it better than seasonality. Working in the car business, I have realized that the seasons play a huge role in the inventory that is purchased. During the wintertime, most people are looking for vehicles that will be able to handle snow, so sports cars will not likely be purchased. Likewise, the summertime is the perfect time to purchase that two-seater that you have always wanted. After reviewing the four tracking tools MAD, MSE, MAPE and seasonality I have determined that MAPE provides the best way to track forecast error. When using the other tools there is a greater risk of making errors using large numbers. MAPE allows you to use smaller numbers and exposes you less to the risk of making errors in your populations. Also, MAPE tells you a percentage error unlike the other ones and people most likely respond high percentages faster than some other numbers.
126 thoughts on “METHODOLOGIES for ERROR MEASUREMENT”
Hey very interesting blog! asmr 0mniartist
Thank you! Do hope the information was helpful.
Pretty section of content. I just stumbled upon your blog and in accession capital to assert
that I acquire actually enjoyed account your blog posts.
Anyway I will be subscribing to your augment and even I achievement
you access consistently rapidly. asmr 0mniartist
Greate pieces. Keep writing such kind of information on your page.
Im really impressed by your blog.
Hello there, You have done an incredible job.
I’ll certainly digg it and in my opinion recommend to my friends.
I’m confident they will be benefited from this website. 0mniartist
asmr
What i don’t understood is in truth how you’re now not really much
more neatly-liked than you may be now. You’re very intelligent.
You already know thus considerably in relation to this subject, made
me personally imagine it from numerous varied angles.
Its like women and men are not fascinated until it’s one thing to accomplish with
Girl gaga! Your individual stuffs outstanding.
All the time care for it up! asmr 0mniartist
Great that you could visit and took the time to respond. Glad you enjoy the content and will continue to provide value to the readers/visitors
After looking over a few of the blog articles on your web page,
I truly like your technique of writing a blog. I saved as a favorite it
to my bookmark site list and will be checking back in the near future.
Please check out my website too and let me know how you feel.
0mniartist asmr
Nice website, Very simple and creative. Love the artwork/characters telling their stories.
Also thanks for visiting my blog and please follow for more informative articles
An interesting discussion is worth comment. I do believe
that you should write more about this subject matter,
it may not be a taboo subject but generally folks don’t speak about such subjects.
To the next! All the best!! asmr 0mniartist
Hi.. good to know you have found interest in this content. I do know that in today’s world everything is done with ease. Regardless, people are still interested in solving errors/problems, and it is so crucial to be able to know how. Thanks for your comment
Hi.. You have to have a backup site or two, and you have to have good SEO hosting company with great malware. You can also regulate spams through your discussion link in the dashboard settings. Just copy the links you do not want in the form provided and whenever such links appear they will automatically go to spam. Hope that helps.
Ahaa, its fastidious conversation about this article here at this web
site, I have read all that, so at this time me also commenting here.
0mniartist asmr
Thank you a bunch for sharing this with all people you actually recognise what you
are speaking approximately! Bookmarked. Please additionally talk over with
my website =). We will have a hyperlink trade contract between us
I have read so many articles or reviews about the blogger lovers however this post is actually a nice piece of
writing, keep it up.
Excellent way of telling, and fastidious article
to take information on the topic of my presentation focus, which
i am going to convey in academy.
Tremendous issues here. I am very satisfied to look your article.
Thank you so much and I’m looking forward to touch you.
Will you please drop me a e-mail?
Hi! glad for your comments and I am enthused that my article was useful to you.
Please follow me on twitter and Facebook also.
Emails: Christopher Thompson @sygyzydesign.com
OR coht187dee@aol.com
Hello there! Do you use Twitter? I’d like to follow you if that
would be okay. I’m definitely enjoying your blog and
look forward to new posts.
Hi! Yes , My blog is also on twitter and Facebook. You can also find me on Google My Business. Good luck!
At this time I am ready to do my breakfast, after having my breakfast coming over again to read more news.
If some one needs to be updated with most recent technologies therefore he must be visit this
web page and be up to date everyday.
Thank you for sharing your info. I really appreciate your efforts
and I am waiting for your further write ups thanks
once again.
Incredible points. Solid arguments. Keep up the amazing effort.
I was suggested this website by my cousin. I’m not sure whether
this post is written by him as nobody else know such detailed about my difficulty.
You’re wonderful! Thanks!
Hey there! This is my first visit to your blog! We are a group of
volunteers and starting a new project in a community in the same niche.
Your blog provided us beneficial information to work on. You have done a outstanding job!
Wow, awesome blog structure! How lengthy have you ever been blogging for?
you make running a blog glance easy. The entire glance of your website is great, as well as the content material!
What a information of un-ambiguity and preserveness of
precious experience regarding unexpected emotions.
These are actually wonderful ideas in on the topic of blogging.
You have touched some nice points here. Any way keep up wrinting.
When I originally commented I clicked the “Notify me when new comments are added” checkbox and
now each time a comment is added I get three e-mails with the same comment.
Is there any way you can remove me from that
service? Many thanks!
Hi! So sorry this is happening. You have to go to your email setting to control this. Reset your default to eliminate this.
Hi! I just wanted to ask if you ever have any problems with hackers?
My last blog (wordpress) was hacked and I ended up
losing months of hard work due to no back up. Do you have any solutions to protect against hackers?
Hello.. I suggest you get a good hosting company to manage you SEO
Oh my goodness! Amazing article dude! Thank you so much, However I am having issues with your RSS.
I don’t understand why I can’t join it. Is there anyone else having
the same RSS problems? Anyone that knows the solution can you kindly respond?
Thanx!!
Hello! Do you know if they make any plugins to safeguard against hackers?
I’m kinda paranoid about losing everything I’ve worked hard on. Any suggestions?
I would suggest you use Updraft and Yoast plugins
You really make it seem so easy with your presentation but I find this topic to be actually something which
I think I would never understand. It seems too complex and extremely broad for me.
I’m looking forward for your next post, I’ll try to get the
hang of it!
Thanks to my father who shared with me regarding
this weblog, this web site is actually awesome.
Hi would you mind letting me know which webhost you’re working with?
I’ve loaded your blog in 3 different internet browsers and I must say this blog loads a lot faster then most.
Can you recommend a good internet hosting provider at a reasonable price?
Thanks, I appreciate it!
Hello and thanks for visiting. Depending on your app/software platform I would recommend Blue Host. They are very reasonable with quality hosting. The packages begin at $2.95/month. The catch is that you will have to lock in for increments of 12 month for a year or pay for 3 years upfront if you can afford it. I am locked in for 3 years with my hosting. Feels better for me. Just remember there is a variety of packages. Go to Bluehost.com and do a search.
Amazing issues here. I’m very satisfied to peer your article.
Thanks so much and I’m looking forward to contact you.
Will you please drop me a e-mail?
Will be happy to drop you a note. Thanks for visiting!
I’m not sure exactly why but this blog is loading very slow for
me. Is anyone else having this problem or is it a issue on my end?
I’ll check back later and see if the problem still exists.
You really make it seem so easy with your presentation but I
find this matter to be actually something which
I think I would never understand. It seems too complicated and very broad for me.
I am looking forward for your next post, I will try to
get the hang of it!
Why users still use to read news papers when in this technological
world all is available on net?
Hello, I think your site might be having browser compatibility issues.
When I look at your blog site in Firefox, it looks fine but when opening in Internet Explorer, it has
some overlapping. I just wanted to give you a quick
heads up! Other then that, awesome blog!
Thanks for visiting and your concerns/comments. Will certainly check this out.
Hi there everybody, here every person is sharing such familiarity, therefore it’s fastidious to read
this website, and I used to go to see this weblog everyday.
I’m not that much of a online reader to be honest but your blogs really
nice, keep it up! I’ll go ahead and bookmark your website to
come back later on. Many thanks
I was recommended this blog by my cousin. I’m not sure whether this post
is written by him as nobody else know such detailed about my trouble.
You are incredible! Thanks!
I love it whenever people come together and share views.
Great website, continue the good work!
Thanks for finally talking about > METHODOLOGIES for ERROR MEASUREMENT – SYGYZY Designs < Liked it!
Quality posts is the secret to be a focus for the people to go to see the
website, that’s what this web page is providing.
It is the best time to make a few plans for the longer term and it’s
time to be happy. I have read this post and if
I may just I want to suggest you few interesting issues or tips.
Perhaps you can write subsequent articles referring to this article.
I desire to read even more issues approximately it!
Quality articles or reviews is the crucial to invite the people to pay a visit the website, that’s what this web site is providing.
I got this web page from my buddy who shared with me about this
website and now this time I am visiting this website and reading very informative content at this place.
scoliosis
After I originally left a comment I seem to have clicked the -Notify me when new comments are added- checkbox and from now on every time
a comment is added I get 4 emails with the exact same
comment. There has to be a means you can remove me from that service?
Thanks a lot! scoliosis
scoliosis
Thank you for the auspicious writeup. It in fact was a amusement account it.
Look advanced to far added agreeable from you! By the way, how could we communicate?
scoliosis
Have you ever thought about including a little bit more than just your articles?
I mean, what you say is fundamental and everything. Nevertheless just imagine if you added some great visuals or videos
to give your posts more, “pop”! Your content is excellent but
with images and videos, this site could definitely be one of the very best in its field.
Amazing blog!
Pretty section of content. I just stumbled upon your blog and
in accession capital to assert that I acquire in fact enjoyed account your blog posts.
Any way I will be subscribing to your feeds and even I achievement you access consistently fast.
Hi there! This post couldn’t be written any better! Reading through this post reminds me of my old room mate!
He always kept talking about this. I will forward this write-up to him.
Pretty sure he will have a good read. Thanks for sharing!
Thank you for sharing your thoughts. I truly appreciate your efforts and
I will be waiting for your further write ups thank you once again.
Sweet blog! I found it while searching on Yahoo News.
Do you have any tips on how to get listed in Yahoo News? I’ve been trying for a while but I never seem to get
there! Thank you
Hmm is anyone else having problems with the images
on this blog loading? I’m trying to determine if its
a problem on my end or if it’s the blog. Any responses would be greatly appreciated.
Hello.. Thanks for visiting. Your concerns are noted and I will certainly take a look. However, I have not had many complaints about the web feed. Some concerns may occur depending on the search engine you are using. The concerns usually relate to Explorer. Please try a different search engine and let me know.
Thanks
Its like you read my mind! You seem to know a lot about this, like you wrote the book in it
or something. I think that you can do with some pics to drive the message home
a bit, but other than that, this is magnificent blog.
A fantastic read. I’ll definitely be back.
Hi there, I check your blogs daily. Your humoristic style is awesome, keep doing what you’re doing!
What’s up friends, its wonderful article concerning
educationand entirely explained, keep it up all the time.
It’s actually a nice and helpful piece of info. I’m happy that
you shared this helpful information with us. Please stay us informed like
this. Thank you for sharing.
I will right away grasp your rss as I can not to find your e-mail subscription hyperlink or newsletter service. Do you’ve any? Please permit me realize so that I may just subscribe. Thanks.
Excellent goods from you, man. I’ve understand your
stuff previous to and you are just too magnificent. I really like what you
have acquired here, really like what you are saying and the way in which you say it.
You make it enjoyable and you still care for to keep it wise.
I cant wait to read far more from you. This is actually a great website.
Also visit my web page :: Nuubu Detox
Thanks for every other excellent post. Where else
could anybody get that type of info in such a perfect way of writing?
I’ve a presentation subsequent week, and I’m on the search for such info.
my homepage :: Premium Shot Keto
Great blog here! Additionally your website quite a
bit up fast! What host are you the usage of? Can I am getting your associate hyperlink in your host?
I wish my site loaded up as quickly as yours lol.
Also visit my site; Premium Shot Keto
I think this is among the most vital info for
me. And i’m glad reading your article. But wanna remark on some general things,
The web site style is great, the articles is really nice : D.
Good job, cheers
my blog Tri-Bol Testo Review
Good day! I could have sworn I’ve been to this site before but after going through a
few of the articles I realized it’s new to me.
Anyways, I’m definitely pleased I came across it and I’ll be book-marking it and checking back
often!
my website – Kodo Detox Patch
I simply wanted to compose a quick remark in order to say thanks
to you for those amazing hints you are posting at this website.
My incredibly long internet lookup has at the end been recognized with professional
tips to share with my partners. I would repeat that many of us site visitors actually are undeniably lucky
to dwell in a fine community with so many special professionals with insightful methods.
I feel rather fortunate to have encountered your entire website page
and look forward to really more fabulous minutes reading here.
Thank you once more for everything.
My blog post – Tri-Bol Testo
I got what you mean, regards for putting up. Woh I
am delighted to find this website through google.
Also visit my web blog :: Boltz Pro
I know this web page offers quality dependent articles or reviews
and other stuff, is there any other website which gives these kinds of information in quality?
Feel free to surf to my web page … Boltz Pro
Some really great articles on this web site, regards for contribution.
Hi there to all, how is the whole thing, I think every one is getting more
from this site, and your views are fastidious in support
of new users.
The other day, while I was at work, my sister stole my iPad
and tested to see if it can survive a 40 foot drop, just so
she can be a youtube sensation. My apple ipad is now broken and she has 83 views.
I know this is completely off topic but I had to share it with someone!
I think that what you composed was very logical. But, what about this?
suppose you added a little information? I ain’t
saying your content isn’t solid, but what if you added a post
title that makes people want more? I mean METHODOLOGIES
for ERROR MEASUREMENT – SYGYZY Designs is kinda boring.
You ought to look at Yahoo’s home page and watch how they create news headlines to get viewers to click.
You might add a related video or a related picture or
two to get people interested about what you’ve
written. Just my opinion, it could make your website
a little bit more interesting.
Ηi tto all, howw is eｖerything, I think eery one іs getting more frkm
thіs website, and your views aгee fastіdious for
new people.
Hello to all, because I am genuinely keen of reading
this webpage’s post to be updated regularly. It carries good material.
Feel free to visit my website … SperMax Control Ingredients
I just like the helpful information you supply to your articles.
I will bookmark your weblog and check once more here frequently.
I am somewhat certain I’ll be informed plenty of new stuff right here!
Best of luck for the following!
Hi, i believe that i noticed you visited my weblog so
i got here to go back the desire?.I am attempting to in finding issues to improve
my website!I suppose its ok to make use of a few of your ideas!!
If some one desires to be updated with newest technologies afterward he must be
visit this site and be up to date everyday.
Also visit my site … Green Naturals CBD Gummimes
Yes! Finally someone writes about website.
Fantástico mas eu era querendo saber se você poderia escrever um pouco
mais sobre este assunto? Eu seria muito grato se você poderia elaborar um pouco mais.
Apreciá-lo !
Outstanding post, I conceive blog owners should learn a lot from this website its very user pleasant.
So much great info on here :D.
Have a look at my web site: shihan.com.ru
Spot on with this write-up, I absolutely feel this website
needs a lot more attention. I’ll probably be back again to read more, thanks for the info!
Highly energetic blog, I enjoyed that a lot. Will there
be a part 2?
My blog – 360X CBD Reviews
As I web site possessor I believe the content material here is rattling excellent , appreciate it for
your hard work. You should keep it up forever! Good Luck.
Feel free to surf to my web blog … Keto Fat Burn Reviews
I’m gone to tell my little brother, that he should also pay a quick visit this
website on regular basis to obtain updated from hottest news.
I was suggested this website by my cousin. I’m not sure whether this post is written by him as nobody else know such
detailed about my problem. You are wonderful! Thanks!
Very good blog! Do you have any hints for aspiring writers?
I’m hoping to start my own blog soon but I’m a little lost on everything.
Would you advise starting with a free platform like WordPress or go for a paid option? There
are so many choices out there that I’m totally confused ..
Any ideas? Many thanks!
Keep on working, great job!
I must thank you for the efforts you’ve put in writing this blog.
I really hope to view the same high-grade blog posts by you in the future
as well. In fact, your creative writing abilities
has encouraged me to get my own blog now 😉
Also visit my web page; Sion Air Cooler Reviews
Fantastic website. A lot of useful info here.
I’m sending it to several pals ans additionally sharing in delicious.
And obviously, thanks to your sweat!
Undeniably believe that which you said. Your favorite justification seemed to be on the net the easiest thing to be aware
of. I say to you, I definitely get irked while people think about
worries that they plainly don’t know about. You managed to hit the
nail upon the top as well as defined out the whole thing without having side-effects
, people could take a signal. Will probably be back to get more.
Thanks
Great info. Lucky me I came across your site by accident
(stumbleupon). I’ve saved as a favorite for later!
It’s hard to find educated people about this subject, but you sound like you
know what you’re talking about! Thanks
Hey there, You’ve done an incredible job. I’ll definitely digg it and personally recommend
to my friends. I’m sure they will be benefited from this web site.
my blog post – Vigra Fast Ingredients
Hi! I’ve been reading your blog for a while now and finally got the bravery to go
ahead and give you a shout out from Porter Tx!
Just wanted to mention keep up the great job!
Also visit my website Vinyasa Cream Review
Undeniably believe that which you stated. Your favourite justification appeared
to be at the net the simplest factor to be aware of.
I say to you, I certainly get annoyed at the same time as folks think about issues that they just
don’t realize about. You managed to hit the nail
upon the highest and also outlined out the whole thing with no need side-effects , folks
can take a signal. Will likely be again to get more. Thanks
I think other web-site proprietors should take this site as an model, very clean and magnificent user genial style and design, let alone the content. You’re an expert in this topic!
Thanks for visiting! I truly depend on my audience to critique and measure my effectiveness in the delivery of releveant content. To you I am grateful that you have found the content relevnt and useful.
Thank you
Your site is amazing. It is definitely fast and also many thanks to get making it possible for remarks.
I take joy in dropping by your wordpress blog. You are actually brilliant as well as compose so properly. All the bests with regards to working doggedly on this particular blog website.
I’m really enjoying the theme/design of your site. Do you ever run into any browser compatibility problems? A small number of my blog audience have complained about my blog not working correctly in Explorer but looks great in Opera. Do you have any advice to help fix this issue?
Thank you so much for giving everyone an exceptionally spectacular chance to read from this blog. It’s always so superb and as well , full of fun for me and my office co-workers to visit your blog at the least 3 times per week to study the newest items you have got. And indeed, I’m at all times fascinated considering the very good creative ideas served by you. Certain 3 facts in this post are easily the most impressive we’ve had.
Very good site you have here but I was curious if you knew of any community forums that cover the same topics talked about here? I’d really like to be a part of group where I can get comments from other knowledgeable people that share the same interest. If you have any recommendations, please let me know. Appreciate it!
Admiring the time and energy you put into your site and in depth information you present. It’s good to come across a blog every once in a while that isn’t the same old rehashed information. Fantastic read! I’ve saved your site and I’m adding your RSS feeds to my Google account.
I have had Firefox, IE, and Google chrome downloaded onto my computer for a while now with no problem. Recently I downloaded YIM and now Firefox doesn’t start up. If I restart my computer a few times, and try to open it a few dozen times each time I restart, it will eventually pop up. How do I fix it? Chrome and IE still work fine but I prefer Firefox..
Hi! Just visited and followed you on Twitter. Can you be so kind as to do the same?
Thanks a bunch!!!
Youyr problem is with your default browser. You will have to decide which it is going to be and adjust that in your settings.
There is apparently a lot to realize about this. I assume you made various nice points in features also.
Hey! Would you mind if I share your blog with my twitter group? There’s a lot of people that I think would really appreciate your content. Please let me know. Thanks
Greetings I am so thrilled I found your webpage, I really found you by accident, while I was researching on Google for something else, Anyhow I am here now and would just like to say thanks a lot for a incredible post and a all round entertaining blog (I also love the theme/design), I dont have time to look over it all at the minute but I have bookmarked it and also added your RSS feeds, so when I have time I will be back to read more, Please do keep up the fantastic job.
My spouse and I stumbled over here from a different page and thought I might as well check things out. I like what I see so now i’m following you. Look forward to finding out about your web page for a second time.
Thanks for a marvelous posting! I seriously enjoyed reading it, you can be a great author.I will make certain to bookmark your blog and will often come back later on. I want to encourage one to continue your great work, have a nice weekend!
Keep functioning ,terrific job!
My brother suggested I might like this blog. He was entirely right. This post truly made my day. You cann’t imagine simply how much time I had spent for this info! Thanks!
Thanks on your marvelous posting! I definitely enjoyed reading it, you might be a great author.I will be sure to bookmark your blog and may come back later in life. I want to encourage yourself to continue your great writing, have a nice morning!
Thank you ever so for you article.Much thanks again. Fantastic.
I’m extremely impressed together with your writing skills and also with the layout on your weblog. Is that this a paid subject or did you customize it your self? Either way keep up the excellent high quality writing, it is rare to peer a great weblog like this one today..
I visited a lot of website but I think this one has got something extra in it in it
Great, thanks for sharing this article. Really Great.